7.3. Other waveshapesAs strange as it may seem, any repeating, nonsinusoidal waveform is actually equivalent to a series of sinusoidal waveforms of different amplitudes and frequencies added together. Square waves are a very common and wellunderstood case, but not the only one. Electronic power control devices such as transistors and siliconcontrolled rectifiers (SCRs) often produce voltage and current waveforms that are essentially choppedup versions of the otherwise “clean” (pure) sinewave 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 nonsinusoidal current through circuit impedances. Circuit components that distort the normal sinewave 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 nonsinusoidal 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 sinewave shape of the AC power line voltage stemming from “switching” loads such as computers and highefficiency 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 the passage of current in one direction. The simplest type of AC/DC rectification is halfwave, where a single diode blocks half of the AC current (over time) from passing through the load. (Figure below) Oddly enough, the conventional diode schematic symbol is drawn such that electrons flow against the direction of the symbol's arrowhead:
Halfwave rectifier. 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 halfwave rectifier
Halfwave rectifier waveforms. V(1)+0.4 shifts the sinewave input V(1) up for clarity. This is not part of the simulation. First, we'll see how SPICE analyzes the source waveform, a pure sine wave voltage: (Figure below) fourier components of transient response v(1) dc component = 8.016E04 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 2.492E03 0.000168 104.347 104.342 3 1.800E+02 6.465E04 0.000044 86.663 86.658 4 2.400E+02 1.132E03 0.000076 61.324 61.319 5 3.000E+02 1.185E03 0.000080 70.091 70.086 6 3.600E+02 1.092E03 0.000074 63.607 63.602 7 4.200E+02 1.220E03 0.000082 56.288 56.283 8 4.800E+02 1.354E03 0.000091 54.669 54.664 9 5.400E+02 1.467E03 0.000099 52.660 52.655
Fourier analysis of the sine wave input. Notice the extremely small harmonic and DC components of this sinusoidal waveform in the table above, though, too small to show on the harmonic plot above. 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 halfwave “rectified” voltage across the load resistor: (Figure below) 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 1.206E01 0.017223 168.005 167.810 4 2.400E+02 5.149E01 0.073556 87.295 87.100 5 3.000E+02 6.382E02 0.009117 152.790 152.595 6 3.600E+02 1.727E01 0.024676 79.362 79.167 7 4.200E+02 4.492E02 0.006417 132.420 132.224 8 4.800E+02 7.493E02 0.010703 61.479 61.284 9 5.400E+02 4.051E02 0.005787 115.085 114.889
Fourier analysis halfwave output. Notice the relatively large evenmultiple harmonics in this analysis. By cutting out half of our AC wave, we've introduced the equivalent of several higherfrequency sinusoidal (actually, cosine) waveforms into our circuit from the original, pure sinewave. 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 backandforth), it behaves a lot more like DC. Another method of AC/DC conversion is called fullwave (Figure below), 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 (Figure below) and its harmonic analysis through SPICE: (Figure below)
Fullwave rectifier circuit. 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
Waveforms for fullwave rectifier 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 7.000E02 1.000000 93.519 0.000 2 1.200E+02 5.997E+00 85.669415 90.230 3.289 3 1.800E+02 7.241E02 1.034465 93.787 0.267 4 2.400E+02 1.013E+00 14.465161 92.492 1.027 5 3.000E+02 7.364E02 1.052023 95.026 1.507 6 3.600E+02 3.337E01 4.767350 100.271 6.752 7 4.200E+02 7.496E02 1.070827 94.023 0.504 8 4.800E+02 1.404E01 2.006043 118.839 25.319 9 5.400E+02 7.457E02 1.065240 90.907 2.612
Fourier analysis of fullwave rectifier output. 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 halfwave 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 sinewave 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.
