AC

Fifth Edition, last update October 31, 2005 2 Lessons In Electric Circuits, Volume II – AC By Tony R. Kuphaldt Fifth Edition, last update October 31, 2005 i c  2000-2005, Tony R. Kuphaldt This book is published under the terms and conditions of the Design Science License. These terms and conditions allow for free copying, distribution, and/or modification of this document by the general public. The full Design Science License text is included in the last chapter. As an open and collaboratively developed text, this book is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MER- CHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the Design Science License for more details. Available in its entirety as part of the Open Book Project collection at http://www.ibiblio.org/obp PRINTING HISTORY • First Edition: Printed in June of 2000. Plain-ASCII illustrations for universal computer readability. • Second Edition: Printed in September of 2000. Illustrations reworked in standard graphic (eps and jpeg) format. Source files translated to Texinfo format for easy online and printed publication. • Third Edition: Equations and tables reworked as graphic images rather than plain-ASCII text. • Fourth Edition: Printed in November 2001. Source files translated to SubML format. SubML is a simple markup language designed to easily convert to other markups like L A T E X, HTML, or DocBook using nothing but search-and-replace substitutions. • Fifth Edition: Printed in November 2002. New sections added, and error corrections made, since the fourth edition. ii 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 effect” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 iii 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.4 Applications of resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.5 Resonance in series-parallel circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7 MIXED-FREQUENCY AC SIGNALS 139 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2 Square wave signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3 Other waveshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.4 More on spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.5 Circuit effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8 FILTERS 179 8.1 What is a filter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.2 Low-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.3 High-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.4 Band-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.5 Band-stop filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.6 Resonant filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.8 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 9 TRANSFORMERS 209 9.1 Mutual inductance and basic operation . . . . . . . . . . . . . . . . . . . . . . . . . . 209 9.2 Step-up and step-down transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.3 Electrical isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 9.4 Phasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.5 Winding configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 9.6 Voltage regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.7 Special transformers and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 CONTENTS v 9.7.1 Impedance matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.7.2 Potential transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.7.3 Current transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 9.7.4 Air core transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 9.7.5 Tesla Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.7.6 Saturable reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.7.7 Scott-T transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 9.7.8 Linear Variable Differential Transformer . . . . . . . . . . . . . . . . . . . . . 255 9.8 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.8.1 Power capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.8.2 Energy losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.8.3 Stray capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.8.4 Core saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 9.8.5 Inrush current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 9.8.6 Heat and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 9.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10 POLYPHASE AC CIRCUITS 269 10.1 Single-phase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 10.2 Three-phase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.3 Phase rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.4 Polyphase motor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.5 Three-phase Y and ∆ configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.6 Three-phase transformer circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.7 Harmonics in polyphase power systems . . . . . . . . . . . . . . . . . . . . . . . . . . 303 10.8 Harmonic phase sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 10.9 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11 POWER FACTOR 331 11.1 Power in resistive and reactive AC circuits . . . . . . . . . . . . . . . . . . . . . . . . 331 11.2 True, Reactive, and Apparent power . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 11.3 Calculating power factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 11.4 Practical power factor correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 11.5 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 12 AC METERING CIRCUITS 347 12.1 AC voltmeters and ammeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.2 Frequency and phase measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 12.3 Power measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 12.4 Power quality measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 12.5 AC bridge circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 12.6 AC instrumentation transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 12.7 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 13 AC MOTORS 385 vi CONTENTS 14 TRANSMISSION LINES 387 14.1 A 50-ohm cable? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 14.2 Circuits and the speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 14.3 Characteristic impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 14.4 Finite-length transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 14.5 ”Long” and ”short” transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . 402 14.6 Standing waves and resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 14.7 Impedance transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 14.8 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 15 ABOUT THIS BOOK 435 15.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 15.2 The use of SPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 15.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 16 CONTRIBUTOR LIST 439 16.1 How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 16.2 Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 16.2.1 Tony R. Kuphaldt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 16.2.2 Jason Starck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 16.2.3 Dennis Crunkilton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 16.2.4 Bill Stoddard, bill@billsclockworks.com . . . . . . . . . . . . . . . . . . . . . 441 16.2.5 Your name here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 16.2.6 Typo corrections and other “minor” contributions . . . . . . . . . . . . . . . 441 17 DESIGN SCIENCE LICENSE 443 17.1 0. Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 17.2 1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 17.3 2. Rights and copyright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 17.4 3. Copying and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 17.5 4. Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 17.6 5. No restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 17.7 6. Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 17.8 7. No warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 17.9 8. Disclaimer of liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 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 flowing in a constant direction, and/or possessing a voltage with constant polarity. DC is the kind of electricity made by a battery (with definite positive and negative terminals), or the kind of charge generated by rubbing certain types of materials against each other. As useful and as easy to understand as DC is, it is not the only ”kind” of electricity in use. Certain sources of electricity (most notably, rotary electro-mechanical generators) naturally produce voltages 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): I I DIRECT CURRENT (DC) ALTERNATING CURRENT (AC) I I Whereas the familiar battery symbol is used as a generic symbol for any DC voltage source, the circle 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 cases AC holds no practical advantage over DC. In applications where electricity is used to dissipate 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 efficient than DC, and so we find 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