Table Of ContentFourth Edition, last update June 29, 2002
2
Lessons In Electric Circuits, Volume II { AC
By Tony R. Kuphaldt
Fourth Edition, last update June 29, 2002
i
(cid:176)c 2000-2002, Tony R. Kuphaldt
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terms and conditions allow for free copying, distribution, and/or modiflcation of this document by
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PRINTING HISTORY
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† Second Edition: Printed in September of 2000. Illustrations reworked in standard graphic
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Contents
1 BASIC AC THEORY 1
1.1 What is alternating current (AC)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AC waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Measurements of AC magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Simple AC circuit calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 AC phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Principles of radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 COMPLEX NUMBERS 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Vectors and AC waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Simple vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Complex vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Polar and rectangular notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 Complex number arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 More on AC "polarity". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.8 Some examples with AC circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 REACTANCE AND IMPEDANCE { INDUCTIVE 53
3.1 AC resistor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 AC inductor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Series resistor-inductor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Parallel resistor-inductor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 Inductor quirks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 More on the "skin efiect" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 REACTANCE AND IMPEDANCE { CAPACITIVE 75
4.1 AC resistor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 AC capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Series resistor-capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Parallel resistor-capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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4.5 Capacitor quirks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 REACTANCE AND IMPEDANCE { R, L, AND C 91
5.1 Review of R, X, and Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Series R, L, and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Parallel R, L, and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Series-parallel R, L, and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Susceptance and Admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 RESONANCE 113
6.1 An electric pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Simple parallel (tank circuit) resonance . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Simple series resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Applications of resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 Resonance in series-parallel circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 MIXED-FREQUENCY AC SIGNALS 137
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Square wave signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.3 Other waveshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.4 More on spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.5 Circuit efiects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8 FILTERS 173
8.1 What is a fllter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.2 Low-pass fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.3 High-pass fllters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.4 Band-pass fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.5 Band-stop fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.6 Resonant fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.8 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9 TRANSFORMERS 199
9.1 Mutual inductance and basic operation. . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.2 Step-up and step-down transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.3 Electrical isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
9.4 Phasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.5 Winding conflgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.6 Voltage regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.7 Special transformers and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . 232
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9.8 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.8.1 Power capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.8.2 Energy losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.8.3 Stray capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.8.4 Core saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.8.5 Inrush current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.8.6 Heat and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10 POLYPHASE AC CIRCUITS 257
10.1 Single-phase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.2 Three-phase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.3 Phase rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.4 Polyphase motor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
10.5 Three-phase Y and ¢ conflgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
10.6 Three-phase transformer circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10.7 Harmonics in polyphase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . 291
10.8 Harmonic phase sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
10.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
11 POWER FACTOR 317
11.1 Power in resistive and reactive AC circuits . . . . . . . . . . . . . . . . . . . . . . . . 317
11.2 True, Reactive, and Apparent power . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
11.3 Calculating power factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
11.4 Practical power factor correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
11.5 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
12 AC METERING CIRCUITS 333
12.1 AC voltmeters and ammeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
12.2 Frequency and phase measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
12.3 Power measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
12.4 Power quality measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
12.5 AC bridge circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
12.6 AC instrumentation transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
12.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
13 AC MOTORS 371
14 TRANSMISSION LINES 373
14.1 A 50-ohm cable? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
14.2 Circuits and the speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
14.3 Characteristic impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
14.4 Finite-length transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
14.5 "Long" and "short" transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . 388
14.6 Standing waves and resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
14.7 Impedance transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
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14.8 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
15 ABOUT THIS BOOK 423
15.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
15.2 The use of SPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
15.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
16 CONTRIBUTOR LIST 427
16.1 How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
16.2 Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
16.2.1 Tony R. Kuphaldt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
16.2.2 Jason Starck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
16.2.3 Your name here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
16.2.4 Typo corrections and other \minor" contributions . . . . . . . . . . . . . . . 429
17 DESIGN SCIENCE LICENSE 431
17.1 0. Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
17.2 1. Deflnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
17.3 2. Rights and copyright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
17.4 3. Copying and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
17.5 4. Modiflcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.6 5. No restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.7 6. Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.8 7. No warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.9 8. Disclaimer of liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Chapter 1
BASIC AC THEORY
1.1 What is alternating current (AC)?
Most students of electricity begin their study with what is known as direct current (DC), which is
electricity (cid:176)owing in a constant direction, and/or possessing a voltage with constant polarity. DC
is the kind of electricity made by a battery (with deflnite positive and negative terminals), or the
kind of charge generated by rubbing certain types of materials against each other.
AsusefulandaseasytounderstandasDCis,itisnottheonly"kind"ofelectricityinuse. Certain
sourcesofelectricity(mostnotably,rotaryelectro-mechanicalgenerators)naturallyproducevoltages
alternating in polarity, reversing positive and negative over time. Either as a voltage switching
polarity or as a current switching direction back and forth, this "kind" of electricity is known as
Alternating Current (AC):
DIRECT CURRENT ALTERNATING CURRENT
(DC) (AC)
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Whereas the familiar battery symbol is used as a generic symbol for any DC voltage source, the
circuit with the wavy line inside is the generic symbol for any AC voltage source.
One might wonder why anyone would bother with such a thing as AC. It is true that in some
casesACholdsnopracticaladvantageoverDC.Inapplicationswhereelectricityisusedtodissipate
energy in the form of heat, the polarity or direction of current is irrelevant, so long as there is
enough voltage and current to the load to produce the desired heat (power dissipation). However,
with AC it is possible to build electric generators, motors and power distribution systems that are
far more e–cient than DC, and so we flnd AC used predominately across the world in high power
applications. To explain the details of why this is so, a bit of background knowledge about AC is
1
