![]() 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: Here we can see the wave becoming flatter at each peak. Sum of 1st, 3rd, 5th, and 7th harmonics approximates square wave. Watch again as we add the next odd harmonic waveform to the mix: 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. The most noticeable change here is how the crests of the wave have flattened even more. Sum of 1st, 3rd and 5th harmonics approximates square wave. ![]() Watch what happens as we add the next odd harmonic frequency: 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. Sum of 1st (50 Hz) and 3rd (150 Hz) harmonics approximates a 50 Hz square wave. Suddenly, it doesn’t look like a clean sine wave any more: 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: 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 plot tran v(2,0) Plot 1st + 3rd 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: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.
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