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textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/03%3A_Electrical_Safety/3.06%3A_Emergency_Response.txt	princeton-nlp/TextbookChapters	If you see someone lying unconscious or “froze on the circuit,” the very first thing to do is shut off the power by opening the appropriate disconnect switch or circuit breaker. If someone touches another person being shocked, there may be enough voltage dropped across the body of the victim to shock the would-be rescuer, thereby “freezing” two people instead of one. Don’t be a hero. Electrons don’t respect heroism. Make sure the situation is safe for you to step into, or else you will be the next victim, and nobody will benefit from your efforts. One problem with this rule is that the source of power may not be known, or easily found in time to save the victim of shock. If a shock victim’s breathing and heartbeat are paralyzed by electric current, their survival time is very limited. If the shock current is of sufficient magnitude, their flesh and internal organs may be quickly roasted by the power the current dissipates as it runs through their body. If the power disconnect switch cannot be located quickly enough, it may be possible to dislodge the victim from the circuit they’re frozen on to by prying them or hitting them away with a dry wooden board or piece of nonmetallic conduit, common items to be found in industrial construction scenes. Another item that could be used to safely drag a “frozen” victim away from contact with power is an extension cord. By looping a cord around their torso and using it as a rope to pull them away from the circuit, their grip on the conductor(s)may be broken. Bear in mind that the victim will be holding on to the conductor with all their strength, so pulling them away probably won’t be easy! Once the victim has been safely disconnected from the source of electric power, the immediate medical concerns for the victim should be respiration and circulation (breathing and pulse). If the rescuer is trained in CPR, they should follow the appropriate steps of checking for breathing and pulse, then applying CPR as necessary to keep the victim’s body from deoxygenating. The cardinal rule of CPR is to keep going until you have been relieved by qualified personnel. If the victim is conscious, it is best to have them lie still until qualified emergency response personnel arrive on the scene. There is the possibility of the victim going into a state of physiological shock—a condition of insufficient blood circulation different from electrical shock—and so they should be kept as warm and comfortable as possible. An electrical shock insufficient to cause immediate interruption of the heartbeat may be strong enough to cause heart irregularities or a heart attack up to several hours later, so the victim should pay close attention to their own condition after the incident, ideally under supervision. Review • A person being shocked needs to be disconnected from the source of electrical power. Locate the disconnecting switch/breaker and turn it off. Alternatively, if the disconnecting device cannot be located, the victim can be pried or pulled from the circuit by an insulated object such as a dry wood board, piece of nonmetallic conduit, or rubber electrical cord. • Victims need immediate medical response: check for breathing and pulse, then apply CPR as necessary to maintain oxygenation. • If a victim is still conscious after having been shocked, they need to be closely monitored and cared for until trained emergency response personnel arrive. There is danger of physiological shock, so keep the victim warm and comfortable. • Shock victims may suffer heart trouble up to several hours after being shocked. The danger of electric shock does not end after the immediate medical attention.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/03%3A_Electrical_Safety/3.07%3A_Common_Sources_of_Hazard.txt	princeton-nlp/TextbookChapters	The easiest way to decrease skin resistance is to get it wet. Therefore, touching electrical devices with wet hands, wet feet, or especially in a sweaty condition (salt water is a much better conductor of electricity than fresh water) is dangerous. In the household, the bathroom is one of the more likely places where wet people may contact electrical appliances, and so shock hazard is a definite threat there. Good bathroom design will locate power receptacles away from bathtubs, showers, and sinks to discourage the use of appliances nearby. Telephones that plug into a wall socket are also sources of hazardous voltage (the open circuit voltage is 48 volts DC, and the ringing signal is 150 volts AC—remember that any voltage over 30 is considered potentially dangerous!). Appliances such as telephones and radios should never, ever be used while sitting in a bathtub. Even battery-powered devices should be avoided. Some battery-operated devices employ voltage-increasing circuitry capable of generating lethal potentials. Swimming pools are another source of trouble, since people often operate radios and other powered appliances nearby. The National Electrical Code requires that special shock-detecting receptacles called Ground-Fault Current Interrupting (GFI or GFCI) be installed in wet and outdoor areas to help prevent shock incidents. More on these devices in a later section of this chapter. These special devices have no doubt saved many lives, but they can be no substitute for common sense and diligent precaution. As with firearms, the best “safety” is an informed and conscientious operator. Extension cords, so commonly used at home and in industry, are also sources of potential hazard. All cords should be regularly inspected for abrasion or cracking of insulation, and repaired immediately. One sure method of removing a damaged cord from service is to unplug it from the receptacle, then cut off that plug (the “male” plug) with a pair of side-cutting pliers to ensure that no one can use it until it is fixed. This is important on jobsites, where many people share the same equipment, and not all people there may be aware of the hazards. Any power tool showing evidence of electrical problems should be immediately serviced as well. I’ve heard several horror stories of people who continue to work with hand tools that periodically shock them. Remember, electricity can kill, and the death it brings can be gruesome. Like extension cords, a bad power tool can be removed from service by unplugging it and cutting off the plug at the end of the cord. Downed power lines are an obvious source of electric shock hazard and should be avoided at all costs. The voltages present between power lines or between a power line and earth ground are typically very high (2400 volts being one of the lowest voltages used in residential distribution systems). If a power line is broken and the metal conductor falls to the ground, the immediate result will usually be a tremendous amount of arcing (sparks produced), often enough to dislodge chunks of concrete or asphalt from the road surface, and reports rivaling that of a rifle or shotgun. To come into direct contact with a downed power line is almost sure to cause death, but other hazards exist which are not so obvious. When a line touches the ground, current travels between that downed conductor and the nearest grounding point in the system, thus establishing a circuit: The earth, being a conductor (if only a poor one), will conduct current between the downed line and the nearest system ground point, which will be some kind of conductor buried in the ground for good contact. Being that the earth is a much poorer conductor of electricity than the metal cables strung along the power poles, there will be substantial voltage dropped between the point of cable contact with the ground and the grounding conductor, and little voltage dropped along the length of the cabling (the following figures are very approximate): If the distance between the two ground contact points (the downed cable and the system ground) is small, there will be substantial voltage dropped along short distances between the two points. Therefore, a person standing on the ground between those two points will be in danger of receiving an electric shock by intercepting a voltage between their two feet! Again, these voltage figures are very approximate, but they serve to illustrate a potential hazard: that a person can become a victim of electric shock from a downed power line without even coming into contact with that line! One practical precaution a person could take if they see a power line falling towards the ground is to only contact the ground at one point, either by running away (when you run, only one foot contacts the ground at any given time), or if there’s nowhere to run, by standing on one foot. Obviously, if there’s somewhere safer to run, running is the best option. By eliminating two points of contact with the ground, there will be no chance of applying deadly voltage across the body through both legs. Review • Wet conditions increase risk of electric shock by lowering skin resistance. • Immediately replace worn or damaged extension cords and power tools. You can prevent innocent use of a bad cord or tool by cutting the male plug off the cord (while its unplugged from the receptacle, of course). • Power lines are very dangerous and should be avoided at all costs. If you see a line about to hit the ground, stand on one foot or run (only one foot contacting the ground) to prevent shock from voltage dropped across the ground between the line and the system ground point.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/03%3A_Electrical_Safety/3.08%3A_Safe_Circuit_Design.txt	princeton-nlp/TextbookChapters	As far as the voltage source and load are concerned, grounding makes no difference at all. It exists purely for the sake of personnel safety, by guaranteeing that at least one point in the circuit will be safe to touch (zero voltage to ground). The “Hot” side of the circuit, named for its potential for shock hazard, will be dangerous to touch unless voltage is secured by proper disconnection from the source (ideally, using a systematic lock-out/tag-out procedure). This imbalance of hazard between the two conductors in a simple power circuit is important to understand. The following series of illustrations are based on common household wiring systems (using DC voltage sources rather than AC for simplicity). If we take a look at a simple, household electrical appliance such as a toaster with a conductive metal case, we can see that there should be no shock hazard when it is operating properly. The wires conducting power to the toaster’s heating element are insulated from touching the metal case (and each other) by rubber or plastic. However, if one of the wires inside the toaster were to accidentally come in contact with the metal case, the case will be made electrically common to the wire, and touching the case will be just as hazardous as touching the wire bare. Whether or not this presents a shock hazard depends on which wire accidentally touches: If the “hot” wire contacts the case, it places the user of the toaster in danger. On the other hand, if the neutral wire contacts the case, there is no danger of shock: To help ensure that the former failure is less likely than the latter, engineers try to design appliances in such a way as to minimize hot conductor contact with the case. Ideally, of course, you don’t want either wire accidentally coming in contact with the conductive case of the appliance, but there are usually ways to design the layout of the parts to make accidental contact less likely for one wire than for the other. However, this preventative measure is effective only if power plug polarity can be guaranteed. If the plug can be reversed, then the conductor more likely to contact the case might very well be the “hot” one: Appliances designed this way usually come with “polarized” plugs, one prong of the plug being slightly narrower than the other. Power receptacles are also designed like this, one slot being narrower than the other. Consequently, the plug cannot be inserted “backwards,” and conductor identity inside the appliance can be guaranteed. Remember that this has no effect whatsoever on the basic function of the appliance: its strictly for the sake of user safety. Some engineers address the safety issue simply by making the outside case of the appliance nonconductive. Such appliances are called double-insulated, since the insulating case serves as a second layer of insulation above and beyond that of the conductors themselves. If a wire inside the appliance accidently comes in contact with the case, there is no danger presented to the user of the appliance. Other engineers tackle the problem of safety by maintaining a conductive case, but using a third conductor to firmly connect that case to ground: The third prong on the power cord provides a direct electrical connection from the appliance case to earth ground, making the two points electrically common with each other. If they’re electrically common, then there cannot be any voltage dropped between them. At least, that’s how it is supposed to work. If the hot conductor accidently touches the metal appliance case, it will create a direct short-circuit back to the voltage source through the ground wire, tripping any overcurrent protection devices. The user of the appliance will remain safe. This is why its so important never to cut the third prong off a power plug when trying to fit it into a two-prong receptacle. If this is done, there will be no grounding of the appliance case to keep the user(s) safe. The appliance will still function properly, but if there is an internal fault bringing the hot wire in contact with the case, the results can be deadly. If a two-prong receptacle must be used, a two- to three-prong receptacle adapter can be installed with a grounding wire attached to the receptacle’s grounded cover screw. This will maintain the safety of the grounded appliance while plugged in to this type of receptacle. Electrically safe engineering doesn’t necessarily end at the load, however. A final safeguard against electrical shock can be arranged on the power supply side of the circuit rather than the appliance itself. This safeguard is called ground-fault detection, and it works like this: In a properly functioning appliance (shown above), the current measured through the hot conductor should be exactly equal to the current through the neutral conductor, because there’s only one path for electrons to flow in the circuit. With no fault inside the appliance, there is no connection between circuit conductors and the person touching the case, and therefore no shock. If, however, the hot wire accidentally contacts the metal case, there will be current through the person touching the case. The presence of a shock current will be manifested as a difference of current between the two power conductors at the receptacle: This difference in current between the “hot” and “neutral” conductors will only exist if there is current through the ground connection, meaning that there is a fault in the system. Therefore, such a current difference can be used as a way to detect a fault condition. If a device is set up to measure this difference of current between the two power conductors, a detection of current imbalance can be used to trigger the opening of a disconnect switch, thus cutting power off and preventing serious shock: Such devices are called Ground Fault Current Interruptors, or GFCIs for short. Outside North America, the GFCI is variously known as a safety switch, a residual current device (RCD), an RCBO or RCD/MCB if combined with a miniature circuit breaker, or earth leakage circuit breaker (ELCB). They are compact enough to be built into a power receptacle. These receptacles are easily identified by their distinctive “Test” and “Reset” buttons. The big advantage with using this approach to ensure safety is that it works regardless of the appliance’s design. Of course, using a double-insulated or grounded appliance in addition to a GFCI receptacle would be better yet, but its comforting to know that something can be done to improve safety above and beyond the design and condition of the appliance. The arc fault circuit interrupter (AFCI), a circuit breaker designed to prevent fires, is designed to open on intermittent resistive short circuits. For example, a normal 15 A breaker is designed to open circuit quickly if loaded well beyond the 15 A rating, more slowly a little beyond the rating. While this protects against direct shorts and several seconds of overload, respectively, it does not protect against arcs– similar to arc-welding. An arc is a highly variable load, repetitively peaking at over 70 A, open circuiting with alternating current zero-crossings. Though, the average current is not enough to trip a standard breaker, it is enough to start a fire. This arc could be created by a metalic short circuit which burns the metal open, leaving a resistive sputtering plasma of ionized gases. The AFCI contains electronic circuitry to sense this intermittent resistive short circuit. It protects against both hot to neutral and hot to ground arcs. The AFCI does not protect against personal shock hazards like a GFCI does. Thus, GFCIs still need to be installed in kitchen, bath, and outdoors circuits. Since the AFCI often trips upon starting large motors, and more generally on brushed motors, its installation is limited to bedroom circuits by the U.S. National Electrical code. Use of the AFCI should reduce the number of electrical fires. However, nuisance-trips when running appliances with motors on AFCI circuits is a problem. Review • Power systems often have one side of the voltage supply connected to earth ground to ensure safety at that point. • The “grounded” conductor in a power system is called the neutral conductor, while the ungrounded conductor is called the hot. • Grounding in power systems exists for the sake of personnel safety, not the operation of the load(s). • Electrical safety of an appliance or other load can be improved by good engineering: polarized plugs, double insulation, and three-prong “grounding” plugs are all ways that safety can be maximized on the load side. • Ground Fault Current Interruptors (GFCIs) work by sensing a difference in current between the two conductors supplying power to the load. There should be no difference in current at all. Any difference means that current must be entering or exiting the load by some means other than the two main conductors, which is not good. A significant current difference will automatically open a disconnecting switch mechanism, cutting power off completely.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/03%3A_Electrical_Safety/3.09%3A_Safe_Meter_Usage.txt	princeton-nlp/TextbookChapters	The most common piece of electrical test equipment is a meter called the multimeter. Multimeters are so named because they have the ability to measure a multiple of variables: voltage, current, resistance, and often many others, some of which cannot be explained here due to their complexity. In the hands of a trained technician, the multimeter is both an efficient work tool and a safety device. In the hands of someone ignorant and/or careless, however, the multimeter may become a source of danger when connected to a “live” circuit. There are many different brands of multimeters, with multiple models made by each manufacturer sporting different sets of features. The multimeter shown here in the following illustrations is a “generic” design, not specific to any manufacturer, but general enough to teach the basic principles of use: You will notice that the display of this meter is of the “digital” type: showing numerical values using four digits in a manner similar to a digital clock. The rotary selector switch (now set in the Off position) has five different measurement positions it can be set in: two “V” settings, two “A” settings, and one setting in the middle with a funny-looking “horseshoe” symbol on it representing “resistance.” The “horseshoe” symbol is the Greek letter “Omega” (Ω), which is the common symbol for the electrical unit of ohms. Of the two “V” settings and two “A” settings, you will notice that each pair is divided into unique markers with either a pair of horizontal lines (one solid, one dashed), or a dashed line with a squiggly curve over it. The parallel lines represent “DC” while the squiggly curve represents “AC.” The “V” of course stands for “voltage” while the “A” stands for “amperage” (current). The meter uses different techniques, internally, to measure DC than it uses to measure AC, and so it requires the user to select which type of voltage (V) or current (A) is to be measured. Although we haven’t discussed alternating current (AC) in any technical detail, this distinction in meter settings is an important one to bear in mind. There are three different sockets on the multimeter face into which we can plug our test leads. Test leads are nothing more than specially-prepared wires used to connect the meter to the circuit under test. The wires are coated in a color-coded (either black or red) flexible insulation to prevent the user’s hands from contacting the bare conductors, and the tips of the probes are sharp, stiff pieces of wire: The black test lead always plugs into the black socket on the multimeter: the one marked “COM” for “common.” The red test lead plugs into either the red socket marked for voltage and resistance, or the red socket marked for current, depending on which quantity you intend to measure with the multimeter. To see how this works, let’s look at a couple of examples showing the meter in use. First, we’ll set up the meter to measure DC voltage from a battery: Note that the two test leads are plugged into the appropriate sockets on the meter for voltage, and the selector switch has been set for DC “V”. Now, we’ll take a look at an example of using the multimeter to measure AC voltage from a household electrical power receptacle (wall socket): The only difference in the setup of the meter is the placement of the selector switch: it is now turned to AC “V”. Since we’re still measuring voltage, the test leads will remain plugged in the same sockets. In both of these examples, it is imperative that you not let the probe tips come in contact with one another while they are both in contact with their respective points on the circuit. If this happens, a short-circuit will be formed, creating a spark and perhaps even a ball of flame if the voltage source is capable of supplying enough current! The following image illustrates the potential for hazard: This is just one of the ways that a meter can become a source of hazard if used improperly. Voltage measurement is perhaps the most common function a multimeter is used for. It is certainly the primary measurement taken for safety purposes (part of the lock-out/tag-out procedure), and it should be well understood by the operator of the meter. Being that voltage is always relative between two points, the meter must be firmly connected to two points in a circuit before it will provide a reliable measurement. That usually means both probes must be grasped by the user’s hands and held against the proper contact points of a voltage source or circuit while measuring. Because a hand-to-hand shock current path is the most dangerous, holding the meter probes on two points in a high-voltage circuit in this manner is always a potential hazard. If the protective insulation on the probes is worn or cracked, it is possible for the user’s fingers to come into contact with the probe conductors during the time of test, causing a bad shock to occur. If it is possible to use only one hand to grasp the probes, that is a safer option. Sometimes it is possible to “latch” one probe tip onto the circuit test point so that it can be let go of and the other probe set in place, using only one hand. Special probe tip accessories such as spring clips can be attached to help facilitate this. Remember that meter test leads are part of the whole equipment package, and that they should be treated with the same care and respect that the meter itself is. If you need a special accessory for your test leads, such as a spring clip or other special probe tip, consult the product catalog of the meter manufacturer or other test equipment manufacturer. Do not try to be creative and make your own test probes, as you may end up placing yourself in danger the next time you use them on a live circuit. Also, it must be remembered that digital multimeters usually do a good job of discriminating between AC and DC measurements, as they are set for one or the other when checking for voltage or current. As we have seen earlier, both AC and DC voltages and currents can be deadly, so when using a multimeter as a safety check device you should always check for the presence of both AC and DC, even if you’re not expecting to find both! Also, when checking for the presence of hazardous voltage, you should be sure to check all pairs of points in question. For example, suppose that you opened up an electrical wiring cabinet to find three large conductors supplying AC power to a load. The circuit breaker feeding these wires (supposedly) has been shut off, locked, and tagged. You double-checked the absence of power by pressing the Start button for the load. Nothing happened, so now you move on to the third phase of your safety check: the meter test for voltage. First, you check your meter on a known source of voltage to see that its working properly. Any nearby power receptacle should provide a convenient source of AC voltage for a test. You do so and find that the meter indicates as it should. Next, you need to check for voltage among these three wires in the cabinet. But voltage is measured between two points, so where do you check? The answer is to check between all combinations of those three points. As you can see, the points are labeled “A”, “B”, and “C” in the illustration, so you would need to take your multimeter (set in the voltmeter mode) and check between points A & B, B & C, and A & C. If you find voltage between any of those pairs, the circuit is not in a Zero Energy State. But wait! Remember that a multimeter will not register DC voltage when its in the AC voltage mode and vice versa, so you need to check those three pairs of points in each mode for a total of six voltage checks in order to be complete! However, even with all that checking, we still haven’t covered all possibilities yet. Remember that hazardous voltage can appear between a single wire and ground (in this case, the metal frame of the cabinet would be a good ground reference point) in a power system. So, to be perfectly safe, we not only have to check between A & B, B & C, and A & C (in both AC and DC modes), but we also have to check between A & ground, B & ground, and C & ground (in both AC and DC modes)! This makes for a grand total of twelve voltage checks for this seemingly simple scenario of only three wires. Then, of course, after we’ve completed all these checks, we need to take our multimeter and re-test it against a known source of voltage such as a power receptacle to ensure that its still in good working order. Using a multimeter to check for resistance is a much simpler task. The test leads will be kept plugged in the same sockets as for the voltage checks, but the selector switch will need to be turned until it points to the “horseshoe” resistance symbol. Touching the probes across the device whose resistance is to be measured, the meter should properly display the resistance in ohms: One very important thing to remember about measuring resistance is that it must only be done on de-energized components! When the meter is in “resistance” mode, it uses a small internal battery to generate a tiny current through the component to be measured. By sensing how difficult it is to move this current through the component, the resistance of that component can be determined and displayed. If there is any additional source of voltage in the meter-lead-component-lead-meter loop to either aid or oppose the resistance-measuring current produced by the meter, faulty readings will result. In a worse-case situation, the meter may even be damaged by the external voltage. The “resistance” mode of a multimeter is very useful in determining wire continuity as well as making precise measurements of resistance. When there is a good, solid connection between the probe tips (simulated by touching them together), the meter shows almost zero Ω. If the test leads had no resistance in them, it would read exactly zero: If the leads are not in contact with each other, or touching opposite ends of a broken wire, the meter will indicate infinite resistance (usually by displaying dashed lines or the abbreviation “O.L.” which stands for “open loop”): By far the most hazardous and complex application of the multimeter is in the measurement of current. The reason for this is quite simple: in order for the meter to measure current, the current to be measured must be forced to go through the meter. This means that the meter must be made part of the current path of the circuit rather than just be connected off to the side somewhere as is the case when measuring voltage. In order to make the meter part of the current path of the circuit, the original circuit must be “broken” and the meter connected across the two points of the open break. To set the meter up for this, the selector switch must point to either AC or DC “A” and the red test lead must be plugged in the red socket marked “A”. The following illustration shows a meter all ready to measure current and a circuit to be tested: Now, the circuit is broken in preparation for the meter to be connected: The next step is to insert the meter in-line with the circuit by connecting the two probe tips to the broken ends of the circuit, the black probe to the negative (-) terminal of the 9-volt battery and the red probe to the loose wire end leading to the lamp: This example shows a very safe circuit to work with. 9 volts hardly constitutes a shock hazard, and so there is little to fear in breaking this circuit open (bare handed, no less!) and connecting the meter in-line with the flow of electrons. However, with higher power circuits, this could be a hazardous endeavor indeed. Even if the circuit voltage was low, the normal current could be high enough that an injurious spark would result the moment the last meter probe connection was established. Another potential hazard of using a multimeter in its current-measuring (“ammeter”) mode is failure to properly put it back into a voltage-measuring configuration before measuring voltage with it. The reasons for this are specific to ammeter design and operation. When measuring circuit current by placing the meter directly in the path of current, it is best to have the meter offer little or no resistance against the flow of electrons. Otherwise, any additional resistance offered by the meter would impede the electron flow and alter the circuits operation. Thus, the multimeter is designed to have practically zero ohms of resistance between the test probe tips when the red probe has been plugged into the red “A” (current-measuring) socket. In the voltage-measuring mode (red lead plugged into the red “V” socket), there are many mega-ohms of resistance between the test probe tips, because voltmeters are designed to have close to infinite resistance (so that they don’t draw any appreciable current from the circuit under test). When switching a multimeter from current- to voltage-measuring mode, its easy to spin the selector switch from the “A” to the “V” position and forget to correspondingly switch the position of the red test lead plug from “A” to “V”. The result—if the meter is then connected across a source of substantial voltage—will be a short-circuit through the meter! To help prevent this, most multimeters have a warning feature by which they beep if ever there’s a lead plugged in the “A” socket and the selector switch is set to “V”. As convenient as features like these are, though, they are still no substitute for clear thinking and caution when using a multimeter. All good-quality multimeters contain fuses inside that are engineered to “blow” in the event of excessive current through them, such as in the case illustrated in the last image. Like all overcurrent protection devices, these fuses are primarily designed to protect the equipment (in this case, the meter itself) from excessive damage, and only secondarily to protect the user from harm. A multimeter can be used to check its own current fuse by setting the selector switch to the resistance position and creating a connection between the two red sockets like this: A good fuse will indicate very little resistance while a blown fuse will always show “O.L.” (or whatever indication that model of multimeter uses to indicate no continuity). The actual number of ohms displayed for a good fuse is of little consequence, so long as its an arbitrarily low figure. So now that we’ve seen how to use a multimeter to measure voltage, resistance, and current, what more is there to know? Plenty! The value and capabilities of this versatile test instrument will become more evident as you gain skill and familiarity using it. There is no substitute for regular practice with complex instruments such as these, so feel free to experiment on safe, battery-powered circuits. Review • A meter capable of checking for voltage, current, and resistance is called a multimeter. • As voltage is always relative between two points, a voltage-measuring meter (“voltmeter”) must be connected to two points in a circuit in order to obtain a good reading. Be careful not to touch the bare probe tips together while measuring voltage, as this will create a short-circuit! • Remember to always check for both AC and DC voltage when using a multimeter to check for the presence of hazardous voltage on a circuit. Make sure you check for voltage between all pair-combinations of conductors, including between the individual conductors and ground! • When in the voltage-measuring (“voltmeter”) mode, multimeters have very high resistance between their leads. • Never try to read resistance or continuity with a multimeter on a circuit that is energized. At best, the resistance readings you obtain from the meter will be inaccurate, and at worst the meter may be damaged and you may be injured. • Current measuring meters (“ammeters”) are always connected in a circuit so the electrons have to flow through the meter. • When in the current-measuring (“ammeter”) mode, multimeters have practically no resistance between their leads. This is intended to allow electrons to flow through the meter with the least possible difficulty. If this were not the case, the meter would add extra resistance in the circuit, thereby affecting the current. 3.10: Electric Shock Data The table found in the Bussmann handbook differs slightly from the one available from MIT: for the DC threshold of perception (men), the MIT table gives 5.2 mA while the Bussmann table gives a slightly greater figure of 6.2 mA. Also, for the “unable to let go” 60 Hz AC threshold (men), the MIT table gives 20 mA while the Bussmann table gives a lesser figure of 16 mA. As I have yet to obtain a primary copy of Dalziel’s research, the figures cited here are conservative: I have listed the lowest values in my table where any data sources differ. These differences, of course, are academic. The point here is that relatively small magnitudes of electric current through the body can be harmful if not lethal. Data regarding the electrical resistance of body contact points was taken from a safety page (document 16.1) from the Lawrence Livermore National Laboratory (website [*]), citing Ralph H. Lee as the data source. Lee’s work was listed here in a document entitled “Human Electrical Sheet,” composed while he was an IEEE Fellow at E.I. duPont de Nemours & Co., and also in an article entitled “Electrical Safety in Industrial Plants” found in the June 1971 issue of IEEE Spectrum magazine. For the morbidly curious, Charles Dalziel’s experimentation conducted at the University of California (Berkeley) began with a state grant to investigate the bodily effects of sub-lethal electric current. His testing method was as follows: healthy male and female volunteer subjects were asked to hold a copper wire in one hand and place their other hand on a round, brass plate. A voltage was then applied between the wire and the plate, causing electrons to flow through the subject’s arms and chest. The current was stopped, then resumed at a higher level. The goal here was to see how much current the subject could tolerate and still keep their hand pressed against the brass plate. When this threshold was reached, laboratory assistants forcefully held the subject’s hand in contact with the plate and the current was again increased. The subject was asked to release the wire they were holding, to see at what current level involuntary muscle contraction (tetanus) prevented them from doing so. For each subject the experiment was conducted using DC and also AC at various frequencies. Over two dozen human volunteers were tested, and later studies on heart fibrillation were conducted using animal subjects.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/04%3A_Scientific_Notation_And_Metric_Prefixes/4.01%3A_Scientific_Notation.txt	princeton-nlp/TextbookChapters	In many disciplines of science and engineering, very large and very small numerical quantities must be managed. Some of these quantities are mind-boggling in their size, either extremely small or extremely large. Take for example the mass of a proton, one of the constituent particles of an atom’s nucleus: Proton mass = 0.00000000000000000000000167 grams Or, consider the number of electrons passing by a point in a circuit every second with a steady electric current of 1 amp: 1 amp = 6,250,000,000,000,000,000 electrons per second A lot of zeros, isn’t it? Obviously, it can get quite confusing to have to handle so many zero digits in numbers such as this, even with the help of calculators and computers. Take note of those two numbers and of the relative sparsity of non-zero digits in them. For the mass of the proton, all we have is a “167” preceded by 23 zeros before the decimal point. For the number of electrons per second in 1 amp, we have “625” followed by 16 zeros. We call the span of non-zero digits (from first to last), plus any zero digits not merely used for placeholding, the “significant digits” of any number. The significant digits in a real-world measurement are typically reflective of the accuracy of that measurement. For example, if we were to say that a car weighs 3,000 pounds, we probably don’t mean that the car in question weighs exactly 3,000 pounds, but that we’ve rounded its weight to a value more convenient to say and remember. That rounded figure of 3,000 has only one significant digit: the “3” in front—the zeros merely serve as placeholders. However, if we were to say that the car weighed 3,005 pounds, the fact that the weight is not rounded to the nearest thousand pounds tells us that the two zeros in the middle aren’t just placeholders, but that all four digits of the number “3,005” are significant to its representative accuracy. Thus, the number “3,005” is said to have four significant figures. In like manner, numbers with many zero digits are not necessarily representative of a real-world quantity all the way to the decimal point. When this is known to be the case, such a number can be written in a kind of mathematical “shorthand” to make it easier to deal with. This “shorthand” is called scientific notation. With scientific notation, a number is written by representing its significant digits as a quantity between 1 and 10 (or -1 and -10, for negative numbers), and the “placeholder” zeros are accounted for by a power-of-ten multiplier. For example: 1 amp = 6,250,000,000,000,000,000 electrons per second . . . can be expressed as . . . 1 amp = 6.25 x 1018 electrons per second 10 to the 18th power (1018) means 10 multiplied by itself 18 times, or a “1” followed by 18 zeros. Multiplied by 6.25, it looks like “625” followed by 16 zeros (take 6.25 and skip the decimal point 18 places to the right). The advantages of scientific notation are obvious: the number isn’t as unwieldy when written on paper, and the significant digits are plain to identify. But what about very small numbers, like the mass of the proton in grams? We can still use scientific notation, except with a negative power-of-ten instead of a positive one, to shift the decimal point to the left instead of to the right: Proton mass = 0.00000000000000000000000167 grams . . . can be expressed as . . . Proton mass = 1.67 x 10-24 grams 10 to the -24th power (10-24) means the inverse (1/x) of 10 multiplied by itself 24 times, or a “1” preceded by a decimal point and 23 zeros. Multiplied by 1.67, it looks like “167” preceded by a decimal point and 23 zeros. Just as in the case with the very large number, it is a lot easier for a human being to deal with this “shorthand” notation. As with the prior case, the significant digits in this quantity are clearly expressed. Because the significant digits are represented “on their own,” away from the power-of-ten multiplier, it is easy to show a level of precision even when the number looks round. Taking our 3,000 pound car example, we could express the rounded number of 3,000 in scientific notation as such: car weight = 3 x 103 pounds If the car actually weighed 3,005 pounds (accurate to the nearest pound) and we wanted to be able to express that full accuracy of measurement, the scientific notation figure could be written like this: car weight = 3.005 x 103 pounds However, what if the car actually did weigh 3,000 pounds, exactly (to the nearest pound)? If we were to write its weight in “normal” form (3,000 lbs), it wouldn’t necessarily be clear that this number was indeed accurate to the nearest pound and not just rounded to the nearest thousand pounds, or to the nearest hundred pounds, or to the nearest ten pounds. Scientific notation, on the other hand, allows us to show that all four digits are significant with no misunderstanding: car weight = 3.000 x 103 pounds Since there would be no point in adding extra zeros to the right of the decimal point (placeholding zeros being unnecessary with scientific notation), we know those zeros must be significant to the precision of the figure.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/04%3A_Scientific_Notation_And_Metric_Prefixes/4.02%3A_Arithmetic_with_Scientific_Notation.txt	princeton-nlp/TextbookChapters	The benefits of scientific notation do not end with ease of writing and expression of accuracy. Such notation also lends itself well to mathematical problems of multiplication and division. Let’s say we wanted to know how many electrons would flow past a point in a circuit carrying 1 amp of electric current in 25 seconds. If we know the number of electrons per second in the circuit (which we do), then all we need to do is multiply that quantity by the number of seconds (25) to arrive at an answer of total electrons: (6,250,000,000,000,000,000 electrons per second) x (25 seconds) = 156,250,000,000,000,000,000 electrons passing by in 25 seconds Using scientific notation, we can write the problem like this: (6.25 x 1018 electrons per second) x (25 seconds) If we take the “6.25” and multiply it by 25, we get 156.25. So, the answer could be written as: 156.25 x 1018 electrons However, if we want to hold to standard convention for scientific notation, we must represent the significant digits as a number between 1 and 10. In this case, we’d say “1.5625” multiplied by some power-of-ten. To obtain 1.5625 from 156.25, we have to skip the decimal point two places to the left. To compensate for this without changing the value of the number, we have to raise our power by two notches (10 to the 20th power instead of 10 to the 18th): 1.5625 x 1020 electrons What if we wanted to see how many electrons would pass by in 3,600 seconds (1 hour)? To make our job easier, we could put the time in scientific notation as well: (6.25 x 1018 electrons per second) x (3.6 x 103 seconds) To multiply, we must take the two significant sets of digits (6.25 and 3.6) and multiply them together; and we need to take the two powers-of-ten and multiply them together. Taking 6.25 times 3.6, we get 22.5. Taking 1018 times 103, we get 1021 (exponents with common base numbers add). So, the answer is: 22.5 x 1021 electrons . . . or more properly . . . 2.25 x 1022 electrons To illustrate how division works with scientific notation, we could figure that last problem “backwards” to find out how long it would take for that many electrons to pass by at a current of 1 amp: (2.25 x 1022 electrons) / (6.25 x 1018 electrons per second) Just as in multiplication, we can handle the significant digits and powers-of-ten in separate steps (remember that you subtract the exponents of divided powers-of-ten): (2.25 / 6.25) x (1022 / 1018) And the answer is: 0.36 x 104, or 3.6 x 103, seconds. You can see that we arrived at the same quantity of time (3600 seconds). Now, you may be wondering what the point of all this is when we have electronic calculators that can handle the math automatically. Well, back in the days of scientists and engineers using “slide rule” analog computers, these techniques were indispensable. The “hard” arithmetic (dealing with the significant digit figures) would be performed with the slide rule while the powers-of-ten could be figured without any help at all, being nothing more than simple addition and subtraction. Review • Significant digits are representative of the real-world accuracy of a number. • Scientific notation is a “shorthand” method to represent very large and very small numbers in easily-handled form. • When multiplying two numbers in scientific notation, you can multiply the two significant digit figures and arrive at a power-of-ten by adding exponents. • When dividing two numbers in scientific notation, you can divide the two significant digit figures and arrive at a power-of-ten by subtracting exponents.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/04%3A_Scientific_Notation_And_Metric_Prefixes/4.03%3A_Metric_Notation.txt	princeton-nlp/TextbookChapters	The metric system, besides being a collection of measurement units for all sorts of physical quantities, is structured around the concept of scientific notation. The primary difference is that the powers-of-ten are represented with alphabetical prefixes instead of by literal powers-of-ten. The following number line shows some of the more common prefixes and their respective powers-of-ten: Looking at this scale, we can see that 2.5 Gigabytes would mean 2.5 x 109 bytes, or 2.5 billion bytes. Likewise, 3.21 picoamps would mean 3.21 x 10-12 amps, or 3.21 1/trillionths of an amp. Other metric prefixes exist to symbolize powers of ten for extremely small and extremely large multipliers. On the extremely small end of the spectrum, femto (f) = 10-15, atto (a) = 10-18, zepto (z) = 10-21, and yocto(y) = 10-24. On the extremely large end of the spectrum, Peta (P) = 1015, Exa (E) = 1018, Zetta (Z) = 1021, and Yotta (Y) = 1024. Because the major prefixes in the metric system refer to powers of 10 that are multiples of 3 (from “kilo” on up, and from “milli” on down), metric notation differs from regular scientific notation in that the mantissa can be anywhere between 1 and 999, depending on which prefix is chosen. For example, if a laboratory sample weighs 0.000267 grams, scientific notation and metric notation would express it differently: 2.67 x 10-4 grams (scientific notation) 267 µgrams (metric notation) The same figure may also be expressed as 0.267 milligrams (0.267 mg), although it is usually more common to see the significant digits represented as a figure greater than 1. In recent years a new style of metric notation for electric quantities has emerged which seeks to avoid the use of the decimal point. Since decimal points (”.”) are easily misread and/or “lost” due to poor print quality, quantities such as 4.7 k may be mistaken for 47 k. The new notation replaces the decimal point with the metric prefix character, so that “4.7 k” is printed instead as “4k7”. Our last figure from the prior example, “0.267 m”, would be expressed in the new notation as “0m267”. Review • The metric system of notation uses alphabetical prefixes to represent certain powers-of-ten instead of the lengthier scientific notation. 4.04: Metric Prefix Conversions To express a quantity in a different metric prefix that what it was originally given, all we need to do is skip the decimal point to the right or to the left as needed. Notice that the metric prefix “number line” in the previous section was laid out from larger to smaller, left to right. This layout was purposely chosen to make it easier to remember which direction you need to skip the decimal point for any given conversion. Example problem: express 0.000023 amps in terms of microamps. 0.000023 amps (has no prefix, just plain unit of amps) From UNITS to micro on the number line is 6 places (powers of ten) to the right, so we need to skip the decimal point 6 places to the right: 0.000023 amps = 23. , or 23 microamps (µA) Example problem: express 304,212 volts in terms of kilovolts. 304,212 volts (has no prefix, just plain unit of volts) From the (none) place to kilo place on the number line is 3 places (powers of ten) to the left, so we need to skip the decimal point 3 places to the left: 304,212. = 304.212 kilovolts (kV) Example problem: express 50.3 Mega-ohms in terms of milli-ohms. 50.3 M ohms (mega = 106) From mega to milli is 9 places (powers of ten) to the right (from 10 to the 6th power to 10 to the -3rd power), so we need to skip the decimal point 9 places to the right: 50.3 M ohms = 50,300,000,000 milli-ohms (mΩ) Review • Follow the metric prefix number line to know which direction you skip the decimal point for conversion purposes. • A number with no decimal point shown has an implicit decimal point to the immediate right of the furthest right digit (i.e. for the number 436 the decimal point is to the right of the 6, as such: 436.) 4.05: Hand Calculator Use Scientific Notation with a Hand Calculator To enter numbers in scientific notation into a hand calculator, there is usually a button marked “E” or “EE” used to enter the correct power of ten. For example, to enter the mass of a proton in grams (1.67 x 10-24grams) into a hand calculator, I would enter the following keystrokes: The [+/-] keystroke changes the sign of the power (24) into a -24. Some calculators allow the use of the subtraction key [-] to do this, but I prefer the “change sign” [+/-] key because its more consistent with the use of that key in other contexts. If I wanted to enter a negative number in scientific notation into a hand calculator, I would have to be careful how I used the [+/-] key, lest I change the sign of the power and not the significant digit value. Pay attention to this example: Number to be entered: -3.221 x 10-15: The first [+/-] keystroke changes the entry from 3.221 to -3.221; the second [+/-] keystroke changes the power from 15 to -15. Metric and Scientific Notation Display Modes Displaying metric and scientific notation on a hand calculator is a different matter. It involves changing the display option from the normal “fixed” decimal point mode to the “scientific” or “engineering” mode. Your calculator manual will tell you how to set each display mode. These display modes tell the calculator how to represent any number on the numerical readout. The actual value of the number is not affected in any way by the choice of display modes—only how the number appears to the calculator user. Likewise, the procedure for entering numbers into the calculator does not change with different display modes either. Powers of ten are usually represented by a pair of digits in the upper-right hand corner of the display, and are visible only in the “scientific” and “engineering” modes. The difference between “scientific” and “engineering” display modes is the difference between scientific and metric notation. In “scientific” mode, the power-of-ten display is set so that the main number on the display is always a value between 1 and 10 (or -1 and -10 for negative numbers). In “engineering” mode, the powers-of-ten are set to display in multiples of 3, to represent the major metric prefixes. All the user has to do is memorize a few prefix/power combinations, and his or her calculator will be “speaking” metric! Review • Use the [EE] key to enter powers of ten. • Use “scientific” or “engineering” to display powers of ten, in scientific or metric notation, respectively.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/04%3A_Scientific_Notation_And_Metric_Prefixes/4.06%3A_Scientific_Notation_in_SPICE.txt	princeton-nlp/TextbookChapters	The SPICE circuit simulation computer program uses scientific notation to display its output information, and can interpret both scientific notation and metric prefixes in the circuit description files. If you are going to be able to successfully interpret the SPICE analyses throughout this book, you must be able to understand the notation used to express variables of voltage, current, etc. in the program. Let’s start with a very simple circuit composed of one voltage source (a battery) and one resistor: To simulate this circuit using SPICE, we first have to designate node numbers for all the distinct points in the circuit, then list the components along with their respective node numbers so the computer knows which component is connected to which, and how. For a circuit of this simplicity, the use of SPICE seems like overkill, but it serves the purpose of demonstrating practical use of scientific notation: Typing out a circuit description file, or netlist, for this circuit, we get this: The line “v1 1 0 dc 24” describes the battery, positioned between nodes 1 and 0, with a DC voltage of 24 volts. The line “r1 1 0 5” describes the 5 Ω resistor placed between nodes 1 and 0. Using a computer to run a SPICE analysis on this circuit description file, we get the following results: SPICE tells us that the voltage “at” node number 1 (actually, this means the voltage between nodes 1 and 0, node 0 being the default reference point for all voltage measurements) is equal to 24 volts. The current through battery “v1” is displayed as -4.800E+00 amps. This is SPICE’s method of denoting scientific notation. What its really saying is “-4.800 x 100 amps,” or simply -4.800 amps. The negative value for current here is due to a quirk in SPICE and does not indicate anything significant about the circuit itself. The “total power dissipation” is given to us as 1.15E+02 watts, which means “1.15 x 102 watts,” or 115 watts. Let’s modify our example circuit so that it has a 5 kΩ (5 kilo-ohm, or 5,000 ohm) resistor instead of a 5 Ω resistor and see what happens. Once again is our circuit description file, or “netlist:” The letter “k” following the number 5 on the resistor’s line tells SPICE that it is a figure of 5 kΩ, not 5 Ω. Let’s see what result we get when we run this through the computer: The battery voltage, of course, hasn’t changed since the first simulation: its still at 24 volts. The circuit current, on the other hand, is much less this time because we’ve made the resistor a larger value, making it more difficult for electrons to flow. SPICE tells us that the current this time is equal to -4.800E-03 amps, or -4.800 x 10-3 amps. This is equivalent to taking the number -4.8 and skipping the decimal point three places to the left. Of course, if we recognize that 10-3 is the same as the metric prefix “milli,” we could write the figure as -4.8 milliamps, or -4.8 mA. Looking at the “total power dissipation” given to us by SPICE on this second simulation, we see that it is 1.15E-01 watts, or 1.15 x 10-1 watts. The power of -1 corresponds to the metric prefix “deci,” but generally we limit our use of metric prefixes in electronics to those associated with powers of ten that are multiples of three (ten to the power of . . . -12, -9, -6, -3, 3, 6, 9, 12, etc.). So, if we want to follow this convention, we must express this power dissipation figure as 0.115 watts or 115 milliwatts (115 mW) rather than 1.15 deciwatts (1.15 dW). Perhaps the easiest way to convert a figure from scientific notation to common metric prefixes is with a scientific calculator set to the “engineering” or “metric” display mode. Just set the calculator for that display mode, type any scientific notation figure into it using the proper keystrokes (see your owner’s manual), press the “equals” or “enter” key, and it should display the same figure in engineering/metric notation. Again, I’ll be using SPICE as a method of demonstrating circuit concepts throughout this book. Consequently, it is in your best interest to understand scientific notation so you can easily comprehend its output data format.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.01%3A_What_are_Series_and_Parallel_Circuits.txt	princeton-nlp/TextbookChapters	Series and Parallel Circuits There are two basic ways in which to connect more than two circuit components: series and parallel. First, an example of a series circuit: Here, we have three resistors (labeled R1, R2, and R3), connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling—those little numbers to the lower-right of the letter “R”—are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The defining characteristic of a series circuit is that there is only one path for electrons to flow. In this circuit the electrons flow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4. Now, let’s look at the other type of circuit, a parallel configuration: Again, we have three resistors, but this time they form more than one continuous path for electrons to flow. There’s one path from 8 to 7 to 2 to 1 and back to 8 again. There’s another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there’s a third path from 8 to 7 to 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through R1, R2, and R3) is called a branch. The defining characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points. And, of course, the complexity doesn’t stop at simple series and parallel either! We can have circuits that are a combination of series and parallel, too: In this circuit, we have two loops for electrons to flow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through R1(from point 2 to point 1). In this configuration, we’d say that R2 and R3 are in parallel with each other, while R1 is in series with the parallel combination of R2 and R3. This is just a preview of things to come. Don’t worry! We’ll explore all these circuit configurations in detail, one at a time! Learn the Basic Ideas of Series and Parallel Connection The basic idea of a “series” connection is that components are connected end-to-end in a line to form a single path for electrons to flow: The basic idea of a “parallel” connection, on the other hand, is that all components are connected across each other’s leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to flow, but only one voltage across all components: Series and parallel resistor configurations have very different electrical properties. We’ll explore the properties of each configuration in the sections to come. Review • In a series circuit, all components are connected end-to-end, forming a single path for electrons to flow. • In a parallel circuit, all components are connected across each other, forming exactly two sets of electrically common points. • A “branch” in a parallel circuit is a path for electric current formed by one of the load components (such as a resistor).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.02%3A_Simple_Series_Circuits.txt	princeton-nlp/TextbookChapters	Let’s start with a series circuit consisting of three resistors and a single battery: The first principle to understand about series circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path for electrons to flow in a series circuit, and because free electrons flow through conductors like marbles in a tube, the rate of flow (marble speed) at any point in the circuit (tube) at any specific point in time must be equal. Using Ohm’s Law in Series Circuits From the way that the 9-volt battery is arranged, we can tell that the electrons in this circuit will flow in a counter-clockwise direction, from point 4 to 3 to 2 to 1 and back to 4. However, we have one source of voltage and three resistances. How do we use Ohm’s Law here? An important caveat to Ohm’s Law is that all quantities (voltage, current, resistance, and power) must relate to each other in terms of the same two points in a circuit. For instance, with a single-battery, single-resistor circuit, we could easily calculate any quantity because they all applied to the same two points in the circuit: Since points 1 and 2 are connected together with wire of negligible resistance, as are points 3 and 4, we can say that point 1 is electrically common to point 2, and that point 3 is electrically common to point 4. Since we know we have 9 volts of electromotive force between points 1 and 4 (directly across the battery), and since point 2 is common to point 1 and point 3 common to point 4, we must also have 9 volts between points 2 and 3 (directly across the resistor). Therefore, we can apply Ohm’s Law (I = E/R) to the current through the resistor, because we know the voltage (E) across the resistor and the resistance (R) of that resistor. All terms (E, I, R) apply to the same two points in the circuit, to that same resistor, so we can use the Ohm’s Law formula with no reservation. Using Ohm’s Law in Circuits with Multiple Resistors However, in circuits containing more than one resistor, we must be careful in how we apply Ohm’s Law. In the three-resistor example circuit below, we know that we have 9 volts between points 1 and 4, which is the amount of electromotive force trying to push electrons through the series combination of R1, R2, and R3. However, we cannot take the value of 9 volts and divide it by 3k, 10k or 5k Ω to try to find a current value, because we don’t know how much voltage is across any one of those resistors, individually. The figure of 9 volts is a total quantity for the whole circuit, whereas the figures of 3k, 10k, and 5k Ω are individual quantities for individual resistors. If we were to plug a figure for total voltage into an Ohm’s Law equation with a figure for individual resistance, the result would not relate accurately to any quantity in the real circuit. For R1, Ohm’s Law will relate the amount of voltage across R1 with the current through R1, given R1‘s resistance, 3kΩ: But, since we don’t know the voltage across R1 (only the total voltage supplied by the battery across the three-resistor series combination) and we don’t know the current through R1, we can’t do any calculations with either formula. The same goes for R2 and R3: we can apply the Ohm’s Law equations if and only if all terms are representative of their respective quantities between the same two points in the circuit. So what can we do? We know the voltage of the source (9 volts) applied across the series combination of R1, R2, and R3, and we know the resistance of each resistor, but since those quantities aren’t in the same context, we can’t use Ohm’s Law to determine the circuit current. If only we knew what the total resistance was for the circuit: then we could calculate total current with our figure for total voltage (I=E/R). This brings us to the second principle of series circuits: the total resistance of any series circuit is equal to the sum of the individual resistances. This should make intuitive sense: the more resistors in series that the electrons must flow through, the more difficult it will be for those electrons to flow. In the example problem, we had a 3 kΩ, 10 kΩ, and 5 kΩ resistor in series, giving us a total resistance of 18 kΩ: In essence, we’ve calculated the equivalent resistance of R1, R2, and R3 combined. Knowing this, we could re-draw the circuit with a single equivalent resistor representing the series combination of R1, R2, and R3: Calculating Circuit Current Now we have all the necessary information to calculate circuit current because we have the voltage between points 1 and 4 (9 volts) and the resistance between points 1 and 4 (18 kΩ): Knowing that current is equal through all components of a series circuit (and we just determined the current through the battery), we can go back to our original circuit schematic and note the current through each component: Now that we know the amount of current through each resistor, we can use Ohm’s Law to determine the voltage drop across each one (applying Ohm’s Law in its proper context): Notice the voltage drops across each resistor, and how the sum of the voltage drops (1.5 + 5 + 2.5) is equal to the battery (supply) voltage: 9 volts. This is the third principle of series circuits: that the supply voltage is equal to the sum of the individual voltage drops. However, the method we just used to analyze this simple series circuit can be streamlined for better understanding. By using a table to list all voltages, currents, and resistance in the circuit, it becomes very easy to see which of those quantities can be properly related in any Ohm’s Law equation: The rule with such a table is to apply Ohm’s Law only to the values within each vertical column. For instance, ER1 only with IR1 and R1; ER2 only with IR2 and R2; etc. You begin your analysis by filling in those elements of the table that are given to you from the beginning: As you can see from the arrangement of the data, we can’t apply the 9 volts of ET (total voltage) to any of the resistances (R1, R2, or R3) in any Ohm’s Law formula because they’re in different columns. The 9 volts of battery voltage is not applied directly across R1, R2, or R3. However, we can use our “rules” of series circuits to fill in blank spots on a horizontal row. In this case, we can use the series rule of resistances to determine a total resistance from the sum of individual resistances: Now, with a value for total resistance inserted into the rightmost (“Total”) column, we can apply Ohm’s Law of I=E/R to total voltage and total resistance to arrive at a total current of 500 µA: Then, knowing that the current is shared equally by all components of a series circuit (another “rule” of series circuits), we can fill in the currents for each resistor from the current figure just calculated: Finally, we can use Ohm’s Law to determine the voltage drop across each resistor, one column at a time: Verifying Calculations with Computer Analysis Just for fun, we can use a computer to analyze this very same circuit automatically. It will be a good way to verify our calculations and also become more familiar with computer analysis. First, we have to describe the circuit to the computer in a format recognizable by the software. The SPICE program we’ll be using requires that all electrically unique points in a circuit be numbered, and component placement is understood by which of those numbered points, or “nodes,” they share. For clarity, I numbered the four corners of our example circuit 1 through 4. SPICE, however, demands that there be a node zero somewhere in the circuit, so I’ll re-draw the circuit, changing the numbering scheme slightly: All I’ve done here is re-numbered the lower-left corner of the circuit 0 instead of 4. Now, I can enter several lines of text into a computer file describing the circuit in terms SPICE will understand, complete with a couple of extra lines of code directing the program to display voltage and current data for our viewing pleasure. This computer file is known as the netlist in SPICE terminology: Now, all I have to do is run the SPICE program to process the netlist and output the results: This printout is telling us the battery voltage is 9 volts, and the voltage drops across R1, R2, and R3 are 1.5 volts, 5 volts, and 2.5 volts, respectively. Voltage drops across any component in SPICE are referenced by the node numbers the component lies between, so v(1,2) is referencing the voltage between nodes 1 and 2 in the circuit, which are the points between which R1 is located. The order of node numbers is important: when SPICE outputs a figure for v(1,2), it regards the polarity the same way as if we were holding a voltmeter with the red test lead on node 1 and the black test lead on node 2. We also have a display showing current (albeit with a negative value) at 0.5 milliamps, or 500 microamps. So our mathematical analysis has been vindicated by the computer. This figure appears as a negative number in the SPICE analysis, due to a quirk in the way SPICE handles current calculations. Review of Basic Series Circuit Characteristics In summary, a series circuit is defined as having only one path for electrons to flow. From this definition, three rules of series circuits follow: all components share the same current; resistances add to equal a larger, total resistance; and voltage drops add to equal a larger, total voltage. All of these rules find root in the definition of a series circuit. If you understand that definition fully, then the rules are nothing more than footnotes to the definition. Review • Components in a series circuit share the same current: ITotal = I1 = I2 = . . . In • Total resistance in a series circuit is equal to the sum of the individual resistances: RTotal = R1 + R2 + . . . Rn • Total voltage in a series circuit is equal to the sum of the individual voltage drops: ETotal = E1 + E2 + . . . En
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.03%3A_Simple_Parallel_Circuits.txt	princeton-nlp/TextbookChapters	Let’s start with a parallel circuit consisting of three resistors and a single battery: The Principle of Parallel Circuits The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across R1 is equal to the voltage across R2 which is equal to the voltage across R3 which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values: Ohm’s Law Applications for Simple Parallel Circuits Just as in the case of series circuits, the same caveat for Ohm’s Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm’s Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor: At this point we still don’t know what the total current or total resistance for this parallel circuit is, so we can’t apply Ohm’s Law to the rightmost (“Total”) column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor (“branch”) currents: As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at point 7 to go up through R1, some more splits off at point 6 to go up through R2, and the remainder goes up through R3. Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The same thing is encountered where the currents through R1, R2, and R3 join to flow back to the positive terminal of the battery (+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through R1, R2, and R3. This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can fill in the IT spot on our table with the sum of IR1, IR2, and IR3: Finally, applying Ohm’s Law to the rightmost (“Total”) column, we can calculate the total circuit resistance: The Equation for Parallel Circuits Please note something very important here. The total circuit resistance is only 625 Ω: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of “rules” for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this: The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit. Just as with the series circuit, we can use computer analysis to double-check our calculations. First, of course, we have to describe our example circuit to the computer in terms it can understand. I’ll start by re-drawing the circuit: How to Alter Parallel Circuit Numbering Schemes for SPICE Once again we find that the original numbering scheme used to identify points in the circuit will have to be altered for the benefit of SPICE. In SPICE, all electrically common points must share identical node numbers. This is how SPICE knows what’s connected to what, and how. In a simple parallel circuit, all points are electrically common in one of two sets of points. For our example circuit, the wire connecting the tops of all the components will have one node number and the wire connecting the bottoms of the components will have the other. Staying true to the convention of including zero as a node number, I choose the numbers 0 and 1: An example like this makes the rationale of node numbers in SPICE fairly clear to understand. By having all components share common sets of numbers, the computer “knows” they’re all connected in parallel with each other. In order to display branch currents in SPICE, we need to insert zero-voltage sources in line (in series) with each resistor, and then reference our current measurements to those sources. For whatever reason, the creators of the SPICE program made it so that current could only be calculated through a voltage source. This is a somewhat annoying demand of the SPICE simulation program. With each of these “dummy” voltage sources added, some new node numbers must be created to connect them to their respective branch resistors: How to Verify Computer Analysis Results The dummy voltage sources are all set at 0 volts so as to have no impact on the operation of the circuit. The circuit description file, or netlist, looks like this: Running the computer analysis, we get these results (I’ve annotated the printout with descriptive labels): These values do indeed match those calculated through Ohm’s Law earlier: 0.9 mA for IR1, 4.5 mA for IR2, and 9 mA for IR3. Being connected in parallel, of course, all resistors have the same voltage dropped across them (9 volts, same as the battery). Three Rules of Parallel Circuits In summary, a parallel circuit is defined as one where all components are connected between the same set of electrically common points. Another way of saying this is that all components are connected across each other’s terminals. From this definition, three rules of parallel circuits follow: all components share the same voltage; resistances diminish to equal a smaller, total resistance; and branch currents add to equal a larger, total current. Just as in the case of series circuits, all of these rules find root in the definition of a parallel circuit. If you understand that definition fully, then the rules are nothing more than footnotes to the definition. Review • Components in a parallel circuit share the same voltage: ETotal = E1 = E2 = . . . En • Total resistance in a parallel circuit is less than any of the individual resistances: RTotal = 1 / (1/R1 + 1/R2+ . . . 1/Rn) • Total current in a parallel circuit is equal to the sum of the individual branch currents: ITotal = I1 + I2 + . . . In.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.04%3A_Conductance.txt	princeton-nlp/TextbookChapters	When students first see the parallel resistance equation, the natural question to ask is, “Where did that thing come from?” It is truly an odd piece of arithmetic, and its origin deserves a good explanation. What is the Difference Between Resistance and Conductance? Resistance, by definition, is the measure of friction a component presents to the flow of electrons through it. Resistance is symbolized by the capital letter “R” and is measured in the unit of “ohm.” However, we can also think of this electrical property in terms of its inverse: how easy it is for electrons to flow through a component, rather than how difficult. If resistance is the word we use to symbolize the measure of how difficult it is for electrons to flow, then a good word to express how easy it is for electrons to flow would be conductance. Mathematically, conductance is the reciprocal, or inverse, of resistance: The greater the resistance, the less the conductance, and vice versa. This should make intuitive sense, resistance and conductance being opposite ways to denote the same essential electrical property. If two components’ resistances are compared and it is found that component “A” has one-half the resistance of component “B,” then we could alternatively express this relationship by saying that component “A” is twiceas conductive as component “B.” If component “A” has but one-third the resistance of component “B,” then we could say it is three times more conductive than component “B,” and so on. Carrying this idea further, a symbol and unit were created to represent conductance. The symbol is the capital letter “G” and the unit is the mho, which is “ohm” spelled backwards (and you didn’t think electronics engineers had any sense of humor!). Despite its appropriateness, the unit of the mho was replaced in later years by the unit of siemens (abbreviated by the capital letter “S”). This decision to change unit names is reminiscent of the change from the temperature unit of degrees Centigrade to degrees Celsius, or the change from the unit of frequency c.p.s. (cycles per second) to Hertz. If you’re looking for a pattern here, Siemens, Celsius, and Hertz are all surnames of famous scientists, the names of which, sadly, tell us less about the nature of the units than the units’ original designations. As a footnote, the unit of siemens is never expressed without the last letter “s.” In other words, there is no such thing as a unit of “siemen” as there is in the case of the “ohm” or the “mho.” The reason for this is the proper spelling of the respective scientists’ surnames. The unit for electrical resistance was named after someone named “Ohm,” whereas the unit for electrical conductance was named after someone named “Siemens,” therefore it would be improper to “singularize” the latter unit as its final “s” does not denote plurality. Back to our parallel circuit example, we should be able to see that multiple paths (branches) for current reduces total resistance for the whole circuit, as electrons are able to flow easier through the whole network of multiple branches than through any one of those branch resistances alone. In terms of resistance, additional branches result in a lesser total (current meets with less opposition). In terms of conductance, however, additional branches results in a greater total (electrons flow with greater conductance): Total Parallel Resistance Total parallel resistance is less than any one of the individual branch resistances because parallel resistors resist less together than they would separately: Total Parallel Conductance Total parallel conductance is greater than any of the individual branch conductances because parallel resistors conduct better together than they would separately: To be more precise, the total conductance in a parallel circuit is equal to the sum of the individual conductances: If we know that conductance is nothing more than the mathematical reciprocal (1/x) of resistance, we can translate each term of the above formula into resistance by substituting the reciprocal of each respective conductance: Solving the above equation for total resistance (instead of the reciprocal of total resistance), we can invert (reciprocate) both sides of the equation: So, we arrive at our cryptic resistance formula at last! Conductance (G) is seldom used as a practical measurement, and so the above formula is a common one to see in the analysis of parallel circuits. Review • Conductance is the opposite of resistance: the measure of how easy it is for electrons to flow through something. • Conductance is symbolized with the letter “G” and is measured in units of mhos or Siemens. • Mathematically, conductance equals the reciprocal of resistance: G = 1/R
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.05%3A_Power_Calculations.txt	princeton-nlp/TextbookChapters	When calculating the power dissipation of resistive components, use any one of the three power equations to derive the answer from values of voltage, current, and/or resistance pertaining to each component: This is easily managed by adding another row to our familiar table of voltages, currents, and resistances: Power for any particular table column can be found by the appropriate Ohm’s Law equation (appropriate based on what figures are present for E, I, and R in that column). An interesting rule for total power versus individual power is that it is additive for any configuration of circuit: series, parallel, series/parallel, or otherwise. Power is a measure of rate of work, and since power dissipated must equal the total power applied by the source(s) (as per the Law of Conservation of Energy in physics), circuit configuration has no effect on the mathematics. Review • Power is additive in any configuration of resistive circuit: PTotal = P1 + P2 + . . . Pn 5.06: Correct use of Ohms Law One of the most common mistakes made by beginning electronics students in their application of Ohm’s Laws is mixing the contexts of voltage, current, and resistance. In other words, a student might mistakenly use a value for I through one resistor and the value for E across a set of interconnected resistors, thinking that they’ll arrive at the resistance of that one resistor. Not so! Remember this important rule: The variables used in Ohm’s Law equations must be common to the same two points in the circuit under consideration. I cannot overemphasize this rule. This is especially important in series-parallel combination circuits where nearby components may have different values for both voltage drop and current. When using Ohm’s Law to calculate a variable pertaining to a single component, be sure the voltage you’re referencing is solely across that single component and the current you’re referencing is solely through that single component and the resistance you’re referencing is solely for that single component. Likewise, when calculating a variable pertaining to a set of components in a circuit, be sure that the voltage, current, and resistance values are specific to that complete set of components only! A good way to remember this is to pay close attention to the two points terminating the component or set of components being analyzed, making sure that the voltage in question is across those two points, that the current in question is the electron flow from one of those points all the way to the other point, that the resistance in question is the equivalent of a single resistor between those two points, and that the power in question is the total power dissipated by all components between those two points. The “table” method presented for both series and parallel circuits in this chapter is a good way to keep the context of Ohm’s Law correct for any kind of circuit configuration. In a table like the one shown below, you are only allowed to apply an Ohm’s Law equation for the values of a single vertical column at a time: Deriving values horizontally across columns is allowable as per the principles of series and parallel circuits: Not only does the “table” method simplify the management of all relevant quantities, it also facilitates cross-checking of answers by making it easy to solve for the original unknown variables through other methods, or by working backwards to solve for the initially given values from your solutions. For example, if you have just solved for all unknown voltages, currents, and resistances in a circuit, you can check your work by adding a row at the bottom for power calculations on each resistor, seeing whether or not all the individual power values add up to the total power. If not, then you must have made a mistake somewhere! While this technique of “cross-checking” your work is nothing new, using the table to arrange all the data for the cross-check(s) results in a minimum of confusion. Review • Apply Ohm’s Law to vertical columns in the table. • Apply rules of series/parallel to horizontal rows in the table. • Check your calculations by working “backwards” to try to arrive at originally given values (from your first calculated answers), or by solving for a quantity using more than one method (from different given values).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.07%3A_Component_Failure_Analysis.txt	princeton-nlp/TextbookChapters	The job of a technician frequently entails “troubleshooting” (locating and correcting a problem) in malfunctioning circuits. Good troubleshooting is a demanding and rewarding effort, requiring a thorough understanding of the basic concepts, the ability to formulate hypotheses (proposed explanations of an effect), the ability to judge the value of different hypotheses based on their probability (how likely one particular cause may be over another), and a sense of creativity in applying a solution to rectify the problem. While it is possible to distill these skills into a scientific methodology, most practiced troubleshooters would agree that troubleshooting involves a touch of art, and that it can take years of experience to fully develop this art. An essential skill to have is a ready and intuitive understanding of how component faults affect circuits in different configurations. We will explore some of the effects of component faults in both series and parallel circuits here, then to a greater degree at the end of the “Series-Parallel Combination Circuits” chapter. Let’s start with a simple series circuit: With all components in this circuit functioning at their proper values, we can mathematically determine all currents and voltage drops: Now let us suppose that R2 fails shorted. Shorted means that the resistor now acts like a straight piece of wire, with little or no resistance. The circuit will behave as though a “jumper” wire were connected across R2(in case you were wondering, “jumper wire” is a common term for a temporary wire connection in a circuit). What causes the shorted condition of R2 is no matter to us in this example; we only care about its effect upon the circuit: With R2 shorted, either by a jumper wire or by an internal resistor failure, the total circuit resistance will decrease. Since the voltage output by the battery is a constant (at least in our ideal simulation here), a decrease in total circuit resistance means that total circuit current must increase: As the circuit current increases from 20 milliamps to 60 milliamps, the voltage drops across R1 and R3(which haven’t changed resistances) increase as well, so that the two resistors are dropping the whole 9 volts. R2, being bypassed by the very low resistance of the jumper wire, is effectively eliminated from the circuit, the resistance from one lead to the other having been reduced to zero. Thus, the voltage drop across R2, even with the increased total current, is zero volts. On the other hand, if R2 were to fail “open”—resistance increasing to nearly infinite levels—it would also create wide-reaching effects in the rest of the circuit: With R2 at infinite resistance and total resistance being the sum of all individual resistances in a series circuit, the total current decreases to zero. With zero circuit current, there is no electron flow to produce voltage drops across R1 or R3. R2, on the other hand, will manifest the full supply voltage across its terminals. We can apply the same before/after analysis technique to parallel circuits as well. First, we determine what a “healthy” parallel circuit should behave like. Supposing that R2 opens in this parallel circuit, here’s what the effects will be: Notice that in this parallel circuit, an open branch only affects the current through that branch and the circuit’s total current. Total voltage—being shared equally across all components in a parallel circuit, will be the same for all resistors. Due to the fact that the voltage source’s tendency is to hold voltage constant, its voltage will not change, and being in parallel with all the resistors, it will hold all the resistors’ voltages the same as they were before: 9 volts. Being that voltage is the only common parameter in a parallel circuit, and the other resistors haven’t changed resistance value, their respective branch currents remain unchanged. This is what happens in a household lamp circuit: all lamps get their operating voltage from power wiring arranged in a parallel fashion. Turning one lamp on and off (one branch in that parallel circuit closing and opening) doesn’t affect the operation of other lamps in the room, only the current in that one lamp (branch circuit) and the total current powering all the lamps in the room: In an ideal case (with perfect voltage sources and zero-resistance connecting wire), shorted resistors in a simple parallel circuit will also have no effect on what’s happening in other branches of the circuit. In real life, the effect is not quite the same, and we’ll see why in the following example: A shorted resistor (resistance of 0 Ω) would theoretically draw infinite current from any finite source of voltage (I=E/0). In this case, the zero resistance of R2 decreases the circuit total resistance to zero Ω as well, increasing total current to a value of infinity. As long as the voltage source holds steady at 9 volts, however, the other branch currents (IR1 and IR3) will remain unchanged. The critical assumption in this “perfect” scheme, however, is that the voltage supply will hold steady at its rated voltage while supplying an infinite amount of current to a short-circuit load. This is simply not realistic. Even if the short has a small amount of resistance (as opposed to absolutely zero resistance), no real voltage source could arbitrarily supply a huge overload current and maintain steady voltage at the same time. This is primarily due to the internal resistance intrinsic to all electrical power sources, stemming from the inescapable physical properties of the materials they’re constructed of: These internal resistances, small as they may be, turn our simple parallel circuit into a series-parallel combination circuit. Usually, the internal resistances of voltage sources are low enough that they can be safely ignored, but when high currents resulting from shorted components are encountered, their effects become very noticeable. In this case, a shorted R2 would result in almost all the voltage being dropped across the internal resistance of the battery, with almost no voltage left over for resistors R1, R2, and R3: Suffice it to say, intentional direct short-circuits across the terminals of any voltage source is a bad idea. Even if the resulting high current (heat, flashes, sparks) causes no harm to people nearby, the voltage source will likely sustain damage, unless it has been specifically designed to handle short-circuits, which most voltage sources are not. Eventually in this book I will lead you through the analysis of circuits without the use of any numbers, that is, analyzing the effects of component failure in a circuit without knowing exactly how many volts the battery produces, how many ohms of resistance is in each resistor, etc. This section serves as an introductory step to that kind of analysis. Whereas the normal application of Ohm’s Law and the rules of series and parallel circuits is performed with numerical quantities (“quantitative”), this new kind of analysis without precise numerical figures is something I like to call qualitative analysis. In other words, we will be analyzing the qualities of the effects in a circuit rather than the precise quantities. The result, for you, will be a much deeper intuitive understanding of electric circuit operation. Review • To determine what would happen in a circuit if a component fails, re-draw that circuit with the equivalent resistance of the failed component in place and re-calculate all values. • The ability to intuitively determine what will happen to a circuit with any given component fault is a crucial skill for any electronics troubleshooter to develop. The best way to learn is to experiment with circuit calculations and real-life circuits, paying close attention to what changes with a fault, what remains the same, and why! • A shorted component is one whose resistance has dramatically decreased. • An open component is one whose resistance has dramatically increased. For the record, resistors tend to fail open more often than fail shorted, and they almost never fail unless physically or electrically overstressed (physically abused or overheated).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/05%3A_Series_And_Parallel_Circuits/5.08%3A_Building_Simple_Resistor_Circuits.txt	princeton-nlp/TextbookChapters	How to Build a Simple Series Circuit If all we wish to construct is a simple single-battery, single-resistor circuit, we may easily use alligator clip jumper wires like this: Jumper wires with “alligator” style spring clips at each end provide a safe and convenient method of electrically joining components together. If we wanted to build a simple series circuit with one battery and three resistors, the same “point-to-point” construction technique using jumper wires could be applied: Using a Solderless Breadboard for More Complex Circuits This technique, however, proves impractical for circuits much more complex than this, due to the awkwardness of the jumper wires and the physical fragility of their connections. A more common method of temporary construction for the hobbyist is the solderless breadboard, a device made of plastic with hundreds of spring-loaded connection sockets joining the inserted ends of components and/or 22-gauge solid wire pieces. A photograph of a real breadboard is shown here, followed by an illustration showing a simple series circuit constructed on one: Underneath each hole in the breadboard face is a metal spring clip, designed to grasp any inserted wire or component lead. These metal spring clips are joined underneath the breadboard face, making connections between inserted leads. The connection pattern joins every five holes along a vertical column (as shown with the long axis of the breadboard situated horizontally): Thus, when a wire or component lead is inserted into a hole on the breadboard, there are four more holes in that column providing potential connection points to other wires and/or component leads. The result is an extremely flexible platform for constructing temporary circuits. For example, the three-resistor circuit just shown could also be built on a breadboard like this: A parallel circuit is also easy to construct on a solderless breadboard: Breadboards have their limitations, though. First and foremost, they are intended for temporary construction only. If you pick up a breadboard, turn it upside-down, and shake it, any components plugged into it are sure to loosen, and may fall out of their respective holes. Also, breadboards are limited to fairly low-current (less than 1 amp) circuits. Those spring clips have a small contact area, and thus cannot support high currents without excessive heating. Soldering or Wire-Wrapping For greater permanence, one might wish to choose soldering or wire-wrapping. These techniques involve fastening the components and wires to some structure providing a secure mechanical location (such as a phenolic or fiberglass board with holes drilled in it, much like a breadboard without the intrinsic spring-clip connections), and then attaching wires to the secured component leads. Soldering is a form of low-temperature welding, using a tin/lead or tin/silver alloy that melts to and electrically bonds copper objects. Wire ends soldered to component leads or to small, copper ring “pads” bonded on the surface of the circuit board serve to connect the components together. In wire wrapping, a small-gauge wire is tightly wrapped around component leads rather than soldered to leads or copper pads, the tension of the wrapped wire providing a sound mechanical and electrical junction to connect components together. An example of a printed circuit board, or PCB, intended for hobbyist use is shown in this photograph: This board appears copper-side-up: the side where all the soldering is done. Each hole is ringed with a small layer of copper metal for bonding to the solder. All holes are independent of each other on this particular board, unlike the holes on a solderless breadboard which are connected together in groups of five. Printed circuit boards with the same 5-hole connection pattern as breadboards can be purchased and used for hobby circuit construction, though. Printed Circuit Boards (PCBs) Production printed circuit boards have traces of copper laid down on the phenolic or fiberglass substrate material to form pre-engineered connection pathways which function as wires in a circuit. An example of such a board is shown here, this unit actually a “power supply” circuit designed to take 120 volt alternating current (AC) power from a household wall socket and transform it into low-voltage direct current (DC). A resistor appears on this board, the fifth component counting up from the bottom, located in the middle-right area of the board. A view of this board’s underside reveals the copper “traces” connecting components together, as well as the silver-colored deposits of solder bonding the component leads to those traces: A soldered or wire-wrapped circuit is considered permanent: that is, it is unlikely to fall apart accidently. However, these construction techniques are sometimes considered too permanent. If anyone wishes to replace a component or change the circuit in any substantial way, they must invest a fair amount of time undoing the connections. Also, both soldering and wire-wrapping require specialized tools which may not be immediately available. Terminal Strips An alternative construction technique used throughout the industrial world is that of the terminal strip. Terminal strips, alternatively called barrier strips or terminal blocks, are comprised of a length of nonconducting material with several small bars of metal embedded within. Each metal bar has at least one machine screw or other fastener under which a wire or component lead may be secured. Multiple wires fastened by one screw are made electrically common to each other, as are wires fastened to multiple screws on the same bar. The following photograph shows one style of terminal strip, with a few wires attached. Another, smaller terminal strip is shown in this next photograph. This type, sometimes referred to as a “European” style, has recessed screws to help prevent accidental shorting between terminals by a screwdriver or other metal object: In the following illustration, a single-battery, three-resistor circuit is shown constructed on a terminal strip: If the terminal strip uses machine screws to hold the component and wire ends, nothing but a screwdriver is needed to secure new connections or break old connections. Some terminal strips use spring-loaded clips—similar to a breadboard’s except for increased ruggedness—engaged and disengaged using a screwdriver as a push tool (no twisting involved). The electrical connections established by a terminal strip are quite robust and are considered suitable for both permanent and temporary construction. Translating a Schematic Diagram to a Circuit Layout One of the essential skills for anyone interested in electricity and electronics is to be able to “translate” a schematic diagram to a real circuit layout where the components may not be oriented the same way. Schematic diagrams are usually drawn for maximum readability (excepting those few noteworthy examples sketched to create maximum confusion!), but practical circuit construction often demands a different component orientation. Building simple circuits on terminal strips is one way to develop the spatial-reasoning skill of “stretching” wires to make the same connection paths. Consider the case of a single-battery, three-resistor parallel circuit constructed on a terminal strip: Progressing from a nice, neat, schematic diagram to the real circuit—especially when the resistors to be connected are physically arranged in a linear fashion on the terminal strip—is not obvious to many, so I’ll outline the process step-by-step. First, start with the clean schematic diagram and all components secured to the terminal strip, with no connecting wires: Next, trace the wire connection from one side of the battery to the first component in the schematic, securing a connecting wire between the same two points on the real circuit. I find it helpful to over-draw the schematic’s wire with another line to indicate what connections I’ve made in real life: Continue this process, wire by wire, until all connections in the schematic diagram have been accounted for. It might be helpful to regard common wires in a SPICE-like fashion: make all connections to a common wire in the circuit as one step, making sure each and every component with a connection to that wire actually has a connection to that wire before proceeding to the next. For the next step, I’ll show how the top sides of the remaining two resistors are connected together, being common with the wire secured in the previous step: With the top sides of all resistors (as shown in the schematic) connected together, and to the battery’s positive (+) terminal, all we have to do now is connect the bottom sides together and to the other side of the battery: Typically in industry, all wires are labeled with number tags, and electrically common wires bear the same tag number, just as they do in a SPICE simulation. In this case, we could label the wires 1 and 2: Another industrial convention is to modify the schematic diagram slightly so as to indicate actual wire connection points on the terminal strip. This demands a labeling system for the strip itself: a “TB” number (terminal block number) for the strip, followed by another number representing each metal bar on the strip. This way, the schematic may be used as a “map” to locate points in a real circuit, regardless of how tangled and complex the connecting wiring may appear to the eyes. This may seem excessive for the simple, three-resistor circuit shown here, but such detail is absolutely necessary for construction and maintenance of large circuits, especially when those circuits may span a great physical distance, using more than one terminal strip located in more than one panel or box. Review • A solderless breadboard is a device used to quickly assemble temporary circuits by plugging wires and components into electrically common spring-clips arranged underneath rows of holes in a plastic board. • Soldering is a low-temperature welding process utilizing a lead/tin or tin/silver alloy to bond wires and component leads together, usually with the components secured to a fiberglass board. • Wire-wrapping is an alternative to soldering, involving small-gauge wire tightly wrapped around component leads rather than a welded joint to connect components together. • A terminal strip, also known as a barrier strip or terminal block is another device used to mount components and wires to build circuits. Screw terminals or heavy spring clips attached to metal bars provide connection points for the wire ends and component leads, these metal bars mounted separately to a piece of nonconducting material such as plastic, bakelite, or ceramic.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/06%3A_Divider_Circuits_and_Kirchhoff's_Laws/6.01%3A_Voltage_Divider_Circuits.txt	princeton-nlp/TextbookChapters	Let’s analyze a simple series circuit, determining the voltage drops across individual resistors: Determine the Total Circuit Resistance From the given values of individual resistances, we can determine a total circuit resistance, knowing that resistances add in series: Use Ohm’s Law to Calculate Electron Flow From here, we can use Ohm’s Law (I=E/R) to determine the total current, which we know will be the same as each resistor current, currents being equal in all parts of a series circuit: Now, knowing that the circuit current is 2 mA, we can use Ohm’s Law (E=IR) to calculate voltage across each resistor: It should be apparent that the voltage drop across each resistor is proportional to its resistance, given that the current is the same through all resistors. Notice how the voltage across R2 is double that of the voltage across R1, just as the resistance of R2 is double that of R1. If we were to change the total voltage, we would find this proportionality of voltage drops remains constant: The voltage across R2 is still exactly twice that of R1‘s drop, despite the fact that the source voltage has changed. The proportionality of voltage drops (ratio of one to another) is strictly a function of resistance values. With a little more observation, it becomes apparent that the voltage drop across each resistor is also a fixed proportion of the supply voltage. The voltage across R1, for example, was 10 volts when the battery supply was 45 volts. When the battery voltage was increased to 180 volts (4 times as much), the voltage drop across R1 also increased by a factor of 4 (from 10 to 40 volts). The ratio between R1‘s voltage drop and total voltage, however, did not change: Likewise, none of the other voltage drop ratios changed with the increased supply voltage either: Voltage Divider Formula For this reason a series circuit is often called a voltage divider for its ability to proportion—or divide—the total voltage into fractional portions of constant ratio. With a little bit of algebra, we can derive a formula for determining series resistor voltage drop given nothing more than total voltage, individual resistance, and total resistance: The ratio of individual resistance to total resistance is the same as the ratio of individual voltage drop to total supply voltage in a voltage divider circuit. This is known as the voltage divider formula, and it is a short-cut method for determining voltage drop in a series circuit without going through the current calculation(s) of Ohm’s Law. Using this formula, we can re-analyze the example circuit’s voltage drops in fewer steps: Voltage dividers find wide application in electric meter circuits, where specific combinations of series resistors are used to “divide” a voltage into precise proportions as part of a voltage measurement device. One device frequently used as a voltage-dividing component is the potentiometer, which is a resistor with a movable element positioned by a manual knob or lever. The movable element, typically called a wiper, makes contact with a resistive strip of material (commonly called the slidewire if made of resistive metal wire) at any point selected by the manual control: The wiper contact is the left-facing arrow symbol drawn in the middle of the vertical resistor element. As it is moved up, it contacts the resistive strip closer to terminal 1 and further away from terminal 2, lowering resistance to terminal 1 and raising resistance to terminal 2. As it is moved down, the opposite effect results. The resistance as measured between terminals 1 and 2 is constant for any wiper position. Rotary vs. Linear Potentiometers Shown here are internal illustrations of two potentiometer types, rotary and linear: Some linear potentiometers are actuated by straight-line motion of a lever or slide button. Others, like the one depicted in the previous illustration, are actuated by a turn-screw for fine adjustment ability. The latter units are sometimes referred to as trimpots, because they work well for applications requiring a variable resistance to be “trimmed” to some precise value. It should be noted that not all linear potentiometers have the same terminal assignments as shown in this illustration. With some, the wiper terminal is in the middle, between the two end terminals. The following photograph shows a real, rotary potentiometer with exposed wiper and slidewire for easy viewing. The shaft which moves the wiper has been turned almost fully clockwise so that the wiper is nearly touching the left terminal end of the slidewire: Here is the same potentiometer with the wiper shaft moved almost to the full-counterclockwise position, so that the wiper is near the other extreme end of travel: If a constant voltage is applied between the outer terminals (across the length of the slidewire), the wiper position will tap off a fraction of the applied voltage, measurable between the wiper contact and either of the other two terminals. The fractional value depends entirely on the physical position of the wiper: The Importance of Potentiometer Application Just like the fixed voltage divider, the potentiometer’s voltage division ratio is strictly a function of resistance and not of the magnitude of applied voltage. In other words, if the potentiometer knob or lever is moved to the 50 percent (exact center) position, the voltage dropped between wiper and either outside terminal would be exactly 1/2 of the applied voltage, no matter what that voltage happens to be, or what the end-to-end resistance of the potentiometer is. In other words, a potentiometer functions as a variable voltage divider where the voltage division ratio is set by wiper position. This application of the potentiometer is a very useful means of obtaining a variable voltage from a fixed-voltage source such as a battery. If a circuit you’re building requires a certain amount of voltage that is less than the value of an available battery’s voltage, you may connect the outer terminals of a potentiometer across that battery and “dial-up” whatever voltage you need between the potentiometer wiper and one of the outer terminals for use in your circuit: When used in this manner, the name potentiometer makes perfect sense: they meter (control) the potential(voltage) applied across them by creating a variable voltage-divider ratio. This use of the three-terminal potentiometer as a variable voltage divider is very popular in circuit design. Shown here are several small potentiometers of the kind commonly used in consumer electronic equipment and by hobbyists and students in constructing circuits: The smaller units on the very left and very right are designed to plug into a solderless breadboard or be soldered into a printed circuit board. The middle units are designed to be mounted on a flat panel with wires soldered to each of the three terminals. Here are three more potentiometers, more specialized than the set just shown: The large “Helipot” unit is a laboratory potentiometer designed for quick and easy connection to a circuit. The unit in the lower-left corner of the photograph is the same type of potentiometer, just without a case or 10-turn counting dial. Both of these potentiometers are precision units, using multi-turn helical-track resistance strips and wiper mechanisms for making small adjustments. The unit on the lower-right is a panel-mount potentiometer, designed for rough service in industrial applications. Review • Series circuits proportion, or divide, the total supply voltage among individual voltage drops, the proportions being strictly dependent upon resistances: ERn = ETotal (Rn / RTotal) • A potentiometer is a variable-resistance component with three connection points, frequently used as an adjustable voltage divider.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/06%3A_Divider_Circuits_and_Kirchhoff's_Laws/6.02%3A_Kirchhoffs_Voltage_Law_%28KVL%29.txt	princeton-nlp/TextbookChapters	What is Kirchhoff’s Voltage Law (KVL)? The principle known as Kirchhoff’s Voltage Law (discovered in 1847 by Gustav R. Kirchhoff, a German physicist) can be stated as such: “The algebraic sum of all voltages in a loop must equal zero” By algebraic, I mean accounting for signs (polarities) as well as magnitudes. By loop, I mean any path traced from one point in a circuit around to other points in that circuit, and finally back to the initial point. Demonstrating Kirchhoff’s Voltage Law in a Series Circuit Let’s take another look at our series circuit from the previous page as an example, this time numbering the points in the circuit for voltage reference: If we were to connect a voltmeter between points 2 and 1, red test lead to point 2 and black test lead to point 1, the meter would register +45 volts. Typically the “+” sign is not shown, but rather implied, for positive readings in digital meter displays. However, for this lesson the polarity of the voltage reading is very important and so I will show positive numbers explicitly: When a voltage is specified with a double subscript (the characters “2-1” in the notation “E2-1”), it means the voltage at the first point (2) as measured in reference to the second point (1). A voltage specified as “Ecd” would mean the voltage as indicated by a digital meter with the red test lead on point “c” and the black test lead on point “d”: the voltage at “c” in reference to “d”. If we were to take that same voltmeter and measure the voltage drop across each resistor, stepping around the circuit in a clockwise direction with the red test lead of our meter on the point ahead and the black test lead on the point behind, we would obtain the following readings: We should already be familiar with the general principle for series circuits stating that individual voltage drops add up to the total applied voltage, but measuring voltage drops in this manner and paying attention to the polarity (mathematical sign) of the readings reveals another facet of this principle: that the voltages measured as such all add up to zero: In the above example, the loop was formed by following points in this order: 1-2-3-4-1. It doesn’t matter which point we start at or which direction we proceed in tracing the loop; the voltage sum will still equal zero. To demonstrate, we can tally up the voltages in loop 3-2-1-4-3 of the same circuit: This may make more sense if we re-draw our example series circuit so that all components are represented in a straight line: It’s still the same series circuit, just with the components arranged in a different form. Notice the polarities of the resistor voltage drops with respect to the battery: the battery’s voltage is negative on the left and positive on the right, whereas all the resistor voltage drops are oriented the other way: positive on the left and negative on the right. This is because the resistors are resisting the flow of electrons being pushed by the battery. In other words, the “push” exerted by the resistors against the flow of electrons must be in a direction opposite the source of electromotive force. Here we see what a digital voltmeter would indicate across each component in this circuit, black lead on the left and red lead on the right, as laid out in horizontal fashion: If we were to take that same voltmeter and read voltage across combinations of components, starting with only R1 on the left and progressing across the whole string of components, we will see how the voltages add algebraically (to zero): The fact that series voltages add up should be no mystery, but we notice that the polarity of these voltages makes a lot of difference in how the figures add. While reading voltage across R1, R1—R2, and R1—R2—R3(I’m using a “double-dash” symbol “—” to represent the series connection between resistors R1, R2, and R3), we see how the voltages measure successively larger (albeit negative) magnitudes, because the polarities of the individual voltage drops are in the same orientation (positive left, negative right). The sum of the voltage drops across R1, R2, and R3 equals 45 volts, which is the same as the battery’s output, except that the battery’s polarity is opposite that of the resistor voltage drops (negative left, positive right), so we end up with 0 volts measured across the whole string of components. That we should end up with exactly 0 volts across the whole string should be no mystery, either. Looking at the circuit, we can see that the far left of the string (left side of R1: point number 2) is directly connected to the far right of the string (right side of battery: point number 2), as necessary to complete the circuit. Since these two points are directly connected, they are electrically common to each other. And, as such, the voltage between those two electrically common points must be zero. Demonstrating Kirchhoff’s Voltage Law in a Parallel Circuit Kirchhoff’s Voltage Law (sometimes denoted as KVL for short) will work for any circuit configuration at all, not just simple series. Note how it works for this parallel circuit: Being a parallel circuit, the voltage across every resistor is the same as the supply voltage: 6 volts. Tallying up voltages around loop 2-3-4-5-6-7-2, we get: Note how I label the final (sum) voltage as E2-2. Since we began our loop-stepping sequence at point 2 and ended at point 2, the algebraic sum of those voltages will be the same as the voltage measured between the same point (E2-2), which of course must be zero. The Validity of Kirchhoff’s Voltage Law, Regardless of Circuit Topology The fact that this circuit is parallel instead of series has nothing to do with the validity of Kirchhoff’s Voltage Law. For that matter, the circuit could be a “black box”—its component configuration completely hidden from our view, with only a set of exposed terminals for us to measure voltage between—and KVL would still hold true: Try any order of steps from any terminal in the above diagram, stepping around back to the original terminal, and you’ll find that the algebraic sum of the voltages always equals zero. Furthermore, the “loop” we trace for KVL doesn’t even have to be a real current path in the closed-circuit sense of the word. All we have to do to comply with KVL is to begin and end at the same point in the circuit, tallying voltage drops and polarities as we go between the next and the last point. Consider this absurd example, tracing “loop” 2-3-6-3-2 in the same parallel resistor circuit: KVL can be used to determine an unknown voltage in a complex circuit, where all other voltages around a particular “loop” are known. Take the following complex circuit (actually two series circuits joined by a single wire at the bottom) as an example: To make the problem simpler, I’ve omitted resistance values and simply given voltage drops across each resistor. The two series circuits share a common wire between them (wire 7-8-9-10), making voltage measurements between the two circuits possible. If we wanted to determine the voltage between points 4 and 3, we could set up a KVL equation with the voltage between those points as the unknown: Stepping around the loop 3-4-9-8-3, we write the voltage drop figures as a digital voltmeter would register them, measuring with the red test lead on the point ahead and black test lead on the point behind as we progress around the loop. Therefore, the voltage from point 9 to point 4 is a positive (+) 12 volts because the “red lead” is on point 9 and the “black lead” is on point 4. The voltage from point 3 to point 8 is a positive (+) 20 volts because the “red lead” is on point 3 and the “black lead” is on point 8. The voltage from point 8 to point 9 is zero, of course, because those two points are electrically common. Our final answer for the voltage from point 4 to point 3 is a negative (-) 32 volts, telling us that point 3 is actually positive with respect to point 4, precisely what a digital voltmeter would indicate with the red lead on point 4 and the black lead on point 3: In other words, the initial placement of our “meter leads” in this KVL problem was “backwards.” Had we generated our KVL equation starting with E3-4 instead of E4-3, stepping around the same loop with the opposite meter lead orientation, the final answer would have been E3-4 = +32 volts: It is important to realize that neither approach is “wrong.” In both cases, we arrive at the correct assessment of voltage between the two points, 3 and 4: point 3 is positive with respect to point 4, and the voltage between them is 32 volts. Review • Kirchhoff’s Voltage Law (KVL): “The algebraic sum of all voltages in a loop must equal zero”
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/06%3A_Divider_Circuits_and_Kirchhoff's_Laws/6.03%3A_Current_Divider_Circuits_and_the_Current_Divider_Formula.txt	princeton-nlp/TextbookChapters	A parallel circuit is often called a current divider for its ability to proportion—or divide—the total current into fractional parts To understand what this means, let’s first analyze a simple parallel circuit, determining the branch currents through individual resistors: Knowing that voltages across all components in a parallel circuit are the same, we can fill in our voltage/current/resistance table with 6 volts across the top row: Using Ohm’s Law (I=E/R) we can calculate each branch current: Knowing that branch currents add up in parallel circuits to equal the total current, we can arrive at total current by summing 6 mA, 2 mA, and 3 mA: The final step, of course, is to figure total resistance. This can be done with Ohm’s Law (R=E/I) in the “total” column, or with the parallel resistance formula from individual resistances. Either way, we’ll get the same answer: Once again, it should be apparent that the current through each resistor is related to its resistance, given that the voltage across all resistors is the same. Rather than being directly proportional, the relationship here is one of inverse proportion. For example, the current through R1 is twice as much as the current through R3, which has twice the resistance of R1. If we were to change the supply voltage of this circuit, we find that (surprise!) these proportional ratios do not change: The current through R1 is still exactly twice that of R3, despite the fact that the source voltage has changed. The proportionality between different branch currents is strictly a function of resistance. Also reminiscent of voltage dividers is the fact that branch currents are fixed proportions of the total current. Despite the fourfold increase in supply voltage, the ratio between any branch current and the total current remains unchanged: Now we can see for ourselves the point we made at the beginning of this page: A parallel circuit is often called a current divider for its ability to proportion—or divide—the total current into fractional parts. The Current Divider Formula With a little bit of algebra, we can derive a formula for determining parallel resistor current given nothing more than total current, individual resistance, and total resistance: The ratio of total resistance to individual resistance is the same ratio as individual (branch) current to total current. This is known as the current divider formula and it is a short-cut method for determining branch currents in a parallel circuit when the total current is known. Current Divider Formula Example Using the original parallel circuit as an example, we can re-calculate the branch currents using this formula, if we start by knowing the total current and total resistance: If you take the time to compare the two divider formulae, you’ll see that they are remarkably similar. Notice, however, that the ratio in the voltage divider formula is Rn (individual resistance) divided by RTotal, and how the ratio in the current divider formula is RTotal divided by Rn: Current Divider Formula vs. Voltage Divider Formula It is quite easy to confuse these two equations, getting the resistance ratios backwards. One way to help remember the proper form is to keep in mind that both ratios in the voltage and current divider equations must equal less than one. After all these are divider equations, not multiplier equations! If the fraction is upside-down, it will provide a ratio greater than one, which is incorrect. Knowing that total resistance in a series (voltage divider) circuit is always greater than any of the individual resistances, we know that the fraction for that formula must be Rn over RTotal. Conversely, knowing that total resistance in a parallel (current divider) circuit is always less then any of the individual resistances, we know that the fraction for that formula must be RTotal over Rn. Current Divider Circuit Example Application: Electric Meter Circuit Current divider circuits find application in electric meter circuits, where a fraction of a measured current is desired to be routed through a sensitive detection device. Using the current divider formula, the proper shunt resistor can be sized to proportion just the right amount of current for the device in any given instance: Current Divider Circuit Review: • Parallel circuits proportion, or “divide,” the total circuit current among individual branch currents, the proportions being strictly dependent upon resistances: In = ITotal (RTotal / Rn)
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/06%3A_Divider_Circuits_and_Kirchhoff's_Laws/6.04%3A_Kirchhoffs_Current_Law_%28KCL%29.txt	princeton-nlp/TextbookChapters	What Is Kirchhoff’s Current Law? Kirchhoff’s Current Law, often shortened to KCL, states that “The algebraic sum of all currents entering and exiting a node must equal zero.” This law is used to describe how a charge enters and leaves a wire junction point or node on a wire. Armed with this information, let’s now take a look at an example of the law in practice, why it’s important, and how it was derived. Parallel Circuit Review Let’s take a closer look at that last parallel example circuit: Solving for all values of voltage and current in this circuit: At this point, we know the value of each branch current and of the total current in the circuit. We know that the total current in a parallel circuit must equal the sum of the branch currents, but there’s more going on in this circuit than just that. Taking a look at the currents at each wire junction point (node) in the circuit, we should be able to see something else: Currents Entering and Exiting a Node At each node on the negative “rail” (wire 8-7-6-5) we have current splitting off the main flow to each successive branch resistor. At each node on the positive “rail” (wire 1-2-3-4) we have current merging together to form the main flow from each successive branch resistor. This fact should be fairly obvious if you think of the water pipe circuit analogy with every branch node acting as a “tee” fitting, the water flow splitting or merging with the main piping as it travels from the output of the water pump toward the return reservoir or sump. If we were to take a closer look at one particular “tee” node, such as node 3, we see that the current entering the node is equal in magnitude to the current exiting the node: From the right and from the bottom, we have two currents entering the wire connection labeled as node 3. To the left, we have a single current exiting the node equal in magnitude to the sum of the two currents entering. To refer to the plumbing analogy: so long as there are no leaks in the piping, what flow enters the fitting must also exit the fitting. This holds true for any node (“fitting”), no matter how many flows are entering or exiting. Mathematically, we can express this general relationship as such: Kirchhoff’s Current Law Mr. Kirchhoff decided to express the above equation in a slightly different form (though mathematically equivalent), calling it Kirchhoff’s Current Law (KCL): Summarized in a phrase, Kirchhoff’s Current Law reads as such: “The algebraic sum of all currents entering and exiting a node must equal zero.” That is, if we assign a mathematical sign (polarity) to each current, denoting whether they enter (+) or exit (-) a node, we can add them together to arrive at a total of zero, guaranteed. Taking our example node (number 3), we can determine the magnitude of the current exiting from the left by setting up a KCL equation with that current as the unknown value: The negative (-) sign on the value of 5 milliamps tells us that the current is exiting the node, as opposed to the 2 milliamp and 3 milliamp currents, which must both be positive (and therefore entering the node). Whether negative or positive denotes current entering or exiting is entirely arbitrary, so long as they are opposite signs for opposite directions and we stay consistent in our notation, KCL will work. Together, Kirchhoff’s Voltage and Current Laws are a formidable pair of tools useful in analyzing electric circuits. Their usefulness will become all the more apparent in a later chapter (“Network Analysis”), but suffice it to say that these Laws deserve to be memorized by the electronics student every bit as much as Ohm’s Law. REVIEW • Kirchhoff’s Current Law (KCL): “The algebraic sum of all currents entering and exiting a node must equal zero”
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/07%3A_Series-parallel_Combination_Circuits/7.01%3A_What_is_a_Series-Parallel_Circuit.txt	princeton-nlp/TextbookChapters	With simple series circuits, all components are connected end-to-end to form only one path for electrons to flow through the circuit: With simple parallel circuits, all components are connected between the same two sets of electrically common points, creating multiple paths for electrons to flow from one end of the battery to the other: With each of these two basic circuit configurations, we have specific sets of rules describing voltage, current, and resistance relationships. • Series Circuits: • Voltage drops add to equal total voltage. • All components share the same (equal) current. • Resistances add to equal total resistance. • Parallel Circuits: • All components share the same (equal) voltage. • Branch currents add to equal total current. • Resistances diminish to equal total resistance. However, if circuit components are series-connected in some parts and parallel in others, we won’t be able to apply a single set of rules to every part of that circuit. Instead, we will have to identify which parts of that circuit are series and which parts are parallel, then selectively apply series and parallel rules as necessary to determine what is happening. Take the following circuit, for instance: This circuit is neither simple series nor simple parallel. Rather, it contains elements of both. The current exits the bottom of the battery, splits up to travel through R3 and R4, rejoins, then splits up again to travel through R1 and R2, then rejoins again to return to the top of the battery. There exists more than one path for current to travel (not series), yet there are more than two sets of electrically common points in the circuit (not parallel). Because the circuit is a combination of both series and parallel, we cannot apply the rules for voltage, current, and resistance “across the table” to begin analysis like we could when the circuits were one way or the other. For instance, if the above circuit were simple series, we could just add up R1 through R4 to arrive at a total resistance, solve for total current, and then solve for all voltage drops. Likewise, if the above circuit were simple parallel, we could just solve for branch currents, add up branch currents to figure the total current, and then calculate total resistance from total voltage and total current. However, this circuit’s solution will be more complex. The table will still help us manage the different values for series-parallel combination circuits, but we’ll have to be careful how and where we apply the different rules for series and parallel. Ohm’s Law, of course, still works just the same for determining values within a vertical column in the table. If we are able to identify which parts of the circuit are series and which parts are parallel, we can analyze it in stages, approaching each part one at a time, using the appropriate rules to determine the relationships of voltage, current, and resistance. The rest of this chapter will be devoted to showing you techniques for doing this. Review • The rules of series and parallel circuits must be applied selectively to circuits containing both types of interconnections.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/07%3A_Series-parallel_Combination_Circuits/7.02%3A_Analysis_Techniques_for_Series_Parallel_Resistor_Circuits.txt	princeton-nlp/TextbookChapters	Process of Series-Parallel Resistor Circuit Analysis The goal of series-parallel resistor circuit analysis is to be able to determine all voltage drops, currents, and power dissipations in a circuit. The general strategy to accomplish this goal is as follows: • Step 1: Assess which resistors in a circuit are connected together in simple series or simple parallel. • Step 2: Re-draw the circuit, replacing each of those series or parallel resistor combinations identified in step 1 with a single, equivalent-value resistor. If using a table to manage variables, make a new table column for each resistance equivalent. • Step 3: Repeat steps 1 and 2 until the entire circuit is reduced to one equivalent resistor. • Step 4: Calculate total current from total voltage and total resistance (I=E/R). • Step 5: Taking total voltage and total current values, go back to last step in the circuit reduction process and insert those values where applicable. • Step 6: From known resistances and total voltage / total current values from step 5, use Ohm’s Law to calculate unknown values (voltage or current) (E=IR or I=E/R). • Step 7: Repeat steps 5 and 6 until all values for voltage and current are known in the original circuit configuration. Essentially, you will proceed step-by-step from the simplified version of the circuit back into its original, complex form, plugging in values of voltage and current where appropriate until all values of voltage and current are known. • Step 8: Calculate power dissipations from known voltage, current, and/or resistance values. This may sound like an intimidating process, but its much easier understood through example than through description. In the example circuit above, R1 and R2 are connected in a simple parallel arrangement, as are R3 and R4. Having been identified, these sections need to be converted into equivalent single resistors, and the circuit re-drawn: The double slash (//) symbols represent “parallel” to show that the equivalent resistor values were calculated using the 1/(1/R) formula. The 71.429 Ω resistor at the top of the circuit is the equivalent of R1and R2 in parallel with each other. The 127.27 Ω resistor at the bottom is the equivalent of R3 and R4 in parallel with each other. Our table can be expanded to include these resistor equivalents in their own columns: It should be apparent now that the circuit has been reduced to a simple series configuration with only two (equivalent) resistances. The final step in reduction is to add these two resistances to come up with a total circuit resistance. When we add those two equivalent resistances, we get a resistance of 198.70 Ω. Now, we can re-draw the circuit as a single equivalent resistance and add the total resistance figure to the rightmost column of our table. Note that the “Total” column has been relabeled (R1//R2—R3//R4) to indicate how it relates electrically to the other columns of figures. The “—” symbol is used here to represent “series,” just as the “//” symbol is used to represent “parallel.” Now, total circuit current can be determined by applying Ohm’s Law (I=E/R) to the “Total” column in the table: Back to our equivalent circuit drawing, our total current value of 120.78 milliamps is shown as the only current here: Now we start to work backwards in our progression of circuit re-drawings to the original configuration. The next step is to go to the circuit where R1//R2 and R3//R4 are in series: Since R1//R2 and R3//R4 are in series with each other, the current through those two sets of equivalent resistances must be the same. Furthermore, the current through them must be the same as the total current, so we can fill in our table with the appropriate current values, simply copying the current figure from the Total column to the R1//R2 and R3//R4 columns: Now, knowing the current through the equivalent resistors R1//R2 and R3//R4, we can apply Ohm’s Law (E=IR) to the two right vertical columns to find voltage drops across them: Because we know R1//R2 and R3//R4 are parallel resistor equivalents, and we know that voltage drops in parallel circuits are the same, we can transfer the respective voltage drops to the appropriate columns on the table for those individual resistors. In other words, we take another step backwards in our drawing sequence to the original configuration, and complete the table accordingly: Finally, the original section of the table (columns R1 through R4) is complete with enough values to finish. Applying Ohm’s Law to the remaining vertical columns (I=E/R), we can determine the currents through R1, R2, R3, and R4 individually: Placing Voltage and Current Values into Diagrams Having found all voltage and current values for this circuit, we can show those values in the schematic diagram as such: As a final check of our work, we can see if the calculated current values add up as they should to the total. Since R1 and R2 are in parallel, their combined currents should add up to the total of 120.78 mA. Likewise, since R3 and R4 are in parallel, their combined currents should also add up to the total of 120.78 mA. You can check for yourself to verify that these figures do add up as expected. A computer simulation can also be used to verify the accuracy of these figures. The following SPICE analysis will show all resistor voltages and currents (note the current-sensing vi1, vi2, . . . “dummy” voltage sources in series with each resistor in the netlist, necessary for the SPICE computer program to track current through each path). These voltage sources will be set to have values of zero volts each so they will not affect the circuit in any way. I’ve annotated SPICE’s output figures to make them more readable, denoting which voltage and current figures belong to which resistors. As you can see, all the figures do agree with our calculated values. Review • To analyze a series-parallel combination circuit, follow these steps: • Reduce the original circuit to a single equivalent resistor, re-drawing the circuit in each step of reduction as simple series and simple parallel parts are reduced to single, equivalent resistors. • Solve for total resistance. • Solve for total current (I=E/R). • Determine equivalent resistor voltage drops and branch currents one stage at a time, working backwards to the original circuit configuration again.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/07%3A_Series-parallel_Combination_Circuits/7.03%3A_Re-drawing_Complex_Schematics.txt	princeton-nlp/TextbookChapters	Typically, complex circuits are not arranged in nice, neat, clean schematic diagrams for us to follow. They are often drawn in such a way that makes it difficult to follow which components are in series and which are in parallel with each other. The purpose of this section is to show you a method useful for re-drawing circuit schematics in a neat and orderly fashion. Like the stage-reduction strategy for solving series-parallel combination circuits, it is a method easier demonstrated than described. Let’s start with the following (convoluted) circuit diagram. Perhaps this diagram was originally drawn this way by a technician or engineer. Perhaps it was sketched as someone traced the wires and connections of a real circuit. In any case, here it is in all its ugliness: With electric circuits and circuit diagrams, the length and routing of wire connecting components in a circuit matters little. (Actually, in some AC circuits it becomes critical, and very long wire lengths can contribute unwanted resistance to both AC and DC circuits, but in most cases wire length is irrelevant.) What this means for us is that we can lengthen, shrink, and/or bend connecting wires without affecting the operation of our circuit. The strategy I have found easiest to apply is to start by tracing the current from one terminal of the battery around to the other terminal, following the loop of components closest to the battery and ignoring all other wires and components for the time being. While tracing the path of the loop, mark each resistor with the appropriate polarity for voltage drop. In this case, I’ll begin my tracing of this circuit at the negative terminal of the battery and finish at the positive terminal, in the same general direction as the electrons would flow. When tracing this direction, I will mark each resistor with the polarity of negative on the entering side and positive on the exiting side, for that is how the actual polarity will be as electrons (negative in charge) enter and exit a resistor: Any components encountered along this short loop are drawn vertically in order: Now, proceed to trace any loops of components connected around components that were just traced. In this case, there’s a loop around R1 formed by R2, and another loop around R3 formed by R4: Tracing those loops, I draw R2 and R4 in parallel with R1 and R3 (respectively) on the vertical diagram. Noting the polarity of voltage drops across R3 and R1, I mark R4 and R2 likewise: Now we have a circuit that is very easily understood and analyzed. In this case, it is identical to the four-resistor series-parallel configuration we examined earlier in the chapter. Let’s look at another example, even uglier than the one before: The first loop I’ll trace is from the negative (-) side of the battery, through R6, through R1, and back to the positive (+) end of the battery: Re-drawing vertically and keeping track of voltage drop polarities along the way, our equivalent circuit starts out looking like this: Next, we can proceed to follow the next loop around one of the traced resistors (R6), in this case, the loop formed by R5 and R7. As before, we start at the negative end of R6 and proceed to the positive end of R6, marking voltage drop polarities across R7 and R5 as we go: Now we add the R5—R7 loop to the vertical drawing. Notice how the voltage drop polarities across R7 and R5 correspond with that of R6, and how this is the same as what we found tracing R7 and R5 in the original circuit: We repeat the process again, identifying and tracing another loop around an already-traced resistor. In this case, the R3—R4 loop around R5 looks like a good loop to trace next: Adding the R3—R4 loop to the vertical drawing, marking the correct polarities as well: With only one remaining resistor left to trace, then next step is obvious: trace the loop formed by R2 around R3: Adding R2 to the vertical drawing, and we’re finished! The result is a diagram that’s very easy to understand compared to the original: This simplified layout greatly eases the task of determining where to start and how to proceed in reducing the circuit down to a single equivalent (total) resistance. Notice how the circuit has been re-drawn, all we have to do is start from the right-hand side and work our way left, reducing simple-series and simple-parallel resistor combinations one group at a time until we’re done. In this particular case, we would start with the simple parallel combination of R2 and R3, reducing it to a single resistance. Then, we would take that equivalent resistance (R2//R3) and the one in series with it (R4), reducing them to another equivalent resistance (R2//R3—R4). Next, we would proceed to calculate the parallel equivalent of that resistance (R2//R3—R4) with R5, then in series with R7, then in parallel with R6, then in series with R1 to give us a grand total resistance for the circuit as a whole. From there we could calculate total current from total voltage and total resistance (I=E/R), then “expand” the circuit back into its original form one stage at a time, distributing the appropriate values of voltage and current to the resistances as we go. Review • Wires in diagrams and in real circuits can be lengthened, shortened, and/or moved without affecting circuit operation. • To simplify a convoluted circuit schematic, follow these steps: • Trace current from one side of the battery to the other, following any single path (“loop”) to the battery. Sometimes it works better to start with the loop containing the most components, but regardless of the path taken the result will be accurate. Mark polarity of voltage drops across each resistor as you trace the loop. Draw those components you encounter along this loop in a vertical schematic. • Mark traced components in the original diagram and trace remaining loops of components in the circuit. Use polarity marks across traced components as guides for what connects where. Document new components in loops on the vertical re-draw schematic as well. • Repeat last step as often as needed until all components in original diagram have been traced.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/07%3A_Series-parallel_Combination_Circuits/7.04%3A_Component_Failure_Analysis_%28Continued%29.txt	princeton-nlp/TextbookChapters	“I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.” —P.A.M Dirac, physicist There is a lot of truth to that quote from Dirac. With a little modification, I can extend his wisdom to electric circuits by saying, “I consider that I understand a circuit when I can predict the approximate effects of various changes made to it without actually performing any calculations.” At the end of the series and parallel circuits chapter, we briefly considered how circuits could be analyzed in a qualitative rather than quantitative manner. Building this skill is an important step towards becoming a proficient troubleshooter of electric circuits. Once you have a thorough understanding of how any particular failure will affect a circuit (i.e. you don’t have to perform any arithmetic to predict the results), it will be much easier to work the other way around: pinpointing the source of trouble by assessing how a circuit is behaving. Also shown at the end of the series and parallel circuits chapter was how the table method works just as well for aiding failure analysis as it does for the analysis of healthy circuits. We may take this technique one step further and adapt it for total qualitative analysis. By “qualitative” I mean working with symbols representing “increase,” “decrease,” and “same” instead of precise numerical figures. We can still use the principles of series and parallel circuits, and the concepts of Ohm’s Law, we’ll just use symbolic qualities instead of numerical quantities. By doing this, we can gain more of an intuitive “feel” for how circuits work rather than leaning on abstract equations, attaining Dirac’s definition of “understanding.” Enough talk. Let’s try this technique on a real circuit example and see how it works: This is the first “convoluted” circuit we straightened out for analysis in the last section. Since you already know how this particular circuit reduces to series and parallel sections, I’ll skip the process and go straight to the final form: R3 and R4 are in parallel with each other; so are R1 and R2. The parallel equivalents of R3//R4 and R1//R2are in series with each other. Expressed in symbolic form, the total resistance for this circuit is as follows: RTotal = (R1//R2)—(R3//R4) First, we need to formulate a table with all the necessary rows and columns for this circuit: Next, we need a failure scenario. Let’s suppose that resistor R2 were to fail shorted. We will assume that all other components maintain their original values. Because we’ll be analyzing this circuit qualitatively rather than quantitatively, we won’t be inserting any real numbers into the table. For any quantity unchanged after the component failure, we’ll use the word “same” to represent “no change from before.” For any quantity that has changed as a result of the failure, we’ll use a down arrow for “decrease” and an up arrow for “increase.” As usual, we start by filling in the spaces of the table for individual resistances and total voltage, our “given” values: The only “given” value different from the normal state of the circuit is R2, which we said was failed shorted (abnormally low resistance). All other initial values are the same as they were before, as represented by the “same” entries. All we have to do now is work through the familiar Ohm’s Law and series-parallel principles to determine what will happen to all the other circuit values. First, we need to determine what happens to the resistances of parallel subsections R1//R2 and R3//R4. If neither R3 nor R4 have changed in resistance value, then neither will their parallel combination. However, since the resistance of R2 has decreased while R1 has stayed the same, their parallel combination must decrease in resistance as well: Now, we need to figure out what happens to the total resistance. This part is easy: when we’re dealing with only one component change in the circuit, the change in total resistance will be in the same direction as the change of the failed component. This is not to say that the magnitude of change between individual component and total circuit will be the same, merely the direction of change. In other words, if any single resistor decreases in value, then the total circuit resistance must also decrease, and vice versa. In this case, since R2 is the only failed component, and its resistance has decreased, the total resistance must decrease: Now we can apply Ohm’s Law (qualitatively) to the Total column in the table. Given the fact that total voltage has remained the same and total resistance has decreased, we can conclude that total current must increase (I=E/R). In case you’re not familiar with the qualitative assessment of an equation, it works like this. First, we write the equation as solved for the unknown quantity. In this case, we’re trying to solve for current, given voltage and resistance: Now that our equation is in the proper form, we assess what change (if any) will be experienced by “I,” given the change(s) to “E” and “R”: If the denominator of a fraction decreases in value while the numerator stays the same, then the overall value of the fraction must increase: Therefore, Ohm’s Law (I=E/R) tells us that the current (I) will increase. We’ll mark this conclusion in our table with an “up” arrow: With all resistance places filled in the table and all quantities determined in the Total column, we can proceed to determine the other voltages and currents. Knowing that the total resistance in this table was the result of R1//R2 and R3//R4 in series, we know that the value of total current will be the same as that in R1//R2 and R3//R4 (because series components share the same current). Therefore, if total current increased, then current through R1//R2 and R3//R4 must also have increased with the failure of R2: Fundamentally, what we’re doing here with a qualitative usage of Ohm’s Law and the rules of series and parallel circuits is no different from what we’ve done before with numerical figures. In fact, its a lot easier because you don’t have to worry about making an arithmetic or calculator keystroke error in a calculation. Instead, you’re just focusing on the principles behind the equations. From our table above, we can see that Ohm’s Law should be applicable to the R1//R2 and R3//R4 columns. For R3//R4, we figure what happens to the voltage, given an increase in current and no change in resistance. Intuitively, we can see that this must result in an increase in voltage across the parallel combination of R3//R4: But how do we apply the same Ohm’s Law formula (E=IR) to the R1//R2 column, where we have resistance decreasing and current increasing? It’s easy to determine if only one variable is changing, as it was with R3//R4, but with two variables moving around and no definite numbers to work with, Ohm’s Law isn’t going to be much help. However, there is another rule we can apply horizontally to determine what happens to the voltage across R1//R2: the rule for voltage in series circuits. If the voltages across R1//R2 and R3//R4 add up to equal the total (battery) voltage and we know that the R3//R4 voltage has increased while total voltage has stayed the same, then the voltage across R1//R2 must have decreased with the change of R2‘s resistance value: Now we’re ready to proceed to some new columns in the table. Knowing that R3 and R4 comprise the parallel subsection R3//R4, and knowing that voltage is shared equally between parallel components, the increase in voltage seen across the parallel combination R3//R4 must also be seen across R3 and R4individually: The same goes for R1 and R2. The voltage decrease seen across the parallel combination of R1 and R2 will be seen across R1 and R2 individually: Applying Ohm’s Law vertically to those columns with unchanged (“same”) resistance values, we can tell what the current will do through those components. Increased voltage across an unchanged resistance leads to increased current. Conversely, decreased voltage across an unchanged resistance leads to decreased current: Once again we find ourselves in a position where Ohm’s Law can’t help us: for R2, both voltage and resistance have decreased, but without knowing how much each one has changed, we can’t use the I=E/R formula to qualitatively determine the resulting change in current. However, we can still apply the rules of series and parallel circuits horizontally. We know that the current through the R1//R2 parallel combination has increased, and we also know that the current through R1 has decreased. One of the rules of parallel circuits is that total current is equal to the sum of the individual branch currents. In this case, the current through R1//R2 is equal to the current through R1 added to the current through R2. If current through R1//R2has increased while current through R1 has decreased, current through R2 must have increased: And with that, our table of qualitative values stands completed. This particular exercise may look laborious due to all the detailed commentary, but the actual process can be performed very quickly with some practice. An important thing to realize here is that the general procedure is little different from quantitative analysis: start with the known values, then proceed to determining total resistance, then total current, then transfer figures of voltage and current as allowed by the rules of series and parallel circuits to the appropriate columns. A few general rules can be memorized to assist and/or to check your progress when proceeding with such an analysis: • For any single component failure (open or shorted), the total resistance will always change in the same direction (either increase or decrease) as the resistance change of the failed component. • When a component fails shorted, its resistance always decreases. Also, the current through it will increase, and the voltage across it may drop. I say “may” because in some cases it will remain the same (case in point: a simple parallel circuit with an ideal power source). • When a component fails open, its resistance always increases. The current through that component will decrease to zero, because it is an incomplete electrical path (no continuity). This may result in an increase of voltage across it. The same exception stated above applies here as well: in a simple parallel circuit with an ideal voltage source, the voltage across an open-failed component will remain unchanged.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/07%3A_Series-parallel_Combination_Circuits/7.05%3A_Building_Series-Parallel_Resistor_Circuits.txt	princeton-nlp/TextbookChapters	Once again, when building battery/resistor circuits, the student or hobbyist is faced with several different modes of construction. Perhaps the most popular is the solderless breadboard: a platform for constructing temporary circuits by plugging components and wires into a grid of interconnected points. A breadboard appears to be nothing but a plastic frame with hundreds of small holes in it. Underneath each hole, though, is a spring clip which connects to other spring clips beneath other holes. The connection pattern between holes is simple and uniform: Suppose we wanted to construct the following series-parallel combination circuit on a breadboard: The recommended way to do so on a breadboard would be to arrange the resistors in approximately the same pattern as seen in the schematic, for ease of relation to the schematic. If 24 volts is required and we only have 6-volt batteries available, four may be connected in series to achieve the same effect: This is by no means the only way to connect these four resistors together to form the circuit shown in the schematic. Consider this alternative layout: If greater permanence is desired without resorting to soldering or wire-wrapping, one could choose to construct this circuit on a terminal strip (also called a barrier strip, or terminal block). In this method, components and wires are secured by mechanical tension underneath screws or heavy clips attached to small metal bars. The metal bars, in turn, are mounted on a nonconducting body to keep them electrically isolated from each other. Building a circuit with components secured to a terminal strip isn’t as easy as plugging components into a breadboard, principally because the components cannot be physically arranged to resemble the schematic layout. Instead, the builder must understand how to “bend” the schematic’s representation into the real-world layout of the strip. Consider one example of how the same four-resistor circuit could be built on a terminal strip: Another terminal strip layout, simpler to understand and relate to the schematic, involves anchoring parallel resistors (R1//R2 and R3//R4) to the same two terminal points on the strip like this: Building more complex circuits on a terminal strip involves the same spatial-reasoning skills, but of course requires greater care and planning. Take for instance this complex circuit, represented in schematic form: The terminal strip used in the prior example barely has enough terminals to mount all seven resistors required for this circuit! It will be a challenge to determine all the necessary wire connections between resistors, but with patience it can be done. First, begin by installing and labeling all resistors on the strip. The original schematic diagram will be shown next to the terminal strip circuit for reference: Next, begin connecting components together wire by wire as shown in the schematic. Over-draw connecting lines in the schematic to indicate completion in the real circuit. Watch this sequence of illustrations as each individual wire is identified in the schematic, then added to the real circuit: Although there are minor variations possible with this terminal strip circuit, the choice of connections shown in this example sequence is both electrically accurate (electrically identical to the schematic diagram) and carries the additional benefit of not burdening any one screw terminal on the strip with more than two wire ends, a good practice in any terminal strip circuit. An example of a “variant” wire connection might be the very last wire added (step 11), which I placed between the left terminal of R2 and the left terminal of R3. This last wire completed the parallel connection between R2 and R3 in the circuit. However, I could have placed this wire instead between the left terminal of R2 and the right terminal of R1, since the right terminal of R1 is already connected to the left terminal of R3(having been placed there in step 9) and so is electrically common with that one point. Doing this, though, would have resulted in three wires secured to the right terminal of R1 instead of two, which is a faux pax in terminal strip etiquette. Would the circuit have worked this way? Certainly! It’s just that more than two wires secured at a single terminal makes for a “messy” connection: one that is aesthetically unpleasing and may place undue stress on the screw terminal. Another variation would be to reverse the terminal connections for resistor R7. As shown in the last diagram, the voltage polarity across R7 is negative on the left and positive on the right (- , +), whereas all the other resistor polarities are positive on the left and negative on the right (+ , -): While this poses no electrical problem, it might cause confusion for anyone measuring resistor voltage drops with a voltmeter, especially an analog voltmeter which will “peg” downscale when subjected to a voltage of the wrong polarity. For the sake of consistency, it might be wise to arrange all wire connections so that all resistor voltage drop polarities are the same, like this: Though electrons do not care about such consistency in component layout, people do. This illustrates an important aspect of any engineering endeavor: the human factor. Whenever a design may be modified for easier comprehension and/or easier maintenance—with no sacrifice of functional performance—it should be done so. Review • Circuits built on terminal strips can be difficult to lay out, but when built they are robust enough to be considered permanent, yet easy to modify. • It is bad practice to secure more than two wire ends and/or component leads under a single terminal screw or clip on a terminal strip. Try to arrange connecting wires so as to avoid this condition. • Whenever possible, build your circuits with clarity and ease of understanding in mind. Even though component and wiring layout is usually of little consequence in DC circuit function, it matters significantly for the sake of the person who has to modify or troubleshoot it later.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.01%3A_What_is_a_Meter.txt	princeton-nlp/TextbookChapters	A meter is any device built to accurately detect and display an electrical quantity in a form readable by a human being. Usually this “readable form” is visual: motion of a pointer on a scale, a series of lights arranged to form a “bargraph,” or some sort of display composed of numerical figures. In the analysis and testing of circuits, there are meters designed to accurately measure the basic quantities of voltage, current, and resistance. There are many other types of meters as well, but this chapter primarily covers the design and operation of the basic three. Most modern meters are “digital” in design, meaning that their readable display is in the form of numerical digits. Older designs of meters are mechanical in nature, using some kind of pointer device to show quantity of measurement. In either case, the principles applied in adapting a display unit to the measurement of (relatively) large quantities of voltage, current, or resistance are the same. The display mechanism of a meter is often referred to as a movement, borrowing from its mechanical nature to move a pointer along a scale so that a measured value may be read. Though modern digital meters have no moving parts, the term “movement” may be applied to the same basic device performing the display function. The design of digital “movements” is beyond the scope of this chapter, but mechanical meter movement designs are very understandable. Most mechanical movements are based on the principle of electromagnetism: that electric current through a conductor produces a magnetic field perpendicular to the axis of electron flow. The greater the electric current, the stronger the magnetic field produced. If the magnetic field formed by the conductor is allowed to interact with another magnetic field, a physical force will be generated between the two sources of fields. If one of these sources is free to move with respect to the other, it will do so as current is conducted through the wire, the motion (usually against the resistance of a spring) being proportional to strength of current. The first meter movements built were known as galvanometers, and were usually designed with maximum sensitivity in mind. A very simple galvanometer may be made from a magnetized needle (such as the needle from a magnetic compass) suspended from a string, and positioned within a coil of wire. Current through the wire coil will produce a magnetic field which will deflect the needle from pointing in the direction of earth’s magnetic field. An antique string galvanometer is shown in the following photograph: Such instruments were useful in their time, but have little place in the modern world except as proof-of-concept and elementary experimental devices. They are highly susceptible to motion of any kind, and to any disturbances in the natural magnetic field of the earth. Now, the term “galvanometer” usually refers to any design of electromagnetic meter movement built for exceptional sensitivity, and not necessarily a crude device such as that shown in the photograph. Practical electromagnetic meter movements can be made now where a pivoting wire coil is suspended in a strong magnetic field, shielded from the majority of outside influences. Such an instrument design is generally known as a permanent-magnet, moving coil, or PMMCmovement: In the picture above, the meter movement “needle” is shown pointing somewhere around 35 percent of full-scale, zero being full to the left of the arc and full-scale being completely to the right of the arc. An increase in measured current will drive the needle to point further to the right and a decrease will cause the needle to drop back down toward its resting point on the left. The arc on the meter display is labeled with numbers to indicate the value of the quantity being measured, whatever that quantity is. In other words, if it takes 50 microamps of current to drive the needle fully to the right (making this a “50 µA full-scale movement”), the scale would have 0 µA written at the very left end and 50 µA at the very right, 25 µA being marked in the middle of the scale. In all likelihood, the scale would be divided into much smaller graduating marks, probably every 5 or 1 µA, to allow whoever is viewing the movement to infer a more precise reading from the needle’s position. The meter movement will have a pair of metal connection terminals on the back for current to enter and exit. Most meter movements are polarity-sensitive, one direction of current driving the needle to the right and the other driving it to the left. Some meter movements have a needle that is spring-centered in the middle of the scale sweep instead of to the left, thus enabling measurements of either polarity: Common polarity-sensitive movements include the D’Arsonval and Weston designs, both PMMC-type instruments. Current in one direction through the wire will produce a clockwise torque on the needle mechanism, while current the other direction will produce a counter-clockwise torque. Some meter movements are polarity-insensitive, relying on the attraction of an unmagnetized, movable iron vane toward a stationary, current-carrying wire to deflect the needle. Such meters are ideally suited for the measurement of alternating current (AC). A polarity-sensitive movement would just vibrate back and forth uselessly if connected to a source of AC. While most mechanical meter movements are based on electromagnetism (electron flow through a conductor creating a perpendicular magnetic field), a few are based on electrostatics: that is, the attractive or repulsive force generated by electric charges across space. This is the same phenomenon exhibited by certain materials (such as wax and wool) when rubbed together. If a voltage is applied between two conductive surfaces across an air gap, there will be a physical force attracting the two surfaces together capable of moving some kind of indicating mechanism. That physical force is directly proportional to the voltage applied between the plates, and inversely proportional to the square of the distance between the plates. The force is also irrespective of polarity, making this a polarity-insensitive type of meter movement: Unfortunately, the force generated by the electrostatic attraction is very small for common voltages. In fact, it is so small that such meter movement designs are impractical for use in general test instruments. Typically, electrostatic meter movements are used for measuring very high voltages (many thousands of volts). One great advantage of the electrostatic meter movement, however, is the fact that it has extremely high resistance, whereas electromagnetic movements (which depend on the flow of electrons through wire to generate a magnetic field) are much lower in resistance. As we will see in greater detail to come, greater resistance (resulting in less current drawn from the circuit under test) makes for a better voltmeter. A much more common application of electrostatic voltage measurement is seen in a device known as a Cathode Ray Tube, or CRT. These are special glass tubes, very similar to television viewscreen tubes. In the cathode ray tube, a beam of electrons traveling in a vacuum are deflected from their course by voltage between pairs of metal plates on either side of the beam. Because electrons are negatively charged, they tend to be repelled by the negative plate and attracted to the positive plate. A reversal of voltage polarity across the two plates will result in a deflection of the electron beam in the opposite direction, making this type of meter “movement” polarity-sensitive: The electrons, having much less mass than metal plates, are moved by this electrostatic force very quickly and readily. Their deflected path can be traced as the electrons impinge on the glass end of the tube where they strike a coating of phosphorus chemical, emitting a glow of light seen outside of the tube. The greater the voltage between the deflection plates, the further the electron beam will be “bent” from its straight path, and the further the glowing spot will be seen from center on the end of the tube. A photograph of a CRT is shown here: In a real CRT, as shown in the above photograph, there are two pairs of deflection plates rather than just one. In order to be able to sweep the electron beam around the whole area of the screen rather than just in a straight line, the beam must be deflected in more than one dimension. Although these tubes are able to accurately register small voltages, they are bulky and require electrical power to operate (unlike electromagnetic meter movements, which are more compact and actuated by the power of the measured signal current going through them). They are also much more fragile than other types of electrical metering devices. Usually, cathode ray tubes are used in conjunction with precise external circuits to form a larger piece of test equipment known as an oscilloscope, which has the ability to display a graph of voltage over time, a tremendously useful tool for certain types of circuits where voltage and/or current levels are dynamically changing. Whatever the type of meter or size of meter movement, there will be a rated value of voltage or current necessary to give full-scale indication. In electromagnetic movements, this will be the “full-scale deflection current” necessary to rotate the needle so that it points to the exact end of the indicating scale. In electrostatic movements, the full-scale rating will be expressed as the value of voltage resulting in the maximum deflection of the needle actuated by the plates, or the value of voltage in a cathode-ray tube which deflects the electron beam to the edge of the indicating screen. In digital “movements,” it is the amount of voltage resulting in a “full-count” indication on the numerical display: when the digits cannot display a larger quantity. The task of the meter designer is to take a given meter movement and design the necessary external circuitry for full-scale indication at some specified amount of voltage or current. Most meter movements (electrostatic movements excepted) are quite sensitive, giving full-scale indication at only a small fraction of a volt or an amp. This is impractical for most tasks of voltage and current measurement. What the technician often requires is a meter capable of measuring high voltages and currents. By making the sensitive meter movement part of a voltage or current divider circuit, the movement’s useful measurement range may be extended to measure far greater levels than what could be indicated by the movement alone. Precision resistors are used to create the divider circuits necessary to divide voltage or current appropriately. One of the lessons you will learn in this chapter is how to design these divider circuits. Review • A “movement” is the display mechanism of a meter. • Electromagnetic movements work on the principle of a magnetic field being generated by electric current through a wire. Examples of electromagnetic meter movements include the D’Arsonval, Weston, and iron-vane designs. • Electrostatic movements work on the principle of physical force generated by an electric field between two plates. • Cathode Ray Tubes (CRT’s) use an electrostatic field to bend the path of an electron beam, providing indication of the beam’s position by light created when the beam strikes the end of the glass tube.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.02%3A_Voltmeter_Design.txt	princeton-nlp/TextbookChapters	As was stated earlier, most meter movements are sensitive devices. Some D’Arsonval movements have full-scale deflection current ratings as little as 50 µA, with an (internal) wire resistance of less than 1000 Ω. This makes for a voltmeter with a full-scale rating of only 50 millivolts (50 µA X 1000 Ω)! In order to build voltmeters with practical (higher voltage) scales from such sensitive movements, we need to find some way to reduce the measured quantity of voltage down to a level the movement can handle. Let’s start our example problems with a D’Arsonval meter movement having a full-scale deflection rating of 1 mA and a coil resistance of 500 Ω: Using Ohm’s Law (E=IR), we can determine how much voltage will drive this meter movement directly to full scale: E = I R E = (1 mA)(500 Ω) E = 0.5 volts If all we wanted was a meter that could measure 1/2 of a volt, the bare meter movement we have here would suffice. But to measure greater levels of voltage, something more is needed. To get an effective voltmeter meter range in excess of 1/2 volt, we’ll need to design a circuit allowing only a precise proportion of measured voltage to drop across the meter movement. This will extend the meter movement’s range to higher voltages. Correspondingly, we will need to re-label the scale on the meter face to indicate its new measurement range with this proportioning circuit connected. But how do we create the necessary proportioning circuit? Well, if our intention is to allow this meter movement to measure a greater voltage than it does now, what we need is a voltage divider circuit to proportion the total measured voltage into a lesser fraction across the meter movement’s connection points. Knowing that voltage divider circuits are built from series resistances, we’ll connect a resistor in series with the meter movement (using the movement’s own internal resistance as the second resistance in the divider): The series resistor is called a “multiplier” resistor because it multiplies the working range of the meter movement as it proportionately divides the measured voltage across it. Determining the required multiplier resistance value is an easy task if you’re familiar with series circuit analysis. For example, let’s determine the necessary multiplier value to make this 1 mA, 500 Ω movement read exactly full-scale at an applied voltage of 10 volts. To do this, we first need to set up an E/I/R table for the two series components: Knowing that the movement will be at full-scale with 1 mA of current going through it, and that we want this to happen at an applied (total series circuit) voltage of 10 volts, we can fill in the table as such: There are a couple of ways to determine the resistance value of the multiplier. One way is to determine total circuit resistance using Ohm’s Law in the “total” column (R=E/I), then subtract the 500 Ω of the movement to arrive at the value for the multiplier: Another way to figure the same value of resistance would be to determine voltage drop across the movement at full-scale deflection (E=IR), then subtract that voltage drop from the total to arrive at the voltage across the multiplier resistor. Finally, Ohm’s Law could be used again to determine resistance (R=E/I) for the multiplier: Either way provides the same answer (9.5 kΩ), and one method could be used as verification for the other, to check accuracy of work. With exactly 10 volts applied between the meter test leads (from some battery or precision power supply), there will be exactly 1 mA of current through the meter movement, as restricted by the “multiplier” resistor and the movement’s own internal resistance. Exactly 1/2 volt will be dropped across the resistance of the movement’s wire coil, and the needle will be pointing precisely at full-scale. Having re-labeled the scale to read from 0 to 10 V (instead of 0 to 1 mA), anyone viewing the scale will interpret its indication as ten volts. Please take note that the meter user does not have to be aware at all that the movement itself is actually measuring just a fraction of that ten volts from the external source. All that matters to the user is that the circuit as a whole functions to accurately display the total, applied voltage. This is how practical electrical meters are designed and used: a sensitive meter movement is built to operate with as little voltage and current as possible for maximum sensitivity, then it is “fooled” by some sort of divider circuit built of precision resistors so that it indicates full-scale when a much larger voltage or current is impressed on the circuit as a whole. We have examined the design of a simple voltmeter here. Ammeters follow the same general rule, except that parallel-connected “shunt” resistors are used to create a current divider circuit as opposed to the series-connected voltage divider “multiplier” resistors used for voltmeter designs. Generally, it is useful to have multiple ranges established for an electromechanical meter such as this, allowing it to read a broad range of voltages with a single movement mechanism. This is accomplished through the use of a multi-pole switch and several multiplier resistors, each one sized for a particular voltage range: The five-position switch makes contact with only one resistor at a time. In the bottom (full clockwise) position, it makes contact with no resistor at all, providing an “off” setting. Each resistor is sized to provide a particular full-scale range for the voltmeter, all based on the particular rating of the meter movement (1 mA, 500 Ω). The end result is a voltmeter with four different full-scale ranges of measurement. Of course, in order to make this work sensibly, the meter movement’s scale must be equipped with labels appropriate for each range. With such a meter design, each resistor value is determined by the same technique, using a known total voltage, movement full-scale deflection rating, and movement resistance. For a voltmeter with ranges of 1 volt, 10 volts, 100 volts, and 1000 volts, the multiplier resistances would be as follows: Note the multiplier resistor values used for these ranges, and how odd they are. It is highly unlikely that a 999.5 kΩ precision resistor will ever be found in a parts bin, so voltmeter designers often opt for a variation of the above design which uses more common resistor values: With each successively higher voltage range, more multiplier resistors are pressed into service by the selector switch, making their series resistances add for the necessary total. For example, with the range selector switch set to the 1000 volt position, we need a total multiplier resistance value of 999.5 kΩ. With this meter design, that’s exactly what we’ll get: RTotal = R4 + R3 + R2 + R1 RTotal = 900 kΩ + 90 kΩ + 9 kΩ + 500 Ω RTotal = 999.5 kΩ The advantage, of course, is that the individual multiplier resistor values are more common (900k, 90k, 9k) than some of the odd values in the first design (999.5k, 99.5k, 9.5k). From the perspective of the meter user, however, there will be no discernible difference in function. Review • Extended voltmeter ranges are created for sensitive meter movements by adding series “multiplier” resistors to the movement circuit, providing a precise voltage division ratio.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.03%3A_Voltmeter_Impact_on_Measured_Circuit.txt	princeton-nlp/TextbookChapters	Every meter impacts the circuit it is measuring to some extent, just as any tire-pressure gauge changes the measured tire pressure slightly as some air is let out to operate the gauge. While some impact is inevitable, it can be minimized through good meter design. Voltage Divider Circuit Since voltmeters are always connected in parallel with the component or components under test, any current through the voltmeter will contribute to the overall current in the tested circuit, potentially affecting the voltage being measured. A perfect voltmeter has infinite resistance, so that it draws no current from the circuit under test. However, perfect voltmeters only exist in the pages of textbooks, not in real life! Take the following voltage divider circuit as an extreme example of how a realistic voltmeter might impact the circuit its measuring: With no voltmeter connected to the circuit, there should be exactly 12 volts across each 250 MΩ resistor in the series circuit, the two equal-value resistors dividing the total voltage (24 volts) exactly in half. However, if the voltmeter in question has a lead-to-lead resistance of 10 MΩ (a common amount for a modern digital voltmeter), its resistance will create a parallel subcircuit with the lower resistor of the divider when connected: This effectively reduces the lower resistance from 250 MΩ to 9.615 MΩ (250 MΩ and 10 MΩ in parallel), drastically altering voltage drops in the circuit. The lower resistor will now have far less voltage across it than before, and the upper resistor far more. Measured Voltage Divider A voltage divider with resistance values of 250 MΩ and 9.615 MΩ will divide 24 volts into portions of 23.1111 volts and 0.8889 volts, respectively. Since the voltmeter is part of that 9.615 MΩ resistance, that is what it will indicate: 0.8889 volts. Now, the voltmeter can only indicate the voltage its connected across. It has no way of “knowing” there was a potential of 12 volts dropped across the lower 250 MΩ resistor before it was connected across it. The very act of connecting the voltmeter to the circuit makes it part of the circuit, and the voltmeter’s own resistance alters the resistance ratio of the voltage divider circuit, consequently affecting the voltage being measured. Imagine using a tire pressure gauge that took so great a volume of air to operate that it would deflate any tire it was connected to. The amount of air consumed by the pressure gauge in the act of measurement is analogous to the current taken by the voltmeter movement to move the needle. The less air a pressure gauge requires to operate, the less it will deflate the tire under test. The less current drawn by a voltmeter to actuate the needle, the less it will burden the circuit under test. This effect is called loading, and it is present to some degree in every instance of voltmeter usage. The scenario shown here is worst-case, with a voltmeter resistance substantially lower than the resistances of the divider resistors. But there always will be some degree of loading, causing the meter to indicate less than the true voltage with no meter connected. Obviously, the higher the voltmeter resistance, the less loading of the circuit under test, and that is why an ideal voltmeter has infinite internal resistance. Voltmeters with electromechanical movements are typically given ratings in “ohms per volt” of range to designate the amount of circuit impact created by the current draw of the movement. Because such meters rely on different values of multiplier resistors to give different measurement ranges, their lead-to-lead resistances will change depending on what range they’re set to. Digital voltmeters, on the other hand, often exhibit a constant resistance across their test leads regardless of range setting (but not always!), and as such are usually rated simply in ohms of input resistance, rather than “ohms per volt” sensitivity. What “ohms per volt” means is how many ohms of lead-to-lead resistance for every volt of range setting on the selector switch. Let’s take our example voltmeter from the last section as an example: On the 1000 volt scale, the total resistance is 1 MΩ (999.5 kΩ + 500Ω), giving 1,000,000 Ω per 1000 volts of range, or 1000 ohms per volt (1 kΩ/V). This ohms-per-volt “sensitivity” rating remains constant for any range of this meter: The astute observer will notice that the ohms-per-volt rating of any meter is determined by a single factor: the full-scale current of the movement, in this case 1 mA. “Ohms per volt” is the mathematical reciprocal of “volts per ohm,” which is defined by Ohm’s Law as current (I=E/R). Consequently, the full-scale current of the movement dictates the Ω/volt sensitivity of the meter, regardless of what ranges the designer equips it with through multiplier resistors. In this case, the meter movement’s full-scale current rating of 1 mA gives it a voltmeter sensitivity of 1000 Ω/V regardless of how we range it with multiplier resistors. To minimize the loading of a voltmeter on any circuit, the designer must seek to minimize the current draw of its movement. This can be accomplished by re-designing the movement itself for maximum sensitivity (less current required for full-scale deflection), but the tradeoff here is typically ruggedness: a more sensitive movement tends to be more fragile. Another approach is to electronically boost the current sent to the movement, so that very little current needs to be drawn from the circuit under test. This special electronic circuit is known as an amplifier, and the voltmeter thus constructed is an amplified voltmeter. The internal workings of an amplifier are too complex to be discussed at this point, but suffice it to say that the circuit allows the measured voltage to control how much battery current is sent to the meter movement. Thus, the movement’s current needs are supplied by a battery internal to the voltmeter and not by the circuit under test. The amplifier still loads the circuit under test to some degree, but generally hundreds or thousands of times less than the meter movement would by itself. Before the advent of semiconductors known as “field-effect transistors,” vacuum tubes were used as amplifying devices to perform this boosting. Such vacuum-tube voltmeters, or (VTVM’s) were once very popular instruments for electronic test and measurement. Here is a photograph of a very old VTVM, with the vacuum tube exposed! Now, solid-state transistor amplifier circuits accomplish the same task in digital meter designs. While this approach (of using an amplifier to boost the measured signal current) works well, it vastly complicates the design of the meter, making it nearly impossible for the beginning electronics student to comprehend its internal workings. A final, and ingenious, solution to the problem of voltmeter loading is that of the potentiometric or null-balance instrument. It requires no advanced (electronic) circuitry or sensitive devices like transistors or vacuum tubes, but it does require greater technician involvement and skill. In a potentiometric instrument, a precision adjustable voltage source is compared against the measured voltage, and a sensitive device called a null detector is used to indicate when the two voltages are equal. In some circuit designs, a precision potentiometer is used to provide the adjustable voltage, hence the label potentiometric. When the voltages are equal, there will be zero current drawn from the circuit under test, and thus the measured voltage should be unaffected. It is easy to show how this works with our last example, the high-resistance voltage divider circuit: The “null detector” is a sensitive device capable of indicating the presence of very small voltages. If an electromechanical meter movement is used as the null detector, it will have a spring-centered needle that can deflect in either direction so as to be useful for indicating a voltage of either polarity. As the purpose of a null detector is to accurately indicate a condition of zero voltage, rather than to indicate any specific (nonzero) quantity as a normal voltmeter would, the scale of the instrument used is irrelevant. Null detectors are typically designed to be as sensitive as possible in order to more precisely indicate a “null” or “balance” (zero voltage) condition. An extremely simple type of null detector is a set of audio headphones, the speakers within acting as a kind of meter movement. When a DC voltage is initially applied to a speaker, the resulting current through it will move the speaker cone and produce an audible “click.” Another “click” sound will be heard when the DC source is disconnected. Building on this principle, a sensitive null detector may be made from nothing more than headphones and a momentary contact switch: If a set of “8 ohm” headphones are used for this purpose, its sensitivity may be greatly increased by connecting it to a device called a transformer. The transformer exploits principles of electromagnetism to “transform” the voltage and current levels of electrical energy pulses. In this case, the type of transformer used is a step-down transformer, and it converts low-current pulses (created by closing and opening the pushbutton switch while connected to a small voltage source) into higher-current pulses to more efficiently drive the speaker cones inside the headphones. An “audio output” transformer with an impedance ratio of 1000:8 is ideal for this purpose. The transformer also increases detector sensitivity by accumulating the energy of a low-current signal in a magnetic field for sudden release into the headphone speakers when the switch is opened. Thus, it will produce louder “clicks” for detecting smaller signals: Connected to the potentiometric circuit as a null detector, the switch/transformer/headphone arrangement is used as such: The purpose of any null detector is to act like a laboratory balance scale, indicating when the two voltages are equal (absence of voltage between points 1 and 2) and nothing more. The laboratory scale balance beam doesn’t actually weigh anything; rather, it simply indicates equality between the unknown mass and the pile of standard (calibrated) masses. Likewise, the null detector simply indicates when the voltage between points 1 and 2 are equal, which (according to Kirchhoff’s Voltage Law) will be when the adjustable voltage source (the battery symbol with a diagonal arrow going through it) is precisely equal in voltage to the drop across R2. To operate this instrument, the technician would manually adjust the output of the precision voltage source until the null detector indicated exactly zero (if using audio headphones as the null detector, the technician would repeatedly press and release the pushbutton switch, listening for silence to indicate that the circuit was “balanced”), and then note the source voltage as indicated by a voltmeter connected across the precision voltage source, that indication being representative of the voltage across the lower 250 MΩ resistor: The voltmeter used to directly measure the precision source need not have an extremely high Ω/V sensitivity, because the source will supply all the current it needs to operate. So long as there is zero voltage across the null detector, there will be zero current between points 1 and 2, equating to no loading of the divider circuit under test. It is worthy to reiterate the fact that this method, properly executed, places almost zero load upon the measured circuit. Ideally, it places absolutely no load on the tested circuit, but to achieve this ideal goal the null detector would have to have absolutely zero voltage across it, which would require an infinitely sensitive null meter and a perfect balance of voltage from the adjustable voltage source. However, despite its practical inability to achieve absolute zero loading, a potentiometric circuit is still an excellent technique for measuring voltage in high-resistance circuits. And unlike the electronic amplifier solution, which solves the problem with advanced technology, the potentiometric method achieves a hypothetically perfect solution by exploiting a fundamental law of electricity (KVL). Review • An ideal voltmeter has infinite resistance. • Too low of an internal resistance in a voltmeter will adversely affect the circuit being measured. • Vacuum tube voltmeters (VTVM’s), transistor voltmeters, and potentiometric circuits are all means of minimizing the load placed on a measured circuit. Of these methods, the potentiometric (“null-balance”) technique is the only one capable of placing zero load on the circuit. • A null detector is a device built for maximum sensitivity to small voltages or currents. It is used in potentiometric voltmeter circuits to indicate the absence of voltage between two points, thus indicating a condition of balance between an adjustable voltage source and the voltage being measured.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.04%3A_Ammeter_Design.txt	princeton-nlp/TextbookChapters	Ammeters Measure Electrical Current A meter designed to measure electrical current is popularly called an “ammeter” because the unit of measurement is “amps.” In ammeter designs, external resistors added to extend the usable range of the movement are connected in parallel with the movement rather than in series as is the case for voltmeters. This is because we want to divide the measured current, not the measured voltage, going to the movement and because current divider circuits are always formed by parallel resistances. Designing an Ammeter Taking the same meter movement as the voltmeter example, we can see that it would make a very limited instrument by itself, full-scale deflection occurring at only 1 mA. As is the case with extending a meter movement’s voltage-measuring ability, we would have to correspondingly re-label the movement’s scale so that it read differently for an extended current range. For example, if we wanted to design an ammeter to have a full-scale range of 5 amps using the same meter movement as before (having an intrinsic full-scale range of only 1 mA), we would have to re-label the movement’s scale to read 0 A on the far left and 5 A on the far right, rather than 0 mA to 1 mA as before. Whatever extended range provided by the parallel-connected resistors, we would have to represent graphically on the meter movement face. Using 5 amps as an extended range for our sample movement, let’s determine the amount of parallel resistance necessary to “shunt,” or bypass, the majority of current so that only 1 mA will go through the movement with a total current of 5 A: From our given values of movement current, movement resistance, and total circuit (measured) current, we can determine the voltage across the meter movement (Ohm’s Law applied to the center column, E=IR): Knowing that the circuit formed by the movement and the shunt is of a parallel configuration, we know that the voltage across the movement, shunt, and test leads (total) must be the same: We also know that the current through the shunt must be the difference between the total current (5 amps) and the current through the movement (1 mA), because branch currents add in a parallel configuration: Then, using Ohm’s Law (R=E/I) in the right column, we can determine the necessary shunt resistance: Of course, we could have calculated the same value of just over 100 milli-ohms (100 mΩ) for the shunt by calculating total resistance (R=E/I; 0.5 volts/5 amps = 100 mΩ exactly), then working the parallel resistance formula backwards, but the arithmetic would have been more challenging: An Ammeter in Real-Life Designs In real life, the shunt resistor of an ammeter will usually be encased within the protective metal housing of the meter unit, hidden from sight. Note the construction of the ammeter in the following photograph: This particular ammeter is an automotive unit manufactured by Stewart-Warner. Although the D’Arsonval meter movement itself probably has a full-scale rating in the range of milliamps, the meter as a whole has a range of +/- 60 amps. The shunt resistor providing this high current range is enclosed within the metal housing of the meter. Note also with this particular meter that the needle centers at zero amps and can indicate either a “positive” current or a “negative” current. Connected to the battery charging circuit of an automobile, this meter is able to indicate a charging condition (electrons flowing from generator to battery) or a discharging condition (electrons flowing from battery to the rest of the car’s loads). Increasing an Ammeter’s Usable Range As is the case with multiple-range voltmeters, ammeters can be given more than one usable range by incorporating several shunt resistors switched with a multi-pole switch: Notice that the range resistors are connected through the switch so as to be in parallel with the meter movement, rather than in series as it was in the voltmeter design. The five-position switch makes contact with only one resistor at a time, of course. Each resistor is sized accordingly for a different full-scale range, based on the particular rating of the meter movement (1 mA, 500 Ω). With such a meter design, each resistor value is determined by the same technique, using a known total current, movement full-scale deflection rating, and movement resistance. For an ammeter with ranges of 100 mA, 1 A, 10 A, and 100 A, the shunt resistances would be as such: Notice that these shunt resistor values are very low! 5.00005 mΩ is 5.00005 milli-ohms, or 0.00500005 ohms! To achieve these low resistances, ammeter shunt resistors often have to be custom-made from relatively large-diameter wire or solid pieces of metal. One thing to be aware of when sizing ammeter shunt resistors is the factor of power dissipation. Unlike the voltmeter, an ammeter’s range resistors have to carry large amounts of current. If those shunt resistors are not sized accordingly, they may overheat and suffer damage, or at the very least lose accuracy due to overheating. For the example meter above, the power dissipations at full-scale indication are (the double-squiggly lines represent “approximately equal to” in mathematics): A 1/8 watt resistor would work just fine for R4, a 1/2 watt resistor would suffice for R3 and a 5 watt for R2(although resistors tend to maintain their long-term accuracy better if not operated near their rated power dissipation, so you might want to over-rate resistors R2 and R3), but precision 50 watt resistors are rare and expensive components indeed. A custom resistor made from metal stock or thick wire may have to be constructed for R1 to meet both the requirements of low resistance and high power rating. Sometimes, shunt resistors are used in conjunction with voltmeters of high input resistance to measure current. In these cases, the current through the voltmeter movement is small enough to be considered negligible, and the shunt resistance can be sized according to how many volts or millivolts of drop will be produced per amp of current: If, for example, the shunt resistor in the above circuit were sized at precisely 1 Ω, there would be 1 volt dropped across it for every amp of current through it. The voltmeter indication could then be taken as a direct indication of current through the shunt. For measuring very small currents, higher values of shunt resistance could be used to generate more voltage drop per given unit of current, thus extending the usable range of the (volt)meter down into lower amounts of current. The use of voltmeters in conjunction with low-value shunt resistances for the measurement of current is something commonly seen in industrial applications. Using a Shunt Resistor and a Voltmeter Instead of an Ammeter The use of a shunt resistor along with a voltmeter to measure current can be a useful trick for simplifying the task of frequent current measurements in a circuit. Normally, to measure current through a circuit with an ammeter, the circuit would have to be broken (interrupted) and the ammeter inserted between the separated wire ends, like this: If we have a circuit where current needs to be measured often, or we would just like to make the process of current measurement more convenient, a shunt resistor could be placed between those points and left there permanently, current readings taken with a voltmeter as needed without interrupting continuity in the circuit: Of course, care must be taken in sizing the shunt resistor low enough so that it doesn’t adversely affect the circuit’s normal operation, but this is generally not difficult to do. This technique might also be useful in computer circuit analysis, where we might want to have the computer display current through a circuit in terms of a voltage (with SPICE, this would allow us to avoid the idiosyncrasy of reading negative current values): We would interpret the voltage reading across the shunt resistor (between circuit nodes 1 and 2 in the SPICE simulation) directly as amps, with 7.999E-04 being 0.7999 mA, or 799.9 µA. Ideally, 12 volts applied directly across 15 kΩ would give us exactly 0.8 mA, but the resistance of the shunt lessens that current just a tiny bit (as it would in real life). However, such a tiny error is generally well within acceptable limits of accuracy for either a simulation or a real circuit, and so shunt resistors can be used in all but the most demanding applications for accurate current measurement. Review • Ammeter ranges are created by adding parallel “shunt” resistors to the movement circuit, providing a precise current division. • Shunt resistors may have high power dissipations, so be careful when choosing parts for such meters! • Shunt resistors can be used in conjunction with high-resistance voltmeters as well as low-resistance ammeter movements, producing accurate voltage drops for given amounts of current. Shunt resistors should be selected for as low a resistance value as possible to minimize their impact upon the circuit under test.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.05%3A_Ammeter_Impact_on_Measured_Circuit.txt	princeton-nlp/TextbookChapters	Just like voltmeters, ammeters tend to influence the amount of current in the circuits they’re connected to. However, unlike the ideal voltmeter, the ideal ammeter has zero internal resistance, so as to drop as little voltage as possible as electrons flow through it. Note that this ideal resistance value is exactly opposite as that of a voltmeter. With voltmeters, we want as little current to be drawn as possible from the circuit under test. With ammeters, we want as little voltage to be dropped as possible while conducting current. Here is an extreme example of an ammeter’s effect upon a circuit: With the ammeter disconnected from this circuit, the current through the 3 Ω resistor would be 666.7 mA, and the current through the 1.5 Ω resistor would be 1.33 amps. If the ammeter had an internal resistance of 1/2 Ω, and it were inserted into one of the branches of this circuit, though, its resistance would seriously affect the measured branch current: Having effectively increased the left branch resistance from 3 Ω to 3.5 Ω, the ammeter will read 571.43 mA instead of 666.7 mA. Placing the same ammeter in the right branch would affect the current to an even greater extent: Now the right branch current is 1 amp instead of 1.333 amps, due to the increase in resistance created by the addition of the ammeter into the current path. When using standard ammeters that connect in series with the circuit being measured, it might not be practical or possible to redesign the meter for a lower input (lead-to-lead) resistance. However, if we were selecting a value of shunt resistor to place in the circuit for a current measurement based on voltage drop, and we had our choice of a wide range of resistances, it would be best to choose the lowest practical resistance for the application. Any more resistance than necessary and the shunt may impact the circuit adversely by adding excessive resistance in the current path. One ingenious way to reduce the impact that a current-measuring device has on a circuit is to use the circuit wire as part of the ammeter movement itself. All current-carrying wires produce a magnetic field, the strength of which is in direct proportion to the strength of the current. By building an instrument that measures the strength of that magnetic field, a no-contact ammeter can be produced. Such a meter is able to measure the current through a conductor without even having to make physical contact with the circuit, much less break continuity or insert additional resistance. Ammeters of this design are made, and are called “clamp-on” meters because they have “jaws” which can be opened and then secured around a circuit wire. Clamp-on ammeters make for quick and safe current measurements, especially on high-power industrial circuits. Because the circuit under test has had no additional resistance inserted into it by a clamp-on meter, there is no error induced in taking a current measurement. The actual movement mechanism of a clamp-on ammeter is much the same as for an iron-vane instrument, except that there is no internal wire coil to generate the magnetic field. More modern designs of clamp-on ammeters utilize a small magnetic field detector device called a Hall-effect sensor to accurately determine field strength. Some clamp-on meters contain electronic amplifier circuitry to generate a small voltage proportional to the current in the wire between the jaws, that small voltage connected to a voltmeter for convenient readout by a technician. Thus, a clamp-on unit can be an accessory device to a voltmeter, for current measurement. A less accurate type of magnetic-field-sensing ammeter than the clamp-on style is shown in the following photograph: The operating principle for this ammeter is identical to the clamp-on style of meter: the circular magnetic field surrounding a current-carrying conductor deflects the meter’s needle, producing an indication on the scale. Note how there are two current scales on this particular meter: +/- 75 amps and +/- 400 amps. These two measurement scales correspond to the two sets of notches on the back of the meter. Depending on which set of notches the current-carrying conductor is laid in, a given strength of magnetic field will have a different amount of effect on the needle. In effect, the two different positions of the conductor relative to the movement act as two different range resistors in a direct-connection style of ammeter. Review • An ideal ammeter has zero resistance. • A “clamp-on” ammeter measures current through a wire by measuring the strength of the magnetic field around it rather than by becoming part of the circuit, making it an ideal ammeter. • Clamp-on meters make for quick and safe current measurements, because there is no conductive contact between the meter and the circuit.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.06%3A_Ohmmeter_Design.txt	princeton-nlp/TextbookChapters	Though mechanical ohmmeter (resistance meter) designs are rarely used today, having largely been superseded by digital instruments, their operation is nonetheless intriguing and worthy of study. The purpose of an ohmmeter, of course, is to measure the resistance placed between its leads. This resistance reading is indicated through a mechanical meter movement which operates on electric current. The ohmmeter must then have an internal source of voltage to create the necessary current to operate the movement, and also have appropriate ranging resistors to allow just the right amount of current through the movement at any given resistance. Starting with a simple movement and battery circuit, let’s see how it would function as an ohmmeter: When there is infinite resistance (no continuity between test leads), there is zero current through the meter movement, and the needle points toward the far left of the scale. In this regard, the ohmmeter indication is “backwards” because maximum indication (infinity) is on the left of the scale, while voltage and current meters have zero at the left of their scales. If the test leads of this ohmmeter are directly shorted together (measuring zero Ω), the meter movement will have a maximum amount of current through it, limited only by the battery voltage and the movement’s internal resistance: With 9 volts of battery potential and only 500 Ω of movement resistance, our circuit current will be 18 mA, which is far beyond the full-scale rating of the movement. Such an excess of current will likely damage the meter. Not only that, but having such a condition limits the usefulness of the device. If full left-of-scale on the meter face represents an infinite amount of resistance, then full right-of-scale should represent zero. Currently, our design “pegs” the meter movement hard to the right when zero resistance is attached between the leads. We need a way to make it so that the movement just registers full-scale when the test leads are shorted together. This is accomplished by adding a series resistance to the meter’s circuit: To determine the proper value for R, we calculate the total circuit resistance needed to limit current to 1 mA (full-scale deflection on the movement) with 9 volts of potential from the battery, then subtract the movement’s internal resistance from that figure: Now that the right value for R has been calculated, we’re still left with a problem of meter range. On the left side of the scale we have “infinity” and on the right side we have zero. Besides being “backwards” from the scales of voltmeters and ammeters, this scale is strange because it goes from nothing to everything, rather than from nothing to a finite value (such as 10 volts, 1 amp, etc.). One might pause to wonder, “what does middle-of-scale represent? What figure lies exactly between zero and infinity?” Infinity is more than just a very big amount: it is an incalculable quantity, larger than any definite number ever could be. If half-scale indication on any other type of meter represents 1/2 of the full-scale range value, then what is half of infinity on an ohmmeter scale? The answer to this paradox is a nonlinear scale. Simply put, the scale of an ohmmeter does not smoothly progress from zero to infinity as the needle sweeps from right to left. Rather, the scale starts out “expanded” at the right-hand side, with the successive resistance values growing closer and closer to each other toward the left side of the scale: Infinity cannot be approached in a linear (even) fashion, because the scale would never get there! With a nonlinear scale, the amount of resistance spanned for any given distance on the scale increases as the scale progresses toward infinity, making infinity an attainable goal. We still have a question of range for our ohmmeter, though. What value of resistance between the test leads will cause exactly 1/2 scale deflection of the needle? If we know that the movement has a full-scale rating of 1 mA, then 0.5 mA (500 µA) must be the value needed for half-scale deflection. Following our design with the 9 volt battery as a source we get: With an internal movement resistance of 500 Ω and a series range resistor of 8.5 kΩ, this leaves 9 kΩ for an external (lead-to-lead) test resistance at 1/2 scale. In other words, the test resistance giving 1/2 scale deflection in an ohmmeter is equal in value to the (internal) series total resistance of the meter circuit. Using Ohm’s Law a few more times, we can determine the test resistance value for 1/4 and 3/4 scale deflection as well: 1/4 scale deflection (0.25 mA of meter current): 3/4 scale deflection (0.75 mA of meter current): So, the scale for this ohmmeter looks something like this: One major problem with this design is its reliance upon a stable battery voltage for accurate resistance reading. If the battery voltage decreases (as all chemical batteries do with age and use), the ohmmeter scale will lose accuracy. With the series range resistor at a constant value of 8.5 kΩ and the battery voltage decreasing, the meter will no longer deflect full-scale to the right when the test leads are shorted together (0 Ω). Likewise, a test resistance of 9 kΩ will fail to deflect the needle to exactly 1/2 scale with a lesser battery voltage. There are design techniques used to compensate for varying battery voltage, but they do not completely take care of the problem and are to be considered approximations at best. For this reason, and for the fact of the nonlinear scale, this type of ohmmeter is never considered to be a precision instrument. One final caveat needs to be mentioned with regard to ohmmeters: they only function correctly when measuring resistance that is not being powered by a voltage or current source. In other words, you cannot measure resistance with an ohmmeter on a “live” circuit! The reason for this is simple: the ohmmeter’s accurate indication depends on the only source of voltage being its internal battery. The presence of any voltage across the component to be measured will interfere with the ohmmeter’s operation. If the voltage is large enough, it may even damage the ohmmeter. Review • Ohmmeters contain internal sources of voltage to supply power in taking resistance measurements. • An analog ohmmeter scale is “backwards” from that of a voltmeter or ammeter, the movement needle reading zero resistance at full-scale and infinite resistance at rest. • Analog ohmmeters also have nonlinear scales, “expanded” at the low end of the scale and “compressed” at the high end to be able to span from zero to infinite resistance. • Analog ohmmeters are not precision instruments. • Ohmmeters should never be connected to an energized circuit (that is, a circuit with its own source of voltage). Any voltage applied to the test leads of an ohmmeter will invalidate its reading.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.07%3A_High_Voltage_Ohmmeters.txt	princeton-nlp/TextbookChapters	Most ohmmeters of the design shown in the previous section utilize a battery of relatively low voltage, usually nine volts or less. This is perfectly adequate for measuring resistances under several mega-ohms (MΩ), but when extremely high resistances need to be measured, a 9 volt battery is insufficient for generating enough current to actuate an electromechanical meter movement. Also, as discussed in an earlier chapter, resistance is not always a stable (linear) quantity. This is especially true of non-metals. Recall the graph of current over voltage for a small air gap (less than an inch): While this is an extreme example of nonlinear conduction, other substances exhibit similar insulating/conducting properties when exposed to high voltages. Obviously, an ohmmeter using a low-voltage battery as a source of power cannot measure resistance at the ionization potential of a gas, or at the breakdown voltage of an insulator. If such resistance values need to be measured, nothing but a high-voltage ohmmeter will suffice. The most direct method of high-voltage resistance measurement involves simply substituting a higher voltage battery in the same basic design of ohmmeter investigated earlier: Knowing, however, that the resistance of some materials tends to change with applied voltage, it would be advantageous to be able to adjust the voltage of this ohmmeter to obtain resistance measurements under different conditions: Unfortunately, this would create a calibration problem for the meter. If the meter movement deflects full-scale with a certain amount of current through it, the full-scale range of the meter in ohms would change as the source voltage changed. Imagine connecting a stable resistance across the test leads of this ohmmeter while varying the source voltage: as the voltage is increased, there will be more current through the meter movement, hence a greater amount of deflection. What we really need is a meter movement that will produce a consistent, stable deflection for any stable resistance value measured, regardless of the applied voltage. Accomplishing this design goal requires a special meter movement, one that is peculiar to megohmmeters, or meggers, as these instruments are known. The numbered, rectangular blocks in the above illustration are cross-sectional representations of wire coils. These three coils all move with the needle mechanism. There is no spring mechanism to return the needle to a set position. When the movement is unpowered, the needle will randomly “float.” The coils are electrically connected like this: With infinite resistance between the test leads (open circuit), there will be no current through coil 1, only through coils 2 and 3. When energized, these coils try to center themselves in the gap between the two magnet poles, driving the needle fully to the right of the scale where it points to “infinity.” Any current through coil 1 (through a measured resistance connected between the test leads) tends to drive the needle to the left of scale, back to zero. The internal resistor values of the meter movement are calibrated so that when the test leads are shorted together, the needle deflects exactly to the 0 Ω position. Because any variations in battery voltage will affect the torque generated by both sets of coils (coils 2 and 3, which drive the needle to the right, and coil 1, which drives the needle to the left), those variations will have no effect of the calibration of the movement. In other words, the accuracy of this ohmmeter movement is unaffected by battery voltage: a given amount of measured resistance will produce a certain needle deflection, no matter how much or little battery voltage is present. The only effect that a variation in voltage will have on meter indication is the degree to which the measured resistance changes with applied voltage. So, if we were to use a megger to measure the resistance of a gas-discharge lamp, it would read very high resistance (needle to the far right of the scale) for low voltages and low resistance (needle moves to the left of the scale) for high voltages. This is precisely what we expect from a good high-voltage ohmmeter: to provide accurate indication of subject resistance under different circumstances. For maximum safety, most meggers are equipped with hand-crank generators for producing the high DC voltage (up to 1000 volts). If the operator of the meter receives a shock from the high voltage, the condition will be self-correcting, as he or she will naturally stop cranking the generator! Sometimes a “slip clutch” is used to stabilize generator speed under different cranking conditions, so as to provide a fairly stable voltage whether it is cranked fast or slow. Multiple voltage output levels from the generator are available by the setting of a selector switch. A simple hand-crank megger is shown in this photograph: Some meggers are battery-powered to provide greater precision in output voltage. For safety reasons these meggers are activated by a momentary-contact pushbutton switch, so the switch cannot be left in the “on” position and pose a significant shock hazard to the meter operator. Real meggers are equipped with three connection terminals, labeled Line, Earth, and Guard. The schematic is quite similar to the simplified version shown earlier: Resistance is measured between the Line and Earth terminals, where current will travel through coil 1. The “Guard” terminal is provided for special testing situations where one resistance must be isolated from another. Take for instance this scenario where the insulation resistance is to be tested in a two-wire cable: To measure insulation resistance from a conductor to the outside of the cable, we need to connect the “Line” lead of the megger to one of the conductors and connect the “Earth” lead of the megger to a wire wrapped around the sheath of the cable: In this configuration the megger should read the resistance between one conductor and the outside sheath. Or will it? If we draw a schematic diagram showing all insulation resistances as resistor symbols, what we have looks like this: Rather than just measure the resistance of the second conductor to the sheath (Rc2-s), what we’ll actually measure is that resistance in parallel with the series combination of conductor-to-conductor resistance (Rc1-c2) and the first conductor to the sheath (Rc1-s). If we don’t care about this fact, we can proceed with the test as configured. If we desire to measure only the resistance between the second conductor and the sheath (Rc2-s), then we need to use the megger’s “Guard” terminal: Now the circuit schematic looks like this: Connecting the “Guard” terminal to the first conductor places the two conductors at almost equal potential. With little or no voltage between them, the insulation resistance is nearly infinite, and thus there will be no current between the two conductors. Consequently, the megger’s resistance indication will be based exclusively on the current through the second conductor’s insulation, through the cable sheath, and to the wire wrapped around, not the current leaking through the first conductor’s insulation. Meggers are field instruments: that is, they are designed to be portable and operated by a technician on the job site with as much ease as a regular ohmmeter. They are very useful for checking high-resistance “short” failures between wires caused by wet or degraded insulation. Because they utilize such high voltages, they are not as affected by stray voltages (voltages less than 1 volt produced by electrochemical reactions between conductors, or “induced” by neighboring magnetic fields) as ordinary ohmmeters. For a more thorough test of wire insulation, another high-voltage ohmmeter commonly called a hi-pot tester is used. These specialized instruments produce voltages in excess of 1 kV, and may be used for testing the insulating effectiveness of oil, ceramic insulators, and even the integrity of other high-voltage instruments. Because they are capable of producing such high voltages, they must be operated with the utmost care, and only by trained personnel. It should be noted that hi-pot testers and even meggers (in certain conditions) are capable of damaging wire insulation if incorrectly used. Once an insulating material has been subjected to breakdown by the application of an excessive voltage, its ability to electrically insulate will be compromised. Again, these instruments are to be used only by trained personnel.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.08%3A_Multimeters.txt	princeton-nlp/TextbookChapters	Seeing as how a common meter movement can be made to function as a voltmeter, ammeter, or ohmmeter simply by connecting it to different external resistor networks, it should make sense that a multi-purpose meter (“multimeter”) could be designed in one unit with the appropriate switch(es) and resistors. For general purpose electronics work, the multimeter reigns supreme as the instrument of choice. No other device is able to do so much with so little an investment in parts and elegant simplicity of operation. As with most things in the world of electronics, the advent of solid-state components like transistors has revolutionized the way things are done, and multimeter design is no exception to this rule. However, in keeping with this chapter’s emphasis on analog (“old-fashioned”) meter technology, I’ll show you a few pre-transistor meters. The unit shown above is typical of a handheld analog multimeter, with ranges for voltage, current, and resistance measurement. Note the many scales on the face of the meter movement for the different ranges and functions selectable by the rotary switch. The wires for connecting this instrument to a circuit (the “test leads”) are plugged into the two copper jacks (socket holes) at the bottom-center of the meter face marked “- TEST +”, black and red. This multimeter (Barnett brand) takes a slightly different design approach than the previous unit. Note how the rotary selector switch has fewer positions than the previous meter, but also how there are many more jacks into which the test leads may be plugged into. Each one of those jacks is labeled with a number indicating the respective full-scale range of the meter. Lastly, here is a picture of a digital multimeter. Note that the familiar meter movement has been replaced by a blank, gray-colored display screen. When powered, numerical digits appear in that screen area, depicting the amount of voltage, current, or resistance being measured. This particular brand and model of digital meter has a rotary selector switch and four jacks into which test leads can be plugged. Two leads—one red and one black—are shown plugged into the meter. A close examination of this meter will reveal one “common” jack for the black test lead and three others for the red test lead. The jack into which the red lead is shown inserted is labeled for voltage and resistance measurement, while the other two jacks are labeled for current (A, mA, and µA) measurement. This is a wise design feature of the multimeter, requiring the user to move a test lead plug from one jack to another in order to switch from the voltage measurement to the current measurement function. It would be hazardous to have the meter set in current measurement mode while connected across a significant source of voltage because of the low input resistance, and making it necessary to move a test lead plug rather than just flip the selector switch to a different position helps ensure that the meter doesn’t get set to measure current unintentionally. Note that the selector switch still has different positions for voltage and current measurement, so in order for the user to switch between these two modes of measurement they must switch the position of the red test lead and move the selector switch to a different position. Also note that neither the selector switch nor the jacks are labeled with measurement ranges. In other words, there are no “100 volt” or “10 volt” or “1 volt” ranges (or any equivalent range steps) on this meter. Rather, this meter is “autoranging,” meaning that it automatically picks the appropriate range for the quantity being measured. Autoranging is a feature only found on digital meters, but not all digital meters. No two models of multimeters are designed to operate exactly the same, even if they’re manufactured by the same company. In order to fully understand the operation of any multimeter, the owner’s manual must be consulted. Here is a schematic for a simple analog volt/ammeter: In the switch’s three lower (most counter-clockwise) positions, the meter movement is connected to the Common and V jacks through one of three different series range resistors (Rmultiplier1 through Rmultiplier3), and so acts as a voltmeter. In the fourth position, the meter movement is connected in parallel with the shunt resistor, and so acts as an ammeter for any current entering the common jack and exiting the A jack. In the last (furthest clockwise) position, the meter movement is disconnected from either red jack, but short-circuited through the switch. This short-circuiting creates a dampening effect on the needle, guarding against mechanical shock damage when the meter is handled and moved. If an ohmmeter function is desired in this multimeter design, it may be substituted for one of the three voltage ranges as such: With all three fundamental functions available, this multimeter may also be known as a volt-ohm-milliammeter. Obtaining a reading from an analog multimeter when there is a multitude of ranges and only one meter movement may seem daunting to the new technician. On an analog multimeter, the meter movement is marked with several scales, each one useful for at least one range setting. Here is a close-up photograph of the scale from the Barnett multimeter shown earlier in this section: Note that there are three types of scales on this meter face: a green scale for resistance at the top, a set of black scales for DC voltage and current in the middle, and a set of blue scales for AC voltage and current at the bottom. Both the DC and AC scales have three sub-scales, one ranging 0 to 2.5, one ranging 0 to 5, and one ranging 0 to 10. The meter operator must choose whichever scale best matches the range switch and plug settings in order to properly interpret the meter’s indication. This particular multimeter has several basic voltage measurement ranges: 2.5 volts, 10 volts, 50 volts, 250 volts, 500 volts, and 1000 volts. With the use of the voltage range extender unit at the top of the multimeter, voltages up to 5000 volts can be measured. Suppose the meter operator chose to switch the meter into the “volt” function and plug the red test lead into the 10 volt jack. To interpret the needle’s position, he or she would have to read the scale ending with the number “10”. If they moved the red test plug into the 250 volt jack, however, they would read the meter indication on the scale ending with “2.5”, multiplying the direct indication by a factor of 100 in order to find what the measured voltage was. If current is measured with this meter, another jack is chosen for the red plug to be inserted into and the range is selected via a rotary switch. This close-up photograph shows the switch set to the 2.5 mA position: Note how all current ranges are power-of-ten multiples of the three scale ranges shown on the meter face: 2.5, 5, and 10. In some range settings, such as the 2.5 mA for example, the meter indication may be read directly on the 0 to 2.5 scale. For other range settings (250 µA, 50 mA, 100 mA, and 500 mA), the meter indication must be read off the appropriate scale and then multiplied by either 10 or 100 to obtain the real figure. The highest current range available on this meter is obtained with the rotary switch in the 2.5/10 amp position. The distinction between 2.5 amps and 10 amps is made by the red test plug position: a special “10 amp” jack next to the regular current-measuring jack provides an alternative plug setting to select the higher range. Resistance in ohms, of course, is read by a nonlinear scale at the top of the meter face. It is “backward,” just like all battery-operated analog ohmmeters, with zero at the right-hand side of the face and infinity at the left-hand side. There is only one jack provided on this particular multimeter for “ohms,” so different resistance-measuring ranges must be selected by the rotary switch. Notice on the switch how five different “multiplier” settings are provided for measuring resistance: Rx1, Rx10, Rx100, Rx1000, and Rx10000. Just as you might suspect, the meter indication is given by multiplying whatever needle position is shown on the meter face by the power-of-ten multiplying factor set by the rotary switch.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.09%3A_Kelvin_%284-wire%29_Resistance_Measurement.txt	princeton-nlp/TextbookChapters	Suppose we wished to measure the resistance of some component located a significant distance away from our ohmmeter. Such a scenario would be problematic, because an ohmmeter measures all resistance in the circuit loop, which includes the resistance of the wires (Rwire) connecting the ohmmeter to the component being measured (Rsubject): Usually, wire resistance is very small (only a few ohms per hundreds of feet, depending primarily on the gauge (size) of the wire), but if the connecting wires are very long, and/or the component to be measured has a very low resistance anyway, the measurement error introduced by wire resistance will be substantial. An ingenious method of measuring the subject resistance in a situation like this involves the use of both an ammeter and a voltmeter. We know from Ohm’s Law that resistance is equal to voltage divided by current (R = E/I). Thus, we should be able to determine the resistance of the subject component if we measure the current going through it and the voltage dropped across it: Current is the same at all points in the circuit, because it is a series loop. Because we’re only measuring voltage dropped across the subject resistance (and not the wires’ resistances), though, the calculated resistance is indicative of the subject component’s resistance (Rsubject) alone. Our goal, though, was to measure this subject resistance from a distance, so our voltmeter must be located somewhere near the ammeter, connected across the subject resistance by another pair of wires containing resistance: At first it appears that we have lost any advantage of measuring resistance this way, because the voltmeter now has to measure voltage through a long pair of (resistive) wires, introducing stray resistance back into the measuring circuit again. However, upon closer inspection it is seen that nothing is lost at all, because the voltmeter’s wires carry miniscule current. Thus, those long lengths of wire connecting the voltmeter across the subject resistance will drop insignificant amounts of voltage, resulting in a voltmeter indication that is very nearly the same as if it were connected directly across the subject resistance: Any voltage dropped across the main current-carrying wires will not be measured by the voltmeter, and so do not factor into the resistance calculation at all. Measurement accuracy may be improved even further if the voltmeter’s current is kept to a minimum, either by using a high-quality (low full-scale current) movement and/or a potentiometric (null-balance) system. This method of measurement which avoids errors caused by wire resistance is called the Kelvin, or 4-wiremethod. Special connecting clips called Kelvin clips are made to facilitate this kind of connection across a subject resistance: In regular, “alligator” style clips, both halves of the jaw are electrically common to each other, usually joined at the hinge point. In Kelvin clips, the jaw halves are insulated from each other at the hinge point, only contacting at the tips where they clasp the wire or terminal of the subject being measured. Thus, current through the “C” (“current”) jaw halves does not go through the “P” (“potential,” or voltage) jaw halves, and will not create any error-inducing voltage drop along their length: The same principle of using different contact points for current conduction and voltage measurement is used in precision shunt resistors for measuring large amounts of current. As discussed previously, shunt resistors function as current measurement devices by dropping a precise amount of voltage for every amp of current through them, the voltage drop being measured by a voltmeter. In this sense, a precision shunt resistor “converts” a current value into a proportional voltage value. Thus, current may be accurately measured by measuring voltage dropped across the shunt: Current measurement using a shunt resistor and voltmeter is particularly well-suited for applications involving particularly large magnitudes of current. In such applications, the shunt resistor’s resistance will likely be in the order of milliohms or microohms, so that only a modest amount of voltage will be dropped at full current. Resistance this low is comparable to wire connection resistance, which means voltage measured across such a shunt must be done so in such a way as to avoid detecting voltage dropped across the current-carrying wire connections, lest huge measurement errors be induced. In order that the voltmeter measure only the voltage dropped by the shunt resistance itself, without any stray voltages originating from wire or connection resistance, shunts are usually equipped with four connection terminals: In metrological (metrology = “the science of measurement”) applications, where accuracy is of paramount importance, highly precise “standard” resistors are also equipped with four terminals: two for carrying the measured current, and two for conveying the resistor’s voltage drop to the voltmeter. This way, the voltmeter only measures voltage dropped across the precision resistance itself, without any stray voltages dropped across current-carrying wires or wire-to-terminal connection resistances. The following photograph shows a precision standard resistor of 1 Ω value immersed in a temperature-controlled oil bath with a few other standard resistors. Note the two large, outer terminals for current, and the two small connection terminals for voltage: Here is another, older (pre-World War II) standard resistor of German manufacture. This unit has a resistance of 0.001 Ω, and again the four terminal connection points can be seen as black knobs (metal pads underneath each knob for direct metal-to-metal connection with the wires), two large knobs for securing the current-carrying wires, and two smaller knobs for securing the voltmeter (“potential”) wires: Appreciation is extended to the Fluke Corporation in Everett, Washington for allowing me to photograph these expensive and somewhat rare standard resistors in their primary standards laboratory. It should be noted that resistance measurement using both an ammeter and a voltmeter is subject to compound error. Because the accuracy of both instruments factors in to the final result, the overall measurement accuracy may be worse than either instrument considered alone. For instance, if the ammeter is accurate to +/- 1% and the voltmeter is also accurate to +/- 1%, any measurement dependent on the indications of both instruments may be inaccurate by as much as +/- 2%. Greater accuracy may be obtained by replacing the ammeter with a standard resistor, used as a current-measuring shunt. There will still be compound error between the standard resistor and the voltmeter used to measure voltage drop, but this will be less than with a voltmeter + ammeter arrangement because typical standard resistor accuracy far exceeds typical ammeter accuracy. Using Kelvin clips to make connection with the subject resistance, the circuit looks something like this: All current-carrying wires in the above circuit are shown in “bold,” to easily distinguish them from wires connecting the voltmeter across both resistances (Rsubject and Rstandard). Ideally, a potentiometric voltmeter is used to ensure as little current through the “potential” wires as possible. The Kelvin measurement can be a practical tool for finding poor connections or unexpected resistance in an electrical circuit. Connect a DC power supply to the circuit and adjust the power supply so that it supplies a constant current to the circuit as shown in the diagram above (within the circuit’s capabilities, of course). With a digital multimeter set to measure DC voltage, measure the voltage drop across various points in the circuit. If you know the wire size, you can estimate the voltage drop you should see and compare this to the voltage drop you measure. This can be a quick and effective method of finding poor connections in wiring exposed to the elements, such as in the lighting circuits of a trailer. It can also work well for unpowered AC conductors (make sure the AC power cannot be turned on). For example, you can measure the voltage drop across a light switch and determine if the wiring connections to the switch or the switch’s contacts are suspect. To be most effective using this technique, you should also measure the same type of circuits after they are newly made so you have a feel for the “correct” values. If you use this technique on new circuits and put the results in a log book, you have valuable information for troubleshooting in the future.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.10%3A_Bridge_Circuits.txt	princeton-nlp/TextbookChapters	No text on electrical metering could be called complete without a section on bridge circuits. These ingenious circuits make use of a null-balance meter to compare two voltages, just like the laboratory balance scale compares two weights and indicates when they’re equal. Unlike the “potentiometer” circuit used to simply measure an unknown voltage, bridge circuits can be used to measure all kinds of electrical values, not the least of which being resistance. The standard bridge circuit, often called a Wheatstone bridge, looks something like this: When the voltage between point 1 and the negative side of the battery is equal to the voltage between point 2 and the negative side of the battery, the null detector will indicate zero and the bridge is said to be “balanced.” The bridge’s state of balance is solely dependent on the ratios of Ra/Rb and R1/R2, and is quite independent of the supply voltage (battery). To measure resistance with a Wheatstone bridge, an unknown resistance is connected in the place of Ra or Rb, while the other three resistors are precision devices of known value. Either of the other three resistors can be replaced or adjusted until the bridge is balanced, and when balance has been reached the unknown resistor value can be determined from the ratios of the known resistances. A requirement for this to be a measurement system is to have a set of variable resistors available whose resistances are precisely known, to serve as reference standards. For example, if we connect a bridge circuit to measure an unknown resistance Rx, we will have to know the exact values of the other three resistors at balance to determine the value of Rx: Each of the four resistances in a bridge circuit are referred to as arms. The resistor in series with the unknown resistance Rx (this would be Ra in the above schematic) is commonly called the rheostat of the bridge, while the other two resistors are called the ratio arms of the bridge. Accurate and stable resistance standards, thankfully, are not that difficult to construct. In fact, they were some of the first electrical “standard” devices made for scientific purposes. Here is a photograph of an antique resistance standard unit: This resistance standard shown here is variable in discrete steps: the amount of resistance between the connection terminals could be varied with the number and pattern of removable copper plugs inserted into sockets. Wheatstone bridges are considered a superior means of resistance measurement to the series battery-movement-resistor meter circuit discussed in the last section. Unlike that circuit, with all its nonlinearities (nonlinear scale) and associated inaccuracies, the bridge circuit is linear (the mathematics describing its operation are based on simple ratios and proportions) and quite accurate. Given standard resistances of sufficient precision and a null detector device of sufficient sensitivity, resistance measurement accuracies of at least +/- 0.05% are attainable with a Wheatstone bridge. It is the preferred method of resistance measurement in calibration laboratories due to its high accuracy. There are many variations of the basic Wheatstone bridge circuit. Most DC bridges are used to measure resistance, while bridges powered by alternating current (AC) may be used to measure different electrical quantities like inductance, capacitance, and frequency. An interesting variation of the Wheatstone bridge is the Kelvin Double bridge, used for measuring very low resistances (typically less than 1/10 of an ohm). Its schematic diagram is as such: The low-value resistors are represented by thick-line symbols, and the wires connecting them to the voltage source (carrying high current) are likewise drawn thickly in the schematic. This oddly-configured bridge is perhaps best understood by beginning with a standard Wheatstone bridge set up for measuring low resistance, and evolving it step-by-step into its final form in an effort to overcome certain problems encountered in the standard Wheatstone configuration. If we were to use a standard Wheatstone bridge to measure low resistance, it would look something like this: When the null detector indicates zero voltage, we know that the bridge is balanced and that the ratios Ra/Rxand RM/RN are mathematically equal to each other. Knowing the values of Ra, RM, and RN therefore provides us with the necessary data to solve for Rx . . . almost. We have a problem, in that the connections and connecting wires between Ra and Rx possess resistance as well, and this stray resistance may be substantial compared to the low resistances of Ra and Rx. These stray resistances will drop substantial voltage, given the high current through them, and thus will affect the null detector’s indication and thus the balance of the bridge: Since we don’t want to measure these stray wire and connection resistances, but only measure Rx, we must find some way to connect the null detector so that it won’t be influenced by voltage dropped across them. If we connect the null detector and RM/RN ratio arms directly across the ends of Ra and Rx, this gets us closer to a practical solution: Now the top two Ewire voltage drops are of no effect to the null detector, and do not influence the accuracy of Rx‘s resistance measurement. However, the two remaining Ewire voltage drops will cause problems, as the wire connecting the lower end of Ra with the top end of Rx is now shunting across those two voltage drops, and will conduct substantial current, introducing stray voltage drops along its own length as well. Knowing that the left side of the null detector must connect to the two near ends of Ra and Rx in order to avoid introducing those Ewire voltage drops into the null detector’s loop, and that any direct wire connecting those ends of Ra and Rx will itself carry substantial current and create more stray voltage drops, the only way out of this predicament is to make the connecting path between the lower end of Ra and the upper end of Rx substantially resistive: We can manage the stray voltage drops between Ra and Rx by sizing the two new resistors so that their ratio from upper to lower is the same ratio as the two ratio arms on the other side of the null detector. This is why these resistors were labeled Rm and Rn in the original Kelvin Double bridge schematic: to signify their proportionality with RM and RN: With ratio Rm/Rn set equal to ratio RM/RN, rheostat arm resistor Ra is adjusted until the null detector indicates balance, and then we can say that Ra/Rx is equal to RM/RN, or simply find Rx by the following equation: The actual balance equation of the Kelvin Double bridge is as follows (Rwire is the resistance of the thick, connecting wire between the low-resistance standard Ra and the test resistance Rx): So long as the ratio between RM and RN is equal to the ratio between Rm and Rn, the balance equation is no more complex than that of a regular Wheatstone bridge, with Rx/Ra equal to RN/RM, because the last term in the equation will be zero, canceling the effects of all resistances except Rx, Ra, RM, and RN. In many Kelvin Double bridge circuits, RM=Rm and RN=Rn. However, the lower the resistances of Rm and Rn, the more sensitive the null detector will be, because there is less resistance in series with it. Increased detector sensitivity is good, because it allows smaller imbalances to be detected, and thus a finer degree of bridge balance to be attained. Therefore, some high-precision Kelvin Double bridges use Rm and Rn values as low as 1/100 of their ratio arm counterparts (RM and RN, respectively). Unfortunately, though, the lower the values of Rm and Rn, the more current they will carry, which will increase the effect of any junction resistances present where Rm and Rn connect to the ends of Ra and Rx. As you can see, high instrument accuracy demands that all error-producing factors be taken into account, and often the best that can be achieved is a compromise minimizing two or more different kinds of errors. Review • Bridge circuits rely on sensitive null-voltage meters to compare two voltages for equality. • A Wheatstone bridge can be used to measure resistance by comparing the unknown resistor against precision resistors of known value, much like a laboratory scale measures an unknown weight by comparing it against known standard weights. • A Kelvin Double bridge is a variant of the Wheatstone bridge used for measuring very low resistances. Its additional complexity over the basic Wheatstone design is necessary for avoiding errors otherwise incurred by stray resistances along the current path between the low-resistance standard and the resistance being measured.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/08%3A_DC_Metering_Circuits/8.11%3A_Wattmeter_Design.txt	princeton-nlp/TextbookChapters	Power in an electric circuit is the product (multiplication) of voltage and current, so any meter designed to measure power must account for both of these variables. A special meter movement designed especially for power measurement is called the dynamometer movement, and is similar to a D’Arsonval or Weston movement in that a lightweight coil of wire is attached to the pointer mechanism. However, unlike the D’Arsonval or Weston movement, another (stationary) coil is used instead of a permanent magnet to provide the magnetic field for the moving coil to react against. The moving coil is generally energized by the voltage in the circuit, while the stationary coil is generally energized by the current in the circuit. A dynamometer movement connected in a circuit looks something like this: The top (horizontal) coil of wire measures load current while the bottom (vertical) coil measures load voltage. Just like the lightweight moving coils of voltmeter movements, the (moving) voltage coil of a dynamometer is typically connected in series with a range resistor so that full load voltage is not applied to it. Likewise, the (stationary) current coil of a dynamometer may have precision shunt resistors to divide the load current around it. With custom-built dynamometer movements, shunt resistors are less likely to be needed because the stationary coil can be constructed with as heavy of wire as needed without impacting meter response, unlike the moving coil which must be constructed of lightweight wire for minimum inertia. Review • Wattmeters are often designed around dynamometer meter movements, which employ both voltage and current coils to move a needle. 8.12: Creating Custom Calibration Resistances Often in the course of designing and building electrical meter circuits, it is necessary to have precise resistances to obtain the desired range(s). More often than not, the resistance values required cannot be found in any manufactured resistor unit and therefore must be built by you. One solution to this dilemma is to make your own resistor out of a length of special high-resistance wire. Usually, a small “bobbin” is used as a form for the resulting wire coil, and the coil is wound in such a way as to eliminate any electromagnetic effects: the desired wire length is folded in half, and the looped wire wound around the bobbin so that current through the wire winds clockwise around the bobbin for half the wire’s length, then counter-clockwise for the other half. This is known as a bifilar winding. Any magnetic fields generated by the current are thus canceled, and external magnetic fields cannot induce any voltage in the resistance wire coil: As you might imagine, this can be a labor-intensive process, especially if more than one resistor must be built! Another, easier solution to the dilemma of a custom resistance is to connect multiple fixed-value resistors together in series-parallel fashion to obtain the desired value of resistance. This solution, although potentially time-intensive in choosing the best resistor values for making the first resistance, can be duplicated much faster for creating multiple custom resistances of the same value: A disadvantage of either technique, though, is the fact that both result in a fixed resistance value. In a perfect world where meter movements never lose magnetic strength of their permanent magnets, where temperature and time have no effect on component resistances, and where wire connections maintain zero resistance forever, fixed-value resistors work quite well for establishing the ranges of precision instruments. However, in the real world, it is advantageous to have the ability to calibrate, or adjust, the instrument in the future. It makes sense, then, to use potentiometers (connected as rheostats, usually) as variable resistances for range resistors. The potentiometer may be mounted inside the instrument case so that only a service technician has access to change its value, and the shaft may be locked in place with thread-fastening compound (ordinary nail polish works well for this!) so that it will not move if subjected to vibration. However, most potentiometers provide too large a resistance span over their mechanically-short movement range to allow for precise adjustment. Suppose you desired a resistance of 8.335 kΩ +/- 1 Ω, and wanted to use a 10 kΩ potentiometer (rheostat) to obtain it. A precision of 1 Ω out of a span of 10 kΩ is 1 part in 10,000, or 1/100 of a percent! Even with a 10-turn potentiometer, it will be very difficult to adjust it to any value this finely. Such a feat would be nearly impossible using a standard 3/4 turn potentiometer. So how can we get the resistance value we need and still have room for adjustment? The solution to this problem is to use a potentiometer as part of a larger resistance network which will create a limited adjustment range. Observe the following example: Here, the 1 kΩ potentiometer, connected as a rheostat, provides by itself a 1 kΩ span (a range of 0 Ω to 1 kΩ). Connected in series with an 8 kΩ resistor, this offsets the total resistance by 8,000 Ω, giving an adjustable range of 8 kΩ to 9 kΩ. Now, a precision of +/- 1 Ω represents 1 part in 1000, or 1/10 of a percent of potentiometer shaft motion. This is ten times better, in terms of adjustment sensitivity, than what we had using a 10 kΩ potentiometer. If we desire to make our adjustment capability even more precise—so we can set the resistance at 8.335 kΩ with even greater precision—we may reduce the span of the potentiometer by connecting a fixed-value resistor in parallel with it: Now, the calibration span of the resistor network is only 500 Ω, from 8 kΩ to 8.5 kΩ. This makes a precision of +/- 1 Ω equal to 1 part in 500, or 0.2 percent. The adjustment is now half as sensitive as it was before the addition of the parallel resistor, facilitating much easier calibration to the target value. The adjustment will not be linear, unfortunately (halfway on the potentiometer’s shaft position will not result in 8.25 kΩ total resistance, but rather 8.333 kΩ). Still, it is an improvement in terms of sensitivity, and it is a practical solution to our problem of building an adjustable resistance for a precision instrument!
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/09%3A_Electrical_Instrumentation_Signals/9.01%3A_Analog_and_Digital_Signals.txt	princeton-nlp/TextbookChapters	• 9.1: Analog and Digital Signals A signal is any kind of physical quantity that conveys information. An analog signal is a kind of signal that is continuously variable, as opposed to having a limited number of steps along its range (called digital). Both analog and digital signals find application in modern electronics. • 9.2: Voltage Signal Systems DC voltage can be used as an analog signal to relay information from one location to another. A major disadvantage of voltage signaling is the possibility that the voltage at the indicator (voltmeter) will be less than the voltage at the signal source, due to line resistance and indicator current draw. This drop in voltage along the conductor length constitutes a measurement error from transmitter to indicator. • 9.3: Current Signal Systems • 9.4: Tachogenerators • 9.5: Thermocouples • 9.6: pH Measurement • 9.7: Strain Gauges If a strip of conductive metal is stretched, it will become skinnier and longer, both changes resulting in an increase of electrical resistance end-to-end. Conversely, if a strip of conductive metal is placed under compressive force (without buckling), it will broaden and shorten. If these stresses are kept within the elastic limit of the metal strip (so that the strip does not permanently deform), the strip can be used as a measuring element for physical force, the amount of applied force infer 09: Electrical Instrumentation Signals Instrumentation is a field of study and work centering on measurement and control of physical processes. These physical processes include pressure, temperature, flow rate, and chemical consistency. An instrument is a device that measures and/or acts to control any kind of physical process. Due to the fact that electrical quantities of voltage and current are easy to measure, manipulate, and transmit over long distances, they are widely used to represent such physical variables and transmit the information to remote locations. A signal is any kind of physical quantity that conveys information. Audible speech is certainly a kind of signal, as it conveys the thoughts (information) of one person to another through the physical medium of sound. Hand gestures are signals, too, conveying information by means of light. This text is another kind of signal, interpreted by your English-trained mind as information about electric circuits. In this chapter, the word signal will be used primarily in reference to an electrical quantity of voltage or current that is used to represent or signify some other physical quantity. An analog signal is a kind of signal that is continuously variable, as opposed to having a limited number of steps along its range (called digital). A well-known example of analog vs. digital is that of clocks: analog being the type with pointers that slowly rotate around a circular scale, and digital being the type with decimal number displays or a “second-hand” that jerks rather than smoothly rotates. The analog clock has no physical limit to how finely it can display the time, as its “hands” move in a smooth, pauseless fashion. The digital clock, on the other hand, cannot convey any unit of time smaller than what its display will allow for. The type of clock with a “second-hand” that jerks in 1-second intervals is a digital device with a minimum resolution of one second. Both analog and digital signals find application in modern electronics, and the distinctions between these two basic forms of information is something to be covered in much greater detail later in this book. For now, I will limit the scope of this discussion to analog signals, since the systems using them tend to be of simpler design. With many physical quantities, especially electrical, analog variability is easy to come by. If such a physical quantity is used as a signal medium, it will be able to represent variations of information with almost unlimited resolution. In the early days of industrial instrumentation, compressed air was used as a signaling medium to convey information from measuring instruments to indicating and controlling devices located remotely. The amount of air pressure corresponded to the magnitude of whatever variable was being measured. Clean, dry air at approximately 20 pounds per square inch (PSI) was supplied from an air compressor through tubing to the measuring instrument and was then regulated by that instrument according to the quantity being measured to produce a corresponding output signal. For example, a pneumatic (air signal) level “transmitter” device set up to measure height of water (the “process variable”) in a storage tank would output a low air pressure when the tank was empty, a medium pressure when the tank was partially full, and a high pressure when the tank was completely full. The “water level indicator” (LI) is nothing more than a pressure gauge measuring the air pressure in the pneumatic signal line. This air pressure, being a signal, is in turn a representation of the water level in the tank. Any variation of level in the tank can be represented by an appropriate variation in the pressure of the pneumatic signal. Aside from certain practical limits imposed by the mechanics of air pressure devices, this pneumatic signal is infinitely variable, able to represent any degree of change in the water’s level, and is therefore analog in the truest sense of the word. Crude as it may appear, this kind of pneumatic signaling system formed the backbone of many industrial measurement and control systems around the world, and still sees use today due to its simplicity, safety, and reliability. Air pressure signals are easily transmitted through inexpensive tubes, easily measured (with mechanical pressure gauges), and are easily manipulated by mechanical devices using bellows, diaphragms, valves, and other pneumatic devices. Air pressure signals are not only useful for measuring physical processes, but for controlling them as well. With a large enough piston or diaphragm, a small air pressure signal can be used to generate a large mechanical force, which can be used to move a valve or other controlling device. Complete automatic control systems have been made using air pressure as the signal medium. They are simple, reliable, and relatively easy to understand. However, the practical limits for air pressure signal accuracy can be too limiting in some cases, especially when the compressed air is not clean and dry, and when the possibility for tubing leaks exist. With the advent of solid-state electronic amplifiers and other technological advances, electrical quantities of voltage and current became practical for use as analog instrument signaling media. Instead of using pneumatic pressure signals to relay information about the fullness of a water storage tank, electrical signals could relay that same information over thin wires (instead of tubing) and not require the support of such expensive equipment as air compressors to operate: Analog electronic signals are still the primary kinds of signals used in the instrumentation world today (January of 2001), but it is giving way to digital modes of communication in many applications (more on that subject later). Despite changes in technology, it is always good to have a thorough understanding of fundamental principles, so the following information will never really become obsolete. One important concept applied in many analog instrumentation signal systems is that of “live zero,” a standard way of scaling a signal so that an indication of 0 percent can be discriminated from the status of a “dead” system. Take the pneumatic signal system as an example: if the signal pressure range for transmitter and indicator was designed to be 0 to 12 PSI, with 0 PSI representing 0 percent of process measurement and 12 PSI representing 100 percent, a received signal of 0 percent could be a legitimate reading of 0 percent measurement or it could mean that the system was malfunctioning (air compressor stopped, tubing broken, transmitter malfunctioning, etc.). With the 0 percent point represented by 0 PSI, there would be no easy way to distinguish one from the other. If, however, we were to scale the instruments (transmitter and indicator) to use a scale of 3 to 15 PSI, with 3 PSI representing 0 percent and 15 PSI representing 100 percent, any kind of a malfunction resulting in zero air pressure at the indicator would generate a reading of -25 percent (0 PSI), which is clearly a faulty value. The person looking at the indicator would then be able to immediately tell that something was wrong. Not all signal standards have been set up with live zero baselines, but the more robust signals standards (3-15 PSI, 4-20 mA) have, and for good reason. REVIEW • A signal is any kind of detectable quantity used to communicate information. • An analog signal is a signal that can be continuously, or infinitely, varied to represent any small amount of change. • Pneumatic, or air pressure, signals used to be used predominately in industrial instrumentation signal systems. This has been largely superseded by analog electrical signals such as voltage and current. • A live zero refers to an analog signal scale using a non-zero quantity to represent 0 percent of real-world measurement so that any system malfunction resulting in a natural “rest” state of zero signal pressure, voltage, or current can be immediately recognized.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/09%3A_Electrical_Instrumentation_Signals/9.02%3A_Voltage_Signal_Systems.txt	princeton-nlp/TextbookChapters	The use of variable voltage for instrumentation signals seems a rather obvious option to explore. Let’s see how a voltage signal instrument might be used to measure and relay information about water tank level: The “transmitter” in this diagram contains its own precision regulated source of voltage, and the potentiometer setting is varied by the motion of a float inside the water tank following the water level. The “indicator” is nothing more than a voltmeter with a scale calibrated to read in some unit height of water (inches, feet, meters) instead of volts. As the water tank level changes, the float will move. As the float moves, the potentiometer wiper will correspondingly be moved, dividing a different proportion of the battery voltage to go across the two-conductor cable and on to the level indicator. As a result, the voltage received by the indicator will be representative of the level of water in the storage tank. This elementary transmitter/indicator system is reliable and easy to understand, but it has its limitations. Perhaps greatest is the fact that the system accuracy can be influenced by excessive cable resistance. Remember that real voltmeters draw small amounts of current, even though it is ideal for a voltmeter not to draw any current at all. This being the case, especially for the kind of heavy, rugged analog meter movement likely used for an industrial-quality system, there will be a small amount of current through the 2-conductor cable wires. The cable, having a small amount of resistance along its length, will consequently drop a small amount of voltage, leaving less voltage across the indicator’s leads than what is across the leads of the transmitter. This loss of voltage, however small, constitutes an error in measurement: Resistor symbols have been added to the wires of the cable to show what is happening in a real system. Bear in mind that these resistances can be minimized with heavy-gauge wire (at additional expense) and/or their effects mitigated through the use of a high-resistance (null-balance?) voltmeter for an indicator (at additional complexity). Despite this inherent disadvantage, voltage signals are still used in many applications because of their extreme design simplicity. One common signal standard is 0-10 volts, meaning that a signal of 0 volts represents 0 percent of measurement, 10 volts represents 100 percent of measurement, 5 volts represents 50 percent of measurement, and so on. Instruments designed to output and/or accept this standard signal range are available for purchase from major manufacturers. A more common voltage range is 1-5 volts, which makes use of the “live zero” concept for circuit fault indication. Review • DC voltage can be used as an analog signal to relay information from one location to another. • A major disadvantage of voltage signaling is the possibility that the voltage at the indicator (voltmeter) will be less than the voltage at the signal source, due to line resistance and indicator current draw. This drop in voltage along the conductor length constitutes a measurement error from transmitter to indicator. 9.03: Current Signal Systems It is possible through the use of electronic amplifiers to design a circuit outputting a constant amount of current rather than a constant amount of voltage. This collection of components is collectively known as a current source, and its symbol looks like this: A current source generates as much or as little voltage as needed across its leads to produce a constant amount of current through it. This is just the opposite of a voltage source (an ideal battery), which will output as much or as little current as demanded by the external circuit in maintaining its output voltage constant. Following the “conventional flow” symbology typical of electronic devices, the arrow points against the direction of electron motion. Apologies for this confusing notation: another legacy of Benjamin Franklin’s false assumption of electron flow! Current sources can be built as variable devices, just like voltage sources, and they can be designed to produce very precise amounts of current. If a transmitter device were to be constructed with a variable current source instead of a variable voltage source, we could design an instrumentation signal system based on current instead of voltage: The internal workings of the transmitter’s current source need not be a concern at this point, only the fact that its output varies in response to changes in the float position, just like the potentiometer setup in the voltage signal system varied voltage output according to float position. Notice now how the indicator is an ammeter rather than a voltmeter (the scale calibrated in inches, feet, or meters of water in the tank, as always). Because the circuit is a series configuration (accounting for the cable resistances), current will be precisely equal through all components. With or without cable resistance, the current at the indicator is exactly the same as the current at the transmitter, and therefore there is no error incurred as there might be with a voltage signal system. This assurance of zero signal degradation is a decided advantage of current signal systems over voltage signal systems. The most common current signal standard in modern use is the 4 to 20 milliamp (4-20 mA) loop, with 4 milliamps representing 0 percent of measurement, 20 milliamps representing 100 percent, 12 milliamps representing 50 percent, and so on. A convenient feature of the 4-20 mA standard is its ease of signal conversion to 1-5 volt indicating instruments. A simple 250 ohm precision resistor connected in series with the circuit will produce 1 volt of drop at 4 milliamps, 5 volts of drop at 20 milliamps, etc: The current loop scale of 4-20 milliamps has not always been the standard for current instruments: for a while there was also a 10-50 milliamp standard, but that standard has since been obsoleted. One reason for the eventual supremacy of the 4-20 milliamp loop was safety: with lower circuit voltages and lower current levels than in 10-50 mA system designs, there was less chance for personal shock injury and/or the generation of sparks capable of igniting flammable atmospheres in certain industrial environments. Review • A current source is a device (usually constructed of several electronic components) that outputs a constant amount of current through a circuit, much like a voltage source (ideal battery) outputting a constant amount of voltage to a circuit. • A current “loop” instrumentation circuit relies on the series circuit principle of current being equal through all components to insure no signal error due to wiring resistance. • The most common analog current signal standard in modern use is the “4 to 20 milliamp current loop.” 9.04: Tachogenerators An electromechanical generator is a device capable of producing electrical power from mechanical energy, usually the turning of a shaft. When not connected to a load resistance, generators will generate voltage roughly proportional to shaft speed. With precise construction and design, generators can be built to produce very precise voltages for certain ranges of shaft speeds, thus making them well-suited as measurement devices for shaft speed in mechanical equipment. A generator specially designed and constructed for this use is called a tachometer or tachogenerator. Often, the word “tach” (pronounced “tack”) is used rather than the whole word. By measuring the voltage produced by a tachogenerator, you can easily determine the rotational speed of whatever its mechanically attached to. One of the more common voltage signal ranges used with tachogenerators is 0 to 10 volts. Obviously, since a tachogenerator cannot produce voltage when its not turning, the zero cannot be “live” in this signal standard. Tachogenerators can be purchased with different “full-scale” (10 volt) speeds for different applications. Although a voltage divider could theoretically be used with a tachogenerator to extend the measurable speed range in the 0-10 volt scale, it is not advisable to significantly overspeed a precision instrument like this, or its life will be shortened. Tachogenerators can also indicate the direction of rotation by the polarity of the output voltage. When a permanent-magnet style DC generator’s rotational direction is reversed, the polarity of its output voltage will switch. In measurement and control systems where directional indication is needed, tachogenerators provide an easy way to determine that. Tachogenerators are frequently used to measure the speeds of electric motors, engines, and the equipment they power: conveyor belts, machine tools, mixers, fans, etc.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/09%3A_Electrical_Instrumentation_Signals/9.05%3A_Thermocouples.txt	princeton-nlp/TextbookChapters	An interesting phenomenon applied in the field of instrumentation is the Seebeck effect, which is the production of a small voltage across the length of a wire due to a difference in temperature along that wire. This effect is most easily observed and applied with a junction of two dissimilar metals in contact, each metal producing a different Seebeck voltage along its length, which translates to a voltage between the two (unjoined) wire ends. Most any pair of dissimilar metals will produce a measurable voltage when their junction is heated, some combinations of metals producing more voltage per degree of temperature than others: The Seebeck effect is fairly linear; that is, the voltage produced by a heated junction of two wires is directly proportional to the temperature. This means that the temperature of the metal wire junction can be determined by measuring the voltage produced. Thus, the Seebeck effect provides for us an electric method of temperature measurement. When a pair of dissimilar metals are joined together for the purpose of measuring temperature, the device formed is called a thermocouple. Thermocouples made for instrumentation use metals of high purity for an accurate temperature/voltage relationship (as linear and as predictable as possible). Seebeck voltages are quite small, in the tens of millivolts for most temperature ranges. This makes them somewhat difficult to measure accurately. Also, the fact that any junction between dissimilar metals will produce temperature-dependent voltage creates a problem when we try to connect the thermocouple to a voltmeter, completing a circuit: The second iron/copper junction formed by the connection between the thermocouple and the meter on the top wire will produce a temperature-dependent voltage opposed in polarity to the voltage produced at the measurement junction. This means that the voltage between the voltmeter’s copper leads will be a function of the difference in temperature between the two junctions, and not the temperature at the measurement junction alone. Even for thermocouple types where copper is not one of the dissimilar metals, the combination of the two metals joining the copper leads of the measuring instrument forms a junction equivalent to the measurement junction: This second junction is called the reference or cold junction, to distinguish it from the junction at the measuring end, and there is no way to avoid having one in a thermocouple circuit. In some applications, a differential temperature measurement between two points is required, and this inherent property of thermocouples can be exploited to make a very simple measurement system. However, in most applications the intent is to measure temperature at a single point only, and in these cases the second junction becomes a liability to function. Compensation for the voltage generated by the reference junction is typically performed by a special circuit designed to measure temperature there and produce a corresponding voltage to counter the reference junction’s effects. At this point you may wonder, “If we have to resort to some other form of temperature measurement just to overcome an idiosyncrasy with thermocouples, then why bother using thermocouples to measure temperature at all? Why not just use this other form of temperature measurement, whatever it may be, to do the job?” The answer is this: because the other forms of temperature measurement used for reference junction compensation are not as robust or versatile as a thermocouple junction, but do the job of measuring room temperature at the reference junction site quite well. For example, the thermocouple measurement junction may be inserted into the 1800 degree (F) flue of a foundry holding furnace, while the reference junction sits a hundred feet away in a metal cabinet at ambient temperature, having its temperature measured by a device that could never survive the heat or corrosive atmosphere of the furnace. The voltage produced by thermocouple junctions is strictly dependent upon temperature. Any current in a thermocouple circuit is a function of circuit resistance in opposition to this voltage (I=E/R). In other words, the relationship between temperature and Seebeck voltage is fixed, while the relationship between temperature and current is variable, depending on the total resistance of the circuit. With heavy enough thermocouple conductors, currents upwards of hundreds of amps can be generated from a single pair of thermocouple junctions! (I’ve actually seen this in a laboratory experiment, using heavy bars of copper and copper/nickel alloy to form the junctions and the circuit conductors.) For measurement purposes, the voltmeter used in a thermocouple circuit is designed to have a very high resistance so as to avoid any error-inducing voltage drops along the thermocouple wire. The problem of voltage drop along the conductor length is even more severe here than with the DC voltage signals discussed earlier, because here we only have a few millivolts of voltage produced by the junction. We simply cannot afford to have even a single millivolt of drop along the conductor lengths without incurring serious temperature measurement errors. Ideally, then, current in a thermocouple circuit is zero. Early thermocouple indicating instruments made use of null-balance potentiometric voltage measurement circuitry to measure the junction voltage. The early Leeds & Northrup “Speedomax” line of temperature indicator/recorders were a good example of this technology. More modern instruments use semiconductor amplifier circuits to allow the thermocouple’s voltage signal to drive an indication device with little or no current drawn in the circuit. Thermocouples, however, can be built from heavy-gauge wire for low resistance, and connected in such a way so as to generate very high currents for purposes other than temperature measurement. One such purpose is electric power generation. By connecting many thermocouples in series, alternating hot/cold temperatures with each junction, a device called a thermopile can be constructed to produce substantial amounts of voltage and current: With the left and right sets of junctions at the same temperature, the voltage at each junction will be equal and the opposing polarities would cancel to a final voltage of zero. However, if the left set of junctions were heated and the right set cooled, the voltage at each left junction would be greater than each right junction, resulting in a total output voltage equal to the sum of all junction pair differentials. In a thermopile, this is exactly how things are set up. A source of heat (combustion, strong radioactive substance, solar heat, etc.) is applied to one set of junctions, while the other set is bonded to a heat sink of some sort (air- or water-cooled). Interestingly enough, as electrons flow through an external load circuit connected to the thermopile, heat energy is transferred from the hot junctions to the cold junctions, demonstrating another thermo-electric phenomenon: the so-called Peltier Effect (electric current transferring heat energy). Another application for thermocouples is in the measurement of average temperature between several locations. The easiest way to do this is to connect several thermocouples in parallel with each other. The millivolt signal produced by each thermocouple will average out at the parallel junction point. The voltage differences between the junctions drop along the resistances of the thermocouple wires: Unfortunately, though, the accurate averaging of these Seebeck voltage potentials relies on each thermocouple’s wire resistances being equal. If the thermocouples are located at different places and their wires join in parallel at a single location, equal wire length will be unlikely. The thermocouple having the greatest wire length from point of measurement to parallel connection point will tend to have the greatest resistance, and will therefore have the least effect on the average voltage produced. To help compensate for this, additional resistance can be added to each of the parallel thermocouple circuit branches to make their respective resistances more equal. Without custom-sizing resistors for each branch (to make resistances precisely equal between all the thermocouples), it is acceptable to simply install resistors with equal values, significantly higher than the thermocouple wires’ resistances so that those wire resistances will have a much smaller impact on the total branch resistance. These resistors are called swamping resistors, because their relatively high values overshadow or “swamp” the resistances of the thermocouple wires themselves: Because thermocouple junctions produce such low voltages, it is imperative that wire connections be very clean and tight for accurate and reliable operation. Also, the location of the reference junction (the place where the dissimilar-metal thermocouple wires join to standard copper) must be kept close to the measuring instrument, to ensure that the instrument can accurately compensate for reference junction temperature. Despite these seemingly restrictive requirements, thermocouples remain one of the most robust and popular methods of industrial temperature measurement in modern use. Review • The Seebeck Effect is the production of a voltage between two dissimilar, joined metals that is proportional to the temperature of that junction. • In any thermocouple circuit, there are two equivalent junctions formed between dissimilar metals. The junction placed at the site of intended measurement is called the measurement junction, while the other (single or equivalent) junction is called the reference junction. • Two thermocouple junctions can be connected in opposition to each other to generate a voltage signal proportional to differential temperature between the two junctions. A collection of junctions so connected for the purpose of generating electricity is called a thermopile. • When electrons flow through the junctions of a thermopile, heat energy is transferred from one set of junctions to the other. This is known as the Peltier Effect. • Multiple thermocouple junctions can be connected in parallel with each other to generate a voltage signal representing the average temperature between the junctions. “Swamping” resistors may be connected in series with each thermocouple to help maintain equality between the junctions, so the resultant voltage will be more representative of a true average temperature. • It is imperative that current in a thermocouple circuit be kept as low as possible for good measurement accuracy. Also, all related wire connections should be clean and tight. Mere millivolts of drop at any place in the circuit will cause substantial measurement errors.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/09%3A_Electrical_Instrumentation_Signals/9.06%3A_pH_Measurement.txt	princeton-nlp/TextbookChapters	A very important measurement in many liquid chemical processes (industrial, pharmaceutical, manufacturing, food production, etc.) is that of pH: the measurement of hydrogen ion concentration in a liquid solution. A solution with a low pH value is called an “acid,” while one with a high pH is called a “caustic.” The common pH scale extends from 0 (strong acid) to 14 (strong caustic), with 7 in the middle representing pure water (neutral): pH is defined as follows: the lower-case letter “p” in pH stands for the negative common (base ten) logarithm, while the upper-case letter “H” stands for the element hydrogen. Thus, pH is a logarithmic measurement of the number of moles of hydrogen ions (H+) per liter of solution. Incidentally, the “p” prefix is also used with other types of chemical measurements where a logarithmic scale is desired, pCO2 (Carbon Dioxide) and pO2 (Oxygen) being two such examples. The logarithmic pH scale works like this: a solution with 10-12 moles of H+ ions per liter has a pH of 12; a solution with 10-3 moles of H+ ions per liter has a pH of 3. While very uncommon, there is such a thing as an acid with a pH measurement below 0 and a caustic with a pH above 14. Such solutions, understandably, are quite concentrated and extremely reactive. While pH can be measured by color changes in certain chemical powders (the “litmus strip” being a familiar example from high school chemistry classes), continuous process monitoring and control of pH requires a more sophisticated approach. The most common approach is the use of a specially-prepared electrode designed to allow hydrogen ions in the solution to migrate through a selective barrier, producing a measurable potential (voltage) difference proportional to the solution’s pH: The design and operational theory of pH electrodes is a very complex subject, explored only briefly here. What is important to understand is that these two electrodes generate a voltage directly proportional to the pH of the solution. At a pH of 7 (neutral), the electrodes will produce 0 volts between them. At a low pH (acid) a voltage will be developed of one polarity, and at a high pH (caustic) a voltage will be developed of the opposite polarity. An unfortunate design constraint of pH electrodes is that one of them (called the measurement electrode) must be constructed of special glass to create the ion-selective barrier needed to screen out hydrogen ions from all the other ions floating around in the solution. This glass is chemically doped with lithium ions, which is what makes it react electrochemically to hydrogen ions. Of course, glass is not exactly what you would call a “conductor;” rather, it is an extremely good insulator. This presents a major problem if our intent is to measure voltage between the two electrodes. The circuit path from one electrode contact, through the glass barrier, through the solution, to the other electrode, and back through the other electrode’s contact, is one of extremely high resistance. The other electrode (called the reference electrode) is made from a chemical solution of neutral (7) pH buffer solution (usually potassium chloride) allowed to exchange ions with the process solution through a porous separator, forming a relatively low resistance connection to the test liquid. At first, one might be inclined to ask: why not just dip a metal wire into the solution to get an electrical connection to the liquid? The reason this will not work is because metals tend to be highly reactive in ionic solutions and can produce a significant voltage across the interface of metal-to-liquid contact. The use of a wet chemical interface with the measured solution is necessary to avoid creating such a voltage, which of course would be falsely interpreted by any measuring device as being indicative of pH. Here is an illustration of the measurement electrode’s construction. Note the thin, lithium-doped glass membrane across which the pH voltage is generated: Here is an illustration of the reference electrode’s construction. The porous junction shown at the bottom of the electrode is where the potassium chloride buffer and process liquid interface with each other: The measurement electrode’s purpose is to generate the voltage used to measure the solution’s pH. This voltage appears across the thickness of the glass, placing the silver wire on one side of the voltage and the liquid solution on the other. The reference electrode’s purpose is to provide the stable, zero-voltage connection to the liquid solution so that a complete circuit can be made to measure the glass electrode’s voltage. While the reference electrode’s connection to the test liquid may only be a few kilo-ohms, the glass electrode’s resistance may range from ten to nine hundred mega-ohms, depending on electrode design! Being that any current in this circuit must travel through both electrodes’ resistances (and the resistance presented by the test liquid itself), these resistances are in series with each other and therefore add to make an even greater total. An ordinary analog or even digital voltmeter has much too low of an internal resistance to measure voltage in such a high-resistance circuit. The equivalent circuit diagram of a typical pH probe circuit illustrates the problem: Even a very small circuit current traveling through the high resistances of each component in the circuit (especially the measurement electrode’s glass membrane), will produce relatively substantial voltage drops across those resistances, seriously reducing the voltage seen by the meter. Making matters worse is the fact that the voltage differential generated by the measurement electrode is very small, in the millivolt range (ideally 59.16 millivolts per pH unit at room temperature). The meter used for this task must be very sensitive and have an extremely high input resistance. The most common solution to this measurement problem is to use an amplified meter with an extremely high internal resistance to measure the electrode voltage, so as to draw as little current through the circuit as possible. With modern semiconductor components, a voltmeter with an input resistance of up to 1017 Ω can be built with little difficulty. Another approach, seldom seen in contemporary use, is to use a potentiometric “null-balance” voltage measurement setup to measure this voltage without drawing anycurrent from the circuit under test. If a technician desired to check the voltage output between a pair of pH electrodes, this would probably be the most practical means of doing so using only standard benchtop metering equipment: As usual, the precision voltage supply would be adjusted by the technician until the null detector registered zero, then the voltmeter connected in parallel with the supply would be viewed to obtain a voltage reading. With the detector “nulled” (registering exactly zero), there should be zero current in the pH electrode circuit, and therefore no voltage dropped across the resistances of either electrode, giving the real electrode voltage at the voltmeter terminals. Wiring requirements for pH electrodes tend to be even more severe than thermocouple wiring, demanding very clean connections and short distances of wire (10 yards or less, even with gold-plated contacts and shielded cable) for accurate and reliable measurement. As with thermocouples, however, the disadvantages of electrode pH measurement are offset by the advantages: good accuracy and relative technical simplicity. Few instrumentation technologies inspire the awe and mystique commanded by pH measurement, because it is so widely misunderstood and difficult to troubleshoot. Without elaborating on the exact chemistry of pH measurement, a few words of wisdom can be given here about pH measurement systems: • All pH electrodes have a finite life, and that lifespan depends greatly on the type and severity of service. In some applications, a pH electrode life of one month may be considered long, and in other applications the same electrode(s) may be expected to last for over a year. • Because the glass (measurement) electrode is responsible for generating the pH-proportional voltage, it is the one to be considered suspect if the measurement system fails to generate sufficient voltage change for a given change in pH (approximately 59 millivolts per pH unit), or fails to respond quickly enough to a fast change in test liquid pH. • If a pH measurement system “drifts,” creating offset errors, the problem likely lies with the reference electrode, which is supposed to provide a zero-voltage connection with the measured solution. • Because pH measurement is a logarithmic representation of ion concentration, there is an incredible range of process conditions represented in the seemingly simple 0-14 pH scale. Also, due to the nonlinear nature of the logarithmic scale, a change of 1 pH at the top end (say, from 12 to 13 pH) does not represent the same quantity of chemical activity change as a change of 1 pH at the bottom end (say, from 2 to 3 pH). Control system engineers and technicians must be aware of this dynamic if there is to be any hope of controlling process pH at a stable value. • The following conditions are hazardous to measurement (glass) electrodes: high temperatures, extreme pH levels (either acidic or alkaline), high ionic concentration in the liquid, abrasion, hydrofluoric acid in the liquid (HF acid dissolves glass!), and any kind of material coating on the surface of the glass. • Temperature changes in the measured liquid affect both the response of the measurement electrode to a given pH level (ideally at 59 mV per pH unit), and the actual pH of the liquid. Temperature measurement devices can be inserted into the liquid, and the signals from those devices used to compensate for the effect of temperature on pH measurement, but this will only compensate for the measurement electrode’s mV/pH response, not the actual pH change of the process liquid! Advances are still being made in the field of pH measurement, some of which hold great promise for overcoming traditional limitations of pH electrodes. One such technology uses a device called a field-effect transistor to electrostatically measure the voltage produced by an ion-permeable membrane rather than measure the voltage with an actual voltmeter circuit. While this technology harbors limitations of its own, it is at least a pioneering concept, and may prove more practical at a later date. Review • pH is a representation of hydrogen ion activity in a liquid. It is the negative logarithm of the amount of hydrogen ions (in moles) per liter of liquid. Thus: 10-11 moles of hydrogen ions in 1 liter of liquid = 11 pH. 10-5.3 moles of hydrogen ions in 1 liter of liquid = 5.3 pH. • The basic pH scale extends from 0 (strong acid) to 7 (neutral, pure water) to 14 (strong caustic). Chemical solutions with pH levels below zero and above 14 are possible, but rare. • pH can be measured by measuring the voltage produced between two special electrodes immersed in the liquid solution. • One electrode, made of a special glass, is called the measurement electrode. It’s job it to generate a small voltage proportional to pH (ideally 59.16 mV per pH unit). • The other electrode (called the reference electrode) uses a porous junction between the measured liquid and a stable, neutral pH buffer solution (usually potassium chloride) to create a zero-voltage electrical connection to the liquid. This provides a point of continuity for a complete circuit so that the voltage produced across the thickness of the glass in the measurement electrode can be measured by an external voltmeter. • The extremely high resistance of the measurement electrode’s glass membrane mandates the use of a voltmeter with extremely high internal resistance, or a null-balance voltmeter, to measure the voltage.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/09%3A_Electrical_Instrumentation_Signals/9.07%3A_Strain_Gauges.txt	princeton-nlp/TextbookChapters	What is a Strain Gauge? Such a device is called a strain gauge. Strain gauges are frequently used in mechanical engineering research and development to measure the stresses generated by machinery. Aircraft component testing is one area of application, tiny strain-gauge strips glued to structural members, linkages, and any other critical component of an airframe to measure stress. Most strain gauges are smaller than a postage stamp, and they look something like this: A strain gauge’s conductors are very thin: if made of round wire, about 1/1000 inch in diameter. Alternatively, strain gauge conductors may be thin strips of metallic film deposited on a nonconducting substrate material called the carrier. The latter form of strain gauge is represented in the previous illustration. The name “bonded gauge” is given to strain gauges that are glued to a larger structure under stress (called the test specimen). The task of bonding strain gauges to test specimens may appear to be very simple, but it is not. “Gauging” is a craft in its own right, absolutely essential for obtaining accurate, stable strain measurements. It is also possible to use an unmounted gauge wire stretched between two mechanical points to measure tension, but this technique has its limitations. Strain Gauge Resistance Typical strain gauge resistances range from 30 Ω to 3 kΩ (unstressed). This resistance may change only a fraction of a percent for the full force range of the gauge, given the limitations imposed by the elastic limits of the gauge material and of the test specimen. Forces great enough to induce greater resistance changes would permanently deform the test specimen and/or the gauge conductors themselves, thus ruining the gauge as a measurement device. Thus, in order to use the strain gauge as a practical instrument, we must measure extremely small changes in resistance with high accuracy. Such demanding precision calls for a bridge measurement circuit. Bridge Measurement Circuit Unlike the Wheatstone bridge shown in the last chapter using a null-balance detector and a human operator to maintain a state of balance, a strain gauge bridge circuit indicates measured strain by the degree of imbalance, and uses a precision voltmeter in the center of the bridge to provide an accurate measurement of that imbalance: Typically, the rheostat arm of the bridge (R2 in the diagram) is set at a value equal to the strain gauge resistance with no force applied. The two ratio arms of the bridge (R1 and R3) are set equal to each other. Thus, with no force applied to the strain gauge, the bridge will be symmetrically balanced and the voltmeter will indicate zero volts, representing zero force on the strain gauge. As the strain gauge is either compressed or tensed, its resistance will decrease or increase, respectively, thus unbalancing the bridge and producing an indication at the voltmeter. This arrangement, with a single element of the bridge changing resistance in response to the measured variable (mechanical force), is known as a quarter-bridge circuit. As the distance between the strain gauge and the three other resistances in the bridge circuit may be substantial, wire resistance has a significant impact on the operation of the circuit. To illustrate the effects of wire resistance, I’ll show the same schematic diagram, but add two resistor symbols in series with the strain gauge to represent the wires: Wire Resistances The strain gauge’s resistance (Rgauge) is not the only resistance being measured: the wire resistances Rwire1and Rwire2, being in series with Rgauge, also contribute to the resistance of the lower half of the rheostat arm of the bridge, and consequently contribute to the voltmeter’s indication. This, of course, will be falsely interpreted by the meter as physical strain on the gauge. While this effect cannot be completely eliminated in this configuration, it can be minimized with the addition of a third wire, connecting the right side of the voltmeter directly to the upper wire of the strain gauge: Because the third wire carries practically no current (due to the voltmeter’s extremely high internal resistance), its resistance will not drop any substantial amount of voltage. Notice how the resistance of the top wire (Rwire1) has been “bypassed” now that the voltmeter connects directly to the top terminal of the strain gauge, leaving only the lower wire’s resistance (Rwire2) to contribute any stray resistance in series with the gauge. Not a perfect solution, of course, but twice as good as the last circuit! There is a way, however, to reduce wire resistance error far beyond the method just described, and also help mitigate another kind of measurement error due to temperature. Resistance Change and Temperature An unfortunate characteristic of strain gauges is that of resistance change with changes in temperature. This is a property common to all conductors, some more than others. Thus, our quarter-bridge circuit as shown (either with two or with three wires connecting the gauge to the bridge) works as a thermometer just as well as it does a strain indicator. If all we want to do is measure strain, this is not good. We can transcend this problem, however, by using a “dummy” strain gauge in place of R2, so that both elements of the rheostat arm will change resistance in the same proportion when temperature changes, thus canceling the effects of temperature change: Resistors R1 and R3 are of equal resistance value, and the strain gauges are identical to one another. With no applied force, the bridge should be in a perfectly balanced condition and the voltmeter should register 0 volts. Both gauges are bonded to the same test specimen, but only one is placed in a position and orientation so as to be exposed to physical strain (the active gauge). The other gauge is isolated from all mechanical stress, and acts merely as a temperature compensation device (the “dummy” gauge). If the temperature changes, both gauge resistances will change by the same percentage, and the bridge’s state of balance will remain unaffected. Only a differential resistance (difference of resistance between the two strain gauges) produced by physical force on the test specimen can alter the balance of the bridge. Wire resistance doesn’t impact the accuracy of the circuit as much as before, because the wires connecting both strain gauges to the bridge are approximately equal length. Therefore, the upper and lower sections of the bridge’s rheostat arm contain approximately the same amount of stray resistance, and their effects tend to cancel: Quarter-Bridge and Half Bridge Circuits Even though there are now two strain gauges in the bridge circuit, only one is responsive to mechanical strain, and thus we would still refer to this arrangement as a quarter-bridge. However, if we were to take the upper strain gauge and position it so that it is exposed to the opposite force as the lower gauge (i.e. when the upper gauge is compressed, the lower gauge will be stretched, and vice versa), we will have both gauges responding to strain, and the bridge will be more responsive to applied force. This utilization is known as a half-bridge. Since both strain gauges will either increase or decrease resistance by the same proportion in response to changes in temperature, the effects of temperature change remain canceled and the circuit will suffer minimal temperature-induced measurement error: An example of how a pair of strain gauges may be bonded to a test specimen so as to yield this effect is illustrated here: With no force applied to the test specimen, both strain gauges have equal resistance and the bridge circuit is balanced. However, when a downward force is applied to the free end of the specimen, it will bend downward, stretching gauge #1 and compressing gauge #2 at the same time: Full-Bridge Circuits In applications where such complementary pairs of strain gauges can be bonded to the test specimen, it may be advantageous to make all four elements of the bridge “active” for even greater sensitivity. This is called a full-bridge circuit: Both half-bridge and full-bridge configurations grant greater sensitivity over the quarter-bridge circuit, but often it is not possible to bond complementary pairs of strain gauges to the test specimen. Thus, the quarter-bridge circuit is frequently used in strain measurement systems. When possible, the full-bridge configuration is the best to use. This is true not only because it is more sensitive than the others, but because it is linear while the others are not. Quarter-bridge and half-bridge circuits provide an output (imbalance) signal that is only approximately proportional to applied strain gauge force. Linearity, or proportionality, of these bridge circuits is best when the amount of resistance change due to applied force is very small compared to the nominal resistance of the gauge(s). With a full-bridge, however, the output voltage is directly proportional to applied force, with no approximation (provided that the change in resistance caused by the applied force is equal for all four strain gauges!). Unlike the Wheatstone and Kelvin bridges, which provide measurement at a condition of perfect balance and therefore function irrespective of source voltage, the amount of source (or “excitation”) voltage matters in an unbalanced bridge like this. Therefore, strain gauge bridges are rated in millivolts of imbalance produced per volt of excitation, per unit measure of force. A typical example for a strain gauge of the type used for measuring force in industrial environments is 15 mV/V at 1000 pounds. That is, at exactly 1000 pounds applied force (either compressive or tensile), the bridge will be unbalanced by 15 millivolts for every volt of excitation voltage. Again, such a figure is precise if the bridge circuit is full-active (four active strain gauges, one in each arm of the bridge), but only approximate for half-bridge and quarter-bridge arrangements. Strain gauges may be purchased as complete units, with both strain gauge elements and bridge resistors in one housing, sealed and encapsulated for protection from the elements, and equipped with mechanical fastening points for attachment to a machine or structure. Such a package is typically called a load cell. Like many of the other topics addressed in this chapter, strain gauge systems can become quite complex, and a full dissertation on strain gauges would be beyond the scope of this book. Review • A strain gauge is a thin strip of metal designed to measure mechanical load by changing resistance when stressed (stretched or compressed within its elastic limit). • Strain gauge resistance changes are typically measured in a bridge circuit, to allow for precise measurement of the small resistance changes, and to provide compensation for resistance variations due to temperature.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.01%3A_What_is_Network_Analysis%3F.txt	princeton-nlp/TextbookChapters	• 10.1: What is Network Analysis? Generally speaking, network analysis is any structured technique used to mathematically analyze a circuit (a “network” of interconnected components). Quite often the technician or engineer will encounter circuits containing multiple sources of power or component configurations which defy simplification by series/parallel analysis techniques. In those cases, he or she will be forced to use other means. This chapter presents a few techniques useful in analyzing such complex circuits. • 10.2: Branch Current Method he first and most straightforward network analysis technique is called the Branch Current Method. In this method, we assume directions of currents in a network, then write equations describing their relationships to each other through Kirchhoff’s and Ohm’s Laws. Once we have one equation for every unknown current, we can solve the simultaneous equations and determine all currents, and therefore all voltage drops in the network. • 10.3: Mesh Current Method and Analysis The Mesh Current Method, also known as the Loop Current Method, is quite similar to the Branch Current method in that it uses simultaneous equations, Kirchhoff’s Voltage Law, and Ohm’s Law to determine unknown currents in a network. It differs from the Branch Current method in that it does not use Kirchhoff’s Current Law, and it is usually able to solve a circuit with less unknown variables and less simultaneous equations, which is especially nice if you’re forced to solve without a calculator. • 10.4: Node Voltage Method The node voltage method of analysis solves for unknown voltages at circuit nodes in terms of a system of KCL equations. This analysis looks strange because it involves replacing voltage sources with equivalent current sources. Also, resistor values in ohms are replaced by equivalent conductances in siemens, G = 1/R. The siemens (S) is the unit of conductance, having replaced the mho unit. In any event S = Ω-1. And S = mho (obsolete). • 10.5: Introduction to Network Theorems • 10.6: Millman’s Theorem In Millman’s Theorem, the circuit is re-drawn as a parallel network of branches, each branch containing a resistor or series battery/resistor combination. Millman’s Theorem is applicable only to those circuits which can be re-drawn accordingly. • 10.7: Superposition Theorem Superposition theorem is one of those strokes of genius that takes a complex subject and simplifies it in a way that makes perfect sense. A theorem like Millman’s certainly works well, but it is not quite obvious why it works so well. Superposition, on the other hand, is obvious. • 10.8: Thevenin’s Theorem Thevenin’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). If we’re dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are som • 10.9: Norton’s Theorem • 10.10: Thevenin-Norton Equivalencies Since Thevenin’s and Norton’s Theorems are two equally valid methods of reducing a complex network down to something simpler to analyze, there must be some way to convert a Thevenin equivalent circuit to a Norton equivalent circuit, and vice versa (just what you were dying to know, right?). Well, the procedure is very simple. • 10.11: Millman’s Theorem Revisited • 10.12: Maximum Power Transfer Theorem The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than the maximum. • 10.13: Δ-Y and Y-Δ Conversions 10: DC Network Analysis To illustrate how even a simple circuit can defy analysis by breakdown into series and parallel portions, take start with this series-parallel circuit: To analyze the above circuit, one would first find the equivalent of R2 and R3 in parallel, then add R1 in series to arrive at a total resistance. Then, taking the voltage of battery B1 with that total circuit resistance, the total current could be calculated through the use of Ohm’s Law (I=E/R), then that current figure used to calculate voltage drops in the circuit. All in all, a fairly simple procedure. However, the addition of just one more battery could change all of that: Resistors R2 and R3 are no longer in parallel with each other, because B2 has been inserted into R3‘s branch of the circuit. Upon closer inspection, it appears there are no two resistors in this circuit directly in series or parallel with each other. This is the crux of our problem: in series-parallel analysis, we started off by identifying sets of resistors that were directly in series or parallel with each other, reducing them to single equivalent resistances. If there are no resistors in a simple series or parallel configuration with each other, then what can we do? It should be clear that this seemingly simple circuit, with only three resistors, is impossible to reduce as a combination of simple series and simple parallel sections: it is something different altogether. However, this is not the only type of circuit defying series/parallel analysis: Here we have a bridge circuit, and for the sake of example we will suppose that it is not balanced (ratio R1/R4 not equal to ratio R2/R5). If it were balanced, there would be zero current through R3, and it could be approached as a series/parallel combination circuit (R1—R4 // R2—R5). However, any current through R3makes a series/parallel analysis impossible. R1 is not in series with R4 because there’s another path for electrons to flow through R3. Neither is R2 in series with R5 for the same reason. Likewise, R1 is not in parallel with R2 because R3 is separating their bottom leads. Neither is R4 in parallel with R5. Aaarrggghhhh! Although it might not be apparent at this point, the heart of the problem is the existence of multiple unknown quantities. At least in a series/parallel combination circuit, there was a way to find total resistance and total voltage, leaving total current as a single unknown value to calculate (and then that current was used to satisfy previously unknown variables in the reduction process until the entire circuit could be analyzed). With these problems, more than one parameter (variable) is unknown at the most basic level of circuit simplification. With the two-battery circuit, there is no way to arrive at a value for “total resistance,” because there are twosources of power to provide voltage and current (we would need two “total” resistances in order to proceed with any Ohm’s Law calculations). With the unbalanced bridge circuit, there is such a thing as total resistance across the one battery (paving the way for a calculation of total current), but that total current immediately splits up into unknown proportions at each end of the bridge, so no further Ohm’s Law calculations for voltage (E=IR) can be carried out. So what can we do when we’re faced with multiple unknowns in a circuit? The answer is initially found in a mathematical process known as simultaneous equations or systems of equations, whereby multiple unknown variables are solved by relating them to each other in multiple equations. In a scenario with only one unknown (such as every Ohm’s Law equation we’ve dealt with thus far), there only needs to be a single equation to solve for the single unknown: However, when we’re solving for multiple unknown values, we need to have the same number of equations as we have unknowns in order to reach a solution. There are several methods of solving simultaneous equations, all rather intimidating and all too complex for explanation in this chapter. However, many scientific and programmable calculators are able to solve for simultaneous unknowns, so it is recommended to use such a calculator when first learning how to analyze these circuits. This is not as scary as it may seem at first. Trust me! Later on we’ll see that some clever people have found tricks to avoid having to use simultaneous equations on these types of circuits. We call these tricks network theorems, and we will explore a few later in this chapter. Review • Some circuit configurations (“networks”) cannot be solved by reduction according to series/parallel circuit rules, due to multiple unknown values. • Mathematical techniques to solve for multiple unknowns (called “simultaneous equations” or “systems”) can be applied to basic Laws of circuits to solve networks.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.02%3A_Branch_Current_Method.txt	princeton-nlp/TextbookChapters	Let’s use this circuit to illustrate the method: The first step is to choose a node (junction of wires) in the circuit to use as a point of reference for our unknown currents. I’ll choose the node joining the right of R1, the top of R2, and the left of R3. At this node, guess which directions the three wires’ currents take, labeling the three currents as I1, I2, and I3, respectively. Bear in mind that these directions of current are speculative at this point. Fortunately, if it turns out that any of our guesses were wrong, we will know when we mathematically solve for the currents (any “wrong” current directions will show up as negative numbers in our solution). Kirchhoff’s Current Law (KCL) tells us that the algebraic sum of currents entering and exiting a node must equal zero, so we can relate these three currents (I1, I2, and I3) to each other in a single equation. For the sake of convention, I’ll denote any current entering the node as positive in sign, and any current exiting the node as negative in sign: The next step is to label all voltage drop polarities across resistors according to the assumed directions of the currents. Remember that the “upstream” end of a resistor will always be negative, and the “downstream” end of a resistor positive with respect to each other, since electrons are negatively charged: The battery polarities, of course, remain as they were according to their symbology (short end negative, long end positive). It is OK if the polarity of a resistor’s voltage drop doesn’t match with the polarity of the nearest battery, so long as the resistor voltage polarity is correctly based on the assumed direction of current through it. In some cases we may discover that current will be forced backwards through a battery, causing this very effect. The important thing to remember here is to base all your resistor polarities and subsequent calculations on the directions of current(s) initially assumed. As stated earlier, if your assumption happens to be incorrect, it will be apparent once the equations have been solved (by means of a negative solution). The magnitude of the solution, however, will still be correct. Kirchhoff’s Voltage Law (KVL) tells us that the algebraic sum of all voltages in a loop must equal zero, so we can create more equations with current terms (I1, I2, and I3) for our simultaneous equations. To obtain a KVL equation, we must tally voltage drops in a loop of the circuit, as though we were measuring with a real voltmeter. I’ll choose to trace the left loop of this circuit first, starting from the upper-left corner and moving counter-clockwise (the choice of starting points and directions is arbitrary). The result will look like this: Having completed our trace of the left loop, we add these voltage indications together for a sum of zero: Of course, we don’t yet know what the voltage is across R1 or R2, so we can’t insert those values into the equation as numerical figures at this point. However, we do know that all three voltages must algebraically add to zero, so the equation is true. We can go a step further and express the unknown voltages as the product of the corresponding unknown currents (I1 and I2) and their respective resistors, following Ohm’s Law (E=IR), as well as eliminate the 0 term: Since we know what the values of all the resistors are in ohms, we can just substitute those figures into the equation to simplify things a bit: You might be wondering why we went through all the trouble of manipulating this equation from its initial form (-28 + ER2 + ER1). After all, the last two terms are still unknown, so what advantage is there to expressing them in terms of unknown voltages or as unknown currents (multiplied by resistances)? The purpose in doing this is to get the KVL equation expressed using the same unknown variables as the KCL equation, for this is a necessary requirement for any simultaneous equation solution method. To solve for three unknown currents (I1, I2, and I3), we must have three equations relating these three currents (not voltages!) together. Applying the same steps to the right loop of the circuit (starting at the chosen node and moving counter-clockwise), we get another KVL equation: Knowing now that the voltage across each resistor can be and should be expressed as the product of the corresponding current and the (known) resistance of each resistor, we can re-write the equation as such: Now we have a mathematical system of three equations (one KCL equation and two KVL equations) and three unknowns: For some methods of solution (especially any method involving a calculator), it is helpful to express each unknown term in each equation, with any constant value to the right of the equal sign, and with any “unity” terms expressed with an explicit coefficient of 1. Re-writing the equations again, we have: Using whatever solution techniques are available to us, we should arrive at a solution for the three unknown current values: So, I1 is 5 amps, I2 is 4 amps, and I3 is a negative 1 amp. But what does “negative” current mean? In this case, it means that our assumed direction for I3 was opposite of its real direction. Going back to our original circuit, we can re-draw the current arrow for I3 (and re-draw the polarity of R3‘s voltage drop to match): Notice how current is being pushed backwards through battery 2 (electrons flowing “up”) due to the higher voltage of battery 1 (whose current is pointed “down” as it normally would)! Despite the fact that battery B2‘s polarity is trying to push electrons down in that branch of the circuit, electrons are being forced backwards through it due to the superior voltage of battery B1. Does this mean that the stronger battery will always “win” and the weaker battery always get current forced through it backwards? No! It actually depends on both the batteries’ relative voltages and the resistor values in the circuit. The only sure way to determine what’s going on is to take the time to mathematically analyze the network. Now that we know the magnitude of all currents in this circuit, we can calculate voltage drops across all resistors with Ohm’s Law (E=IR): Let us now analyze this network using SPICE to verify our voltage figures. We could analyze current as well with SPICE, but since that requires the insertion of extra components into the circuit, and because we know that if the voltages are all the same and all the resistances are the same, the currents must all be the same, I’ll opt for the less complex analysis. Here’s a re-drawing of our circuit, complete with node numbers for SPICE to reference: Sure enough, the voltage figures all turn out to be the same: 20 volts across R1 (nodes 1 and 2), 8 volts across R2 (nodes 2 and 0), and 1 volt across R3 (nodes 2 and 3). Take note of the signs of all these voltage figures: they’re all positive values! SPICE bases its polarities on the order in which nodes are listed, the first node being positive and the second node negative. For example, a figure of positive (+) 20 volts between nodes 1 and 2 means that node 1 is positive with respect to node 2. If the figure had come out negative in the SPICE analysis, we would have known that our actual polarity was “backwards” (node 1 negative with respect to node 2). Checking the node orders in the SPICE listing, we can see that the polarities all match what we determined through the Branch Current method of analysis. Review • Steps to follow for the “Branch Current” method of analysis: (1) Choose a node and assume directions of currents. (2) Write a KCL equation relating currents at the node. (3) Label resistor voltage drop polarities based on assumed currents. (4) Write KVL equations for each loop of the circuit, substituting the product IR for E in each resistor term of the equations. (5) Solve for unknown branch currents (simultaneous equations). (6) If any solution is negative, then the assumed direction of current for that solution is wrong! (7) Solve for voltage drops across all resistors (E=IR).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.03%3A_Mesh_Current_Method_and_Analysis.txt	princeton-nlp/TextbookChapters	Mesh Current, Conventional Method Let’s see how this method works on the same example problem: Identifying Loops in a Circuit The first step in the Mesh Current method is to identify “loops” within the circuit encompassing all components. In our example circuit, the loop formed by B1, R1, and R2 will be the first while the loop formed by B2, R2, and R3 will be the second. The strangest part of the Mesh Current method is envisioning circulating currents in each of the loops. In fact, this method gets its name from the idea of these currents meshing together between loops like sets of spinning gears: The choice of each current’s direction is entirely arbitrary, just as in the Branch Current method, but the resulting equations are easier to solve if the currents are going the same direction through intersecting components (note how currents I1 and I2 are both going “up” through resistor R2, where they “mesh,” or intersect). If the assumed direction of a mesh current is wrong, the answer for that current will have a negative value. Label the Voltage Drop Polarities The next step is to label all voltage drop polarities across resistors according to the assumed directions of the mesh currents. Remember that the “upstream” end of a resistor will always be negative, and the “downstream” end of a resistor positive with respect to each other, since electrons are negatively charged. The battery polarities, of course, are dictated by their symbol orientations in the diagram, and may or may not “agree” with the resistor polarities (assumed current directions): Using Kirchhoff’s Voltage Law, we can now step around each of these loops, generating equations representative of the component voltage drops and polarities. As with the Branch Current method, we will denote a resistor’s voltage drop as the product of the resistance (in ohms) and its respective mesh current (that quantity being unknown at this point). Where two currents mesh together, we will write that term in the equation with resistor current being the sum of the two meshing currents. Tracing the Left Loop of the Circuit with Equations Tracing the left loop of the circuit, starting from the upper-left corner and moving counter-clockwise (the choice of starting points and directions is ultimately irrelevant), counting polarity as if we had a voltmeter in hand, red lead on the point ahead and black lead on the point behind, we get this equation: Notice that the middle term of the equation uses the sum of mesh currents I1 and I2 as the current through resistor R2. This is because mesh currents I1 and I2 are going the same direction through R2, and thus complement each other. Distributing the coefficient of 2 to the I1 and I2 terms, and then combining I1 terms in the equation, we can simplify as such: At this time we have one equation with two unknowns. To be able to solve for two unknown mesh currents, we must have two equations. If we trace the other loop of the circuit, we can obtain another KVL equation and have enough data to solve for the two currents. Creature of habit that I am, I’ll start at the upper-left hand corner of the right loop and trace counter-clockwise: Simplifying the equation as before, we end up with: Now, with two equations, we can use one of several methods to mathematically solve for the unknown currents I1 and I2: Knowing that these solutions are values for mesh currents, not branch currents, we must go back to our diagram to see how they fit together to give currents through all components: The solution of -1 amp for I2 means that our initially assumed direction of current was incorrect. In actuality, I2 is flowing in a counter-clockwise direction at a value of (positive) 1 amp: This change of current direction from what was first assumed will alter the polarity of the voltage drops across R2 and R3 due to current I2. From here, we can say that the current through R1 is 5 amps, with the voltage drop across R1 being the product of current and resistance (E=IR), 20 volts (positive on the left and negative on the right). Also, we can safely say that the current through R3 is 1 amp, with a voltage drop of 1 volt (E=IR), positive on the left and negative on the right. But what is happening at R2? Mesh current I1 is going “up” through R2, while mesh current I2 is going “down” through R2. To determine the actual current through R2, we must see how mesh currents I1 and I2 interact (in this case they’re in opposition), and algebraically add them to arrive at a final value. Since I1 is going “up” at 5 amps, and I2 is going “down” at 1 amp, the real current through R2 must be a value of 4 amps, going “up:” A current of 4 amps through R2‘s resistance of 2 Ω gives us a voltage drop of 8 volts (E=IR), positive on the top and negative on the bottom. Advantage of Mesh Current Analysis The primary advantage of Mesh Current analysis is that it generally allows for the solution of a large network with fewer unknown values and fewer simultaneous equations. Our example problem took three equations to solve the Branch Current method and only two equations using the Mesh Current method. This advantage is much greater as networks increase in complexity: To solve this network using Branch Currents, we’d have to establish five variables to account for each and every unique current in the circuit (I1 through I5). This would require five equations for solution, in the form of two KCL equations and three KVL equations (two equations for KCL at the nodes, and three equations for KVL in each loop): I suppose if you have nothing better to do with your time than to solve for five unknown variables with five equations, you might not mind using the Branch Current method of analysis for this circuit. For those of us who have better things to do with our time, the Mesh Current method is a whole lot easier, requiring only three unknowns and three equations to solve: Less equations to work with is a decided advantage, especially when performing simultaneous equation solution by hand (without a calculator). Another type of circuit that lends itself well to Mesh Current is the unbalanced Wheatstone Bridge. Take this circuit, for example: Since the ratios of R1/R4 and R2/R5 are unequal, we know that there will be voltage across resistor R3, and some amount of current through it. As discussed at the beginning of this chapter, this type of circuit is irreducible by normal series-parallel analysis, and may only be analyzed by some other method. We could apply the Branch Current method to this circuit, but it would require six currents (I1 through I6), leading to a very large set of simultaneous equations to solve. Using the Mesh Current method, though, we may solve for all currents and voltages with much fewer variables. The first step in the Mesh Current method is to draw just enough mesh currents to account for all components in the circuit. Looking at our bridge circuit, it should be obvious where to place two of these currents: The directions of these mesh currents, of course, is arbitrary. However, two mesh currents is not enough in this circuit, because neither I1 nor I2 goes through the battery. So, we must add a third mesh current, I3: Here, I have chosen I3 to loop from the bottom side of the battery, through R4, through R1, and back to the top side of the battery. This is not the only path I could have chosen for I3, but it seems the simplest. Now, we must label the resistor voltage drop polarities, following each of the assumed currents’ directions: Notice something very important here: at resistor R4, the polarities for the respective mesh currents do not agree. This is because those mesh currents (I2 and I3) are going through R4 in different directions. This does not preclude the use of the Mesh Current method of analysis, but it does complicate it a bit. Though later, we will show how to avoid the R4 current clash. (See Example below) Generating a KVL equation for the top loop of the bridge, starting from the top node and tracing in a clockwise direction: In this equation, we represent the common directions of currents by their sums through common resistors. For example, resistor R3, with a value of 100 Ω, has its voltage drop represented in the above KVL equation by the expression 100(I1 + I2), since both currents I1 and I2 go through R3 from right to left. The same may be said for resistor R1, with its voltage drop expression shown as 150(I1 + I3), since both I1 and I3 go from bottom to top through that resistor, and thus work together to generate its voltage drop. Generating a KVL equation for the bottom loop of the bridge will not be so easy, since we have two currents going against each other through resistor R4. Here is how I do it (starting at the right-hand node, and tracing counter-clockwise): Note how the second term in the equation’s original form has resistor R4‘s value of 300 Ω multiplied by the difference between I2 and I3 (I2 - I3). This is how we represent the combined effect of two mesh currents going in opposite directions through the same component. Choosing the appropriate mathematical signs is very important here: 300(I2 - I3) does not mean the same thing as 300(I3 - I2). I chose to write 300(I2 - I3) because I was thinking first of I2‘s effect (creating a positive voltage drop, measuring with an imaginary voltmeter across R4, red lead on the bottom and black lead on the top), and secondarily of I3‘s effect (creating a negative voltage drop, red lead on the bottom and black lead on the top). If I had thought in terms of I3‘s effect first and I2‘s effect secondarily, holding my imaginary voltmeter leads in the same positions (red on bottom and black on top), the expression would have been -300(I3 - I2). Note that this expression is mathematically equivalent to the first one: +300(I2 - I3). Well, that takes care of two equations, but I still need a third equation to complete my simultaneous equation set of three variables, three equations. This third equation must also include the battery’s voltage, which up to this point does not appear in either two of the previous KVL equations. To generate this equation, I will trace a loop again with my imaginary voltmeter starting from the battery’s bottom (negative) terminal, stepping clockwise (again, the direction in which I step is arbitrary, and does not need to be the same as the direction of the mesh current in that loop): Solving for I1, I2, and I3 using whatever simultaneous equation method we prefer: Example: Use Octave to find the solution for I1, I2, and I3 from the above simplified form of equations. Solution: In Octave, an open source Matlab® clone, enter the coefficients into the A matrix between square brackets with column elements comma separated, and rows semicolon separated.Enter the voltages into the column vector: b. The unknown currents: I1, 2, and I3 are calculated by the command: x=A\b. These are contained within the x column vector. The negative value arrived at for I1 tells us that the assumed direction for that mesh current was incorrect. Thus, the actual current values through each resistor is as such: Calculating voltage drops across each resistor: A SPICE simulation confirms the accuracy of our voltage calculations: Example: (a) Find a new path for current I3 that does not produce a conflicting polarity on any resistor compared to I1or I2. R4 was the offending component. (b) Find values for I1, I2, and I3. (c) Find the five resistor currents and compare to the previous values. Solution: (a) Route I3 through R5, R3 and R1 as shown: Note that the conflicting polarity on R4 has been removed. Moreover, none of the other resistors have conflicting polarities. (b) Octave, an open source (free) matlab clone, yields a mesh current vector at “x”: Not all currents I1, I2, and I3 are the same (I2) as the previous bridge because of different loop paths However, the resistor currents compare to the previous values: Since the resistor currents are the same as the previous values, the resistor voltages will be identical and need not be calculated again. Review • Steps to follow for the “Mesh Current” method of analysis: (1) Draw mesh currents in loops of circuit, enough to account for all components. (2) Label resistor voltage drop polarities based on assumed directions of mesh currents. (3) Write KVL equations for each loop of the circuit, substituting the product IR for E in each resistor term of the equation. Where two mesh currents intersect through a component, express the current as the algebraic sum of those two mesh currents (i.e. I1 + I2) if the currents go in the same direction through that component. If not, express the current as the difference (i.e. I1 - I2). (4) Solve for unknown mesh currents (simultaneous equations). (5) If any solution is negative, then the assumed current direction is wrong! (6) Algebraically add mesh currents to find current in components sharing multiple mesh currents. (7) Solve for voltage drops across all resistors (E=IR). Mesh Current by Inspection We take a second look at the “mesh current method” with all the currents running counterclockwise (ccw). The motivation is to simplify the writing of mesh equations by ignoring the resistor voltage drop polarity. Though, we must pay attention to the polarity of voltage sources with respect to assumed current direction. The sign of the resistor voltage drops will follow a fixed pattern. If we write a set of conventional mesh-current equations for the circuit below, where we do pay attention to the signs of the voltage drop across the resistors, we may rearrange the coefficients into a fixed pattern: Once rearranged, we may write equations by inspection. The signs of the coefficients follow a fixed pattern in the pair above, or the set of three in the rules below. Mesh Current Rules: This method assumes electron flow (not conventional current flow) voltage sources. Replace any current source in parallel with a resistor with an equivalent voltage source in series with an equivalent resistance. Ignoring current direction or voltage polarity on resistors, draw counterclockwise current loops traversing all components. Avoid nested loops. Write voltage-law equations in terms of unknown currents: I1, I2, and I3. Equation 1 coefficient 1, equation 2, coefficient 2, and equation 3 coefficient 3 are the positive sums of resistors around the respective loops. All other coefficients are negative, representative of the resistance common to a pair of loops. Equation 1 coefficient 2 is the resistor common to loops 1 and 2, coefficient 3 the resistor common to loops 1 an 3. Repeat for other equations and coefficients. The right-hand side of the equations is equal to any electron current flow voltage source. A voltage rise with respect to the counterclockwise assumed current is positive, and 0 for no voltage source. Solve equations for mesh currents:I1, I2, and I3 . Solve for currents through individual resistors with KCL. Solve for voltages with Ohms Law and KVL. While the above rules are specific for a three mesh circuit, the rules may be extended to smaller or larger meshes. The figure below illustrates the application of the rules. The three currents are all drawn in the same direction, counterclockwise. One KVL equation is written for each of the three loops. Note that there is no polarity drawn on the resistors. We do not need it to determine the signs of the coefficients. Though we do need to pay attention to the polarity of the voltage source with respect to current direction. The I3counterclockwise current traverses the 24V source from (+) to (-). This is a voltage rise for electron current flow. Therefore, the third equation right-hand side is +24V. In Octave, enter the coefficients into the A matrix with column elements comma separated, and rows semicolon separated. Enter the voltages into the column vector b. Solve for the unknown currents: I1, I2, and I3 with the command: x=A\b. These currents are contained within the x column vector. The positive values indicate that the three mesh currents all flow in the assumed counterclockwise direction. The mesh currents match the previous solution by a different mesh current method.. The calculation of resistor voltages and currents will be identical to the previous solution. No need to repeat here. Note that electrical engineering texts are based on conventional current flow. The loop-current, mesh-current method in those text will run the assumed mesh currents clockwise.The conventional current flows out the (+) terminal of the battery through the circuit, returning to the (-) terminal. A conventional current voltage rise corresponds to tracing the assumed current from (-) to (+) through any voltage sources. One more example of a previous circuit follows. The resistance around loop 1 is 6 Ω, around loop 2: 3 Ω. The resistance common to both loops is 2 Ω. Note the coefficients of I1 and I2 in the pair of equations. Tracing the assumed counterclockwise loop 1 current through B1 from (+) to (-) corresponds to an electron current flow voltage rise. Thus, the sign of the 28 V is positive. The loop 2 counter clockwise assumed current traces (-) to (+) through B2, a voltage drop. Thus, the sign of B2 is negative, -7 in the 2nd mesh equation. Once again, there are no polarity markings on the resistors. Nor do they figure into the equations. The currents I1 = 5 A, and I2 = 1 A are both positive. They both flow in the direction of the counterclockwise loops. This compares with previous results. Summary: • The modified mesh-current method avoids having to determine the signs of the equation coefficients by drawing all mesh currents counterclockwise for electron current flow. • However, we do need to determine the sign of any voltage sources in the loop. The voltage source is positive if the assumed ccw current flows with the battery (source). The sign is negative if the assumed ccw current flows against the battery. • See rules above for details.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.04%3A_Node_Voltage_Method.txt	princeton-nlp/TextbookChapters	Method for Node Voltage Calculation We start with a circuit having conventional voltage sources. A common node E0 is chosen as a reference point. The node voltages E1 and E2 are calculated with respect to this point. A voltage source in series with a resistance must be replaced by an equivalent current source in parallel with the resistance. We will write KCL equations for each node. The right hand side of the equation is the value of the current source feeding the node. Replacing voltage sources and associated series resistors with equivalent current sources and parallel resistors yields the modified circuit. Substitute resistor conductances in siemens for resistance in ohms. The parallel conductances (resistors) may be combined by addition of the conductances. Though, we will not redraw the circuit. The circuit is ready for application of the node voltage method. Deriving a general node voltage method, we write a pair of KCL equations in terms of unknown node voltages V1 and V2 this one time. We do this to illustrate a pattern for writing equations by inspection. The coefficients of the last pair of equations above have been rearranged to show a pattern. The sum of conductances connected to the first node is the positive coefficient of the first voltage in equation (1). The sum of conductances connected to the second node is the positive coefficient of the second voltage in equation (2). The other coefficients are negative, representing conductances between nodes. For both equations, the right hand side is equal to the respective current source connected to the node. This pattern allows us to quickly write the equations by inspection. This leads to a set of rules for the node voltage method of analysis. Node Voltage Rules: • Convert voltage sources in series with a resistor to an equivalent current source with the resistor in parallel. • Change resistor values to conductances. • Select a reference node(E0) • Assign unknown voltages (E1)(E2) ... (EN)to remaining nodes. • Write a KCL equation for each node 1,2, ... N. The positive coefficient of the first voltage in the first equation is the sum of conductances connected to the node. The coefficient for the second voltage in the second equation is the sum of conductances connected to that node. Repeat for coefficient of third voltage, third equation, and other equations. These coefficients fall on a diagonal. • All other coefficients for all equations are negative, representing conductances between nodes. The first equation, second coefficient is the conductance from node 1 to node 2, the third coefficient is the conductance from node 1 to node 3. Fill in negative coefficients for other equations. • The right hand side of the equations is the current source connected to the respective nodes. • Solve system of equations for unknown node voltages. Example: Set up the equations and solve for the node voltages using the numerical values in the above figure. Solution: The solution of two equations can be performed with a calculator, or with octave (not shown).[octav] The solution is verified with SPICE based on the original schematic diagram with voltage sources.[spi] .Though, the circuit with the current sources could have been simulated. One more example. This one has three nodes. We do not list the conductances on the schematic diagram. However, G1 = 1/R1, etc. There are three nodes to write equations for by inspection. Note that the coefficients are positive for equation (1) E1, equation (2) E2, and equation (3) E3. These are the sums of all conductances connected to the nodes. All other coefficients are negative, representing a conductance between nodes. The right hand side of the equations is the associated current source, 0.136092 A for the only current source at node 1. The other equations are zero on the right hand side for lack of current sources. We are too lazy to calculate the conductances for the resistors on the diagram. Thus, the subscripted G’s are the coefficients. We are so lazy that we enter reciprocal resistances and sums of reciprocal resistances into the octave “A” matrix, letting octave compute the matrix of conductances after “A=”. [octav] The initial entry line was so long that it was split into three rows. This is different than previous examples. The entered “A” matrix is delineated by starting and ending square brackets. Column elements are space separated. Rows are “new line” separated. Commas and semicolons are not need as separators. Though, the current vector at “b” is semicolon separated to yield a column vector of currents. Note that the “A” matrix diagonal coefficients are positive, That all other coefficients are negative. The solution as a voltage vector is at “x”. E1 = 24.000 V, E2 = 17.655 V, E3 = 19.310 V. These three voltages compare to the previous mesh current and SPICE solutions to the unbalanced bridge problem. This is no coincidence, for the 0.13609 A current source was purposely chosen to yield the 24 V used as a voltage source in that problem. Summary • Given a network of conductances and current sources, the node voltage method of circuit analysis solves for unknown node voltages from KCL equations. • See rules above for details in writing the equations by inspection.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.05%3A_Introduction_to_Network_Theorems.txt	princeton-nlp/TextbookChapters	Anyone who’s studied geometry should be familiar with the concept of a theorem: a relatively simple rule used to solve a problem, derived from a more intensive analysis using fundamental rules of mathematics. At least hypothetically, any problem in math can be solved just by using the simple rules of arithmetic (in fact, this is how modern digital computers carry out the most complex mathematical calculations: by repeating many cycles of additions and subtractions!), but human beings aren’t as consistent or as fast as a digital computer. We need “shortcut” methods in order to avoid procedural errors. In electric network analysis, the fundamental rules are Ohm’s Law and Kirchhoff’s Laws. While these humble laws may be applied to analyze just about any circuit configuration (even if we have to resort to complex algebra to handle multiple unknowns), there are some “shortcut” methods of analysis to make the math easier for the average human. As with any theorem of geometry or algebra, these network theorems are derived from fundamental rules. In this chapter, I’m not going to delve into the formal proofs of any of these theorems. If you doubt their validity, you can always empirically test them by setting up example circuits and calculating values using the “old” (simultaneous equation) methods versus the “new” theorems, to see if the answers coincide. They always should! 10.06: Millman’s Theorem In Millman’s Theorem, the circuit is re-drawn as a parallel network of branches, each branch containing a resistor or series battery/resistor combination. Millman’s Theorem is applicable only to those circuits which can be re-drawn accordingly. Here again is our example circuit used for the last two analysis methods: And here is that same circuit, re-drawn for the sake of applying Millman’s Theorem: By considering the supply voltage within each branch and the resistance within each branch, Millman’s Theorem will tell us the voltage across all branches. Please note that I’ve labeled the battery in the rightmost branch as “B3” to clearly denote it as being in the third branch, even though there is no “B2” in the circuit! Millman’s Theorem is nothing more than a long equation, applied to any circuit drawn as a set of parallel-connected branches, each branch with its own voltage source and series resistance: Substituting actual voltage and resistance figures from our example circuit for the variable terms of this equation, we get the following expression: The final answer of 8 volts is the voltage seen across all parallel branches, like this: The polarity of all voltages in Millman’s Theorem are referenced to the same point. In the example circuit above, I used the bottom wire of the parallel circuit as my reference point, and so the voltages within each branch (28 for the R1 branch, 0 for the R2 branch, and 7 for the R3 branch) were inserted into the equation as positive numbers. Likewise, when the answer came out to 8 volts (positive), this meant that the top wire of the circuit was positive with respect to the bottom wire (the original point of reference). If both batteries had been connected backwards (negative ends up and positive ends down), the voltage for branch 1 would have been entered into the equation as a -28 volts, the voltage for branch 3 as -7 volts, and the resulting answer of -8 volts would have told us that the top wire was negative with respect to the bottom wire (our initial point of reference). To solve for resistor voltage drops, the Millman voltage (across the parallel network) must be compared against the voltage source within each branch, using the principle of voltages adding in series to determine the magnitude and polarity of voltage across each resistor: To solve for branch currents, each resistor voltage drop can be divided by its respective resistance (I=E/R): The direction of current through each resistor is determined by the polarity across each resistor, not by the polarity across each battery, as current can be forced backwards through a battery, as is the case with B3 in the example circuit. This is important to keep in mind, since Millman’s Theorem doesn’t provide as direct an indication of “wrong” current direction as does the Branch Current or Mesh Current methods. You must pay close attention to the polarities of resistor voltage drops as given by Kirchhoff’s Voltage Law, determining direction of currents from that. Millman’s Theorem is very convenient for determining the voltage across a set of parallel branches, where there are enough voltage sources present to preclude solution via regular series-parallel reduction method. It also is easy in the sense that it doesn’t require the use of simultaneous equations. However, it is limited in that it only applied to circuits which can be re-drawn to fit this form. It cannot be used, for example, to solve an unbalanced bridge circuit. And, even in cases where Millman’s Theorem can be applied, the solution of individual resistor voltage drops can be a bit daunting to some, the Millman’s Theorem equation only providing a single figure for branch voltage. As you will see, each network analysis method has its own advantages and disadvantages. Each method is a tool, and there is no tool that is perfect for all jobs. The skilled technician, however, carries these methods in his or her mind like a mechanic carries a set of tools in his or her tool box. The more tools you have equipped yourself with, the better prepared you will be for any eventuality. Review • Millman’s Theorem treats circuits as a parallel set of series-component branches. • All voltages entered and solved for in Millman’s Theorem are polarity-referenced at the same point in the circuit (typically the bottom wire of the parallel network).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.07%3A_Superposition_Theorem.txt	princeton-nlp/TextbookChapters	Series/Parallel Analysis The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active. Let’s look at our example circuit again and apply Superposition Theorem to it: Since we have two sources of power in this circuit, we will have to calculate two sets of values for voltage drops and/or currents, one for the circuit with only the 28-volt battery in effect. . . . . . and one for the circuit with only the 7-volt battery in effect: When re-drawing the circuit for series/parallel analysis with one source, all other voltage sources are replaced by wires (shorts), and all current sources with open circuits (breaks). Since we only have voltage sources (batteries) in our example circuit, we will replace every inactive source during analysis with a wire. Analyzing the circuit with only the 28-volt battery, we obtain the following values for voltage and current: Analyzing the circuit with only the 7-volt battery, we obtain another set of values for voltage and current: When superimposing these values of voltage and current, we have to be very careful to consider polarity (voltage drop) and direction (electron flow), as the values have to be added algebraically. Applying these superimposed voltage figures to the circuit, the end result looks something like this: Currents add up algebraically as well and can either be superimposed as done with the resistor voltage drops or simply calculated from the final voltage drops and respective resistances (I=E/R). Either way, the answers will be the same. Here I will show the superposition method applied to current: Once again applying these superimposed figures to our circuit: Prerequisites for the Superposition Theorem Quite simple and elegant, don’t you think? It must be noted, though, that the Superposition Theorem works only for circuits that are reducible to series/parallel combinations for each of the power sources at a time (thus, this theorem is useless for analyzing an unbalanced bridge circuit), and it only works where the underlying equations are linear (no mathematical powers or roots). The requisite of linearity means that Superposition Theorem is only applicable for determining voltage and current, not power!!! Power dissipations, being nonlinear functions, do not algebraically add to an accurate total when only one source is considered at a time. The need for linearity also means this Theorem cannot be applied in circuits where the resistance of a component changes with voltage or current. Hence, networks containing components like lamps (incandescent or gas-discharge) or varistors could not be analyzed. Another prerequisite for Superposition Theorem is that all components must be “bilateral,” meaning that they behave the same with electrons flowing in either direction through them. Resistors have no polarity-specific behavior, and so the circuits we’ve been studying so far all meet this criterion. The Superposition Theorem finds use in the study of alternating current (AC) circuits, and semiconductor (amplifier) circuits, where sometimes AC is often mixed (superimposed) with DC. Because AC voltage and current equations (Ohm’s Law) are linear just like DC, we can use Superposition to analyze the circuit with just the DC power source, then just the AC power source, combining the results to tell what will happen with both AC and DC sources in effect. For now, though, Superposition will suffice as a break from having to do simultaneous equations to analyze a circuit. Review • The Superposition Theorem states that a circuit can be analyzed with only one source of power at a time, the corresponding component voltages and currents algebraically added to find out what they’ll do with all power sources in effect. • To negate all but one power source for analysis, replace any source of voltage (batteries) with a wire; replace any current source with an open (break).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.08%3A_Thevenin%E2%80%99s_Theorem.txt	princeton-nlp/TextbookChapters	Thevenin’s Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it. Let’s take another look at our example circuit: Let’s suppose that we decide to designate R2 as the “load” resistor in this circuit. We already have four methods of analysis at our disposal (Branch Current, Mesh Current, Millman’s Theorem, and Superposition Theorem) to use in determining voltage across R2 and current through R2, but each of these methods are time-consuming. Imagine repeating any of these methods over and over again to find what would happen if the load resistance changed (changing load resistance is very common in power systems, as multiple loads get switched on and off as needed. the total resistance of their parallel connections changing depending on how many are connected at a time). This could potentially involve a lot of work! Thevenin’s Theorem makes this easy by temporarily removing the load resistance from the original circuit and reducing what’s left to an equivalent circuit composed of a single voltage source and series resistance. The load resistance can then be re-connected to this “Thevenin equivalent circuit” and calculations carried out as if the whole network were nothing but a simple series circuit: after Thevenin conversion . . . The “Thevenin Equivalent Circuit” is the electrical equivalent of B1, R1, R3, and B2 as seen from the two points where our load resistor (R2) connects. The Thevenin equivalent circuit, if correctly derived, will behave exactly the same as the original circuit formed by B1, R1, R3, and B2. In other words, the load resistor (R2) voltage and current should be exactly the same for the same value of load resistance in the two circuits. The load resistor R2 cannot “tell the difference” between the original network of B1, R1, R3, and B2, and the Thevenin equivalent circuit of EThevenin, and RThevenin, provided that the values for EThevenin and RThevenin have been calculated correctly. The advantage in performing the “Thevenin conversion” to the simpler circuit, of course, is that it makes load voltage and load current so much easier to solve than in the original network. Calculating the equivalent Thevenin source voltage and series resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit, replaced with a break (open circuit): Next, the voltage between the two points where the load resistor used to be attached is determined. Use whatever analysis methods are at your disposal to do this. In this case, the original circuit with the load resistor removed is nothing more than a simple series circuit with opposing batteries, and so we can determine the voltage across the open load terminals by applying the rules of series circuits, Ohm’s Law, and Kirchhoff’s Voltage Law: The voltage between the two load connection points can be figured from the one of the battery’s voltage and one of the resistor’s voltage drops, and comes out to 11.2 volts. This is our “Thevenin voltage” (EThevenin) in the equivalent circuit: To find the Thevenin series resistance for our equivalent circuit, we need to take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure the resistance from one load terminal to the other: With the removal of the two batteries, the total resistance measured at this location is equal to R1 and R3 in parallel: 0.8 Ω. This is our “Thevenin resistance” (RThevenin) for the equivalent circuit: With the load resistor (2 Ω) attached between the connection points, we can determine voltage across it and current through it as though the whole network were nothing more than a simple series circuit: Notice that the voltage and current figures for R2 (8 volts, 4 amps) are identical to those found using other methods of analysis. Also notice that the voltage and current figures for the Thevenin series resistance and the Thevenin source (total) do not apply to any component in the original, complex circuit. Thevenin’s Theorem is only useful for determining what happens to a single resistor in a network: the load. The advantage, of course, is that you can quickly determine what would happen to that single resistor if it were of a value other than 2 Ω without having to go through a lot of analysis again. Just plug in that other value for the load resistor into the Thevenin equivalent circuit and a little bit of series circuit calculation will give you the result. Review • Thevenin’s Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load. • Steps to follow for Thevenin’s Theorem: (1) Find the Thevenin source voltage by removing the load resistor from the original circuit and calculating voltage across the open connection points where the load resistor used to be. (2) Find the Thevenin resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points. (3) Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the Thevenin resistance. The load resistor re-attaches between the two open points of the equivalent circuit. (4) Analyze voltage and current for the load resistor following the rules for series circuits.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.09%3A_Norton%E2%80%99s_Theorem.txt	princeton-nlp/TextbookChapters	What is Norton’s Theorem? Norton’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin’s Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots). Simplifying Linear Circuits Contrasting our original example circuit against the Norton equivalent: it looks something like this: after Norton conversion . . . Remember that a current source is a component whose job is to provide a constant amount of current, outputting as much or as little voltage necessary to maintain that constant current. Thevenin’s Theorem vs. Norton’s Theorem As with Thevenin’s Theorem, everything in the original circuit except the load resistance has been reduced to an equivalent circuit that is simpler to analyze. Also similar to Thevenin’s Theorem are the steps used in Norton’s Theorem to calculate the Norton source current (INorton) and Norton resistance (RNorton). As before, the first step is to identify the load resistance and remove it from the original circuit: Then, to find the Norton current (for the current source in the Norton equivalent circuit), place a direct wire (short) connection between the load points and determine the resultant current. Note that this step is exactly opposite the respective step in Thevenin’s Theorem, where we replaced the load resistor with a break (open circuit): With zero voltage dropped between the load resistor connection points, the current through R1 is strictly a function of B1‘s voltage and R1‘s resistance: 7 amps (I=E/R). Likewise, the current through R3 is now strictly a function of B2‘s voltage and R3‘s resistance: 7 amps (I=E/R). The total current through the short between the load connection points is the sum of these two currents: 7 amps + 7 amps = 14 amps. This figure of 14 amps becomes the Norton source current (INorton) in our equivalent circuit: Remember, the arrow notation for a current source points in the direction opposite that of electron flow. Again, apologies for the confusion. For better or for worse, this is standard electronic symbol notation. Blame Mr. Franklin again! To calculate the Norton resistance (RNorton), we do the exact same thing as we did for calculating Thevenin resistance (RThevenin): take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure total resistance from one load connection point to the other: Now our Norton equivalent circuit looks like this: If we re-connect our original load resistance of 2 Ω, we can analyze the Norton circuit as a simple parallel arrangement: As with the Thevenin equivalent circuit, the only useful information from this analysis is the voltage and current values for R2; the rest of the information is irrelevant to the original circuit. However, the same advantages seen with Thevenin’s Theorem apply to Norton’s as well: if we wish to analyze load resistor voltage and current over several different values of load resistance, we can use the Norton equivalent circuit again and again, applying nothing more complex than simple parallel circuit analysis to determine what’s happening with each trial load. Review • Norton’s Theorem is a way to reduce a network to an equivalent circuit composed of a single current source, parallel resistance, and parallel load. • Steps to follow for Norton’s Theorem: (1) Find the Norton source current by removing the load resistor from the original circuit and calculating current through a short (wire) jumping across the open connection points where the load resistor used to be. (2) Find the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points. (3) Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance. The load resistor re-attaches between the two open points of the equivalent circuit. (4) Analyze voltage and current for the load resistor following the rules for parallel circuits.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.10%3A_Thevenin-Norton_Equivalencies.txt	princeton-nlp/TextbookChapters	You may have noticed that the procedure for calculating Thevenin resistance is identical to the procedure for calculating Norton resistance: remove all power sources and determine resistance between the open load connection points. As such, Thevenin and Norton resistances for the same original network must be equal. Using the example circuits from the last two sections, we can see that the two resistances are indeed equal: Considering the fact that both Thevenin and Norton equivalent circuits are intended to behave the same as the original network in supplying voltage and current to the load resistor (as seen from the perspective of the load connection points), these two equivalent circuits, having been derived from the same original network should behave identically. This means that both Thevenin and Norton equivalent circuits should produce the same voltage across the load terminals with no load resistor attached. With the Thevenin equivalent, the open-circuited voltage would be equal to the Thevenin source voltage (no circuit current present to drop voltage across the series resistor), which is 11.2 volts in this case. With the Norton equivalent circuit, all 14 amps from the Norton current source would have to flow through the 0.8 Ω Norton resistance, producing the exact same voltage, 11.2 volts (E=IR). Thus, we can say that the Thevenin voltage is equal to the Norton current times the Norton resistance: So, if we wanted to convert a Norton equivalent circuit to a Thevenin equivalent circuit, we could use the same resistance and calculate the Thevenin voltage with Ohm’s Law. Conversely, both Thevenin and Norton equivalent circuits should generate the same amount of current through a short circuit across the load terminals. With the Norton equivalent, the short-circuit current would be exactly equal to the Norton source current, which is 14 amps in this case. With the Thevenin equivalent, all 11.2 volts would be applied across the 0.8 Ω Thevenin resistance, producing the exact same current through the short, 14 amps (I=E/R). Thus, we can say that the Norton current is equal to the Thevenin voltage divided by the Thevenin resistance: This equivalence between Thevenin and Norton circuits can be a useful tool in itself, as we shall see in the next section. Review • Thevenin and Norton resistances are equal. • Thevenin voltage is equal to Norton current times Norton resistance. • Norton current is equal to Thevenin voltage divided by Thevenin resistance. 10.11: Millman’s Theorem Revisited You may have wondered where we got that strange equation for the determination of “Millman Voltage” across parallel branches of a circuit where each branch contains a series resistance and voltage source: Parts of this equation seem familiar to equations we’ve seen before. For instance, the denominator of the large fraction looks conspicuously like the denominator of our parallel resistance equation. And, of course, the E/R terms in the numerator of the large fraction should give figures for current, Ohm’s Law being what it is (I=E/R). Now that we’ve covered Thevenin and Norton source equivalencies, we have the tools necessary to understand Millman’s equation. What Millman’s equation is actually doing is treating each branch (with its series voltage source and resistance) as a Thevenin equivalent circuit and then converting each one into equivalent Norton circuits. Thus, in the circuit above, battery B1 and resistor R1 are seen as a Thevenin source to be converted into a Norton source of 7 amps (28 volts / 4 Ω) in parallel with a 4 Ω resistor. The rightmost branch will be converted into a 7 amp current source (7 volts / 1 Ω) and 1 Ω resistor in parallel. The center branch, containing no voltage source at all, will be converted into a Norton source of 0 amps in parallel with a 2 Ω resistor: Since current sources directly add their respective currents in parallel, the total circuit current will be 7 + 0 + 7, or 14 amps. This addition of Norton source currents is what’s being represented in the numerator of the Millman equation: All the Norton resistances are in parallel with each other as well in the equivalent circuit, so they diminish to create a total resistance. This diminishing of source resistances is what’s being represented in the denominator of the Millman’s equation: In this case, the resistance total will be equal to 571.43 milliohms (571.43 mΩ). We can re-draw our equivalent circuit now as one with a single Norton current source and Norton resistance: Ohm’s Law can tell us the voltage across these two components now (E=IR): Let’s summarize what we know about the circuit thus far. We know that the total current in this circuit is given by the sum of all the branch voltages divided by their respective resistances. We also know that the total resistance is found by taking the reciprocal of all the branch resistance reciprocals. Furthermore, we should be well aware of the fact that total voltage across all the branches can be found by multiplying total current by total resistance (E=IR). All we need to do is put together the two equations we had earlier for total circuit current and total resistance, multiplying them to find total voltage: The Millman’s equation is nothing more than a Thevenin-to-Norton conversion matched together with the parallel resistance formula to find total voltage across all the branches of the circuit. So, hopefully some of the mystery is gone now!
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.12%3A_Maximum_Power_Transfer_Theorem.txt	princeton-nlp/TextbookChapters	This is essentially what is aimed for in radio transmitter design, where the antenna or transmission line “impedance” is matched to final power amplifier “impedance” for maximum radio frequency power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance. Maximum Power Transfer Example Taking our Thevenin equivalent example circuit, the Maximum Power Transfer Theorem tells us that the load resistance resulting in greatest power dissipation is equal in value to the Thevenin resistance (in this case, 0.8 Ω): With this value of load resistance, the dissipated power will be 39.2 watts: If we were to try a lower value for the load resistance (0.5 Ω instead of 0.8 Ω, for example), our power dissipated by the load resistance would decrease: Power dissipation increased for both the Thevenin resistance and the total circuit, but it decreased for the load resistor. Likewise, if we increase the load resistance (1.1 Ω instead of 0.8 Ω, for example), power dissipation will also be less than it was at 0.8 Ω exactly: If you were designing a circuit for maximum power dissipation at the load resistance, this theorem would be very useful. Having reduced a network down to a Thevenin voltage and resistance (or Norton current and resistance), you simply set the load resistance equal to that Thevenin or Norton equivalent (or vice versa) to ensure maximum power dissipation at the load. Practical applications of this might include radio transmitter final amplifier stage design (seeking to maximize power delivered to the antenna or transmission line), a grid-tied inverter loading a solar array, or electric vehicle design (seeking to maximize power delivered to drive motor). Maximum Power Doesn’t Mean Maximum Efficiency Maximum power transfer does not coincide with maximum efficiency. Application of The Maximum Power Transfer theorem to AC power distribution will not result in maximum or even high efficiency. The goal of high efficiency is more important for AC power distribution, which dictates a relatively low generator impedance compared to the load impedance. Similar to AC power distribution, high fidelity audio amplifiers are designed for a relatively low output impedance and a relatively high speaker load impedance. As a ratio, “output impedance” : “load impedance” is known as damping factor, typically in the range of 100 to 1000. Maximum power transfer does not coincide with the goal of lowest noise. For example, the low-level radio frequency amplifier between the antenna and a radio receiver is often designed for lowest possible noise. This often requires a mismatch of the amplifier input impedance to the antenna as compared with that dictated by the maximum power transfer theorem. Review • The Maximum Power Transfer Theorem states that the maximum amount of power will be dissipated by a load resistance if it is equal to the Thevenin or Norton resistance of the network supplying power. • The Maximum Power Transfer Theorem does not satisfy the goal of maximum efficiency.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/10%3A_DC_Network_Analysis/10.13%3A_%CE%94-Y_and_Y-%CE%94_Conversions.txt	princeton-nlp/TextbookChapters	In many circuit applications, we encounter components connected together in one of two ways to form a three-terminal network: the “Delta,” or Δ (also known as the “Pi,” or π) configuration, and the “Y” (also known as the “T”) configuration. It is possible to calculate the proper values of resistors necessary to form one kind of network (Δ or Y) that behaves identically to the other kind, as analyzed from the terminal connections alone. That is, if we had two separate resistor networks, one Δ and one Y, each with its resistors hidden from view, with nothing but the three terminals (A, B, and C) exposed for testing, the resistors could be sized for the two networks so that there would be no way to electrically determine one network apart from the other. In other words, equivalent Δ and Y networks behave identically. There are several equations used to convert one network to the other: Δ and Y networks are seen frequently in 3-phase AC power systems (a topic covered in volume II of this book series), but even then they’re usually balanced networks (all resistors equal in value) and conversion from one to the other need not involve such complex calculations. When would the average technician ever need to use these equations? A prime application for Δ-Y conversion is in the solution of unbalanced bridge circuits, such as the one below: Solution of this circuit with Branch Current or Mesh Current analysis is fairly involved, and neither the Millman nor Superposition Theorems are of any help, since there’s only one source of power. We could use Thevenin’s or Norton’s Theorem, treating R3 as our load, but what fun would that be? If we were to treat resistors R1, R2, and R3 as being connected in a Δ configuration (Rab, Rac, and Rbc, respectively) and generate an equivalent Y network to replace them, we could turn this bridge circuit into a (simpler) series/parallel combination circuit: After the Δ-Y conversion . . . If we perform our calculations correctly, the voltages between points A, B, and C will be the same in the converted circuit as in the original circuit, and we can transfer those values back to the original bridge configuration. Resistors R4 and R5, of course, remain the same at 18 Ω and 12 Ω, respectively. Analyzing the circuit now as a series/parallel combination, we arrive at the following figures: We must use the voltage drops figures from the table above to determine the voltages between points A, B, and C, seeing how the add up (or subtract, as is the case with voltage between points B and C): Now that we know these voltages, we can transfer them to the same points A, B, and C in the original bridge circuit: Voltage drops across R4 and R5, of course, are exactly the same as they were in the converted circuit. At this point, we could take these voltages and determine resistor currents through the repeated use of Ohm’s Law (I=E/R): A quick simulation with SPICE will serve to verify our work: The voltage figures, as read from left to right, represent voltage drops across the five respective resistors, R1 through R5. I could have shown currents as well, but since that would have required insertion of “dummy” voltage sources in the SPICE netlist, and since we’re primarily interested in validating the Δ-Y conversion equations and not Ohm’s Law, this will suffice. Review • “Delta” (Δ) networks are also known as “Pi” (π) networks. • “Y” networks are also known as “T” networks. • Δ and Y networks can be converted to their equivalent counterparts with the proper resistance equations. By “equivalent,” I mean that the two networks will be electrically identical as measured from the three terminals (A, B, and C). • A bridge circuit can be simplified to a series/parallel circuit by converting half of it from a Δ to a Y network. After voltage drops between the original three connection points (A, B, and C) have been solved for, those voltages can be transferred back to the original bridge circuit, across those same equivalent points.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/11%3A_Batteries_And_Power_Systems/11.01%3A_Electron_Activity_in_Chemical_Reactions.txt	princeton-nlp/TextbookChapters	• 11.1: Electron Activity in Chemical Reactions So far in our discussions on electricity and electric circuits, we have not discussed in any detail how batteries function. Rather, we have simply assumed that they produce constant voltage through some sort of mysterious process. Here, we will explore that process to some degree and cover some of the practical considerations involved with real batteries and their use in power systems. • 11.2: Battery Construction The word battery simply means a group of similar components. In military vocabulary, a “battery” refers to a cluster of guns. In electricity, a “battery” is a set of voltaic cells designed to provide greater voltage and/or current than is possible with one cell alone. • 11.3: Battery Ratings Because batteries create electron flow in a circuit by exchanging electrons in ionic chemical reactions, and there is a limited number of molecules in any charged battery available to react, there must be a limited amount of total electrons that any battery can motivate through a circuit before its energy reserves are exhausted. Battery capacity could be measured in terms of total number of electrons, but this would be a huge number. We could use the unit of the coulomb (equal to 6.25 x 1018 ele • 11.4: Special-purpose Batteries Back in the early days of electrical measurement technology, a special type of battery known as a mercury standard cell was popularly used as a voltage calibration standard. The output of a mercury cell was 1.0183 to 1.0194 volts DC (depending on the specific design of cell), and was extremely stable over time. Advertised drift was around 0.004 percent of rated voltage per year. Mercury standard cells were sometimes known as Weston cells or cadmium cells. • 11.5: Practical Considerations - Batteries When connecting batteries together to form larger “banks” (a battery of batteries?), the constituent batteries must be matched to each other so as to not cause problems. 11: Batteries And Power Systems In the first chapter of this book, the concept of an atom was discussed, as being the basic building-block of all material objects. Atoms, in turn, are composed of even smaller pieces of matter called particles. Electrons, protons, and neutrons are the basic types of particles found in atoms. Each of these particle types plays a distinct role in the behavior of an atom. While electrical activity involves the motion of electrons, the chemical identity of an atom (which largely determines how conductive the material will be) is determined by the number of protons in the nucleus (center). The protons in an atom’s nucleus are extremely difficult to dislodge, and so the chemical identity of any atom is very stable. One of the goals of the ancient alchemists (to turn lead into gold) was foiled by this sub-atomic stability. All efforts to alter this property of an atom by means of heat, light, or friction were met with failure. The electrons of an atom, however, are much more easily dislodged. As we have already seen, friction is one way in which electrons can be transferred from one atom to another (glass and silk, wax and wool), as well as heat (generating voltage by heating a junction of dissimilar metals, as in the case of thermocouples). Electrons can do much more than just move around and between atoms: they can also serve to link different atoms together. This linking of atoms by electrons is called a chemical bond. A crude (and simplified) representation of such a bond between two atoms might look like this: There are several types of chemical bonds, the one shown above being representative of a covalent bond, where electrons are shared between atoms. Because chemical bonds are based on links formed by electrons, these bonds are only as strong as the immobility of the electrons forming them. That is to say, chemical bonds can be created or broken by the same forces that force electrons to move: heat, light, friction, etc. When atoms are joined by chemical bonds, they form materials with unique properties known as molecules. The dual-atom picture shown above is an example of a simple molecule formed by two atoms of the same type. Most molecules are unions of different types of atoms. Even molecules formed by atoms of the same type can have radically different physical properties. Take the element carbon, for instance: in one form, graphite, carbon atoms link together to form flat “plates” which slide against one another very easily, giving graphite its natural lubricating properties. In another form, diamond, the same carbon atoms link together in a different configuration, this time in the shapes of interlocking pyramids, forming a material of exceeding hardness. In yet another form, Fullerene, dozens of carbon atoms form each molecule, which looks something like a soccer ball. Fullerene molecules are very fragile and lightweight. The airy soot formed by excessively rich combustion of acetylene gas (as in the initial ignition of an oxy-acetylene welding/cutting torch) contains many Fullerene molecules. When alchemists succeeded in changing the properties of a substance by heat, light, friction, or mixture with other substances, they were really observing changes in the types of molecules formed by atoms breaking and forming bonds with other atoms. Chemistry is the modern counterpart to alchemy and concerns itself primarily with the properties of these chemical bonds and the reactions associated with them. A type of chemical bond of particular interest to our study of batteries is the so-called ionic bond, and it differs from the covalent bond in that one atom of the molecule possesses an excess of electrons while another atom lacks electrons, the bonds between them being a result of the electrostatic attraction between the two unlike charges. When ionic bonds are formed from neutral atoms, there is a transfer of electrons between the positively and negatively charged atoms. An atom that gains an excess of electrons is said to be reduced; an atom with a deficiency of electrons is said to be oxidized. A mnemonic to help remember the definitions is OIL RIG (oxidized is less; reduced is gained). It is important to note that molecules will often contain both ionic and covalent bonds. Sodium hydroxide (lye, NaOH) has an ionic bond between the sodium atom (positive) and the hydroxyl ion (negative). The hydroxyl ion has a covalent bond (shown as a bar) between the hydrogen and oxygen atoms: Na+ O—H- Sodium only loses one electron, so its charge is +1 in the above example. If an atom loses more than one electron, the resulting charge can be indicated as +2, +3, +4, etc. or by a Roman numeral in parentheses showing the oxidation state, such as (I), (II), (IV), etc. Some atoms can have multiple oxidation states, and it is sometimes important to include the oxidation state in the molecular formula to avoid ambiguity. The formation of ions and ionic bonds from neutral atoms or molecules (or vice versa) involves the transfer of electrons. That transfer of electrons can be harnessed to generate an electric current.A device constructed to do just this is called a voltaic cell, or cell for short, usually consisting of two metal electrodes immersed in a chemical mixture (called an electrolyte) designed to facilitate such an electrochemical (oxidation/reduction) reaction: In the common “lead-acid” cell (the kind commonly used in automobiles), the negative electrode is made of lead (Pb) and the positive is made of lead (IV) dioxide (PbO2), both metallic substances. It is important to note that lead dioxide is metallic and is an electrical conductor, unlike other metal oxides that are usually insulators. (note: Table below) The electrolyte solution is a dilute sulfuric acid (H2SO4 + H2O). If the electrodes of the cell are connected to an external circuit, such that electrons have a place to flow from one to the other, lead(IV) atoms in the positive electrode (PbO2) will gain two electrons each to produce Pb(II)O. The oxygen atoms which are “left over” combine with positively charged hydrogen ions (H)+to form water (H2O). This flow of electrons into into the lead dioxide (PbO2) electrode, gives it a positive electrical charge. Consequently, lead atoms in the negative electrode give up two electrons each to produce lead Pb(II), which combines with sulfate ions (SO4-2) produced from the disassociation of the hydrogen ions (H+) from the sulfuric acid (H2SO4) to form lead sulfate (PbSO4). The flow of electrons out of the lead electrode gives it a negative electrical charge. These reactions are shown diagrammatically below: Note on lead oxide nomenclature: This process of the cell providing electrical energy to supply a load is called discharging since it is depleting its internal chemical reserves. Theoretically, after all of the sulfuric acid has been exhausted, the result will be two electrodes of lead sulfate (PbSO4) and an electrolyte solution of pure water (H2O), leaving no more capacity for additional ionic bonding. In this state, the cell is said to be fully discharged. In a lead-acid cell, the state of charge can be determined by an analysis of acid strength. This is easily accomplished with a device called a hydrometer, which measures the specific gravity (density) of the electrolyte. Sulfuric acid is denser than water, so the greater the charge of a cell, the greater the acid concentration, and thus a denser electrolyte solution. There is no single chemical reaction representative of all voltaic cells, so any detailed discussion of chemistry is bound to have limited application. The important thing to understand is that electrons are motivated to and/or from the cell’s electrodes via ionic reactions between the electrode molecules and the electrolyte molecules. The reaction is enabled when there is an external path for electric current and ceases when that path is broken. Being that the motivation for electrons to move through a cell is chemical in nature, the amount of voltage (electromotive force) generated by any cell will be specific to the particular chemical reaction for that cell type. For instance, the lead-acid cell just described has a nominal voltage of 2.04 volts per cell, based on a fully “charged” cell (acid concentration strong) in good physical condition. There are other types of cells with different specific voltage outputs. The Edison cell, for example, with a positive electrode made of nickel oxide, a negative electrode made of iron, and an electrolyte solution of potassium hydroxide (a caustic, not acid, substance) generates a nominal voltage of only 1.2 volts, due to the specific differences in chemical reaction with those electrode and electrolyte substances. The chemical reactions of some types of cells can be reversed by forcing electric current backward through the cell (in the negative electrode and out the positive electrode). This process is called charging. Any such (rechargeable) cell is called a secondary cell. A cell whose chemistry cannot be reversed by a reverse current is called a primary cell. When a lead-acid cell is charged by an external current source, the chemical reactions experienced during discharge are reversed: Review • Atoms bound together by electrons are called molecules. • Ionic bonds are molecular unions formed when an electron-deficient atom (a positive ion) joins with an electron-excessive atom (a negative ion). • Electrochemical reactions involve the transfer of electrons between atoms. This transfer can be harnessed to form an electric current. • A cell is a device constructed to harness such chemical reactions to generate electric current. • A cell is said to be discharged when its internal chemical reserves have been depleted through use. • A secondary cell’s chemistry can be reversed (recharged) by forcing current backward through it. • A primary cell cannot be practically recharged. • Lead-acid cell charge can be assessed with an instrument called a hydrometer, which measures the density of the electrolyte liquid. The denser the electrolyte, the stronger the acid concentration, and the greater charge state of the cell.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/11%3A_Batteries_And_Power_Systems/11.02%3A_Battery_Construction.txt	princeton-nlp/TextbookChapters	The symbol for a cell is very simple, consisting of one long line and one short line, parallel to each other, with connecting wires: The symbol for a battery is nothing more than a couple of cell symbols stacked in series: As was stated before, the voltage produced by any particular kind of cell is determined strictly by the chemistry of that cell type. The size of the cell is irrelevant to its voltage. To obtain greater voltage than the output of a single cell, multiple cells must be connected in series. The total voltage of a battery is the sum of all cell voltages. A typical automotive lead-acid battery has six cells, for a nominal voltage output of 6 x 2.0 or 12.0 volts: The cells in an automotive battery are contained within the same hard rubber housing, connected together with thick, lead bars instead of wires. The electrodes and electrolyte solutions for each cell are contained in separate, partitioned sections of the battery case. In large batteries, the electrodes commonly take the shape of thin metal grids or plates, and are often referred to as plates instead of electrodes. For the sake of convenience, battery symbols are usually limited to four lines, alternating long/short, although the real battery it represents may have many more cells than that. On occasion, however, you might come across a symbol for a battery with unusually high voltage, intentionally drawn with extra lines. The lines, of course, are representative of the individual cell plates: If the physical size of a cell has no impact on its voltage, then what does it affect? The answer is resistance, which in turn affects the maximum amount of current that a cell can provide. Every voltaic cell contains some amount of internal resistance due to the electrodes and the electrolyte. The larger a cell is constructed, the greater the electrode contact area with the electrolyte, and thus the less internal resistance it will have. Although we generally consider a cell or battery in a circuit to be a perfect source of voltage (absolutely constant), the current through it dictated solely by the external resistance of the circuit to which it is attached, this is not entirely true in real life. Since every cell or battery contains some internal resistance, that resistance must affect the current in any given circuit: The real battery shown above within the dotted lines has an internal resistance of 0.2 Ω, which affects its ability to supply current to the load resistance of 1 Ω. The ideal battery on the left has no internal resistance, and so our Ohm’s Law calculations for current (I=E/R) give us a perfect value of 10 amps for current with the 1 ohm load and 10 volt supply. The real battery, with its built-in resistance further impeding the flow of electrons, can only supply 8.333 amps to the same resistance load. The ideal battery, in a short circuit with 0 Ω resistance, would be able to supply an infinite amount of current. The real battery, on the other hand, can only supply 50 amps (10 volts / 0.2 Ω) to a short circuit of 0 Ω resistance, due to its internal resistance. The chemical reaction inside the cell may still be providing exactly 10 volts, but voltage is dropped across that internal resistance as electrons flow through the battery, which reduces the amount of voltage available at the battery terminals to the load. Since we live in an imperfect world, with imperfect batteries, we need to understand the implications of factors such as internal resistance. Typically, batteries are placed in applications where their internal resistance is negligible compared to that of the circuit load (where their short-circuit current far exceeds their usual load current), and so the performance is very close to that of an ideal voltage source. If we need to construct a battery with lower resistance than what one cell can provide (for greater current capacity), we will have to connect the cells together in parallel: Essentially, what we have done here is determine the Thevenin equivalent of the five cells in parallel (an equivalent network of one voltage source and one series resistance). The equivalent network has the same source voltage but a fraction of the resistance of any individual cell in the original network. The overall effect of connecting cells in parallel is to decrease the equivalent internal resistance, just as resistors in parallel diminish in total resistance. The equivalent internal resistance of this battery of 5 cells is 1/5 that of each individual cell. The overall voltage stays the same: 2.0 volts. If this battery of cells were powering a circuit, the current through each cell would be 1/5 of the total circuit current, due to the equal split of current through equal-resistance parallel branches. Review • A battery is a cluster of cells connected together for greater voltage and/or current capacity. • Cells connected together in series (polarities aiding) results in greater total voltage. • Physical cell size impacts cell resistance, which in turn impacts the ability for the cell to supply current to a circuit. Generally, the larger the cell, the less its internal resistance. • Cells connected together in parallel results in less total resistance, and potentially greater total current.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/11%3A_Batteries_And_Power_Systems/11.03%3A_Battery_Ratings.txt	princeton-nlp/TextbookChapters	A battery with a capacity of 1 amp-hour should be able to continuously supply a current of 1 amp to a load for exactly 1 hour, or 2 amps for 1/2 hour, or 1/3 amp for 3 hours, etc., before becoming completely discharged. In an ideal battery, this relationship between continuous current and discharge time is stable and absolute, but real batteries don’t behave exactly as this simple linear formula would indicate. Therefore, when amp-hour capacity is given for a battery, it is specified at either a given current, given time, or assumed to be rated for a time period of 8 hours (if no limiting factor is given). For example, an average automotive battery might have a capacity of about 70 amp-hours, specified at a current of 3.5 amps. This means that the amount of time this battery could continuously supply a current of 3.5 amps to a load would be 20 hours (70 amp-hours / 3.5 amps). But let’s suppose that a lower-resistance load were connected to that battery, drawing 70 amps continuously. Our amp-hour equation tells us that the battery should hold out for exactly 1 hour (70 amp-hours / 70 amps), but this might not be true in real life. With higher currents, the battery will dissipate more heat across its internal resistance, which has the effect of altering the chemical reactions taking place within. Chances are, the battery would fully discharge some time before the calculated time of 1 hour under this greater load. Conversely, if a very light load (1 mA) were to be connected to the battery, our equation would tell us that the battery should provide power for 70,000 hours, or just under 8 years (70 amp-hours / 1 milliamp), but the odds are that much of the chemical energy in a real battery would have been drained due to other factors (evaporation of electrolyte, deterioration of electrodes, leakage current within battery) long before 8 years had elapsed. Therefore, we must take the amp-hour relationship as being an ideal approximation of battery life, the amp-hour rating trusted only near the specified current or timespan given by the manufacturer. Some manufacturers will provide amp-hour derating factors specifying reductions in total capacity at different levels of current and/or temperature. For secondary cells, the amp-hour rating provides a rule for necessary charging time at any given level of charge current. For example, the 70 amp-hour automotive battery in the previous example should take 10 hours to charge from a fully-discharged state at a constant charging current of 7 amps (70 amp-hours / 7 amps). Approximate amp-hour capacities of some common batteries are given here: • Typical automotive battery: 70 amp-hours @ 3.5 A (secondary cell) • D-size carbon-zinc battery: 4.5 amp-hours @ 100 mA (primary cell) • 9 volt carbon-zinc battery: 400 milliamp-hours @ 8 mA (primary cell) As a battery discharges, not only does it diminish its internal store of energy, but its internal resistance also increases (as the electrolyte becomes less and less conductive), and its open-circuit cell voltage decreases (as the chemicals become more and more dilute). The most deceptive change that a discharging battery exhibits is increased resistance. The best check for a battery’s condition is a voltage measurement under load, while the battery is supplying a substantial current through a circuit. Otherwise, a simple voltmeter check across the terminals may falsely indicate a healthy battery (adequate voltage) even though the internal resistance has increased considerably. What constitutes a “substantial current” is determined by the battery’s design parameters. A voltmeter check revealing too low of a voltage, of course, would positively indicate a discharged battery: Fully charged battery: Now, if the battery discharges a bit . . . and discharges a bit further . . . and a bit further until its dead. Notice how much better the battery’s true condition is revealed when its voltage is checked under load as opposed to without a load. Does this mean that its pointless to check a battery with just a voltmeter (no load)? Well, no. If a simple voltmeter check reveals only 7.5 volts for a 13.2 volt battery, then you know without a doubt that its dead. However, if the voltmeter were to indicate 12.5 volts, it may be near full charge or somewhat depleted—you couldn’t tell without a load check. Bear in mind also that the resistance used to place a battery under load must be rated for the amount of power expected to be dissipated. For checking large batteries such as an automobile (12 volt nominal) lead-acid battery, this may mean a resistor with a power rating of several hundred watts. Review • The amp-hour is a unit of battery energy capacity, equal to the amount of continuous current multiplied by the discharge time, that a battery can supply before exhausting its internal store of chemical energy. • An amp-hour battery rating is only an approximation of the battery’s charge capacity, and should be trusted only at the current level or time specified by the manufacturer. Such a rating cannot be extrapolated for very high currents or very long times with any accuracy. • Discharged batteries lose voltage and increase in resistance. The best check for a dead battery is a voltage test under load.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/11%3A_Batteries_And_Power_Systems/11.04%3A_Special-purpose_Batteries.txt	princeton-nlp/TextbookChapters	Back in the early days of electrical measurement technology, a special type of battery known as a mercury standard cell was popularly used as a voltage calibration standard. The output of a mercury cell was 1.0183 to 1.0194 volts DC (depending on the specific design of cell), and was extremely stable over time. Advertised drift was around 0.004 percent of rated voltage per year. Mercury standard cells were sometimes known as Weston cells or cadmium cells. Unfortunately, mercury cells were rather intolerant of any current drain and could not even be measured with an analog voltmeter without compromising accuracy. Manufacturers typically called for no more than 0.1 mA of current through the cell, and even that figure was considered a momentary, or surge maximum! Consequently, standard cells could only be measured with a potentiometric (null-balance) device where current drain is almost zero. Short-circuiting a mercury cell was prohibited, and once short-circuited, the cell could never be relied upon again as a standard device. Mercury standard cells were also susceptible to slight changes in voltage if physically or thermally disturbed. Two different types of mercury standard cells were developed for different calibration purposes: saturated and unsaturated. Saturated standard cells provided the greatest voltage stability over time, at the expense of thermal instability. In other words, their voltage drifted very little with the passage of time (just a few microvolts over the span of a decade!), but tended to vary with changes in temperature (tens of microvolts per degree Celsius). These cells functioned best in temperature-controlled laboratory environments where long-term stability is paramount. Unsaturated cells provided thermal stability at the expense of stability over time, the voltage remaining virtually constant with changes in temperature but decreasing steadily by about 100 µV every year. These cells functioned best as “field” calibration devices where ambient temperature is not precisely controlled. Nominal voltage for a saturated cell was 1.0186 volts, and 1.019 volts for an unsaturated cell. Modern semiconductor voltage (zener diode regulator) references have superseded standard cell batteries as laboratory and field voltage standards. A fascinating device closely related to primary-cell batteries is the fuel cell, so-called because it harnesses the chemical reaction of combustion to generate an electric current. The process of chemical oxidation (oxygen ionically bonding with other elements) is capable of producing an electron flow between two electrodes just as well as any combination of metals and electrolytes. A fuel cell can be thought of as a battery with an externally supplied chemical energy source. To date, the most successful fuel cells constructed are those which run on hydrogen and oxygen, although much research has been done on cells using hydrocarbon fuels. While “burning” hydrogen, a fuel cell’s only waste byproducts are water and a small amount of heat. When operating on carbon-containing fuels, carbon dioxide is also released as a byproduct. Because the operating temperature of modern fuel cells is far below that of normal combustion, no oxides of nitrogen (NOx) are formed, making it far less polluting, all other factors being equal. The efficiency of energy conversion in a fuel cell from chemical to electrical far exceeds the theoretical Carnot efficiency limit of any internal-combustion engine, which is an exciting prospect for power generation and hybrid electric automobiles. Another type of “battery” is the solar cell, a by-product of the semiconductor revolution in electronics. The photoelectric effect, whereby electrons are dislodged from atoms under the influence of light, has been known in physics for many decades, but it has only been with recent advances in semiconductor technology that a device existed capable of harnessing this effect to any practical degree. Conversion efficiencies for silicon solar cells are still quite low, but their benefits as power sources are legion: no moving parts, no noise, no waste products or pollution (aside from the manufacture of solar cells, which is still a fairly “dirty” industry), and indefinite life. Specific cost of solar cell technology (dollars per kilowatt) is still very high, with little prospect of significant decrease barring some kind of revolutionary advance in technology. Unlike electronic components made from semiconductor material, which can be made smaller and smaller with less scrap as a result of better quality control, a single solar cell still takes the same amount of ultra-pure silicon to make as it did thirty years ago. Superior quality control fails to yield the same production gain seen in the manufacture of chips and transistors (where isolated specks of impurity can ruin many microscopic circuits on one wafer of silicon). The same number of impure inclusions does little to impact the overall efficiency of a 3-inch solar cell. Yet another type of special-purpose “battery” is the chemical detection cell. Simply put, these cells chemically react with specific substances in the air to create a voltage directly proportional to the concentration of that substance. A common application for a chemical detection cell is in the detection and measurement of oxygen concentration. Many portable oxygen analyzers have been designed around these small cells. Cell chemistry must be designed to match the specific substance(s) to be detected, and the cells do tend to “wear out,” as their electrode materials deplete or become contaminated with use. Review • mercury standard cells are special types of batteries which were once used as voltage calibration standards before the advent of precision semiconductor reference devices. • A fuel cell is a kind of battery that uses a combustible fuel and oxidizer as reactants to generate electricity. They are promising sources of electrical power in the future, “burning” fuels with very low emissions. • A solar cell uses ambient light energy to motivate electrons from one electrode to the other, producing voltage (and current, providing an external circuit). • A chemical detection cell is a special type of voltaic cell which produces voltage proportional to the concentration of an applied substance (usually a specific gas in ambient air).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/11%3A_Batteries_And_Power_Systems/11.05%3A_Practical_Considerations_-_Batteries.txt	princeton-nlp/TextbookChapters	When connecting batteries together to form larger “banks” (a battery of batteries?), the constituent batteries must be matched to each other so as to not cause problems. First we will consider connecting batteries in series for greater voltage: We know that the current is equal at all points in a series circuit, so whatever amount of current there is in any one of the series-connected batteries must be the same for all the others as well. For this reason, each battery must have the same amp-hour rating, or else some of the batteries will become depleted sooner than others, compromising the capacity of the whole bank. Please note that the total amp-hour capacity of this series battery bank is not affected by the number of batteries. Next, we will consider connecting batteries in parallel for greater current capacity (lower internal resistance), or greater amp-hour capacity: We know that the voltage is equal across all branches of a parallel circuit, so we must be sure that these batteries are of equal voltage. If not, we will have relatively large currents circulating from one battery through another, the higher-voltage batteries overpowering the lower-voltage batteries. This is not good. On this same theme, we must be sure that any overcurrent protection (circuit breakers or fuses) are installed in such a way as to be effective. For our series battery bank, one fuse will suffice to protect the wiring from excessive current, since any break in a series circuit stops current through all parts of the circuit: With a parallel battery bank, one fuse is adequate for protecting the wiring against load overcurrent (between the parallel-connected batteries and the load), but we have other concerns to protect against as well. Batteries have been known to internally short-circuit, due to electrode separator failure, causing a problem not unlike that where batteries of unequal voltage are connected in parallel: the good batteries will overpower the failed (lower voltage) battery, causing relatively large currents within the batteries’ connecting wires. To guard against this eventuality, we should protect each and every battery against overcurrent with individual battery fuses, in addition to the load fuse: When dealing with secondary-cell batteries, particular attention must be paid to the method and timing of charging. Different types and construction of batteries have different charging needs, and the manufacturer’s recommendations are probably the best guide to follow when designing or maintaining a system. Two distinct concerns of battery charging are cycling and overcharging. Cycling refers to the process of charging a battery to a “full” condition and then discharging it to a lower state. All batteries have a finite (limited) cycle life, and the allowable “depth” of cycle (how far it should be discharged at any time) varies from design to design. Overcharging is the condition where current continues to be forced backwards through a secondary cell beyond the point where the cell has reached full charge. With lead-acid cells in particular, overcharging leads to electrolysis of the water (“boiling” the water out of the battery) and shortened life. Any battery containing water in the electrolyte is subject to the production of hydrogen gas due to electrolysis. This is especially true for overcharged lead-acid cells, but not exclusive to that type. Hydrogen is an extremely flammable gas (especially in the presence of free oxygen created by the same electrolysis process), odorless and colorless. Such batteries pose an explosion threat even under normal operating conditions, and must be treated with respect. The author has been a firsthand witness to a lead-acid battery explosion, where a spark created by the removal of a battery charger (small DC power supply) from an automotive battery ignited hydrogen gas within the battery case, blowing the top off the battery and splashing sulfuric acid everywhere. This occurred in a high school automotive shop, no less. If it were not for all the students nearby wearing safety glasses and buttoned-collar overalls, significant injury could have occurred. When connecting and disconnecting charging equipment to a battery, always make the last connection (or first disconnection) at a location away from the battery itself (such as at a point on one of the battery cables, at least a foot away from the battery), so that any resultant spark has little or no chance of igniting hydrogen gas. In large, permanently installed battery banks, batteries are equipped with vent caps above each cell, and hydrogen gas is vented outside of the battery room through hoods immediately over the batteries. Hydrogen gas is very light and rises quickly. The greatest danger is when it is allowed to accumulate in an area, awaiting ignition. More modern lead-acid battery designs are sealed, fabricated to re-combine the electrolyzed hydrogen and oxygen back into water, inside the battery case itself. Adequate ventilation might still be a good idea, just in case a battery were to develop a leak. Review • Connecting batteries in series increases voltage, but does not increase overall amp-hour capacity. • All batteries in a series bank must have the same amp-hour rating. • Connecting batteries in parallel increases total current capacity by decreasing total resistance, and it also increases overall amp-hour capacity. • All batteries in a parallel bank must have the same voltage rating. • Batteries can be damaged by excessive cycling and overcharging. • Water-based electrolyte batteries are capable of generating explosive hydrogen gas, which must not be allowed to accumulate in an area.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.01%3A_12.1_Introduction_to_Conductance_and_Conductors.txt	princeton-nlp/TextbookChapters	• 12.1: 12.1 Introduction to Conductance and Conductors By now you should be well aware of the correlation between electrical conductivity and certain types of materials. Those materials allowing for easy passage of free electrons are called conductors, while those materials impeding the passage of free electrons are called insulators. • 12.2: Conductor Size It should be common-sense knowledge that liquids flow through large-diameter pipes easier than they do through small-diameter pipes (if you would like a practical illustration, try drinking a liquid through straws of different diameters). The same general principle holds for the flow of electrons through conductors: the broader the cross-sectional area (thickness) of the conductor, the more room for electrons to flow, and consequently, the easier it is for flow to occur (less resistance). • 12.3: Conductor Ampacity The smaller the wire, the greater the resistance for any given length, all other factors being equal. A wire with greater resistance will dissipate a greater amount of heat energy for any given amount of current, the power being equal to P=I2R. Dissipated power in a resistance manifests itself in the form of heat, and excessive heat can be damaging to a wire (not to mention objects near the wire!), especially considering the fact that most wires are insulated with a plastic or rubber coating, w • 12.4: Fuses Normally, the ampacity rating of a conductor is a circuit design limit never to be intentionally exceeded, but there is an application where ampacity exceedance is expected: in the case of fuses. • 12.5: Specific Resistance • 12.6: Temperature Coefficient of Resistance You might have noticed on the table for specific resistances that all figures were specified at a temperature of 20o Celsius. If you suspected that this meant specific resistance of a material may change with temperature, you were right! • 12.7: Superconductivity Conductors lose all of their electrical resistance when cooled to super-low temperatures (near absolute zero, about -273o Celsius). It must be understood that superconductivity is not merely an extrapolation of most conductors’ tendency to gradually lose resistance with decreasing temperature; rather, it is a sudden, quantum leap in resistivity from finite to nothing. A superconducting material has absolutely zero electrical resistance, not just some small amount. Superconductivity was first di • 12.8: Insulator Breakdown Voltage The atoms in insulating materials have very tightly-bound electrons, resisting free electron flow very well. However, insulators cannot resist indefinite amounts of voltage. With enough voltage applied, any insulating material will eventually succumb to the electrical “pressure” and electron flow will occur. However, unlike the situation with conductors where current is in a linear proportion to applied voltage (given a fixed resistance), current through an insulator is quite nonlinear: for volt 12: Physics of Conductors and Insulators Unfortunately, the scientific theories explaining why certain materials conduct and others don’t are quite complex, rooted in quantum mechanical explanations in how electrons are arranged around the nuclei of atoms. Contrary to the well-known “planetary” model of electrons whirling around an atom’s nucleus as well-defined chunks of matter in circular or elliptical orbits, electrons in “orbit” don’t really act like pieces of matter at all. Rather, they exhibit the characteristics of both particle and wave, their behavior constrained by placement within distinct zones around the nucleus referred to as “shells” and “subshells.” Electrons can occupy these zones only in a limited range of energies depending on the particular zone and how occupied that zone is with other electrons. If electrons really did act like tiny planets held in orbit around the nucleus by electrostatic attraction, their actions described by the same laws describing the motions of real planets, there could be no real distinction between conductors and insulators, and chemical bonds between atoms would not exist in the way they do now. It is the discrete, “quantitized” nature of electron energy and placement described by quantum physics that gives these phenomena their regularity. When an electron is free to assume higher energy states around an atom’s nucleus (due to its placement in a particular “shell”), it may be free to break away from the atom and comprise part of an electric current through the substance. If the quantum limitations imposed on an electron deny it this freedom, however, the electron is considered to be “bound” and cannot break away (at least not easily) to constitute a current. The former scenario is typical of conducting materials, while the latter is typical of insulating materials. Some textbooks will tell you that an element’s conductivity or nonconductivity is exclusively determined by the number of electrons residing in the atoms’ outer “shell” (called the valence shell), but this is an oversimplification, as any examination of conductivity versus valence electrons in a table of elements will confirm. The true complexity of the situation is further revealed when the conductivity of molecules (collections of atoms bound to one another by electron activity) is considered. A good example of this is the element carbon, which comprises materials of vastly differing conductivity: graphite and diamond. Graphite is a fair conductor of electricity, while diamond is practically an insulator (stranger yet, it is technically classified as a semiconductor, which in its pure form acts as an insulator, but can conduct under high temperatures and/or the influence of impurities). Both graphite and diamond are composed of the exact same types of atoms: carbon, with 6 protons, 6 neutrons and 6 electrons each. The fundamental difference between graphite and diamond being that graphite molecules are flat groupings of carbon atoms while diamond molecules are tetrahedral (pyramid-shaped) groupings of carbon atoms. If atoms of carbon are joined to other types of atoms to form compounds, electrical conductivity becomes altered once again. Silicon carbide, a compound of the elements silicon and carbon, exhibits nonlinear behavior: its electrical resistance decreases with increases in applied voltage! Hydrocarbon compounds (such as the molecules found in oils) tend to be very good insulators. As you can see, a simple count of valence electrons in an atom is a poor indicator of a substance’s electrical conductivity. All metallic elements are good conductors of electricity, due to the way the atoms bond with each other. The electrons of the atoms comprising a mass of metal are so uninhibited in their allowable energy states that they float freely between the different nuclei in the substance, readily motivated by any electric field. The electrons are so mobile, in fact, that they are sometimes described by scientists as an electron gas, or even an electron sea in which the atomic nuclei rest. This electron mobility accounts for some of the other common properties of metals: good heat conductivity, malleability and ductility (easily formed into different shapes), and a lustrous finish when pure. Thankfully, the physics behind all this is mostly irrelevant to our purposes here. Suffice it to say that some materials are good conductors, some are poor conductors, and some are in between. For now it is good enough to simply understand that these distinctions are determined by the configuration of the electrons around the constituent atoms of the material. An important step in getting electricity to do our bidding is to be able to construct paths for electrons to flow with controlled amounts of resistance. It is also vitally important that we be able to prevent electrons from flowing where we don’t want them to, by using insulating materials. However, not all conductors are the same, and neither are all insulators. We need to understand some of the characteristics of common conductors and insulators, and be able to apply these characteristics to specific applications. Almost all conductors possess a certain, measurable resistance (special types of materials called superconductors possess absolutely no electrical resistance, but these are not ordinary materials, and they must be held in special conditions in order to be super conductive). Typically, we assume the resistance of the conductors in a circuit to be zero, and we expect that current passes through them without producing any appreciable voltage drop. In reality, however, there will almost always be a voltage drop along the (normal) conductive pathways of an electric circuit, whether we want a voltage drop to be there or not: In order to calculate what these voltage drops will be in any particular circuit, we must be able to ascertain the resistance of ordinary wire, knowing the wire size and diameter. Some of the following sections of this chapter will address the details of doing this. Review • Electrical conductivity of a material is determined by the configuration of electrons in that materials atoms and molecules (groups of bonded atoms). • All normal conductors possess resistance to some degree. • Electrons flowing through a conductor with (any) resistance will produce some amount of voltage drop across the length of that conductor.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.02%3A_Conductor_Size.txt	princeton-nlp/TextbookChapters	Two Basic Varieties of Electrical Wire: Solid and Stranded Electrical wire is usually round in cross-section (although there are some unique exceptions to this rule), and comes in two basic varieties: solid and stranded. Solid copper wire is just as it sounds: a single, solid strand of copper the whole length of the wire. Stranded wire is composed of smaller strands of solid copper wire twisted together to form a single, larger conductor. The greatest benefit of stranded wire is its mechanical flexibility, being able to withstand repeated bending and twisting much better than solid copper (which tends to fatigue and break after time). Wire size can be measured in several ways. We could speak of a wire’s diameter, but since its really the cross-sectional area that matters most regarding the flow of electrons, we are better off designating wire size in terms of area. The wire cross-section picture shown above is, of course, not drawn to scale. The diameter is shown as being 0.1019 inches. Calculating the area of the cross-section with the formula Area = πr2, we get an area of 0.008155 square inches: These are fairly small numbers to work with, so wire sizes are often expressed in measures of thousandths-of-an-inch, or mils. For the illustrated example, we would say that the diameter of the wire was 101.9 mils (0.1019 inch times 1000). We could also, if we wanted, express the area of the wire in the unit of square mils, calculating that value with the same circle-area formula, Area = πr2: Calculating the Circular-mil Area of a Wire However, electricians and others frequently concerned with wire size use another unit of area measurement tailored specifically for wire’s circular cross-section. This special unit is called the circular mil (sometimes abbreviated cmil). The sole purpose for having this special unit of measurement is to eliminate the need to invoke the factor π (3.1415927 . . .) in the formula for calculating area, plus the need to figure wire radiuswhen you’ve been given diameter. The formula for calculating the circular-mil area of a circular wire is very simple: Because this is a unit of area measurement, the mathematical power of 2 is still in effect (doubling the width of a circle will always quadruple its area, no matter what units are used, or if the width of that circle is expressed in terms of radius or diameter). To illustrate the difference between measurements in square mils and measurements in circular mils, I will compare a circle with a square, showing the area of each shape in both unit measures: And for another size of wire: Obviously, the circle of a given diameter has less cross-sectional area than a square of width and height equal to the circle’s diameter: both units of area measurement reflect that. However, it should be clear that the unit of “square mil” is really tailored for the convenient determination of a square’s area, while “circular mil” is tailored for the convenient determination of a circle’s area: the respective formula for each is simpler to work with. It must be understood that both units are valid for measuring the area of a shape, no matter what shape that may be. The conversion between circular mils and square mils is a simple ratio: there are π (3.1415927 . . .) square mils to every 4 circular mils. Measuring Cross-Sectional Wire Area with Gauge Another measure of cross-sectional wire area is the gauge. The gauge scale is based on whole numbers rather than fractional or decimal inches. The larger the gauge number, the skinnier the wire; the smaller the gauge number, the fatter the wire. For those acquainted with shotguns, this inversely-proportional measurement scale should sound familiar. The table at the end of this section equates gauge with inch diameter, circular mils, and square inches for solid wire. The larger sizes of wire reach an end of the common gauge scale (which naturally tops out at a value of 1), and are represented by a series of zeros. “3/0” is another way to represent “000,” and is pronounced “triple-ought.” Again, those acquainted with shotguns should recognize the terminology, strange as it may sound. To make matters even more confusing, there is more than one gauge “standard” in use around the world. For electrical conductor sizing, the American Wire Gauge (AWG), also known as the Brown and Sharpe (B&S) gauge, is the measurement system of choice. In Canada and Great Britain, the British Standard Wire Gauge (SWG) is the legal measurement system for electrical conductors. Other wire gauge systems exist in the world for classifying wire diameter, such as the Stubs steel wire gauge and the Steel Music Wire Gauge (MWG), but these measurement systems apply to non-electrical wire use. The American Wire Gauge (AWG) measurement system, despite its oddities, was designed with a purpose: for every three steps in the gauge scale, wire area (and weight per unit length) approximately doubles. This is a handy rule to remember when making rough wire size estimations! For very large wire sizes (fatter than 4/0), the wire gauge system is typically abandoned for cross-sectional area measurement in thousands of circular mils (MCM), borrowing the old Roman numeral “M” to denote a multiple of “thousand” in front of “CM” for “circular mils.” The following table of wire sizes does not show any sizes bigger than 4/0 gauge, because solid copper wire becomes impractical to handle at those sizes. Stranded wire construction is favored, instead. For some high-current applications, conductor sizes beyond the practical size limit of round wire are required. In these instances, thick bars of solid metal called busbars are used as conductors. Busbars are usually made of copper or aluminum, and are most often uninsulated. They are physically supported away from whatever framework or structure is holding them by insulator standoff mounts. Although a square or rectangular cross-section is very common for busbar shape, other shapes are used as well. Cross-sectional area for busbars is typically rated in terms of circular mils (even for square and rectangular bars!), most likely for the convenience of being able to directly equate busbar size with round wire. REVIEW: • Electrons flow through large-diameter wires easier than small-diameter wires, due to the greater cross-sectional area they have in which to move. • Rather than measure small wire sizes in inches, the unit of “mil” (1/1000 of an inch) is often employed. • The cross-sectional area of a wire can be expressed in terms of square units (square inches or square mils), circular mils, or “gauge” scale. • Calculating square-unit wire area for a circular wire involves the circle area formula: • Calculating circular-mil wire area for a circular wire is much simpler, due to the fact that the unit of “circular mil” was sized just for this purpose: to eliminate the “pi” and the d/2 (radius) factors in the formula. • There are π (3.1416) square mils for every 4 circular mils. • The gauge system of wire sizing is based on whole numbers, larger numbers representing smaller-area wires and vice versa. Wires thicker than 1 gauge are represented by zeros: 0, 00, 000, and 0000 (spoken “single-ought,” “double-ought,” “triple-ought,” and “quadruple-ought.” • Very large wire sizes are rated in thousands of circular mils (MCM’s), typical for busbars and wire sizes beyond 4/0. • Busbars are solid bars of copper or aluminum used in high-current circuit construction. Connections made to busbars are usually welded or bolted, and the busbars are often bare (uninsulated), supported away from metal frames through the use of insulating standoffs.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.03%3A_Conductor_Ampacity.txt	princeton-nlp/TextbookChapters	Primarily for reasons of safety, certain standards for electrical wiring have been established within the United States, and are specified in the National Electrical Code (NEC). Typical NEC wire ampacity tables will show allowable maximum currents for different sizes and applications of wire. Though the melting point of copper theoretically imposes a limit on wire ampacity, the materials commonly employed for insulating conductors melt at temperatures far below the melting point of copper, and so practical ampacity ratings are based on the thermal limits of the insulation. Voltage dropped as a result of excessive wire resistance is also a factor in sizing conductors for their use in circuits, but this consideration is better assessed through more complex means (which we will cover in this chapter). A table derived from an NEC listing is shown for example: Notice the substantial ampacity differences between same-size wires with different types of insulation. This is due, again, to the thermal limits (60o, 75o, 90o) of each type of insulation material. These ampacity ratings are given for copper conductors in “free air” (maximum typical air circulation), as opposed to wires placed in conduit or wire trays. As you will notice, the table fails to specify ampacities for small wire sizes. This is because the NEC concerns itself primarily with power wiring (large currents, big wires) rather than with wires common to low-current electronic work. There is meaning in the letter sequences used to identify conductor types, and these letters usually refer to properties of the conductor’s insulating layer(s). Some of these letters symbolize individual properties of the wire while others are simply abbreviations. For example, the letter “T” by itself means “thermoplastic” as an insulation material, as in “TW” or “THHN.” However, the three-letter combination “MTW” is an abbreviation for Machine Tool Wire, a type of wire whose insulation is made to be flexible for use in machines experiencing significant motion or vibration. Therefore, a “THWN” conductor has Thermoplastic insulation, is Heat resistant to 75o Celsius, is rated for Wet conditions, and comes with a Nylon outer jacketing. Letter codes like these are only used for general-purpose wires such as those used in households and businesses. For high-power applications and/or severe service conditions, the complexity of conductor technology defies classification according to a few letter codes. Overhead power line conductors are typically bare metal, suspended from towers by glass, porcelain, or ceramic mounts known as insulators. Even so, the actual construction of the wire to withstand physical forces both static (dead weight) and dynamic (wind) loading can be complex, with multiple layers and different types of metals wound together to form a single conductor. Large, underground power conductors are sometimes insulated by paper, then enclosed in a steel pipe filled with pressurized nitrogen or oil to prevent water intrusion. Such conductors require support equipment to maintain fluid pressure throughout the pipe. Other insulating materials find use in small-scale applications. For instance, the small-diameter wire used to make electromagnets (coils producing a magnetic field from the flow of electrons) are often insulated with a thin layer of enamel. The enamel is an excellent insulating material and is very thin, allowing many “turns” of wire to be wound in a small space. Review • Wire resistance creates heat in operating circuits. This heat is a potential fire ignition hazard. • Skinny wires have a lower allowable current (“ampacity”) than fat wires, due to their greater resistance per unit length, and consequently greater heat generation per unit current. • The National Electrical Code (NEC) specifies ampacities for power wiring based on allowable insulation temperature and wire application.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.04%3A_Fuses.txt	princeton-nlp/TextbookChapters	What is a Fuse? A fuse is nothing more than a short length of wire designed to melt and separate in the event of excessive current. Fuses are always connected in series with the component(s) to be protected from overcurrent, so that when the fuse blows (opens) it will open the entire circuit and stop current through the component(s). A fuse connected in one branch of a parallel circuit, of course, would not affect current through any of the other branches. Normally, the thin piece of fuse wire is contained within a safety sheath to minimize hazards of arc blast if the wire burns open with violent force, as can happen in the case of severe overcurrents. In the case of small automotive fuses, the sheath is transparent so that the fusible element can be visually inspected. Residential wiring used to commonly employ screw-in fuses with glass bodies and a thin, narrow metal foil strip in the middle. A photograph showing both types of fuses is shown here: Cartridge type fuses are popular in automotive applications, and in industrial applications when constructed with sheath materials other than glass. Because fuses are designed to “fail” open when their current rating is exceeded, they are typically designed to be replaced easily in a circuit. This means they will be inserted into some type of holder rather than being directly soldered or bolted to the circuit conductors. The following is a photograph showing a couple of glass cartridge fuses in a multi-fuse holder: The fuses are held by spring metal clips, the clips themselves being permanently connected to the circuit conductors. The base material of the fuse holder (or fuse block as they are sometimes called) is chosen to be a good insulator. Another type of fuse holder for cartridge-type fuses is commonly used for installation in equipment control panels, where it is desirable to conceal all electrical contact points from human contact. Unlike the fuse block just shown, where all the metal clips are openly exposed, this type of fuse holder completely encloses the fuse in an insulating housing: The most common device in use for overcurrent protection in high-current circuits today is the circuit breaker. What is a Circuit Breaker? Circuit breakers are specially designed switches that automatically open to stop current in the event of an overcurrent condition. Small circuit breakers, such as those used in residential, commercial and light industrial service are thermally operated. They contain a bimetallic strip (a thin strip of two metals bonded back-to-back) carrying circuit current, which bends when heated. When enough force is generated by the bimetallic strip (due to overcurrent heating of the strip), the trip mechanism is actuated and the breaker will open. Larger circuit breakers are automatically actuated by the strength of the magnetic field produced by current-carrying conductors within the breaker, or can be triggered to trip by external devices monitoring the circuit current (those devices being called protective relays). Because circuit breakers don’t fail when subjected to overcurrent conditions—rather, they merely open and can be re-closed by moving a lever—they are more likely to be found connected to a circuit in a more permanent manner than fuses. A photograph of a small circuit breaker is shown here: From outside appearances, it looks like nothing more than a switch. Indeed, it could be used as such. However, its true function is to operate as an overcurrent protection device. It should be noted that some automobiles use inexpensive devices known as fusible links for overcurrent protection in the battery charging circuit, due to the expense of a properly-rated fuse and holder. A fusible link is a primitive fuse, being nothing more than a short piece of rubber-insulated wire designed to melt open in the event of overcurrent, with no hard sheathing of any kind. Such crude and potentially dangerous devices are never used in industry or even residential power use, mainly due to the greater voltage and current levels encountered. As far as this author is concerned, their application even in automotive circuits is questionable. The electrical schematic drawing symbol for a fuse is an S-shaped curve: Fuse Ratings Fuses are primarily rated, as one might expect, in the unit for current: amps. Although their operation depends on the self-generation of heat under conditions of excessive current by means of the fuse’s own electrical resistance, they are engineered to contribute a negligible amount of extra resistance to the circuits they protect. This is largely accomplished by making the fuse wire as short as is practically possible. Just as a normal wire’s ampacity is not related to its length (10-gauge solid copper wire will handle 40 amps of current in free air, regardless of how long or short of a piece it is), a fuse wire of certain material and gauge will blow at a certain current no matter how long it is. Since length is not a factor in current rating, the shorter it can be made, the less resistance it will have end-to-end. However, the fuse designer also has to consider what happens after a fuse blows: the melted ends of the once-continuous wire will be separated by an air gap, with full supply voltage between the ends. If the fuse isn’t made long enough on a high-voltage circuit, a spark may be able to jump from one of the melted wire ends to the other, completing the circuit again: Consequently, fuses are rated in terms of their voltage capacity as well as the current level at which they will blow. Some large industrial fuses have replaceable wire elements, to reduce the expense. The body of the fuse is an opaque, reusable cartridge, shielding the fuse wire from exposure and shielding surrounding objects from the fuse wire. There’s more to the current rating of a fuse than a single number. If a current of 35 amps is sent through a 30 amp fuse, it may blow suddenly or delay before blowing, depending on other aspects of its design. Some fuses are intended to blow very fast, while others are designed for more modest “opening” times, or even for a delayed action depending on the application. The latter fuses are sometimes called slow-blow fuses due to their intentional time-delay characteristics. A classic example of a slow-blow fuse application is in electric motor protection, where inrush currents of up to ten times normal operating current are commonly experienced every time the motor is started from a dead stop. If fast-blowing fuses were to be used in an application like this, the motor could never get started because the normal inrush current levels would blow the fuse(s) immediately! The design of a slow-blow fuse is such that the fuse element has more mass (but no more ampacity) than an equivalent fast-blow fuse, meaning that it will heat up slower (but to the same ultimate temperature) for any given amount of current. On the other end of the fuse action spectrum, there are so-called semiconductor fuses designed to open very quickly in the event of an overcurrent condition. Semiconductor devices such as transistors tend to be especially intolerant of overcurrent conditions, and as such require fast-acting protection against overcurrents in high-power applications. Fuses are always supposed to be placed on the “hot” side of the load in systems that are grounded. The intent of this is for the load to be completely de-energized in all respects after the fuse opens. To see the difference between fusing the “hot” side versus the “neutral” side of a load, compare these two circuits: In either case, the fuse successfully interrupted current to the load, but the lower circuit fails to interrupt potentially dangerous voltage from either side of the load to ground, where a person might be standing. The first circuit design is much safer. As it was said before, fuses are not the only type of overcurrent protection device in use. Switch-like devices called circuit breakers are often (and more commonly) used to open circuits with excessive current, their popularity due to the fact that they don’t destroy themselves in the process of breaking the circuit as fuses do. In any case, though, placement of the overcurrent protection device in a circuit will follow the same general guidelines listed above: namely, to “fuse” the side of the power supply not connected to ground. Although overcurrent protection placement in a circuit may determine the relative shock hazard of that circuit under various conditions, it must be understood that such devices were never intended to guard against electric shock. Neither fuses nor circuit breakers were designed to open in the event of a person getting shocked; rather, they are intended to open only under conditions of potential conductor overheating. Overcurrent devices primarily protect the conductors of a circuit from over temperature damage (and the fire hazards associated with overly hot conductors), and secondarily protect specific pieces of equipment such as loads and generators (some fast-acting fuses are designed to protect electronic devices particularly susceptible to current surges). Since the current levels necessary for electric shock or electrocution are much lower than the normal current levels of common power loads, a condition of overcurrent is not indicative of shock occurring. There are other devices designed to detect certain shock conditions (ground-fault detectors being the most popular), but these devices strictly serve that one purpose and are uninvolved with protection of the conductors against overheating. Review • A fuse is a small, thin conductor designed to melt and separate into two pieces for the purpose of breaking a circuit in the event of excessive current. • A circuit breaker is a specially designed switch that automatically opens to interrupt circuit current in the event of an overcurrent condition. They can be “tripped” (opened) thermally, by magnetic fields, or by external devices called “protective relays,” depending on the design of breaker, its size, and the application. • Fuses are primarily rated in terms of maximum current, but are also rated in terms of how much voltage drop they will safely withstand after interrupting a circuit. • Fuses can be designed to blow fast, slow, or anywhere in between for the same maximum level of current. • The best place to install a fuse in a grounded power system is on the ungrounded conductor path to the load. That way, when the fuse blows there will only be the grounded (safe) conductor still connected to the load, making it safer for people to be around.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.05%3A_Specific_Resistance.txt	princeton-nlp/TextbookChapters	Designing Wire Resistance Conductor ampacity rating is a crude assessment of resistance based on the potential for current to create a fire hazard. However, we may come across situations where the voltage drop created by wire resistance in a circuit poses concerns other than fire avoidance. For instance, we may be designing a circuit where voltage across a component is critical, and must not fall below a certain limit. If this is the case, the voltage drops resulting from wire resistance may cause an engineering problem while being well within safe (fire) limits of ampacity: The Resistance Formula If the load in the above circuit will not tolerate less than 220 volts, given a source voltage of 230 volts, then we’d better be sure that the wiring doesn’t drop more than 10 volts along the way. Counting both the supply and return conductors of this circuit, this leaves a maximum tolerable drop of 5 volts along the length of each wire. Using Ohm’s Law (R=E/I), we can determine the maximum allowable resistance for each piece of wire: We know that the wire length is 2300 feet for each piece of wire, but how do we determine the amount of resistance for a specific size and length of wire? To do that, we need another formula: This formula relates the resistance of a conductor with its specific resistance (the Greek letter “rho” (ρ), which looks similar to a lower-case letter “p”), its length (“l”), and its cross-sectional area (“A”). Notice that with the length variable on the top of the fraction, the resistance value increases as the length increases (analogy: it is more difficult to force liquid through a long pipe than a short one), and decreases as cross-sectional area increases (analogy: liquid flows easier through a fat pipe than through a skinny one). Specific resistance is a constant for the type of conductor material being calculated. The specific resistances of several conductive materials can be found in the following table. We find copper near the bottom of the table, second only to silver in having low specific resistance (good conductivity): Notice that the figures for specific resistance in the above table are given in the very strange unit of “ohms-cmil/ft” (Ω-cmil/ft), This unit indicates what units we are expected to use in the resistance formula (R=ρl/A). In this case, these figures for specific resistance are intended to be used when length is measured in feet and cross-sectional area is measured in circular mils. The metric unit for specific resistance is the ohm-meter (Ω-m), or ohm-centimeter (Ω-cm), with 1.66243 x 10-9 Ω-meters per Ω-cmil/ft (1.66243 x 10-7 Ω-cm per Ω-cmil/ft). In the Ω-cm column of the table, the figures are actually scaled as µΩ-cm due to their very small magnitudes. For example, iron is listed as 9.61 µΩ-cm, which could be represented as 9.61 x 10-6 Ω-cm. When using the unit of Ω-meter for specific resistance in the R=ρl/A formula, the length needs to be in meters and the area in square meters. When using the unit of Ω-centimeter (Ω-cm) in the same formula, the length needs to be in centimeters and the area in square centimeters. All these units for specific resistance are valid for any material (Ω-cmil/ft, Ω-m, or Ω-cm). One might prefer to use Ω-cmil/ft, however, when dealing with round wire where the cross-sectional area is already known in circular mils. Conversely, when dealing with odd-shaped busbar or custom busbar cut out of metal stock, where only the linear dimensions of length, width, and height are known, the specific resistance units of Ω-meter or Ω-cm may be more appropriate. Solving Going back to our example circuit, we were looking for wire that had 0.2 Ω or less of resistance over a length of 2300 feet. Assuming that we’re going to use copper wire (the most common type of electrical wire manufactured), we can set up our formula as such: Algebraically solving for A, we get a value of 116,035 circular mils. Referencing our solid wire size table, we find that “double-ought” (2/0) wire with 133,100 cmils is adequate, whereas the next lower size, “single-ought” (1/0), at 105,500 cmils is too small. Bear in mind that our circuit current is a modest 25 amps. According to our ampacity table for copper wire in free air, 14 gauge wire would have sufficed (as far as notstarting a fire is concerned). However, from the standpoint of voltage drop, 14 gauge wire would have been very unacceptable. Just for fun, let’s see what 14 gauge wire would have done to our power circuit’s performance. Looking at our wire size table, we find that 14 gauge wire has a cross-sectional area of 4,107 circular mils. If we’re still using copper as a wire material (a good choice, unless we’re really rich and can afford 4600 feet of 14 gauge silver wire!), then our specific resistance will still be 10.09 Ω-cmil/ft: Remember that this is 5.651 Ω per 2300 feet of 14-gauge copper wire, and that we have two runs of 2300 feet in the entire circuit, so each wire piece in the circuit has 5.651 Ω of resistance: Our total circuit wire resistance is 2 times 5.651, or 11.301 Ω. Unfortunately, this is far too much resistance to allow 25 amps of current with a source voltage of 230 volts. Even if our load resistance was 0 Ω, our wiring resistance of 11.301 Ω would restrict the circuit current to a mere 20.352 amps! As you can see, a “small” amount of wire resistance can make a big difference in circuit performance, especially in power circuits where the currents are much higher than typically encountered in electronic circuits. Let’s do an example resistance problem for a piece of custom-cut busbar. Suppose we have a piece of solid aluminum bar, 4 centimeters wide by 3 centimeters tall by 125 centimeters long, and we wish to figure the end-to-end resistance along the long dimension (125 cm). First, we would need to determine the cross-sectional area of the bar: We also need to know the specific resistance of aluminum, in the unit proper for this application (Ω-cm). From our table of specific resistances, we see that this is 2.65 x 10-6 Ω-cm. Setting up our R=ρl/A formula, we have: As you can see, the sheer thickness of a busbar makes for very low resistances compared to that of standard wire sizes, even when using a material with a greater specific resistance. The procedure for determining busbar resistance is not fundamentally different than for determining round wire resistance. We just need to make sure that cross-sectional area is calculated properly and that all the units correspond to each other as they should. Review • Conductor resistance increases with increased length and decreases with increased cross-sectional area, all other factors being equal. • Specific Resistance (”ρ”) is a property of any conductive material, a figure used to determine the end-to-end resistance of a conductor given length and area in this formula: R = ρl/A • Specific resistance for materials are given in units of Ω-cmil/ft or Ω-meters (metric). Conversion factor between these two units is 1.66243 x 10-9 Ω-meters per Ω-cmil/ft, or 1.66243 x 10-7 Ω-cm per Ω-cmil/ft. • If wiring voltage drop in a circuit is critical, exact resistance calculations for the wires must be made before wire size is chosen.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.06%3A_Temperature_Coefficient_of_Resistance.txt	princeton-nlp/TextbookChapters	Resistance values for conductors at any temperature other than the standard temperature (usually specified at 20 Celsius) on the specific resistance table must be determined through yet another formula: The “alpha” (α) constant is known as the temperature coefficient of resistance, and symbolizes the resistance change factor per degree of temperature change. Just as all materials have a certain specific resistance (at 20o C), they also change resistance according to temperature by certain amounts. For pure metals, this coefficient is a positive number, meaning that resistance increases with increasing temperature. For the elements carbon, silicon, and germanium, this coefficient is a negative number, meaning that resistance decreases with increasing temperature. For some metal alloys, the temperature coefficient of resistance is very close to zero, meaning that the resistance hardly changes at all with variations in temperature (a good property if you want to build a precision resistor out of metal wire!). The following table gives the temperature coefficients of resistance for several common metals, both pure and alloy: Let’s take a look at an example circuit to see how temperature can affect wire resistance, and consequently circuit performance: This circuit has a total wire resistance (wire 1 + wire 2) of 30 Ω at standard temperature. Setting up a table of voltage, current, and resistance values we get: At 20o Celsius, we get 12.5 volts across the load and a total of 1.5 volts (0.75 + 0.75) dropped across the wire resistance. If the temperature were to rise to 35o Celsius, we could easily determine the change of resistance for each piece of wire. Assuming the use of copper wire (α = 0.004041) we get: Recalculating our circuit values, we see what changes this increase in temperature will bring: As you can see, voltage across the load went down (from 12.5 volts to 12.42 volts) and voltage drop across the wires went up (from 0.75 volts to 0.79 volts) as a result of the temperature increasing. Though the changes may seem small, they can be significant for power lines stretching miles between power plants and substations, substations and loads. In fact, power utility companies often have to take line resistance changes resulting from seasonal temperature variations into account when calculating allowable system loading. Review • Most conductive materials change specific resistance with changes in temperature. This is why figures of specific resistance are always specified at a standard temperature (usually 20o or 25o Celsius). • The resistance-change factor per degree Celsius of temperature change is called the temperature coefficient of resistance. This factor is represented by the Greek lower-case letter “alpha” (α). • A positive coefficient for a material means that its resistance increases with an increase in temperature. Pure metals typically have positive temperature coefficients of resistance. Coefficients approaching zero can be obtained by alloying certain metals. • A negative coefficient for a material means that its resistance decreases with an increase in temperature. Semiconductor materials (carbon, silicon, germanium) typically have negative temperature coefficients of resistance. • The formula used to determine the resistance of a conductor at some temperature other than what is specified in a resistance table is as follows:
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/12%3A_Physics_of_Conductors_and_Insulators/12.07%3A_Superconductivity.txt	princeton-nlp/TextbookChapters	There is some debate over exactly how and why superconducting materials superconduct. One theory holds that electrons group together and travel in pairs (called Cooper pairs) within a superconductor rather than travel independently, and that has something to do with their frictionless flow. Interestingly enough, another phenomenon of super-cold temperatures, superfluidity, happens with certain liquids (especially liquid helium), resulting in frictionless flow of molecules. Superconductivity promises extraordinary capabilities for electric circuits. If conductor resistance could be eliminated entirely, there would be no power losses or inefficiencies in electric power systems due to stray resistances. Electric motors could be made almost perfectly (100%) efficient. Components such as capacitors and inductors, whose ideal characteristics are normally spoiled by inherent wire resistances, could be made ideal in a practical sense. Already, some practical superconducting conductors, motors, and capacitors have been developed, but their use at this present time is limited due to the practical problems intrinsic to maintaining super-cold temperatures. The threshold temperature for a superconductor to switch from normal conduction to superconductivity is called the transition temperature. Transition temperatures for “classic” superconductors are in the cryogenic range (near absolute zero), but much progress has been made in developing “high-temperature” superconductors which superconduct at warmer temperatures. One type is a ceramic mixture of yttrium, barium, copper, and oxygen which transitions at a relatively balmy -160o Celsius. Ideally, a superconductor should be able to operate within the range of ambient temperatures, or at least within the range of inexpensive refrigeration equipment. The critical temperatures for a few common substances are shown here in this table. Temperatures are given in kelvins, which has the same incremental span as degrees Celsius (an increase or decrease of 1 kelvin is the same amount of temperature change as 1o Celsius), only offset so that 0 K is absolute zero. This way, we don’t have to deal with a lot of negative figures. Superconducting materials also interact in interesting ways with magnetic fields. While in the superconducting state, a superconducting material will tend to exclude all magnetic fields, a phenomenon known as the Meissner effect. However, if the magnetic field strength intensifies beyond a critical level, the superconducting material will be rendered non-superconductive. In other words, superconducting materials will lose their superconductivity (no matter how cold you make them) if exposed to too strong of a magnetic field. In fact, the presence of any magnetic field tends to lower the critical temperature of any superconducting material: the more magnetic field present, the colder you have to make the material before it will superconduct. This is another practical limitation to superconductors in circuit design, since electric current through any conductor produces a magnetic field. Even though a superconducting wire would have zero resistance to oppose current, there will still be a limit of how much current could practically go through that wire due to its critical magnetic field limit. There are already a few industrial applications of superconductors, especially since the recent (1987) advent of the yttrium-barium-copper-oxygen ceramic, which only requires liquid nitrogen to cool, as opposed to liquid helium. It is even possible to order superconductivity kits from educational suppliers which can be operated in high school labs (liquid nitrogen not included). Typically, these kits exhibit superconductivity by the Meissner effect, suspending a tiny magnet in mid-air over a superconducting disk cooled by a bath of liquid nitrogen. The zero resistance offered by superconducting circuits leads to unique consequences. In a superconducting short-circuit, it is possible to maintain large currents indefinitely with zero applied voltage! Rings of superconducting material have been experimentally proven to sustain continuous current for years with no applied voltage. So far as anyone knows, there is no theoretical time limit to how long an unaided current could be sustained in a superconducting circuit. If you’re thinking this appears to be a form of perpetual motion, you’re correct! Contrary to popular belief, there is no law of physics prohibiting perpetual motion; rather, the prohibition stands against any machine or system generating more energy than it consumes (what would be referred to as an over-unity device). At best, all a perpetual motion machine (like the superconducting ring) would be good for is to store energy, not generate it freely! Superconductors also offer some strange possibilities having nothing to do with Ohm’s Law. One such possibility is the construction of a device called a Josephson Junction, which acts as a relay of sorts, controlling one current with another current (with no moving parts, of course). The small size and fast switching time of Josephson Junctions may lead to new computer circuit designs: an alternative to using semiconductor transistors. Review • Superconductors are materials which have absolutely zero electrical resistance. • All presently known superconductive materials need to be cooled far below ambient temperature to superconduct. The maximum temperature at which they do so is called the transition temperature. 12.08: Insulator Breakdown Voltage Once current is forced through an insulating material, breakdown of that material’s molecular structure has occurred. After breakdown, the material may or may not behave as an insulator any more, the molecular structure having been altered by the breach. There is usually a localized “puncture” of the insulating medium where the electrons flowed during breakdown. Thickness of an insulating material plays a role in determining its breakdown voltage, otherwise known as dielectric strength. Specific dielectric strength is sometimes listed in terms of volts per mil (1/1000 of an inch), or kilovolts per inch (the two units are equivalent), but in practice it has been found that the relationship between breakdown voltage and thickness is not exactly linear. An insulator three times as thick has a dielectric strength slightly less than 3 times as much. However, for rough estimation use, volt-per-thickness ratings are fine. Review • With a high enough applied voltage, electrons can be freed from the atoms of insulating materials, resulting in current through that material. • The minimum voltage required to “violate” an insulator by forcing current through it is called the breakdown voltage, or dielectric strength. • The thicker a piece of insulating material, the higher the breakdown voltage, all other factors being equal. • Specific dielectric strength is typically rated in one of two equivalent units: volts per mil, or kilovolts per inch.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/13%3A_Capacitors/13.01%3A_Electric_Fields_and_Capacitance.txt	princeton-nlp/TextbookChapters	Admittedly, the concept of a “field” is somewhat abstract. At least with electric current it isn’t too difficult to envision tiny particles called electrons moving their way between the nuclei of atoms within a conductor, but a “field” doesn’t even have mass, and need not exist within matter at all. Despite its abstract nature, almost every one of us has direct experience with fields, at least in the form of magnets. Have you ever played with a pair of magnets, noticing how they attract or repel each other depending on their relative orientation? There is an undeniable force between a pair of magnets, and this force is without “substance.” It has no mass, no color, no odor, and if not for the physical force exerted on the magnets themselves, it would be utterly insensible to our bodies. Physicists describe the interaction of magnets in terms of magnetic fields in the space between them. If iron filings are placed near a magnet, they orient themselves along the lines of the field, visually indicating its presence. The Electric Fields The subject of this chapter is electric fields (and devices called capacitors that exploit them), not magneticfields, but there are many similarities. Most likely you have experienced electric fields as well. Chapter 1 of this book began with an explanation of static electricity, and how materials such as wax and wool—when rubbed against each other—produced a physical attraction. Again, physicists would describe this interaction in terms of electric fields generated by the two objects as a result of their electron imbalances. Suffice it to say that whenever a voltage exists between two points, there will be an electric field manifested in the space between those points. The Field Force and the Field Flux Fields have two measures: a field force and a field flux. The field force is the amount of “push” that a field exerts over a certain distance. The field flux is the total quantity, or effect, of the field through space. Field force and flux are roughly analogous to voltage (“push”) and current (flow) through a conductor, respectively, although field flux can exist in totally empty space (without the motion of particles such as electrons) whereas current can only take place where there are free electrons to move. Field flux can be opposed in space, just as the flow of electrons can be opposed by resistance. The amount of field flux that will develop in space is proportional to the amount of field force applied, divided by the amount of opposition to flux. Just as the type of conducting material dictates that conductor’s specific resistance to electric current, the type of insulating material separating two conductors dictates the specific opposition to field flux. Normally, electrons cannot enter a conductor unless there is a path for an equal amount of electrons to exit (remember the marble-in-tube analogy?). This is why conductors must be connected together in a circular path (a circuit) for continuous current to occur. Oddly enough, however, extra electrons can be “squeezed” into a conductor without a path to exit if an electric field is allowed to develop in space relative to another conductor. The number of extra free electrons added to the conductor (or free electrons taken away) is directly proportional to the amount of field flux between the two conductors. The Capacitors Electric Field Capacitors are components designed to take advantage of this phenomenon by placing two conductive plates (usually metal) in close proximity with each other. There are many different styles of capacitor construction, each one suited for particular ratings and purposes. For very small capacitors, two circular plates sandwiching an insulating material will suffice. For larger capacitor values, the “plates” may be strips of metal foil, sandwiched around a flexible insulating medium and rolled up for compactness. The highest capacitance values are obtained by using a microscopic-thickness layer of insulating oxide separating two conductive surfaces. In any case, though, the general idea is the same: two conductors, separated by an insulator. The schematic symbol for a capacitor is quite simple, being little more than two short, parallel lines (representing the plates) separated by a gap. Wires attach to the respective plates for connection to other components. An older, obsolete schematic symbol for capacitors showed interleaved plates, which is actually a more accurate way of representing the real construction of most capacitors: When a voltage is applied across the two plates of a capacitor, a concentrated field flux is created between them, allowing a significant difference of free electrons (a charge) to develop between the two plates: As the electric field is established by the applied voltage, extra free electrons are forced to collect on the negative conductor, while free electrons are “robbed” from the positive conductor. This differential charge equates to a storage of energy in the capacitor, representing the potential charge of the electrons between the two plates. The greater the difference of electrons on opposing plates of a capacitor, the greater the field flux, and the greater “charge” of energy the capacitor will store. Because capacitors store the potential energy of accumulated electrons in the form of an electric field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. Energy storage in a capacitor is a function of the voltage between the plates, as well as other factors which we will discuss later in this chapter. A capacitor’s ability to store energy as a function of voltage (potential difference between the two leads) results in a tendency to try to maintain voltage at a constant level. In other words, capacitors tend to resist changes in voltage drop. When voltage across a capacitor is increased or decreased, the capacitor “resists” the change by drawing current from or supplying current to the source of the voltage change, in opposition to the change. To store more energy in a capacitor, the voltage across it must be increased. This means that more electrons must be added to the (-) plate and more taken away from the (+) plate, necessitating a current in that direction. Conversely, to release energy from a capacitor, the voltage across it must be decreased. This means some of the excess electrons on the (-) plate must be returned to the (+) plate, necessitating a current in the other direction. Just as Isaac Newton’s first Law of Motion (“an object in motion tends to stay in motion; an object at rest tends to stay at rest”) describes the tendency of a mass to oppose changes in velocity, we can state a capacitor’s tendency to oppose changes in voltage as such: “A charged capacitor tends to stay charged; a discharged capacitor tends to stay discharged.” Hypothetically, a capacitor left untouched will indefinitely maintain whatever state of voltage charge that its been left it. Only an outside source (or drain) of current can alter the voltage charge stored by a perfect capacitor: Practically speaking, however, capacitors will eventually lose their stored voltage charges due to internal leakage paths for electrons to flow from one plate to the other. Depending on the specific type of capacitor, the time it takes for a stored voltage charge to self-dissipate can be a long time (several years with the capacitor sitting on a shelf!). When the voltage across a capacitor is increased, it draws current from the rest of the circuit, acting as a power load. In this condition the capacitor is said to be charging, because there is an increasing amount of energy being stored in its electric field. Note the direction of electron current with regard to the voltage polarity: Conversely, when the voltage across a capacitor is decreased, the capacitor supplies current to the rest of the circuit, acting as a power source. In this condition the capacitor is said to be discharging. Its store of energy—held in the electric field—is decreasing now as energy is released to the rest of the circuit. Note the direction of electron current with regard to the voltage polarity: If a source of voltage is suddenly applied to an uncharged capacitor (a sudden increase of voltage), the capacitor will draw current from that source, absorbing energy from it, until the capacitor’s voltage equals that of the source. Once the capacitor voltage reached this final (charged) state, its current decays to zero. Conversely, if a load resistance is connected to a charged capacitor, the capacitor will supply current to the load, until it has released all its stored energy and its voltage decays to zero. Once the capacitor voltage reaches this final (discharged) state, its current decays to zero. In their ability to be charged and discharged, capacitors can be thought of as acting somewhat like secondary-cell batteries. The choice of insulating material between the plates, as was mentioned before, has a great impact upon how much field flux (and therefore how much charge) will develop with any given amount of voltage applied across the plates. Because of the role of this insulating material in affecting field flux, it has a special name: dielectric. Not all dielectric materials are equal: the extent to which materials inhibit or encourage the formation of electric field flux is called the permittivity of the dielectric. The measure of a capacitor’s ability to store energy for a given amount of voltage drop is called capacitance. Not surprisingly, capacitance is also a measure of the intensity of opposition to changes in voltage (exactly how much current it will produce for a given rate of change in voltage). Capacitance is symbolically denoted with a capital “C,” and is measured in the unit of the Farad, abbreviated as “F.” Convention, for some odd reason, has favored the metric prefix “micro” in the measurement of large capacitances, and so many capacitors are rated in terms of confusingly large microFarad values: for example, one large capacitor I have seen was rated 330,000 microFarads!! Why not state it as 330 milliFarads? I don’t know. The Capacitor’s Obsolete Name An obsolete name for a capacitor is condenser or condensor. These terms are not used in any new books or schematic diagrams (to my knowledge), but they might be encountered in older electronics literature. Perhaps the most well-known usage for the term “condenser” is in automotive engineering, where a small capacitor called by that name was used to mitigate excessive sparking across the switch contacts (called “points”) in electromechanical ignition systems. Review • Capacitors react against changes in voltage by supplying or drawing current in the direction necessary to oppose the change. • When a capacitor is faced with an increasing voltage, it acts as a load: drawing current as it absorbs energy (current going in the negative side and out the positive side, like a resistor). • When a capacitor is faced with a decreasing voltage, it acts as a source: supplying current as it releases stored energy (current going out the negative side and in the positive side, like a battery). • The ability of a capacitor to store energy in the form of an electric field (and consequently to oppose changes in voltage) is called capacitance. It is measured in the unit of the Farad (F). • Capacitors used to be commonly known by another term: condenser (alternatively spelled “condensor”).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/13%3A_Capacitors/13.02%3A_Capacitors_and_Calculus.txt	princeton-nlp/TextbookChapters	Capacitors do not have a stable “resistance” as conductors do. However, there is a definite mathematical relationship between voltage and current for a capacitor, as follows: The lower-case letter “i” symbolizes instantaneous current, which means the amount of current at a specific point in time. This stands in contrast to constant current or average current (capital letter “I”) over an unspecified period of time. The expression “dv/dt” is one borrowed from calculus, meaning the instantaneous rate of voltage change over time, or the rate of change of voltage (volts per second increase or decrease) at a specific point in time, the same specific point in time that the instantaneous current is referenced at. For whatever reason, the letter v is usually used to represent instantaneous voltage rather than the letter e. However, it would not be incorrect to express the instantaneous voltage rate-of-change as “de/dt” instead. In this equation we see something novel to our experience thus far with electric circuits: the variable of time. When relating the quantities of voltage, current, and resistance to a resistor, it doesn’t matter if we’re dealing with measurements taken over an unspecified period of time (E=IR; V=IR), or at a specific moment in time (e=ir; v=ir). The same basic formula holds true, because time is irrelevant to voltage, current, and resistance in a component like a resistor. In a capacitor, however, time is an essential variable, because current is related to how rapidly voltage changes over time. To fully understand this, a few illustrations may be necessary. Suppose we were to connect a capacitor to a variable-voltage source, constructed with a potentiometer and a battery: If the potentiometer mechanism remains in a single position (wiper is stationary), the voltmeter connected across the capacitor will register a constant (unchanging) voltage, and the ammeter will register 0 amps. In this scenario, the instantaneous rate of voltage change (dv/dt) is equal to zero, because the voltage is unchanging. The equation tells us that with 0 volts per second change for a dv/dt, there must be zero instantaneous current (i). From a physical perspective, with no change in voltage, there is no need for any electron motion to add or subtract charge from the capacitor’s plates, and thus there will be no current. Now, if the potentiometer wiper is moved slowly and steadily in the “up” direction, a greater voltage will gradually be imposed across the capacitor. Thus, the voltmeter indication will be increasing at a slow rate: If we assume that the potentiometer wiper is being moved such that the rate of voltage increase across the capacitor is steady (for example, voltage increasing at a constant rate of 2 volts per second), the dv/dt term of the formula will be a fixed value. According to the equation, this fixed value of dv/dt, multiplied by the capacitor’s capacitance in Farads (also fixed), results in a fixed current of some magnitude. From a physical perspective, an increasing voltage across the capacitor demands that there be an increasing charge differential between the plates. Thus, for a slow, steady voltage increase rate, there must be a slow, steady rate of charge building in the capacitor, which equates to a slow, steady flow rate of electrons, or current. In this scenario, the capacitor is acting as a load, with electrons entering the negative plate and exiting the positive, accumulating energy in the electric field. If the potentiometer is moved in the same direction, but at a faster rate, the rate of voltage change (dv/dt) will be greater and so will be the capacitor’s current: When mathematics students first study calculus, they begin by exploring the concept of rates of change for various mathematical functions. The derivative, which is the first and most elementary calculus principle, is an expression of one variable’s rate of change in terms of another. Calculus students have to learn this principle while studying abstract equations. You get to learn this principle while studying something you can relate to: electric circuits! To put this relationship between voltage and current in a capacitor in calculus terms, the current through a capacitor is the derivative of the voltage across the capacitor with respect to time. Or, stated in simpler terms, a capacitor’s current is directly proportional to how quickly the voltage across it is changing. In this circuit where capacitor voltage is set by the position of a rotary knob on a potentiometer, we can say that the capacitor’s current is directly proportional to how quickly we turn the knob. If we were to move the potentiometer’s wiper in the same direction as before (“up”), but at varying rates, we would obtain graphs that looked like this: Note how that at any given point in time, the capacitor’s current is proportional to the rate-of-change, or slope of the capacitor’s voltage plot. When the voltage plot line is rising quickly (steep slope), the current will likewise be great. Where the voltage plot has a mild slope, the current is small. At one place in the voltage plot where it levels off (zero slope, representing a period of time when the potentiometer wasn’t moving), the current falls to zero. If we were to move the potentiometer wiper in the “down” direction, the capacitor voltage would decreaserather than increase. Again, the capacitor will react to this change of voltage by producing a current, but this time the current will be in the opposite direction. A decreasing capacitor voltage requires that the charge differential between the capacitor’s plates be reduced, and the only way that can happen is if the electrons reverse their direction of flow, the capacitor discharging rather than charging. In this condition, with electrons exiting the negative plate and entering the positive, the capacitor will act as a source, like a battery, releasing its stored energy to the rest of the circuit. Again, the amount of current through the capacitor is directly proportional to the rate of voltage change across it. The only difference between the effects of a decreasing voltage and an increasing voltage is the direction of electron flow. For the same rate of voltage change over time, either increasing or decreasing, the current magnitude (amps) will be the same. Mathematically, a decreasing voltage rate-of-change is expressed as a negative dv/dt quantity. Following the formula i = C(dv/dt), this will result in a current figure (i) that is likewise negative in sign, indicating a direction of flow corresponding to discharge of the capacitor.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/13%3A_Capacitors/13.03%3A_Factors_Affecting_Capacitance.txt	princeton-nlp/TextbookChapters	There are three basic factors of capacitor construction determining the amount of capacitance created. These factors all dictate capacitance by affecting how much electric field flux (relative difference of electrons between plates) will develop for a given amount of electric field force (voltage between the two plates): PLATE AREA: All other factors being equal, greater plate area gives greater capacitance; less plate area gives less capacitance. Explanation: Larger plate area results in more field flux (charge collected on the plates) for a given field force (voltage across the plates). PLATE SPACING: All other factors being equal, further plate spacing gives less capacitance; closer plate spacing gives greater capacitance. Explanation: Closer spacing results in a greater field force (voltage across the capacitor divided by the distance between the plates), which results in a greater field flux (charge collected on the plates) for any given voltage applied across the plates. DIELECTRIC MATERIAL: All other factors being equal, greater permittivity of the dielectric gives greater capacitance; less permittivity of the dielectric gives less capacitance. Explanation: Although its complicated to explain, some materials offer less opposition to field flux for a given amount of field force. Materials with a greater permittivity allow for more field flux (offer less opposition), and thus a greater collected charge, for any given amount of field force (applied voltage). “Relative” permittivity means the permittivity of a material, relative to that of a pure vacuum. The greater the number, the greater the permittivity of the material. Glass, for instance, with a relative permittivity of 7, has seven times the permittivity of a pure vacuum, and consequently will allow for the establishment of an electric field flux seven times stronger than that of a vacuum, all other factors being equal. The following is a table listing the relative permittivities (also known as the “dielectric constant”) of various common substances: An approximation of capacitance for any pair of separated conductors can be found with this formula: A capacitor can be made variable rather than fixed in value by varying any of the physical factors determining capacitance. One relatively easy factor to vary in capacitor construction is that of plate area, or more properly, the amount of plate overlap. The following photograph shows an example of a variable capacitor using a set of interleaved metal plates and an air gap as the dielectric material: As the shaft is rotated, the degree to which the sets of plates overlap each other will vary, changing the effective area of the plates between which a concentrated electric field can be established. This particular capacitor has a capacitance in the picofarad range, and finds use in radio circuitry. 13.04: Series and Parallel Capacitors When capacitors are connected in series, the total capacitance is less than any one of the series capacitors’ individual capacitances. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. As we’ve just seen, an increase in plate spacing, with all other factors unchanged, results in decreased capacitance. Thus, the total capacitance is less than any one of the individual capacitors’ capacitances. The formula for calculating the series total capacitance is the same form as for calculating parallel resistances: When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors’ capacitances. If two or more capacitors are connected in parallel, the overall effect is that of a single equivalent capacitor having the sum total of the plate areas of the individual capacitors. As we’ve just seen, an increase in plate area, with all other factors unchanged, results in increased capacitance. Thus, the total capacitance is more than any one of the individual capacitors’ capacitances. The formula for calculating the parallel total capacitance is the same form as for calculating series resistances: As you will no doubt notice, this is exactly opposite of the phenomenon exhibited by resistors. With resistors, series connections result in additive values while parallel connections result in diminished values. With capacitors, its the reverse: parallel connections result in additive values while series connections result in diminished values. Review • Capacitances diminish in series. • Capacitances add in parallel.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/13%3A_Capacitors/13.05%3A_Practical_Considerations_-_Capacitors.txt	princeton-nlp/TextbookChapters	Capacitor Working Voltage Working voltage: Since capacitors are nothing more than two conductors separated by an insulator (the dielectric), you must pay attention to the maximum voltage allowed across it. If too much voltage is applied, the “breakdown” rating of the dielectric material may be exceeded, resulting in the capacitor internally short-circuiting. Capacitor Polarity Polarity: Some capacitors are manufactured so they can only tolerate applied voltage in one polarity but not the other. This is due to their construction: the dielectric is a microscopically thin layer of insulation deposited on one of the plates by a DC voltage during manufacture. These are called electrolytic capacitors, and their polarity is clearly marked. Reversing voltage polarity to an electrolytic capacitor may result in the destruction of that super-thin dielectric layer, thus ruining the device. However, the thinness of that dielectric permits extremely high values of capacitance in a relatively small package size. For the same reason, electrolytic capacitors tend to be low in voltage rating as compared with other types of capacitor construction. Capacitor Equivalent Circuit Equivalent circuit: Since the plates in a capacitor have some resistance, and since no dielectric is a perfect insulator, there is no such thing as a “perfect” capacitor. In real life, a capacitor has both a series resistance and a parallel (leakage) resistance interacting with its purely capacitive characteristics: Fortunately, it is relatively easy to manufacture capacitors with very small series resistances and very high leakage resistances! Capacitor Physical Size Physical Size: For most applications in electronics, minimum size is the goal for component engineering. The smaller components can be made, the more circuitry can be built into a smaller package, and usually weight is saved as well. With capacitors, there are two major limiting factors to the minimum size of a unit: working voltage and capacitance. And these two factors tend to be in opposition to each other. For any given choice in dielectric materials, the only way to increase the voltage rating of a capacitor is to increase the thickness of the dielectric. However, as we have seen, this has the effect of decreasing capacitance. Capacitance can be brought back up by increasing plate area. but this makes for a larger unit. This is why you cannot judge a capacitor’s rating in Farads simply by size. A capacitor of any given size may be relatively high in capacitance and low in working voltage, vice versa, or some compromise between the two extremes. Take the following two photographs for example: This is a fairly large capacitor in physical size, but it has quite a low capacitance value: only 2 µF. However, its working voltage is quite high: 2000 volts! If this capacitor were re-engineered to have a thinner layer of dielectric between its plates, at least a hundredfold increase in capacitance might be achievable, but at a cost of significantly lowering its working voltage. Compare the above photograph with the one below. The capacitor shown in the lower picture is an electrolytic unit, similar in size to the one above, but with very different values of capacitance and working voltage: The thinner dielectric layer gives it a much greater capacitance (20,000 µF) and a drastically reduced working voltage (35 volts continuous, 45 volts intermittent). Samples of Different Capacitor Types Here are some samples of different capacitor types, all smaller than the units shown previously: The electrolytic and tantalum capacitors are polarized (polarity sensitive), and are always labeled as such. The electrolytic units have their negative (-) leads distinguished by arrow symbols on their cases. Some polarized capacitors have their polarity designated by marking the positive terminal. The large, 20,000 µF electrolytic unit shown in the upright position has its positive (+) terminal labeled with a “plus” mark. Ceramic, mylar, plastic film, and air capacitors do not have polarity markings, because those types are nonpolarized (they are not polarity sensitive). Capacitors are very common components in electronic circuits. Take a close look at the following photograph—every component marked with a “C” designation on the printed circuit board is a capacitor: Some of the capacitors shown on this circuit board are standard electrolytic: C30 (top of board, center) and C36 (left side, 1/3 from the top). Some others are a special kind of electrolytic capacitor called tantalum, because this is the type of metal used to make the plates. Tantalum capacitors have relatively high capacitance for their physical size. The following capacitors on the circuit board shown above are tantalum: C14 (just to the lower-left of C30), C19 (directly below R10, which is below C30), C24 (lower-left corner of board), and C22 (lower-right). Examples of even smaller capacitors can be seen in this photograph: The capacitors on this circuit board are “surface mount devices” as are all the resistors, for reasons of saving space. Following component labeling convention, the capacitors can be identified by labels beginning with the letter “C”.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/14%3A_Magnetism_and_Electromagnetism/14.01%3A_Permanent_Magnets.txt	princeton-nlp/TextbookChapters	• 14.1: Permanent Magnets Centuries ago, it was discovered that certain types of mineral rock possessed unusual properties of attraction to the metal iron. One particular mineral, called lodestone, or magnetite, is found mentioned in very old historical records (about 2500 years ago in Europe, and much earlier in the Far East) as a subject of curiosity. Later, it was employed in the aid of navigation, as it was found that a piece of this unusual rock would tend to orient itself in a north-south direction if left free to • 14.2: Electromagnetism The discovery of the relationship between magnetism and electricity was, like so many other scientific discoveries, stumbled upon almost by accident. The Danish physicist Hans Christian Oersted was lecturing one day in 1820 on the possibility of electricity and magnetism being related to one another, and in the process demonstrated it conclusively by experiment in front of his whole class! By passing an electric current through a metal wire suspended above a magnetic compass, Oersted was able to • 14.3: Magnetic Units of Measurement If the burden of two systems of measurement for common quantities (English vs. metric) throws your mind into confusion, this is not the place for you! Due to an early lack of standardization in the science of magnetism, we have been plagued with no less than three complete systems of measurement for magnetic quantities. • 14.4: Permeability and Saturation • 14.5: Electromagnetic Induction While Oersted’s surprising discovery of electromagnetism paved the way for more practical applications of electricity, it was Michael Faraday who gave us the key to the practical generation of electricity: electromagnetic induction. Faraday discovered that a voltage would be generated across a length of wire if that wire was exposed to a perpendicular magnetic field flux of changing intensity. • 14.6: Mutual Inductance If two coils of wire are brought into close proximity with each other so the magnetic field from one links with the other, a voltage will be generated in the second coil as a result. This is called mutual inductance: when voltage impressed upon one coil induces a voltage in another. 14: Magnetism and Electromagnetism Unlike electric charges (such as those observed when amber is rubbed against cloth), magnetic objects possessed two poles of opposite effect, denoted “north” and “south” after their self-orientation to the earth. As Peregrinus found, it was impossible to isolate one of these poles by itself by cutting a piece of lodestone in half: each resulting piece possessed its own pair of poles: Like electric charges, there were only two types of poles to be found: north and south (by analogy, positive and negative). Just as with electric charges, same poles repel one another, while opposite poles attract. This force, like that caused by static electricity, extended itself invisibly over space, and could even pass through objects such as paper and wood with little effect upon strength. The philosopher-scientist Rene Descartes noted that this invisible “field” could be mapped by placing a magnet underneath a flat piece of cloth or wood and sprinkling iron filings on top. The filings will align themselves with the magnetic field, “mapping” its shape. The result shows how the field continues unbroken from one pole of a magnet to the other: As with any kind of field (electric, magnetic, gravitational), the total quantity, or effect, of the field is referred to as a flux, while the “push” causing the flux to form in space is called a force. Michael Faraday coined the term “tube” to refer to a string of magnetic flux in space (the term “line” is more commonly used now). Indeed, the measurement of magnetic field flux is often defined in terms of the number of flux lines, although it is doubtful that such fields exist in individual, discrete lines of constant value. Modern theories of magnetism maintain that a magnetic field is produced by an electric charge in motion, and thus it is theorized that the magnetic field of a so-called “permanent” magnets such as lodestone is the result of electrons within the atoms of iron spinning uniformly in the same direction. Whether or not the electrons in a material’s atoms are subject to this kind of uniform spinning is dictated by the atomic structure of the material (not unlike how electrical conductivity is dictated by the electron binding in a material’s atoms). Thus, only certain types of substances react with magnetic fields, and even fewer have the ability to permanently sustain a magnetic field. Iron is one of those types of substances that readily magnetizes. If a piece of iron is brought near a permanent magnet, the electrons within the atoms in the iron orient their spins to match the magnetic field force produced by the permanent magnet, and the iron becomes “magnetized.” The iron will magnetize in such a way as to incorporate the magnetic flux lines into its shape, which attracts it toward the permanent magnet, no matter which pole of the permanent magnet is offered to the iron: The previously unmagnetized iron becomes magnetized as it is brought closer to the permanent magnet. No matter what pole of the permanent magnet is extended toward the iron, the iron will magnetize in such a way as to be attracted toward the magnet: Referencing the natural magnetic properties of iron (Latin = “ferrum”), a ferromagnetic material is one that readily magnetizes (its constituent atoms easily orient their electron spins to conform to an external magnetic field force). All materials are magnetic to some degree, and those that are not considered ferromagnetic (easily magnetized) are classified as either paramagnetic (slightly magnetic) or diamagnetic(tend to exclude magnetic fields). Of the two, diamagnetic materials are the strangest. In the presence of an external magnetic field, they actually become slightly magnetized in the opposite direction, so as to repel the external field! If a ferromagnetic material tends to retain its magnetization after an external field is removed, it is said to have good retentivity. This, of course, is a necessary quality for a permanent magnet. Review • Lodestone (also called Magnetite) is a naturally-occurring “permanent” magnet mineral. By “permanent,” it is meant that the material maintains a magnetic field with no external help. The characteristic of any magnetic material to do so is called retentivity. • Ferromagnetic materials are easily magnetized. • Paramagnetic materials are magnetized with more difficulty. • Diamagnetic materials actually tend to repel external magnetic fields by magnetizing in the opposite direction.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/14%3A_Magnetism_and_Electromagnetism/14.02%3A_Electromagnetism.txt	princeton-nlp/TextbookChapters	Detailed experiments showed that the magnetic field produced by an electric current is always oriented perpendicular to the direction of flow. A simple method of showing this relationship is called the left-hand rule. Simply stated, the left-hand rule says that the magnetic flux lines produced by a current-carrying wire will be oriented the same direction as the curled fingers of a person’s left hand (in the “hitchhiking” position), with the thumb pointing in the direction of electron flow: The magnetic field encircles this straight piece of current-carrying wire, the magnetic flux lines having no definite “north” or “south’ poles. While the magnetic field surrounding a current-carrying wire is indeed interesting, it is quite weak for common amounts of current, able to deflect a compass needle and not much more. To create a stronger magnetic field force (and consequently, more field flux) with the same amount of electric current, we can wrap the wire into a coil shape, where the circling magnetic fields around the wire will join to create a larger field with a definite magnetic (north and south) polarity: The amount of magnetic field force generated by a coiled wire is proportional to the current through the wire multiplied by the number of “turns” or “wraps” of wire in the coil. This field force is called magnetomotive force (mmf), and is very much analogous to electromotive force (E) in an electric circuit. An electromagnet is a piece of wire intended to generate a magnetic field with the passage of electric current through it. Though all current-carrying conductors produce magnetic fields, an electromagnet is usually constructed in such a way as to maximize the strength of the magnetic field it produces for a special purpose. Electromagnets find frequent application in research, industry, medical, and consumer products. As an electrically-controllable magnet, electromagnets find application in a wide variety of “electromechanical” devices: machines that effect mechanical force or motion through electrical power. Perhaps the most obvious example of such a machine is the electric motor. Another example is the relay, an electrically-controlled switch. If a switch contact mechanism is built so that it can be actuated (opened and closed) by the application of a magnetic field, and an electromagnet coil is placed in the near vicinity to produce that requisite field, it will be possible to open and close the switch by the application of a current through the coil. In effect, this gives us a device that enables electricity to control electricity: Relays can be constructed to actuate multiple switch contacts, or operate them in “reverse” (energizing the coil will open the switch contact, and unpowering the coil will allow it to spring closed again). Review • When electrons flow through a conductor, a magnetic field will be produced around that conductor. • The left-hand rule states that the magnetic flux lines produced by a current-carrying wire will be oriented the same direction as the curled fingers of a person’s left hand (in the “hitchhiking” position), with the thumb pointing in the direction of electron flow. • The magnetic field force produced by a current-carrying wire can be greatly increased by shaping the wire into a coil instead of a straight line. If wound in a coil shape, the magnetic field will be oriented along the axis of the coil’s length. • The magnetic field force produced by an electromagnet (called the magnetomotive force, or mmf), is proportional to the product (multiplication) of the current through the electromagnet and the number of complete coil “turns” formed by the wire.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/14%3A_Magnetism_and_Electromagnetism/14.03%3A_Magnetic_Units_of_Measurement.txt	princeton-nlp/TextbookChapters	First, we need to become acquainted with the various quantities associated with magnetism. There are quite a few more quantities to be dealt with in magnetic systems than for electrical systems. With electricity, the basic quantities are Voltage (E), Current (I), Resistance (R), and Power (P). The first three are related to one another by Ohm’s Law (E=IR ; I=E/R ; R=E/I), while Power is related to voltage, current, and resistance by Joule’s Law (P=IE ; P=I2R ; P=E2/R). With magnetism, we have the following quantities to deal with: Magnetomotive Force—The quantity of magnetic field force, or “push.” Analogous to electric voltage (electromotive force). Field Flux—The quantity of total field effect, or “substance” of the field. Analogous to electric current. Field Intensity—The amount of field force (mmf) distributed over the length of the electromagnet. Sometimes referred to as Magnetizing Force. Flux Density—The amount of magnetic field flux concentrated in a given area. Reluctance—The opposition to magnetic field flux through a given volume of space or material. Analogous to electrical resistance. Permeability—The specific measure of a material’s acceptance of magnetic flux, analogous to the specific resistance of a conductive material (ρ), except inverse (greater permeability means easier passage of magnetic flux, whereas greater specific resistance means more difficult passage of electric current). But wait . . . the fun is just beginning! Not only do we have more quantities to keep track of with magnetism than with electricity, but we have several different systems of unit measurement for each of these quantities. As with common quantities of length, weight, volume, and temperature, we have both English and metric systems. However, there is actually more than one metric system of units, and multiple metric systems are used in magnetic field measurements! One is called the cgs, which stands for Centimeter-Gram-Second, denoting the root measures upon which the whole system is based. The other was originally known as the mks system, which stood for Meter-Kilogram-Second, which was later revised into another system, called rmks, standing for Rationalized Meter-Kilogram-Second. This ended up being adopted as an international standard and renamed SI (Systeme International). And yes, the µ symbol is really the same as the metric prefix “micro.” I find this especially confusing, using the exact same alphabetical character to symbolize both a specific quantity and a general metric prefix! As you might have guessed already, the relationship between field force, field flux, and reluctance is much the same as that between the electrical quantities of electromotive force (E), current (I), and resistance (R). This provides something akin to an Ohm’s Law for magnetic circuits: And, given that permeability is inversely analogous to specific resistance, the equation for finding the reluctance of a magnetic material is very similar to that for finding the resistance of a conductor: In either case, a longer piece of material provides a greater opposition, all other factors being equal. Also, a larger cross-sectional area makes for less opposition, all other factors being equal. The major caveat here is that the reluctance of a material to magnetic flux actually changes with the concentration of flux going through it. This makes the “Ohm’s Law” for magnetic circuits nonlinear and far more difficult to work with than the electrical version of Ohm’s Law. It would be analogous to having a resistor that changed resistance as the current through it varied (a circuit composed of varistors instead of resistors).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/14%3A_Magnetism_and_Electromagnetism/14.04%3A_Permeability_and_Saturation.txt	princeton-nlp/TextbookChapters	The nonlinearity of material permeability may be graphed for better understanding. We’ll place the quantity of field intensity (H), equal to field force (mmf) divided by the length of the material, on the horizontal axis of the graph. On the vertical axis, we’ll place the quantity of flux density (B), equal to total flux divided by the cross-sectional area of the material. We will use the quantities of field intensity (H) and flux density (B) instead of field force (mmf) and total flux (Φ) so that the shape of our graph remains independent of the physical dimensions of our test material. What we’re trying to do here is show a mathematical relationship between field force and flux for any chunk of a particular substance, in the same spirit as describing a material’s specific resistance in ohm-cmil/ft instead of its actual resistance in ohms. This is called the normal magnetization curve, or B-H curve, for any particular material. Notice how the flux density for any of the above materials (cast iron, cast steel, and sheet steel) levels off with increasing amounts of field intensity. This effect is known as saturation. When there is little applied magnetic force (low H), only a few atoms are in alignment, and the rest are easily aligned with additional force. However, as more flux gets crammed into the same cross-sectional area of a ferromagnetic material, fewer atoms are available within that material to align their electrons with additional force, and so it takes more and more force (H) to get less and less “help” from the material in creating more flux density (B). To put this in economic terms, we’re seeing a case of diminishing returns (B) on our investment (H). Saturation is a phenomenon limited to iron-core electromagnets. Air-core electromagnets don’t saturate, but on the other hand they don’t produce nearly as much magnetic flux as a ferromagnetic core for the same number of wire turns and current. Another quirk to confound our analysis of magnetic flux versus force is the phenomenon of magnetic hysteresis. As a general term, hysteresis means a lag between input and output in a system upon a change in direction. Anyone who’s ever driven an old automobile with “loose” steering knows what hysteresis is: to change from turning left to turning right (or vice versa), you have to rotate the steering wheel an additional amount to overcome the built-in “lag” in the mechanical linkage system between the steering wheel and the front wheels of the car. In a magnetic system, hysteresis is seen in a ferromagnetic material that tends to stay magnetized after an applied field force has been removed (see “retentivity” in the first section of this chapter), if the force is reversed in polarity. Let’s use the same graph again, only extending the axes to indicate both positive and negative quantities. First we’ll apply an increasing field force (current through the coils of our electromagnet). We should see the flux density increase (go up and to the right) according to the normal magnetization curve: Next, we’ll stop the current going through the coil of the electromagnet and see what happens to the flux, leaving the first curve still on the graph: Due to the retentivity of the material, we still have a magnetic flux with no applied force (no current through the coil). Our electromagnet core is acting as a permanent magnet at this point. Now we will slowly apply the same amount of magnetic field force in the opposite direction to our sample: The flux density has now reached a point equivalent to what it was with a full positive value of field intensity (H), except in the negative, or opposite, direction. Let’s stop the current going through the coil again and see how much flux remains: Once again, due to the natural retentivity of the material, it will hold a magnetic flux with no power applied to the coil, except this time its in a direction opposite to that of the last time we stopped current through the coil. If we re-apply power in a positive direction again, we should see the flux density reach its prior peak in the upper-right corner of the graph again: The “S”-shaped curve traced by these steps form what is called the hysteresis curve of a ferromagnetic material for a given set of field intensity extremes (-H and +H). If this doesn’t quite make sense, consider a hysteresis graph for the automobile steering scenario described earlier, one graph depicting a “tight” steering system and one depicting a “loose” system: Just as in the case of automobile steering systems, hysteresis can be a problem. If you’re designing a system to produce precise amounts of magnetic field flux for given amounts of current, hysteresis may hinder this design goal (due to the fact that the amount of flux density would depend on the current and how strongly it was magnetized before!). Similarly, a loose steering system is unacceptable in a race car, where precise, repeatable steering response is a necessity. Also, having to overcome prior magnetization in an electromagnet can be a waste of energy if the current used to energize the coil is alternating back and forth (AC). The area within the hysteresis curve gives a rough estimate of the amount of this wasted energy. Other times, magnetic hysteresis is a desirable thing. Such is the case when magnetic materials are used as a means of storing information (computer disks, audio and video tapes). In these applications, it is desirable to be able to magnetize a speck of iron oxide (ferrite) and rely on that material’s retentivity to “remember” its last magnetized state. Another productive application for magnetic hysteresis is in filtering high-frequency electromagnetic “noise” (rapidly alternating surges of voltage) from signal wiring by running those wires through the middle of a ferrite ring. The energy consumed in overcoming the hysteresis of ferrite attenuates the strength of the “noise” signal. Interestingly enough, the hysteresis curve of ferrite is quite extreme: Review • The permeability of a material changes with the amount of magnetic flux forced through it. • The specific relationship of force to flux (field intensity H to flux density B) is graphed in a form called the normal magnetization curve. • It is possible to apply so much magnetic field force to a ferromagnetic material that no more flux can be crammed into it. This condition is known as magnetic saturation. • When the retentivity of a ferromagnetic substance interferes with its re-magnetization in the opposite direction, a condition known as hysteresis occurs.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/14%3A_Magnetism_and_Electromagnetism/14.05%3A_Electromagnetic_Induction.txt	princeton-nlp/TextbookChapters	An easy way to create a magnetic field of changing intensity is to move a permanent magnet next to a wire or coil of wire. Remember: the magnetic field must increase or decrease in intensity perpendicular to the wire (so that the lines of flux “cut across” the conductor), or else no voltage will be induced: Faraday was able to mathematically relate the rate of change of the magnetic field flux with induced voltage (note the use of a lower-case letter “e” for voltage. This refers to instantaneous voltage, or voltage at a specific point in time, rather than a steady, stable voltage.): The “d” terms are standard calculus notation, representing rate-of-change of flux over time. “N” stands for the number of turns, or wraps, in the wire coil (assuming that the wire is formed in the shape of a coil for maximum electromagnetic efficiency). This phenomenon is put into obvious practical use in the construction of electrical generators, which use mechanical power to move a magnetic field past coils of wire to generate voltage. However, this is by no means the only practical use for this principle. If we recall that the magnetic field produced by a current-carrying wire was always perpendicular to that wire, and that the flux intensity of that magnetic field varied with the amount of current through it, we can see that a wire is capable of inducing a voltage along its own length simply due to a change in current through it. This effect is called self-induction: a changing magnetic field produced by changes in current through a wire inducing voltage along the length of that same wire. If the magnetic field flux is enhanced by bending the wire into the shape of a coil, and/or wrapping that coil around a material of high permeability, this effect of self-induced voltage will be more intense. A device constructed to take advantage of this effect is called an inductor, and will be discussed in greater detail in the next chapter. Review • A magnetic field of changing intensity perpendicular to a wire will induce a voltage along the length of that wire. The amount of voltage induced depends on the rate of change of the magnetic field flux and the number of turns of wire (if coiled) exposed to the change in flux. • Faraday’s equation for induced voltage: e = N(dΦ/dt) • A current-carrying wire will experience an induced voltage along its length if the current changes (thus changing the magnetic field flux perpendicular to the wire, thus inducing voltage according to Faraday’s formula). A device built specifically to take advantage of this effect is called an inductor. 14.06: Mutual Inductance A device specifically designed to produce the effect of mutual inductance between two or more coils is called a transformer. The device shown in the above photograph is a kind of transformer, with two concentric wire coils. It is actually intended as a precision standard unit for mutual inductance, but for the purposes of illustrating what the essence of a transformer is, it will suffice. The two wire coils can be distinguished from each other by color: the bulk of the tube’s length is wrapped in green-insulated wire (the first coil) while the second coil (wire with bronze-colored insulation) stands in the middle of the tube’s length. The wire ends run down to connection terminals at the bottom of the unit. Most transformer units are not built with their wire coils exposed like this. Because magnetically-induced voltage only happens when the magnetic field flux is changing in strength relative to the wire, mutual inductance between two coils can only happen with alternating (changing—AC) voltage, and not with direct (steady—DC) voltage. The only applications for mutual inductance in a DC system is where some means is available to switch power on and off to the coil (thus creating a pulsing DC voltage), the induced voltage peaking at every pulse. A very useful property of transformers is the ability to transform voltage and current levels according to a simple ratio, determined by the ratio of input and output coil turns. If the energized coil of a transformer is energized by an AC voltage, the amount of AC voltage induced in the unpowered coil will be equal to the input voltage multiplied by the ratio of output to input wire turns in the coils. Conversely, the current through the windings of the output coil compared to the input coil will follow the opposite ratio: if the voltage is increased from input coil to output coil, the current will be decreased by the same proportion. This action of the transformer is analogous to that of mechanical gear, belt sheave, or chain sprocket ratios: A transformer designed to output more voltage than it takes in across the input coil is called a “step-up” transformer, while one designed to do the opposite is called a “step-down,” in reference to the transformation of voltage that takes place. The current through each respective coil, of course, follows the exact opposite proportion. Review • Mutual inductance is where the magnetic field generated by a coil of wire induces voltage in an adjacent coil of wire. • A transformer is a device constructed of two or more coils in close proximity to each other, with the express purpose of creating a condition of mutual inductance between the coils. • Transformers only work with changing voltages, not steady voltages. Thus, they may be classified as an AC device and not a DC device.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/15%3A_Inductors/15.01%3A_Magnetic_Fields_and_Inductance.txt	princeton-nlp/TextbookChapters	• 15.1: Magnetic Fields and Inductance Whenever electrons flow through a conductor, a magnetic field will develop around that conductor. This effect is called electromagnetism. Magnetic fields effect the alignment of electrons in an atom, and can cause physical force to develop between atoms across space just as with electric fields developing force between electrically charged particles. Like electric fields, magnetic fields can occupy completely empty space, and affect matter at a distance. • 15.2: Inductors and Calculus • 15.3: Factors Affecting Inductance There are four basic factors of inductor construction determining the amount of inductance created. These factors all dictate inductance by affecting how much magnetic field flux will develop for a given amount of magnetic field force (current through the inductor’s wire coil): • 15.4: Series and Parallel Inductors When inductors are connected in series, the total inductance is the sum of the individual inductors’ inductances. When inductors are connected in parallel, the total inductance is less than any one of the parallel inductors’ inductances. • 15.5: Practical Considerations - Inductors Inductors, like all electrical components, have limitations which must be respected for the sake of reliability and proper circuit operation. 15: Inductors Field Force and Field Flux Fields have two measures: a field force and a field flux. The field force is the amount of “push” that a field exerts over a certain distance. The field flux is the total quantity, or effect, of the field through space. Field force and flux are roughly analogous to voltage (“push”) and current (flow) through a conductor, respectively, although field flux can exist in totally empty space (without the motion of particles such as electrons) whereas current can only take place where there are free electrons to move. Field flux can be opposed in space, just as the flow of electrons can be opposed by resistance. The amount of field flux that will develop in space is proportional to the amount of field force applied, divided by the amount of opposition to flux. Just as the type of conducting material dictates that conductor’s specific resistance to electric current, the type of material occupying the space through which a magnetic field force is impressed dictates the specific opposition to magnetic field flux. Whereas an electric field flux between two conductors allows for an accumulation of free electron charge within those conductors, a magnetic field flux allows for a certain “inertia” to accumulate in the flow of electrons through the conductor producing the field. Stronger Magnetic Fields With Inductors Inductors are components designed to take advantage of this phenomenon by shaping the length of conductive wire in the form of a coil. This shape creates a stronger magnetic field than what would be produced by a straight wire. Some inductors are formed with wire wound in a self-supporting coil. Others wrap the wire around a solid core material of some type. Sometimes the core of an inductor will be straight, and other times it will be joined in a loop (square, rectangular, or circular) to fully contain the magnetic flux. These design options all have an effect on the performance and characteristics of inductors. The schematic symbol for an inductor, like the capacitor, is quite simple, being little more than a coil symbol representing the coiled wire. Although a simple coil shape is the generic symbol for any inductor, inductors with cores are sometimes distinguished by the addition of parallel lines to the axis of the coil. A newer version of the inductor symbol dispenses with the coil shape in favor of several “humps” in a row: As the electric current produces a concentrated magnetic field around the coil, this field flux equates to a storage of energy representing the kinetic motion of the electrons through the coil. The more current in the coil, the stronger the magnetic field will be, and the more energy the inductor will store. Because inductors store the kinetic energy of moving electrons in the form of a magnetic field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. Energy storage in an inductor is a function of the amount of current through it. An inductor’s ability to store energy as a function of current results in a tendency to try to maintain current at a constant level. In other words, inductors tend to resist changes in current. When current through an inductor is increased or decreased, the inductor “resists” the change by producing a voltage between its leads in opposing polarity to the change. To store more energy in an inductor, the current through it must be increased. This means that its magnetic field must increase in strength, and that change in field strength produces the corresponding voltage according to the principle of electromagnetic self-induction. Conversely, to release energy from an inductor, the current through it must be decreased. This means that the inductor’s magnetic field must decrease in strength, and that change in field strength self-induces a voltage drop of just the opposite polarity. Just as Isaac Newton’s first Law of Motion (“an object in motion tends to stay in motion; an object at rest tends to stay at rest”) describes the tendency of a mass to oppose changes in velocity, we can state an inductor’s tendency to oppose changes in current as such: “Electrons moving through an inductor tend to stay in motion; electrons at rest in an inductor tend to stay at rest.” Hypothetically, an inductor left short-circuited will maintain a constant rate of current through it with no external assistance: Practically speaking, however, the ability for an inductor to self-sustain current is realized only with superconductive wire, as the wire resistance in any normal inductor is enough to cause current to decay very quickly with no external source of power. When the current through an inductor is increased, it drops a voltage opposing the direction of electron flow, acting as a power load. In this condition, the inductor is said to be charging, because there is an increasing amount of energy being stored in its magnetic field. Note the polarity of the voltage with regard to the direction of current: Conversely, when the current through the inductor is decreased, it drops a voltage aiding the direction of electron flow, acting as a power source. In this condition, the inductor is said to be discharging, because its store of energy is decreasing as it releases energy from its magnetic field to the rest of the circuit. Note the polarity of the voltage with regard to the direction of current. If a source of electric power is suddenly applied to an unmagnetized inductor, the inductor will initially resist the flow of electrons by dropping the full voltage of the source. As current begins to increase, a stronger and stronger magnetic field will be created, absorbing energy from the source. Eventually the current reaches a maximum level, and stops increasing. At this point, the inductor stops absorbing energy from the source, and is dropping minimum voltage across its leads, while the current remains at a maximum level. As an inductor stores more energy, its current level increases, while its voltage drop decreases. Note that this is precisely the opposite of capacitor behavior, where the storage of energy results in an increased voltage across the component! Whereas capacitors store their energy charge by maintaining a static voltage, inductors maintain their energy “charge” by maintaining a steady current through the coil. The type of material the wire is coiled around greatly impacts the strength of the magnetic field flux (and therefore the amount of stored energy) generated for any given amount of current through the coil. Coil cores made of ferromagnetic materials (such as soft iron) will encourage stronger field fluxes to develop with a given field force than nonmagnetic substances such as aluminum or air. What is Inductance? The measure of an inductor’s ability to store energy for a given amount of current flow is called inductance. Not surprisingly, inductance is also a measure of the intensity of opposition to changes in current (exactly how much self-induced voltage will be produced for a given rate of change of current). Inductance is symbolically denoted with a capital “L,” and is measured in the unit of the Henry, abbreviated as “H.” Choke Vs. Inductor An obsolete name for an inductor is choke, so called for its common usage to block (“choke”) high-frequency AC signals in radio circuits. Another name for an inductor, still used in modern times, is reactor, especially when used in large power applications. Both of these names will make more sense after you’ve studied alternating current (AC) circuit theory, and especially a principle known as inductive reactance. Review • Inductors react against changes in current by dropping voltage in the polarity necessary to oppose the change. • When an inductor is faced with an increasing current, it acts as a load: dropping voltage as it absorbs energy (negative on the current entry side and positive on the current exit side, like a resistor). • When an inductor is faced with a decreasing current, it acts as a source: creating voltage as it releases stored energy (positive on the current entry side and negative on the current exit side, like a battery). • The ability of an inductor to store energy in the form of a magnetic field (and consequently to oppose changes in current) is called inductance. It is measured in the unit of the Henry (H). • Inductors used to be commonly known by another term: choke. In large power applications, they are sometimes referred to as reactors.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/15%3A_Inductors/15.02%3A_Inductors_and_Calculus.txt	princeton-nlp/TextbookChapters	Inductors do not have a stable “resistance” as conductors do. However, there is a definite mathematical relationship between voltage and current for an inductor, as follows: You should recognize the form of this equation from the capacitor chapter. It relates one variable (in this case, inductor voltage drop) to a rate of change of another variable (in this case, inductor current). Both voltage (v) and rate of current change (di/dt) are instantaneous: that is, in relation to a specific point in time, thus the lower-case letters “v” and “i”. As with the capacitor formula, it is convention to express instantaneous voltage as v rather than e, but using the latter designation would not be wrong. Current rate-of-change (di/dt) is expressed in units of amps per second, a positive number representing an increase and a negative number representing a decrease. Like a capacitor, an inductor’s behavior is rooted in the variable of time. Aside from any resistance intrinsic to an inductor’s wire coil (which we will assume is zero for the sake of this section), the voltage dropped across the terminals of an inductor is purely related to how quickly its current changes over time. Suppose we were to connect a perfect inductor (one having zero ohms of wire resistance) to a circuit where we could vary the amount of current through it with a potentiometer connected as a variable resistor: If the potentiometer mechanism remains in a single position (wiper is stationary), the series-connected ammeter will register a constant (unchanging) current, and the voltmeter connected across the inductor will register 0 volts. In this scenario, the instantaneous rate of current change (di/dt) is equal to zero, because the current is stable. The equation tells us that with 0 amps per second change for a di/dt, there must be zero instantaneous voltage (v) across the inductor. From a physical perspective, with no current change, there will be a steady magnetic field generated by the inductor. With no change in magnetic flux (dΦ/dt = 0 Webers per second), there will be no voltage dropped across the length of the coil due to induction. If we move the potentiometer wiper slowly in the “up” direction, its resistance from end to end will slowly decrease. This has the effect of increasing current in the circuit, so the ammeter indication should be increasing at a slow rate: Assuming that the potentiometer wiper is being moved such that the rate of current increase through the inductor is steady, the di/dt term of the formula will be a fixed value. This fixed value, multiplied by the inductor’s inductance in Henrys (also fixed), results in a fixed voltage of some magnitude. From a physical perspective, the gradual increase in current results in a magnetic field that is likewise increasing. This gradual increase in magnetic flux causes a voltage to be induced in the coil as expressed by Michael Faraday’s induction equation e = N(dΦ/dt). This self-induced voltage across the coil, as a result of a gradual change in current magnitude through the coil, happens to be of a polarity that attempts to oppose the change in current. In other words, the induced voltage polarity resulting from an increase in current will be oriented in such a way as to push against the direction of current, to try to keep the current at its former magnitude. This phenomenon exhibits a more general principle of physics known as Lenz’s Law, which states that an induced effect will always be opposed to the cause producing it. In this scenario, the inductor will be acting as a load, with the negative side of the induced voltage on the end where electrons are entering, and the positive side of the induced voltage on the end where electrons are exiting. Changing the rate of current increase through the inductor by moving the potentiometer wiper “up” at different speeds results in different amounts of voltage being dropped across the inductor, all with the same polarity (opposing the increase in current): Here again we see the derivative function of calculus exhibited in the behavior of an inductor. In calculus terms, we would say that the induced voltage across the inductor is the derivative of the current through the inductor: that is, proportional to the current’s rate-of-change with respect to time. Reversing the direction of wiper motion on the potentiometer (going “down” rather than “up”) will result in its end-to-end resistance increasing. This will result in circuit current decreasing (a negative figure for di/dt). The inductor, always opposing any change in current, will produce a voltage drop opposed to the direction of change: How much voltage the inductor will produce depends, of course, on how rapidly the current through it is decreased. As described by Lenz’s Law, the induced voltage will be opposed to the change in current. With a decreasing current, the voltage polarity will be oriented so as to try to keep the current at its former magnitude. In this scenario, the inductor will be acting as a source, with the negative side of the induced voltage on the end where electrons are exiting, and the positive side of the induced voltage on the end where electrons are entering. The more rapidly current is decreased, the more voltage will be produced by the inductor, in its release of stored energy to try to keep current constant. Again, the amount of voltage across a perfect inductor is directly proportional to the rate of current change through it. The only difference between the effects of a decreasing current and an increasing current is the polarity of the induced voltage. For the same rate of current change over time, either increasing or decreasing, the voltage magnitude (volts) will be the same. For example, a di/dt of -2 amps per second will produce the same amount of induced voltage drop across an inductor as a di/dt of +2 amps per second, just in the opposite polarity. If current through an inductor is forced to change very rapidly, very high voltages will be produced. Consider the following circuit: In this circuit, a lamp is connected across the terminals of an inductor. A switch is used to control current in the circuit, and power is supplied by a 6 volt battery. When the switch is closed, the inductor will briefly oppose the change in current from zero to some magnitude, but will drop only a small amount of voltage. It takes about 70 volts to ionize the neon gas inside a neon bulb like this, so the bulb cannot be lit on the 6 volts produced by the battery, or the low voltage momentarily dropped by the inductor when the switch is closed: When the switch is opened, however, it suddenly introduces an extremely high resistance into the circuit (the resistance of the air gap between the contacts). This sudden introduction of high resistance into the circuit causes the circuit current to decrease almost instantly. Mathematically, the di/dt term will be a very large negative number. Such a rapid change of current (from some magnitude to zero in very little time) will induce a very high voltage across the inductor, oriented with negative on the left and positive on the right, in an effort to oppose this decrease in current. The voltage produced is usually more than enough to light the neon lamp, if only for a brief moment until the current decays to zero: For maximum effect, the inductor should be sized as large as possible (at least 1 Henry of inductance).
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/15%3A_Inductors/15.03%3A_Factors_Affecting_Inductance.txt	princeton-nlp/TextbookChapters	NUMBER OF WIRE WRAPS, OR “TURNS” IN THE COIL: All other factors being equal, a greater number of turns of wire in the coil results in greater inductance; fewer turns of wire in the coil results in less inductance. Explanation: More turns of wire means that the coil will generate a greater amount of magnetic field force (measured in amp-turns!), for a given amount of coil current. COIL AREA: All other factors being equal, greater coil area (as measured looking lengthwise through the coil, at the cross-section of the core) results in greater inductance; less coil area results in less inductance. Explanation: Greater coil area presents less opposition to the formation of magnetic field flux, for a given amount of field force (amp-turns). COIL LENGTH: All other factors being equal, the longer the coil’s length, the less inductance; the shorter the coil’s length, the greater the inductance. Explanation: A longer path for the magnetic field flux to take results in more opposition to the formation of that flux for any given amount of field force (amp-turns). CORE MATERIAL: All other factors being equal, the greater the magnetic permeability of the core which the coil is wrapped around, the greater the inductance; the less the permeability of the core, the less the inductance. Explanation: A core material with greater magnetic permeability results in greater magnetic field flux for any given amount of field force (amp-turns). An approximation of inductance for any coil of wire can be found with this formula: It must be understood that this formula yields approximate figures only. One reason for this is the fact that permeability changes as the field intensity varies (remember the nonlinear “B/H” curves for different materials). Obviously, if permeability (µ) in the equation is unstable, then the inductance (L) will also be unstable to some degree as the current through the coil changes in magnitude. If the hysteresis of the core material is significant, this will also have strange effects on the inductance of the coil. Inductor designers try to minimize these effects by designing the core in such a way that its flux density never approaches saturation levels, and so the inductor operates in a more linear portion of the B/H curve. If an inductor is designed so that any one of these factors may be varied at will, its inductance will correspondingly vary. Variable inductors are usually made by providing a way to vary the number of wire turns in use at any given time, or by varying the core material (a sliding core that can be moved in and out of the coil). An example of the former design is shown in this photograph: This unit uses sliding copper contacts to tap into the coil at different points along its length. The unit shown happens to be an air-core inductor used in early radio work. A fixed-value inductor is shown in the next photograph, another antique air-core unit built for radios. The connection terminals can be seen at the bottom, as well as the few turns of relatively thick wire: Here is another inductor (of greater inductance value), also intended for radio applications. Its wire coil is wound around a white ceramic tube for greater rigidity: Inductors can also be made very small for printed circuit board applications. Closely examine the following photograph and see if you can identify two inductors near each other: The two inductors on this circuit board are labeled L1 and L2, and they are located to the right-center of the board. Two nearby components are R3 (a resistor) and C16 (a capacitor). These inductors are called “toroidal” because their wire coils are wound around donut-shaped (“torus”) cores. Like resistors and capacitors, inductors can be packaged as “surface mount devices” as well. The following photograph shows just how small an inductor can be when packaged as such: A pair of inductors can be seen on this circuit board, to the right and center, appearing as small black chips with the number “100” printed on both. The upper inductor’s label can be seen printed on the green circuit board as L5. Of course these inductors are very small in inductance value, but it demonstrates just how tiny they can be manufactured to meet certain circuit design needs.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_I_-_Direct_Current_(Kuphaldt)/15%3A_Inductors/15.04%3A_Series_and_Parallel_Inductors.txt	princeton-nlp/TextbookChapters	When inductors are connected in series, the total inductance is the sum of the individual inductors’ inductances. To understand why this is so, consider the following: the definitive measure of inductance is the amount of voltage dropped across an inductor for a given rate of current change through it. If inductors are connected together in series (thus sharing the same current, and seeing the same rate of change in current), then the total voltage dropped as the result of a change in current will be additive with each inductor, creating a greater total voltage than either of the individual inductors alone. Greater voltage for the same rate of change in current means greater inductance. Thus, the total inductance for series inductors is more than any one of the individual inductors’ inductances. The formula for calculating the series total inductance is the same form as for calculating series resistances: $\underbrace{L_{total} = L_1 + L_2 + \cdots + L_m}_{\text{Series Inductaces}}$ When inductors are connected in parallel, the total inductance is less than any one of the parallel inductors’ inductances. Again, remember that the definitive measure of inductance is the amount of voltage dropped across an inductor for a given rate of current change through it. Since the current through each parallel inductor will be a fraction of the total current, and the voltage across each parallel inductor will be equal, a change in total current will result in less voltage dropped across the parallel array than for any one of the inductors considered separately. In other words, there will be less voltage dropped across parallel inductors for a given rate of change in current than for any of those inductors considered separately, because total current divides among parallel branches. Less voltage for the same rate of change in current means less inductance. Thus, the total inductance is less than any one of the individual inductors’ inductances. The formula for calculating the parallel total inductance is the same form as for calculating parallel resistances: Review • Inductances add in series. • Inductances diminish in parallel. 15.05: Practical Considerations - Inductors Rated current: Since inductors are constructed of coiled wire, and any wire will be limited in its current-carrying capacity by its resistance and ability to dissipate heat, you must pay attention to the maximum current allowed through an inductor. Equivalent circuit: Since inductor wire has some resistance, and circuit design constraints typically demand the inductor be built to the smallest possible dimensions, there is no such thing as a “perfect” inductor. Inductor coil wire usually presents a substantial amount of series resistance, and the close spacing of wire from one coil turn to another (separated by insulation) may present measurable amounts of stray capacitance to interact with its purely inductive characteristics. Unlike capacitors, which are relatively easy to manufacture with negligible stray effects, inductors are difficult to find in “pure” form. In certain applications, these undesirable characteristics may present significant engineering problems. Inductor size: Inductors tend to be much larger, physically, than capacitors are for storing equivalent amounts of energy. This is especially true considering the recent advances in electrolytic capacitor technology, allowing incredibly large capacitance values to be packed into a small package. If a circuit designer needs to store a large amount of energy in a small volume and has the freedom to choose either capacitors or inductors for the task, he or she will most likely choose a capacitor. A notable exception to this rule is in applications requiring huge amounts of either capacitance or inductance to store electrical energy: inductors made of superconducting wire (zero resistance) are more practical to build and safely operate than capacitors of equivalent value, and are probably smaller too. Interference: Inductors may affect nearby components on a circuit board with their magnetic fields, which can extend significant distances beyond the inductor. This is especially true if there are other inductors nearby on the circuit board. If the magnetic fields of two or more inductors are able to “link” with each others’ turns of wire, there will be mutual inductance present in the circuit as well as self-inductance, which could very well cause unwanted effects. This is another reason why circuit designers tend to choose capacitors over inductors to perform similar tasks: capacitors inherently contain their respective electric fields neatly within the component package and therefore do not typically generate any “mutual” effects with other components.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/01%3A_Introduction/1.01%3A_Electronics_as_Science.txt	princeton-nlp/TextbookChapters	Electronics is a science, and a very accessible science at that. With other areas of scientific study, expensive equipment is generally required to perform any non-trivial experiments. Not so with electronics. Many advanced concepts may be explored using parts and equipment totaling under a few hundred US dollars. This is good, because hands-on experimentation is vital to gaining scientific knowledge about any subject. When I started writing Lessons In Electric Circuits, my intent was to create a textbook suitable for introductory college use. However, being mostly self-taught in electronics myself, I knew the value of a good textbook to hobbyists and experimenters not enrolled in any formal electronics course. Many people selflessly volunteered their time and expertise in helping me learn electronics when I was younger, and my intent is to honor their service and love by giving back to the world what they gave to me. In order for someone to teach themselves a science such as electronics, they must engage in hands-on experimentation. Knowledge gleaned from books alone has limited use, especially in scientific endeavors. If my contribution to society is to be complete, I must include a guide to experimentation along with the text(s) on theory, so that the individual learning on their own has a resource to guide their experimental adventures. A formal laboratory course for college electronics study requires an enormous amount of work to prepare and usually must be based on specific parts and equipment so that the experiments will be sufficiently detailed, with results sufficiently precise to allow for rigorous comparison between experimental and theoretical data. A process of assessment, articulated through a qualified instructor, is also vital to guarantee that a certain level of learning has taken place. Peer review (comparison of experimental results with the work of others) is another important component of college-level laboratory study and helps to improve the quality of learning. Since I cannot meet these criteria through the medium of a book, it is impractical for me to present a complete laboratory course here. In the interest of keeping this experiment guide reasonably low-cost for people to follow, and practical for deployment over the internet, I am forced to design the experiments at a lower level than what would be expected for a college lab course. The experiments in this volume begin at a level appropriate for someone with no electronics knowledge, and progress to higher levels. They stress qualitative knowledge over quantitative knowledge, although they could serve as templates for more rigorous coursework. If there is any portion of Lessons In Electric Circuitsthat will remain “incomplete,” it is this one: I fully intend to continue adding experiments ad infinitum so as to provide the experimenter or hobbyist with a wealth of ideas to explore the science of electronics. This volume of the book series is also the easiest to contribute to, for those who would like to help me in providing free information to people learning electronics. It doesn’t take a tremendous effort to describe an experiment or two, and I will gladly include it if you email it to me, giving you full credit for the work. Refer to Appendix 2 for details on contributing to this book. When performing these experiments, feel free to explore by trying different circuit construction and measurement techniques. If something isn’t working as the text describes it should, don’t give up! It’s probably due to a simple problem in construction (loose wire, wrong component value) or test equipment setup. It can be frustrating working through these problems on your own, but the knowledge gained by “troubleshooting” a circuit yourself is at least as important as the knowledge gained by a properly functioning experiment. This is one of the most important reasons why experimentation is so vital to your scientific education: the real problems you will invariably encounter in experimentation challenge you to develop practical problem-solving skills. In many of these experiments, I offer part numbers for Radio Shack brand components. This is not an endorsement of Radio Shack, but simply a convenient reference to an electronic supply company well-known in North America. Often times, components of better quality and lower price may be obtained through mail-order companies and other, lesser-known supply houses. I strongly recommend that experimenters obtain some of the more expensive components such as transformers (see the AC chapter) by salvaging them from discarded electrical appliances, both for economic and ecological reasons. All experiments shown in this book are designed with safety in mind. It is nearly impossible to shock or otherwise hurt yourself by battery-powered experiments or other circuits of low voltage. However, hazards do exist building anything with your own two hands. Where there is a greater-than-normal level of danger in an experiment, I take efforts to direct the reader’s attention toward it. However, it is, unfortunately, necessary in this litigious society to disclaim any and all liability for the outcome of any experiment presented here. Neither I nor any contributors bear responsibility for injuries resulting from the construction or use of any of these projects, from the mishandling of electricity by the experimenter, or from any other unsafe practices leading to injury. Perform these experiments at your own risk!
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/01%3A_Introduction/1.02%3A_Setting_Up_a_Home_Lab.txt	princeton-nlp/TextbookChapters	In order to build the circuits described in this volume, you will need a small work area, as well as a few tools and critical supplies. This section describes the setup of a home electronics laboratory. Work area A work area should consist of a large workbench, desk, or table (preferably wooden) for performing circuit assembly, with household electrical power (120 volts AC) readily accessible to power soldering equipment, power supplies, and any test equipment. Inexpensive desks intended for computer use function very well for this purpose. Avoid a metal-surface desk, as the electrical conductivity of a metal surface creates both a shock hazard and the very distinct possibility of unintentional “short circuits” developing from circuit components touching the metal tabletop. Vinyl and plastic bench surfaces are to be avoided for their ability to generate and store large static-electric charges, which may damage sensitive electronic components. Also, these materials melt easily when exposed to hot soldering irons and molten solder droplets. If you cannot obtain a wooden-surface workbench, you may turn any form of table or desk into one by laying a piece of plywood on top. If you are reasonably skilled with woodworking tools, you may construct your own desk using plywood and 2x4 boards. The work area should be well-lit and comfortable. I have a small radio set up on my own workbench for listening to music or news as I experiment. My own workbench has a “power strip” receptacle and switch assembly mounted to the underside, into which I plug all 120 volt devices. It is convenient to have a single switch for shutting off all power in case of an accidental short-circuit! Tools A few tools are required for basic electronics work. Most of these tools are inexpensive and easy to obtain. If you desire to keep the cost as low as possible, you might want to search for them at thrift stores and pawn shops before buying them new. As you can tell from the photographs, some of my own tools are rather old but function well nonetheless. First and foremost in your tool collection is a multimeter. This is an electrical instrument designed to measure voltage, current, resistance, and often other variables as well. Multimeters are manufactured in both digital and analog form. A digital multimeter is preferred for precision work, but analog meters are also useful for gaining an intuitive understanding of instrument sensitivity and range. My own digital multimeter is a Fluke model 27, purchased in 1987: Digital multimeter Most analog multimeters sold today are quite inexpensive, and not necessarily precision test instruments. I recommend having both digital and analog meter types in your tool collection, spending as little money as possible on the analog multimeter and investing in a good-quality digital multimeter (I highly recommend the Fluke brand). ====================================== A test instrument I have found indispensable in my home work is a sensitive voltage detector, or sensitive audio detector, described in nearly identical experiments in two chapters of this book volume. It is nothing more than a sensitized set of audio headphones, equipped with an attenuator (volume control) and limiting diodes to limit sound intensity from strong signals. Its purpose is to audibly indicate the presence of low-intensity voltage signals, DC or AC. In the absence of an oscilloscope, this is a most valuable tool, because it allows you to listen to an electronic signal, and thereby determine something of its nature. Few tools engender an intuitive comprehension of frequency and amplitude as this! I cite its use in many of the experiments shown in this volume, so I strongly encourage that you build your own. Second, only to a multimeter, it is the most useful piece of test equipment in the collection of the budget electronics experimenter. Sensitive voltage/audio detector As you can see, I built my detector using scrap parts (household electrical switch/receptacle box for the enclosure, section of brown lamp cord for the test leads). Even some of the internal components were salvaged from scrap (the step-down transformer and headphone jack were taken from an old radio, purchased in non-working condition from a thrift store). The entire thing, including the headphones, purchased second-hand, cost no more than \$15 to build. Of course, one could take much greater care in choosing construction materials (metal box, shielded test probe cable), but it probably wouldn’t improve its performance significantly. The single most influential component with regard to detector sensitivity is the headphone assembly: generally speaking, the greater the “dB” rating of the headphones, the better they will function for this purpose. Since the headphones need not be modified for use in the detector circuit, and they can be unplugged from it, you might justify the purchase of more expensive, high-quality headphones by using them as part of a home entertainment (audio/video) system. ====================================== Also essential is a solderless breadboard, sometimes called a prototyping board or proto-board. This device allows you to quickly join electronic components to one another without having to solder component terminals and wires together. Solderless breadboard ====================================== When working with wire, you need a tool to “strip” the plastic insulation off the ends so that bare copper metal is exposed. This tool is called a wire stripper, and it is a special form of plier with several knife-edged holes in the jaw area sized just right for cutting through the plastic insulation and not the copper, for a multitude of wire sizes, or gauges. Shown here are two different sizes of wire stripping pliers: Wire stripping pliers ====================================== In order to make quick, temporary connections between some electronic components, you need jumper wires with small “alligator-jaw” clips at each end. These may be purchased complete, or assembled from clips and wires. Jumper wires (as sold by Radio Shack) Jumper wires (home-made) The home-made jumper wires with large, uninsulated (bare metal) alligator clips are okay to use so long as care is taken to avoid any unintentional contact between the bare clips and any other wires or components. For use in crowded breadboard circuits, jumper wires with insulated (rubber-covered) clips like the jumper shown from Radio Shack are much preferred. ====================================== Needle-nose pliers are designed to grasp small objects, and are especially useful for pushing wires into stubborn breadboard holes. Needle-nose pliers ====================================== No tool set would be complete without screwdrivers, and I recommend a complementary pair (3/16 inch slotted and #2 Phillips) as the starting point for your collection. You may later find it useful to invest in a set of jeweler’s screwdrivers for work with very small screws and screw-head adjustments. Screwdrivers ====================================== For projects involving printed-circuit board assembly or repair, a small soldering iron and a spool of “rosin-core” solder are essential tools. I recommend a 25 watt soldering iron, no larger for printed circuit board work, and the thinnest solder you can find. Do not use “acid-core” solder! Acid-core solder is intended for the soldering of copper tubes (plumbing), where a small amount of acid helps to clean the copper of surface impurities and provide a stronger bond. If used for electrical work, the residual acid will cause wires to corrode. Also, you should avoid solder containing the metal lead, opting instead for silver-alloy solder. If you do not already wear glasses, a pair of safety glasses is highly recommended while soldering, to prevent bits of molten solder from accidently landing in your eye should a wire release from the joint during the soldering process and fling bits of solder toward you. Soldering iron and solder (“rosin core”) ====================================== Projects requiring the joining of large wires by soldering will necessitate a more powerful heat source than a 25 watt soldering iron. A soldering gun is a practical option. Soldering gun ====================================== Knives, like screwdrivers, are essential tools for all kinds of work. For safety’s sake, I recommend a “utility” knife with retracting blade. These knives are also advantageous to have for their ability to accept replacement blades. Utility knife ====================================== Pliers other than the needle-nose type are useful for the assembly and disassembly of electronic device chassis. Two types I recommend are slip-joint and adjustable-joint (“Channel-lock”). Slip-joint pliers Adjustable-joint pliers ====================================== Drilling may be required for the assembly of large projects. Although power drills work well, I have found that a simple hand-crank drill does a remarkable job drilling through plastic, wood, and most metals. It is certainly safer and quieter than a power drill, and costs quite a bit less. Hand drill As the wear on my drill indicates, it is an often-used tool around my home! ====================================== Some experiments will require a source of audio-frequency voltage signals. Normally, this type of signal is generated in an electronics laboratory by a device called a signal generator or function generator. While building such a device is not impossible (nor difficult!), it often requires the use of an oscilloscope to fine-tune, and oscilloscopes are usually outside the budgetary range of the home experimenter. A relatively inexpensive alternative to a commercial signal generator is an electronic keyboard of the musical type. You need not be a musician to operate one for the purposes of generating an audio signal (just press any key on the board!), and they may be obtained quite readily at second-hand stores for substantially less than new price. The electronic signal generated by the keyboard is conducted to your circuit via a headphone cable plugged into the “headphones” jack. More details regarding the use of a “Musical Keyboard as a Signal Generator” may be found in the experiment of that name in chapter 4 (AC). Supplies Wire used in solderless breadboards must be 22-gauge, solid copper. Spools of this wire are available from electronic supply stores and some hardware stores, in different insulation colors. Insulation color has no bearing on the wire’s performance, but different colors are sometimes useful for “color-coding” wire functions in a complex circuit. Spool of 22-gauge, solid copper wire Note how the last 1/4 inch or so of the copper wire protruding from the spool has been “stripped” of its plastic insulation. ====================================== An alternative to solderless breadboard circuit construction is wire-wrap, where 30-gauge (very thin!) solid copper wire is tightly wrapped around the terminals of components inserted through the holes of a fiberglass board. No soldering is required, and the connections made are at least as durable as soldered connections, perhaps more. Wire-wrapping requires a spool of this very thin wire, and a special wrapping tool, the simplest kind resembling a small screwdriver. Wire-wrap wire and wrapping tool ====================================== Large wire (14 gauge and bigger) may be needed for building circuits that carry significant levels of current. Though electrical wire of practically any gauge may be purchased on spools, I have found a very inexpensive source of stranded (flexible), copper wire, available at any hardware store: cheap extension cords. Typically comprised of three wires colored white, black, and green, extension cords are often sold at prices less than the retail cost of the constituent wire alone. This is especially true if the cord is purchased on sale! Also, an extension cord provides you with a pair of 120 volt connectors: male (plug) and female (receptacle) that may be used for projects powered by 120 volts. Extension cord, in package To extract the wires, carefully cut the outer layer of plastic insulation away using a utility knife. With practice, you may find you can peel away the outer insulation by making a short cut in it at one end of the cable, then grasping the wires with one hand and the insulation with the other and pulling them apart. This is, of course, much preferable to slicing the entire length of the insulation with a knife, both for safety’s sake and for the sake of avoiding cuts in the individual wires’ insulation. ====================================== During the course of building many circuits, you will accumulate a large number of small components. One technique for keeping these components organized is to keep them in a plastic “organizer” box like the type used for fishing tackle. Component box In this view of one of my component boxes, you can see plenty of 1/8 watt resistors, transistors, diodes, and even a few 8-pin integrated circuits (“chips”). Labels for each compartment were made with a permanent ink marker.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.01%3A_Voltmeter_Usage.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Multimeter, digital or analog • Assorted batteries • One light-emitting diode (Radio Shack catalog # 276-026 or equivalent) • Small “hobby” motor, permanent-magnet type (Radio Shack catalog # 273-223 or equivalent) • Two jumper wires with “alligator clip” ends (Radio Shack catalog # 278-1156, 278-1157, or equivalent) A multimeter is an electrical instrument capable of measuring voltage, current, and resistance. Digitalmultimeters have numerical displays, like digital clocks, for indicating the quantity of voltage, current, or resistance. Analog multimeters indicate these quantities by means of a moving pointer over a printed scale. Analog multimeters tend to be less expensive than digital multimeters, and more beneficial as learning tools for the first-time student of electricity. I strongly recommend purchasing an analog multimeter before purchasing a digital multimeter, but to eventually have both in your tool kit for these experiments. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 1: “Basic Concepts of Electricity” Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” LEARNING OBJECTIVES • How to measure voltage • Characteristics of voltage: existing between two points • Selection of proper meter range ILLUSTRATION INSTRUCTIONS In all the experiments in this book, you will be using some sort of test equipment to measure aspects of electricity you cannot directly see, feel, hear, taste, or smell. Electricity—at least in small, safe quantities—is insensible by our human bodies. Your most fundamental “eyes” in the world of electricity and electronics will be a device called a multimeter. Multimeters indicate the presence of, and measure the quantity of, electrical properties such as voltage, current, and resistance. In this experiment, you will familiarize yourself with the measurement of voltage. Voltage is the measure of electrical “push” ready to motivate electrons to move through a conductor. In scientific terms, it is the specific energy per unit charge, mathematically defined as joules per coulomb. It is analogous to pressure in a fluid system: the force that moves fluid through a pipe, and is measured in the unit of the Volt (V). Your multimeter should come with some basic instructions. Read them well! If your multimeter is digital, it will require a small battery to operate. If it is analog, it does not need a battery to measure voltage. Some digital multimeters are autoranging. An autoranging meter has only a few selector switch (dial) positions. Manual-ranging meters have several different selector positions for each basic quantity: several for voltage, several for current, and several for resistance. Autoranging is usually found on only the more expensive digital meters, and is to manual ranging as an automatic transmission is to a manual transmission in a car. An autoranging meter “shifts gears” automatically to find the best measurement range to display the particular quantity being measured. Set your multimeter’s selector switch to the highest-value “DC volt” position available. Autoranging multimeters may only have a single position for DC voltage, in which case you need to set the switch to that one position. Touch the red test probe to the positive (+) side of a battery, and the black test probe to the negative (-) side of the same battery. The meter should now provide you with some sort of indication. Reverse the test probe connections to the battery if the meter’s indication is negative (on an analog meter, a negative value is indicated by the pointer deflecting left instead of right). If your meter is a manual-range type, and the selector switch has been set to a high-range position, the indication will be small. Move the selector switch to the next lower DC voltage range setting and reconnect to the battery. The indication should be stronger now, as indicated by a greater deflection of the analog meter pointer (needle), or more active digits on the digital meter display. For the best results, move the selector switch to the lowest-range setting that does not “over-range” the meter. An over-ranged analog meter is said to be “pegged,” as the needle will be forced all the way to the right-hand side of the scale, past the full-range scale value. An over-ranged digital meter sometimes displays the letters “OL”, or a series of dashed lines. This indication is manufacturer-specific. What happens if you only touch one meter test probe to one end of a battery? How does the meter have to connect to the battery in order to provide an indication? What does this tell us about voltmeter use and the nature of voltage? Is there such a thing as voltage “at” a single point? Be sure to measure more than one size of battery, and learn how to select the best voltage range on the multimeter to give you maximum indication without over-ranging. Now switch your multimeter to the lowest DC voltage range available, and touch the meter’s test probes to the terminals (wire leads) of the light-emitting diode (LED). An LED is designed to produce light when powered by a small amount of electricity, but LEDs also happen to generate DC voltage when exposed to light, somewhat like a solar cell. Point the LED toward a bright source of light with your multimeter connected to it, and note the meter’s indication: Batteries develop electrical voltage through chemical reactions. When a battery “dies,” it has exhausted its original store of chemical “fuel.” The LED, however, does not rely on an internal “fuel” to generate voltage; rather, it converts optical energy into electrical energy. So long as there is light to illuminate the LED, it will produce voltage. Another source of voltage through energy conversion a generator. The small electric motor specified in the “Parts and Materials” list functions as an electrical generator if its shaft is turned by a mechanical force. Connect your voltmeter (your multimeter, set to the “volt” function) to the motor’s terminals just as you connected it to the LED’s terminals, and spin the shaft with your fingers. The meter should indicate voltage by means of needle deflection (analog) or numerical readout (digital). If you find it difficult to maintain both meter test probes in connection with the motor’s terminals while simultaneously spinning the shaft with your fingers, you may use alligator clip “jumper” wires like this: Determine the relationship between voltage and generator shaft speed? Reverse the generator’s direction of rotation and note the change in meter indication. When you reverse shaft rotation, you change the polarity of the voltage created by the generator. The voltmeter indicates polarity by direction of needle direction (analog) or sign of numerical indication (digital). When the red test lead is positive (+) and the black test lead negative (-), the meter will register voltage in the normal direction. If the applied voltage is of the reverse polarity (negative on red and positive on black), the meter will indicate “backwards.”
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.02%3A_Ohmmeter_Usage.txt	princeton-nlp/TextbookChapters	Parts and Materials • Multimeter, digital or analog • Assorted resistors (Radio Shack catalog # 271-312 is a 500-piece assortment) • Rectifying diode (1N4001 or equivalent; Radio Shack catalog # 276-1101) • Cadmium Sulphide photocell (Radio Shack catalog # 276-1657) • Breadboard (Radio Shack catalog # 276-174 or equivalent) • Jumper wires • Paper • Pencil • Glass of water • Table salt This experiment describes how to measure the electrical resistance of several objects. You need not possess all items listed above in order to effectively learn about resistance. Conversely, you need not limit your experiments to these items. However, be sure to never measure the resistance of any electrically “live” object or circuit. In other words, do not attempt to measure the resistance of a battery or any other source of substantial voltage using a multimeter set to the resistance (“ohms”) function. Failing to heed this warning will likely result in meter damage and even personal injury. Learning Objectives • Determination and comprehension of “electrical continuity” • Determination and comprehension of “electrically common points” • How to measure resistance • Characteristics of resistance: existing between two points • Selection of proper meter range • Relative conductivity of various components and materials Instructions Resistance is the measure of electrical “friction” as electrons move through a conductor. It is measured in the unit of the “Ohm,” that unit symbolized by the capital Greek letter omega (Ω). Set your multimeter to the highest resistance range available. The resistance function is usually denoted by the unit symbol for resistance: the Greek letter omega (Ω), or sometimes by the word “ohms.” Touch the two test probes of your meter together. When you do, the meter should register 0 ohms of resistance. If you are using an analog meter, you will notice the needle deflect full-scale when the probes are touched together, and return to its resting position when the probes are pulled apart. The resistance scale on an analog multimeter is reverse-printed from the other scales: zero resistance in indicated at the far right-hand side of the scale, and infinite resistance is indicated at the far left-hand side. There should also be a small adjustment knob or “wheel” on the analog multimeter to calibrate it for “zero” ohms of resistance. Touch the test probes together and move this adjustment until the needle exactly points to zero at the right-hand end of the scale. Although your multimeter is capable of providing quantitative values of measured resistance, it is also useful for qualitative tests of continuity: whether or not there is a continuous electrical connection from one point to another. You can, for instance, test the continuity of a piece of wire by connecting the meter probes to opposite ends of the wire and checking to see the the needle moves full-scale. What would we say about a piece of wire if the ohmmeter needle didn’t move at all when the probes were connected to opposite ends? How to Measure Resistance Digital multimeters set to the “resistance” mode indicate non-continuity by displaying some non-numerical indication on the display. Some models say “OL” (Open-Loop), while others display dashed lines. Use your meter to determine continuity between the holes on a breadboard: a device used for temporary construction of circuits, where component terminals are inserted into holes on a plastic grid, metal spring clips underneath each hole connecting certain holes to others. Use small pieces of 22-gauge solid copper wire, inserted into the holes of the breadboard, to connect the meter to these spring clips so that you can test for continuity: An important concept in electricity, closely related to electrical continuity, is that of points being electrically common to each other. Electrically common points are points of contact on a device or in a circuit that have negligible (extremely small) resistance between them. We could say, then, that points within a breadboard column (vertical in the illustrations) are electrically common to each other, because there is electrical continuity between them. Conversely, breadboard points within a row (horizontal in the illustrations) are not electrically common, because there is no continuity between them. Continuity describes what is between points of contact, while commonality describes how the points themselves relate to each other. Like continuity, commonality is a qualitative assessment, based on a relative comparison of resistance between other points in a circuit. It is an important concept to grasp, because there are certain facts regarding voltage in relation to electrically common points that are valuable in circuit analysis and troubleshooting, the first one being that there will never be substantial voltage dropped between points that are electrically common to each other. Select a 10,000 ohm (10 kΩ) resistor from your parts assortment. This resistance value is indicated by a series of color bands: Brown, Black, Orange, and then another color representing the precision of the resistor, Gold (+/- 5%) or Silver (+/- 10%). Some resistors have no color for precision, which marks them as +/- 20%. Other resistors use five color bands to denote their value and precision, in which case the colors for a 10 kΩ resistor will be Brown, Black, Black, Red, and a fifth color for precision. Connect the meter’s test probes across the resistor as such, and note its indication on the resistance scale: If the needle points very close to zero, you need to select a lower resistance range on the meter, just as you needed to select an appropriate voltage range when reading the voltage of a battery. If you are using a digital multimeter, you should see a numerical figure close to 10 shown on the display, with a small “k” symbol on the right-hand side denoting the metric prefix for “kilo” (thousand). Some digital meters are manually-ranged, and require appropriate range selection just as the analog meter. If yours is like this, experiment with different range switch positions and see which one gives you the best indication. Try reversing the test probe connections on the resistor. Does this change the meter’s indication at all? What does this tell us about the resistance of a resistor? What happens when you only touch one probe to the resistor? What does this tell us about the nature of resistance, and how it is measured? How does this compare with voltage measurement, and what happened when we tried to measure battery voltage by touching only one probe to the battery? When you touch the meter probes to the resistor terminals, try not to touch both probe tips to your fingers. If you do, you will be measuring the parallel combination of the resistor and your own body, which will tend to make the meter indication lower than it should be! When measuring a 10 kΩ resistor, this error will be minimal, but it may be more severe when measuring other values of resistor. You may safely measure the resistance of your own body by holding one probe tip with the fingers of one hand, and the other probe tip with the fingers of the other hand. Note: be very careful with the probes, as they are often sharpened to a needle-point. Hold the probe tips along their length, not at the very points! You may need to adjust the meter range again after measuring the 10 kΩ resistor, as your body resistance tends to be greater than 10,000 ohms hand-to-hand. Try wetting your fingers with water and re-measuring resistance with the meter. What impact does this have on the indication? Try wetting your fingers with saltwater prepared using the glass of water and table salt, and re-measuring resistance. What impact does this have on your body’s resistance as measured by the meter? Resistance is the measure of friction to electron flow through an object. The more resistance there is between two points, the harder it is for electrons to move (flow) between those two points. Given that electric shock is caused by a large flow of electrons through a person’s body, and increased body resistance acts as a safeguard by making it more difficult for electrons to flow through us, what can we ascertain about electrical safety from the resistance readings obtained with wet fingers? Does water increase or decrease shock hazard to people? Measure the resistance of a rectifying diode with an analog meter. Try reversing the test probe connections to the diode and re-measure resistance. What strikes you as being remarkable about the diode, especially in contrast to the resistor? Take a piece of paper and draw a very heavy black mark on it with a pencil (not a pen!). Measure resistance on the black strip with your meter, placing the probe tips at each end of the mark like this: Move the probe tips closer together on the black mark and note the change in resistance value. Does it increase or decrease with decreased probe spacing? If the results are inconsistent, you need to redraw the mark with more and heavier pencil strokes, so that it is consistent in its density. What does this teach you about resistance versus length of a conductive material? Connect your meter to the terminals of a cadmium-sulphide (CdS) photocell and measure the change in resistance created by differences in light exposure. Just as with the light-emitting diode (LED) of the voltmeter experiment, you may want to use alligator-clip jumper wires to make connection with the component, leaving your hands free to hold the photocell to a light source and/or change meter ranges: Experiment with measuring the resistance of several different types of materials, just be sure not to try measure anything that produces substantial voltage, like a battery. Suggestions for materials to measure are: fabric, plastic, wood, metal, clean water, dirty water, salt water, glass, diamond (on a diamond ring or other piece of jewelry), paper, rubber, and oil.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.03%3A_A_Very_Simple_Circuit.txt	princeton-nlp/TextbookChapters	Parts and Materials • 6-volt battery • 6-volt incandescent lamp • Jumper wires • Breadboard • Terminal strip From this experiment on, a multimeter is assumed to be necessary and will not be included in the required list of parts and materials. In all subsequent illustrations, a digital multimeter will be shown instead of an analog meter unless there is some particular reason to use an analog meter. You are encouraged to use both types of meters to gain familiarity with the operation of each in these experiments. Cross-references Lessons In Electric Circuits, Volume 1, chapter 1: “Basic Concepts of Electricity” Learning Objectives • Essential configuration needed to make a circuit • Normal voltage drops in an operating circuit • Importance of continuity to a circuit • Working definitions of “open” and “short” circuits • Breadboard usage • Terminal strip usage Creating a Simple Circuit This is the simplest complete circuit in this collection of experiments: a battery and an incandescent lamp. Connect the lamp to the battery as shown in the illustration, and the lamp should light, assuming the battery and lamp are both in good condition and they are matched to one another in terms of voltage. If there is a “break” (discontinuity) anywhere in the circuit, the lamp will fail to light. It does not matter where such a break occurs! Many students assume that because electrons leave the negative (-) side of the battery and continue through the circuit to the positive (+) side, that the wire connecting the negative terminal of the battery to the lamp is more important to circuit operation than the other wire providing a return path for electrons back to the battery. This is not true! Using your multimeter set to the appropriate “DC volt” range, measure voltage across the battery, across the lamp, and across each jumper wire. Familiarize yourself with the normal voltages in a functioning circuit. Now, “break” the circuit at one point and re-measure voltage between the same sets of points, additionally measuring voltage across the break like this: What voltages measure the same as before? What voltages are different since introducing the break? How much voltage is manifested, or dropped across the break? What is the polarity of the voltage drop across the break, as indicated by the meter? Re-connect the jumper wire to the lamp, and break the circuit in another place. Measure all voltage “drops” again, familiarizing yourself with the voltages of an “open” circuit. Construct the same circuit on a breadboard, taking care to place the lamp and wires into the breadboard in such a way that continuity will be maintained. The example shown here is only that: an example, not the only way to build a circuit on a breadboard: Experiment with different configurations on the breadboard, plugging the lamp into different holes. If you encounter a situation where the lamp refuses to light up and the connecting wires are getting warm, you probably have a situation known as a short circuit, where a lower-resistance path than the lamp bypasses current around the lamp, preventing enough voltage from being dropped across the lamp to light it up. Here is an example of a short circuit made on a breadboard: An Example of an Accidental Short Circuit Here is an example of an accidental short circuit of the type typically made by students unfamiliar with breadboard usage: Here there is no “shorting” wire present on the breadboard, yet there is a short circuit, and the lamp refuses to light. Based on your understanding of breadboard hole connections, can you determine where the “short” is in this circuit? Tips to Avoid Short Circuits Short circuits are generally to be avoided, as they result in very high rates of electron flow, causing wires to heat up and battery power sources to deplete. If the power source is substantial enough, a short circuit may cause heat of explosive proportions to manifest, causing equipment damage and hazard to nearby personnel. This is what happens when a tree limb “shorts” across wires on a power line: the limb—being composed of wet wood—acts as a low-resistance path to electric current, resulting in heat and sparks. You may also build the battery/lamp circuit on a terminal strip: a length of insulating material with metal bars and screws to attach wires and component terminals to. Here is an example of how this circuit might be constructed on a terminal strip:
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.04%3A_How_to_Use_an_Ammeter_to_Measure_Current.txt	princeton-nlp/TextbookChapters	Parts and Materials • 6-volt battery • 6-volt incandescent lamp Basic circuit construction components such as breadboard, terminal strip, and jumper wires are also assumed to be available from now on, leaving only components and materials unique to the project listed under “Parts and Materials”. Further Reading Lessons In Electric Circuits, Volume 1, chapter 1: “Basic Concepts of Electricity” Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” Ammeter Usage Learning Objectives • How to measure current with a multimeter • How to check a multimeter’s internal fuse • Selection of proper meter range Experiment Instructions Current is the measure of the rate of electron “flow” in a circuit. It is measured in the unit of the Ampere, simply called “Amp,” (A). The most common way to measure current in a circuit is to break the circuit open and insert an “ammeter” in series (in-line) with the circuit so that all electrons flowing through the circuit also have to go through the meter. Because measuring current in this manner requires the meter be made part of the circuit, it is a more difficult type of measurement to make than either voltage or resistance. Some digital meters, like the unit shown in the illustration, have a separate jack to insert the red test lead plug when measuring current. Other meters, like most inexpensive analog meters, use the same jacks for measuring voltage, resistance, and current. Consult your owner’s manual on the particular model of meter you own for details on measuring current. When an ammeter is placed in series with a circuit, it ideally drops no voltage as current goes through it. In other words, it acts very much like a piece of wire, with very little resistance from one test probe to the other. Consequently, an ammeter will act as a short circuit if placed in parallel (across the terminals of) a substantial source of voltage. If this is done, a surge in current will result, potentially damaging the meter: Ammeters are generally protected from excessive current by means of a small fuse located inside the meter housing. If the ammeter is accidently connected across a substantial voltage source, the resultant surge in current will “blow” the fuse and render the meter incapable of measuring current until the fuse is replaced. Be very careful to avoid this scenario! You may test the condition of a multimeter’s fuse by switching it to the resistance mode and measuring continuity through the test leads (and through the fuse). On a meter where the same test lead jacks are used for both resistance and current measurement, simply leave the test lead plugs where they are and touch the two probes together. On a meter where different jacks are used, this is how you insert the test lead plugs to check the fuse: Build the one-battery, one-lamp circuit using jumper wires to connect the battery to the lamp, and verify that the lamp lights up before connecting the meter in series with it. Then, break the circuit open at any point and connect the meter’s test probes to the two points of the break to measure current. As usual, if your meter is manually-ranged, begin by selecting the highest range for current, then move the selector switch to lower range positions until the strongest indication is obtained on the meter display without over-ranging it. If the meter indication is “backwards,” (left motion on analog needle, or negative reading on a digital display), then reverse the test probe connections and try again. When the ammeter indicates a normal reading (not “backwards”), electrons are entering the black test lead and exiting the red. This is how you determine direction of current using a meter. For a 6-volt battery and a small lamp, the circuit current will be in the range of thousandths of an amp, or milliamps. Digital meters often show a small letter “m” in the right-hand side of the display to indicate this metric prefix. Try breaking the circuit at some other point and inserting the meter there instead. What do you notice about the amount of current measured? Why do you think this is? Re-construct the circuit on a breadboard like this: Connecting an Ammeter to a Breadboard Circuit: Tips and Tricks Students often get confused when connecting an ammeter to a breadboard circuit. How can the meter be connected so as to intercept all the circuit’s current and not create a short circuit? One easy method that guarantees success is this: • • Identify what wire or component terminal you wish to measure current through. • • Pull that wire or terminal out of the breadboard hole. Leave it hanging in mid-air. • • Insert a spare piece of wire into the hole you just pulled the other wire or terminal out of. Leave the other end of this wire hanging in mid-air. • • Connect the ammeter between the two unconnected wire ends (the two that were hanging in mid-air). You are now assured of measuring current through the wire or terminal initially identified. Again, measure current through different wires in this circuit, following the same connection procedure outlined above. What do you notice about these current measurements? The results in the breadboard circuit should be the same as the results in the free-form (no breadboard) circuit. Experiment Results Building the same circuit on a terminal strip should also yield similar results: The current figure of 24.70 milliamps (24.70 mA) shown in the illustrations is an arbitrary quantity, reasonable for a small incandescent lamp. If the current for your circuit is a different value, that is okay, so long as the lamp is functioning when the meter is connected. If the lamp refuses to light when the meter is connected to the circuit, and the meter registers a much greater reading, you probably have a short-circuit condition through the meter. If your lamp refuses to light when the meter is connected in the circuit, and the meter registers zero current, you’ve probably blown the fuse inside the meter. Check the condition of your meter’s fuse as described previously in this section and replace the fuse if necessary.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.05%3A_Ohms_Law.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Calculator (or pencil and paper for doing arithmetic) • 6-volt battery • Assortment of resistors between 1 KΩ and 100 kΩ in value I’m purposely restricting the resistance values between 1 kΩ and 100 kΩ for the sake of obtaining accurate voltage and current readings with your meter. With very low resistance values, the internal resistance of the ammeter has a significant impact on measurement accuracy. Very high resistance values can cause problems for voltage measurement, the internal resistance of the voltmeter substantially changing circuit resistance when it is connected in parallel with a high-value resistor. At the recommended resistance values, there will still be a small amount of measurement error due to the “impact” of the meter, but not enough to cause serious disagreement with calculated values. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 2: “Ohm’s Law” LEARNING OBJECTIVES • Voltmeter use • Ammeter use • Ohmmeter use • Use of Ohm’s Law SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Select a resistor from the assortment, and measure its resistance with your multimeter set to the appropriate resistance range. Be sure not to hold the resistor terminals when measuring resistance, or else your hand-to-hand body resistance will influence the measurement! Record this resistance value for future use. Build a one-battery, one-resistor circuit. A terminal strip is shown in the illustration, but any form of circuit construction is okay. Set your multimeter to the appropriate voltage range and measure voltage across the resistor as it is being powered by the battery. Record this voltage value along with the resistance value previously measured. Set your multimeter to the highest current range available. Break the circuit and connect the ammeter within that break, so it becomes a part of the circuit, in series with the battery and resistor. Select the best current range: whichever one gives the strongest meter indication without over-ranging the meter. If your multimeter is autoranging, of course, you need not bother with setting ranges. Record this current value along with the resistance and voltage values previously recorded. Taking the measured figures for voltage and resistance, use the Ohm’s Law equation to calculate circuit current. Compare this calculated figure with the measured figure for circuit current: Taking the measured figures for voltage and current, use the Ohm’s Law equation to calculate circuit resistance. Compare this calculated figure with the measured figure for circuit resistance: Finally, taking the measured figures for resistance and current, use the Ohm’s Law equation to calculate circuit voltage. Compare this calculated figure with the measured figure for circuit voltage: There should be close agreement between all measured and all calculated figures. Any differences in respective quantities of voltage, current, or resistance are most likely due to meter inaccuracies. These differences should be rather small, no more than several percent. Some meters, of course, are more accurate than others! Substitute different resistors in the circuit and re-take all resistance, voltage, and current measurements. Re-calculate these figures and check for agreement with the experimental data (measured quantities). Also note the simple mathematical relationship between changes in resistor value and changes in circuit current. Voltage should remain approximately the same for any resistor size inserted into the circuit, because it is the nature of a battery to maintain voltage at a constant level.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.06%3A_Nonlinear_Resistance.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Calculator (or pencil and paper for doing arithmetic) • 6-volt battery • Low-voltage incandescent lamp (Radio Shack catalog # 272-1130 or equivalent) CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 2: “Ohm’s Law” LEARNING OBJECTIVES • Voltmeter use • Ammeter use • Ohmmeter use • Use of Ohm’s Law • Realization that some resistances are unstable! • Scientific method SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Measure the resistance of the lamp with your multimeter. This resistance figure is due to the thin metal “filament” inside the lamp. It has substantially more resistance than a jumper wire, but less than any of the resistors from the last experiment. Record this resistance value for future use. Build a one-battery, one-lamp circuit. Set your multimeter to the appropriate voltage range and measure voltage across the lamp as it is energized (lit). Record this voltage value along with the resistance value previously measured. Set your multimeter to the highest current range available. Break the circuit and connect the ammeter within that break, so it becomes a part of the circuit, in series with the battery and lamp. Select the best current range: whichever one gives the strongest meter indication without over-ranging the meter. If your multimeter is autoranging, of course, you need not bother with setting ranges. Record this current value along with the resistance and voltage values previously recorded. Taking the measured figures for voltage and resistance, use the Ohm’s Law equation to calculate circuit current. Compare this calculated figure with the measured figure for circuit current: What you should find is a marked difference between measured current and calculated current: the calculated figure is much greater. Why is this? To make things more interesting, try measuring the lamp’s resistance again, this time using a different model of meter. You will need to disconnect the lamp from the battery circuit in order to obtain a resistance reading, because voltages outside of the meter interfere with resistance measurement. This is a general rule that should be remembered: measure resistance only on an unpowered component! Using a different ohmmeter, the lamp will probably register as a different value of resistance. Usually, analog meters give higher lamp resistance readings than digital meters. This behavior is very different from that of the resistors in the last experiment. Why? What factor(s) might influence the resistance of the lamp filament, and how might those factors be different between conditions of lit and unlit, or between resistance measurements taken with different types of meters? This problem is a good test case for the application of scientific method. Once you’ve thought of a possible reason for the lamp’s resistance changing between lit and unlit conditions, try to duplicate that cause by some other means. For example, if you think the lamp resistance might change as it is exposed to light (its own light, when lit), and that this accounts for the difference between the measured and calculated circuit currents, try exposing the lamp to an external source of light while measuring its resistance. If you measure substantial resistance change as a result of light exposure, then your hypothesis has some evidential support. If not, then your hypothesis has been falsified, and another cause must be responsible for the change in circuit current.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.07%3A_Power_Dissipation.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Calculator (or pencil and paper for doing arithmetic) • 6 volt battery • Two 1/4 watt resistors: 10 Ω and 330 Ω. • Small thermometer The resistor values need not be exact, but within five percent of the figures specified (+/- 0.5 Ω for the 10 Ω resistor; +/- 16.5 Ω for the 330 Ω resistor). Color codes for 5% tolerance 10 Ω and 330 Ω resistors are as follows: Brown, Black, Black, Gold (10, +/- 5%), and Orange, Orange, Brown, Gold (330, +/- 5%). Do not use any battery size other than 6 volts for this experiment. The thermometer should be as small as possible, to facilitate rapid detection of heat produced by the resistor. I recommend a medical thermometer, the type used to take body temperature. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 2: “Ohm’s Law” LEARNING OBJECTIVES • Voltmeter use • Ammeter use • Ohmmeter use • Use of Joule’s Law • Importance of component power ratings • Significance of electrically common points SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Measure each resistor’s resistance with your ohmmeter, noting the exact values on a piece of paper for later reference. Connect the 330 Ω resistor to the 6 volt battery using a pair of jumper wires as shown in the illustration. Connect the jumper wires to the resistor terminals before connecting the other ends to the battery. This will ensure your fingers are not touching the resistor when battery power is applied. You might be wondering why I advise no bodily contact with the powered resistor. This is because it will become hot when powered by the battery. You will use the thermometer to measure the temperature of each resistor when powered. With the 330 Ω resistor connected to the battery, measure voltage with a voltmeter. In measuring voltage, there is more than one way to obtain a proper reading. Voltage may be measured directly across the battery, or directly across the resistor. Battery voltage is the same as resistor voltage in this circuit, since those two components share the same set of electrically common points: one side of the resistor is directly connected to one side of the battery, and the other side of the resistor is directly connected to the other side of the battery. All points of contact along the upper wire in the illustration (colored red) are electrically common to each other. All points of contact along the lower wire (colored black) are likewise electrically common to each other. Voltage measured between any point on the upper wire and any point on the lower wire should be the same. Voltage measured between any two common points, however, should be zero. Using an ammeter, measure current through the circuit. Again, there is no one “correct” way to measure current, so long as the ammeter is placed within the flow-path of electrons through the resistor and not across a source of voltage. To do this, make a break in the circuit, and place the ammeter within that break: connect the two test probes to the two wire or terminal ends left open from the break. One viable option is shown in the following illustration: Now that you’ve measured and recorded resistor resistance, circuit voltage, and circuit current, you are ready to calculate power dissipation. Whereas voltage is the measure of electrical “push” motivating electrons to move through a circuit, and current is the measure of electron flow rate, power is the measure of work-rate: how fast work is being done in the circuit. It takes a certain amount of work to push electrons through a resistance, and power is a description of how rapidly that work is taking place. In mathematical equations, power is symbolized by the letter “P” and measured in the unit of the Watt (W). Power may be calculated by any one of three equations—collectively referred to as Joule’s Law—given any two out of three quantities of voltage, current, and resistance: Try calculating power in this circuit, using the three measured values of voltage, current, and resistance. Any way you calculate it, the power dissipation figure should be roughly the same. Assuming a battery with 6.000 volts and a resistor of exactly 330 Ω, the power dissipation will be 0.1090909 watts, or 109.0909 milli-watts (mW), to use a metric prefix. Since the resistor has a power rating of 1/4 watt (0.25 watts, or 250 mW), it is more than capable of sustaining this level of power dissipation. Because the actual power level is almost half the rated power, the resistor should become noticeably warm but it should not overheat. Touch the thermometer end to the middle of the resistor and see how warm it gets. The power rating of any electrical component does not tell us how much power it will dissipate, but simply how much power it may dissipate without sustaining damage. If the actual amount of dissipated power exceeds a component’s power rating, that component will increase temperature to the point of damage. To illustrate, disconnect the 330 Ω resistor and replace it with the 10 Ω resistor. Again, avoid touching the resistor once the circuit is complete, as it will heat up rapidly. The safest way to do this is to disconnect one jumper wire from a battery terminal, then disconnect the 330 Ω resistor from the two alligator clips, then connect the 10 Ω resistor between the two clips, and finally reconnect the jumper wire back to the battery terminal. Caution: keep the 10 Ω resistor away from any flammable materials when it is powered by the battery! You may not have enough time to take voltage and current measurements before the resistor begins to smoke. At the first sign of distress, disconnect one of the jumper wires from a battery terminal to interrupt circuit current, and give the resistor a few moments to cool down. With power still disconnected, measure the resistor’s resistance with an ohmmeter and note any substantial deviation from its original value. If the resistor still measures within +/- 5% of its advertised value (between 9.5 and 10.5 Ω), re-connect the jumper wire and let it smoke a bit more. What trend do you notice with the resistor’s value as it is damaged more and more by overpowering? It is typical of resistors to fail with a greater-than-normal resistance when overheated. This is often a self-protective mode of failure, as an increased resistance results in less current and (generally) less power dissipation, cooling it down again. However, the resistor’s normal resistance value will not return if sufficiently damaged. Performing some Joule’s Law calculations for resistor power again, we find that a 10 Ω resistor connected to a 6 volt battery dissipates about 3.6 watts of power, about 14.4 times its rated power dissipation. Little wonder it smokes so quickly after connection to the battery!
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/02%3A_Basic_Concepts_and_Test_Equipment/2.08%3A_Circuit_With_a_Switch.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • 6-volt battery • Low-voltage incandescent lamp (Radio Shack catalog # 272-1130 or equivalent) • Long lengths of wire, 22-gauge or larger • Household light switch (these are readily available at any hardware store) Household light switches are a bargain for students of basic electricity. They are readily available, very inexpensive, and almost impossible to damage with battery power. Do not get “dimmer” switches, just the simple on-off “toggle” variety used for ordinary household wall-mounted light controls. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 1: “Basic Concepts of Electricity” LEARNING OBJECTIVES • Switch behavior • Using an ohmmeter to check switch action SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Build a one-battery, one-switch, one-lamp circuit as shown in the schematic diagram and in the illustration. This circuit is most impressive when the wires are long, as it shows how the switch is able to control circuit current no matter how physically large the circuit may be. Measure voltage across the battery, across the switch (measure from one screw terminal to another with the voltmeter), and across the lamp with the switch in both positions. When the switch is turned off, it is said to be open, and the lamp will go out just the same as if a wire were pulled loose from a terminal. As before, any break in the circuit at any location causes the lamp to immediately de-energize (darken). 2.09: Electromagnetism Experiment PARTS AND MATERIALS • 6-volt battery • Magnetic compass • Small permanent magnet • Spool of 28-gauge magnet wire • Large bolt, nail, or steel rod • Electrical tape Magnet wire is a term for thin-gauge copper wire with enamel insulation instead of rubber or plastic insulation. Its small size and very thin insulation allow for many “turns” to be wound in a compact coil. You will need enough magnet wire to wrap hundreds of turns around the bolt, nail, or other rod-shaped steel form. Be sure to select a bolt, nail, or rod that is magnetic. Stainless steel, for example, is non-magnetic and will not function for the purpose of an electromagnet coil! The ideal material for this experiment is soft iron, but any commonly available steel will suffice. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 14: “Magnetism and Electromagnetism” LEARNING OBJECTIVES • Application of the left-hand rule • Electromagnet construction SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Wrap a single layer of electrical tape around the steel bar (or bolt, or mail) to protect the wire from abrasion. Proceed to wrap several hundred turns of wire around the steel bar, making the coil as even as possible. It is okay to overlap wire, and it is okay to wrap in the same style that a fishing reel wraps line around the spool. The only rule you must follow is that all turns must be wrapped around the bar in the same direction (no reversing from clockwise to counter-clockwise!). I find that a drill press works as a great tool for coil winding: clamp the rod in the drill’s chuck as if it were a drill bit, then turn the drill motor on at a slow speed and let it do the wrapping! This allows you to feed wire onto the rod in a very steady, even manner. After you’ve wrapped several hundred turns of wire around the rod, wrap a layer or two of electrical tape over the wire coil to secure the wire in place. Scrape the enamel insulation off the ends of the coil wires for connection to jumper leads, then connect the coil to a battery. When electric current goes through the coil, it will produce a strong magnetic field: one “pole” at each end of the rod. This phenomenon is known as electromagnetism. The magnetic compass is used to identify the “North” and “South” poles of the electromagnet. With the electromagnet energized (connected to the battery), place a permanent magnet near one pole and note whether there is an attractive or repulsive force. Reverse the orientation of the permanent magnet and note the difference in force. Electromagnetism has many applications, including relays, electric motors, solenoids, doorbells, buzzers, computer printer mechanisms, and magnetic media “write” heads (tape recorders, disk drives). You might notice a significant spark whenever the battery is disconnected from the electromagnet coil: much greater than the spark produced if the battery is simply short-circuited. This spark is the result of a high-voltage surge created whenever current is suddenly interrupted through the coil. The effect is known as inductive “kickback” and is capable of delivering a small but harmless electric shock! To avoid receiving this shock, do not place your body across the break in the circuit when de-energizing! Use one hand at a time when un-powering the coil and you’ll be perfectly safe. This phenomenon will be explored in greater detail in the next chapter (DC Circuits). 2.10: Electromagnetic Induction Experiment PARTS AND MATERIALS • Electromagnet from previous experiment • Permanent magnet See previous experiment for instructions on electromagnet construction. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 14: “Magnetism and Electromagnetism” LEARNING OBJECTIVES • Relationship between magnetic field strength and induced voltage SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Electromagnetic induction is the complementary phenomenon to electromagnetism. Instead of producing a magnetic field from electricity, we produce electricity from a magnetic field. There is one important difference, though: whereas electromagnetism produces a steady magnetic field from a steady electric current, electromagnetic induction requires motion between the magnet and the coil to produce a voltage. Connect the multimeter to the coil, and set it to the most sensitive DC voltage range available. Move the magnet slowly to and from one end of the electromagnet, noting the polarity and magnitude of the induced voltage. Experiment with moving the magnet, and discover for yourself what factor(s) determine the amount of voltage induced. Try the other end of the coil and compare results. Try the other end of the permanent magnet and compare. If using an analog multimeter, be sure to use long jumper wires and locate the meter far away from the coil, as the magnetic field from the permanent magnet may affect the meter’s operation and produce false readings. Digital meters are unaffected by magnetic fields.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.01%3A_Introduction_to_DC_Circuits.txt	princeton-nlp/TextbookChapters	“DC” stands for Direct Current, which can refer to either voltage or current in a constant polarity or direction, respectively. These experiments are designed to introduce you to several important concepts of electricity related to DC circuits. 3.02: Series Batteries PARTS AND MATERIALS • Two 6-volt batteries • One 9-volt battery Actually, any size batteries will suffice for this experiment, but it is recommended to have at least two different voltages available to make it more interesting. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 5: “Series and Parallel Circuits” Lessons In Electric Circuits, Volume 1, chapter 11: “Batteries and Power Systems” LEARNING OBJECTIVES • How to connect batteries to obtain different voltage levels SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Connecting components in series means to connect them in-line with each other, so that there is but a single path for electrons to flow through them all. If you connect batteries so that the positive of one connects to the negative of the other, you will find that their respective voltages add. Measure the voltage across each battery individually as they are connected, then measure the total voltage across them both, like this: Try connecting batteries of different sizes in series with each other, for instance a 6-volt battery with a 9-volt battery. What is the total voltage in this case? Try reversing the terminal connections of just one of these batteries, so that they are opposing each other like this: How does the total voltage compare in this situation to the previous one with both batteries “aiding?” Note the polarity of the total voltage as indicated by the voltmeter indication and test probe orientation. Remember, if the meter’s digital indication is a positive number, the red probe is positive (+) and the black probe negative (-); if the indication is a negative number, the polarity is “backward” (red=negative, black=positive). Analog meters simply will not read properly if reverse-connected, because the needle tries to move the wrong direction (left instead of right). Can you predict what the overall voltage polarity will be, knowing the polarities of the individual batteries and their respective strengths. 3.03: Parallel Batteries PARTS AND MATERIALS • Four 6-volt batteries • 12-volt light bulb, 25 or 50 watt • Lamp socket High-wattage 12-volt lamps may be purchased from recreational vehicle (RV) and boating supply stores. Common sizes are 25 watt and 50 watt. This lamp will be used as a “heavy” load for your batteries (heavyload = one that draws substantial current). A regular household (120 volt) lamp socket will work just fine for these low-voltage “RV” lamps. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 5: “Series and Parallel Circuits” Lessons In Electric Circuits, Volume 1, chapter 11: “Batteries and Power Systems” LEARNING OBJECTIVES • Voltage source regulation • Boosting current capacity through parallel connections SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Begin this experiment by connecting one 6-volt battery to the lamp. The lamp, designed to operate on 12 volts, should glow dimly when powered by the 6-volt battery. Use your voltmeter to read voltage across the lamp like this: The voltmeter should register a voltage lower than the usual voltage of the battery. If you use your voltmeter to read the voltage directly at the battery terminals, you will measure a low voltage there as well. Why is this? The large current drawn by the high-power lamp causes the voltage at the battery terminals to “sag” or “droop,” due to voltage dropped across resistance internal to the battery. We may overcome this problem by connecting batteries in parallel with each other, so that each battery only has to supply a fraction of the total current demanded by the lamp. Parallel connections involve making all the positive (+) battery terminals electrically common to each other by connection through jumper wires, and all negative (-) terminals common to each other as well. Add one battery at a time in parallel, noting the lamp voltage with the addition of each new, parallel-connected battery: There should also be a noticeable difference in light intensity as the voltage “sag” is improved. Try measuring the current of one battery and comparing it to the total current (light bulb current). Shown here is the easiest way to measure single-battery current: By breaking the circuit for just one battery, and inserting our ammeter within that break, we intercept the current of that one battery and are therefore able to measure it. Measuring total current involves a similar procedure: make a break somewhere in the path that total current must take, then insert the ammeter within than break: Note the difference in current between the single-battery and total measurements. To obtain maximum brightness from the light bulb, a series-parallel connection is required. Two 6-volt batteries connected series-aiding will provide 12 volts. Connecting two of these series-connected battery pairs in parallel improves their current-sourcing ability for minimum voltage sag:
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.04%3A_Voltage_Divider.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Calculator (or pencil and paper for doing arithmetic) • 6-volt battery • Assortment of resistors between 1 KΩ and 100 kΩ in value I’m purposely restricting the resistance values between 1 kΩ and 100 kΩ for the sake of obtaining accurate voltage and current readings with your meter. With very low resistance values, the internal resistance of the ammeter has a significant impact on measurement accuracy. Very high resistance values may cause problems for voltage measurement, the internal resistance of the voltmeter substantially changing circuit resistance when it is connected in parallel with a high-value resistor. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 6: “Divider Circuits and Kirchhoff’s Laws” LEARNING OBJECTIVES • Voltmeter use • Ammeter use • Ohmmeter use • Use of Ohm’s Law • Use of Kirchhoff’s Voltage Law (“KVL”) • Voltage divider design SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Shown here are three different methods of circuit construction: on a breadboard, on a terminal strip, and “free-form.” Try building the same circuit each way to familiarize yourself with the different construction techniques and their respective merits. The “free-form” method—where all components are connected together with “alligator-” style jumper wires—is the least professional, but appropriate for a simple experiment such as this. Breadboard construction is versatile and allows for high component density (many parts in a small space), but is quite temporary. Terminal strips offer a much more permanent form of construction at the cost of low component density. Select three resistors from your resistor assortment and measure the resistance of each one with an ohmmeter. Note these resistance values with pen and paper, for reference in your circuit calculations. Connect the three resistors in series, and to the 6-volt battery, as shown in the illustrations. Measure battery voltage with a voltmeter after the resistors have been connected to it, noting this voltage figure on paper as well. It is advisable to measure battery voltage while its powering the resistor circuit because this voltage may differ slightly from a no-load condition. We saw this effect exaggerated in the “parallel battery” experiment while powering a high-wattage lamp: battery voltage tends to “sag” or “droop” under load. Although this three-resistor circuit should not present a heavy enough load (not enough current drawn) to cause significant voltage “sag,” measuring battery voltage under load is a good scientific practice because it provides more realistic data. Use Ohm’s Law (I=E/R) to calculate circuit current, then verify this calculated value by measuring current with an ammeter like this (“terminal strip” version of the circuit shown as an arbitrary choice in construction method): If your resistor values are indeed between 1 kΩ and 100 kΩ, and the battery voltage approximately 6 volts, the current should be a very small value, in the milliamp (mA) or microamp (µA) range. When you measure current with a digital meter, the meter may show the appropriate metric prefix symbol (m or µ) in some corner of the display. These metric prefix telltales are easy to overlook when reading the display of a digital meter, so pay close attention! The measured value of current should agree closely with your Ohm’s Law calculation. Now, take that calculated value for current and multiply it by the respective resistances of each resistor to predict their voltage drops (E=IR). Switch you multimeter to the “voltage” mode and measure the voltage dropped across each resistor, verifying the accuracy of your predictions. Again, there should be close agreement between the calculated and measured voltage figures. Each resistor voltage drop will be some fraction or percentage of the total voltage, hence the name voltage divider given to this circuit. This fractional value is determined by the resistance of the particular resistor and the total resistance. If a resistor drops 50% of the total battery voltage in a voltage divider circuit, that proportion of 50% will remain the same as long as the resistor values are not altered. So, if the total voltage is 6 volts, the voltage across that resistor will be 50% of 6, or 3 volts. If the total voltage is 20 volts, that resistor will drop 10 volts, or 50% of 20 volts. The next part of this experiment is a validation of Kirchhoff’s Voltage Law. For this, you need to identify each unique point in the circuit with a number. Points that are electrically common (directly connected to each other with insignificant resistance between) must bear the same number. An example using the numbers 0 through 3 is shown here in both illustrative and schematic form. In the illustration, I show how points in the circuit may be labeled with small pieces of tape, numbers written on the tape: Using a digital voltmeter (this is important!), measure voltage drops around the loop formed by the points 0-1-2-3-0. Write on paper each of these voltages, along with its respective sign as indicated by the meter. In other words, if the voltmeter registers a negative voltage such as -1.325 volts, you should write that figure as a negative number. Do not reverse the meter probe connections with the circuit to make the number read “correctly.” Mathematical sign is very significant in this phase of the experiment! Here is a sequence of illustrations showing how to “step around” the circuit loop, starting and ending at point 0: Using the voltmeter to “step” around the circuit in this manner yields three positive voltage figures and one negative: These figures, algebraically added (“algebraically” = respecting the signs of the numbers), should equal zero. This is the fundamental principle of Kirchhoff’s Voltage Law: that the algebraic sum of all voltage drops in a “loop” add to zero. It is important to realize that the “loop” stepped around does not have to be the same path that current takes in the circuit, or even a legitimate current path at all. The loop in which we tally voltage drops can be any collection of points, so long as it begins and ends with the same point. For example, we may measure and add the voltages in the loop 1-2-3-1, and they will form a sum of zero as well: Try stepping between any set of points, in any order, around your circuit and see for yourself that the algebraic sum always equals zero. This Law holds true no matter what the configuration of the circuit: series, parallel, series-parallel, or even an irreducible network. Kirchhoff’s Voltage Law is a powerful concept, allowing us to predict the magnitude and polarity of voltages in a circuit by developing mathematical equations for analysis based on the truth of all voltages in a loop adding up to zero. This experiment is intended to give empirical evidence for and a deep understanding of Kirchhoff’s Voltage Law as a general principle. COMPUTER SIMULATION Netlist (make a text file containing the following text, verbatim): This computer simulation is based on the point numbers shown in the previous diagrams for illustrating Kirchhoff’s Voltage Law (points 0 through 3). Resistor values were chosen to provide 50%, 30%, and 20% proportions of total voltage across R1, R2, and R3, respectively. Feel free to modify the voltage source value (in the “.dc” line, shown here as 6 volts), and/or the resistor values. When run, SPICE will print a line of text containing four voltage figures, then another line of text containing three voltage figures, along with lots of other text lines describing the analysis process. Add the voltage figures in each line to see that the sum is zero.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.05%3A_Current_Divider.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Calculator (or pencil and paper for doing arithmetic) • 6-volt battery • Assortment of resistors between 1 KΩ and 100 kΩ in value CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 6: “Divider Circuits and Kirchhoff’s Laws” LEARNING OBJECTIVES • Voltmeter use • Ammeter use • Ohmmeter use • Use of Ohm’s Law • Use of Kirchhoff’s Current Law (KCL) • Current divider design SCHEMATIC DIAGRAM ILLUSTRATION Normally, it is considered improper to secure more than two wires under a single terminal strip screw. In this illustration, I show three wires joining at the top screw of the rightmost lug used on this strip. This is done for the ease of proving a concept (of current summing at a circuit node), and does not represent professional assembly technique. The non-professional nature of the “free-form” construction method merits no further comment. INSTRUCTIONS Once again, I show different methods of constructing the same circuit: breadboard, terminal strip, and “free-form.” Experiment with all these construction formats and become familiar with their respective advantages and disadvantages. Select three resistors from your resistor assortment and measure the resistance of each one with an ohmmeter. Note these resistance values with pen and paper, for reference in your circuit calculations. Connect the three resistors in parallel to and each other, and with the 6-volt battery, as shown in the illustrations. Measure battery voltage with a voltmeter after the resistors have been connected to it, noting this voltage figure on paper as well. It is advisable to measure battery voltage while its powering the resistor circuit because this voltage may differ slightly from a no-load condition. Measure voltage across each of the three resistors. What do you notice? In a series circuit, current is equal through all components at any given time. In a parallel circuit, voltage is the common variable between all components. Use Ohm’s Law (I=E/R) to calculate current through each resistor, then verify this calculated value by measuring current with a digital ammeter. Place the red probe of the ammeter at the point where the positive (+) ends of the resistors connect to each other and lift one resistor wire at a time, connecting the meter’s black probe to the lifted wire. In this manner, measure each resistor current, noting both the magnitude of the current and the polarity. In these illustrations, I show an ammeter used to measure the current through R1: Measure current for each of the three resistors, comparing with the current figures calculated previously. With the digital ammeter connected as shown, all three indications should be positive, not negative. Now, measure total circuit current, keeping the ammeter’s red probe on the same point of the circuit, but disconnecting the wire leading to the positive (+) side of the battery and touching the black probe to it: Note both the magnitude and the sign of the current as indicated by the ammeter. Add this figure (algebraically) to the three resistor currents. What do you notice about the result that is similar to the Kirchhoff’s Voltage Law experiment? Kirchhoff’s Current Law is to currents “summing” at a point (node) in a circuit, just as Kirchhoff’s Voltage Law is to voltages adding in a series loop: in both cases, the algebraic sum is equal to zero. This Law is also very useful in the mathematical analysis of circuits. Along with Kirchhoff’s Voltage Law, it allows us to generate equations describing several variables in a circuit, which may then be solved using a variety of mathematical techniques. Now consider the four current measurements as all positive numbers: the first three representing the current through each resistor, and the fourth representing total circuit current as a positive sum of the three “branch” currents. Each resistor (branch) current is a fraction, or percentage, of the total current. This is why a parallel resistor circuit is often called a current divider. Disconnect the battery from the rest of the circuit, and measure resistance across the parallel resistors. You may read total resistance across any of the individual resistors’ terminals and obtain the same indication: it will be a value less than any of the individual resistor values. This is often surprising to new students of electricity, that you read the exact same (total) resistance figure when connecting an ohmmeter across any one of a set of parallel-connected resistors. It makes sense, though, if you consider the points in a parallel circuit in terms of electrical commonality. All parallel components are connected between two sets of electrically common points. Since the meter cannot distinguish between points common to each other by way of direct connection, to read resistance across one resistor is to read the resistance of them all. The same is true for voltage, which is why battery voltage could be read across any one of the resistors as easily as it could be read across the battery terminals directly. If you divide the battery voltage (previously measured) by this total resistance figure, you should obtain a figure for total current (I=E/R) closely matching the measured figure. The ratio of resistor current to total current is the same as the ratio of total resistance to individual resistance. For example, if a 10 kΩ resistor is part of a current divider circuit with a total resistance of 1 kΩ, that resistor will conduct 1/10 of the total current, whatever value that current total happens to be. COMPUTER SIMULATION Schematic with SPICE node numbers: Ammeters in SPICE simulations are actually zero-voltage sources inserted in the paths of electron flow. You will notice the voltage sources Vir1, Vir2, and Vir3 are set to 0 volts in the netlist. When electrons enter the negative side of one of these “dummy” batteries and out the positive, the battery’s current indication will be a positive number. In other words, these 0-volt sources are to be regarded as ammeters with the red probe on the long-line side of the battery symbol and the black probe on the short-line side. Netlist (make a text file containing the following text, verbatim): When run, SPICE will print a line of text containing four current figures, the first current representing the total as a negative quantity, and the other three representing currents for resistors R1, R2, and R3. When algebraically added, the one negative figure and the three positive figures will form a sum of zero, as described by Kirchhoff’s Current Law.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.06%3A_Potentiometer_as_a_Voltage_Divider.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Two 6-volt batteries • Carbon pencil “lead” for a mechanical-style pencil • Potentiometer, single turn, 5 kΩ to 50 kΩ, linear taper (Radio Shack catalog # 271-1714 through 271-1716) • Potentiometer, multi turn, 1 kΩ to 20 kΩ, (Radio Shack catalog # 271-342, 271-343, 900-8583, or 900-8587 through 900-8590) Potentiometers are variable voltage dividers with a shaft or slide control for setting the division ratio. They are manufactured in panel-mount as well as breadboard (printed-circuit board) mount versions. Any style of potentiometer will suffice for this experiment. If you salvage a potentiometer from an old radio or other audio device, you will likely be getting what is called an audio taper potentiometer. These potentiometers exhibit a logarithmic relationship between division ratio and shaft position. By contrast, a linear potentiometer exhibits a direct correlation between shaft position and voltage division ratio. I highly recommend a linear potentiometer for this experiment, and for most experiments in general. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 6: “Divider Circuits and Kirchhoff’s Laws” LEARNING OBJECTIVES • Voltmeter use • Ohmmeter use • Voltage divider design and function • How voltages add in series SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Begin this experiment with the pencil “lead” circuit. Pencils use a rod made of a graphite-clay mixture, not lead (the metal), to make black marks on paper. Graphite, being a mediocre electrical conductor, acts as a resistor connected across the battery by the two alligator-clip jumper wires. Connect the voltmeter as shown and touch the red test probe to the graphite rod. Move the red probe along the length of the rod and notice the voltmeter’s indication change. What probe position gives the greatest voltage indication? Essentially, the rod acts as a pair of resistors, the ratio between the two resistances established by the position of the red test probe along the rod’s length: Now, change the voltmeter connection to the circuit so as to measure voltage across the “upper resistor” of the pencil lead, like this: Move the black test probe position along the length of the rod, noting the voltmeter indication. Which position gives the greatest voltage drop for the meter to measure? Does this differ from the previous arrangement? Why? Manufactured potentiometers enclose a resistive strip inside a metal or plastic housing, and provide some kind of mechanism for moving a “wiper” across the length of that resistive strip. Here is an illustration of a rotary potentiometer’s construction: Some rotary potentiometers have a spiral resistive strip, and a wiper that moves axially as it rotates, so as to require multiple turns of the shaft to drive the wiper from one end of the potentiometer’s range to the other. Multi-turn potentiometers are used in applications where precise setting is important. Linear potentiometers also contain a resistive strip, the only difference being the wiper’s direction of travel. Some linear potentiometers use a slide mechanism to move the wiper, while others a screw, to facilitate multiple-turn operation: It should be noted that not all linear potentiometers have the same pin assignments. On some, the middle pin is the wiper. Set up a circuit using a manufactured potentiometer, not the “home-made” one made from a pencil lead. You may use any form of construction that is convenient. Measure battery voltage while powering the potentiometer, and make note of this voltage figure on paper. Measure voltage between the wiper and the potentiometer end connected to the negative (-) side of the battery. Adjust the potentiometer mechanism until the voltmeter registers exactly 1/3 of total voltage. For a 6-volt battery, this will be approximately 2 volts. Now, connect two batteries in a series-aiding configuration, to provide approximately 12 volts across the potentiometer. Measure the total battery voltage, and then measure the voltage between the same two points on the potentiometer (wiper and negative side). Divide the potentiometer’s measured output voltage by the measured total voltage. The quotient should be 1/3, the same voltage division ratio as was set previously:
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.07%3A_Potentiometer_as_a_Rheostat.txt	princeton-nlp/TextbookChapters	Learning Objectives • Rheostat use • Wiring a potentiometer as a rheostat • Simple motor speed control • Use of voltmeter over ammeter to verify a continuous circuit Parts and Materials • 6-volt battery • Potentiometer, single turn, 5 kΩ, linear taper (Radio Shack catalog # 271-1714) • Small “hobby” motor, permanent-magnet type (Radio Shack catalog # 273-223 or equivalent) For this experiment, you will need a relatively low-value potentiometer, certainly not more than 5 kΩ. Instructions for Potentiometer Wiring Potentiometers find their most sophisticated application as voltage dividers, where shaft position determines a specific voltage division ratio. However, there are applications where we don’t necessarily need a variable voltage divider, but merely a variable resistor: a two-terminal device. Technically, a variable resistor is known as a rheostat, but potentiometers can be made to function as rheostats quite easily. In its simplest configuration, a potentiometer may be used as a rheostat by simply using the wiper terminal and one of the other terminals, the third terminal left unconnected and unused: Moving the potentiometer control in the direction that brings the wiper closest to the other used terminal results in a lower resistance. The direction of motion required to increase or decrease resistance may be changed by using a different set of terminals: Be careful, though, that you don’t use the two outer terminals, as this will result in no change in resistance as the potentiometer shaft is turned. In other words, it will no longer function as a variable resistance: Build the circuit as shown in the schematic and illustration, using just two terminals on the potentiometer, and see how motor speed may be controlled by adjusting shaft position. Experiment with different terminal connections on the potentiometer, noting the changes in motor speed control. If your potentiometer has a high resistance (as measured between the two outer terminals), the motor might not move at all until the wiper is brought very close to the connected outer terminal. As you can see, motor speed may be made variable using a series-connected rheostat to change total circuit resistance and limit total current. This simple method of motor speed control, however, is inefficient, as it results in substantial amounts of power being dissipated (wasted) by the rheostat. A much more efficient means of motor control relies on fast “pulsing” of power to the motor, using a high-speed switching device such as a transistor. A similar method of power control is used in household light “dimmer” switches. Unfortunately, these techniques are much too sophisticated to explore at this point in the experiments. When a potentiometer is used as a rheostat, the “unused” terminal is often connected to the wiper terminal, like this: At first, this seems rather pointless, as it has no impact on resistance control. You may verify this fact for yourself by inserting another wire in your circuit and comparing motor behavior before and after the change: If the potentiometer is in good working order, this additional wire makes no difference whatsoever. However, if the wiper ever loses contact with the resistive strip inside the potentiometer, this connection ensures the circuit does not completely open: that there will still be a resistive path for current through the motor. In some applications, this may be an important. Old potentiometers tend to suffer from intermittent losses of contact between the wiper and the resistive strip, and if a circuit cannot tolerate the complete loss of continuity (infinite resistance) created by this condition, that “extra” wire provides a measure of protection by maintaining circuit continuity. You may simulate such a wiper contact “failure” by disconnecting the potentiometer’s middle terminal from the terminal strip, measuring voltage across the motor to ensure there is still power getting to it, however small: Using Motor Voltage is a Safer Alternative to Measuring Circuit Current It would have been valid to measure circuit current instead of motor voltage to verify a completed circuit, but this is a safer method because it does not involve breaking the circuit to insert an ammeter in series. Whenever an ammeter is used, there is risk of causing a short circuit by connecting it across a substantial voltage source, possibly resulting in instrument damage or personal injury. Voltmeters lack this inherent safety risk, and so whenever a voltage measurement may be made instead of a current measurement to verify the same thing, it is the wiser choice.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.08%3A_Precision_Potentiometer.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Two single-turn, linear-taper potentiometers, 5 kΩ each (Radio Shack catalog # 271-1714) • One single-turn, linear-taper potentiometer, 50 kΩ (Radio Shack catalog # 271-1716) • Plastic or metal mounting box • Three “banana” jack style binding posts, or other terminal hardware, for connection to potentiometer circuit (Radio Shack catalog # 274-662 or equivalent) This is a project useful to those who want a precision potentiometer without spending a lot of money. Ordinarily, multi-turn potentiometers are used to obtain precise voltage division ratios, but a cheaper alternative exists using multiple, single-turn (sometimes called “3/4-turn”) potentiometers connected together in a compound divider network. Because this is a useful project, I recommend building it in permanent form using some form of project enclosure. Suppliers such as Radio Shack offer nice project boxes, but boxes purchased at a general hardware store are much less expensive, if a bit ugly. The ultimate in low cost for a new box are the plastic boxes sold as light switch and receptacle boxes for household electrical wiring. “Banana” jacks allow for the temporary connection of test leads and jumper wires equipped with matching “banana” plug ends. Most multimeter test leads have this style of plug for insertion into the meter jacks. Banana plugs are so named because of their oblong appearance formed by spring steel strips, which maintain firm contact with the jack walls when inserted. Some banana jacks are called binding postsbecause they also allow plain wires to be firmly attached. Binding posts have screw-on sleeves that fit over a metal post. The sleeve is used as a nut to secure a wire wrapped around the post, or inserted through a perpendicular hole drilled through the post. A brief inspection of any binding post will clarify this verbal description. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 6: “Divider Circuits and Kirchhoff’s Laws” LEARNING OBJECTIVES • Soldering practice • Potentiometer function and operation SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS It is essential that the connecting wires be soldered to the potentiometer terminals, not twisted or taped. Since potentiometer action is dependent on resistance, the resistance of all wiring connections must be carefully controlled to a bare minimum. Soldering ensures a condition of low resistance between joined conductors, and also provides very good mechanical strength for the connections. When the circuit is assembled, connect a 6-volt battery to the outer two binding posts. Connect a voltmeter between the “wiper” post and the battery’s negative (-) terminal. This voltmeter will measure the “output” of the circuit. The circuit works on the principle of compressed range: the voltage output range of this circuit available by adjusting potentiometer R3 is restricted between the limits set by potentiometers R1 and R2. In other words, if R1 and R2 were set to output 5 volts and 3 volts, respectively, from a 6-volt battery, the range of output voltages obtainable by adjusting R3 would be restricted from 3 to 5 volts for the full rotation of that potentiometer. If only a single potentiometer were used instead of this three-potentiometer circuit, full rotation would produce an output voltage from 0 volts to full battery voltage. The “range compression” afforded by this circuit allows for more precise voltage adjustment than would be normally obtainable using a single potentiometer. Operating this potentiometer network is more complex than using a single potentiometer. To begin, turn the R3 potentiometer fully clockwise, so that its wiper is in the full “up” position as referenced to the schematic diagram (electrically “closest” to R1‘s wiper terminal). Adjust potentiometer R1 until the upper voltage limit is reached, as indicated by the voltmeter. Turn the R3 potentiometer fully counter-clockwise, so that its wiper is in the full “down” position as referenced to the schematic diagram (electrically “closest” to R2‘s wiper terminal). Adjust potentiometer R2until the lower voltage limit is reached, as indicated by the voltmeter. When either the R1 or the R2 potentiometer is adjusted, it interferes with the prior setting of the other. In other words, if R1 is initially adjusted to provide an upper voltage limit of 5.000 volts from a 6 volt battery, and then R2 is adjusted to provide some lower limit voltage different from what it was before, R1 will no longer be set to 5.000 volts. To obtain precise upper and lower voltage limits, turn R3 fully clockwise to read and adjust the voltage of R1, then turn R3 fully counter-clockwise to read and adjust the voltage of R2, repeating as necessary. Technically, this phenomenon of one adjustment affecting the other is known as interaction, and it is usually undesirable due to the extra effort required to set and re-set the adjustments. The reason that R1 and R2were specified as 10 times less resistance than R3 is to minimize this effect. If all three potentiometers were of equal resistance value, the interaction between R1 and R2 would be more severe, though manageable with patience. Bear in mind that the upper and lower voltage limits need not be set precisely in order for this circuit to achieve its goal of increased precision. So long as R3‘s adjustment range is compressed to some lesser value than full battery voltage, we will enjoy greater precision than a single potentiometer could provide. Once the upper and lower voltage limits have been set, potentiometer R3 may be adjusted to produce an output voltage anywhere between those limits.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.09%3A_Rheostat_Range_Limiting.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Several 10 kΩ resistors • One 10 kΩ potentiometer, linear taper (Radio Shack catalog # 271-1715) CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 5: “Series and Parallel Circuits” Lessons In Electric Circuits, Volume 1, chapter 7: “Series-Parallel Combination Circuits” Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” LEARNING OBJECTIVES • Series-parallel resistances • Calibration theory and practice SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS This experiment explores the different resistance ranges obtainable from combining fixed-value resistors with a potentiometer connected as a rheostat. To begin, connect a 10 kΩ potentiometer as a rheostat with no other resistors connected. Adjusting the potentiometer through its full range of travel should result in a resistance that varies smoothly from 0 Ω to 10,000 Ω: Suppose we wanted to elevate the lower end of this resistance range so that we had an adjustable range from 10 kΩ to 20 kΩ with a full sweep of the potentiometer’s adjustment. This could be easily accomplished by adding a 10 kΩ resistor in series with the potentiometer. Add one to the circuit as shown and re-measure total resistance while adjusting the potentiometer: A shift in the low end of an adjustment range is called a zero calibration, in metrological terms. With the addition of a series 10 kΩ resistor, the “zero point” was shifted upward by 10,000 Ω. The difference between high and low ends of a range—called the span of the circuit—has not changed, though: a range of 10 kΩ to 20 kΩ has the same 10,000 Ω span as a range of 0 Ω to 10 kΩ. If we wish to shift the span of this rheostat circuit as well, we must change the range of the potentiometer itself. We could replace the potentiometer with one of another value, or we could simulate a lower-value potentiometer by placing a resistor in parallel with it, diminishing its maximum obtainable resistance. This will decrease the span of the circuit from 10 kΩ to something less. Add a 10 kΩ resistor in parallel with the potentiometer, to reduce the span to one-half of its former value: from 10 KΩ to 5 kΩ. Now the calibrated resistance range of this circuit will be 10 kΩ to 15 kΩ: There is nothing we can do to increase the span of this rheostat circuit, short of replacing the potentiometer with another of greater total resistance. Adding resistors in parallel can only decrease the span. However, there is no such restriction with calibrating the zero point of this circuit, as it began at 0 Ω and may be made as great as we wish by adding resistance in series. A multitude of resistance ranges may be obtained using only 10 KΩ fixed-value resistors, if we are creative with series-parallel combinations of them. For instance, we can create a range of 7.5 kΩ to 10 kΩ by building the following circuit: Creating a custom resistance range from fixed-value resistors and a potentiometer is a very useful technique for producing precise resistances required for certain circuits, especially meter circuits. In many electrical instruments—multimeters especially—resistance is the determining factor for the instrument’s range of measurement. If an instrument’s internal resistance values are not precise, neither will its indications be. Finding a fixed-value resistor of just the right resistance for placement in an instrument circuit design is unlikely, so custom resistance “networks” may need to be built to provide the desired resistance. Having a potentiometer as part of the resistor network provides a means of correction if the network’s resistance should “drift” from its original value. Designing the network for minimum span ensures that the potentiometer’s effect will be small, so that precise adjustment is possible and so that accidental movement of its mechanism will not result in severe calibration errors. Experiment with different resistor “networks” and note the effects on total resistance range. 3.10: Thermoelectricity PARTS AND MATERIALS • Length of bare (uninsulated) copper wire • Length of bare (uninsulated) iron wire • Candle • Ice cubes Iron wire may be obtained from a hardware store. If some cannot be found, aluminum wire also works. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 9: “Electrical Instrumentation Signals” LEARNING OBJECTIVES • Thermocouple function and purpose SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Twist one end of the iron wire together with one end of the copper wire. Connect the free ends of these wires to respective terminals on a terminal strip. Set your voltmeter to its most sensitive range and connect it to the terminals where the wires attach. The meter should indicate nearly zero voltage. What you have just constructed is a thermocouple: a device which generates a small voltage proportional to the temperature difference between the tip and the meter connection points. When the tip is at a temperature equal to the terminal strip, there will be no voltage produced, and thus no indication seen on the voltmeter. Light a candle and insert the twisted-wire tip into the flame. You should notice an indication on your voltmeter. Remove the thermocouple tip from the flame and let cool until the voltmeter indication is nearly zero again. Now, touch the thermocouple tip to an ice cube and note the voltage indicated by the meter. Is it a greater or lesser magnitude than the indication obtained with the flame? How does the polarity of this voltage compare with that generated by the flame? After touching the thermocouple tip to the ice cube, warm it by holding it between your fingers. It may take a short while to reach body temperature, so be patient while observing the voltmeter’s indication. A thermocouple is an application of the Seebeck effect: the production of a small voltage proportional to a temperature gradient along the length of a wire. This voltage is dependent upon the magnitude of the temperature difference and the type of wire. Directly measuring the Seebeck voltage produced along a length of continuous wire from a temperature gradient is quite difficult, and so will not be attempted in this experiment. Thermocouples, being made of two dissimilar metals joined at one end, produce a voltage proportional to the temperature of the junction. The temperature gradient along both wires resulting from a constant temperature at the junction produces different Seebeck voltages along those wires’ lengths, because the wires are made of different metals. The resultant voltage between the two free wire ends is the difference between the two Seebeck voltages: Thermocouples are widely used as temperature-sensing devices because the mathematical relationship between temperature difference and resultant voltage is both repeatable and fairly linear. By measuring voltage, it is possible to infer temperature. Different ranges of temperature measurement are possible by selecting different metal pairs to be joined together.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.11%3A_Make_Your_Own_Multimeter.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Sensitive meter movement (Radio Shack catalog # 22-410) • Selector switch, single-pole, multi-throw, break-before-make (Radio Shack catalog # 275-1386 is a 2-pole, 6-position unit that works well) • Multi-turn potentiometers, PCB mount (Radio Shack catalog # 271-342 and 271-343 are 15-turn, 1 kΩ and 10 kΩ “trimmer” units, respectively) • Assorted resistors, preferably high-precision metal film or wire-wound types (Radio Shack catalog # 271-309 is an assortment of metal-film resistors, +/- 1% tolerance) • Plastic or metal mounting box • Three “banana” jack style binding posts, or other terminal hardware, for connection to potentiometer circuit (Radio Shack catalog # 274-662 or equivalent) The most important and expensive component in a meter is the movement: the actual needle-and-scale mechanism whose task it is to translate an electrical current into mechanical displacement where it may be visually interpreted. The ideal meter movement is physically large (for ease of viewing) and as sensitive as possible (requires minimal current to produce full-scale deflection of the needle). High-quality meter movements are expensive, but Radio Shack carries some of acceptable quality that are reasonably priced. The model recommended in the parts list is sold as a voltmeter with a 0-15 volt range, but is actually a milliammeter with a range (“multiplier”) resistor included separately. It may be cheaper to purchase an inexpensive analog meter and disassemble it for the meter movement alone. Although the thought of destroying a working multimeter in order to have parts to make your own may sound counter-productive, the goal here is learning, not meter function. I cannot specify resistor values for this experiment, as these depend on the particular meter movement and measurement ranges chosen. Be sure to use high-precision fixed-value resistors rather than carbon-composition resistors. Even if you happen to find carbon-composition resistors of just the right value(s), those values will change or “drift” over time due to aging and temperature fluctuations. Of course, if you don’t care about the long-term stability of this meter but are building it just for the learning experience, resistor precision matters little. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” LEARNING OBJECTIVES • Voltmeter design and use • Ammeter design and use • Rheostat range limiting • Calibration theory and practice • Soldering practice SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS First, you need to determine the characteristics of your meter movement. Most important is to know the full-scale deflection in milliamps or microamps. To determine this, connect the meter movement, a potentiometer, battery, and digital ammeter in series. Adjust the potentiometer until the meter movement is deflected exactly to full-scale. Read the ammeter’s display to find the full-scale current value: Be very careful not to apply too much current to the meter movement, as movements are very sensitive devices and easily damaged by overcurrent. Most meter movements have full-scale deflection current ratings of 1 mA or less, so choose a potentiometer value high enough to limit current appropriately, and begin testing with the potentiometer turned to maximum resistance. The lower the full-scale current rating of a movement, the more sensitive it is. After determining the full-scale current rating of your meter movement, you must accurately measure its internal resistance. To do this, disconnect all components from the previous testing circuit and connect your digital ohmmeter across the meter movement terminals. Record this resistance figure along with the full-scale current figure obtained in the last procedure. Perhaps the most challenging portion of this project is determining the proper range resistance values and implementing those values in the form of rheostat networks. The calculations are outlined in chapter 8 of volume 1 (“Metering Circuits”), but an example is given here. Suppose your meter movement had a full-scale rating of 1 mA and an internal resistance of 400 Ω. If we wanted to determine the necessary range resistance (“Rmultiplier”) to give this movement a range of 0 to 15 volts, we would have to divide 15 volts (total applied voltage) by 1 mA (full-scale current) to obtain the total probe-to-probe resistance of the voltmeter (R=E/I). For this example, that total resistance is 15 kΩ. From this total resistance figure, we subtract the movement’s internal resistance, leaving 14.6 kΩ for the range resistor value. A simple rheostat network to produce 14.6 kΩ (adjustable) would be a 10 kΩ potentiometer in parallel with a 10 kΩ fixed resistor, all in series with another 10 kΩ fixed resistor: One position of the selector switch directly connects the meter movement between the black Common binding post and the red V/mA binding post. In this position, the meter is a sensitive ammeter with a range equal to the full-scale current rating of the meter movement. The far clockwise position of the switch disconnects the positive (+) terminal of the movement from either red binding post and shorts it directly to the negative (-) terminal. This protects the meter from electrical damage by isolating it from the red test probe, and it “dampens” the needle mechanism to further guard against mechanical shock. The shunt resistor (Rshunt) necessary for a high-current ammeter function needs to be a low-resistance unit with a high power dissipation. You will definitely not be using any 1/4 watt resistors for this unless you form a resistance network with several smaller resistors in parallel combination. If you plan on having an ammeter range in excess of 1 amp, I recommend using a thick piece of wire or even a skinny piece of sheet metal as the “resistor,” suitably filed or notched to provide just the right amount of resistance. To calibrate a home-made shunt resistor, you will need to connect the your multimeter assembly to a calibrated source of high current, or a high-current source in series with a digital ammeter for reference. Use a small metal file to shave off shunt wire thickness or to notch the sheet metal strip in small, careful amounts. The resistance of your shunt will increase with every stroke of the file, causing the meter movement to deflect more strongly. Remember that you can always approach the exact value in slower and slower steps (file strokes), but you cannot go “backward” and decrease the shunt resistance! Build the multimeter circuit on a breadboard first while determining proper range resistance values, and perform all calibration adjustments there. For final construction, solder the components onto a printed circuit board. Radio Shack sells printed circuit boards that have the same layout as a breadboard, for convenience (catalog # 276-170). Feel free to alter the component layout from what is shown. I strongly recommend that you mount the circuit board and all components in a sturdy box so that the meter is durably finished. Despite the limitations of this multimeter (no resistance function, inability to measure alternating current, and lower precision than most purchased analog multimeters), it is an excellent project to assist learning fundamental instrument principles and circuit function. A far more accurate and versatile multimeter may be constructed using many of the same parts if an amplifier circuit is added to it, so save the parts and pieces for a later experiment!
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.12%3A_Sensitive_Voltage_Detector.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • High-quality “closed-cup” audio headphones • Headphone jack: female receptacle for headphone plug (Radio Shack catalog # 274-312) • Small step-down power transformer (Radio Shack catalog # 273-1365 or equivalent, using the 6-volt secondary winding tap) • Two 1N4001 rectifying diodes (Radio Shack catalog # 276-1101) • 1 kΩ resistor • 100 kΩ potentiometer (Radio Shack catalog # 271-092) • Two “banana” jack style binding posts, or other terminal hardware, for connection to potentiometer circuit (Radio Shack catalog # 274-662 or equivalent) • Plastic or metal mounting box Regarding the headphones, the higher the “sensitivity” rating in decibels (dB), the better, but listening is believing: if you’re serious about building a detector with maximum sensitivity for small electrical signals, you should try a few different headphone models at a high-quality audio store and “listen” for which ones produce an audible sound for the lowest volume setting on a radio or CD player. Beware, as you could spend hundreds of dollars on a pair of headphones to get the absolute best sensitivity! Take heart, though: I’ve used an old pair of Radio Shack “Realistic” brand headphones with perfectly adequate results, so you don’t need to buy the best. A transformer is a device normally used with alternating current (“AC”) circuits, used to convert high-voltage AC power into low-voltage AC power, and for many other purposes. It is not important that you understand its intended function in this experiment, other than it makes the headphones become more sensitive to low-current electrical signals. Normally, the transformer used in this type of application (audio speaker impedance matching) is called an “audio transformer,” with its primary and secondary windings represented by impedance values (1000 Ω : 8 Ω) instead of voltages. An audio transformer will work, but I’ve found small step-down power transformers of 120/6 volt ratio to be perfectly adequate for the task, cheaper (especially when taken from an old thrift-store alarm clock radio), and far more rugged. The tolerance (precision) rating for the 1 kΩ resistor is irrelevant. The 100 kΩ potentiometer is a recommended option for incorporation into this project, as it gives the user control over the loudness for any given signal. Even though an audio-taper potentiometer would be appropriate for this application, it is not necessary. A linear-taper potentiometer works quite well. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” Lessons In Electric Circuits, Volume 1, chapter 10: “DC Network Analysis” (in regard to the Maximum Power Transfer Theorem) Lessons In Electric Circuits, Volume 2, chapter 9: “Transformers” Lessons In Electric Circuits, Volume 2, chapter 12: “AC Metering Circuits” LEARNING OBJECTIVES • Soldering practice • Detection of extremely small electrical signals • Using a potentiometer as a voltage divider/signal attenuator • Using diodes to “clip” voltage at some maximum level SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS The headphones, most likely being stereo units (separate left and right speakers) will have a three-contact plug. You will be connecting to only two of those three contact points. If you only have a “mono” headphone set with a two-contact plug, just connect to those two contact points. You may either connect the two stereo speakers in series or in parallel. I’ve found the series connection to work best, that is, to produce the most sound from a small signal: Solder all wire connections well. This detector system is extremely sensitive, and any loose wire connections in the circuit will add unwanted noise to the sounds produced by the measured voltage signal. The two diodes (arrow-like component symbols) connected in parallel with the transformer’s primary winding, along with the series-connected 1 kΩ resistor, work together to prevent any more than about 0.7 volts from being dropped across the primary coil of the transformer. This does one thing and one thing only: limit the amount of sound the headphones can produce. The system will work without the diodes and resistor in place, but there will be no limit to sound volume in the circuit, and the resulting sound caused by accidently connecting the test leads across a substantial voltage source (like a battery) can be deafening! Binding posts provide points of connection for a pair of test probes with banana-style plugs, once the detector components are mounted inside a box. You may use ordinary multimeter probes, or make your own probes with alligator clips at the ends for a secure connection to a circuit. Detectors are intended to be used for balancing bridge measurement circuits, potentiometric (null-balance) voltmeter circuits, and detect extremely low-amplitude AC (“alternating current”) signals in the audio frequency range. It is a valuable piece of test equipment, especially for the low-budget experimenter without an oscilloscope. It is also valuable in that it allows you to use a different bodily sense in interpreting the behavior of a circuit. For connection across any non-trivial source of voltage (1 volt and greater), the detector’s extremely high sensitivity should be attenuated. This may be accomplished by connecting a voltage divider to the “front” of the circuit: SCHEMATIC DIAGRAM ILLUSTRATION Adjust the 100 kΩ voltage divider potentiometer to about mid-range when initially sensing a voltage signal of unknown magnitude. If the sound is too loud, turn the potentiometer down and try again. If too soft, turn it up and try again. The detector produces a “click” sound whenever the test leads make or break contact with the voltage source under test. With my cheap headphones, I’ve been able to detect currents of less than 1/10 of a microamp ( < 0.1 µA). A good demonstration of the detector’s sensitivity is to touch both tests leads to the end of your tongue, with the sensitivity adjustment set to maximum. The voltage produced by metal-to-electrolyte contact (called galvanic voltage) is very small, but enough to produce soft “clicking” sounds every time the leads make and break contact on the wet skin of your tongue. Try unplugging the headphone plug from the jack (receptacle) and similarly touching it to the end of your tongue. You should still hear soft clicking sounds, but they will be much smaller in amplitude. Headphone speakers are “low impedance” devices: they require low voltage and “high” current to deliver substantial sound power. Impedance is a measure of opposition to any and all forms of electric current, including alternating current (AC). Resistance, by comparison, is a strict measure of opposition to direct current (DC). Like resistance, impedance is measured in the unit of the Ohm (Ω), but it is symbolized in equations by the capital letter “Z” rather than the capital letter “R”. We use the term “impedance” to describe the headphone’s opposition to current because it is primarily AC signals that headphones are normally subjected to, not DC. Most small signal sources have high internal impedances, some much higher than the nominal 8 Ω of the headphone speakers. This is a technical way of saying that they are incapable of supplying substantial amounts of current. As the Maximum Power Transfer Theorem predicts, maximum sound power will be delivered by the headphone speakers when their impedance is “matched” to the impedance of the voltage source. The transformer does this. The transformer also helps aid the detection of small DC signals by producing inductive “kickback” every time the test lead circuit is broken, thus “amplifying” the signal by magnetically storing up electrical energy and suddenly releasing it to the headphone speakers. I recommend building this detector in a permanent fashion (mounting all components inside of a box and providing nice test lead wires) so it may be easily used in the future. Constructed as such, it might look something like this:
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.13%3A_Potentiometric_Voltmeter.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Two 6 volt batteries • One potentiometer, single turn, 10 kΩ, linear taper (Radio Shack catalog # 271-1715) • Two high-value resistors (at least 1 MΩ each) • Sensitive voltage detector (from the previous experiment) • Analog voltmeter (from the previous experiment) The potentiometer value is not critical: anything from 1 kΩ to 100 kΩ is acceptable. If you have built the “precision potentiometer” described earlier in this chapter, it is recommended that you use it in this experiment. Likewise, the actual values of the resistors are not critical. In this particular experiment, the greater the value, the better the results. They need not be precisely equal value, either. If you have not yet built the sensitive voltage detector, it is recommended that you build one before proceeding with this experiment! It is a very useful, yet simple, a piece of test equipment that you should not be without. You can use a digital multimeter set to the “DC millivolt” (DC mV) range in lieu of a voltage detector, but the headphone-based voltage detector is more appropriate because it demonstrates how you can make precise voltage measurements without using expensive or advanced meter equipment. I recommend using your home-made multimeter for the same reason, although any voltmeter will suffice for this experiment. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” LEARNING OBJECTIVES • Voltmeter loading: its causes and its solution • Using a potentiometer as a source of variable voltage • Potentiometric method of voltage measurement SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Build the two-resistor voltage divider circuit shown on the left of the schematic diagram and of the illustration. If the two high-value resistors are of equal value, the battery’s voltage should be split in half, with approximately 3 volts dropped across each resistor. Measure the battery voltage directly with a voltmeter, then measure each resistor’s voltage drop. Do you notice anything unusual about the voltmeter’s readings? Normally, series voltage drops add to equal the total applied voltage, but in this case, you will notice a serious discrepancy. Is Kirchhoff’s Voltage Law untrue? Is this an exception to one of the most fundamental laws of electric circuits? No! What is happening is this: when you connect a voltmeter across either resistor, the voltmeter itself alters the circuit so that the voltage is not the same as with no meter connected. I like to use the analogy of an air pressure gauge used to check the pressure of a pneumatic tire. When a gauge is connected to the tire’s fill valve, it releases some air out of the tire. This affects the pressure in the tire, and so the gauge reads a slightly lower pressure than what was in the tire before the gauge was connected. In other words, the act of measuring tire pressure alters the tire’s pressure. Hopefully, though, there is so little air released from the tire during the act of measurement that the reduction in pressure is negligible. Voltmeters similarly impact the voltage they measure, by bypassing some current around the component whose voltage drop is being measured. This affects the voltage drop, but the effect is so slight that you usually don’t notice it. In this circuit, though, the effect is very pronounced. Why is this? Try replacing the two high-value resistors with two of 100 kΩ value each and repeat the experiment. Replace those resistors with two 10 KΩ units and repeat. What do you notice about the voltage readings with lower-value resistors? What does this tell you about voltmeter “impact” on a circuit in relation to that circuit’s resistance? Replace any low-value resistors with the original, high-value (>= 1 MΩ) resistors before proceeding. Try measuring voltage across the two high-value resistors—one at a time—with a digital voltmeter instead of an analog voltmeter. What do you notice about the digital meter’s readings versus the analog meter’s? Digital voltmeters typically have greater internal (probe-to-probe) resistance, meaning they draw less current than a comparable analog voltmeter when measuring the same voltage source. An ideal voltmeter would draw zero current from the circuit under test, and thus suffer no voltage “impact” problems. If you happen to have two voltmeters, try this: connect one voltmeter across one resistor, and the other voltmeter across the other resistor. The voltage readings you get will add up to the total voltage this time, no matter what the resistor values are, even though they’re different from the readings obtained from a single meter used twice. Unfortunately, though, it is unlikely that the voltage readings obtained this way are equal to the true voltage drops with no meters connected, and so it is not a practical solution to the problem. Is there any way to make a “perfect” voltmeter: one that has infinite resistance and draws no current from the circuit under test? Modern laboratory voltmeters approach this goal by using semiconductor “amplifier” circuits, but this method is too technologically advanced for the student or hobbyist to duplicate. A much simpler and much older technique is called the potentiometric or null-balance method. This involves using an adjustable voltage source to “balance” the measured voltage. When the two voltages are equal, as indicated by a very sensitive null detector, the adjustable voltage source is measured with an ordinary voltmeter. Because the two voltage sources are equal to each other, measuring the adjustable source is the same as measuring across the test circuit, except that there is no “impact” error because the adjustable source provides any current needed by the voltmeter. Consequently, the circuit under test remains unaffected, allowing measurement of its true voltage drop. Examine the following schematic to see how the potentiometric voltmeter method is implemented: The circle symbol with the word “null” written inside represents the null detector. This can be any arbitrarily sensitive meter movement or voltage indicator. Its sole purpose in this circuit is to indicate when there is zero voltage: when the adjustable voltage source (potentiometer) is precisely equal to the voltage drop in the circuit under test. The more sensitive this null detector is, the more precisely the adjustable source may be adjusted to equal the voltage under test, and the more precisely that test voltage may be measured. Build this circuit as shown in the illustration and test its operation measuring the voltage drop across one of the high-value resistors in the test circuit. It may be easier to use a regular multimeter as a null detector at first, until you become familiar with the process of adjusting the potentiometer for a “null” indication, then reading the voltmeter connected across the potentiometer. If you are using the headphone-based voltage detector as your null meter, you will need to intermittently make and break contact with the circuit under test and listen for “clicking” sounds. Do this by firmly securing one of the test probes to the test circuit and momentarily touching the other test probe to the other point in the test circuit again and again, listening for sounds in the headphones indicating a difference of voltage between the test circuit and the potentiometer. Adjust the potentiometer until no clicking sounds can be heard from the headphones. This indicates a “null” or “balanced” condition, and you may read the voltmeter indication to see how much voltage is dropped across the test circuit resistor. Unfortunately, the headphone-based null detector provides no indication of whether the potentiometer voltage is greater than, or less than the test circuit voltage, so you will have to listen for decreasing “click” intensity while turning the potentiometer to determine if you need to adjust the voltage higher or lower. You may find that a single-turn (“3/4 turn”) potentiometer is too coarse of an adjustment device to accurately “null” the measurement circuit. A multi-turn potentiometer may be used instead of the single-turn unit for greater adjustment precision, or the “precision potentiometer” circuit described in an earlier experiment may be used. Prior to the advent of amplified voltmeter technology, the potentiometric method was the only method for making highly accurate voltage measurements. Even now, electrical standards laboratories make use of this technique along with the latest meter technology to minimize meter “impact” errors and maximize measurement accuracy. Although the potentiometric method requires more skill to use than simply connecting a modern digital voltmeter across a component, and is considered obsolete for all but the most precise measurement applications, it is still a valuable learning process for the new student of electronics, and a useful technique for the hobbyist who may lack expensive instrumentation in their home laboratory. COMPUTER SIMULATION Schematic with SPICE node numbers: Netlist (make a text file containing the following text, verbatim): This SPICE simulation shows the actual voltage across R2 of the test circuit, the null detector’s voltage, and the voltage across the adjustable voltage source, as that source is adjusted from 0 volts to 6 volts in 0.5 volt steps. In the output of this simulation, you will notice that the voltage across R2 is impacted significantly when the measurement circuit is unbalanced, returning to its true voltage only when there is practically zero voltage across the null detector. At that point, of course, the adjustable voltage source is at a value of 3.000 volts: precisely equal to the (unaffected) test circuit voltage drop. What is the lesson to be learned from this simulation? That a potentiometric voltmeter avoids impacting the test circuit only when it is in a condition of perfect balance (“null”) with the test circuit !
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.14%3A_4-wire_Resistance_Measurement.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • 6-volt battery • Electromagnet made from experiment in previous chapter, or a large spool of wire It would be ideal in this experiment to have two meters: one voltmeter and one ammeter. For experimenters on a budget, this may not be possible. Whatever ammeter is used should be capable measuring at least a few amps of current. A 6-volt “lantern” battery essentially short-circuited by a long piece of wire may produce currents of this magnitude, and your ammeter needs to be capable of measuring it without blowing a fuse or sustaining other damage. Make sure the highest current range on the meter is at least 5 amps! CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 8: “DC Metering Circuits” LEARNING OBJECTIVES • Operating principle of Kelvin (4-wire) resistance measurement • How to measure low resistances with common test equipment SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Although this experiment is best performed with two meters, and indeed is shown as such in the schematic diagram and illustration, one multimeter is sufficient. Most ohmmeters operate on the principle of applying a small voltage across an unknown resistance (Runknown) and inferring resistance from the amount of current drawn by it. Except in special cases such as the megger, both the voltage and current quantities employed by the meter are quite small. This presents a problem for measurement of low resistances, as a low resistance specimen may be of much smaller resistance value than the meter circuitry itself. Imagine trying to measure the diameter of a cotton thread with a yardstick, or measuring the weight of a coin with a scale built for weighing freight trucks, and you will appreciate the problem at hand. One of the many sources of error in measuring small resistances with an ordinary ohmmeter is the resistance of the ohmmeter’s own test leads. Being part of the measurement circuit, the test leads may contain more resistance than the resistance of the test specimen, incurring significant measurement error by their presence: One solution is called the Kelvin, or 4-wire, resistance measurement method. It involves the use of an ammeter and voltmeter, determining specimen resistance by Ohm’s Law calculation. A current is passed through the unknown resistance and measured. The voltage dropped across the resistance is measured by the voltmeter, and resistance calculated using Ohm’s Law (R=E/I). Very small resistances may be measured easily by using large current, providing a more easily measured voltage drop from which to infer resistance than if a small current were used. Because only the voltage dropped by the unknown resistance is factored into the calculation—not the voltage dropped across the ammeter’s test leads or any other connecting wires carrying the main current—errors otherwise caused by these stray resistances are completely eliminated. First, select a suitably low resistance specimen to use in this experiment. I suggest the electromagnet coil specified in the last chapter or a spool of wire where both ends may be accessed. Connect a 6-volt battery to this specimen, with an ammeter connected in series. WARNING: the ammeter used should be capable of measuring at least 5 amps of current so that it will not be damaged by the (possibly) high current generated in this near-short circuit condition. If you have a second meter, use it to measure the voltage across the specimen’s connection points, as shown in the illustration, and record both meters’ indications. If you have only one meter, use it to measure current first, recording its indication as quickly as possible, then immediately opening (breaking) the circuit. Switch the meter to its voltage mode, connect it across the specimen’s connection points, and re-connect the battery, quickly noting the voltage indication. You don’t want to leave the battery connected to the specimen for any longer than necessary for obtaining meter measurements, as it will begin to rapidly discharge due to the high circuit current, thus compromising measurement accuracy when the meter is re-configured and the circuit closed once more for the next measurement. When two meters are used, this is not as significant an issue, because the current and voltage indications may be recorded simultaneously. Take the voltage measurement and divide it by the current measurement. The quotient will be equal to the specimen’s resistance in ohms.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.15%3A_A_Very_Simple_Computer.txt	princeton-nlp/TextbookChapters	PARTS AND MATERIALS • Three batteries, each one with a different voltage • Three equal-value resistors, between 10 kΩ and 47 kΩ each When selecting resistors, measure each one with an ohmmeter and choose three that are the closest in value to each other. Precision is very important for this experiment! CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 10: “DC Network Analysis” LEARNING OBJECTIVES • How a resistor network can function as a voltage signal averager • Application of Millman’s Theorem SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS This deceptively crude circuit performs the function of mathematically averaging three voltage signals together and so fulfills a specialized computational role. In other words, it is a computer that can only do one mathematical operation: averaging three quantities together. Build this circuit as shown and measure all battery voltages with a voltmeter. Write these voltage figures on paper and average them together (E1 + E2 + E3, divided by three). When you measure each battery voltage, keep the black test probe connected to the “ground” point (the side of the battery directly joined to the other batteries by jumper wires), and touch the red probe to the other battery terminal. Polarity is important here! You will notice one battery in the schematic diagram connected “backward” to the other two, negative side “up.” This battery’s voltage should read as a negative quantity when measured by a properly connected digital meter, the other batteries measuring positive. When the voltmeter is connected to the circuit at the point shown in the schematic and illustrations, it should register the algebraic average of the three batteries’ voltages. If the resistor values are chosen to match each other very closely, the “output” voltage of this circuit should match the calculated average very closely as well. If one battery is disconnected, the output voltage will equal the average voltage of the remaining batteries. If the jumper wires formerly connecting the removed battery to the average circuit are connected to each other, the circuit will average the two remaining voltages together with 0 volts, producing a smaller output signal: The sheer simplicity of this circuit deters most people from calling it a “computer,” but it undeniably performs the mathematical function of averaging. Not only does it perform this function, but it performs it much faster than any modern digital computer can! Digital computers, such as personal computers (PCs) and pushbutton calculators, perform mathematical operations in a series of discrete steps. Analog computers perform calculations in continuous fashion, exploiting Ohm’s and Kirchhoff’s Laws for an arithmetic purpose, the “answer” computed as fast as voltage propagates through the circuit (ideally, at the speed of light!). With the addition of circuits called amplifiers, voltage signals in analog computer networks may be boosted and re-used in other networks to perform a wide variety of mathematical functions. Such analog computers excel at performing the calculus operations of numerical differentiation and integration, and as such may be used to simulate the behavior of complex mechanical, electrical, and even chemical systems. At one time, analog computers were the ultimate tool for engineering research, but since then have been largely supplanted by digital computer technology. Digital computers enjoy the advantage of performing mathematical operations with much better precision than analog computers, albeit at much slower theoretical speeds. COMPUTER SIMULATION Schematic with SPICE node numbers: Netlist (make a text file containing the following text, verbatim): With this SPICE netlist, we can force a digital computer to simulate an analog computer, which averages three numbers together. Obviously, we aren’t doing this for the practical task of averaging numbers, but rather to learn more about circuits and more about computer simulation of circuits! 3.16: Potato Battery PARTS AND MATERIALS • One large potato • One lemon (optional) • Strip of zinc, or galvanized metal • Piece of thick copper wire The basic experiment is based on the use of a potato, but many fruits and vegetables work as potential batteries! For the zinc electrode, a large galvanized nail works well. Nails with a thick, rough zinc texture are preferable to galvanized nails that are smooth. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 11: “Batteries and Power Systems” LEARNING OBJECTIVES • The importance of chemical activity in battery operation • How electrode surface area affects battery operation ILLUSTRATION INSTRUCTIONS Push both the nail and the wire deep into the potato. Measure voltage output by the potato battery with a voltmeter. Now, wasn’t that easy? Seriously, though, experiment with different metals, electrode depths, and electrode spacings to obtain the greatest voltage possible from the potato. Try other vegetables or fruits and compare voltage output with the same electrode metals. It can be difficult to power a load with a single “potato” battery, so don’t expect to light up an incandescent lamp or power a hobby motor or do anything like that. Even if the voltage output is adequate, a potato battery has a fairly high internal resistance which causes its voltage to “sag” badly under even a light load. With multiple potato batteries connected in series, parallel, or series-parallel arrangement, though, it is possible to obtain enough voltage and current capacity to power a small load.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/03%3A_DC_Circuits/3.17%3A_Capacitor_Charging_and_Discharging.txt	princeton-nlp/TextbookChapters	In this experiment, we will aim to learn about the following concepts: • Capacitor charging action • Capacitor discharging action • Time constant calculation • Series and parallel capacitance Capacitor Charging and Discharging Experiment Parts and Materials To do this experiment, you will need the following: • 6-volt battery • Two large electrolytic capacitors, 1000 µF minimum (Radio Shack catalog # 272-1019, 272-1032, or equivalent) • Two 1 kΩ resistors • One toggle switch, SPST (“Single-Pole, Single-Throw”) Large-value capacitors are required for this experiment to produce time constants slow enough to track with a voltmeter and stopwatch. Be warned that most large capacitors are of the “electrolytic” type, and they are polarity sensitive! One terminal of each capacitor should be marked with a definite polarity sign. Usually, capacitors of the size specified to have a negative (-) marking or series of negative markings pointing toward the negative terminal. Very large capacitors are often polarity-labeled by a positive (+) marking next to one terminal. Failure to heed proper polarity will almost surely result in capacitor failure, even with a source voltage as low as 6 volts. When electrolytic capacitors fail, they typically explode, spewing caustic chemicals and emitting foul odors. Please, try to avoid this! I recommend a household light switch for the “SPST toggle switch” specified in the parts list. Further Reading • Lessons In Electric Circuits, Volume 1, chapter 13: “Capacitors” • Lessons In Electric Circuits, Volume 1, chapter 16: “RC and L/R Time Constants” Experiment Instructions Measuring Voltage of Your Circuit Build the “charging” circuit and measure voltage across the capacitor when the switch is closed. Notice how it increases slowly over time, rather than suddenly as would be the case with a resistor. You can “reset” the capacitor back to a voltage of zero by shorting across its terminals with a piece of wire. The “time constant” (τ) of a resistor-capacitor circuit is calculated by taking the circuit resistance and multiplying it by the circuit capacitance. For a 1 kΩ resistor and a 1000 µF capacitor, the time constant should be 1 second. This is the amount of time it takes for the capacitor voltage to increase approximately 63.2% from its present value to its final value: the voltage of the battery. It is educational to plot the voltage of a charging capacitor over time on a sheet of graph paper, to see how the inverse exponential curve develops. In order to plot the action of this circuit, though, we must find a way of slowing it down. A one-second time constant doesn’t provide much time to take voltmeter readings! Changing a Circuit’s Time Constant We can increase this circuit’s time constant two different ways: • Changing the total circuit resistance, and/or • Changing the total circuit capacitance. Given a pair of identical resistors and a pair of identical capacitors, experiment with various series and parallel combinations to obtain the slowest charging action. You should already know by now how multiple resistors need to be connected to form a greater total resistance, but what about capacitors? This circuit will demonstrate to you how capacitance changes with series and parallel capacitor connections. Just be sure that you insert the capacitor(s) in the proper direction: with the ends labeled negative (-) electrically “closest” to the battery’s negative terminal! The discharging circuit provides the same kind of changing capacitor voltage, except this time the voltage jumps to full battery voltage when the switch closes and slowly falls when the switch is opened. Experiment once again with different combinations of resistors and capacitors, making sure as always that the capacitor’s polarity is correct. Computer Simulation Schematic with SPICE node numbers: Netlist (make a text file containing the following text, verbatim): 3.18: Rate-of-change Indicator PARTS AND MATERIALS • Two 6 volt batteries • Capacitor, 0.1 µF (Radio Shack catalog # 272-135) • 1 MΩ resistor • Potentiometer, single turn, 5 kΩ, linear taper (Radio Shack catalog # 271-1714) The potentiometer value is not especially critical, although lower-resistance units will, in theory, work better for this experiment than high-resistance units. I’ve used a 10 kΩ potentiometer for this circuit with excellent results. CROSS-REFERENCES Lessons In Electric Circuits, Volume 1, chapter 13: “Capacitors” LEARNING OBJECTIVES • How to build a differentiator circuit • Obtain an empirical understanding of the derivative calculus function SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Measure voltage between the potentiometer’s wiper terminal and the “ground” point shown in the schematic diagram (the negative terminal of the lower 6-volt battery). This is the input voltage for the circuit, and you can see how it smoothly varies between zero and 12 volts as the potentiometer control is turned full-range. Since the potentiometer is used here as a voltage divider, this behavior should be unsurprising to you. Now, measure voltage across the 1 MΩ resistor while moving the potentiometer control. A digital voltmeter is highly recommended, and I advise setting it to a very sensitive (millivolt) range to obtain the strongest indications. What does the voltmeter indicate while the potentiometer is not being moved? Turn the potentiometer slowly clockwise and note the voltmeter’s indication. Turn the potentiometer slowly counter-clockwise and note the voltmeter’s indication. What difference do you see between the two different directions of potentiometer control motion? Try moving the potentiometer in such a way that the voltmeter gives a steady, small indication. What kind of potentiometer motion provides the steadiest voltage across the 1 MΩ resistor? In calculus, a function representing the rate of change of one variable as compared to another is called the derivative. This simple circuit illustrates the concept of the derivative by producing an output voltage proportional to the input voltage’s rate of change over time. Because this circuit performs the calculus function of differentiation with respect to time (outputting the time-derivative of an incoming signal), it is called a differentiator circuit. Like the average circuit shown earlier in this chapter, the differentiator circuit is a kind of analog computer. Differentiation is a far more complex mathematical function than averaging, especially when implemented in a digital computer, so this circuit is an excellent demonstration of the elegance of analog circuitry in performing mathematical computations. More accurate differentiator circuits may be built by combining resistor-capacitor networks with electronic amplifier circuits. For more detail on computational circuitry, go to the “Analog Integrated Circuits” chapter in this Experiments volume.
textbooks/workforce/Electronics_Technology/Book%3A_Electric_Circuits_VI_-_Experiments_(Kuphaldt)/04%3A_AC_Circuits/4.01%3A_Introduction_to_AC_Circuits.txt	princeton-nlp/TextbookChapters	“AC” stands for Alternating Current, which can refer to either voltage or current that alternates in polarity or direction, respectively. These experiments are designed to introduce you to several important concepts specific to AC. A convenient source of AC voltage is household wall-socket power, which presents significant shock hazard. In order to minimize this hazard while taking advantage of the convenience of this source of AC, a small power supply will be the first project, consisting of a transformer that steps the hazardous voltage (110 to 120 volts AC, RMS) down to 12 volts or less. The title of “power supply” is somewhat misleading. This device does not really act as a source or supply of power, but rather as a power converter, to reduce the hazardous voltage of wall-socket power to a much safer level. 4.02: Transformer—Power Supply PARTS AND MATERIALS • Power transformer, 120VAC step-down to 12VAC, with the center-tapped secondary winding (Radio Shack catalog # 273-1365, 273-1352, or 273-1511). • Terminal strip with at least three terminals. • Household wall-socket power plug and cord. • Line cord switch. • Box (optional). • Fuse and fuse holder (optional). Power transformers may be obtained from old radios, which can usually be obtained from a thrift store for a few dollars (or less!). The radio would also provide the power cord and plug necessary for this project. Line cord switches may be obtained from a hardware store. If you want to be absolutely sure what kind of transformer you’re getting, though, you should purchase one from an electronics supply store. If you decide to equip your power supply with a fuse, be sure to get a slow-acting, or slow-blow fuse. Transformers may draw high “surge” currents when initially connected to an AC source, and these transient currents will blow a fast-acting fuse. Determine the proper current rating of the fuse by dividing the transformer’s “VA” rating by 120 volts: in other words, calculate the full allowable primary winding current and size the fuse accordingly. CROSS-REFERENCES Lessons In Electric Circuits, Volume 2, chapter 1: “Basic AC Theory” Lessons In Electric Circuits, Volume 2, chapter 9: “Transformers” LEARNING OBJECTIVES • Transformer voltage step-down behavior. • Purpose of tapped windings. • Safe wiring techniques for power cords. SCHEMATIC DIAGRAM ILLUSTRATION INSTRUCTIONS Warning! This project involves the use of dangerous voltages. You must make sure all high-voltage (120-volt household power) conductors are safely insulated from accidental contact. No bare wires should be seen anywhere on the “primary” side of the transformer circuit. Be sure to solder all wire connections so that they’re secure, and use real electrical tape (not duct tape, scotch tape, packing tape, or any other kind!) to insulate your soldered connections. If you wish to enclose the transformer inside of a box, you may use an electrical “junction” box, obtained from a hardware store or electrical supply house. If the enclosure used is metal rather than plastic, a three-prong plug should be used, with the “ground” prong (the longest one on the plug) connected directly to the metal case for maximum safety. Before plugging the plug into a wall socket, do a safety check with an ohmmeter. With the line switch in the “on” position, measure resistance between either plug prong and the transformer case. There should be infinite (maximum) resistance. If the meter registers continuity (some resistance value less than infinity), then you have a “short” between one of the power conductors and the case, which is dangerous! Next, check the transformer windings themselves for continuity. With the line switch in the “on” position, there should be a small amount of resistance between the two plug prongs. When the switch is turned “off,” the resistance indication should increase to infinity (open circuit—no continuity). Measure resistance between pairs of wires on the secondary side. These secondary windings should register much lower resistance than the primary. Why is this? Plug the cord into a wall socket and turn the switch on. You should be able to measure AC voltage at the secondary side of the transformer, between pairs of terminals. Between two of these terminals, you should measure about 12 volts. Between either of these two terminals and the third terminal, you should measure half that. This third wire is the “center-tap” wire of the secondary winding. It would be advisable to keep this project assembled for use in powering other experiments shown in this book. From here on, I will designate this “low-voltage AC power supply” using this illustration: COMPUTER SIMULATION Schematic with SPICE node numbers: Netlist (make a text file containing the following text, verbatim):

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