AC Volume II Service Manual
Fifth 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
c
° 2000-2004, 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 MERCHANTABILITY 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.
i
• 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 LATEX, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.6 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8 FILTERS 175
8.1 What is a filter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2 Low-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.3 High-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.4 Band-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.5 Band-stop filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.6 Resonant filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∆ configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
vi CONTENTS
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. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
17.3 2. Rights and copyright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
17.4 3. Copying and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
17.5 4. Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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):
DIRECT CURRENT
(DC)
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
ALTERNATING CURRENT
(AC)
I
I
1
2 CHAPTER 1. BASIC AC THEORY
necessary.
If a machine is constructed to rotate a magnetic field around a set of stationary wire coils with
the turning of a shaft, AC voltage will be produced across the wire coils as that shaft is rotated, in
accordance with Faraday’s Law of electromagnetic induction. This is the basic operating principle
of an AC generator, also known as an alternator :
Alternator operation
Step #1 Step #2
S
N S
N
no current!
Load
Step #3 Step #4
N
S
no current!
Load
+ -
Load
S
- +
I I
Load
II
N
Notice how the polarity of the voltage across the wire coils reverses as the opposite poles of the
rotating magnet pass by. Connected to a load, this reversing voltage polarity will create a reversing
current direction in the circuit. The faster the alternator’s shaft is turned, the faster the magnet
will spin, resulting in an alternating voltage and current that switches directions more often in a
given amount of time.
While DC generators work on the same general principle of electromagnetic induction, their
construction is not as simple as their AC counterparts. With a DC generator, the coil of wire is
mounted in the shaft where the magnet is on the AC alternator, and electrical connections are
made to this spinning coil via stationary carbon ”brushes” contacting copper strips on the rotating
shaft. All this is necessary to switch the coil’s changing output polarity to the external circuit so
the external circuit sees a constant polarity:
1.1. WHAT IS ALTERNATING CURRENT (AC)? 3
(DC) Generator operation
Step #1
N S SN
Step #2
N S N S
- +
+-
I
Load
Load
Step #3 Step #4
N S SN
N S
-
SN
-
+
+
I
Load
The generator shown above will produce two pulses of voltage per revolution of the shaft, both
pulses in the same direction (polarity). In order for a DC generator to produce constant voltage,
rather than brief pulses of voltage once every 1/2 revolution, there are multiple sets of coils making
intermittent contact with the brushes. The diagram shown above is a bit more simplified than what
you would see in real life.
The problems involved with making and breaking electrical contact with a moving coil should
be obvious (sparking and heat), especially if the shaft of the generator is revolving at high speed.
If the atmosphere surrounding the machine contains flammable or explosive vapors, the practical
problems of spark-producing brush contacts are even greater. An AC generator (alternator) does
not require brushes and commutators to work, and so is immune to these problems experienced by
DC generators.
The benefits of AC over DC with regard to generator design is also reflected in electric motors.
While DC motors require the use of brushes to make electrical contact with moving coils of wire, AC
motors do not. In fact, AC and DC motor designs are very similar to their generator counterparts
(identical for the sake of this tutorial), the AC motor being dependent upon the reversing magnetic
field produced by alternating current through its stationary coils of wire to rotate the rotating
magnet around on its shaft, and the DC motor being dependent on the brush contacts making and
breaking connections to reverse current through the rotating coil every 1/2 rotation (180 degrees).
So we know that AC generators and AC motors tend to be simpler than DC generators and DC
motors. This relative simplicity translates into greater reliability and lower cost of manufacture.
But what else is AC good for? Surely there must be more to it than design details of generators and
Load
4 CHAPTER 1. BASIC AC THEORY
motors! Indeed there is. There is an effect of electromagnetism known as mutual induction, whereby
two or more coils of wire placed so that the changing magnetic field created by one induces a voltage
in the other. If we have two mutually inductive coils and we energize one coil with AC, we will
create an AC voltage in the other coil. When used as such, this device is known as a transformer:
Transformer
AC
voltage
source
Induced AC
voltage
The fundamental significance of a transformer is its ability to step voltage up or down from the
powered coil to the unpowered coil. The AC voltage induced in the unpowered (”secondary”) coil
is equal to the AC voltage across the powered (”primary”) coil multiplied by the ratio of secondary
coil turns to primary coil turns. If the secondary coil is powering a load, the current through
the secondary coil is just the opposite: primary coil current multiplied by the ratio of primary to
secondary turns. This relationship has a very close mechanical analogy, using torque and speed to
represent voltage and current, respectively:
Speed multiplication geartrain
Large gear
(many teeth)
Small gear
(few teeth)
high torque
low speed
+
+
low torque
high speed
"Step-down" transformer
high voltage
AC voltage
source
many
turns
low current
low voltage
few turns
high current
Load
If the winding ratio is reversed so that the primary coil has less turns than the secondary coil,
the transformer ”steps up” the voltage from the source level to a higher level at the load:
1.1. WHAT IS ALTERNATING CURRENT (AC)? 5
Speed reduction geartrain
Large gear
(many teeth)
Small gear
(few teeth)
low torque
high speed
AC voltage
source Load
+
"Step-up" transformer
low voltage
few turns
high current
+
high voltage
many turns
low current
high torque
low speed
The transformer’s ability to step AC voltage up or down with ease gives AC an advantage
unmatched by DC in the realm of power distribution. When transmitting electrical power over
long distances, it is far more efficient to do so with stepped-up voltages and stepped-down currents
(smaller-diameter wire with less resistive power losses), then step the voltage back down and the
current back up for industry, business, or consumer use use.
high voltage
Power Plant
low voltage
Step-up
Step-down
Home or
Business
. . . to other customers
low voltage
Transformer technology has made long-range electric power distribution practical. Without the
ability to efficiently step voltage up and down, it would be cost-prohibitive to construct power
systems for anything but close-range (within a few miles at most) use.
As useful as transformers are, they only work with AC, not DC. Because the phenomenon of
mutual inductance relies on changing magnetic fields, and direct current (DC) can only produce
steady magnetic fields, transformers simply will not work with direct current. Of course, direct
6 CHAPTER 1. BASIC AC THEORY
current may be interrupted (pulsed) through the primary winding of a transformer to create a
changing magnetic field (as is done in automotive ignition systems to produce high-voltage spark
plug power from a low-voltage DC battery), but pulsed DC is not that different from AC. Perhaps
more than any other reason, this is why AC finds such widespread application in power systems.
• REVIEW:
• DC stands for ”Direct Current,” meaning voltage or current that maintains constant polarity
or direction, respectively, over time.
• AC stands for ”Alternating Current,” meaning voltage or current that changes polarity or
direction, respectively, over time.
• AC electromechanical generators, known as alternators, are of simpler construction than DC
electromechanical generators.
• AC and DC motor design follows respective generator design principles very closely.
• A transformer is a pair of mutually-inductive coils used to convey AC power from one coil to
the other. Often, the number of turns in each coil is set to create a voltage increase or decrease
from the powered (primary) coil to the unpowered (secondary) coil.
• Secondary voltage = Primary voltage (secondary turns / primary turns)
• Secondary current = Primary current (primary turns / secondary turns)
1.2 AC waveforms
When an alternator produces AC voltage, the voltage switches polarity over time, but does so in a
very particular manner. When graphed over time, the ”wave” traced by this voltage of alternating
polarity from an alternator takes on a distinct shape, known as a sine wave :
Graph of AC voltage over time
(the sine wave)
+
Time
In the voltage plot from an electromechanical alternator, the change from one polarity to the
other is a smooth one, the voltage level changing most rapidly at the zero (”crossover”) point and
most slowly at its peak. If we were to graph the trigonometric function of ”sine” over a horizontal
range of 0 to 360 degrees, we would find the exact same pattern:
1.2. AC WAVEFORMS 7
Angle Sine(angle)
in degrees
0 ............... 0.0000 -- zero
15 ............... 0.2588
30 ............... 0.5000
45 ............... 0.7071
60 ............... 0.8660
75 ............... 0.9659
90 ............... 1.0000 -- positive peak
105 .............. 0.9659
120 .............. 0.8660
135 .............. 0.7071
150 .............. 0.5000
165 .............. 0.2588
180 .............. 0.0000 -- zero
195 .............. -0.2588
210 .............. -0.5000
225 .............. -0.7071
240 .............. -0.8660
255 .............. -0.9659
270 .............. -1.0000 -- negative peak
285 .............. -0.9659
300 .............. -0.8660
315 .............. -0.7071
330 .............. -0.5000
345 .............. -0.2588
360 .............. 0.0000 -- zero
The reason why an electromechanical alternator outputs sine-wave AC is due to the physics of
its operation. The voltage produced by the stationary coils by the motion of the rotating magnet is
proportional to the rate at which the magnetic flux is changing perpendicular to the coils (Faraday’s
Law of Electromagnetic Induction). That rate is greatest when the magnet poles are closest to the
coils, and least when the magnet poles are furthest away from the coils. Mathematically, the rate
of magnetic flux change due to a rotating magnet follows that of a sine function, so the voltage
produced by the coils follows that same function.
If we were to follow the changing voltage produced by a coil in an alternator from any point
on the sine wave graph to that point when the wave shape begins to repeat itself, we would have
marked exactly one cycle of that wave. This is most easily shown by spanning the distance between
identical peaks, but may be measured between any corresponding points on the graph. The degree
marks on the horizontal axis of the graph represent the domain of the trigonometric sine function,
and also the angular position of our simple two-pole alternator shaft as it rotates:
8 CHAPTER 1. BASIC AC THEORY
one wave cycle
0 90 180 270 360
(0)
90 180 270 360
(0)
one wave cycle
Alternator shaft
position (degrees)
Since the horizontal axis of this graph can mark the passage of time as well as shaft position in
degrees, the dimension marked for one cycle is often measured in a unit of time, most often seconds
or fractions of a second. When expressed as a measurement, this is often called the period of a wave.
The period of a wave in degrees is always 360, but the amount of time one period occupies depends
on the rate voltage oscillates back and forth.
A more popular measure for describing the alternating rate of an AC voltage or current wave
than period is the rate of that back-and-forth oscillation. This is called frequency. The modern unit
for frequency is the Hertz (abbreviated Hz), which represents the number of wave cycles completed
during one second of time. In the United States of America, the standard power-line frequency is
60 Hz, meaning that the AC voltage oscillates at a rate of 60 complete back-and-forth cycles every
second. In Europe, where the power system frequency is 50 Hz, the AC voltage only completes 50
cycles every second. A radio station transmitter broadcasting at a frequency of 100 MHz generates
an AC voltage oscillating at a rate of 100 million cycles every second.
Prior to the canonization of the Hertz unit, frequency was simply expressed as ”cycles per
second.” Older meters and electronic equipment often bore frequency units of ”CPS” (Cycles Per
Second) instead of Hz. Many people believe the change from self-explanatory units like CPS to
Hertz constitutes a step backward in clarity. A similar change occurred when the unit of ”Celsius”
replaced that of ”Centigrade” for metric temperature measurement. The name Centigrade was
based on a 100-count (”Centi-”) scale (”-grade”) representing the melting and boiling points of
H2O, respectively. The name Celsius, on the other hand, gives no hint as to the unit’s origin or
meaning.
Period and frequency are mathematical reciprocals of one another. That is to say, if a wave has
a period of 10 seconds, its frequency will be 0.1 Hz, or 1/10 of a cycle per second:
Frequency in Hertz =
1
Period in seconds
An instrument called an oscilloscope is used to display a changing voltage over time on a graphical
screen. You may be familiar with the appearance of an ECG or EKG (electrocardiograph) machine,
used by physicians to graph the oscillations of a patient’s heart over time. The ECG is a specialpurpose oscilloscope expressly designed for medical use. General-purpose oscilloscopes have the
ability to display voltage from virtually any voltage source, plotted as a graph with time as the
independent variable. The relationship between period and frequency is very useful to know when
displaying an AC voltage or current waveform on an oscilloscope screen. By measuring the period
of the wave on the horizontal axis of the oscilloscope screen and reciprocating that time value (in
seconds), you can determine the frequency in Hertz.
1.2. AC WAVEFORMS 9
OSCILLOSCOPE
vertical
Y
AC
V/div
GNDDC
trigger
16 divisions
@ 1ms/div =
a period of 16 ms
Frequency =
1 1
period
= = 62.5 Hz
16 ms
timebase
s/div
1m
X
DC GND AC
Voltage and current are by no means the only physical variables subject to variation over time.
Much more common to our everyday experience is sound, which is nothing more than the alternating
compression and decompression (pressure waves) of air molecules, interpreted by our ears as a physical sensation. Because alternating current is a wave phenomenon, it shares many of the properties
of other wave phenomena, like sound. For this reason, sound (especially structured music) provides
an excellent analogy for relating AC concepts.
In musical terms, frequency is equivalent to pitch. Low-pitch notes such as those produced by
a tuba or bassoon consist of air molecule vibrations that are relatively slow (low frequency). Highpitch notes such as those produced by a flute or whistle consist of the same type of vibrations in
the air, only vibrating at a much faster rate (higher frequency). Here is a table showing the actual
frequencies for a range of common musical notes:
10 CHAPTER 1. BASIC AC THEORY
Note Musical designation
A A
A sharp (or B flat) A# or B
B
C (middle)
C sharp (or D flat) C# or D
1
b
B
1
C
b
D D
D sharp (or E flat) D# or E
b
E E
F F
F sharp (or G flat) F# or G
b
G G
G sharp (or A flat) G# or A
b
A A
A sharp (or B flat) A# or B
b
B B
1
C
C
Frequency (in hertz)
220.00
233.08
246.94
261.63
277.18
293.66
311.13
329.63
349.23
369.99
392.00
415.30
440.00
466.16
493.88
523.25
Astute observers will notice that all notes on the table bearing the same letter designation are
related by a frequency ratio of 2:1. For example, the first frequency shown (designated with the
letter ”A”) is 220 Hz. The next highest ”A” note has a frequency of 440 Hz – exactly twice as many
sound wave cycles per second. The same 2:1 ratio holds true for the first A sharp (233.08 Hz) and
the next A sharp (466.16 Hz), and for all note pairs found in the table.
Audibly, two notes whose frequencies are exactly double each other sound remarkably similar.
This similarity in sound is musically recognized, the shortest span on a musical scale separating such
note pairs being called an octave. Following this rule, the next highest ”A” note (one octave above
440 Hz) will be 880 Hz, the next lowest ”A” (one octave below 220 Hz) will be 110 Hz. A view of a
piano keyboard helps to put this scale into perspective:
#
D
C
b
b
E
D
F
G
A
G
b
b
b
B
A
#
#
#
#
#
D
C
b
b
E
D
F
G
A
G
b
b
b
B
A
D
C
b
b
E
D
F
G
A
G
b
b
b
B
A
#
#
#
#
#
#
#
#
#
C D E F G A B C D E F G A BC D E F G A B
one octave
As you can see, one octave is equal to eight white keys’ worth of distance on a piano keyboard.
1.2. AC WAVEFORMS 11
The familiar musical mnemonic (doe-ray-mee-fah-so-lah-tee-doe) – yes, the same pattern immortalized in the whimsical Rodgers and Hammerstein song sung in The Sound of Music
– covers one
octave from C to C.
While electromechanical alternators and many other physical phenomena naturally produce sine
waves, this is not the only kind of alternating wave in existence. Other ”waveforms” of AC are
commonly produced within electronic circuitry. Here are but a few sample waveforms and their
common designations:
Square wave Triangle wave
one wave cycle one wave cycle
Sawtooth wave
These waveforms are by no means the only kinds of waveforms in existence. They’re simply a
few that are common enough to have been given distinct names. Even in circuits that are supposed
to manifest ”pure” sine, square, triangle, or sawtooth voltage/current waveforms, the real-life result
is often a distorted version of the intended waveshape. Some waveforms are so complex that they
defy classification as a particular ”type” (including waveforms associated with many kinds of musical
instruments). Generally speaking, any waveshape bearing close resemblance to a perfect sine wave
is termed sinusoidal, anything different being labeled as non-sinusoidal. Being that the waveform of
an AC voltage or current is crucial to its impact in a circuit, we need to be aware of the fact that
AC waves come in a variety of shapes.
• REVIEW:
• AC produced by an electromechanical alternator follows the graphical shape of a sine wave.
• One cycle of a wave is one complete evolution of its shape until the point that it is ready to
repeat itself.
• The period of a wave is the amount of time it takes to complete one cycle.
• Frequency is the number of complete cycles that a wave completes in a given amount of time.
Usually measured in Hertz (Hz), 1 Hz being equal to one complete wave cycle per second.
• Frequency = 1/(period in seconds)
12 CHAPTER 1. BASIC AC THEORY
1.3 Measurements of AC magnitude
So far we know that AC voltage alternates in polarity and AC current alternates in direction. We
also know that AC can alternate in a variety of different ways, and by tracing the alternation over
time we can plot it as a ”waveform.” We can measure the rate of alternation by measuring the time
it takes for a wave to evolve before it repeats itself (the ”period”), and express this as cycles per
unit time, or ”frequency.” In music, frequency is the same as pitch, which is the essential property
distinguishing one note from another.
However, we encounter a measurement problem if we try to express how large or small an AC
quantity is. With DC, where quantities of voltage and current are generally stable, we have little
trouble expressing how much voltage or current we have in any part of a circuit. But how do you
grant a single measurement of magnitude to something that is constantly changing?
One way to express the intensity, or magnitude (also called the amplitude), of an AC quantity
is to measure its peak height on a waveform graph. This is known as the peak or crest value of an
AC waveform:
Peak
Time
Another way is to measure the total height between opposite peaks. This is known as the
peak-to-peak (P-P) value of an AC waveform:
Peak-to-Peak
Time
Unfortunately, either one of these expressions of waveform amplitude can be misleading when
comparing two different types of waves. For example, a square wave peaking at 10 volts is obviously
a greater amount of voltage for a greater amount of time than a triangle wave peaking at 10 volts.
The effects of these two AC voltages powering a load would be quite different:
1.3. MEASUREMENTS OF AC MAGNITUDE 13
10 V
Time
10 V
(peak)
more heat energy
dissipated
(same load resistance)
10 V
(peak)
One way of expressing the amplitude of different waveshapes in a more equivalent fashion is to
mathematically average the values of all the points on a waveform’s graph to a single, aggregate
number. This amplitude measure is known simply as the average value of the waveform. If we
average all the points on the waveform algebraically (that is, to consider their sign, either positive
or negative), the average value for most waveforms is technically zero, because all the positive points
cancel out all the negative points over a full cycle:
less heat energy
dissipated
+++
+
+
+
+
+
+
-
-
-
-
-
-
-
-
-
True average value of all points
(considering their signs) is zero!
This, of course, will be true for any waveform having equal-area portions above and below the
”zero” line of a plot. However, as a practical measure of a waveform’s aggregate value, ”average” is
usually defined as the mathematical mean of all the points’ absolute values over a cycle. In other
words, we calculate the practical average value of the waveform by considering all points on the wave
as positive quantities, as if the waveform looked like this:
14 CHAPTER 1. BASIC AC THEORY
+++
+
+
+
+
++++
+
+++
+
+
+
Practical average of points, all
values assumed to be positive.
Polarity-insensitive mechanical meter movements (meters designed to respond equally to the
positive and negative half-cycles of an alternating voltage or current) register in proportion to
the waveform’s (practical) average value, because the inertia of the pointer against the tension of
the spring naturally averages the force produced by the varying voltage/current values over time.
Conversely, polarity-sensitive meter movements vibrate uselessly if exposed to AC voltage or current,
their needles oscillating rapidly about the zero mark, indicating the true (algebraic) average value of
zero for a symmetrical waveform. When the ”average” value of a waveform is referenced in this text,
it will be assumed that the ”practical” definition of average is intended unless otherwise specified.
Another method of deriving an aggregate value for waveform amplitude is based on the waveform’s ability to do useful work when applied to a load resistance. Unfortunately, an AC measurement based on work performed by a waveform is not the same as that waveform’s ”average”
value, because the power dissipated by a given load (work performed per unit time) is not directly
proportional to the magnitude of either the voltage or current impressed upon it. Rather, power is
proportional to the square of the voltage or current applied to a resistance (P = E2/R, and P =
I2R). Although the mathematics of such an amplitude measurement might not be straightforward,
the utility of it is.
Consider a bandsaw and a jigsaw, two pieces of modern woodworking equipment. Both types of
saws cut with a thin, toothed, motor-powered metal blade to cut wood. But while the bandsaw uses
a continuous motion of the blade to cut, the jigsaw uses a back-and-forth motion. The comparison
of alternating current (AC) to direct current (DC) may be likened to the comparison of these two
saw types:
Bandsaw
Jigsaw
blade
motion
wood
(analogous to DC)
wood
blade
motion
(analogous to AC)
The problem of trying to describe the changing quantities of AC voltage or current in a single,
aggregate measurement is also present in this saw analogy: how might we express the speed of a
jigsaw blade? A bandsaw blade moves with a constant speed, similar to the way DC voltage pushes
1.3. MEASUREMENTS OF AC MAGNITUDE 15
or DC current moves with a constant magnitude. A jigsaw blade, on the other hand, moves back
and forth, its blade speed constantly changing. What is more, the back-and-forth motion of any two
jigsaws may not be of the same type, depending on the mechanical design of the saws. One jigsaw
might move its blade with a sine-wave motion, while another with a triangle-wave motion. To rate
a jigsaw based on its peak blade speed would be quite misleading when comparing one jigsaw to
another (or a jigsaw with a bandsaw!). Despite the fact that these different saws move their blades
in different manners, they are equal in one respect: they all cut wood, and a quantitative comparison
of this common function can serve as a common basis for which to rate blade speed.
Picture a jigsaw and bandsaw side-by-side, equipped with identical blades (same tooth pitch,
angle, etc.), equally capable of cutting the same thickness of the same type of wood at the same
rate. We might say that the two saws were equivalent or equal in their cutting capacity. Might this
comparison be used to assign a ”bandsaw equivalent” blade speed to the jigsaw’s back-and-forth
blade motion; to relate the wood-cutting effectiveness of one to the other? This is the general idea
used to assign a ”DC equivalent” measurement to any AC voltage or current: whatever magnitude
of DC voltage or current would produce the same amount of heat energy dissipation through an
equal resistance:
5 A RMS
50 W
10 V
RMS
2 Ω
5 A RMS
power
dissipated
Equal power dissipated through
equal resistance loads
5 A
50 W
10 V
2 Ω
5 A
power
dissipated
In the two circuits above, we have the same amount of load resistance (2 Ω) dissipating the same
amount of power in the form of heat (50 watts), one powered by AC and the other by DC. Because
the AC voltage source pictured above is equivalent (in terms of power delivered to a load) to a 10 volt
DC battery, we would call this a ”10 volt” AC source. More specifically, we would denote its voltage
value as being 10 volts RMS. The qualifier ”RMS” stands for Root Mean Square, the algorithm used
to obtain the DC equivalent value from points on a graph (essentially, the procedure consists of
squaring all the positive and negative points on a waveform graph, averaging those squared values,
then taking the square root of that average to obtain the final answer). Sometimes the alternative
terms equivalent or DC equivalent are used instead of ”RMS,” but the quantity and principle are
both the same.
RMS amplitude measurement is the best way to relate AC quantities to DC quantities, or other
AC quantities of differing waveform shapes, when dealing with measurements of electric power. For
other considerations, peak or peak-to-peak measurements may be the best to employ. For instance,
when determining the proper size of wire (ampacity) to conduct electric power from a source to
a load, RMS current measurement is the best to use, because the principal concern with current
16 CHAPTER 1. BASIC AC THEORY
is overheating of the wire, which is a function of power dissipation caused by current through the
resistance of the wire. However, when rating insulators for service in high-voltage AC applications,
peak voltage measurements are the most appropriate, because the principal concern here is insulator
”flashover” caused by brief spikes of voltage, irrespective of time.
Peak and peak-to-peak measurements are best performed with an oscilloscope, which can capture
the crests of the waveform with a high degree of accuracy due to the fast action of the cathoderay-tube in response to changes in voltage. For RMS measurements, analog meter movements
(D’Arsonval, Weston, iron vane, electrodynamometer) will work so long as they have been calibrated
in RMS figures. Because the mechanical inertia and dampening effects of an electromechanical meter
movement makes the deflection of the needle naturally proportional to the average value of the AC,
not the true RMS value, analog meters must be specifically calibrated (or mis-calibrated, depending
on how you look at it) to indicate voltage or current in RMS units. The accuracy of this calibration
depends on an assumed waveshape, usually a sine wave.
Electronic meters specifically designed for RMS measurement are best for the task. Some instrument manufacturers have designed ingenious methods for determining the RMS value of any
waveform. One such manufacturer produces ”True-RMS” meters with a tiny resistive heating element powered by a voltage proportional to that being measured. The heating effect of that resistance
element is measured thermally to give a true RMS value with no mathematical calculations whatsoever, just the laws of physics in action in fulfillment of the definition of RMS. The accuracy of this
type of RMS measurement is independent of waveshape.
For ”pure” waveforms, simple conversion coefficients exist for equating Peak, Peak-to-Peak, Average (practical, not algebraic), and RMS measurements to one another:
RMS = 0.707 (Peak)
AVG = 0.637 (Peak)
P-P = 2 (Peak)
RMS = Peak
AVG = Peak
P-P = 2 (Peak)
RMS = 0.577 (Peak)
AVG = 0.5 (Peak)
P-P = 2 (Peak)
In addition to RMS, average, peak (crest), and peak-to-peak measures of an AC waveform, there
1.3. MEASUREMENTS OF AC MAGNITUDE 17
are ratios expressing the proportionality between some of these fundamental measurements. The
crest factor of an AC waveform, for instance, is the ratio of its peak (crest) value divided by its RMS
value. The form factor of an AC waveform is the ratio of its peak value divided by its average value.
Square-shaped waveforms always have crest and form factors equal to 1, since the peak is the same
as the RMS and average values. Sinusoidal waveforms have crest factors of 1.414 (the square root
of 2) and form factors of 1.571 (π/2). Triangle- and sawtooth-shaped waveforms have crest values
of 1.732 (the square root of 3) and form factors of 2.
Bear in mind that the conversion constants shown here for peak, RMS, and average amplitudes
of sine waves, square waves, and triangle waves hold true only for pure forms of these waveshapes.
The RMS and average values of distorted waveshapes are not related by the same ratios:
RMS = ???
AVG = ???
P-P = 2 (Peak)
This is a very important concept to understand when using an analog meter movement to measure
AC voltage or current. An analog movement, calibrated to indicate sine-wave RMS amplitude, will
only be accurate when measuring pure sine waves. If the waveform of the voltage or current being
measured is anything but a pure sine wave, the indication given by the meter will not be the true
RMS value of the waveform, because the degree of needle deflection in an analog meter movement is
proportional to the average value of the waveform, not the RMS. RMS meter calibration is obtained
by ”skewing” the span of the meter so that it displays a small multiple of the average value, which
will be equal to be the RMS value for a particular waveshape and a particular waveshape only.
Since the sine-wave shape is most common in electrical measurements, it is the waveshape assumed for analog meter calibration, and the small multiple used in the calibration of the meter is
1.1107 (the form factor π/2 divided by the crest factor 1.414: the ratio of RMS divided by average
for a sinusoidal waveform). Any waveshape other than a pure sine wave will have a different ratio
of RMS and average values, and thus a meter calibrated for sine-wave voltage or current will not
indicate true RMS when reading a non-sinusoidal wave. Bear in mind that this limitation applies
only to simple, analog AC meters not employing ”True-RMS” technology.
• REVIEW:
• The amplitude of an AC waveform is its height as depicted on a graph over time. An amplitude
measurement can take the form of peak, peak-to-peak, average, or RMS quantity.
• Peak amplitude is the height of an AC waveform as measured from the zero mark to the highest
positive or lowest negative point on a graph. Also known as the crest amplitude of a wave.
• Peak-to-peak amplitude is the total height of an AC waveform as measured from maximum
positive to maximum negative peaks on a graph. Often abbreviated as ”P-P”.
• Average amplitude is the mathematical ”mean” of all a waveform’s points over the period of
one cycle. Technically, the average amplitude of any waveform with equal-area portions above
and below the ”zero” line on a graph is zero. However, as a practical measure of amplitude,
a waveform’s average value is often calculated as the mathematical mean of all the points’
18 CHAPTER 1. BASIC AC THEORY
absolute values (taking all the negative values and considering them as positive). For a sine
wave, the average value so calculated is approximately 0.637 of its peak value.
• ”RMS” stands for Root Mean Square, and is a way of expressing an AC quantity of voltage or
current in terms functionally equivalent to DC. For example, 10 volts AC RMS is the amount
of voltage that would produce the same amount of heat dissipation across a resistor of given
value as a 10 volt DC power supply. Also known as the ”equivalent” or ”DC equivalent” value
of an AC voltage or current. For a sine wave, the RMS value is approximately 0.707 of its
peak value.
• The crest factor of an AC waveform is the ratio of its peak (crest) to its RMS value.
• The form factor of an AC waveform is the ratio of its peak (crest) value to its average value.
• Analog, electromechanical meter movements respond proportionally to the average value of
an AC voltage or current. When RMS indication is desired, the meter’s calibration must be
”skewed” accordingly. This means that the accuracy of an electromechanical meter’s RMS
indication is dependent on the purity of the waveform: whether it is the exact same waveshape
as the waveform used in calibrating.
1.4 Simple AC circuit calculations
Over the course of the next few chapters, you will learn that AC circuit measurements and calculations can get very complicated due to the complex nature of alternating current in circuits with
inductance and capacitance. However, with simple circuits involving nothing more than an AC
power source and resistance, the same laws and rules of DC apply simply and directly.
R
1
100 Ω
R
10 V
R
3
400 Ω
2
500 Ω
1.4. SIMPLE AC CIRCUIT CALCULATIONS 19
R
= R1 + R2 + R
total
R
total
I
total
I
total
I
total
= 1 kΩ
E
total
=
R
total
10 V
=
1 kΩ
= 10 mA
3
ER1 = I
totalR1
ER2 = I
totalR2
ER3 = I
totalR3
ER1 = 1 V ER2 = 5 V ER3 = 4 V
Series resistances still add, parallel resistances still diminish, and the Laws of Kirchhoff and
Ohm still hold true. Actually, as we will discover later on, these rules and laws always hold true,
it’s just that we have to express the quantities of voltage, current, and opposition to current in more
advanced mathematical forms. With purely resistive circuits, however, these complexities of AC are
of no practical consequence, and so we can treat the numbers as though we were dealing with simple
DC quantities.
Because all these mathematical relationships still hold true, we can make use of our familiar
”table” method of organizing circuit values just as with DC:
R
1
E
I
R
One major caveat needs to be given here: all measurements of AC voltage and current must be
expressed in the same terms (peak, peak-to-peak, average, or RMS). If the source voltage is given in
peak AC volts, then all currents and voltages subsequently calculated are cast in terms of peak units.
If the source voltage is given in AC RMS volts, then all calculated currents and voltages are cast in
AC RMS units as well. This holds true for any calculation based on Ohm’s Laws, Kirchhoff’s Laws,
etc. Unless otherwise stated, all values of voltage and current in AC circuits are generally assumed to
be RMS rather than peak, average, or peak-to-peak. In some areas of electronics, peak measurements
are assumed, but in most applications (especially industrial electronics) the assumption is RMS.
1
10m 10m 10m 10m
100
R
2
5 4
400500
R
3
Total
10
Volts
Amps
1k
Ohms
• REVIEW:
• All the old rules and laws of DC (Kirchhoff’s Voltage and Current Laws, Ohm’s Law) still hold
20 CHAPTER 1. BASIC AC THEORY
true for AC. However, with more complex circuits, we may need to represent the AC quantities
in more complex form. More on this later, I promise!
• The ”table” method of organizing circuit values is still a valid analysis tool for AC circuits.
1.5 AC phase
Things start to get complicated when we need to relate two or more AC voltages or currents that are
out of step with each other. By ”out of step,” I mean that the two waveforms are not synchronized:
that their peaks and zero points do not match up at the same points in time. The following graph
illustrates an example of this:
A B
A B
A B
A B
A B
The two waves shown above (A versus B) are of the same amplitude and frequency, but they
are out of step with each other. In technical terms, this is called a phase shift. Earlier we saw
how we could plot a ”sine wave” by calculating the trigonometric sine function for angles ranging
from 0 to 360 degrees, a full circle. The starting point of a sine wave was zero amplitude at zero
degrees, progressing to full positive amplitude at 90 degrees, zero at 180 degrees, full negative at 270
degrees, and back to the starting point of zero at 360 degrees. We can use this angle scale along the
horizontal axis of our waveform plot to express just how far out of step one wave is with another:
degrees
(0)
0 90 180 270 360
A
A B
90 180 270 360
A B
(0)
0 90 180 270 360
B
degrees
The shift between these two waveforms is about 45 degrees, the ”A” wave being ahead of the
”B” wave. A sampling of different phase shifts is given in the following graphs to better illustrate
this concept:
90 180 270 360
(0)
(0)
1.5. AC PHASE 21
Phase shift = 90 degrees
A B
A is ahead of B
(A "leads" B)
Phase shift = 90 degrees
B A
B is ahead of A
(B "leads" A)
A
Phase shift = 180 degrees
A and B waveforms are
mirror-images of each other
B
Phase shift = 0 degrees
A B
A and B waveforms are
in perfect step with each other
Because the waveforms in the above examples are at the same frequency, they will be out of step
by the same angular amount at every point in time. For this reason, we can express phase shift for
two or more waveforms of the same frequency as a constant quantity for the entire wave, and not
just an expression of shift between any two particular points along the waves. That is, it is safe to
say something like, ”voltage ’A’ is 45 degrees out of phase with voltage ’B’.” Whichever waveform
is ahead in its evolution is said to be leading and the one behind is said to be lagging.
Phase shift, like voltage, is always a measurement relative between two things. There’s really no
such thing as a waveform with an absolute phase measurement because there’s no known universal
reference for phase. Typically in the analysis of AC circuits, the voltage waveform of the power
supply is used as a reference for phase, that voltage stated as ”xxx volts at 0 degrees.” Any other
AC voltage or current in that circuit will have its phase shift expressed in terms relative to that
source voltage.
This is what makes AC circuit calculations more complicated than DC. When applying Ohm’s
Law and Kirchhoff’s Laws, quantities of AC voltage and current must reflect phase shift as well
as amplitude. Mathematical operations of addition, subtraction, multiplication, and division must
operate on these quantities of phase shift as well as amplitude. Fortunately, there is a mathematical
system of quantities called complex numbers ideally suited for this task of representing amplitude
and phase.
Because the subject of complex numbers is so essential to the understanding of AC circuits, the
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