Table Of ContentFifth Edition, last update January 10, 2004
2
Lessons In Electric Circuits, Volume II { AC
By Tony R. Kuphaldt
Fifth Edition, last update January 10, 2004
i
(cid:176)c 2000-2004, Tony R. Kuphaldt
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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 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Vectors and AC waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Simple vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Complex vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Polar and rectangular notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Complex number arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 More on AC "polarity". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.8 Some examples with AC circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 REACTANCE AND IMPEDANCE { INDUCTIVE 55
3.1 AC resistor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 AC inductor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Series resistor-inductor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Parallel resistor-inductor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Inductor quirks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 More on the "skin efiect" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 REACTANCE AND IMPEDANCE { CAPACITIVE 77
4.1 AC resistor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 AC capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Series resistor-capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Parallel resistor-capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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iv CONTENTS
4.5 Capacitor quirks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 REACTANCE AND IMPEDANCE { R, L, AND C 93
5.1 Review of R, X, and Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Series R, L, and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Parallel R, L, and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Series-parallel R, L, and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Susceptance and Admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 RESONANCE 115
6.1 An electric pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Simple parallel (tank circuit) resonance . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Simple series resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Applications of resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5 Resonance in series-parallel circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7 MIXED-FREQUENCY AC SIGNALS 139
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 Square wave signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3 Other waveshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 More on spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.5 Circuit efiects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8 FILTERS 175
8.1 What is a fllter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2 Low-pass fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.3 High-pass fllters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.4 Band-pass fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.5 Band-stop fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.6 Resonant fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.8 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9 TRANSFORMERS 201
9.1 Mutual inductance and basic operation. . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2 Step-up and step-down transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.3 Electrical isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.4 Phasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.5 Winding conflgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.6 Voltage regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.7 Special transformers and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . 234
CONTENTS v
9.8 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.8.1 Power capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.8.2 Energy losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.8.3 Stray capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . 249
9.8.4 Core saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.8.5 Inrush current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.8.6 Heat and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
10 POLYPHASE AC CIRCUITS 259
10.1 Single-phase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
10.2 Three-phase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.3 Phase rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
10.4 Polyphase motor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
10.5 Three-phase Y and ¢ conflgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
10.6 Three-phase transformer circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
10.7 Harmonics in polyphase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.8 Harmonic phase sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
11 POWER FACTOR 319
11.1 Power in resistive and reactive AC circuits . . . . . . . . . . . . . . . . . . . . . . . . 319
11.2 True, Reactive, and Apparent power . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
11.3 Calculating power factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
11.4 Practical power factor correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
11.5 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
12 AC METERING CIRCUITS 335
12.1 AC voltmeters and ammeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
12.2 Frequency and phase measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
12.3 Power measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
12.4 Power quality measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
12.5 AC bridge circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
12.6 AC instrumentation transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
12.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
13 AC MOTORS 373
14 TRANSMISSION LINES 375
14.1 A 50-ohm cable? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
14.2 Circuits and the speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
14.3 Characteristic impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
14.4 Finite-length transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
14.5 "Long" and "short" transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . 390
14.6 Standing waves and resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
14.7 Impedance transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
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14.8 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
15 ABOUT THIS BOOK 425
15.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
15.2 The use of SPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
15.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
16 CONTRIBUTOR LIST 429
16.1 How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
16.2 Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
16.2.1 Tony R. Kuphaldt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
16.2.2 Jason Starck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
16.2.3 Your name here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
16.2.4 Typo corrections and other \minor" contributions . . . . . . . . . . . . . . . 431
17 DESIGN SCIENCE LICENSE 433
17.1 0. Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.2 1. Deflnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.3 2. Rights and copyright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
17.4 3. Copying and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
17.5 4. Modiflcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
17.6 5. No restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
17.7 6. Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
17.8 7. No warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
17.9 8. Disclaimer of liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
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
