Lessons In Electric Circuits Volume I – DC Book

Lessons In Electric Circuits Volume I – DC Book
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Lessons In Electric Circuits Volume I – DC Book

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    Fifth Edition, last update October 18, 2006

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    Lessons In Electric Circuits, Volume I – DCBy Tony R. KuphaldtFifth Edition, last update October 18, 2006

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    ic 2000-2015, Tony R. KuphaldtThis book is published under the terms and conditions of the Design Science License. Theseterms and conditions allow for free copying, distribution, and/or modification of this documentby the general public. The full Design Science License text is included in the la...

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    ii

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    Contents 11,1 11,BASIC 11,CONCEPTS 11,OF 11,ELECTRICITY1 11,1.1 11,Static 11,electricity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 18,1.2 18,Conductors, 18,insulators, 18,and 18,electron 18,flow . . . . . . . . . . . . . . . . . . . . . . .8 22,1.3 22,Ele...

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    ivCONTENTS 127,Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 129,4 129,SCIENTIFIC 129,NOTATION 129,AND 129,METRIC 129,PREFIXES119 129,4.1 129,Scientific 129,notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

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    CONTENTSv 292,8.9 292,Kelvin 292,(4-wire) 292,resistance 292,measurement . . . . . . . . . . . . . . . . . . . . . . . 282 299,8.10 299,Bridge 299,circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 306,8.11 306,Wattmeter 306,design . . . . . . . . . ...

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    viCONTENTS 419,12 419,PHYSICS 419,OF 419,CONDUCTORS 419,AND 419,INSULATORS409 419,12.1 419,Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 421,12.2 421,Conductor 421,size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

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    CONTENTSvii 534,16.9 534,Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 535,A-1 535,ABOUT 535,THIS 535,BOOK525 539,A-2 539,CONTRIBUTOR 539,LIST529 547,A-3 547,DESIGN 547,SCIENCE 547,LICENSE537 551,INDEX541

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    Chapter 1BASIC CONCEPTS OFELECTRICITYContents 11,1.1 11,Static 11,electricity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 18,1.2 18,Conductors, 18,insulators, 18,and 18,electron 18,flow . . . . . . . . . . . . . . . . . . .8 22,1.3 22,Electric 22,circuits . . . . ...

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    2CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYGlass rodSilk clothattractionGlass and silk aren’t the only materials known to behave like this. Anyone who has everbrushed up against a latex balloon only to find that it tries to stick to them has experiencedthis same phenomenon. Paraffin wax and woo...

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    1.1. STATIC ELECTRICITY3Glass rodGlass rodrepulsionWaxrepulsionWaxIt was also noted that when a piece of glass rubbed with silk was exposed to a piece of waxrubbed with wool, the two materials would attract one another:Glass rodWaxattractionFurthermore, it was found that any material demonstratin...

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    4CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYSilk clothSilk clothrepulsionrepulsionWool clothWool clothNow, this was really strange to witness. After all, none of these objects were visibly alteredby the rubbing, yet they definitely behaved differently than before they were rubbed. Whateverchange to...

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    1.1. STATIC ELECTRICITY5Following Franklin’s speculation of the wool rubbing something off of the wax, the typeof charge that was associated with rubbed wax became known as ”negative” (because it wassupposed to have a deficiency of fluid) while the type of charge associated with the rubbi...

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    6CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY= electron= proton= neutroneNPPPPPPPNNN NNNeeeeeeEven though each atom in a piece of material tends to hold together as a unit, there’sactually a lot of empty space between the electrons and the cluster of protons and neutronsresiding in the middle.This ...

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    1.1. STATIC ELECTRICITY7of this attraction/repulsion behavior between individual particles, electrons and protons aresaid to have opposite electric charges. That is, each electron has a negative charge, and eachproton a positive charge. In equal numbers within an atom, they counteract each other...

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    8CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY1.2Conductors, insulators, and electron flowThe electrons of different types of atoms have different degrees of freedom to move around.With some types of materials, such as metals, the outermost electrons in the atoms are soloosely bound that they chaotic...

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    1.2. CONDUCTORS, INSULATORS, AND ELECTRON FLOW9• rubber• oil• asphalt• fiberglass• porcelain• ceramic• quartz• (dry) cotton• (dry) paper• (dry) wood• plastic• air• diamond• pure waterIt must be understood that not all conductive materials have the same level of conduc...

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    10CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYemptiness of a pipe, electrons are able to move within the empty space within and betweenthe atoms of a conductor. The conductor may appear to be solid to our eyes, but any materialcomposed of atoms is mostly empty space! The liquid-flow analogy is so ...

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    1.2. CONDUCTORS, INSULATORS, AND ELECTRON FLOW11However, the flow will be interrupted if the conductive path formed by the wire is broken:ElectronElectronSourceDestinationno flow!no flow!(break)Since air is an insulating material, and an air gap separates the two pieces of wire, the once-continu...

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    12CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY1.3Electric circuitsYou might have been wondering how electrons can continuously flow in a uniform directionthrough wires without the benefit of these hypothetical electron Sources and Destinations.In order for the Source-and-Destination scheme to work,...

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    1.3. ELECTRIC CIRCUITS13(break)electron flow cannot in a "broken" circuit!no flow!no flow!no flow!occur anywherecontinuousAn important principle to realize here is that it doesn’t matter where the break occurs. Anydiscontinuity in the circuit will prevent electron flow throughout the...

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    14CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY1.4Voltage and currentAs was previously mentioned, we need more than just a continuous path (circuit) before a con-tinuous flow of electrons will occur: we also need some means to push these electrons aroundthe circuit. Just like marbles in a tube or wat...

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    1.4. VOLTAGE AND CURRENT15PumpPondReservoirEnergy stored Water flowThe influence of gravity on the water in the reservoir creates a force that attempts to movethe water down to the lower level again. If a suitable pipe is run from the reservoir back to thepond, water will flow under the infl...

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    16CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYIf the water is pumped to an even higher level, it will take even more energy to do so, thusmore energy will be stored, and more energy released if the water is allowed to flow through apipe back down again:ReservoirPumpPondEnergy storedMore energy relea...

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    1.4. VOLTAGE AND CURRENT17away from their normal ”levels,” creating a condition where a force exists between the waxand wool, as the electrons seek to re-establish their former positions (and balance within theirrespective atoms). The force attracting electrons back to their original position...

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    18CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYReservoirLocation #1Location #2DropDropBecause of the difference in the height of the drop, there’s potential for much more energyto be released from the reservoir through the piping to location 2 than to location 1. Theprinciple can be intuitively unde...

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    1.4. VOLTAGE AND CURRENT19Battery-+12Any source of voltage, including batteries, have two points for electrical contact. In thiscase, we have point 1 and point 2 in the above diagram. The horizontal lines of varying lengthindicate that this is a battery, and they further indicate the direction wh...

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    20CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYpipe back to the pond, stored energy in the reservoir cannot be released in the form of waterflow. Once the reservoir is completely filled up, no flow can occur, no matter how much pressurethe pump may generate. There needs to be a complete path (circu...

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    1.4. VOLTAGE AND CURRENT21isn’t broken, electrons will continue to flow in the circuit. Following the metaphor of watermoving through a pipe, this continuous, uniform flow of electrons through the circuit is calleda current. So long as the voltage source keeps ”pushing” in the same direct...

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    22CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYBattery-+12(break)no flow!no flow!-+34With the circuit’s continuity broken between points 2 and 3, the polarity of the voltagedropped between points 2 and 3 is ”-” for point 2 and ”+” for point 3. The battery’s polarity (1”-” and 4 ”+”...

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    1.5. RESISTANCE23• When a voltage source is connected to a circuit, the voltage will cause a uniform flow ofelectrons through that circuit called a current.• In a single (one loop) circuit, the amount of current at any point is the same as the amountof current at any other point.• If a cir...

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    24CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYthe ”short circuit” where we had nothing but a wire joining one end of the voltage source(battery) to the other.When electrons move against the opposition of resistance, ”friction” is generated. Just likemechanical friction, the friction produced ...

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    1.5. RESISTANCE25Battery-+switchIt doesn’t matter how twisted orconvoluted a route the wires takeconducting current, so long as theyform a complete, uninterrupted loop (circuit).This is how a switch mounted on the wall of a house can control a lamp that is mounteddown a long hallway, or even in...

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    26CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYThe particular knife switch shown here has one ”blade” but two stationary contacts, mean-ing that it can make or break more than one circuit. For now this is not terribly important tobe aware of, just the basic concept of what a switch is and how it w...

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    1.5. RESISTANCE27In keeping with the ”open” and ”closed” terminology of circuits, a switch that is makingcontact from one connection terminal to the other (example: a knife switch with the blade fullytouching the stationary contact point) provides continuity for electrons to flow through...

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    28CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY• The terms ”open”and ”closed”refer to switches as well as entire circuits. An open switchis one without continuity: electrons cannot flow through it. A closed switch is one thatprovides a direct (low resistance) path for electrons to flow thr...

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    1.7. CONVENTIONAL VERSUS ELECTRON FLOW29PumpPondReservoirWaterwheel(energy released)(energy stored)1234Between points 2 and 3, where the falling water is releasing energy at the water-wheel,there is a difference of pressure between the two points, reflecting the opposition to the flowof water t...

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    30CHAPTER 1. BASIC CONCEPTS OF ELECTRICITYreferring to ”excess” charge. You see, the terms ”positive” and ”negative” are human inventions,and as such have no absolute meaning beyond our own conventions of language and scientificdescription. Franklin could have just as easily referred...

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    1.7. CONVENTIONAL VERSUS ELECTRON FLOW31books (this one included) and in the writings of professional scientists, especially solid-statephysicists who are concerned with the actual motion of electrons in substances. These pref-erences are cultural, in the sense that certain groups of people have ...

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    32CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY+-Diode operationCurrent permitted+-Current prohibitedWhen the diode is facing in the proper direction to permit current, the lamp glows. Other-wise, the diode blocks all electron flow just like a break in the circuit, and the lamp will notglow.If we lab...

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    1.8. CONTRIBUTORS33selves, ”just remember the electrons are actuallymoving the other way” whenever the truedirection of electron motion becomes an issue.In this series of textbooks, I have committed to using electron flow notation. Ironically, thiswas not my first choice. I found it much ea...

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    34CHAPTER 1. BASIC CONCEPTS OF ELECTRICITY

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    Chapter 2OHM’s LAWContents 45,2.1 45,How 45,voltage, 45,current, 45,and 45,resistance 45,relate . . . . . . . . . . . . . . . . . . 35 50,2.2 50,An 50,analogy 50,for 50,Ohm’s 50,Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 52,2.3 52,Power 52,in 52,electric 52,c...

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    36CHAPTER 2. OHM’S LAWFree electrons tend to move through conductors with some degree of friction, or oppositionto motion. This opposition to motion is more properly called resistance. The amount of currentin a circuit depends on the amount of voltage available to motivate the electrons, and al...

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    2.1. HOW VOLTAGE, CURRENT, AND RESISTANCE RELATE37One foundational unit of electrical measurement, often taught in the beginnings of electron-ics courses but used infrequently afterwards, is the unit of the coulomb, which is a measure ofelectric charge proportional to the number of electrons in a...

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    38CHAPTER 2. OHM’S LAWBattery-+electron flowelectron flowElectric lamp (glowing)In the above circuit, there is only one source of voltage (the battery, on the left) and only onesource of resistance to current (the lamp, on the right). This makes it very easy to apply Ohm’sLaw. If we know the ...

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    2.1. HOW VOLTAGE, CURRENT, AND RESISTANCE RELATE39Battery-+LampE = 36 VI = 4 AI = 4 AR = ???What is the amount of resistance (R) offered by the lamp?ER===I36 V4 A9 ΩIn the last example, we will calculate the amount of voltage supplied by a battery, givenvalues of current (I) and resistance (R):...

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    40CHAPTER 2. OHM’S LAWEIRIf you know E and I, and wish to determine R, just eliminate R from the picture and seewhat’s left:EIREIR =If you know E and R, and wish to determine I, eliminate I and see what’s left:EIREI =RLastly, if you know I and R, and wish to determine E, eliminate E and see...

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    2.2. AN ANALOGY FOR OHM’S LAW41a restriction (resistance), we can model how the three variables interrelate. If the resistance towater flow stays the same and the pump pressure increases, the flow rate must also increase.PressureFlow rateResistance ===VoltageCurrentResistance ===increasesamei...

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    42CHAPTER 2. OHM’S LAW• With voltage steady, changes in current and resistance are opposite (an increase in cur-rent means a decrease in resistance, and vice versa).• With current steady, voltage follows resistance (an increase in resistance means an in-crease in voltage).2.3Power in electr...

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    2.3. POWER IN ELECTRIC CIRCUITS43S THorsepowerThis symbol means"proportional to"Because the unit of the ”horsepower” doesn’t coincide exactly with speed in revolutions perminute multiplied by torque in pound-feet, we can’t say that horsepower equalsST. However,they are proportio...

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    44CHAPTER 2. OHM’S LAWthis rating would be the norm rather than the exception.• REVIEW:• Power is the measure of how much work can be done in a given amount of time.• Mechanical power is commonly measured (in America) in ”horsepower.”• Electrical power is almost always measured in ...

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    2.4. CALCULATING ELECTRIC POWER45Let’s try taking that same circuit and increasing the battery voltage to see what happens.Intuition should tell us that the circuit current will increase as the voltage increases and thelamp resistance stays the same. Likewise, the power will increase as well:Ba...

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    46CHAPTER 2. OHM’S LAWIf,I=ERandP = I EThen,P =orP =R2II R()IA historical note: it was James Prescott Joule, not Georg Simon Ohm, who first discoveredthe mathematical relationship between power dissipation and current through a resistance.This discovery, published in 1841, followed the form of...

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    2.5. RESISTORS47with a resistance valueof 150 ohms.with a resistance valueof 25 ohms.R1R215025This is resistor "R1"This is resistor "R2"Real resistors look nothing like the zig-zag symbol. Instead, they look like small tubes orcylinders with two wires protruding for connection...

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    48CHAPTER 2. OHM’S LAWVariable resistors must have some physical means of adjustment, either a rotating shaftor lever that can be moved to vary the amount of electrical resistance. Here is a photographshowing some devices called potentiometers, which can be used as variable resistors:Because re...

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    2.5. RESISTORS49This particular circuit board is a computer accessory called a ”modem,” which allows digitalinformation transfer over telephone lines. There are at least a dozen resistors (all rated at1/4 watt power dissipation) that can be seen on this modem’s board. Every one of the black...

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    50CHAPTER 2. OHM’S LAWThere are over one hundred surface-mount resistors on this circuit board, and this countof course does not include the number of resistors internal to the black ”chips.” These twophotographs should convince anyone that resistors – devices that ”merely” oppose the...

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    2.6. NONLINEAR CONDUCTION51BatteryE = 10 VI = 2 AR = ???P = ???All we’ve been given here to start with is the battery voltage (10 volts) and the circuitcurrent (2 amps). We don’t know the resistor’s resistance in ohms or the power dissipated byit in watts. Surveying our array of Ohm’s Law...

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    52CHAPTER 2. OHM’S LAWOhm’s Law is a simple and powerful mathematical tool for helping us analyze electric cir-cuits, but it has limitations, and we must understand these limitations in order to properlyapply it to real circuits. For most conductors, resistance is a rather stable property, la...

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    2.6. NONLINEAR CONDUCTION53resistance are small enough to be ignored. In the application of metal lamp filaments, thechange happens to be quite large.This is just one example of ”nonlinearity” in electric circuits. It is by no means the onlyexample. A ”linear” function in mathematics is ...

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    54CHAPTER 2. OHM’S LAWIf we try to apply Ohm’s Law to find the resistance of this lamp circuit with the voltageand current values plotted above, we arrive at several different values. We could say that theresistance here is nonlinear, increasing with increasing current and voltage. The nonli...

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    2.6. NONLINEAR CONDUCTION55voltage will have been decreased to some lower level, which may allow breakdown to occurmore easily in the future. This is a common mode of failure in high-voltage wiring: insulationdamage due to breakdown. Such failures may be detected through the use of special resist...

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    56CHAPTER 2. OHM’S LAWI(current)E(voltage)negativeresistanceregion ofMost notably, high-vacuum electron tubes known as tetrodesand semiconductor diodesknown as Esakior tunneldiodes exhibit negative resistance for certain ranges of applied volt-age.Ohm’s Law is not very useful for analyzing th...

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    2.7. CIRCUIT WIRING572.7Circuit wiringSo far, we’ve been analyzing single-battery, single-resistor circuits with no regard for the con-necting wires between the components, so long as a complete circuit is formed. Does the wirelength or circuit ”shape” matter to our calculations? Let’s lo...

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    58CHAPTER 2. OHM’S LAWbe the same. That is, the voltage between points 1 and 4 (directly across the battery) will bethe same as the voltage between points 2 and 3 (directly across the resistor). Take a close lookat the following circuit, and try to determine which points are common to each othe...

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    2.7. CIRCUIT WIRING59E = I RE = (2 A)(0 Ω)E = 0 VIt should be obvious that the calculated voltage drop across any uninterrupted length ofwire in a circuit where wire is assumed to have zero resistance will always be zero, no matterwhat the magnitude of current, since zero multiplied by anything...

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    60CHAPTER 2. OHM’S LAWto create a small voltage across the length of it as current is conducted through. So long as youunderstand that these rules are based upon idealconditions, you won’t be perplexed when youcome across some condition appearing to be an exception to the rule.• REVIEW:• ...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS61We could make our table of voltages a little more complete by marking the polarity of thevoltage for each pair of points in this circuit:Between points 1 (+) and 4 (-) = 10 voltsBetween points 2 (+) and 4 (-) = 10 voltsBetween points 3 (+) and 4 (-) ...

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    62CHAPTER 2. OHM’S LAW5 (”Reference”) of this book series for those wanting more information. Here, I’ll just introducethe basic concepts and then apply SPICE to the analysis of these simple circuits we’ve beenreading about.First, we need to have SPICE installed on our computer. As a fr...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS63Creating a text file in a computer involves the use of a program called a text editor. Similarto a word processor, a text editor allows you to type text and record what you’ve typed in theform of a file stored on the computer’s hard disk. Text ...

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    64CHAPTER 2. OHM’S LAWThis line of text tells SPICE that we have a voltage source connected between nodes 1 and0, direct current (DC), 10 volts. That’s all the computer needs to know regarding the battery.Now we turn to the resistor: SPICE requires that resistors be described with a letter ...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS65Now, SPICE will know there is a resistor connected between nodes 1 and 0 with a value of5 Ω. This very brief line of text tells the computer we have a resistor (”r”) connected betweenthe same two nodes as the battery (1 and 0), with a resistanc...

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    66CHAPTER 2. OHM’S LAWOnce we have finished typing all the necessary SPICE commands, we need to ”save” them toa file on the computer’s hard disk so that SPICE has something to reference to when invoked.Since this is my first SPICE netlist, I’ll save it under the filename ”circuit1...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS67As soon as you press the [Enter] key to issue this command, text from SPICE’s outputshould scroll by on the computer screen. Here is a screenshot showing what SPICE outputson my computer (I’ve lengthened the ”terminal” window to show you the ...

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    68CHAPTER 2. OHM’S LAWSPICE begins with a reiteration of the netlist, complete with title line and .endstatement.About halfway through the simulation it displays the voltage at all nodes with reference tonode 0. In this example, we only have one node other than node 0, so it displays the voltag...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS69value of 2 amps is output as a negative (-) 2 amps.The last line of text in the computer’s analysis report is ”total power dissipation,” which inthis case is given as ”2.00E+01” watts: 2.00 x 101, or 20 watts. SPICE outputs most figuresin ...

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    70CHAPTER 2. OHM’S LAWNow, I may freely edit this file, deleting any extraneous text (such as the ”banners” showingdate and time), leaving only the text that I feel to be pertinent to my circuit’s analysis:

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS71Once suitably edited and re-saved under the same filename (output.txt in this example),the text may be pasted into any kind of document, ”plain text” being a universal file formatfor almost all computer systems. I can even include it directly i...

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    72CHAPTER 2. OHM’S LAWediting and processing a text file is one familiar to every computer programmer. One of thereasons I like to teach SPICE is that it prepares the learner to think and work like a computerprogrammer, which is good because computer programming is a significant area of advan...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS73.dc v 0 100 5.print dc v(1) i(v).end

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    74CHAPTER 2. OHM’S LAWThe .printcommand in this SPICE netlist instructs SPICE to print columns of numberscorresponding to each step in the analysis:vi(v)0.000E+000.000E+005.000E+00-1.000E+001.000E+01-2.000E+001.500E+01-3.000E+002.000E+01-4.000E+002.500E+01-5.000E+003.000E+01-6.000E+003.500E+01-...

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    2.9. COMPUTER SIMULATION OF ELECTRIC CIRCUITS75If I re-edit the netlist file, changing the .printcommand into a .plotcommand, SPICEwill output a crude graph made up of text characters:Legend:+ = v#branch------------------------------------------------------------------------sweepv#branch-2.00e+0...

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    76CHAPTER 2. OHM’S LAWNote how Nutmeg plots the resistor voltage v(1)(voltage between node 1 and the impliedreference point of node 0) as a line with a positive slope (from lower-left to upper-right).Whether or not you ever become proficient at using SPICE is not relevant to its applicationin ...

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    Chapter 3ELECTRICAL SAFETYContents 87,3.1 87,The 87,importance 87,of 87,electrical 87,safety. . . . . . . . . . . . . . . . . . . . . . . 77 88,3.2 88,Physiological 88,effects 88,of 88,electricity . . . . . . . . . . . . . . . . . . . . . . . . 78 90,3.3 90,Shock 90,current 90,path . . ....

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    78CHAPTER 3. ELECTRICAL SAFETYthis chapter. Its placement after the first two chapters is intentional: in order for the con-cepts of electrical safety to make the most sense, some foundational knowledge of electricity isnecessary.Another benefit of including a detailed lesson on electrical safe...

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    3.2. PHYSIOLOGICAL EFFECTS OF ELECTRICITY79completely unable to let go of the wire.Medically, this condition of involuntary muscle contraction is called tetanus. Electriciansfamiliar with this effect of electric shock often refer to an immobilized victim of electric shockas being ”froze on the ...

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    80CHAPTER 3. ELECTRICAL SAFETY• Electric current is capable of producing deep and severe burns in the body due to powerdissipation across the body’s electrical resistance.• Tetanusis the condition where muscles involuntarily contract due to the passage of ex-ternal electric current through ...

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    3.3. SHOCK CURRENT PATH81ground when they contact a ”live” wire. Many times, one side of a power system will be inten-tionally connected to earth ground, and so the person touching a single wire is actually makingcontact between two points in the circuit (the wire and earth ground):High volta...

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    82CHAPTER 3. ELECTRICAL SAFETYHigh voltageacross sourceand loadbird (not shocked)person (not shocked)no current!Because the bottom side of the circuit is firmly connected to ground through the groundingpoint on the lower-left of the circuit, the lower conductor of the circuit is made electricall...

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    3.3. SHOCK CURRENT PATH83High voltageacross sourceand loadbird (not shocked)person (SHOCKED!)accidental ground path through tree (touching wire) completes the circuitfor shock current through the victim.Such an accidental connection between a power system conductor and the earth (ground) iscalled...

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    84CHAPTER 3. ELECTRICAL SAFETYHigh voltageacross sourceand loadbird (not shocked)person (SHOCKED!)accidental ground path through tree (touching wire) completes the circuitfor shock current through the victim.person (not shocked)With a tree branch contacting the top wire, that wire becomes the gro...

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    3.3. SHOCK CURRENT PATH85dangerous: the voltage between any point in the circuit and ground (earth) is unpredictable,because a ground fault could appear at any point in the circuit at any time. The only characterguaranteed to be safe in these scenarios is the bird, who has no connection to earth ...

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    86CHAPTER 3. ELECTRICAL SAFETY• Special, insulated shoes and mats are made to protect persons from shock via groundconduction, but even these pieces of gear must be in clean, dry condition to be effective.Normal footwear is not good enough to provide protection from shock by insulating itsweare...

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    3.4. OHM’S LAW (AGAIN!)87Measuring electrical resistance with a sensitive meter, I measure approximately 1 millionohms of resistance (1 MΩ) between my two hands, holding on to the meter’s metal probesbetween my fingers. The meter indicates less resistance when I squeeze the probes tightly ...

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    88CHAPTER 3. ELECTRICAL SAFETYrefers to alternating current that completes ten thousand (10,000) back-and-forth cycles eachand every second.Keep in mind that these figures are only approximate, as individuals with different bodychemistry may react differently. It has been suggested that an acros...

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    3.4. OHM’S LAW (AGAIN!)89E = (20 mA)(1 kΩ)E = 20 voltsNotice that in this condition, 20 volts is enough to produce a current of 20 milliamps througha person: enough to induce tetanus. Remember, it has been suggested a current of only 17milliamps may induce ventricular (heart) fibrillation. W...

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    90CHAPTER 3. ELECTRICAL SAFETY1.5" metal pipe2 kΩWith two hands, the bodily contact area is twice as great as with one hand. This is an im-portant lesson to learn: electrical resistance between any contacting objects diminishes withincreased contact area, all other factors being equal. Wit...

  • Page 101

    3.4. OHM’S LAW (AGAIN!)91Thankfully, nothing bad happened, but had the engine been running and the shock felt atmy hand instead of my leg, I might have reflexively jerked my arm into the path of the rotatingfan, or dropped the metal wrench across the battery terminals (producing largeamounts o...

  • Page 102

    92CHAPTER 3. ELECTRICAL SAFETYBody resistanceIIGlove resistanceBoot resistanceI =ERbodyPerson wearing insulating gloves and boots:current now limited by total circuit resistance.RgloveRboot++Because electric current must pass through the boot andthe body andthe glove to completeits circuit back t...

  • Page 103

    3.5. SAFE PRACTICES933.5Safe practicesIf at all possible, shut off the power to a circuit before performing any work on it. You mustsecure all sources of harmful energy before a system may be considered safe to work on. Inindustry, securing a circuit, device, or system in this condition is common...

  • Page 104

    94CHAPTER 3. ELECTRICAL SAFETYWith the disconnect switch in the ”open” position as shown (no continuity), the circuit isbroken and no current will exist. There will be zero voltage across the load, and the full voltageof the source will be dropped across the open contacts of the disconnect sw...

  • Page 105

    3.5. SAFE PRACTICES95the load are kept safe.It would be good to mention at this point that overcurrent devices are not intended toprovide protection against electric shock. Rather, they exist solely to protect conductors fromoverheating due to excessive currents. The temporary shorting wires just...

  • Page 106

    96CHAPTER 3. ELECTRICAL SAFETYa circuit to see if it was ”dead.” Had I not used other means to check for the presence of voltage,I might not be alive today to write this. There’s always the chance that your voltage meter willbe defective just when you need it to check for a dangerous condit...

  • Page 107

    3.6. EMERGENCY RESPONSE97If you see someone lying unconscious or ”froze on the circuit,” the very first thing to do isshut off the power by opening the appropriate disconnect switch or circuit breaker. If someonetouches another person being shocked, there may be enough voltage dropped across...

  • Page 108

    98CHAPTER 3. ELECTRICAL SAFETY• Shock victims may suffer heart trouble up to several hours after being shocked. Thedanger of electric shock does not end after the immediate medical attention.3.7Common sources of hazardOf course there is danger of electrical shock when directly performing manual...

  • Page 109

    3.7. COMMON SOURCES OF HAZARD99the immediate result will usually be a tremendous amount of arcing (sparks produced), oftenenough to dislodge chunks of concrete or asphalt from the road surface, and reports rivalingthat of a rifle or shotgun. To come into direct contact with a downed power line i...

  • Page 110

    100CHAPTER 3. ELECTRICAL SAFETYdowned power linecurrent through the earth2400volts2390volts10volts250 voltsperson(SHOCKED!)Again, these voltage figures are very approximate, but they serve to illustrate a potentialhazard: that a person can become a victim of electric shock from a downed power li...

  • Page 111

    3.8. SAFE CIRCUIT DESIGN101SourceLoad"Hot" conductor"Neutral" conductorGround pointAs 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 thecircuit...

  • Page 112

    102CHAPTER 3. ELECTRICAL SAFETYSourceGround point"Hot""Neutral"120 Vplugaccidentalcontactvoltage betweencase and ground!If the ”hot” wire contacts the case, it places the user of the toaster in danger. On the otherhand, if the neutral wire contacts the case, there is no da...

  • Page 113

    3.8. SAFE CIRCUIT DESIGN103conductor identity inside the appliance can be guaranteed. Remember that this has no effectwhatsoever 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 applia...

  • Page 114

    104CHAPTER 3. ELECTRICAL SAFETYSourceGround point"Hot""Neutral"120 Vno voltagebetween case and groundIIIn a properly functioning appliance (shown above), the current measured through the hotconductor should be exactly equal to the current through the neutral conductor, because...

  • Page 115

    3.8. SAFE CIRCUIT DESIGN105Source"Hot""Neutral"120 VIIswitches open automaticallyif the difference between thetwo currents becomes toogreat.Such devices are called Ground Fault Current Interruptors, or GFCIs for short. OutsideNorth America, the GFCI is variously known as a saf...

  • Page 116

    106CHAPTER 3. ELECTRICAL SAFETY• Electrical safety of an appliance or other load can be improved by good engineering: polar-ized plugs, double insulation, and three-prong ”grounding” plugs are all ways that safetycan be maximized on the load side.• Ground Fault Current Interruptors(GFCIs)...

  • Page 117

    3.9. SAFE METER USAGE107(now set in the Offposition) 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 t...

  • Page 118

    108CHAPTER 3. ELECTRICAL SAFETYmeasure 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:COMAVVAAOFF-+9voltsNote that the two test leads are plugged into the appropriate soc...

  • Page 119

    3.9. SAFE METER USAGE109COMAVVAAOFFlarge sparkfrom short-circuit!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. Itis certainly the primary measurement taken for safety pur...

  • Page 120

    110CHAPTER 3. ELECTRICAL SAFETYStartbutton for the load. Nothing happened, so now you move on to the third phase of yoursafety 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 c...

  • Page 121

    3.9. SAFE METER USAGE111be turned until it points to the ”horseshoe” resistance symbol. Touching the probes across thedevice whose resistance is to be measured, the meter should properly display the resistance inohms:COMAVVAAOFFkcarbon-compositionresistorOne very important thing to remember a...

  • Page 122

    112CHAPTER 3. ELECTRICAL SAFETYthe meter will indicate infinite resistance (usually by displaying dashed lines or the abbrevia-tion ”O.L.” which stands for ”open loop”):COMAVVAAOFFBy far the most hazardous and complex application of the multimeter is in the measure-ment of current. The r...

  • Page 123

    3.9. SAFE METER USAGE113COMAVVAAOFF-+9voltslamp goes outThe next step is to insert the meter in-line with the circuit by connecting the two probetips to the broken ends of the circuit, the black probe to the negative (-) terminal of the 9-voltbattery and the red probe to the loose wire end leadin...

  • Page 124

    114CHAPTER 3. ELECTRICAL SAFETYresistance 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 testprobe tips when the red probe has been plugged into the red ”A” (current...

  • Page 125

    3.9. SAFE METER USAGE115COMAVVAAOFFtouch probe tipstogetherCOMAVVAAOFFtouch probe tipstogetherIndication with a good fuseIndication with a "blown" fuseA good fuse will indicate very little resistance while a blown fuse will always show ”O.L.”(or whatever indication that model of mul...

  • Page 126

    116CHAPTER 3. ELECTRICAL SAFETY• When in the current-measuring (”ammeter”) mode, multimeters have practically no resis-tance between their leads. This is intended to allow electrons to flow through the meterwith the least possible difficulty. If this were not the case, the meter would add...

  • Page 127

    3.11. CONTRIBUTORS1173.11ContributorsContributors to this chapter are listed in chronological order of their contributions, from mostrecent to first. See Appendix 2 (Contributor List) for dates and contact information.Jason Starck (June 2000): HTML document formatting, which led to a much better...

  • Page 128

    118CHAPTER 3. ELECTRICAL SAFETY

  • Page 129

    Chapter 4SCIENTIFIC NOTATION ANDMETRIC PREFIXESContents 129,4.1 129,Scientific 129,notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 131,4.2 131,Arithmetic 131,with 131,scientific 131,notation . . . . . . . . . . . . . . . . . . . . . . . . 121 133,4.3 133,Met...

  • Page 130

    120CHAPTER 4. SCIENTIFIC NOTATION AND METRIC PREFIXESTake note of those two numbers and of the relative sparsity of non-zero digits in them. Forthe mass of the proton, all we have is a ”167” preceded by 23 zeros before the decimal point. Forthe number of electrons per second in 1 amp, we have...

  • Page 131

    4.2. ARITHMETIC WITH SCIENTIFIC NOTATION121easier for a human being to deal with this ”shorthand” notation. As with the prior case, thesignificant digits in this quantity are clearly expressed.Because the significant digits are represented ”on their own,” away from the power-of-tenmulti...

  • Page 132

    122CHAPTER 4. SCIENTIFIC NOTATION AND METRIC PREFIXES156.25 x 1018 electronsHowever, if we want to hold to standard convention for scientific notation, we must rep-resent the significant digits as a number between 1 and 10. In this case, we’d say ”1.5625”multiplied by some power-of-ten. T...

  • Page 133

    4.3. METRIC NOTATION123• 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 num-bers in easily-handled form.• When multiplying two numbers in scientific notation, you can mu...

  • Page 134

    124CHAPTER 4. SCIENTIFIC NOTATION AND METRIC PREFIXESIn recent years a new style of metric notation for electric quantities has emerged whichseeks to avoid the use of the decimal point. Since decimal points (”.”) are easily misread and/or”lost” due to poor print quality, quantities such a...

  • Page 135

    4.5. HAND CALCULATOR USE125• REVIEW:• Follow the metric prefix number line to know which direction you skip the decimal pointfor conversion purposes.• A number with no decimal point shown has an implicit decimal point to the immediateright of the furthest right digit (i.e. for the number 4...

  • Page 136

    126CHAPTER 4. SCIENTIFIC NOTATION AND METRIC PREFIXESPOWERMETRIC PREFIX------------------12 ......... Tera(T)9 .......... Giga(G)6 .......... Mega(M)3 .......... Kilo(k)0 .......... UNITS (plain)-3 ......... milli (m)-6 ......... micro (u)-9 ......... nano(n)-12 ........ pico(p)• REVIEW:• Use...

  • Page 137

    4.6. SCIENTIFIC NOTATION IN SPICE12724 V11005 ΩTyping out a circuit description file, or netlist, for this circuit, we get this:simple circuitv1 1 0 dc 24r1 1 0 5.endThe line ”v1 1 0 dc 24” describes the battery, positioned between nodes 1 and 0, with aDC voltage of 24 volts. The line ”r...

  • Page 138

    128CHAPTER 4. SCIENTIFIC NOTATION AND METRIC PREFIXESOnce again is our circuit description file, or ”netlist:”simple circuitv1 1 0 dc 24r1 1 0 5k.endThe letter ”k” following the number 5 on the resistor’s line tells SPICE that it is a figure of5 kΩ, not 5 Ω. Let’s see what resul...

  • Page 139

    Chapter 5SERIES AND PARALLELCIRCUITSContents 139,5.1 139,What 139,are 139,”series” 139,and 139,”parallel” 139,circuits?. . . . . . . . . . . . . . . . . . . 129 142,5.2 142,Simple 142,series 142,circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 149,5.3 14...

  • Page 140

    130CHAPTER 5. SERIES AND PARALLEL CIRCUITS1234+-R1R2R3SeriesHere, we have three resistors (labeled R1, R2, and R3), connected in a long chain from oneterminal of the battery to the other. (It should be noted that the subscript labeling – thoselittle numbers to the lower-right of the letter ”R...

  • Page 141

    5.1. WHAT ARE ”SERIES” AND ”PARALLEL” CIRCUITS?1311+-23456R1R2R3Series-parallelIn this circuit, we have two loops for electrons to flow through: one from 6 to 5 to 2 to 1 andback to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how bothcurrent paths go ...

  • Page 142

    132CHAPTER 5. SERIES AND PARALLEL CIRCUITSThese points are electrically commonThese points are electrically commonR1R2R3R4Parallel connectionSeries and parallel resistor configurations have very different electrical properties. We’llexplore the properties of each configuration in the sections...

  • Page 143

    5.2. SIMPLE SERIES CIRCUITS133From the way that the 9 volt battery is arranged, we can tell that the electrons in this circuitwill 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 ...

  • Page 144

    134CHAPTER 5. SERIES AND PARALLEL CIRCUITS1234+-9 VR1R2R33 kΩ10 kΩ5 kΩThe figure of 9 volts is a totalquantity for the whole circuit, whereas the figures of 3k, 10k,and 5k Ω are individualquantities for individual resistors. If we were to plug a figure for totalvoltage into an Ohm’s ...

  • Page 145

    5.2. SIMPLE SERIES CIRCUITS13514+-R1 + R2 + R3 =18 kΩ9 VNow we have all the necessary information to calculate circuit current, because we havethe voltage between points 1 and 4 (9 volts) and the resistance between points 1 and 4 (18 kΩ):=9 volts=18 kΩ500 µAItotalItotal =EtotalRtotalKnowin...

  • Page 146

    136CHAPTER 5. SERIES AND PARALLEL CIRCUITScircuit, it becomes very easy to see which of those quantities can be properly related in anyOhm’s Law equation:EIRVoltsAmpsOhmsR1R2R3TotalOhm’s LawOhm’s LawOhm’s LawOhm’s LawThe rule with such a table is to apply Ohm’s Law only to the values ...

  • Page 147

    5.2. SIMPLE SERIES CIRCUITS137EIRVoltsAmpsOhmsR1R2R3TotalOhm’s Law3k10k5k18k9500µThen, 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 currentfigure just calculated...

  • Page 148

    138CHAPTER 5. SERIES AND PARALLEL CIRCUITS123+-9 V0R1R2R33 kΩ10 kΩ5 kΩAll 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 SPICEwill understand, complete with a co...

  • Page 149

    5.3. SIMPLE PARALLEL CIRCUITS139• 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 ...

  • Page 150

    140CHAPTER 5. SERIES AND PARALLEL CIRCUITSIR1 =ER1R1IR2 =ER2R2IR3 =ER3R3IR1 =9 V10 kΩ= 0.9 mAIR2 =9 V=2 kΩ4.5 mAIR3 =9 V=1 kΩ9 mAEIRVoltsAmpsOhmsR1R2R3Total999910k2k1k0.9m4.5m9mOhm’sLawOhm’sLawOhm’sLawAt this point we still don’t know what the total current or total resistance for t...

  • Page 151

    5.3. SIMPLE PARALLEL CIRCUITS141This is the second principle of parallel circuits: the total circuit current is equal to the sumof the individual branch currents. Using this principle, we can fill in the IT spot on our tablewith the sum of IR1, IR2, and IR3:EIRVoltsAmpsOhmsR1R2R3Total999910k2k1k...

  • Page 152

    142CHAPTER 5. SERIES AND PARALLEL CIRCUITS1+-2345678R1R2R310 kΩ2 kΩ1 kΩ9 VOnce again we find that the original numbering scheme used to identify points in the circuitwill have to be altered for the benefit of SPICE. In SPICE, all electrically common points mustshare identical node numbers...

  • Page 153

    5.3. SIMPLE PARALLEL CIRCUITS1431+-0000111234vr1vr2vr3NOTE: vr1, vr2, and vr3 are all"dummy" voltage sources with values of 0 volts each!!R1R2R310 kΩ2 kΩ1 kΩ9 VThe dummy voltage sources are all set at 0 volts so as to have no impact on the operationof the circuit. The circuit desc...

  • Page 154

    144CHAPTER 5. SERIES AND PARALLEL CIRCUITS4.5 mA for IR2, and 9 mA for IR3. Being connected in parallel, of course, all resistors have thesame voltage dropped across them (9 volts, same as the battery).In summary, a parallel circuit is defined as one where all components are connected betweenthe...

  • Page 155

    5.4. CONDUCTANCE145capital letter ”S”). This decision to change unit names is reminiscent of the change from thetemperature unit of degrees Centigradeto degrees Celsius, or the change from the unit offrequency c.p.s.(cycles per second) to Hertz. If you’re looking for a pattern here, Siemens...

  • Page 156

    146CHAPTER 5. SERIES AND PARALLEL CIRCUITSR1R2R3111++1Rtotal=+1R4Solving the above equation for total resistance (instead of the reciprocal of total resistance),we can invert (reciprocate) both sides of the equation:Rtotal = R1R2R3111++11+R4So, we arrive at our cryptic resistance formula at last!...

  • Page 157

    5.6. CORRECT USE OF OHM’S LAW147An interesting rule for total power versus individual power is that it is additive for anyconfiguration of circuit: series, parallel, series/parallel, or otherwise. Power is a measure ofrate of work, and since power dissipated mustequal the total power applied b...

  • Page 158

    148CHAPTER 5. SERIES AND PARALLEL CIRCUITSEIRVoltsAmpsOhmsR1R2R3TotalPWattsOhm’sLawOhm’sLawOhm’sLawOhm’sLawDeriving values horizontallyacross columns is allowable as per the principles of series andparallel circuits:EIRVoltsAmpsOhmsR1R2R3TotalPWattsFor series circuits:AddEqualAddAddEtotal...

  • Page 159

    5.7. COMPONENT FAILURE ANALYSIS149EIRVoltsAmpsOhmsR1R2R3TotalPWattsEqualAddAddPtotal = P1 + P2 + P3For parallel circuits:DiminishEtotal = E1 = E2 = E3Itotal = I1 + I2 + I3Rtotal = R1R2R3111++1Not only does the ”table” method simplify the management of all relevant quantities, it alsofacilitat...

  • Page 160

    150CHAPTER 5. SERIES AND PARALLEL CIRCUITSon their probability (how likely one particular cause may be over another), and a sense of cre-ativity in applying a solution to rectify the problem. While it is possible to distill these skillsinto a scientific methodology, most practiced troubleshooter...

  • Page 161

    5.7. COMPONENT FAILURE ANALYSIS151must increase:EIRVoltsAmpsOhmsR1R2R3Total100509Shortedresistor060m60m60m60m150063As the circuit current increases from 20 milliamps to 60 milliamps, the voltage drops acrossR1 and R3 (which haven’t changed resistances) increase as well, so that the two resistor...

  • Page 162

    152CHAPTER 5. SERIES AND PARALLEL CIRCUITS+-R1R2R390 Ω45 Ω180 Ω9 VEIRVoltsAmpsOhmsR1R2R3Total9999904518025.714350m100m200m50mSupposing that R2 opens in this parallel circuit, here’s what the effects will be:+-R1R2R390 Ω45 Ω180 Ω9 VEIRVoltsAmpsOhmsR1R2R3Total999990180100m50m0150m60Op...

  • Page 163

    5.7. COMPONENT FAILURE ANALYSIS153+-120VIn an ideal case (with perfect voltage sources and zero-resistance connecting wire), shortedresistors in a simple parallel circuit will also have no effect on what’s happening in otherbranches of the circuit. In real life, the effect is not quite the same...

  • Page 164

    154CHAPTER 5. SERIES AND PARALLEL CIRCUITS+-9 VRinternalBatteryThese internal resistances, small as they may be, turn our simple parallel circuit into aseries-parallel combination circuit. Usually, the internal resistances of voltage sources arelow enough that they can be safely ignored, but when...

  • Page 165

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS155numerical figures is something I like to call qualitativeanalysis. In other words, we will beanalyzing the qualitiesof the effects in a circuit rather than the precise quantities. The result,for you, will be a much deeper intuitive understanding of electr...

  • Page 166

    156CHAPTER 5. SERIES AND PARALLEL CIRCUITSBatteryResistor+-SchematicdiagramReal circuit using jumper wiresJumper wires with ”alligator” style spring clips at each end provide a safe and convenientmethod of electrically joining components together.If we wanted to build a simple series circuit ...

  • Page 167

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS157This technique, however, proves impractical for circuits much more complex than this, dueto the awkwardness of the jumper wires and the physical fragility of their connections. A morecommon method of temporary construction for the hobbyist is the solderles...

  • Page 168

    158CHAPTER 5. SERIES AND PARALLEL CIRCUITSboard face, making connections between inserted leads. The connection pattern joins everyfive holes along a vertical column (as shown with the long axis of the breadboard situatedhorizontally):Lines show common connectionsunderneath board between holesTh...

  • Page 169

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS159Battery+-SchematicdiagramReal circuit using a solderless breadboardBreadboards have their limitations, though. First and foremost, they are intended for tem-poraryconstruction only. If you pick up a breadboard, turn it upside-down, and shake it, anycompone...

  • Page 170

    160CHAPTER 5. SERIES AND PARALLEL CIRCUITSThis board appears copper-side-up: the side where all the soldering is done. Each hole isringed with a small layer of copper metal for bonding to the solder. All holes are independentof each other on this particular board, unlike the holes on a solderless...

  • Page 171

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS161A soldered or wire-wrapped circuit is considered permanent: that is, it is unlikely to fallapart accidently. However, these construction techniques are sometimes considered tooper-manent. If anyone wishes to replace a component or change the circuit in any...

  • Page 172

    162CHAPTER 5. SERIES AND PARALLEL CIRCUITSbetween terminals by a screwdriver or other metal object:In the following illustration, a single-battery, three-resistor circuit is shown constructed ona terminal strip:+-Series circuit constructed on a terminal stripIf the terminal strip uses machine scr...

  • Page 173

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS163circuit construction often demands a different component orientation. Building simple circuitson terminal strips is one way to develop the spatial-reasoning skill of ”stretching” wires tomake the same connection paths. Consider the case of a single-bat...

  • Page 174

    164CHAPTER 5. SERIES AND PARALLEL CIRCUITS+-Schematic diagramReal circuit using a terminal stripNext, trace the wire connection from one side of the battery to the first component in theschematic, securing a connecting wire between the same two points on the real circuit. I findit helpful to ov...

  • Page 175

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS165+-Schematic diagramReal circuit using a terminal stripContinue this process, wire by wire, until all connections in the schematic diagram havebeen accounted for. It might be helpful to regard common wires in a SPICE-like fashion: makeall connections to a c...

  • Page 176

    166CHAPTER 5. SERIES AND PARALLEL CIRCUITS+-Schematic diagramReal circuit using a terminal stripWith the top sides of all resistors (as shown in the schematic) connected together, and tothe battery’s positive (+) terminal, all we have to do now is connect the bottom sides togetherand to the oth...

  • Page 177

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS167+-Schematic diagramReal circuit using a terminal stripTypically in industry, all wires are labeled with number tags, and electrically common wiresbear the same tag number, just as they do in a SPICE simulation. In this case, we could labelthe wires 1 and 2:

  • Page 178

    168CHAPTER 5. SERIES AND PARALLEL CIRCUITS+-1111111111111222222222222111222Common wire numbers representingelectrically common points12Another industrial convention is to modify the schematic diagram slightly so as to indicateactual wire connection points on the terminal strip. This demands a lab...

  • Page 179

    5.8. BUILDING SIMPLE RESISTOR CIRCUITS169+-1111111111111222222222222111222121 23456789101112131415TB1TB1-1TB1-5TB1-6TB1-10TB1-11TB1-15Terminal strip bars labeled and connection points referenced in diagramThis way, the schematic may be used as a ”map” to locate points in a real circuit, regar...

  • Page 180

    170CHAPTER 5. SERIES AND PARALLEL CIRCUITS5.9ContributorsContributors to this chapter are listed in chronological order of their contributions, from mostrecent to first. See Appendix 2 (Contributor List) for dates and contact information.Jason Starck (June 2000): HTML document formatting, which ...

  • Page 181

    Chapter 6DIVIDER CIRCUITS ANDKIRCHHOFF’S LAWSContents 181,6.1 181,Voltage 181,divider 181,circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 189,6.2 189,Kirchhoff’s 189,Voltage 189,Law 189,(KVL) . . . . . . . . . . . . . . . . . . . . . . . . . . 179 200,6.3 200,C...

  • Page 182

    172CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSEIRVoltsAmpsOhmsR1R2R3Total5k10k7.5k45From the given values of individual resistances, we can determine a total circuit resistance,knowing that resistances add in series:EIRVoltsAmpsOhmsR1R2R3Total5k10k7.5k22.5k45From here, we can use Ohm’s L...

  • Page 183

    6.1. VOLTAGE DIVIDER CIRCUITS173function of resistance values.With a little more observation, it becomes apparent that the voltage drop across each re-sistor is also a fixed proportion of the supply voltage. The voltage across R1, for example, was10 volts when the battery supply was 45 volts. Wh...

  • Page 184

    174CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSwithout 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:+-R1R2R35 kΩ7.5 kΩ10 kΩ45 VER1 =5 kΩ22.5 kΩ= 10 V45 VER2 =45 V22.5 kΩ=10 kΩ...

  • Page 185

    6.1. VOLTAGE DIVIDER CIRCUITS17512wiper contactPotentiometerThe wiper contact is the left-facing arrow symbol drawn in the middle of the vertical resistorelement. As it is moved up, it contacts the resistive strip closer to terminal 1 and further awayfrom terminal 2, lowering resistance to termin...

  • Page 186

    176CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSResistive stripWiperTerminalsLinear potentiometer constructionSome 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 fo...

  • Page 187

    6.1. VOLTAGE DIVIDER CIRCUITS177If a constant voltage is applied between the outer terminals (across the length of theslidewire), the wiper position will tap off a fraction of the applied voltage, measurable betweenthe wiper contact and either of the other two terminals. The fractional value depe...

  • Page 188

    178CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSCircuit requiringless voltage thanwhat the batteryprovides+V-Adjust potentiometerto obtain desired voltageBatteryWhen used in this manner, the name potentiometermakes perfect sense: they meter(con-trol) the potential(voltage) applied across the...

  • Page 189

    6.2. KIRCHHOFF’S VOLTAGE LAW (KVL)179The large ”Helipot” unit is a laboratory potentiometer designed for quick and easy connec-tion to a circuit. The unit in the lower-left corner of the photograph is the same type of poten-tiometer, just without a case or 10-turn counting dial. Both of the...

  • Page 190

    180CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWS+-1234+++---R1R2R35 kΩ10 kΩ7.5 k Ω45 VIf we were to connect a voltmeter between points 2 and 1, red test lead to point 2 and blacktest lead to point 1, the meter would register +45 volts. Typically the ”+” sign is not shown,but rather...

  • Page 191

    6.2. KIRCHHOFF’S VOLTAGE LAW (KVL)181E3-2 = -10 VE4-3 = -20 VE1-4 = -15 V+-1234+++---R1R2R35 kΩ10 kΩ7.5 k Ω45 VV ΩCOMAV ΩCOMAV ΩCOMAV ΩCOMAE2-1E3-2E4-3E1-4+45-10-20-15We should already be familiar with the general principle for series circuits stating thatindividual voltage drops ...

  • Page 192

    182CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSin this order: 1-2-3-4-1. It doesn’t matter which point we start at or which direction we proceedin tracing the loop; the voltage sum will still equal zero. To demonstrate, we can tally up thevoltages in loop 3-2-1-4-3 of the same circuit:0 V...

  • Page 193

    6.2. KIRCHHOFF’S VOLTAGE LAW (KVL)183starting with only R1 on the left and progressing across the whole string of components, we willsee how the voltages add algebraically (to zero):+1234+++---2-currentR1R2R345 V5 kΩ10 kΩ7.5 kΩV ΩCOMA-10V ΩCOMAV ΩCOMAV ΩCOMA-20-15+45V ΩCOMAV Ω...

  • Page 194

    184CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSKirchhoff’s Voltage Law (sometimes denoted as KVLfor short) will work for anycircuitconfiguration at all, not just simple series. Note how it works for this parallel circuit:+-+-+-+-12345678R1R2R36 VBeing a parallel circuit, the voltage acro...

  • Page 195

    6.2. KIRCHHOFF’S VOLTAGE LAW (KVL)185+-+--++--++-+-5 V8 V3 V11 V8 V10 V2 VTry any order of steps from any terminal in the above diagram, stepping around back to theoriginal terminal, and you’ll find that the algebraic sum of the voltages alwaysequals zero.Furthermore, the ”loop” we trace...

  • Page 196

    186CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSKVL can be used to determine an unknown voltage in a complex circuit, where all othervoltages around a particular ”loop” are known. Take the following complex circuit (actuallytwo series circuits joined by a single wire at the bottom) as an...

  • Page 197

    6.2. KIRCHHOFF’S VOLTAGE LAW (KVL)18712345678910+-+-+-+-+-+-35 V15 V20 V13 V12 V25 VMeasuring voltage from point 4 to point 3 (unknown amount)V ΩCOMAE4-3???12345678910+-+-+-+-+-+-35 V15 V20 V13 V12 V25 VV ΩCOMAMeasuring voltage from point 9 to point 4 (+12 volts)E4-3 + 12+12

  • Page 198

    188CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWS12345678910+-+-+-+-+-+-35 V15 V20 V13 V12 V25 VV ΩCOMA0E4-3 + 12 + 0Measuring voltage from point 8 to point 9 (0 volts)12345678910+-+-+-+-+-+-35 V15 V20 V13 V12 V25 VV ΩCOMA+20E4-3 + 12 + 0 + 20 = 0Measuring voltage from point 3 to point 8 ...

  • Page 199

    6.2. KIRCHHOFF’S VOLTAGE LAW (KVL)189would indicate with the red lead on point 4 and the black lead on point 3:12345678910+-+-+-+-+-+-35 V15 V20 V13 V12 V25 VV ΩCOMA-32E4-3 = -32 In other words, the initial placement of our ”meter leads” in this KVL problem was ”back-wards.” Had we ge...

  • Page 200

    190CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWS6.3Current divider circuitsLet’s analyze a simple parallel circuit, determining the branch currents through individualresistors:+-+-+-+-R1R2R31 kΩ3 kΩ2 kΩ6 VKnowing that voltages across all components in a parallel circuit are the same,...

  • Page 201

    6.3. CURRENT DIVIDER CIRCUITS191Once again, it should be apparent that the current through each resistor is related to its re-sistance, given that the voltage across all resistors is the same. Rather than being directly pro-portional, the relationship here is one of inverse proportion. For exampl...

  • Page 202

    192CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWSCurrent through any resistorEnRnIn =Voltage in a parallel circuitEtotal = En = Itotal RtotalSubstituting . . .Itotal Rtotal for En in the first equation . . .Current through any parallel resistorIn =RnItotal Rtotal. . . or . . .In = ItotalRnRto...

  • Page 203

    6.4. KIRCHHOFF’S CURRENT LAW (KCL)193RtotalRnEtotalEn = In = ItotalRnRtotalVoltage dividerformulaformulaCurrent dividerIt is quite easy to confuse these two equations, getting the resistance ratios backwards. Oneway to help remember the proper form is to keep in mind that both ratios in the vol...

  • Page 204

    194CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWS+-+-+-+-12345678ItotalItotal6 VR1R2R31 kΩ3 kΩ2 kΩIR1IR2IR3Solving for all values of voltage and current in this circuit:EIRVoltsAmpsOhmsR1R2R3Total66661k3k2k6m2m3m11m545.45At this point, we know the value of each branch current and of the...

  • Page 205

    6.4. KIRCHHOFF’S CURRENT LAW (KCL)195+-3IR2 + IR3IR3R23 kΩIR2From the right and from the bottom, we have two currents entering the wire connectionlabeled as node 3. To the left, we have a single current exiting the node equal in magnitude tothe sum of the two currents entering. To refer to th...

  • Page 206

    196CHAPTER 6. DIVIDER CIRCUITS AND KIRCHHOFF’S LAWS(”Network Analysis”), but suffice it to say that these Laws deserve to be memorized by theelectronics student every bit as much as Ohm’s Law.• REVIEW:• Kirchhoff’s Current Law (KCL): ”The algebraic sum of all currents entering an...

  • Page 207

    Chapter 7SERIES-PARALLELCOMBINATION CIRCUITSContents 207,7.1 207,What 207,is 207,a 207,series-parallel 207,circuit? . . . . . . . . . . . . . . . . . . . . . . . . . 197 210,7.2 210,Analysis 210,technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 218,7.3 218,Re-...

  • Page 208

    198CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSWith simple parallel circuits, all components are connected between the same two sets ofelectrically common points, creating multiple paths for electrons to flow from one end of thebattery to the other:1+-2345678R1R2R3ParallelWith each of these t...

  • Page 209

    7.1. WHAT IS A SERIES-PARALLEL CIRCUIT?199R1R2R3R4100 Ω250 Ω200 Ω350 Ω24 VA series-parallel combination circuitEIRVoltsAmpsOhmsR1R2R3TotalR424100250350200This circuit is neither simple series nor simple parallel. Rather, it contains elements of both.The current exits the bottom of the bat...

  • Page 210

    200CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS• REVIEW:• The rules of series and parallel circuits must be applied selectively to circuits containingboth types of interconnections.7.2Analysis techniqueThe goal of series-parallel resistor circuit analysis is to be able to determine all vol...

  • Page 211

    7.2. ANALYSIS TECHNIQUE201R1R2R3R4100 Ω250 Ω200 Ω350 Ω24 VA series-parallel combination circuitEIRVoltsAmpsOhmsR1R2R3TotalR424100250350200In 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 t...

  • Page 212

    202CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSequivalent of R3 and R4 in parallel with each other.Our table can be expanded to include these resistor equivalents in their own columns:EIRVoltsAmpsOhmsR1R2R3TotalR424100250350200R1 // R2R3 // R471.429127.27It should be apparent now that the circ...

  • Page 213

    7.2. ANALYSIS TECHNIQUE203EIRVoltsAmpsOhmsR1R2R3TotalR424100250350200R1 // R2R3 // R471.429127.27R3 // R4R1 // R2--198.70120.78mBack to our equivalent circuit drawing, our total current value of 120.78 milliamps is shownas the only current here:I = 120.78 mAI = 120.78 mA24 V198.70 ΩR1 // R2 --...

  • Page 214

    204CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSEIRVoltsAmpsOhmsR1R2R3TotalR424100250350200R1 // R2R3 // R471.429127.27R3 // R4R1 // R2--198.70120.78m120.78m120.78mNow, knowing the current through the equivalent resistors R1//R2 and R3//R4, we can applyOhm’s Law (E=IR) to the two right vertic...

  • Page 215

    7.2. ANALYSIS TECHNIQUE205++--I = 120.78 mAI = 120.78 mA100 Ω350 Ω250 Ω200 ΩR1R2R3R48.6275 V15.373 V24 VEIRVoltsAmpsOhmsR1R2R3TotalR424100250350200R1 // R2R3 // R471.429127.27R3 // R4R1 // R2--198.70120.78m120.78m120.78m8.627515.3738.62758.627515.37315.373Finally, the original section of ...

  • Page 216

    206CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS++--I = 120.78 mAI = 120.78 mA43.922 mA76.863 mA350 Ω200 Ω250 Ω100 Ω86.275 mA34.510 mAR1R2R3R48.6275 V15.373 V24 VAs a final check of our work, we can see if the calculated current values add up as theyshould to the total. Since R1 and R2...

  • Page 217

    7.2. ANALYSIS TECHNIQUE207vi1vi2vi3vi4NOTE: voltage sources vi1,vi2, vi3, and vi4 are "dummy"sources set at zero volts each.11112344456000024 VR1R2R3R4100 Ω250 Ω200 Ω350 Ωseries-parallel circuitv1 1 0vi1 1 2 dc 0vi2 1 3 dc 0r1 2 4 100r2 3 4 250vi3 4 5 dc 0vi4 4 6 dc 0r3 5 0 350r...

  • Page 218

    208CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSvoltagev1i(vi1)i(vi2)i(vi3)i(vi4)2.400E+018.627E-023.451E-024.392E-027.686E-02BatteryR1 currentR2 currentR3 currentR4 currentvoltageAs you can see, all the figures do agree with the our calculated values.• REVIEW:• To analyze a series-paralle...

  • Page 219

    7.3. RE-DRAWING COMPLEX SCHEMATICS209With electric circuits and circuit diagrams, the length and routing of wire connecting com-ponents in a circuit matters little. (Actually, in some AC circuits it becomes critical, and verylong wire lengths can contribute unwanted resistance to both AC and DC c...

  • Page 220

    210CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS+-++--R1R3Now, proceed to trace any loops of components connected around components that were justtraced. In this case, there’s a loop around R1 formed by R2, and another loop around R3 formedby R4:+-++--R1R3R2R4loops aroundR2R1loops aroundR4R3T...

  • Page 221

    7.3. RE-DRAWING COMPLEX SCHEMATICS211+-++--+-+-R1R2R3R4Now we have a circuit that is very easily understood and analyzed. In this case, it isidentical to the four-resistor series-parallel configuration we examined earlier in the chapter.Let’s look at another example, even uglier than the one b...

  • Page 222

    212CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSR1R2R3R4R5R6R7+-+-+-Re-drawing vertically and keeping track of voltage drop polarities along the way, our equiv-alent circuit starts out looking like this:+-++--R1R6Next, we can proceed to follow the next loop around one of the traced resistors (R...

  • Page 223

    7.3. RE-DRAWING COMPLEX SCHEMATICS213R1R2R3R4R5R6R7+-+-+-R5 and R7loop aroundR6+-+-Now we add the R5−−R7 loop to the vertical drawing. Notice how the voltage drop polaritiesacross R7 and R5 correspond with that of R6, and how this is the same as what we found tracingR7 and R5 in the original ...

  • Page 224

    214CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSR1R2R3R4R5R6R7+-+-+-andloop around+-+-R3R4R5+-+-Adding the R3−−R4 loop to the vertical drawing, marking the correct polarities as well:+-++--+-+-+-+-R1R6R5R7R3R4With only one remaining resistor left to trace, then next step is obvious: trace t...

  • Page 225

    7.3. RE-DRAWING COMPLEX SCHEMATICS215R1R2R3R4R5R6R7+-+-+-+-+-+-+-loops aroundR2R3+-Adding R2 to the vertical drawing, and we’re finished! The result is a diagram that’s veryeasy to understand compared to the original:+-++--+-+-+-+-+-R1R6R5R7R4R3R2This simplified layout greatly eases the tas...

  • Page 226

    216CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSreducing it to a single resistance. Then, we would take that equivalent resistance (R2//R3) andthe one in series with it (R4), reducing them to another equivalent resistance (R2//R3−−R4).Next, we would proceed to calculate the parallel equival...

  • Page 227

    7.4. COMPONENT FAILURE ANALYSIS217Also shown at the end of the series and parallel circuits chapter was how the table methodworks 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...

  • Page 228

    218CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSEIRVoltsAmpsOhmsR1R2R3TotalR4R1 // R2R3 // R4Next, we need a failure scenario. Let’s suppose that resistor R2 were to fail shorted. We willassume that all other components maintain their original values. Because we’ll be analyzingthis circuit ...

  • Page 229

    7.4. COMPONENT FAILURE ANALYSIS219that total voltage has remained the same and total resistance has decreased, we can concludethat total current must increase (I=E/R).In case you’re not familiar with the qualitative assessment of an equation, it works likethis. First, we write the equation as s...

  • Page 230

    220CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSEIRVoltsAmpsOhmsR1R2R3TotalR4R1 // R2R3 // R4samesamesamesamesameBut how do we apply the same Ohm’s Law formula (E=IR) to the R1//R2 column, where wehave resistance decreasing andcurrent increasing? It’s easy to determine if only one variable ...

  • Page 231

    7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS221voltage and resistance have decreased, but without knowing how mucheach 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 circui...

  • Page 232

    222CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSLines show common connectionsunderneath board between holesSuppose we wanted to construct the following series-parallel combination circuit on a bread-board:R1R2R3R4100 Ω250 Ω200 Ω350 Ω24 VA series-parallel combination circuitThe recommend...

  • Page 233

    7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS223R1R2R3R4+-+-+-+-6 volts6 volts6 volts6 voltsThis is by no means the only way to connect these four resistors together to form the circuitshown in the schematic. Consider this alternative layout:R1R2R3R4+-+-+-+-6 volts6 volts6 volts6 voltsIf greate...

  • Page 234

    224CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSBuilding a circuit with components secured to a terminal strip isn’t as easy as pluggingcomponents into a breadboard, principally because the components cannot be physically ar-ranged to resemble the schematic layout. Instead, the builder must u...

  • Page 235

    7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS225R1R2R3R4R5R6R7The terminal strip used in the prior example barely has enough terminals to mount allseven resistors required for this circuit! It will be a challenge to determine all the necessarywire connections between resistors, but with patienc...

  • Page 236

    226CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 1:+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 2:

  • Page 237

    7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS227+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 3:+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 4:

  • Page 238

    228CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 5:+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 6:

  • Page 239

    7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS229+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 7:+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 8:

  • Page 240

    230CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 9:+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 10:

  • Page 241

    7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS231+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Step 11:Although there are minor variations possible with this terminal strip circuit, the choice ofconnections shown in this example sequence is both electrically accurate (electrically identicalto the schematic diag...

  • Page 242

    232CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITS+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7While this poses no electrical problem, it might cause confusion for anyone measuring resis-tor voltage drops with a voltmeter, especially an analog voltmeter which will ”peg” downscalewhen subjected to a voltage ...

  • Page 243

    7.6. CONTRIBUTORS233+-R1R2R3R4R5R6R7R1R2R3R4R5R6R7Wires movedThough electrons do not care about such consistency in component layout, people do. Thisillustrates an important aspect of any engineering endeavor: the human factor. Whenever adesign may be modified for easier comprehension and/or eas...

  • Page 244

    234CHAPTER 7. SERIES-PARALLEL COMBINATION CIRCUITSJason Starck (June 2000): HTML document formatting, which led to a much better-looking second edition.Ron LaPlante (October 1998): helped create ”table” method of series and parallel circuitanalysis.

  • Page 245

    Chapter 8DC METERING CIRCUITSContents 245,8.1 245,What 245,is 245,a 245,meter?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 251,8.2 251,Voltmeter 251,design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 256,8.3 256,Voltmeter 256,impact ...

  • Page 246

    236CHAPTER 8. DC METERING CIRCUITSadapting 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 itsmechanical nature to movea pointer along a scale so t...

  • Page 247

    8.1. WHAT IS A METER?237wire coilmeter terminalconnectionsmagnetmagnet"needle"050100current through wire coilcauses needle to deflectPermanent magnet, moving coil (PMMC) meter movementIn the picture above, the meter movement ”needle” is shown pointing somewhere around35 percent of f...

  • Page 248

    238CHAPTER 8. DC METERING CIRCUITS0100-100A "zero-center" meter movementCommon polarity-sensitive movements include the D’Arsonval and Weston designs, bothPMMC-type instruments. Current in one direction through the wire will produce a clockwisetorque on the needle mechanism, while cur...

  • Page 249

    8.1. WHAT IS A METER?239forceVoltage to be measuredElectrostatic meter movementUnfortunately, the force generated by the electrostatic attraction is verysmall for commonvoltages. In fact, it is so small that such meter movement designs are impractical for use ingeneral test instruments. Typically...

  • Page 250

    240CHAPTER 8. DC METERING CIRCUITSfurther the electron beam will be ”bent” from its straight path, and the further the glowingspot 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 defle...

  • Page 251

    8.2. VOLTMETER DESIGN241• A ”movement” is the display mechanism of a meter.• Electromagnetic movements work on the principle of a magnetic field being generated byelectric current through a wire. Examples of electromagnetic meter movements includethe D’Arsonval, Weston, and iron-vane d...

  • Page 252

    242CHAPTER 8. DC METERING CIRCUITSallowing 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 willneed to re-label the scale on the meter face to indicate its new measurement range w...

  • Page 253

    8.2. VOLTMETER DESIGN243is to determine total circuit resistance using Ohm’s Law in the ”total” column (R=E/I), thensubtract the 500 Ω of the movement to arrive at the value for the multiplier:EIRVoltsAmpsOhmsTotalMovementRmultiplier101m1m1m50010k9.5kAnother way to figure the same value ...

  • Page 254

    244CHAPTER 8. DC METERING CIRCUITSmeter user does not have to be aware at all that the movement itself is actually measuring justa fraction of that ten volts from the external source. All that matters to the user is that thecircuit as a whole functions to accurately display the total, applied vol...

  • Page 255

    8.2. VOLTMETER DESIGN245black testleadleadred test+-range selectorswitchoffR1 = 999.5 kΩR2 = 99.5 kΩR3 = 9.5 kΩR4 = 500 Ω500 Ω F.S. = 1 mAR1R2R3R41000 V100 V10 V1 VNote the multiplier resistor values used for these ranges, and how odd they are. It is highlyunlikely that a 999.5 kΩ p...

  • Page 256

    246CHAPTER 8. DC METERING CIRCUITS• 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.8.3Voltmeter impact on measured circuitEvery meter impacts the circui...

  • Page 257

    8.3. VOLTMETER IMPACT ON MEASURED CIRCUIT247+V-voltmeter250 MΩ250 MΩ(10 MΩ)24 VThis effectively reduces the lower resistance from 250 MΩ to 9.615 MΩ (250 MΩ and 10 MΩin parallel), drastically altering voltage drops in the circuit. The lower resistor will now havefar less voltage acr...

  • Page 258

    248CHAPTER 8. DC METERING CIRCUITSthan 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 t...

  • Page 259

    8.3. VOLTMETER IMPACT ON MEASURED CIRCUIT249regardless 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 1000Ω/V regardless of how we range it with multiplier resisto...

  • Page 260

    250CHAPTER 8. DC METERING CIRCUITSNow, solid-state transistor amplifier circuits accomplish the same task in digital meter de-signs. While this approach (of using an amplifier to boost the measured signal current) workswell, it vastly complicates the design of the meter, making it nearly imposs...

  • Page 261

    8.3. VOLTMETER IMPACT ON MEASURED CIRCUIT251HeadphonesTestleadsPushbuttonswitchIf a set of ”8 ohm” headphones are used for this purpose, its sensitivity may be greatlyincreased by connecting it to a device called a transformer. The transformer exploits principlesof electromagnetism to ”tran...

  • Page 262

    252CHAPTER 8. DC METERING CIRCUITSadjustablevoltagesource12250 MΩ250 MΩR1R224 VPush button totest for balanceThe purpose of any null detector is to act like a laboratory balance scale, indicating whenthe two voltages are equal (absence of voltage between points 1 and 2) and nothing more.The l...

  • Page 263

    8.4. AMMETER DESIGN253adjustablevoltagesource12+V-250 MΩ250 MΩR1R224 Vnull"null" detectorAdjust voltage source until null detector registers zero.Then, read voltmeter indication for voltage across R2.The voltmeter used to directly measure the precision source need not have an extrem...

  • Page 264

    254CHAPTER 8. DC METERING CIRCUITSIn ammeter designs, external resistors added to extend the usable range of the movementare connected in parallelwith 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,...

  • Page 265

    8.4. AMMETER DESIGN255EIRVoltsAmpsOhmsTotalMovementRshunt51m500From our given values of movement current, movement resistance, and total circuit (mea-sured) current, we can determine the voltage across the meter movement (Ohm’s Law appliedto the center column, E=IR):EIRVoltsAmpsOhmsTotalMovemen...

  • Page 266

    256CHAPTER 8. DC METERING CIRCUITSworking the parallel resistance formula backwards, but the arithmetic would have been morechallenging:Rshunt =111100m500Rshunt = 100.02 mΩ-In real life, the shunt resistor of an ammeter will usually be encased within the protectivemetal housing of the meter uni...

  • Page 267

    8.4. AMMETER DESIGN257black testleadleadred test+-range selectorswitch500 Ω F.S. = 1 mAR1R2R3R4A multirange ammeteroffNotice that the range resistors are connected through the switch so as to be in parallelwith the meter movement, rather than in series as it was in the voltmeter design. The ...

  • Page 268

    258CHAPTER 8. DC METERING CIRCUITSpower dissipations at full-scale indication are (the double-squiggly lines represent ”approxi-mately equal to” in mathematics):PR1 =E2R1=5.00005 mΩ(0.5 V)250 WE2=(0.5 V)2PR2 =R250.005 mΩ5 WE2=(0.5 V)2E2=(0.5 V)2PR3 =PR4 =R3R4500.5 mΩ0.5 W5.05 Ω49.5 mW...

  • Page 269

    8.4. AMMETER DESIGN259per given unit of current, thus extending the usable range of the (volt)meter down into loweramounts of current. The use of voltmeters in conjunction with low-value shunt resistances forthe measurement of current is something commonly seen in industrial applications.The use ...

  • Page 270

    260CHAPTER 8. DC METERING CIRCUITS1200Rshunt1 ΩRload15 kΩ12 Vshunt resistor example circuitv1 1 0rshunt 1 2 1rload 2 0 15k.dc v1 12 12 1.print dc v(1,2).endv1v(1,2)1.200E+017.999E-04We would interpret the voltage reading across the shunt resistor (between circuit nodes 1and 2 in the SPICE sim...

  • Page 271

    8.5. AMMETER IMPACT ON MEASURED CIRCUIT261resistance value is exactly opposite as that of a voltmeter. With voltmeters, we want as littlecurrent to be drawn as possible from the circuit under test. With ammeters, we want as littlevoltage to be dropped as possible while conducting current.Here is ...

  • Page 272

    262CHAPTER 8. DC METERING CIRCUITS+A-R1R23 Ω1.5 Ω2 V666.7 mA1 ARinternal0.5 ΩNow the right branch current is 1 amp instead of 1.333 amps, due to the increase in resis-tance created by the addition of the ammeter into the current path.When using standard ammeters that connect in series with ...

  • Page 273

    8.5. AMMETER IMPACT ON MEASURED CIRCUIT263current to bemeasuredmagnetic fieldencircling the current-carryingconductorclamp-onammeterAmmeters 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-...

  • Page 274

    264CHAPTER 8. DC METERING CIRCUITSwire between the jaws, that small voltage connected to a voltmeter for convenient readout bya technician. Thus, a clamp-on unit can be an accessory device to a voltmeter, for currentmeasurement.A less accurate type of magnetic-field-sensing ammeter than the clam...

  • Page 275

    8.6. OHMMETER DESIGN265Starting with a simple movement and battery circuit, let’s see how it would function as anohmmeter:black testleadleadred test+-500 Ω F.S. = 1 mA9 VA simple ohmmeterWhen there is infinite resistance (no continuity between test leads), there is zero currentthrough the ...

  • Page 276

    266CHAPTER 8. DC METERING CIRCUITSblack testleadleadred test+-500 Ω F.S. = 1 mA9 VRTo determine the proper value for R, we calculate the total circuit resistance needed tolimit current to 1 mA (full-scale deflection on the movement) with 9 volts of potential from thebattery, then subtract th...

  • Page 277

    8.6. OHMMETER DESIGN2670300751001507501.5k15kAn ohmmeter’s logarithmic scaleInfinity cannot be approached in a linear (even) fashion, because the scale would nevergetthere! With a nonlinear scale, the amount of resistance spanned for any given distance on thescale increases as the scale progre...

  • Page 278

    268CHAPTER 8. DC METERING CIRCUITSRtotal =EI=9 VRtotal = 36 kΩ250 µARtest = Rtotal - RinternalRtest = 36 kΩ - 9 kΩRtest = 27 kΩ3/4 scale deflection (0.75 mA of meter current):Rtotal =EI=9 VRtotal =Rtest = Rtotal - Rinternal750 µA12 kΩRtest = 12 kΩ - 9 kΩRtest = 3 kΩSo, the scal...

  • Page 279

    8.7. HIGH VOLTAGE OHMMETERS269resistance reading. If the battery voltage decreases (as all chemical batteries do with age anduse), the ohmmeter scale will lose accuracy. With the series range resistor at a constant valueof 8.5 kΩ and the battery voltage decreasing, the meter will no longer de...

  • Page 280

    270CHAPTER 8. DC METERING CIRCUITSI(current)E(voltage)ionization potential050100150200250300350400While this is an extreme example of nonlinear conduction, other substances exhibit similarinsulating/conducting properties when exposed to high voltages. Obviously, an ohmmeter usinga low-voltage bat...

  • Page 281

    8.7. HIGH VOLTAGE OHMMETERS271black testleadleadred test+-Unfortunately, this would create a calibration problem for the meter. If the meter movementdeflects full-scale with a certain amount of current through it, the full-scale range of the meterin ohms would change as the source voltage change...

  • Page 282

    272CHAPTER 8. DC METERING CIRCUITS231Test leadsRedBlackHigh voltageWith infinite resistance between the test leads (open circuit), there will be no currentthrough coil 1, only through coils 2 and 3. When energized, these coils try to center them-selves in the gap between the two magnet poles, dr...

  • Page 283

    8.7. HIGH VOLTAGE OHMMETERS273coils (coils 2 and 3, which drive the needle to the right, and coil 1, which drives the needle to theleft), 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...

  • Page 284

    274CHAPTER 8. DC METERING CIRCUITS231High voltageEarthLineGuardResistance is measured between the Line and Earth terminals, where current will travelthrough coil 1. The ”Guard” terminal is provided for special testing situations where one re-sistance must be isolated from another. Take for in...

  • Page 285

    8.7. HIGH VOLTAGE OHMMETERS275wire wrappedaroundcable sheathLEGIn this configuration the megger should read the resistance between one conductor and theoutside sheath. Or will it? If we draw a schematic diagram showing all insulation resistancesas resistor symbols, what we have looks like this:c...

  • Page 286

    276CHAPTER 8. DC METERING CIRCUITSthe resistance between the second conductor and the sheath (Rc2−s ), then we need to use themegger’s ”Guard” terminal:wire wrappedaroundcable sheathLEGMegger with "Guard"connectedNow the circuit schematic looks like this:conductor1conductor2Rc1-...

  • Page 287

    8.8. MULTIMETERS277megger’s resistance indication will be based exclusively on the current through the secondconductor’s insulation, through the cable sheath, and to the wire wrapped around, not thecurrent leaking through the first conductor’s insulation.Meggers are field instruments: tha...

  • Page 288

    278CHAPTER 8. DC METERING CIRCUITSThe 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 movementfor the different ranges and functions selectable by the rotary switch. The wires f...

  • Page 289

    8.8. MULTIMETERS279A close examination of this meter will reveal one ”common” jack for the black test lead andthree others for the red test lead. The jack into which the red lead is shown inserted is labeledfor voltage and resistance measurement, while the other two jacks are labeled for curr...

  • Page 290

    280CHAPTER 8. DC METERING CIRCUITSthe switch. This short-circuiting creates a dampening effect on the needle, guarding againstmechanical shock damage when the meter is handled and moved.If an ohmmeter function is desired in this multimeter design, it may be substituted for oneof the three voltage...

  • Page 291

    8.8. MULTIMETERS281Note that there are three types of scales on this meter face: a green scale for resistance atthe top, a set of black scales for DC voltage and current in the middle, and a set of blue scalesfor AC voltage and current at the bottom. Both the DC and AC scales have three sub-scale...

  • Page 292

    282CHAPTER 8. DC METERING CIRCUITSNote how all current ranges are power-of-ten multiples of the three scale ranges shown onthe meter face: 2.5, 5, and 10. In some range settings, such as the 2.5 mA for example, themeter indication may be read directly on the 0 to 2.5 scale. For other range settin...

  • Page 293

    8.9. KELVIN (4-WIRE) RESISTANCE MEASUREMENT283Usually, wire resistance is very small (only a few ohms per hundreds of feet, dependingprimarily on the gauge (size) of the wire), but if the connecting wires are very long, and/or thecomponent to be measured has a very low resistance anyway, the meas...

  • Page 294

    284CHAPTER 8. DC METERING CIRCUITSThus, those long lengths of wire connecting the voltmeter across the subject resistance willdrop insignificant amounts of voltage, resulting in a voltmeter indication that is very nearlythe same as if it were connected directly across the subject resistance:Rsub...

  • Page 295

    8.9. KELVIN (4-WIRE) RESISTANCE MEASUREMENT285Rsubject4-wire cableCPCPclipclipAVRsubject =Voltmeter indicationAmmeter indicationThe same principle of using different contact points for current conduction and voltagemeasurement is used in precision shunt resistors for measuring large amounts of cu...

  • Page 296

    286CHAPTER 8. DC METERING CIRCUITSVoltmeterShuntMeasured currentMeasured currentIn metrological (metrology = ”the science of measurement”) applications, where accuracy isof paramount importance, highly precise ”standard” resistors are also equipped with four ter-minals: two for carrying t...

  • Page 297

    8.9. KELVIN (4-WIRE) RESISTANCE MEASUREMENT287as black knobs (metal pads underneath each knob for direct metal-to-metal connection withthe wires), two large knobs for securing the current-carrying wires, and two smaller knobs forsecuring the voltmeter (”potential”) wires:Appreciation is exten...

  • Page 298

    288CHAPTER 8. DC METERING CIRCUITSRsubjectCPCPclipclipVRstandardAll current-carrying wires in the above circuit are shown in ”bold,” to easily distinguishthem from wires connecting the voltmeter across both resistances (Rsubject and Rstandard). Ide-ally, a potentiometric voltmeter is used to ...

  • Page 299

    8.10. BRIDGE CIRCUITS2898.10Bridge circuitsNo 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 likethe laboratory balance scale compares two weights and indicates when ...

  • Page 300

    290CHAPTER 8. DC METERING CIRCUITSRaR1R212RxRaRx=R1R2Bridge circuit isnullbalanced when:Each of the four resistances in a bridge circuit are referred to as arms. The resistor in serieswith the unknown resistance Rx (this would be Ra in the above schematic) is commonly calledthe rheostatof the bri...

  • Page 301

    8.10. BRIDGE CIRCUITS291measuring very low resistances (typically less than 1/10 of an ohm). Its schematic diagram isas such:RaRxnullRa and Rx are low-value resistancesRMRNRmRnKelvin Double bridgeThe low-value resistors are represented by thick-line symbols, and the wires connectingthem to the vo...

  • Page 302

    292CHAPTER 8. DC METERING CIRCUITSRaRxnullRNRMWhen the null detector indicates zero voltage, we know that the bridge is balanced and thatthe ratios Ra/Rx and RM /RN are mathematically equal to each other. Knowing the values ofRa, RM , and RN therefore provides us with the necessary data to solve ...

  • Page 303

    8.10. BRIDGE CIRCUITS293RaRxnullRNRMERaERxEwireEwireEwireEwireStray Ewire voltages will corrupt the accuracy of Rx’s measurementSince we don’t want to measure these stray wire and connection resistances, but only mea-sure Rx, we must find some way to connect the null detector so that it won...

  • Page 304

    294CHAPTER 8. DC METERING CIRCUITSNow the top two Ewire voltage drops are of no effect to the null detector, and do not influencethe accuracy of Rx’s resistance measurement. However, the two remaining Ewire voltage dropswill cause problems, as the wire connecting the lower end of Ra with the t...

  • Page 305

    8.10. BRIDGE CIRCUITS295RaRxnullRa and Rx are low-value resistancesRMRNRmRnKelvin Double bridgeWith ratio Rm/Rn set equal to ratio RM /RN , rheostat arm resistor Ra is adjusted until thenull detector indicates balance, and then we can say that Ra/Rx is equal to RM /RN , or simplyfind Rx by the f...

  • Page 306

    296CHAPTER 8. DC METERING CIRCUITSinstrument accuracy demands that allerror-producing factors be taken into account, and oftenthe 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 comp...

  • Page 307

    8.12. CREATING CUSTOM CALIBRATION RESISTANCES297Electrodynamometer movementcurrentcoilvoltagecoil(stationary)(moving)RshuntRmultiplier• REVIEW:• Wattmeters are often designed around dynamometer meter movements, which employboth voltage and current coils to move a needle.8.12Creating custom ca...

  • Page 308

    298CHAPTER 8. DC METERING CIRCUITSSpecial resistancewireBobbinCompleted resistorBefore winding coilAs you might imagine, this can be a labor-intensive process, especially if more than oneresistor must be built! Another, easier solution to the dilemma of a custom resistance is toconnect multiple ...

  • Page 309

    8.12. CREATING CUSTOM CALIBRATION RESISTANCES299potentiometer, it will be very difficult to adjust it to any value this finely. Such a feat would benearly impossible using a standard 3/4 turn potentiometer. So how can we get the resistancevalue we need and still have room for adjustment?The sol...

  • Page 310

    300CHAPTER 8. DC METERING CIRCUITS8.13ContributorsContributors to this chapter are listed in chronological order of their contributions, from mostrecent to first. See Appendix 2 (Contributor List) for dates and contact information.Jason Starck (June 2000): HTML document formatting, which led to ...

  • Page 311

    Chapter 9ELECTRICALINSTRUMENTATION SIGNALSContents 311,9.1 311,Analog 311,and 311,digital 311,signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 314,9.2 314,Voltage 314,signal 314,systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 316,9.3 316,Curre...

  • Page 312

    302CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSAn analogsignal is a kind of signal that is continuously variable, as opposed to havinga limited number of steps along its range (called digital). A well-known example of analogvs. digital is that of clocks: analog being the type with pointers that ...

  • Page 313

    9.1. ANALOG AND DIGITAL SIGNALS303limits imposed by the mechanics of air pressure devices, this pneumatic signal is infinitelyvariable, able to represent any degree of change in the water’s level, and is therefore analoginthe truest sense of the word.Crude as it may appear, this kind of pneuma...

  • Page 314

    304CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSreceived signal of 0 percent could be a legitimate reading of 0 percent measurement orit couldmean that the system was malfunctioning (air compressor stopped, tubing broken, transmittermalfunctioning, etc.). With the 0 percent point represented by 0...

  • Page 315

    9.2. VOLTAGE SIGNAL SYSTEMS305the water level. The ”indicator” is nothing more than a voltmeter with a scale calibrated toread 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 potentiometerwipe...

  • Page 316

    306CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALS• A major disadvantage of voltage signaling is the possibility that the voltage at the indi-cator (voltmeter) will be less than the voltage at the signal source, due to line resistanceand indicator current draw. This drop in voltage along the cond...

  • Page 317

    9.3. CURRENT SIGNAL SYSTEMS307floatLevel transmitterLevel indicatorfloat position changesoutput of current sourcevoltage dropvoltage drop--++Being a simple seriescircuit, current is equalat all points, regardlessof any voltage drops!+A-The internal workings of the transmitter’s current source n...

  • Page 318

    308CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALS-++A-Transmitter+V-Indicator (1-5 V instrument)250 Ω4 - 20 mA current signalIndicator(4-20 mA instrument)----------------------------------------| Percent of|4-20 mA|1-5 V|| measurement|signal|signal|----------------------------------------|0|4.0 ...

  • Page 319

    9.4. TACHOGENERATORS309----------------------------------------The current loop scale of 4-20 milliamps has not always been thestandard for current in-struments: for a while there was also a 10-50 milliamp standard, but that standard has sincebeen obsoleted. One reason for the eventual supremacy ...

  • Page 320

    310CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSTachogenerators can also indicate the direction of rotation by the polarity of the outputvoltage. When a permanent-magnet style DC generator’s rotational direction is reversed, thepolarity of its output voltage will switch. In measurement and cont...

  • Page 321

    9.5. THERMOCOUPLES311the voltmeter’s copper leads will be a function of the differencein temperature between thetwo junctions, and not the temperature at the measurement junction alone. Even for ther-mocouple types where copper is not one of the dissimilar metals, the combination of the twometa...

  • Page 322

    312CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALStotal resistance of the circuit. With heavy enough thermocouple conductors, currents upwardsof hundreds of amps can be generated from a single pair of thermocouple junctions! (I’veactually seen this in a laboratory experiment, using heavy bars of ...

  • Page 323

    9.5. THERMOCOUPLES313tion would be greater than each right junction, resulting in a total output voltage equal to thesum 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 appl...

  • Page 324

    314CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSiron wirejunction+V-copper wirecopper wire+-constantan wire#1junction+-constantan wireiron wirejunction+-constantan wireiron wirejunction+-constantan wireiron wire#2#3#4The meter will registera more realistic averageof all junction temperatureswith ...

  • Page 325

    9.6. PH MEASUREMENT3159.6pH measurementA very important measurement in many liquid chemical processes (industrial, pharmaceutical,manufacturing, food production, etc.) is that of pH: the measurement of hydrogen ion concen-tration in a liquid solution. A solution with a low pH value is called an ...

  • Page 326

    316CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSliquid solutionelectrodeselectrodes is proportionalto the pH of the solutionVoltage produced betweenThe design and operational theory of pH electrodes is a very complex subject, explored onlybriefly here. What is important to understand is that the...

  • Page 327

    9.6. PH MEASUREMENT317silver chloridetipsealsilverwirevery thin glass bulb,chemically "doped" withlithium ions so as to reactwith hydrogen ions outsidethe bulb.bulb filled withpotassium chloride- ++++++++++++++----- - -------voltage producedacross thickness ofglass membrane"buffer&...

  • Page 328

    318CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSsilver chloridetipsilverwirepotassium chloride"buffer" solutionwire connection pointELECTRODEREFERENCEglass or plastic bodyporous junctionfilled with The measurement electrode’s purpose is to generate the voltage used to measure the solu...

  • Page 329

    9.6. PH MEASUREMENT319+V-precision voltmeterRmeasurement electrodeRreference electrodevoltage produced byelectrodes400 MΩ3 kΩEven a very small circuit current traveling through the high resistances of each componentin the circuit (especially the measurement electrode’s glass membrane), will...

  • Page 330

    320CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSsurement, because it is so widely misunderstood and difficult to troubleshoot. Without elabo-rating on the exact chemistry of pH measurement, a few words of wisdom can be given hereabout pH measurement systems:• All pH electrodes have a finite l...

  • Page 331

    9.7. STRAIN GAUGES321• The basic pH scale extends from 0 (strong acid) to 7 (neutral, pure water) to 14 (strongcaustic). Chemical solutions with pH levels below zero and above 14 are possible, butrare.• pH can be measured by measuring the voltage produced between two special electrodesimmerse...

  • Page 332

    322CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSon a nonconducting substrate material called the carrier. The latter form of strain gauge isrepresented in the previous illustration. The name ”bonded gauge” is given to strain gaugesthat are glued to a larger structure under stress (called the ...

  • Page 333

    9.7. STRAIN GAUGES323circuit may be substantial, wire resistance has a significant impact on the operation of thecircuit. To illustrate the effects of wire resistance, I’ll show the same schematic diagram, butadd two resistor symbols in series with the strain gauge to represent the wires:VR1R2...

  • Page 334

    324CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSthe resistance of the top wire (Rwire1) has been ”bypassed” now that the voltmeter connectsdirectly 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...

  • Page 335

    9.7. STRAIN GAUGES325amount of stray resistance, and their effects tend to cancel:VR1R3strain gauge(unstressed)(stressed)strain gaugeRwire1Rwire2Rwire3Even though there are now two strain gauges in the bridge circuit, only one is responsiveto mechanical strain, and thus we would still refer to th...

  • Page 336

    326CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALSthis effect is illustrated here:Test specimenStrain gauge #1Strain gauge #2RRRgauge#1Rgauge#2V(+)(-)Bridge balancedWith no force applied to the test specimen, both strain gauges have equal resistance andthe bridge circuit is balanced. However, when ...

  • Page 337

    9.7. STRAIN GAUGES327Vstrain gauge(stressed)strain gauge(stressed)strain gauge(stressed)strain gauge(stressed)Full-bridge strain gauge circuitBoth half-bridge and full-bridge configurations grant greater sensitivity over the quarter-bridge circuit, but often it is not possible to bond complement...

  • Page 338

    328CHAPTER 9. ELECTRICAL INSTRUMENTATION SIGNALS• REVIEW:• A strain gauge is a thin strip of metal designed to measure mechanical load by changingresistance when stressed (stretched or compressed within its elastic limit).• Strain gauge resistance changes are typically measured in a bridge ...

  • Page 339

    Chapter 10DC NETWORK ANALYSISContents 339,10.1 339,What 339,is 339,network 339,analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 342,10.2 342,Branch 342,current 342,method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 351,10.3 351,Mesh 351,curren...

  • Page 340

    330CHAPTER 10. DC NETWORK ANALYSISTo illustrate how even a simple circuit can defy analysis by breakdown into series andparallel portions, take start with this series-parallel circuit:R1R3R2B1To analyze the above circuit, one would first find the equivalent of R2 and R3 in parallel,then add R1 ...

  • Page 341

    10.1. WHAT IS NETWORK ANALYSIS?331R1R2R3R4R5Here we have a bridge circuit, and for the sake of example we will suppose that it is notbalanced (ratio R1/R4 not equal to ratio R2/R5). If it were balanced, there would be zero currentthrough R3, and it could be approached as a series/parallel combina...

  • Page 342

    332CHAPTER 10. DC NETWORK ANALYSISEI R=is unknown;are known()EandIR. . . or . . .I =ER(is unknown; andare known )IER. . . or . . .I=ER(is unknown; andare known )REIHowever, when we’re solving for multiple unknown values, we need to have the same num-ber of equations as we have unknowns in order...

  • Page 343

    10.2. BRANCH CURRENT METHOD33328 V7 V2 ΩR2R1R34 Ω1 ΩB1B2The first step is to choose a node (junction of wires) in the circuit to use as a point ofreference for our unknown currents. I’ll choose the node joining the right of R1, the top of R2,and the left of R3.28 V7 V2 ΩR2R1R34 Ω1 ...

  • Page 344

    334CHAPTER 10. DC NETWORK ANALYSIS- I1 + I2 - I3 = 0Kirchhoff’s Current Law (KCL)applied to currents at nodeThe next step is to label all voltage drop polarities across resistors according to the assumeddirections of the currents. Remember that the “upstream” end of a resistor will always b...

  • Page 345

    10.2. BRANCH CURRENT METHOD33528 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates: -28 V28 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates:0 V28 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates: a positive voltage+ ER2

  • Page 346

    336CHAPTER 10. DC NETWORK ANALYSIS28 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates: a positive voltage+ ER2Having completed our trace of the left loop, we add these voltage indications together for asum of zero:Kirchhoff’s Voltage Law (KVL)applied to voltage drops in left loop- 28 + 0 + ER2 ...

  • Page 347

    10.2. BRANCH CURRENT METHOD33728 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates: a negative voltage- ER228 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates: 0 V28 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates: + 7 V

  • Page 348

    338CHAPTER 10. DC NETWORK ANALYSIS28 V7 VR2R1R3+-+-+++---VredblackVoltmeter indicates:a negative voltage- ER3Kirchhoff’s Voltage Law (KVL)applied to voltage drops in right loop- ER2 + 0 + 7 - ER3 = 0Knowing now that the voltage across each resistor can be and should beexpressed as theproduct of...

  • Page 349

    10.2. BRANCH CURRENT METHOD339the three unknown current values:Solutions:I1 = 5 AI2 = 4 AI3 = -1 ASo, I1 is 5 amps, I2 is 4 amps, and I3 is a negative 1 amp. But what does “negative” currentmean? In this case, it means that our assumeddirection for I3 was opposite of its realdirection.Going b...

  • Page 350

    340CHAPTER 10. DC NETWORK ANALYSIS28 V7 V2 ΩR2R1R34 Ω1 Ω123000B1B2network analysis examplev1 1 0v2 3 0 dc 7r1 1 2 4r2 2 0 2r3 2 3 1.dc v1 28 28 1.print dc v(1,2) v(2,0) v(2,3).endv1v(1,2)v(2)v(2,3)2.800E+012.000E+018.000E+001.000E+00Sure enough, the voltage figures all turn out to be the s...

  • Page 351

    10.3. MESH CURRENT METHOD341• (5) Solve for unknown branch currents (simultaneous equations).• (6) If any solution is negative, then the assumed direction of current for that solution iswrong!• (7) Solve for voltage drops across all resistors (E=IR).10.3Mesh current methodThe Mesh Current M...

  • Page 352

    342CHAPTER 10. DC NETWORK ANALYSISThe choice of each current’s direction is entirely arbitrary, just as in the Branch Currentmethod, but the resulting equations are easier to solve if the currents are going the samedirection through intersecting components (note how currents I1 and I2 are both ...

  • Page 353

    10.3. MESH CURRENT METHOD343- 28 + 2(I1 + I2) + 4I1 = 0Original form of equation. . . distributing to terms within parentheses . . .. . . combining like terms . . .- 28 + 6I1 + 2I2 = 0- 28 + 2I1 + 2I2 + 4I1 = 0Simplified form of equationAt this time we have one equation with two unknowns. To be a...

  • Page 354

    344CHAPTER 10. DC NETWORK ANALYSIS28 V7 V2 ΩR2R1R34 Ω1 ΩI1I2+-+-++---+5 A-1 AB1B2The solution of -1 amp for I2 means that our initially assumed direction of current wasincorrect. In actuality, I2 is flowing in a counter-clockwise direction at a value of (positive) 1amp:28 V7 V2 ΩR2R1R34 ...

  • Page 355

    10.3. MESH CURRENT METHOD345A 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.The primary advantage of Mesh Current analysis is that it generally allows for the solutionof a large network with fewer unkn...

  • Page 356

    346CHAPTER 10. DC NETWORK ANALYSISR1R2R3R4R5I1I2I3+-+-+-+-+-+--+B1B2- EB1 + R2(I1 + I2) + I1R1 = 0- R2(I2 + I1) - R4(I2 + I3) - I2R3 = 0R4(I3 + I2) + EB2 + I3R5 = 0Kirchhoff’s Voltage LawKirchhoff’s Voltage LawKirchhoff’s Voltage Lawin left loopin middle loopin right loopLess equations to w...

  • Page 357

    10.3. MESH CURRENT METHOD347place two of these currents:R1R2R3R4R524 V+-100 Ω300 Ω250 Ω150 Ω50 ΩI1I2The directions of these mesh currents, of course, is arbitrary. However, two mesh currentsis not enough in this circuit, because neither I1 nor I2 goes through the battery. So, we mustadd...

  • Page 358

    348CHAPTER 10. DC NETWORK ANALYSISR1R2R3R4R524 V+-100 Ω300 Ω250 Ω150 Ω50 ΩI1I2I3+--++--++--++-+-Notice something very important here: at resistor R4, the polarities for the respective meshcurrents do not agree. This is because those mesh currents (I2 and I3) are going through R4 indiffe...

  • Page 359

    10.3. MESH CURRENT METHOD349Original form of equation. . . distributing to terms within parentheses . . .. . . combining like terms . . .Simplified form of equation100(I1 + I2) + 300(I2 - I3) + 250I2 = 0100I1 + 100I2 + 300I2 - 300I3 + 250I2 = 0100I1 + 650I2 - 300I3 = 0Note how the second term in ...

  • Page 360

    350CHAPTER 10. DC NETWORK ANALYSIS-150I1 + 300I2 - 450I3 = -24100I1 + 650I2 - 300I3 = 0300I1 + 100I2 + 150I3 = 0Solutions:I1 = -93.793 mAI2 = 77.241 mAI3 = 136.092 mAExample:Use Octave to find the solution for I1, I2, and I3 from the above simplified form of equations. 400,[4]Solution:In Octave...

  • Page 361

    10.3. MESH CURRENT METHOD351I1I2I3IR1IR2IR3IR4IR5IR2 = I1 = 93.793 mAIR1 = I3 - I1 = 136.092 mA - 93.793 mA = 42.299 mAIR3 = I1 - I2 = 93.793 mA - 77.241 mA = 16.552 mAIR4 = I3 - I2 = 136.092 mA - 77.241 mA = 58.851 mAI3 > I1 > I2IR5 = I2 = 77.241 mACalculating voltage drops across each res...

  • Page 362

    352CHAPTER 10. DC NETWORK ANALYSISR1R2R3R4R524 V+-100 Ω300 Ω250 Ω150 Ω50 Ω110023unbalanced wheatstone bridgev1 1 0r1 1 2 150r2 1 3 50r3 2 3 100r4 2 0 300r5 3 0 250.dc v1 24 24 1.print dc v(1,2) v(1,3) v(3,2) v(2,0) v(3,0).endv1v(1,2)v(1,3)v(3,2)v(2)v(3)2.400E+016.345E+004.690E+001.655E+...

  • Page 363

    10.3. MESH CURRENT METHOD353R1R2R3R4R524 V+-100 Ω300 Ω250 Ω150 Ω50 ΩI1I2I3-++--++--++-+-+-50I1 + 100(I1 + I2 + I3) + 150(I1 + I3) = 0300I2 + 250(I2 + I3) + 100(I1 + I2 + I3) = 024 - 250(I2 + I3) - 100(I1 + I2 + I3) - 150(I1+I3) = 0 300I1 + 100I2 + 250I3 = 0 100I1 + 650I2 + 350I3 = 0-250...

  • Page 364

    354CHAPTER 10. DC NETWORK 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 ...

  • Page 365

    10.3. MESH CURRENT METHOD355• Write voltage-law equations in terms of unknown currents currents: I1, I2, and I3. Equa-tion 1 coefficient 1, equation 2, coefficient 2, and equation 3 coefficient 3 are the positivesums of resistors around the respective loops.• All other coefficients are ne...

  • Page 366

    356CHAPTER 10. DC NETWORK ANALYSIS300-100-150-100650-300-150-300450octave:3> b=[0;0;24]b =0024octave:4> x=A\bx =0.0937930.0772410.136092The mesh currents match the previous solution by a different mesh current method.. Thecalculation of resistor voltages and currents will be identical to th...

  • Page 367

    10.4. NODE VOLTAGE METHOD357• See rules above for details.10.4Node voltage methodThe node voltage method of analysis solves for unknown voltages at circuit nodes in terms ofa system of KCL equations. This analysis looks strange because it involves replacing voltagesources with equivalent curren...

  • Page 368

    358CHAPTER 10. DC NETWORK ANALYSISG1G2G3G4G5+-+-I1I25A4ΑE1E20.5 S0.4 S1 S0.2 S0.25 SE0GAGBThe 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 voltagemethod.GA = G1 + G2 = 0....

  • Page 369

    10.4. NODE VOLTAGE METHOD359• All other coefficients for all equations are negative, representing conductances betweennodes. The first equation, second coefficient is the conductance from node 1 to node 2, thethird coefficient is the conductance from node 1 to node 3. Fill in negative coef...

  • Page 370

    360CHAPTER 10. DC NETWORK ANALYSISR1R2R3R4R5+-100 Ω300 Ω250 Ω150 Ω50 ΩI=0.136092E1E2E3E0There are three nodes to write equations for by inspection. Note that the coefficients arepositive for equation (1) E1, equation (2) E2, and equation (3) E3. These are the sums of allconductances co...

  • Page 371

    10.5. INTRODUCTION TO NETWORK THEOREMS36124.00017.65519.310Note that the “A” matrix diagonal coefficients are positive, That all other coefficients arenegative.The solution as a voltage vector is at “x”. E1 = 24.000 V, E2 = 17.655 V, E3 = 19.310 V. Thesethree voltages compare to the pre...

  • Page 372

    362CHAPTER 10. DC NETWORK ANALYSIS28 V7 V2 ΩR2R1R34 Ω1 ΩB1B2And here is that same circuit, re-drawn for the sake of applying Millman’s Theorem:+-+-R1R2R34 Ω2 Ω1 Ω28 V7 VB1B3By considering the supply voltage within each branch and the resistance within each branch,Millman’s Theorem...

  • Page 373

    10.6. MILLMAN’S THEOREM363+-+-+-R1R2R328 V7 V8 V+-8 V1 V+-20 V-+B1B3The polarity of all voltages in Millman’s Theorem are referenced to the same point. In theexample circuit above, I used the bottom wire of the parallel circuit as my reference point,and so the voltages within each branch (28 ...

  • Page 374

    364CHAPTER 10. DC NETWORK ANALYSISthe Branch Current or Mesh Current methods. You must pay close attention to the polaritiesof resistor voltage drops as given by Kirchhoff’s Voltage Law, determining direction of currentsfrom that.+-+-+-+-+-R1R2R328 V7 V8 V20 V1 VIR3IR2IR15 A4 A1 AB1B3Millman’...

  • Page 375

    10.7. SUPERPOSITION THEOREM365voltage drops/currents with all sources active. Let’s look at our example circuit again andapply Superposition Theorem to it:28 V7 V2 ΩR2R1R34 Ω1 ΩB1B2Since we have two sources of power in this circuit, we will have to calculate two sets ofvalues for voltage ...

  • Page 376

    366CHAPTER 10. DC NETWORK ANALYSISEIRVoltsAmpsOhmsR1R2R3R2//R3R2//R3R1 +Total28244444210.6674.66766624R128 VR2R3+-+-+-+-24 V4 V4 V6 A4 A2 AB1Analyzing the circuit with only the 7 volt battery, we obtain another set of values for voltageand current:EIRVoltsAmpsOhmsR1R2R3+Total421R1//R2R1//R2R371.3...

  • Page 377

    10.7. SUPERPOSITION THEOREM367+-+-+-+-+--+++--With 28 VbatteryWith 7 VbatteryWith both batteries24 VER1ER2ER34 V20 V24 V - 4 V = 20 V4 V+-4 V8 V4 V + 4 V = 8 V4 V3 V1 V4 V - 3 V = 1 VER1ER1ER3ER3ER2ER2Applying these superimposed voltage figures to the circuit, the end result looks somethinglike ...

  • Page 378

    368CHAPTER 10. DC NETWORK ANALYSISWith 28 VbatteryWith 7 VbatteryWith both batteries6 AIR1IR11 A6 A - 1 A = 5 AIR15 A2 A2 A4 AIR2IR2IR22 A + 2 A = 4 A4 A3 AIR3IR3IR31 A4 A - 3 A = 1 AOnce again applying these superimposed figures to our circuit:+-+-R1R2R328 V7 V5 A4 A1 AB1B2Quite simple and eleg...

  • Page 379

    10.8. THEVENIN’S THEOREM369semiconductor (amplifier) circuits, where sometimes AC is often mixed (superimposed) withDC. Because AC voltage and current equations (Ohm’s Law) are linear just like DC, we canuse Superposition to analyze the circuit with just the DC power source, then just the AC...

  • Page 380

    370CHAPTER 10. DC NETWORK ANALYSISload resistance is verycommon in power systems, as multiple loads get switched on and off asneeded. the total resistance of their parallel connections changing depending on how many areconnected at a time). This could potentially involve a lotof work!Thevenin’s...

  • Page 381

    10.8. THEVENIN’S THEOREM371break (open circuit):R1R328 V7 VLoad resistorremoved4 Ω1 ΩB1B2Next, the voltage between the two points where the load resistor used to be attached isdetermined. Use whatever analysis methods are at your disposal to do this. In this case, theoriginal circuit with t...

  • Page 382

    372CHAPTER 10. DC NETWORK ANALYSISRTheveninR2(Load)ETheveninThevenin Equivalent Circuit2 Ω11.2 VTo find the Thevenin series resistance for our equivalent circuit, we need to take the originalcircuit (with the load resistor still removed), remove the power sources (in the same style as wedid wi...

  • Page 383

    10.9. NORTON’S THEOREM373With the load resistor (2 Ω) attached between the connection points, we can determinevoltage across it and current through it as though the whole network were nothing more thana simple series circuit:EIRVoltsAmpsOhmsTotalRTheveninRLoad11.242.80.82443.28Notice that the...

  • Page 384

    374CHAPTER 10. DC NETWORK ANALYSISContrasting our original example circuit against the Norton equivalent: it looks somethinglike this:R1R2(Load)R328 V7 V4 Ω2 Ω1 ΩB1B2. . . after Norton conversion . . .INortonRNortonR22 Ω(Load)Norton Equivalent CircuitRemember that a current sourceis a com...

  • Page 385

    10.9. NORTON’S THEOREM375place a direct wire (short) connection between the load points and determine the resultantcurrent. 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):R1R328 V7 V4 Ω1 Ω7 ...

  • Page 386

    376CHAPTER 10. DC NETWORK ANALYSISR1R34 Ω1 Ω0.8 ΩNow our Norton equivalent circuit looks like this:INortonRNortonR22 Ω(Load)14 A0.8 ΩNorton Equivalent CircuitIf we re-connect our original load resistance of 2 Ω, we can analyze the Norton circuit as asimple parallel arrangement:EIRVolt...

  • Page 387

    10.10. THEVENIN-NORTON EQUIVALENCIES377• (1) Find the Norton source current by removing the load resistor from the original circuitand calculating current through a short (wire) jumping across the open connection pointswhere the load resistor used to be.• (2) Find the Norton resistance by rem...

  • Page 388

    378CHAPTER 10. DC NETWORK ANALYSISINortonRNortonR22 Ω(Load)14 A0.8 ΩNorton Equivalent CircuitRThevenin = RNortonConsidering the fact that both Thevenin and Norton equivalent circuits are intended tobehave the same as the original network in supplying voltage and current to the load resistor(a...

  • Page 389

    10.11. MILLMAN’S THEOREM REVISITED379• Thevenin voltage is equal to Norton current times Norton resistance.• Norton current is equal to Thevenin voltage divided by Thevenin resistance.10.11Millman’s Theorem revisitedYou may have wondered where we got that strange equation for the determin...

  • Page 390

    380CHAPTER 10. DC NETWORK ANALYSIS7 A4 Ω0 A2 Ω7 A1 ΩSince current sources directly add their respective currents in parallel, the total circuitcurrent will be 7 + 0 + 7, or 14 amps. This addition of Norton source currents is what’s beingrepresented in the numerator of the Millman equation...

  • Page 391

    10.12. MAXIMUM POWER TRANSFER THEOREM381+-14 A571.43 mΩ 8 VLet’s summarize what we know about the circuit thus far. We know that the total current inthis circuit is given by the sum of all the branch voltages divided by their respective resistances.We also know that the total resistance is fo...

  • Page 392

    382CHAPTER 10. DC NETWORK ANALYSIStransmission line “impedance” is matched to final power amplifier “impedance” for maximumradio frequency power output. Impedance, the overall opposition to AC and DC current, is verysimilar to resistance, and must be equal between source and load for th...

  • Page 393

    10.13. ∆-Y AND Y-∆ CONVERSIONS383EIRVoltsAmpsOhmsTotalRLoadRTheveninPWatts11.20.81.11.95.8955.8955.8954.7166.48427.8038.2266.02If you were designing a circuit for maximum power dissipation at the load resistance, thistheorem would be very useful. Having reduced a network down to a Thevenin vo...

  • Page 394

    384CHAPTER 10. DC NETWORK ANALYSISABCBCARACRABRBCRARCRBDelta (∆) networkWye (Y) networkACBACBRACRABRBCRARCRBTee (T) networkPi (π) networkIt is possible to calculate the proper values of resistors necessary to form one kind of net-work (∆ or Y) that behaves identically to the other kind, as a...

  • Page 395

    10.13. ∆-Y AND Y-∆ CONVERSIONS385equal in value) and conversion from one to the other need not involve such complex calcula-tions. When would the average technician ever need to use these equations?A prime application for ∆-Y conversion is in the solution of unbalanced bridge circuits, such...

  • Page 396

    386CHAPTER 10. DC NETWORK ANALYSISABC10 VRARBRCR4R518 Ω12 Ω∆ converted to a YIf we perform our calculations correctly, the voltages between points A, B, and C will be thesame in the converted circuit as in the original circuit, and we can transfer those values backto the original bridge con...

  • Page 397

    10.13. ∆-Y AND Y-∆ CONVERSIONS387Resistors R4 and R5, of course, remain the same at 18 Ω and 12 Ω, respectively. Analyzingthe circuit now as a series/parallel combination, we arrive at the following figures:EIRVoltsAmpsOhmsRARBRCR4R56231812EIRVoltsAmpsOhmsRB + R4RC + R5RC + R5//RB + R4To...

  • Page 398

    388CHAPTER 10. DC NETWORK ANALYSIS10 VR1R2R3R4 R54.706V5.294V5.294V4.706V0.588 VVoltage drops across R4 and R5, of course, are exactly the same as they were in the convertedcircuit.At this point, we could take these voltages and determine resistor currents through therepeated use of Ohm’s Law (...

  • Page 399

    10.14. CONTRIBUTORS389r1 1 2 12r2 1 3 18r3 2 3 6r4 2 0 18r5 3 0 12.dc v1 10 10 1.print dc v(1,2) v(1,3) v(2,3) v(2,0) v(3,0).endv1v(1,2)v(1,3)v(2,3)v(2)v(3)1.000E+014.706E+005.294E+005.882E-015.294E+004.706E+00The voltage figures, as read from left to right, represent voltage drops across the ...

  • Page 400

    390CHAPTER 10. DC NETWORK ANALYSISBibliography[1] A.E. Fitzergerald, David E. Higginbotham, Arvin Grabel, Basic Electrical Engineering,(McGraw-Hill, 1975).[2] TonyKuphaldt,UsingtheSpiceCircuitSimulationProgram,in“LessonsinElectricity,Reference”,Volume5,Chapter7,atactionURI(http://www.ibiblio....

  • Page 401

    Chapter 11BATTERIES AND POWERSYSTEMSContents 401,11.1 401,Electron 401,activity 401,in 401,chemical 401,reactions . . . . . . . . . . . . . . . . . . . . . 391 407,11.2 407,Battery 407,construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 410,11.3 410,Battery ...

  • Page 402

    392CHAPTER 11. BATTERIES AND POWER SYSTEMS= electron= proton= neutroneNPPPPPPPNNN NNNeeeeeeThe protons in an atom’s nucleus are extremely difficult to dislodge, and so the chemicalidentity of any atom is very stable. One of the goals of the ancient alchemists (to turn leadinto gold) was foiled...

  • Page 403

    11.1. ELECTRON ACTIVITY IN CHEMICAL REACTIONS393covalentbond, where electrons are shared between atoms. Because chemical bonds are basedon links formed by electrons, these bonds are only as strong as the immobility of the electronsforming them. That is to say, chemical bonds can be created or bro...

  • Page 404

    394CHAPTER 11. BATTERIES AND POWER SYSTEMSelectrolyte solutionelectrodes+-The two electrodes are made of different materials,both of which chemically react with the electrolytein some form of ionic bonding.Voltaic cellIn the common ”lead-acid” cell (the kind commonly used in automobiles), the...

  • Page 405

    11.1. ELECTRON ACTIVITY IN CHEMICAL REACTIONS395+-Pb electrodePbO2 electrodeH2SO4 + H2Oelectrolyte:-+loadIPb(IV)O2 + 3H+ + HSO4- + 2e- Pb(II)SO4 + 2H2OPb + HSO4-Pb(II)SO4 + H+ + 2e- electronsLead-acid cell dischargingPbO2 + Pb + 2H2SO42PbSO4 + 2H2O At (+) electrode: At (-) electrode:Overall cell:...

  • Page 406

    396CHAPTER 11. BATTERIES AND POWER SYSTEMSis that electrons are motivated to and/or from the cell’s electrodes via ionic reactions betweenthe electrode molecules and the electrolyte molecules. The reaction is enabled when there isan external path for electric current, and ceases when that path ...

  • Page 407

    11.2. BATTERY CONSTRUCTION397• Electrochemical reactions involve the transfer of electrons between atoms. This transfercan be harnessed to form an electric current.• A cellis a device constructed to harness such chemical reactions to generate electric cur-rent.• A cell is said to be dischar...

  • Page 408

    398CHAPTER 11. BATTERIES AND POWER SYSTEMS+--+-+-+-+-+-+2.0 V12.0 V2.0 V2.0 V2.0 V2.0 V2.0 VThe 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 solu-tions for each cell are c...

  • Page 409

    11.2. BATTERY CONSTRUCTION399which affects its ability to supply current to the load resistance of 1 Ω. The ideal battery on theleft has no internal resistance, and so our Ohm’s Law calculations for current (I=E/R) give usa perfect value of 10 amps for current with the 1 ohm load and 10 volt ...

  • Page 410

    400CHAPTER 11. BATTERIES AND POWER SYSTEMS• Cells connected together in parallel results in less total resistance, and potentially greatertotal current.11.3Battery ratingsBecause batteries create electron flow in a circuit by exchanging electrons in ionic chemicalreactions, and there is a limi...

  • Page 411

    11.3. BATTERY RATINGS401Approximate 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 (prim...

  • Page 412

    402CHAPTER 11. BATTERIES AND POWER SYSTEMS. . . and a bit further until its dead.+V-Voltmeter indication:+V-Voltmeter indication:No loadUnder load50 Ω7.5 V50 Ω7.5 V7.5 V100 Ω5 VScenario for a dead batteryNotice how much better the battery’s true condition is revealed when its voltage is c...

  • Page 413

    11.4. SPECIAL-PURPOSE BATTERIES403cork washercork washermercurycadmium amalgammercurous sulphatecadmium sulphatesolutioncadmium sulphatesolutionwirewireglass bulb+-Mercury "standard" cellHg2SO4CdSO4CdSO4Unfortunately, mercury cells were rather intolerant of any current drain and could n...

  • Page 414

    404CHAPTER 11. BATTERIES AND POWER SYSTEMSenergy source.water outoxygen inhydrogen inelectrolytemembranes-+electrodesload--+H2H2H2H2H2H2O2O2O2O2O2O2H+H+H+H+e-e-e-e-e-e-Hydrogen/Oxygen fuel cellTo date, the most successful fuel cells constructed are those which run on hydrogen and oxy-gen, althoug...

  • Page 415

    11.4. SPECIAL-PURPOSE BATTERIES405thin, round wafer ofcrystalline siliconwiresschematic symbolSolar cellSpecific cost of solar cell technology (dollars per kilowatt) is still very high, with littleprospect of significant decrease barring some kind of revolutionary advance in technology. Un-like...

  • Page 416

    406CHAPTER 11. BATTERIES AND POWER SYSTEMS11.5Practical considerationsWhen connecting batteries together to form larger ”banks” (a batteryof batteries?), the con-stituent batteries must be matched to each other so as to not cause problems. First we willconsider connecting batteries in series ...

  • Page 417

    11.5. PRACTICAL CONSIDERATIONS407cerns to protect against as well. Batteries have been known to internally short-circuit, dueto electrode separator failure, causing a problem not unlike that where batteries of unequalvoltage are connected in parallel: the good batteries will overpower the failed ...

  • Page 418

    408CHAPTER 11. BATTERIES AND POWER SYSTEMSMore modern lead-acid battery designs are sealed, fabricated to re-combine the electrolyzedhydrogen and oxygen back into water, inside the battery case itself. Adequate ventilation mightstill be a good idea, just in case a battery were to develop a leak. ...

  • Page 419

    Chapter 12PHYSICS OF CONDUCTORSAND INSULATORSContents 419,12.1 419,Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 421,12.2 421,Conductor 421,size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 427,12.3 427,Conductor 42...

  • Page 420

    410CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSthese zones only in a limited range of energies depending on the particular zone and how oc-cupied that zone is with other electrons. If electrons really did act like tiny planets held inorbit around the nucleus by electrostatic attraction, their...

  • Page 421

    12.2. CONDUCTOR SIZE411for electrons to flow with controlled amounts of resistance. It is also vitally important that webe able to prevent electrons from flowing where we don’t want them to, by using insulatingmaterials. However, not all conductors are the same, and neither are all insulators...

  • Page 422

    412CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSsounds: a single, solid strand of copper the whole length of the wire. Stranded wire is composedof smaller strands of solid copper wire twisted together to form a single, larger conductor. Thegreatest benefit of stranded wire is its mechanical ...

  • Page 423

    12.2. CONDUCTOR SIZE413end-view ofsolid round wireCross-sectional area101.9milsis 8155.27 square milsA = πr2A = (3.1416)22101.9A = 8155.27 square milsmilsHowever, electricians and others frequently concerned with wire size use another unit ofarea measurement tailored specifically for wire’s c...

  • Page 424

    414CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSAnd for another size of wire:2 milsArea = 4 circular milsArea = 3.1416 square mils2 milsArea = 5.0930 circular milsArea = 4 square milsObviously, the circle of a given diameter has less cross-sectional area than a square ofwidth and height equal ...

  • Page 425

    12.2. CONDUCTOR SIZE415Roman numeral ”M” to denote a multiple of ”thousand” in front of ”CM” for ”circular mils.” Thefollowing table of wire sizes does not show any sizes bigger than 4/0 gauge, because solidcop-per wire becomes impractical to handle at those sizes. Stranded wire c...

  • Page 426

    416CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORS33------- 0.007080 ------- 50.13 ----- 0.00003937 ---- 0.151734------- 0.006305 ------- 39.75 ----- 0.00003122 ---- 0.120335------- 0.005615 ------- 31.52 ----- 0.00002476 --- 0.0954236------- 0.005000 ------- 25.00 ----- 0.00001963 --- 0.0756737...

  • Page 427

    12.3. CONDUCTOR AMPACITY417• Very large wire sizes are rated in thousands of circular mils (MCM’s), typical for busbarsand wire sizes beyond 4/0.• Busbarsare solid bars of copper or aluminum used in high-current circuit construction.Connections made to busbars are usually welded or bolted, ...

  • Page 428

    418CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORS2 --------- 140 ------------ 170 ------------ 1901 --------- 165 ------------ 195 ------------ 2201/0 ------- 195 ------------ 230 ------------ 2602/0 ------- 225 ------------ 265 ------------ 3003/0 ------- 260 ------------ 310 ------------ 3504...

  • Page 429

    12.4. FUSES419OUTER COVERING ("JACKET")=========================N = NylonSPECIAL SERVICE CONDITIONS==========================U = UndergroundW = Wet-2 = 90 degrees Celsius and wetTherefore, a ”THWN” conductor has Thermoplastic insulation, is Heat resistant to 75oCelsius, is rated for...

  • Page 430

    420CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSprotected from overcurrent, so that when the fuse blows(opens) it will open the entire circuitand 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 o...

  • Page 431

    12.4. FUSES421The fuses are held by spring metal clips, the clips themselves being permanently connectedto the circuit conductors. The base material of the fuse holder (or fuse blockas they are some-times called) is chosen to be a good insulator.Another type of fuse holder for cartridge-type fuse...

  • Page 432

    422CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSThe most common device in use for overcurrent protection in high-current circuits todayis the circuit breaker. Circuit breakers are specially designed switches that automaticallyopen to stop current in the event of an overcurrent condition. Small...

  • Page 433

    12.4. FUSES423From outside appearances, it looks like nothing more than a switch. Indeed, it could beused 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 linksforovercurrent pro...

  • Page 434

    424CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSthe fuse wire as short as is practically possible. Just as a normal wire’s ampacity is not relatedto its length (10-gauge solid copper wire will handle 40 amps of current in free air, regardlessof how long or short of a piece it is), a fuse wir...

  • Page 435

    12.4. FUSES425latter fuses are sometimes called slow-blowfuses due to their intentional time-delay charac-teristics.A classic example of a slow-blow fuse application is in electric motor protection, where in-rushcurrents of up to ten times normal operating current are commonly experienced everyti...

  • Page 436

    426CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSloadblown fuse"Hot""Neutral"voltage present between either sideof load and ground!In either case, the fuse successfully interrupted current to the load, but the lower circuitfails to interrupt potentially dangerous voltage fro...

  • Page 437

    12.5. SPECIFIC RESISTANCE427• Fuses are primarily rated in terms of maximum current, but are also rated in terms ofhow 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 maximumlevel of curr...

  • Page 438

    428CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSR =ρlAThis formula relates the resistance of a conductor with its specific resistance (the Greekletter ”rho” (ρ), which looks similar to a lower-case letter ”p”), its length (”l”), and its cross-sectional area (”A”). Notice tha...

  • Page 439

    12.5. SPECIFIC RESISTANCE429(Ω-cm) in the same formula, the length needs to be in centimeters and the area in squarecentimeters.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 rou...

  • Page 440

    430CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSwire resistancewire resistanceLoad(requires at least 220 V)2300 feet230 V5.651 Ω5.651 ΩOur total circuit wire resistance is 2 times 5.651, or 11.301 Ω. Unfortunately, this is fartoomuch resistance to allow 25 amps of current with a source v...

  • Page 441

    12.6. TEMPERATURE COEFFICIENT OF RESISTANCE431• Specific Resistance(”ρ”) is a property of any conductive material, a figure used to deter-mine 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...

  • Page 442

    432CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSMaterialElement/Alloy"alpha" per degree Celsius==========================================================Nickel -------- Element --------------- 0.005866Iron ---------- Element --------------- 0.005671Molybdenum ---- Element -----------...

  • Page 443

    12.6. TEMPERATURE COEFFICIENT OF RESISTANCE433R = Rref [1 + α(T - Tref)]R = 15.909 ΩR = (15 Ω)[1 + 0.004041(35o - 20o)]Recalculating our circuit values, we see what changes this increase in temperature willbring:EIRVoltsAmpsOhmsTotalWire1Wire2Load2501415.90915.909281.8249.677m49.677m49.677m4...

  • Page 444

    434CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORS•R = Rref [1 + α(T - Tref)]Where,R = Conductor resistance at temperature "T"Rref = Conductor resistance at reference temperatureα = Temperature coefficient of resistance for theconductor material.T = Conductor temperature in degrees...

  • Page 445

    12.7. SUPERCONDUCTIVITY435in developing ”high-temperature” superconductors which superconduct at warmer tempera-tures. One type is a ceramic mixture of yttrium, barium, copper, and oxygen which transitionsat a relatively balmy -160o Celsius. Ideally, a superconductor should be able to operate...

  • Page 446

    436CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORSsuperconducting wireelectrons will flow unimpeded byresistance, continuing to flowforever!Rings of superconducting material have been experimentally proven to sustain continuouscurrent for years with no applied voltage. So far as anyone knows, th...

  • Page 447

    12.8. INSULATOR BREAKDOWN VOLTAGE437and electron flow will occur. However, unlike the situation with conductors where current is ina linear proportion to applied voltage (given a fixed resistance), current through an insulatoris quite nonlinear: for voltages below a certain threshold level, vir...

  • Page 448

    438CHAPTER 12. PHYSICS OF CONDUCTORS AND INSULATORS12.9DataTables of specific resistance and temperature coefficient of resistance for elemental materi-als (not alloys) were derived from figures found in the 78th edition of the CRC Handbook ofChemistry and Physics.Table of superconductor criti...

  • Page 449

    Chapter 13CAPACITORSContents 449,13.1 449,Electric 449,fields 449,and 449,capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . 439 454,13.2 454,Capacitors 454,and 454,calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 459,13.3 459,Factors 459,affecting...

  • Page 450

    440CHAPTER 13. CAPACITORSThe subject of this chapter is electricfields (and devices called capacitorsthat exploit them),not magneticfields, but there are many similarities. Most likely you have experienced electricfields as well. Chapter 1 of this book began with an explanation of static elect...

  • Page 451

    13.1. ELECTRIC FIELDS AND CAPACITANCE441modernobsoleteCapacitor symbolsWhen a voltage is applied across the two plates of a capacitor, a concentrated field flux iscreated between them, allowing a significant difference of free electrons (a charge) to developbetween the two plates:-++ + ++ + +-...

  • Page 452

    442CHAPTER 13. CAPACITORScapacitor tends to stay charged; a discharged capacitor tends to stay discharged.” Hypothet-ically, a capacitor left untouched will indefinitely maintain whatever state of voltage chargethat its been left it. Only an outside source (or drain) of current can alter the v...

  • Page 453

    13.1. ELECTRIC FIELDS AND CAPACITANCE443C+-. . .. . .. . . to the rest of the circuitIIvoltageThe capacitor acts as a SOURCEEnergy being released by thecapacitor to the rest of the circuitdecreasingIf a source of voltage is suddenly applied to an uncharged capacitor (a sudden increase ofvoltage),...

  • Page 454

    444CHAPTER 13. CAPACITORS• When a capacitor is faced with an increasing voltage, it acts as a load: drawing currentas it absorbs energy (current going in the negative side and out the positive side, like aresistor).• When a capacitor is faced with a decreasing voltage, it acts as a source: su...

  • Page 455

    13.2. CAPACITORS AND CALCULUS445In a capacitor, however, time is an essential variable, because current is related to howrapidlyvoltage changes over time. To fully understand this, a few illustrations may be neces-sary. Suppose we were to connect a capacitor to a variable-voltage source, construc...

  • Page 456

    446CHAPTER 13. CAPACITORS+-+-+V-Potentiometer wiper movingslowly in the "up" directionIncreasingSteady currentvoltageIf we assume that the potentiometer wiper is being moved such that the rateof voltageincrease across the capacitor is steady (for example, voltage increasing at a constan...

  • Page 457

    13.2. CAPACITORS AND CALCULUS447+-+-+V-Potentiometer wiper movingIncreasingSteady currentvoltagequickly in the "up" direction(greater)(faster)TimeTimeCapacitorvoltageCapacitorcurrentECICVoltagechangeTimePotentiometer wiper moving quickly "up"When mathematics students first st...

  • Page 458

    448CHAPTER 13. CAPACITORSIf we were to move the potentiometer’s wiper in the same direction as before (”up”), but atvarying rates, we would obtain graphs that looked like this:TimeTimeCapacitorvoltageCapacitorcurrentECICPotentiometer wiper moving "up" atdifferent ratesNote how tha...

  • Page 459

    13.3. FACTORS AFFECTING CAPACITANCE449+-+-+V-Potentiometer wiper movingvoltagein the "down" directionDecreasingAgain, the amount of current through the capacitor is directly proportional to the rate ofvoltage change across it. The only difference between the effects of a decreasingvolta...

  • Page 460

    450CHAPTER 13. CAPACITORSless capacitancemore capacitanceDIELECTRIC MATERIAL: All other factors being equal, greater permittivity of the di-electric gives greater capacitance; less permittivity of the dielectric gives less capacitance.Explanation:Although its complicated to explain, some material...

  • Page 461

    13.3. FACTORS AFFECTING CAPACITANCE451Castor oil --------------------- 5.0Wood (Birch) ------------------- 5.2Mica, muscovite ---------------- 5.0 to 8.7Glass-bonded mica -------------- 6.3 to 9.3Porcelain, Steatite ------------ 6.5Alumina ------------------------ 8.0 to 10.0Distilled water -----...

  • Page 462

    452CHAPTER 13. CAPACITORSAs 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 beestablished. This particular capacitor has a capacitance in the picofarad range, and ...

  • Page 463

    13.5. PRACTICAL CONSIDERATIONS453parallel resistances:Series CapacitancesCtotal = C1C2Cn1+1+ . . .11When capacitors are connected in parallel, the total capacitance is the sum of the individualcapacitors’ capacitances. If two or more capacitors are connected in parallel, the overall effectis th...

  • Page 464

    454CHAPTER 13. CAPACITORSPolarity: Some capacitors are manufactured so they can only tolerate applied voltage in onepolarity but not the other. This is due to their construction: the dielectric is a microscopicallythin layer of insulation deposited on one of the plates by a DC voltage during manu...

  • Page 465

    13.5. PRACTICAL CONSIDERATIONS455you cannot judge a capacitor’s rating in Farads simply by size. A capacitor of any given size maybe relatively high in capacitance and low in working voltage, vice versa, or some compromisebetween the two extremes. Take the following two photographs for example:...

  • Page 466

    456CHAPTER 13. CAPACITORSThe thinner dielectric layer gives it a much greater capacitance (20,000 µF) and a drasti-cally reduced working voltage (35 volts continuous, 45 volts intermittent).Here are some samples of different capacitor types, all smaller than the units shown previ-ously:

  • Page 467

    13.5. PRACTICAL CONSIDERATIONS457The electrolytic and tantalum capacitors are polarized(polarity sensitive), and are alwayslabeled as such. The electrolytic units have their negative (-) leads distinguished by arrowsymbols on their cases. Some polarized capacitors have their polarity designated b...

  • Page 468

    458CHAPTER 13. CAPACITORScapacitors do not have polarity markings, because those types are nonpolarized(they are notpolarity sensitive).Capacitors are very common components in electronic circuits. Take a close look at thefollowing photograph – every component marked with a ”C” designation ...

  • Page 469

    13.6. CONTRIBUTORS459The 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 beidentified by labels beginning with the letter ”C”.13.6ContributorsContributors to thi...

  • Page 470

    460CHAPTER 13. CAPACITORS

  • Page 471

    Chapter 14MAGNETISM ANDELECTROMAGNETISMContents 471,14.1 471,Permanent 471,magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 475,14.2 475,Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 477,14.3 477,Magnetic 477,units 477,...

  • Page 472

    462CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMNSmagnetNNSSmagnetmagnet. . . after breaking in half . . .Like electric charges, there were only two types of poles to be found: north and south (byanalogy, positive and negative). Just as with electric charges, same poles repel one another,while oppos...

  • Page 473

    14.1. PERMANENT MAGNETS463term ”line” is more commonly used now). Indeed, the measurement of magnetic field flux isoften defined in terms of the number of flux lines, although it is doubtful that such fields existin individual, discrete lines of constant value.Modern theories of magnetis...

  • Page 474

    464CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMNSmagnetironNSattractionReferencing the natural magnetic properties of iron (Latin = ”ferrum”), a ferromagneticmaterial is one that readily magnetizes (its constituent atoms easily orient their electron spinsto conform to an external magnetic fiel...

  • Page 475

    14.2. ELECTROMAGNETISM465• 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 externalhelp. The characteristic of any magnetic material to do so is called retentivity.• ...

  • Page 476

    466CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMmore. To create a stronger magnetic field force (and consequently, more field flux) with thesame amount of electric current, we can wrap the wire into a coil shape, where the circlingmagnetic fields around the wire will join to create a larger fie...

  • Page 477

    14.3. MAGNETIC UNITS OF MEASUREMENT467RelayApplying current through the coilcauses the switch to close.Relays can be constructed to actuate multiple switch contacts, or operate them in ”reverse”(energizing the coil will openthe switch contact, and unpowering the coil will allow it to springcl...

  • Page 478

    468CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMization in the science of magnetism, we have been plagued with no less than three completesystems of measurement for magnetic quantities.First, we need to become acquainted with the various quantities associated with mag-netism. There are quite a few m...

  • Page 479

    14.3. MAGNETIC UNITS OF MEASUREMENT469Quantity SymbolMeasurementUnit ofand abbreviationField ForceField FluxFieldIntensityFluxDensityCGSSIEnglishReluctancePermeabilitymmfΦHBµGilbert (Gb)Amp-turnAmp-turnOersted (Oe)Maxwell (Mx) Weber (Wb)LineGauss (G)Tesla (T)per meterper inchLines persquare inc...

  • Page 480

    470CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMThe major caveat here is that the reluctance of a material to magnetic flux actually changeswith the concentration of flux going through it. This makes the ”Ohm’s Law” for magneticcircuits nonlinear and far more difficult to work with than the...

  • Page 481

    14.4. PERMEABILITY AND SATURATION471in 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 viceversa), you have to rotate the steering wheel an additional amount to ov...

  • Page 482

    472CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMDue 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 atthis point. Now we will slowly apply the same amount of magnetic field ...

  • Page 483

    14.4. PERMEABILITY AND SATURATION473Flux density(B)Field intensity (H)The ”S”-shaped curve traced by these steps form what is called the hysteresis curveof aferromagnetic material for a given set of field intensity extremes (-H and +H). If this doesn’tquite make sense, consider a hysteresi...

  • Page 484

    474CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMangle of front wheelsrotation of steering wheel(right)(left)(CCW)(CW)amount of "looseness"in the steering mechanismA "loose" steering systemJust as in the case of automobile steering systems, hysteresis can be a problem. If you’re...

  • Page 485

    14.5. ELECTROMAGNETIC INDUCTION475Field intensity (H)Flux density(B)Hysteresis curve for ferrite• REVIEW:• The permeability of a material changes with the amount of magnetic flux forced throughit.• The specific relationship of force to flux (field intensity H to flux density B) is grap...

  • Page 486

    476CHAPTER 14. MAGNETISM AND ELECTROMAGNETISM+V-magnet movedback and forthNS-+voltage changes polaritywith change in magnet motionwith change in magnet motioncurrent changes directionElectromagnetic inductionFaraday was able to mathematically relate the rate of change of the magnetic field flux...

  • Page 487

    14.6. MUTUAL INDUCTANCE477self-induced voltage will be more intense. A device constructed to take advantage of this effectis 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...

  • Page 488

    478CHAPTER 14. MAGNETISM AND ELECTROMAGNETISMThe device shown in the above photograph is a kind of transformer, with two concentricwire coils. It is actually intended as a precision standard unit for mutual inductance, but forthe purposes of illustrating what the essence of a transformer is, it w...

  • Page 489

    14.6. MUTUAL INDUCTANCE479coil to output coil, the current will be decreased by the same proportion. This action of thetransformer is analogous to that of mechanical gear, belt sheave, or chain sprocket ratios:++Large gearSmall gear(many teeth)(few teeth)high torque, low speedlow torque, high spe...

  • Page 490

    480CHAPTER 14. MAGNETISM AND ELECTROMAGNETISM14.7ContributorsContributors to this chapter are listed in chronological order of their contributions, from mostrecent to first. See Appendix 2 (Contributor List) for dates and contact information.Jason Starck (June 2000): HTML document formatting, wh...

  • Page 491

    Chapter 15INDUCTORSContents 491,15.1 491,Magnetic 491,fields 491,and 491,inductance . . . . . . . . . . . . . . . . . . . . . . . . . . 481 495,15.2 495,Inductors 495,and 495,calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 501,15.3 501,Factors 501,affecting 5...

  • Page 492

    482CHAPTER 15. INDUCTORSWhereas an electric field flux between two conductors allows for an accumulation of freeelectron charge within those conductors, a magnetic field flux allows for a certain ”inertia” toaccumulate in the flow of electrons through the conductor producing the field.I...

  • Page 493

    15.1. MAGNETIC FIELDS AND INDUCTANCE483to maintain current at a constant level. In other words, inductors tend to resist changesincurrent. When current through an inductor is increased or decreased, the inductor ”resists”the changeby producing a voltage between its leads in opposing polarity ...

  • Page 494

    484CHAPTER 15. INDUCTORSthe direction of electron flow, acting as a power source. In this condition the inductor is said tobe discharging, because its store of energy is decreasing as it releases energy from its magneticfield to the rest of the circuit. Note the polarity of the voltage with reg...

  • Page 495

    15.2. INDUCTORS AND CALCULUS485• Inductors react against changes in current by dropping voltage in the polarity necessaryto oppose the change.• When an inductor is faced with an increasing current, it acts as a load: dropping voltageas it absorbs energy (negative on the current entry side and...

  • Page 496

    486CHAPTER 15. INDUCTORSthis section), the voltage dropped across the terminals of an inductor is purely related to howquickly its current changes over time.Suppose we were to connect a perfect inductor (one having zero ohms of wire resistance) toa circuit where we could vary the amount of curren...

  • Page 497

    15.2. INDUCTORS AND CALCULUS487TimeTimePotentiometer wiper not movingInductorcurrentILInductorvoltageELIf we move the potentiometer wiper slowly in the ”up” direction, its resistance from end toend will slowly decrease. This has the effect of increasing current in the circuit, so the ammeteri...

  • Page 498

    488CHAPTER 15. INDUCTORStion e = N(dΦ/dt). This self-induced voltage across the coil, as a result of a gradual change incurrent magnitude through the coil, happens to be of a polarity that attempts to oppose thechange in current. In other words, the induced voltage polarity resulting from an inc...

  • Page 499

    15.2. INDUCTORS AND CALCULUS489TimeTimeInductorcurrentILInductorvoltageELPotentiometer wiper moving "up" atdifferent ratesHere again we see the derivativefunction of calculus exhibited in the behavior of an in-ductor. In calculus terms, we would say that the induced voltage across the i...

  • Page 500

    490CHAPTER 15. INDUCTORSthrough it is decreased. As described by Lenz’s Law, the induced voltage will be opposed to thechange in current. With a decreasingcurrent, the voltage polarity will be oriented so as to tryto keep the current at its former magnitude. In this scenario, the inductor will ...

  • Page 501

    15.3. FACTORS AFFECTING INDUCTANCE491When the switch is opened, however, it suddenly introduces an extremely high resistanceinto the circuit (the resistance of the air gap between the contacts). This sudden introductionof high resistance into the circuit causes the circuit current to decrease alm...

  • Page 502

    492CHAPTER 15. INDUCTORSExplanation:Greater coil area presents less opposition to the formation of magnetic fieldflux, for a given amount of field force (amp-turns).less inductancemore inductanceCOIL LENGTH: All other factors being equal, the longer the coil’s length, the less induc-tance; t...

  • Page 503

    15.3. FACTORS AFFECTING INDUCTANCE493Where,N = Number of turns in wire coil (straight wire = 1)L =N2µAlL =µ =A =l =Inductance of coil in HenrysPermeability of core material (absolute, not relative)Area of coil in square meters = πr2Average length of coil in metersµ = µr µ0µr =µ0 =Relative...

  • Page 504

    494CHAPTER 15. INDUCTORSThis 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 builtfor radio...

  • Page 505

    15.3. FACTORS AFFECTING INDUCTANCE495Inductors can also be made very small for printed circuit board applications. Closely exam-ine the following photograph and see if you can identify two inductors near each other:

  • Page 506

    496CHAPTER 15. INDUCTORSThe two inductors on this circuit board are labeled L1 and L2, and they are located to theright-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-sh...

  • Page 507

    15.4. SERIES AND PARALLEL INDUCTORS497A pair of inductors can be seen on this circuit board, to the right and center, appearingas small black chips with the number ”100” printed on both. The upper inductor’s label canbe seen printed on the green circuit board as L5. Of course these inductor...

  • Page 508

    498CHAPTER 15. INDUCTORSincrease in currentL1L2-+-+voltagedroptotal voltage drop-+voltagedropThus, the total inductance for series inductors is more than any one of the individual in-ductors’ inductances. The formula for calculating the series total inductance is the same formas for calculating...

  • Page 509

    15.5. PRACTICAL CONSIDERATIONS49915.5Practical considerationsInductors, like all electrical components, have limitations which must be respected for the sakeof reliability and proper circuit operation.Rated current:Since inductors are constructed of coiled wire, and any wire will be limitedin its...

  • Page 510

    500CHAPTER 15. INDUCTORS

  • Page 511

    Chapter 16RC AND L/R TIME CONSTANTSContents 511,16.1 511,Electrical 511,transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 511,16.2 511,Capacitor 511,transient 511,response . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 514,16.3 514,Inductor 514,tra...

  • Page 512

    502CHAPTER 16. RC AND L/R TIME CONSTANTSdischarged capacitor, having a terminal voltage of zero, will initially act as a short-circuit whenattached to a source of voltage, drawing maximum current as it begins to build a charge. Overtime, the capacitor’s terminal voltage rises to meet the applie...

  • Page 513

    16.2. CAPACITOR TRANSIENT RESPONSE503|-------------------------------------------||0.5|15 V|5.902 V| 909.8 uA||-------------------------------------------||1|15 V|9.482 V| 551.8 uA||-------------------------------------------||2|15 V|12.970 V| 203.0 uA||-------------------------------------------...

  • Page 514

    504CHAPTER 16. RC AND L/R TIME CONSTANTS2.500E+001.377E+01 .+..*.3.000E+001.426E+01 . +..* .3.500E+001.455E+01 .+..*.4.000E+001.473E+01 .+..*.4.500E+001.484E+01 +..*5.000E+001.490E+01 +..*5.500E+001.494E+01 +..*6.000E+001.496E+01 +..*6.500E+001.498E+01 +..*7.000E+001.499E+01 +..*7.500E+001.499E+0...

  • Page 515

    16.3. INDUCTOR TRANSIENT RESPONSE505inductor tries to maintain a constant current through its windings. Because of this, inductorsoppose changes in current, and act precisely the opposite of capacitors, which oppose changesin voltage. A fully discharged inductor (no magnetic field), having zero ...

  • Page 516

    506CHAPTER 16. RC AND L/R TIME CONSTANTS|Time| Battery| Inductor| Current||(seconds) | voltage|voltage|||-------------------------------------------||0|15 V|15 V|0||-------------------------------------------||0.5|15 V|9.098 V|5.902 A||-------------------------------------------||1|15 V|5.518 V|9...

  • Page 517

    16.4. VOLTAGE AND CURRENT CALCULATIONS5075.000E-019.119E+00 .. +* ..1.000E+005.526E+00 ..*+..1.500E+003.343E+00 .*..+.2.000E+002.026E+00 .*..+.2.500E+001.226E+00 .*..+.3.000E+007.429E-01 . *..+ .3.500E+004.495E-01 .*..+.4.000E+002.724E-01 .*..+.4.500E+001.648E-01 *..+5.000E+009.987E-02 *..+5.500E...

  • Page 518

    508CHAPTER 16. RC AND L/R TIME CONSTANTSbe after an infinite amount of time. This can be determined by analyzing a capacitive circuit asthough the capacitor was an open-circuit, and an inductive circuit as though the inductor wasa short-circuit, because that is what these components behave as wh...

  • Page 519

    16.4. VOLTAGE AND CURRENT CALCULATIONS5091 - 1e10x 100% = 99.995%The more time that passes since the transient application of voltage from the battery, thelarger the value of the denominator in the fraction, which makes for a smaller value for thewhole fraction, which makes for a grand total (1 m...

  • Page 520

    510CHAPTER 16. RC AND L/R TIME CONSTANTSτ = RCτ = (10 kΩ)(100 µF)τ = 1 secondIf the capacitor starts in a totally discharged state (0 volts), then we can use that value ofvoltage for a ”starting” value. The final value, of course, will be the battery voltage (15 volts).Our universal fo...

  • Page 521

    16.4. VOLTAGE AND CURRENT CALCULATIONS5111 - 1Change = - 1.4989 mAChange = (0 mA - 1.5 mA)e7.25/1Change = (0 mA - 1.5 mA)(0.99929)Note that the figure obtained for change is negative, not positive! This tells us that currenthas decreasedrather than increased with the passage of time. Since we st...

  • Page 522

    512CHAPTER 16. RC AND L/R TIME CONSTANTSBecause this is an inductive circuit, and we know that inductors oppose change in current,we’ll set up our time constant formula for starting and final values of current. If we start withthe switch in the open position, the current will be equal to zero,...

  • Page 523

    16.5. WHY L/R AND NOT LR?513•1 - 1(Final-Start)Change =Universal Time Constant FormulaWhere,Final =Start =e =t = Value of calculated variable after infinite time(its ultimate value)Initial value of calculated variableEuler’s number ( 2.7182818)Time in secondsTime constant for circuit in seco...

  • Page 524

    514CHAPTER 16. RC AND L/R TIME CONSTANTSquantities of energy, the capacitor storing energy in the medium of an electric field and theinductor storing energy in the medium of a magnetic field. A capacitor’s electrostatic energystorage manifests itself in the tendency to maintain a constant vol...

  • Page 525

    16.5. WHY L/R AND NOT LR?515while kinetic energy can be illustrated by a moving mass. Consider the following illustrationas an analogy of a capacitor:gravityCartslopePotential energy storageand releaseThe cart, sitting at the top of a slope, possesses potential energy due to the influence ofgrav...

  • Page 526

    516CHAPTER 16. RC AND L/R TIME CONSTANTS16.6Complex voltage and current calculationsThere are circumstances when you may need to analyze a DC reactive circuit when the startingvalues of voltage and current are not respective of a fully ”discharged” state. In other words,the capacitor might st...

  • Page 527

    16.7. COMPLEX CIRCUITS517So, the inductor in this circuit has a starting current of 5 amps and an ending current of7.5 amps. Since the ”timing” will take place during the time that the switch is closed and R2is shorted past, we need to calculate our time constant from L1 and R1: 1 Henry divid...

  • Page 528

    518CHAPTER 16. RC AND L/R TIME CONSTANTSSwitch20 V2 kΩR1R2500 ΩR33 kΩC100 µFThe simple time constant formula (τ =RC) is based on a simple series resistance connected tothe capacitor. For that matter, the time constant formula for an inductive circuit (τ =L/R) is alsobased on the assumpti...

  • Page 529

    16.7. COMPLEX CIRCUITS519Now, to solve for our Thevenin resistance, we need to eliminate all power sources in theoriginal circuit and calculate resistance as seen from the load terminals:Switch(closed)TheveninresistanceR1R2500 ΩR33 kΩ2 kΩ454.545 Ω=RThevenin = R2 // (R1 -- R3)RThevenin = 5...

  • Page 530

    520CHAPTER 16. RC AND L/R TIME CONSTANTSChange = (Final - Start)1 -1Change = (1.8182 V - 0 V)1 -1e60m/45.4545mChange = (1.8182 V)(0.73286)Change = 1.3325 Vet/τAgain, because our starting value for capacitor voltage was assumed to be zero, the actualvoltage across the capacitor at 60 milliseconds...

  • Page 531

    16.7. COMPLEX CIRCUITS5213.500E-029.747E-019.747E-014.000E-021.064E+001.064E+004.500E-021.142E+001.142E+005.000E-021.212E+001.212E+005.500E-021.276E+001.276E+006.000E-021.333E+001.333E+006.500E-021.383E+001.383E+007.000E-021.429E+001.429E+007.500E-021.470E+001.470E+008.000E-021.505E+001.505E+008....

  • Page 532

    522CHAPTER 16. RC AND L/R TIME CONSTANTS2.650E-011.813E+001.813E+002.700E-011.813E+001.813E+002.750E-011.814E+001.814E+002.800E-011.814E+001.814E+002.850E-011.815E+001.815E+002.900E-011.815E+001.815E+002.950E-011.815E+001.815E+003.000E-011.816E+001.816E+003.050E-011.816E+001.816E+003.100E-011.816...

  • Page 533

    16.8. SOLVING FOR UNKNOWN TIME523However, we want to solve for time, not the amount of change. To do this, we algebraicallymanipulate the formula so that time is all by itself on one side of the equal sign, with all therest on the other side:Change 1 - = (Final-Start)e-t/τChange Final-Start= e-t...

  • Page 534

    524CHAPTER 16. RC AND L/R TIME CONSTANTSln 1 -t = -(1 second)12.970 V15 V - 0 Vt =t = 2 seconds-(1 second)t = (1 second)(2)(ln 0.13534))Indeed, we end up with a value of 2 seconds for the time it takes to go from 0 to 12.970 voltsacross the capacitor. This variation of the universal time constant...

  • Page 535

    Appendix A-1ABOUT THIS BOOKA-1.1PurposeThey say that necessity is the mother of invention. At least in the case of this book, that adageis true. As an industrial electronics instructor, I was forced to use a sub-standard textbookduring my first year of teaching. My students were daily frustrated...

  • Page 536

    526APPENDIX A-1.ABOUT THIS BOOKsystem, whose fame is growing even as I write). The goal was to copyright the text – so as toprotect my authorship – but expressly allow anyone to distribute and/or modify the text to suittheir own needs with a minimum of legal encumbrance. This willful and form...

  • Page 537

    A-1.3. ACKNOWLEDGEMENTS527seem to approach the task of education from a deductive perspective: tell the student howthings are supposed to work, then apply those principles to specific instances that the studentmay or may not be able to explore by themselves. The inductive approach, as useful as ...

  • Page 538

    528APPENDIX A-1.ABOUT THIS BOOK• TEX text processing system – Donald Knuth and others.• Texinfodocument formatting system – Free Software Foundation.• LATEX document formatting system – Leslie Lamport and others.• Gimpimage manipulation program – too many contributors to mention.A...

  • Page 539

    Appendix A-2CONTRIBUTOR LISTA-2.1How to contribute to this bookAs a copylefted work, this book is open to revision and expansion by any interested parties.The only ”catch” is that credit must be given where credit is due. This isa copyrighted work:it is notin the public domain!If you wish to ...

  • Page 540

    530APPENDIX A-2.CONTRIBUTOR LISTproducing a derivative work, and to distribute the derivative workunder the terms described in the section for distribution above,provided that the following terms are met:(a) The new, derivative work is published under the terms of thisLicense.(b) The derivative w...

  • Page 541

    A-2.2. CREDITS531A-2.2.2Benjamin Crowell, Ph.D.• Date(s) of contribution(s): January 2001• Nature of contribution: Suggestions on improving technical accuracy of electric fieldand charge explanations in the first two chapters.• Contact at: crowell01@lightandmatter.comA-2.2.3Dennis Crunkil...

  • Page 542

    532APPENDIX A-2.CONTRIBUTOR LISTA-2.2.7Ray A. Rayburn• Date(s) of contribution(s): September 2009• Nature of contribution: Nonapplicability of Maximum Power Transfer Theorem to Hi-Fi audio amplifier.• Contact at: http://forum.allaboutcircuits.com/member.php?u=54720A-2.2.8Jason Starck• Da...

  • Page 543

    A-2.2. CREDITS533• Dejan Budimir (January 2003) Clarification of Mesh Current method explanation.• Sridhar Chitta, Assoc. Professor, Dept. of Instrumentation and Control Engg., VignanInstitute of Technology and Science, Deshmukhi Village, Pochampally Mandal, NalgondaDistt, Andhra Pradesh, In...

  • Page 544

    534APPENDIX A-2.CONTRIBUTOR LIST• David M. St. Pierre (November 2007): Corrected spelling error in Andrew Tanenbaum’sname (from the title page of his book).• Geoffrey Lessel,Thompsons Station, TN (June 2005): Corrected typo error in Ch 1 ”Ifthis charge (static electricity) is stationary, ...

  • Page 545

    A-2.2. CREDITS535• Cory Benjamin (November 2007) Ch 3 s/on hand/one hand.• Larry Weber (Feb 2008) Ch 3 s/on hand/one hand.• trunks14@allaboutrcircuits.com (Feb 2008) Ch 15 s/of of/of .• Greg Herrington (Feb 2008) Ch 1, Clarification: no neutron in hydrogen atom.• mark44 (Feb 2008) Ch 1...

  • Page 546

    536APPENDIX A-2.CONTRIBUTOR LIST• vspriyan@allaboutcircuits.com (Jan 2013) Ch 10, Near: voltages divided by theirs/currents/resistances/ .• Eugene Smirnoff (Jan 2013) Ch1, s/an hypothetical/a hypothetical/ . Ch 2 s/An his-toric/A historic/ .• Gulliveig@allaboutcircuits.com (Jan 2014) Ch4, s...

  • Page 547

    Appendix A-3DESIGN SCIENCE LICENSECopyright c 1999-2000 Michael Stutz stutz@dsl.orgVerbatim copying of this document is permitted, in any medium.A-3.10. PreambleCopyright law gives certain exclusive rights to the author of a work, including the rightsto copy, modify and distribute the work (the ...

  • Page 548

    538APPENDIX A-3.DESIGN SCIENCE LICENSE”Object Form” shall mean an executable or performable form of the Work, being an embod-iment of the Work in some tangible medium.”Source Data” shall mean the origin of the Object Form, being the entire, machine-readable,preferred form of the Work for ...

  • Page 549

    A-3.5. 4. MODIFICATION539(c) A third party’s written offer for obtaining the Source Data at no cost, as described inparagraph (b) above, is included with the distribution. This option is valid only if you are anon-commercial party, and only if you received the Object Form of the Work along with...

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    540APPENDIX A-3.DESIGN SCIENCE LICENSEA-3.87. No warrantyTHE WORK IS PROVIDED ”AS IS,” AND COMES WITH ABSOLUTELY NO WARRANTY,EXPRESS OR IMPLIED, TO THE EXTENT PERMITTED BY APPLICABLE LAW, INCLUD-ING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY ORFITNESS FOR A PARTICULAR PURPOS...

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    A-3.9. 8. DISCLAIMER OF LIABILITY541

  • Page 552

    Index10-50 milliamp signal, 319,3093-15 PSI signal, 314,3044-20 milliamp signal, 317,3074-wire resistance measurement, 294,284AC, 30,20, 89,79Acid, 325,315AFCI, 115,105Algebraic sum, 191,181Alligator clips, 294,284Alternating current, 30,20, 89,79Ammeter, 123,113, 263,253Ammeter imp...

  • Page 553

    INDEX543Capacitor, tantalum, 468,458Capacitor, variable, 461,451Capacitors, nonpolarized, 467,457Capacitors, polarized, 467,457Capacitors, series and parallel, 462,452Capacity, battery, 410,400Cardio-Pulmonary Resuscitation, 107,97Carrier, strain gauge, 331,321Cathode Ray Tube, 249,239Ca...

  • Page 554

    544INDEXDynamic electricity, 19,9Dynamometer meter movement, 306,296e, symbol for Euler’s constant, 518,508e, symbol for instantaneous voltage, 46,36, 454,444, 486,476, 495,485E, symbol for voltage, 46,36Edison cell, 406,396Effect, Meissner, 445,435Effect, Peltier, 322,312Effect, Seeb...

  • Page 555

    INDEX545Hydrometer, 405,395Hysteresis, 480,470I, symbol for current, 46,36i, symbol for instantaneous current, 46,36, 454,444, 495,485IC, 59,49Impedance, 391,381Indicator, 312,302, 314,304Inductance, 494,484Inductance, mutual, 487,477, 509,499Induction, electromagnetic, 485,475Induct...

  • Page 556

    546INDEXMillman’s Theorem, 371,361, 389,379Mks, metric system, 478,468Molecule, 403,393Motion, perpetual, 446,436Motor, electric, 476,466Movement, meter, 246,236Multimeter, 116,106, 287,277Multiplier, 252,242Mutual inductance, 487,477, 509,499MWG (Steel Music Wire Gauge), 424,414Na...

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    INDEX547Q, symbol for electric charge, 46,36Qualitative analysis, 164,154, 226,216Quantum physics, 419,409Quarter-bridge circuit, 332,322R, symbol for resistance, 46,36Radioactivity, 16,6Ratio arm, Wheatstone bridge, 300,290Re-drawing schematic diagrams, 218,208Reactance, inductive, 494...

  • Page 558

    548INDEXSuperconductor, 444,434Superfluidity, 444,434Superposition Theorem, 374,364Surface-mount device, 59,49SWG (British Standard Wire Gauge), 424,414Switch, 34,24Switch, closed, 37,27Switch, open, 37,27Switch, safety disconnect, 103,93System, metric, 133,123Systems of equations, 34...

  • Page 559

    INDEX549Voltage, between common points, 69,59Voltage, potential, 45,35Voltage, precise definition, 27,17, 53,43Voltage, sources, 28,18Voltmeter, 120,110, 251,241Voltmeter impact, 256,246Voltmeter loading, 257,247Voltmeter, amplified, 259,249Voltmeter, null-balance, 260,250, 329,319V...

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