Introduction
In our study of AC circuits thus far,
we've explored circuits powered by a single-frequency sine voltage
waveform. In many applications of electronics, though, single-frequency
signals are the exception rather than the rule. Quite often we may
encounter circuits where multiple frequencies of voltage coexist
simultaneously. Also, circuit waveforms may be something other than
sine-wave shaped, in which case we call them non-sinusoidal waveforms.
Additionally, we may encounter situations
where DC is mixed with AC: where a waveform is superimposed on a steady
(DC) signal. The result of such a mix is a signal varying in intensity,
but never changing polarity, or changing polarity asymmetrically
(spending more time positive than negative, for example). Since DC does
not alternate as AC does, its "frequency" is said to be zero, and any
signal containing DC along with a signal of varying intensity (AC) may
be rightly called a mixed-frequency signal as well. In any of these
cases where there is a mix of frequencies in the same circuit, analysis
is more complex than what we've seen up to this point.
Sometimes mixed-frequency voltage and
current signals are created accidentally. This may be the result of
unintended connections between circuits -- called coupling --
made possible by stray capacitance and/or inductance between the
conductors of those circuits. A classic example of coupling phenomenon
is seen frequently in industry where DC signal wiring is placed in close
proximity to AC power wiring. The nearby presence of high AC voltages
and currents may cause "foreign" voltages to be impressed upon the
length of the signal wiring. Stray capacitance formed by the electrical
insulation separating power conductors from signal conductors may cause
voltage (with respect to earth ground) from the power conductors to be
impressed upon the signal conductors, while stray inductance formed by
parallel runs of wire in conduit may cause current from the power
conductors to electromagnetically induce voltage along the signal
conductors. The result is a mix of DC and AC at the signal load. The
following schematic shows how an AC "noise" source may "couple" to a DC
circuit through mutual inductance (Mstray) and capacitance (Cstray)
along the length of the conductors.
When stray AC voltages from a "noise"
source mix with DC signals conducted along signal wiring, the results
are usually undesirable. For this reason, power wiring and low-level
signal wiring should always be routed through separated,
dedicated metal conduit, and signals should be conducted via 2-conductor
"twisted pair" cable rather than through a single wire and ground
connection:
The grounded cable shield -- a wire braid
or metal foil wrapped around the two insulated conductors -- isolates
both conductors from electrostatic (capacitive) coupling by blocking any
external electric fields, while the parallal proximity of the two
conductors effectively cancels any electromagnetic (mutually inductive)
coupling because any induced noise voltage will be approximately equal
in magnitude and opposite in phase along both conductors, canceling each
other at the receiving end for a net (differential) noise voltage of
almost zero. Polarity marks placed near each inductive portion of signal
conductor length shows how the induced voltages are phased in such a way
as to cancel one another.
Coupling may also occur between two sets
of conductors carrying AC signals, in which case both signals may become
"mixed" with each other:
Coupling is but one example of how
signals of different frequencies may become mixed. Whether it be AC
mixed with DC, or two AC signals mixing with each other, signal coupling
via stray inductance and capacitance is usually accidental and
undesired. In other cases, mixed-frequency signals are the result of
intentional design or they may be an intrinsic quality of a signal. It
is generally quite easy to create mixed-frequency signal sources.
Perhaps the easiest way is to simply connect voltage sources in series:
Some computer communications networks
operate on the principle of superimposing high-frequency voltage signals
along 60 Hz power-line conductors, so as to convey computer data along
existing lengths of power cabling. This technique has been used for
years in electric power distribution networks to communicate load data
along high-voltage power lines. Certainly these are examples of
mixed-frequency AC voltages, under conditions that are deliberately
established.
In some cases, mixed-frequency signals
may be produced by a single voltage source. Such is the case with
microphones, which convert audio-frequency air pressure waves into
corresponding voltage waveforms. The particular mix of frequencies in
the voltage signal output by the microphone is dependent on the sound
being reproduced. If the sound waves consist of a single, pure note or
tone, the voltage waveform will likewise be a sine wave at a single
frequency. If the sound wave is a chord or other harmony of several
notes, the resulting voltage waveform produced by the microphone will
consist of those frequencies mixed together. Very few natural sounds
consist of single, pure sine wave vibrations but rather are a mix of
different frequency vibrations at different amplitudes.
Musical chords are produced by
blending one frequency with other frequencies of particular fractional
multiples of the first. However, investigating a little further, we find
that even a single piano note (produced by a plucked string) consists of
one predominant frequency mixed with several other frequencies, each
frequency a whole-number multiple of the first (called harmonics,
while the first frequency is called the fundamental). An
illustration of these terms is shown below with a fundamental frequency
of 1000 Hz (an arbitrary figure chosen for this example), each of the
frequency multiples appropriately labeled:
FOR A "BASE" FREQUENCY OF 1000 Hz:
Frequency (Hz) Term
-------------------------------------------
1000 --------- 1st harmonic, or fundamental
2000 --------- 2nd harmonic
3000 --------- 3rd harmonic
4000 --------- 4th harmonic
5000 --------- 5th harmonic
6000 --------- 6th harmonic
7000 --------- 7th harmonic
ad infinitum
Sometimes the term "overtone" is used to
describe the a harmonic frequency produced by a musical instrument. The
"first" overtone is the first harmonic frequency greater than the
fundamental. If we had an instrument producing the entire range of
harmonic frequencies shown in the table above, the first overtone would
be 2000 Hz (the 2nd harmonic), while the second overtone would be 3000
Hz (the 3rd harmonic), etc. However, this application of the term
"overtone" is specific to particular instruments.
It so happens that certain instruments
are incapable of producing certain types of harmonic frequencies. For
example, an instrument made from a tube that is open on one end and
closed on the other (such as a bottle, which produces sound when air is
blown across the opening) is incapable of producing even-numbered
harmonics. Such an instrument set up to produce a fundamental frequency
of 1000 Hz would also produce frequencies of 3000 Hz, 5000 Hz, 7000 Hz,
etc, but would not produce 2000 Hz, 4000 Hz, 6000 Hz, or any
other even-multiple frequencies of the fundamental. As such, we would
say that the first overtone (the first frequency greater than the
fundamental) in such an instrument would be 3000 Hz (the 3rd harmonic),
while the second overtone would be 5000 Hz (the 5th harmonic), and so
on.
A pure sine wave (single frequency),
being entirely devoid of any harmonics, sounds very "flat" and
"featureless" to the human ear. Most musical instruments are incapable
of producing sounds this simple. What gives each instrument its
distinctive tone is the same phenomenon that gives each person a
distinctive voice: the unique blending of harmonic waveforms with each
fundamental note, described by the physics of motion for each unique
object producing the sound.
Brass instruments do not possess the same
"harmonic content" as woodwind instruments, and neither produce the same
harmonic content as stringed instruments. A distinctive blend of
frequencies is what gives a musical instrument its characteristic tone.
As anyone who has played guitar can tell you, steel strings have a
different sound than nylon strings. Also, the tone produced by a guitar
string changes depending on where along its length it is plucked. These
differences in tone, as well, are a result of different harmonic content
produced by differences in the mechanical vibrations of an instrument's
parts. All these instruments produce harmonic frequencies (whole-number
multiples of the fundamental frequency) when a single note is played,
but the relative amplitudes of those harmonic frequencies are different
for different instruments. In musical terms, the measure of a tone's
harmonic content is called timbre or color.
Musical tones become even more complex
when the resonating element of an instrument is a two-dimensional
surface rather than a one-dimensional string. Instruments based on the
vibration of a string (guitar, piano, banjo, lute, dulcimer, etc.) or of
a column of air in a tube (trumpet, flute, clarinet, tuba, pipe organ,
etc.) tend to produce sounds composed of a single frequency (the
"fundamental") and a mix of harmonics. Instruments based on the
vibration of a flat plate (steel drums, and some types of bells),
however, produce a much broader range of frequencies, not limited to
whole-number multiples of the fundamental. The result is a distinctive
tone that some people find acoustically offensive.
As you can see, music provides a rich
field of study for mixed frequencies and their effects. Later sections
of this chapter will refer to musical instruments as sources of
waveforms for analysis in more detail.
- REVIEW:
- A sinusoidal waveform is one
shaped exactly like a sine wave.
- A non-sinusoidal waveform can
be anything from a distorted sine-wave shape to something completely
different like a square wave.
- Mixed-frequency waveforms can be
accidently created, purposely created, or simply exist out of
necessity. Most musical tones, for instance, are not composed of a
single frequency sine-wave, but are rich blends of different
frequencies.
- When multiple sine waveforms are mixed
together (as is often the case in music), the lowest frequency
sine-wave is called the fundamental, and the other sine-waves
whose frequencies are whole-number multiples of the fundamental wave
are called harmonics.
- An overtone is a harmonic
produced by a particular device. The "first" overtone is the first
frequency greater than the fundamental, while the "second" overtone is
the next greater frequency produced. Successive overtones may or may
not correspond to incremental harmonics, depending on the device
producing the mixed frequencies. Some devices and systems do not
permit the establishment of certain harmonics, and so their overtones
would only include some (not all) harmonic frequencies.
Square wave signals
It has been found that any
repeating, non-sinusoidal waveform can be equated to a combination of DC
voltage, sine waves, and/or cosine waves (sine waves with a 90 degree
phase shift) at various amplitudes and frequencies. This is true no
matter how strange or convoluted the waveform in question may be. So
long as it repeats itself regularly over time, it is reducible to this
series of sinusoidal waves. In particular, it has been found that square
waves are mathematically equivalent to the sum of a sine wave at that
same frequency, plus an infinite series of odd-multiple frequency sine
waves at diminishing amplitude:
This truth about waveforms at first may
seem too strange to believe. However, if a square wave is actually an
infinite series of sine wave harmonics added together, it stands to
reason that we should be able to prove this by adding together several
sine wave harmonics to produce a close approximation of a square wave.
This reasoning is not only sound, but easily demonstrated with SPICE.
The circuit we'll be simulating is
nothing more than several sine wave AC voltage sources of the proper
amplitudes and frequencies connected together in series. We'll use SPICE
to plot the voltage waveforms across successive additions of voltage
sources, like this:
In this particular SPICE simulation, I've
summed the 1st, 3rd, 5th, 7th, and 9th harmonic voltage sources in
series for a total of five AC voltage sources. The fundamental frequency
is 50 Hz and each harmonic is, of course, an integer multiple of that
frequency. The amplitude (voltage) figures are not random numbers;
rather, they have been arrived at through the equations shown in the
frequency series (the fraction 4/π multiplied by 1, 1/3, 1/5, 1/7, etc.
for each of the increasing odd harmonics).
building a squarewave
v1 1 0 sin (0 1.27324 50 0 0) 1st harmonic (50 Hz)
v3 2 1 sin (0 424.413m 150 0 0) 3rd harmonic
v5 3 2 sin (0 254.648m 250 0 0) 5th harmonic
v7 4 3 sin (0 181.891m 350 0 0) 7th harmonic
v9 5 4 sin (0 141.471m 450 0 0) 9th harmonic
r1 5 0 10k
.tran 1m 20m
.plot tran v(1,0) Plot 1st harmonic
.plot tran v(2,0) Plot 1st + 3rd harmonics
.plot tran v(3,0) Plot 1st + 3rd + 5th harmonics
.plot tran v(4,0) Plot 1st + 3rd + 5th + 7th harmonics
.plot tran v(5,0) Plot 1st + . . . + 9th harmonics
.end
I'll narrate the analysis step by step
from here, explaining what it is we're looking at. In this first plot,
we see the fundamental-frequency sine-wave of 50 Hz by itself. It is
nothing but a pure sine shape, with no additional harmonic content. This
is the kind of waveform produced by an ideal AC power source:
time v(1) -1.000E+00 0.000E+00 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . * . .
1.000E-03 3.915E-01 . . . * . .
2.000E-03 7.414E-01 . . . * . .
3.000E-03 1.020E+00 . . . * .
4.000E-03 1.199E+00 . . . . * .
5.000E-03 1.261E+00 . . . . * .
6.000E-03 1.199E+00 . . . . * .
7.000E-03 1.020E+00 . . . * .
8.000E-03 7.405E-01 . . . * . .
9.000E-03 3.890E-01 . . . * . .
1.000E-02 -5.819E-04 . . * . .
1.100E-02 -3.901E-01 . . * . . .
1.200E-02 -7.414E-01 . . * . . .
1.300E-02 -1.020E+00 . * . . .
1.400E-02 -1.199E+00 . * . . . .
1.500E-02 -1.261E+00 . * . . . .
1.600E-02 -1.199E+00 . * . . . .
1.700E-02 -1.020E+00 . * . . .
1.800E-02 -7.405E-01 . . * . . .
1.900E-02 -3.890E-01 . . * . . .
2.000E-02 5.819E-04 . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Next, we see what happens when this clean
and simple waveform is combined with the third harmonic (three times 50
Hz, or 150 Hz). Suddenly, it doesn't look like a clean sine wave any
more:
time v(2) -1.000E+00 0.000E+00 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . * . .
1.000E-03 7.199E-01 . . . * . .
2.000E-03 1.108E+00 . . . . * .
3.000E-03 1.135E+00 . . . . * .
4.000E-03 9.672E-01 . . . * .
5.000E-03 8.731E-01 . . . * . .
6.000E-03 9.751E-01 . . . * .
7.000E-03 1.144E+00 . . . . * .
8.000E-03 1.111E+00 . . . . * .
9.000E-03 6.995E-01 . . . * . .
1.000E-02 -5.697E-03 . . * . .
1.100E-02 -7.066E-01 . . * . . .
1.200E-02 -1.108E+00 . * . . . .
1.300E-02 -1.135E+00 . * . . . .
1.400E-02 -9.672E-01 . * . . .
1.500E-02 -8.731E-01 . . * . . .
1.600E-02 -9.751E-01 . * . . .
1.700E-02 -1.144E+00 . * . . . .
1.800E-02 -1.111E+00 . * . . . .
1.900E-02 -6.995E-01 . . * . . .
2.000E-02 5.697E-03 . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The rise and fall times between positive
and negative cycles are much steeper now, and the crests of the wave are
closer to becoming flat like a squarewave. Watch what happens as we add
the next odd harmonic frequency:
time v(3)
time v(3) -1.000E+00 0.000E+00 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . * . .
1.000E-03 9.436E-01 . . . *. .
2.000E-03 1.095E+00 . . . .* .
3.000E-03 9.388E-01 . . . *. .
4.000E-03 9.807E-01 . . . * .
5.000E-03 1.069E+00 . . . .* .
6.000E-03 9.616E-01 . . . *. .
7.000E-03 9.479E-01 . . . *. .
8.000E-03 1.124E+00 . . . . * .
9.000E-03 8.957E-01 . . . *. .
1.000E-02 -1.925E-02 . . * . .
1.100E-02 -9.029E-01 . .* . . .
1.200E-02 -1.095E+00 . *. . . .
1.300E-02 -9.388E-01 . .* . . .
1.400E-02 -9.807E-01 . * . . .
1.500E-02 -1.069E+00 . *. . . .
1.600E-02 -9.616E-01 . .* . . .
1.700E-02 -9.479E-01 . .* . . .
1.800E-02 -1.124E+00 . * . . . .
1.900E-02 -8.957E-01 . .* . . .
2.000E-02 1.925E-02 . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The most noticeable change here is how
the crests of the wave have flattened even more. There are more several
dips and crests at each end of the wave, but those dips and crests are
smaller in amplitude than they were before. Watch again as we add the
next odd harmonic waveform to the mix:
time v(4) -1.000E+00 0.000E+00 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . * . .
1.000E-03 1.055E+00 . . . .* .
2.000E-03 9.861E-01 . . . * .
3.000E-03 9.952E-01 . . . * .
4.000E-03 1.023E+00 . . . * .
5.000E-03 9.631E-01 . . . *. .
6.000E-03 1.044E+00 . . . .* .
7.000E-03 9.572E-01 . . . *. .
8.000E-03 1.031E+00 . . . * .
9.000E-03 9.962E-01 . . . * .
1.000E-02 -4.396E-02 . . *. . .
1.100E-02 -9.743E-01 . * . . .
1.200E-02 -9.861E-01 . * . . .
1.300E-02 -9.952E-01 . * . . .
1.400E-02 -1.023E+00 . * . . .
1.500E-02 -9.631E-01 . .* . . .
1.600E-02 -1.044E+00 . *. . . .
1.700E-02 -9.572E-01 . .* . . .
1.800E-02 -1.031E+00 . * . . .
1.900E-02 -9.962E-01 . * . . .
2.000E-02 4.396E-02 . . .* . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Here we can see the wave becoming flatter
at each peak. Finally, adding the 9th harmonic, the fifth sine wave
voltage source in our circuit, we obtain this result:
time v(5) -1.000E+00 0.000E+00 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . * . .
1.000E-03 1.079E+00 . . . .* .
2.000E-03 9.845E-01 . . . * .
3.000E-03 1.017E+00 . . . * .
4.000E-03 9.835E-01 . . . * .
5.000E-03 1.017E+00 . . . * .
6.000E-03 9.814E-01 . . . * .
7.000E-03 1.023E+00 . . . * .
8.000E-03 9.691E-01 . . . * .
9.000E-03 1.048E+00 . . . .* .
1.000E-02 -8.103E-02 . . *. . .
1.100E-02 -9.557E-01 . .* . . .
1.200E-02 -9.845E-01 . * . . .
1.300E-02 -1.017E+00 . * . . .
1.400E-02 -9.835E-01 . * . . .
1.500E-02 -1.017E+00 . * . . .
1.600E-02 -9.814E-01 . * . . .
1.700E-02 -1.023E+00 . * . . .
1.800E-02 -9.691E-01 . * . . .
1.900E-02 -1.048E+00 . *. . . .
2.000E-02 8.103E-02 . . .* . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The end result of adding the first five
odd harmonic waveforms together (all at the proper amplitudes, of
course) is a close approximation of a square wave. The point in doing
this is to illustrate how we can build a square wave up from multiple
sine waves at different frequencies, to prove that a pure square wave is
actually equivalent to a series of sine waves. When a square wave
AC voltage is applied to a circuit with reactive components (capacitors
and inductors), those components react as if they were being exposed to
several sine wave voltages of different frequencies, which in fact they
are.
The fact that repeating, non-sinusoidal
waves are equivalent to a definite series of additive DC voltage, sine
waves, and/or cosine waves is a consequence of how waves work: a
fundamental property of all wave-related phenomena, electrical or
otherwise. The mathematical process of reducing a non-sinusoidal wave
into these constituent frequencies is called Fourier analysis,
the details of which are well beyond the scope of this text. However,
computer algorithms have been created to perform this analysis at high
speeds on real waveforms, and its application in AC power quality and
signal analysis is widespread.
SPICE has the ability to sample a
waveform and reduce it into its constituent sine wave harmonics by way
of a Fourier Transform algorithm, outputting the frequency
analysis as a table of numbers. Let's try this on a square wave, which
we already know is composed of odd-harmonic sine waves:
squarewave analysis netlist
v1 1 0 pulse (-1 1 0 .1m .1m 10m 20m)
r1 1 0 10k
.tran 1m 40m
.plot tran v(1,0)
.four 50 v(1,0)
.end
The pulse option in the
netlist line describing voltage source v1
instructs SPICE to simulate a square-shaped "pulse" waveform, in this
case one that is symmetrical (equal time for each half-cycle) and has a
peak amplitude of 1 volt. First we'll plot the square wave to be
analyzed:
time v(1) -1 -0.5 0 0.5 1
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 -1.000E+00 * . . . .
1.000E-03 1.000E+00 . . . . *
2.000E-03 1.000E+00 . . . . *
3.000E-03 1.000E+00 . . . . *
4.000E-03 1.000E+00 . . . . *
5.000E-03 1.000E+00 . . . . *
6.000E-03 1.000E+00 . . . . *
7.000E-03 1.000E+00 . . . . *
8.000E-03 1.000E+00 . . . . *
9.000E-03 1.000E+00 . . . . *
1.000E-02 1.000E+00 . . . . *
1.100E-02 -1.000E+00 * . . . .
1.200E-02 -1.000E+00 * . . . .
1.300E-02 -1.000E+00 * . . . .
1.400E-02 -1.000E+00 * . . . .
1.500E-02 -1.000E+00 * . . . .
1.600E-02 -1.000E+00 * . . . .
1.700E-02 -1.000E+00 * . . . .
1.800E-02 -1.000E+00 * . . . .
1.900E-02 -1.000E+00 * . . . .
2.000E-02 -1.000E+00 * . . . .
2.100E-02 1.000E+00 . . . . *
2.200E-02 1.000E+00 . . . . *
2.300E-02 1.000E+00 . . . . *
2.400E-02 1.000E+00 . . . . *
2.500E-02 1.000E+00 . . . . *
2.600E-02 1.000E+00 . . . . *
2.700E-02 1.000E+00 . . . . *
2.800E-02 1.000E+00 . . . . *
2.900E-02 1.000E+00 . . . . *
3.000E-02 1.000E+00 . . . . *
3.100E-02 -1.000E+00 * . . . .
3.200E-02 -1.000E+00 * . . . .
3.300E-02 -1.000E+00 * . . . .
3.400E-02 -1.000E+00 * . . . .
3.500E-02 -1.000E+00 * . . . .
3.600E-02 -1.000E+00 * . . . .
3.700E-02 -1.000E+00 * . . . .
3.800E-02 -1.000E+00 * . . . .
3.900E-02 -1.000E+00 * . . . .
4.000E-02 -1.000E+00 * . . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Next, we'll print the Fourier analysis
generated by SPICE for this square wave:
fourier components of transient response v(1)
dc component = -2.439E-02
harmonic frequency fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 5.000E+01 1.274E+00 1.000000 -2.195 0.000
2 1.000E+02 4.892E-02 0.038415 -94.390 -92.195
3 1.500E+02 4.253E-01 0.333987 -6.585 -4.390
4 2.000E+02 4.936E-02 0.038757 -98.780 -96.585
5 2.500E+02 2.562E-01 0.201179 -10.976 -8.780
6 3.000E+02 5.010E-02 0.039337 -103.171 -100.976
7 3.500E+02 1.841E-01 0.144549 -15.366 -13.171
8 4.000E+02 5.116E-02 0.040175 -107.561 -105.366
9 4.500E+02 1.443E-01 0.113316 -19.756 -17.561
total harmonic distortion = 43.805747 percent
Here, SPICE has broken the waveform
down into a spectrum of sinusoidal frequencies up to the ninth harmonic,
plus a small DC voltage labelled DC
component. I had to inform SPICE of the
fundamental frequency (for a square wave with a 20 millisecond period,
this frequency is 50 Hz), so it knew how to classify the harmonics. Note
how small the figures are for all the even harmonics (2nd, 4th, 6th,
8th), and how the amplitudes of the odd harmonics diminish (1st is
largest, 9th is smallest).
This same technique of "Fourier
Transformation" is often used in computerized power instrumentation,
sampling the AC waveform(s) and determining the harmonic content
thereof. A common computer algorithm (sequence of program steps to
perform a task) for this is the Fast Fourier Transform or FFT
function. You need not be concerned with exactly how these computer
routines work, but be aware of their existence and application.
This same mathematical technique used in
SPICE to analyze the harmonic content of waves can be applied to the
technical analysis of music: breaking up any particular sound into its
constituent sine-wave frequencies. In fact, you may have already seen a
device designed to do just that without realizing what it was! A
graphic equalizer is a piece of high-fidelity stereo equipment that
controls (and sometimes displays) the nature of music's harmonic
content. Equipped with several knobs or slide levers, the equalizer is
able to selectively attenuate (reduce) the amplitude of certain
frequencies present in music, to "customize" the sound for the
listener's benefit. Typically, there will be a "bar graph" display next
to each control lever, displaying the amplitude of each particular
frequency.
A device built strictly to display -- not
control -- the amplitudes of each frequency range for a mixed-frequency
signal is typically called a spectrum analyzer. The design of
spectrum analyzers may be as simple as a set of "filter" circuits (see
the next chapter for details) designed to separate the different
frequencies from each other, or as complex as a special-purpose digital
computer running an FFT algorithm to mathematically split the signal
into its harmonic components. Spectrum analyzers are often designed to
analyze extremely high-frequency signals, such as those produced by
radio transmitters and computer network hardware. In that form, they
often have an appearance like that of an oscilloscope:
Like an oscilloscope, the spectrum
analyzer uses a CRT (or a computer display mimicking a CRT) to display a
plot of the signal. Unlike an oscilloscope, this plot is amplitude over
frequency rather than amplitude over time. In essence, a
frequency analyzer gives the operator a Bode plot of the signal:
something an engineer might call a frequency-domain rather than a
time-domain analysis.
The term "domain" is mathematical: a
sophisticated word to describe the horizontal axis of a graph. Thus, an
oscilloscope's plot of amplitude (vertical) over time (horizontal) is a
"time-domain" analysis, whereas a spectrum analyzer's plot of amplitude
(vertical) over frequency (horizontal) is a "frequency-domain" analysis.
When we use SPICE to plot signal amplitude (either voltage or current
amplitude) over a range of frequencies, we are performing
frequency-domain analysis.
Please take note of how the Fourier
analysis from the last SPICE simulation isn't "perfect." Ideally, the
amplitudes of all the even harmonics should be absolutely zero, and so
should the DC component. Again, this is not so much a quirk of SPICE as
it is a property of waveforms in general. A waveform of infinite
duration (infinite number of cycles) can be analyzed with absolute
precision, but the less cycles available to the computer for analysis,
the less precise the analysis. It is only when we have an equation
describing a waveform in its entirety that Fourier analysis can reduce
it to a definite series of sinusoidal waveforms. The fewer times that a
wave cycles, the less certain its frequency is. Taking this concept to
its logical extreme, a short pulse -- a waveform that doesn't even
complete a cycle -- actually has no frequency, but rather acts as
an infinite range of frequencies. This principle is common to all
wave-based phenomena, not just AC voltages and currents.
Suffice it to say that the number of
cycles and the certainty of a waveform's frequency component(s) are
directly related. We could improve the precision of our analysis here by
letting the wave oscillate on and on for many cycles, and the result
would be a spectrum analysis more consistent with the ideal. In the
following analysis, I've omitted the waveform plot for brevity's sake --
it's just a really long square wave:
squarewave
v1 1 0 pulse (-1 1 0 .1m .1m 10m 20m)
r1 1 0 10k
.option limpts=1001
.tran 1m 1
.plot tran v(1,0)
.four 50 v(1,0)
.end
fourier components of transient response v(1)
dc component = 9.999E-03
harmonic frequency fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 5.000E+01 1.273E+00 1.000000 -1.800 0.000
2 1.000E+02 1.999E-02 0.015704 86.382 88.182
3 1.500E+02 4.238E-01 0.332897 -5.400 -3.600
4 2.000E+02 1.997E-02 0.015688 82.764 84.564
5 2.500E+02 2.536E-01 0.199215 -9.000 -7.200
6 3.000E+02 1.994E-02 0.015663 79.146 80.946
7 3.500E+02 1.804E-01 0.141737 -12.600 -10.800
8 4.000E+02 1.989E-02 0.015627 75.529 77.329
9 4.500E+02 1.396E-01 0.109662 -16.199 -14.399
Notice how this analysis shows less of a
DC component voltage and lower amplitudes for each of the even harmonic
frequency sine waves, all because we let the computer sample more cycles
of the wave. Again, the imprecision of the first analysis is not so much
a flaw in SPICE as it is a fundamental property of waves and of signal
analysis.
- REVIEW:
- Square waves are equivalent to a sine
wave at the same (fundamental) frequency added to an infinite series
of odd-multiple sine-wave harmonics at decreasing amplitudes.
- Computer algorithms exist which are
able to sample waveshapes and determine their constituent sinusoidal
components. The Fourier Transform algorithm (particularly the
Fast Fourier Transform, or FFT) is commonly used in
computer circuit simulation programs such as SPICE and in electronic
metering equipment for determining power quality.
Other waveshapes
As strange as it may seem, any
repeating, non-sinusoidal waveform is actually equivalent to a series of
sinusoidal waveforms of different amplitudes and frequencies added
together. Square waves are a very common and well-understood case, but
not the only one.
Electronic power control devices such as
transistors and silicon-controlled rectifiers (SCRs) often
produce voltage and current waveforms that are essentially chopped-up
versions of the otherwise "clean" (pure) sine-wave AC from the power
supply. These devices have the ability to suddenly change their
resistance with the application of a control signal voltage or current,
thus "turning on" or "turning off" almost instantaneously, producing
current waveforms bearing little resemblance to the source voltage
waveform powering the circuit. These current waveforms then produce
changes in the voltage waveform to other circuit components, due to
voltage drops created by the non-sinusoidal current through circuit
impedances.
Circuit components that distort the
normal sine-wave shape of AC voltage or current are called nonlinear.
Nonlinear components such as SCRs find popular use in power electronics
due to their ability to regulate large amounts of electrical power
without dissipating much heat. While this is an advantage from the
perspective of energy efficiency, the waveshape distortions they
introduce can cause problems.
These non-sinusoidal waveforms,
regardless of their actual shape, are equivalent to a series of
sinusoidal waveforms of higher (harmonic) frequencies. If not taken into
consideration by the circuit designer, these harmonic waveforms created
by electronic switching components may cause erratic circuit behavior.
It is becoming increasingly common in the electric power industry to
observe overheating of transformers and motors due to distortions in the
sine-wave shape of the AC power line voltage stemming from "switching"
loads such as computers and high-efficiency lights. This is no
theoretical exercise: it is very real and potentially very troublesome.
In this section, I will investigate a few
of the more common waveshapes and show their harmonic components by way
of Fourier analysis using SPICE.
One very common way harmonics are
generated in an AC power system is when AC is converted, or "rectified"
into DC. This is generally done with components called diodes,
which only allow passage current in one direction. The simplest type of
AC/DC rectification is half-wave, where a single diode blocks
half of the AC current (over time) from passing through the load. Oddly
enough, the conventional diode schematic symbol is drawn such that
electrons flow against the direction of the symbol's arrowhead:
halfwave rectifier
v1 1 0 sin(0 15 60 0 0)
rload 2 0 10k
d1 1 2 mod1
.model mod1 d
.tran .5m 17m
.plot tran v(1,0) v(2,0)
.four 60 v(1,0) v(2,0)
.end
legend:
*: v(1)
+: v(2)
time v(1)
(*)---------- -20 -10 0 10 20
(+)---------- -5 0 5 10 15
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . + * . .
5.000E-04 2.806E+00 . . + . * . .
1.000E-03 5.483E+00 . . + * . .
1.500E-03 7.929E+00 . . . + *. .
2.000E-03 1.013E+01 . . . +* .
2.500E-03 1.198E+01 . . . . *+ .
3.000E-03 1.338E+01 . . . . * + .
3.500E-03 1.435E+01 . . . . * + .
4.000E-03 1.476E+01 . . . . * +.
4.500E-03 1.470E+01 . . . . * +.
5.000E-03 1.406E+01 . . . . * + .
5.500E-03 1.299E+01 . . . . * + .
6.000E-03 1.139E+01 . . . . x .
6.500E-03 9.455E+00 . . . + *. .
7.000E-03 7.113E+00 . . . + * . .
7.500E-03 4.591E+00 . . + . * . .
8.000E-03 1.841E+00 . . + . * . .
8.500E-03 -9.177E-01 . + *. . .
9.000E-03 -3.689E+00 . + * . . .
9.500E-03 -6.298E+00 . + * . . .
1.000E-02 -8.701E+00 . +* . . .
1.050E-02 -1.079E+01 . *+ . . .
1.100E-02 -1.249E+01 . * + . . .
1.150E-02 -1.377E+01 . * + . . .
1.200E-02 -1.453E+01 . * + . . .
1.250E-02 -1.482E+01 .* + . . .
1.300E-02 -1.452E+01 . * + . . .
1.350E-02 -1.378E+01 . * + . . .
1.400E-02 -1.248E+01 . * + . . .
1.450E-02 -1.081E+01 . *+ . . .
1.500E-02 -8.681E+00 . +* . . .
1.550E-02 -6.321E+00 . + * . . .
1.600E-02 -3.666E+00 . + * . . .
1.650E-02 -9.432E-01 . . + *. . .
1.700E-02 1.865E+00 . . + . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
First, we'll see how SPICE analyzes the
source waveform, a pure sine wave voltage:
fourier components of transient response v(1)
dc component = 8.016E-04
harmonic frequency fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.000E+01 1.482E+01 1.000000 -0.005 0.000
2 1.200E+02 2.492E-03 0.000168 -104.347 -104.342
3 1.800E+02 6.465E-04 0.000044 -86.663 -86.658
4 2.400E+02 1.132E-03 0.000076 -61.324 -61.319
5 3.000E+02 1.185E-03 0.000080 -70.091 -70.086
6 3.600E+02 1.092E-03 0.000074 -63.607 -63.602
7 4.200E+02 1.220E-03 0.000082 -56.288 -56.283
8 4.800E+02 1.354E-03 0.000091 -54.669 -54.664
9 5.400E+02 1.467E-03 0.000099 -52.660 -52.655
Notice the extremely small harmonic and
DC components of this sinusoidal waveform. Ideally, there would be
nothing but the fundamental frequency showing (being a perfect sine
wave), but our Fourier analysis figures aren't perfect because SPICE
doesn't have the luxury of sampling a waveform of infinite duration.
Next, we'll compare this with the Fourier analysis of the half-wave
"rectified" voltage across the load resistor:
fourier components of transient response v(2)
dc component = 4.456E+00
harmonic frequency fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.000E+01 7.000E+00 1.000000 -0.195 0.000
2 1.200E+02 3.016E+00 0.430849 -89.765 -89.570
3 1.800E+02 1.206E-01 0.017223 -168.005 -167.810
4 2.400E+02 5.149E-01 0.073556 -87.295 -87.100
5 3.000E+02 6.382E-02 0.009117 -152.790 -152.595
6 3.600E+02 1.727E-01 0.024676 -79.362 -79.167
7 4.200E+02 4.492E-02 0.006417 -132.420 -132.224
8 4.800E+02 7.493E-02 0.010703 -61.479 -61.284
9 5.400E+02 4.051E-02 0.005787 -115.085 -114.889
Notice the relatively large even-multiple
harmonics in this analysis. By cutting out half of our AC wave, we've
introduced the equivalent of several higher-frequency sinusoidal
(actually, cosine) waveforms into our circuit from the original, pure
sine-wave. Also take note of the large DC component: 4.456 volts.
Because our AC voltage waveform has been "rectified" (only allowed to
push in one direction across the load rather than back-and-forth), it
behaves a lot more like DC.
Another method of AC/DC conversion is
called full-wave, which as you may have guessed utilizes the full
cycle of AC power from the source, reversing the polarity of half the AC
cycle to get electrons to flow through the load the same direction all
the time. I won't bore you with details of exactly how this is done, but
we can examine the waveform and its harmonic analysis through SPICE:
fullwave bridge rectifier
v1 1 0 sin(0 15 60 0 0)
rload 2 3 10k
d1 1 2 mod1
d2 0 2 mod1
d3 3 1 mod1
d4 3 0 mod1
.model mod1 d
.tran .5m 17m
.plot tran v(1,0) v(2,3)
.four 60 v(2,3)
.end
legend:
*: v(1)
+: v(2,3)
time v(1)
(*)---------- -20 -10 0.000E+00 1.000E+01
(+)---------- 0.000E+00 5.000E+00 1.000E+01 1.500E+01
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 + . * . .
5.000E-04 2.806E+00 . + . . * . .
1.000E-03 5.483E+00 . +. . * . .
1.500E-03 7.929E+00 . . + . * . .
2.000E-03 1.013E+01 . . + . * .
2.500E-03 1.198E+01 . . . + . * .
3.000E-03 1.338E+01 . . . + . * .
3.500E-03 1.435E+01 . . . + . *.
4.000E-03 1.476E+01 . . . + . *
4.500E-03 1.470E+01 . . . + . *.
5.000E-03 1.406E+01 . . . + . * .
5.500E-03 1.299E+01 . . . + . * .
6.000E-03 1.139E+01 . . + .* .
6.500E-03 9.455E+00 . . + . *. .
7.000E-03 7.113E+00 . . + . * . .
7.500E-03 4.591E+00 . + . . * . .
8.000E-03 1.841E+00 . + . . * . .
8.500E-03 -9.177E-01 . + . *. . .
9.000E-03 -3.689E+00 . + . * . . .
9.500E-03 -6.298E+00 . + * . . .
1.000E-02 -8.701E+00 . . * + . . .
1.050E-02 -1.079E+01 . *. +. . .
1.100E-02 -1.249E+01 . * . . + . .
1.150E-02 -1.377E+01 . * . . + . .
1.200E-02 -1.453E+01 . * . . + . .
1.250E-02 -1.482E+01 . * . . + . .
1.300E-02 -1.452E+01 . * . . + . .
1.350E-02 -1.378E+01 . * . . + . .
1.400E-02 -1.248E+01 . * . . + . .
1.450E-02 -1.081E+01 . *. +. . .
1.500E-02 -8.681E+00 . . * + . . .
1.550E-02 -6.321E+00 . + * . . .
1.600E-02 -3.666E+00 . + . * . . .
1.650E-02 -9.432E-01 . + . *. . .
1.700E-02 1.865E+00 . + . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
fourier components of transient response v(2,3)
dc component = 8.273E+00
harmonic frequency fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.000E+01 7.000E-02 1.000000 -93.519 0.000
2 1.200E+02 5.997E+00 85.669415 -90.230 3.289
3 1.800E+02 7.241E-02 1.034465 -93.787 -0.267
4 2.400E+02 1.013E+00 14.465161 -92.492 1.027
5 3.000E+02 7.364E-02 1.052023 -95.026 -1.507
6 3.600E+02 3.337E-01 4.767350 -100.271 -6.752
7 4.200E+02 7.496E-02 1.070827 -94.023 -0.504
8 4.800E+02 1.404E-01 2.006043 -118.839 -25.319
9 5.400E+02 7.457E-02 1.065240 -90.907 2.612
What a difference! According to SPICE's
Fourier transform, we have a 2nd harmonic component to this waveform
that's over 85 times the amplitude of the original AC source frequency!
The DC component of this wave shows up as being 8.273 volts (almost
twice what is was for the half-wave rectifier circuit) while the second
harmonic is almost 6 volts in amplitude. Notice all the other harmonics
further on down the table. The odd harmonics are actually stronger at
some of the higher frequencies than they are at the lower frequencies,
which is interesting.
As you can see, what may begin as a neat,
simple AC sine-wave may end up as a complex mess of harmonics after
passing through just a few electronic components. While the complex
mathematics behind all this Fourier transformation is not necessary for
the beginning student of electric circuits to understand, it is of the
utmost importance to realize the principles at work and to grasp the
practical effects that harmonic signals may have on circuits. The
practical effects of harmonic frequencies in circuits will be explored
in the last section of this chapter, but before we do that we'll take a
closer look at waveforms and their respective harmonics.
- REVIEW:
- Any
waveform at all, so long as it is repetitive, can be reduced to a
series of sinusoidal waveforms added together. Different waveshapes
consist of different blends of sine-wave harmonics.
- Rectification of AC to DC is a very
common source of harmonics within industrial power systems.
More on spectrum analysis
Computerized Fourier analysis,
particularly in the form of the FFT algorithm, is a powerful tool
for furthering our understanding of waveforms and their related spectral
components. This same mathematical routine programmed into the SPICE
simulator as the .fourier
option is also programmed into a variety of electronic test instruments
to perform real-time Fourier analysis on measured signals. This section
is devoted to the use of such tools and the analysis of several
different waveforms.
First we have a simple sine wave at a
frequency of 523.25 Hz. This particular frequency value is a "C" pitch
on a piano keyboard, one octave above "middle C". Actually, the signal
measured for this demonstration was created by an electronic keyboard
set to produce the tone of a panflute, the closest instrument "voice" I
could find resembling a perfect sine wave. The plot below was taken from
an oscilloscope display, showing signal amplitude (voltage) over time:
Viewed with an oscilloscope, a sine wave
looks like a wavy curve traced horizontally on the screen. The
horizontal axis of this oscilloscope display is marked with the word
"Time" and an arrow pointing in the direction of time's progression. The
curve itself, of course, represents the cyclic increase and decrease of
voltage over time.
Close observation reveals imperfections
in the sine-wave shape. This, unfortunately, is a result of the specific
equipment used to analyze the waveform. Characteristics like these due
to quirks of the test equipment are technically known as artifacts:
phenomena existing solely because of a peculiarity in the equipment used
to perform the experiment.
If we view this same AC voltage on a
spectrum analyzer, the result is quite different:
As you can see, the horizontal axis of
the display is marked with the word "Frequency," denoting the domain of
this measurement. The single peak on the curve represents the
predominance of a single frequency within the range of frequencies
covered by the width of the display. If the scale of this analyzer
instrument were marked with numbers, you would see that this peak occurs
at 523.25 Hz. The height of the peak represents the signal amplitude
(voltage).
If we mix three different sine-wave tones
together on the electronic keyboard (C-E-G, a C-major chord) and measure
the result, both the oscilloscope display and the spectrum analyzer
display reflect this increased complexity:
The oscilloscope display (time-domain)
shows a waveform with many more peaks and valleys than before, a direct
result of the mixing of these three frequencies. As you will notice,
some of these peaks are higher than the peaks of the original
single-pitch waveform, while others are lower. This is a result of the
three different waveforms alternately reinforcing and canceling each
other as their respective phase shifts change in time.
The spectrum display (frequency-domain)
is much easier to interpret: each pitch is represented by its own peak
on the curve. The difference in height between these three peaks is
another artifact of the test equipment: a consequence of limitations
within the equipment used to generate and analyze these waveforms, and
not a necessary characteristic of the musical chord itself.
As was stated before, the device used to
generate these waveforms is an electronic keyboard: a musical instrument
designed to mimic the tones of many different instruments. The panflute
"voice" was chosen for the first demonstrations because it most closely
resembled a pure sine wave (a single frequency on the spectrum analyzer
display). Other musical instrument "voices" are not as simple as this
one, though. In fact, the unique tone produced by any instrument
is a function of its waveshape (or spectrum of frequencies). For
example, let's view the signal for a trumpet tone:
The fundamental frequency of this tone is
the same as in the first panflute example: 523.25 Hz, one octave above
"middle C." The waveform itself is far from a pure and simple sine-wave
form. Knowing that any repeating, non-sinusoidal waveform is equivalent
to a series of sinusoidal waveforms at different amplitudes and
frequencies, we should expect to see multiple peaks on the spectrum
analyzer display:
Indeed we do! The fundamental frequency
component of 523.25 Hz is represented by the left-most peak, with each
successive harmonic represented as its own peak along the width of the
analyzer screen. The second harmonic is twice the frequency of the
fundamental (1046.5 Hz), the third harmonic three times the fundamental
(1569.75 Hz), and so on. This display only shows the first six
harmonics, but there are many more comprising this complex tone.
Trying a different instrument voice (the
accordion) on the keyboard, we obtain a similarly complex oscilloscope
(time-domain) plot and spectrum analyzer (frequency-domain) display:
Note the differences in relative harmonic
amplitudes (peak heights) on the spectrum displays for trumpet and
accordion. Both instrument tones contain harmonics all the way from 1st
(fundamental) to 6th (and beyond!), but the proportions aren't the same.
Each instrument has a unique harmonic "signature" to its tone. Bear in
mind that all this complexity is in reference to a single note
played with these two instrument "voices." Multiple notes played on an
accordion, for example, would create a much more complex mixture of
frequencies than what is seen here.
The analytical power of the oscilloscope
and spectrum analyzer permit us to derive general rules about waveforms
and their harmonic spectra from real waveform examples. We already know
that any deviation from a pure sine-wave results in the equivalent of a
mixture of multiple sine-wave waveforms at different amplitudes and
frequencies. However, close observation allows us to be more specific
than this. Note, for example, the time- and frequency-domain plots for a
waveform approximating a square wave:
According to the spectrum analysis, this
waveform contains no even harmonics, only odd. Although this
display doesn't show frequencies past the sixth harmonic, the pattern of
odd-only harmonics in descending amplitude continues indefinitely. This
should come as no surprise, as we've already seen with SPICE that a
square wave is comprised of an infinitude of odd harmonics. The trumpet
and accordion tones, however, contained both even and odd
harmonics. This difference in harmonic content is noteworthy. Let's
continue our investigation with an analysis of a triangle wave:
In this waveform there are practically no
even harmonics: the only significant frequency peaks on the spectrum
analyzer display belong to odd-numbered multiples of the fundamental
frequency. Tiny peaks can be seen for the second, fourth, and sixth
harmonics, but this is due to imperfections in this particular triangle
waveshape (once again, artifacts of the test equipment used in this
analysis). A perfect triangle waveshape produces no even harmonics, just
like a perfect square wave. It should be obvious from inspection that
the harmonic spectrum of the triangle wave is not identical to the
spectrum of the square wave: the respective harmonic peaks are of
different heights. However, the two different waveforms are common in
their lack of even harmonics.
Let's examine another waveform, this one
very similar to the triangle wave, except that its rise-time is not the
same as its fall-time. Known as a sawtooth wave, its oscilloscope
plot reveals it to be aptly named:
When the spectrum analysis of this
waveform is plotted, we see a result that is quite different from that
of the regular triangle wave, for this analysis shows the strong
presence of even-numbered harmonics (second and fourth):
The distinction between a waveform having
even harmonics versus no even harmonics resides in the difference
between a triangle waveshape and a sawtooth waveshape. That difference
is symmetry above and below the horizontal centerline of the
wave. A waveform that is symmetrical above and below its centerline (the
shape on both sides mirror each other precisely) will contain no
even-numbered harmonics.
Square waves, triangle waves, and pure
sine waves all exhibit this symmetry, and all are devoid of even
harmonics. Waveforms like the trumpet tone, the accordion tone, and the
sawtooth wave are unsymmetrical around their centerlines and therefore
do contain even harmonics.
This principle of centerline symmetry
should not be confused with symmetry around the zero line. In the
examples shown, the horizontal centerline of the waveform happens to be
zero volts on the time-domain graph, but this has nothing to do with
harmonic content. This rule of harmonic content (even harmonics only
with unsymmetrical waveforms) applies whether or not the waveform is
shifted above or below zero volts with a "DC component." For further
clarification, I will show the same sets of waveforms, shifted with DC
voltage, and note that their harmonic contents are unchanged.
Again, the amount of DC voltage present
in a waveform has nothing to do with that waveform's harmonic frequency
content.
Why is this harmonic rule-of-thumb an
important rule to know? It can help us comprehend the relationship
between harmonics in AC circuits and specific circuit components. Since
most sources of sine-wave distortion in AC power circuits tend to be
symmetrical, even-numbered harmonics are rarely seen in those
applications. This is good to know if you're a power system designer and
are planning ahead for harmonic reduction: you only have to concern
yourself with mitigating the odd harmonic frequencies, even harmonics
being practically nonexistent. Also, if you happen to measure even
harmonics in an AC circuit with a spectrum analyzer or frequency meter,
you know that something in that circuit must be unsymmetrically
distorting the sine-wave voltage or current, and that clue may be
helpful in locating the source of a problem (look for components or
conditions more likely to distort one half-cycle of the AC waveform more
than the other).
Now that we have this rule to guide our
interpretation of nonsinusoidal waveforms, it makes more sense that a
waveform like that produced by a rectifier circuit should contain such
strong even harmonics, there being no symmetry at all above and below
center.
- REVIEW:
- Waveforms that are symmetrical above
and below their horizontal centerlines contain no even-numbered
harmonics.
- The amount of DC "bias" voltage
present (a waveform's "DC component") has no impact on that wave's
harmonic frequency content.
Circuit effects
The principle of non-sinusoidal,
repeating waveforms being equivalent to a series of sine waves at
different frequencies is a fundamental property of waves in general and
it has great practical import in the study of AC circuits. It means that
any time we have a waveform that isn't perfectly sine-wave-shaped, the
circuit in question will react as though it's having an array of
different frequency voltages imposed on it at once.
When an AC circuit is subjected to a
source voltage consisting of a mixture of frequencies, the components in
that circuit respond to each constituent frequency in a different way.
Any reactive component such as a capacitor or an inductor will
simultaneously present a unique amount of impedance to each and every
frequency present in a circuit. Thankfully, the analysis of such
circuits is made relatively easy by applying the Superposition
Theorem, regarding the multiple-frequency source as a set of
single-frequency voltage sources connected in series, and analyzing the
circuit for one source at a time, summing the results at the end to
determine the aggregate total:
Analyzing circuit for 60 Hz source alone:
Analyzing the circuit for 90 Hz source
alone:
Superimposing the voltage drops across R
and C, we get:
Because the two voltages across each
component are at different frequencies, we cannot consolidate them into
a single voltage figure as we could if we were adding together two
voltages of different amplitude and/or phase angle at the same
frequency. Complex number notation give us the ability to represent
waveform amplitude (polar magnitude) and phase angle (polar angle), but
not frequency.
What we can tell from this application of
the superposition theorem is that there will be a greater 60 Hz voltage
dropped across the capacitor than a 90 Hz voltage. Just the opposite is
true for the resistor's voltage drop. This is worthy to note, especially
in light of the fact that the two source voltages are equal. It is this
kind of unequal circuit response to signals of differing frequency that
will be our specific focus in the next chapter.
We can also apply the superposition
theorem to the analysis of a circuit powered by a non-sinusoidal
voltage, such as a square wave. If we know the Fourier series (multiple
sine/cosine wave equivalent) of that wave, we can regard it as
originating from a series-connected string of multiple sinusoidal
voltage sources at the appropriate amplitudes, frequencies, and phase
shifts. Needless to say, this can be a laborious task for some waveforms
(an accurate square-wave Fourier Series is considered to be expressed
out to the ninth harmonic, or five sine waves in all!), but it is
possible. I mention this not to scare you, but to inform you of the
potential complexity lurking behind seemingly simple waveforms. A
real-life circuit will respond just the same to being powered by a
square wave as being powered by an infinite series of sine waves
of odd-multiple frequencies and diminishing amplitudes. This has been
known to translate into unexpected circuit resonances, transformer and
inductor core overheating due to eddy currents, electromagnetic noise
over broad ranges of the frequency spectrum, and the like. Technicians
and engineers need to be made aware of the potential effects of
non-sinusoidal waveforms in reactive circuits.
Harmonics are known to manifest their
effects in the form of electromagnetic radiation as well. Studies have
been performed on the potential hazards of using portable computers
aboard passenger aircraft, citing the fact that computers' high
frequency square-wave "clock" voltage signals are capable of generating
radio waves that could interfere with the operation of the aircraft's
electronic navigation equipment. It's bad enough that typical
microprocessor clock signal frequencies are within the range of aircraft
radio frequency bands, but worse yet is the fact that the harmonic
multiples of those fundamental frequencies span an even larger range,
due to the fact that clock signal voltages are square-wave in shape and
not sine-wave.
Electromagnetic "emissions" of this
nature can be a problem in industrial applications, too, with harmonics
abounding in very large quantities due to (nonlinear) electronic control
of motor and electric furnace power. The fundamental power line
frequency may only be 60 Hz, but those harmonic frequency multiples
theoretically extend into infinitely high frequency ranges. Low
frequency power line voltage and current doesn't radiate into space very
well as electromagnetic energy, but high frequencies do.
Also, capacitive and inductive "coupling"
caused by close-proximity conductors is usually more severe at high
frequencies. Signal wiring nearby power wiring will tend to "pick up"
harmonic interference from the power wiring to a far greater extent than
pure sine-wave interference. This problem can manifest itself in
industry when old motor controls are replaced with new, solid-state
electronic motor controls providing greater energy efficiency. Suddenly
there may be weird electrical noise being impressed upon signal wiring
that never used to be there, because the old controls never generated
harmonics, and those high-frequency harmonic voltages and currents tend
to inductively and capacitively "couple" better to nearby conductors
than any 60 Hz signals from the old controls used to.
- REVIEW:
- Any regular (repeating),
non-sinusoidal waveform is equivalent to a particular series of
sine/cosine waves of different frequencies, phases, and amplitudes,
plus a DC offset voltage if necessary. The mathematical process for
determining the sinusoidal waveform equivalent for any waveform is
called Fourier analysis.
- Multiple-frequency voltage sources can
be simulated for analysis by connecting several single-frequency
voltage sources in series. Analysis of voltages and currents is
accomplished by using the superposition theorem. NOTE: superimposed
voltages and currents of different frequencies cannot be added
together in complex number form, since complex numbers only account
for amplitude and phase shift, not frequency!
- Harmonics can cause problems by
impressing unwanted ("noise") voltage signals upon nearby circuits.
These unwanted signals may come by way of capacitive coupling,
inductive coupling, electromagnetic radiation, or a combination
thereof.
Contributors
Contributors to this chapter are listed
in chronological order of their contributions, from most recent to
first.
Jason Starck
(June 2000): HTML document formatting, which led to a much
better-looking second edition.
|
