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. 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.
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. 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 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. 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. Sum of 1st, 3rd, 5th, 7th and 9th harmonics approximates square wave. Finally, adding the 9th harmonic, the fifth sine wave voltage source in our circuit, we obtain this result: (Figure below) Here we can see the wave becoming flatter at each peak. Sum of 1st, 3rd, 5th, and 7th harmonics approximates square wave. Suddenly, it doesn\’t look like a clean sine wave any more: (Figure below) Next, we see what happens when this clean and simple waveform is combined with the third harmonic (three times 50 Hz, or 150 Hz). This is the kind of waveform produced by an ideal AC power source: (Figure below) It is nothing but a pure sine shape, with no additional harmonic content. In this first plot, we see the fundamental-frequency sine-wave of 50 Hz by itself. I\’ll narrate the analysis step by step from here, explaining what it is we\’re looking at.
plot tran v(4,0) Plot 1st 3rd 5th 7th harmonics plot tran v(3,0) Plot 1st 3rd 5th harmonics for each of the increasing odd harmonics).
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. The fundamental frequency is 50 Hz and each harmonic is, of course, an integer multiple of that frequency. 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. We\’ll use SPICE to plot the voltage waveforms across successive additions of voltage sources, like this: (Figure below)Ī square wave is approximated by the sum of harmonics. 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. This reasoning is not only sound, but easily demonstrated with SPICE. 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 truth about waveforms at first may seem too strange to believe.
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: So long as it repeats itself regularly over time, it is reducible to this series of sinusoidal waves. This is true no matter how strange or convoluted the waveform in question may be. 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.