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.
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