Eg. Fundamental is 5 MHz & you get harmonics at 10 MHz, 20MHz, etc.
How is the diode’s non-linearity causing this? I can’t find a way to visualize it.
In: 10
You may try to visualize a square wave at 5MHz. The issue with square waves is that they are not actually a single signal but it is a sum of a number of different sine waves at different frequencies, amplitudes and phase. The first is pretty obvious, it is the 5MHz sine wave which forms the majority of the square wave. But then you end up with lobes in the corners that is not covered by this sine wave. If you try to find a sine wave that covers these loabs you end up with a 15MHz wave. But again it does not fit perfectly so you need a 25MHz wave to cover the remaining parts. And then a 35MHz wave, etc. In fact a square wave is a special case here as it only have odd harmonics.
Diodes do not exactly create a square wave from a sine wave. But because of their non-linearity they do make the wave more square. They cut off the tops and bottoms of the waves and squashes them a bit. And what this means in practice is that there are harmonics added to the signal. Odd harmonics is created by the symmetric distortion and even harmonics by the asymmetric distortion.
A diode’s IV curve can be modeled as an exponential. So a change in voltage about the bias point will result in a change in current. However, that same change in voltage about a higher voltage point will result in a larger change in current. Overall, this behavior squishes peaks and accentuates low values (appears to bring everything towards a central value). This results in a more square wave.
A diode is a nonn-linear device – it has approximately linear behaviour in forward bias, and is essentially open circuit in reverse bias. There is a huge non-linearity at the zero bias point.
Take a sine wave and rectify it with a diode. Now you have a half wave waveform. This waveform contains lots of harmonics because it is a mixture of the original sine wave and a square wave (you have effectively got a square wave which switches the sine wave on and off). The square wave contains lots of harmonics and the process of mixing waveforms retains the harmonics.
Check out these graphs.
The first one shows the output of a diode subjected to a sinusoidal input. As you can see, the bottom half of the sine wave is cut off by the action of the diode.
https://www.wolframalpha.com/input?i=plot%28max%28sin%282+pi+t%29%2C0%29%29+with+0%3Ct%3C2
The next one shows the [best possible representation](https://en.wikipedia.org/wiki/Trigonometric_series) of the diode’s signal, using a constant output plus an oscillating output at the *same frequency* as the input.
https://www.wolframalpha.com/input?i=plot%28max%28sin%282+pi+t%29%2C0%29%2C1%2Fpi+%2B+%281%2F2%29+sin%282+pi+t%29%29+with+0%3Ct%3C2
You can see that because the clipped sine wave generated by the diode isn’t a pure sine wave, this isn’t a perfect representation. Over the course of one cycle, the “best representation” is too high, too low, too high, too low: it’s off by an amount that cycles twice for every cycle of the input wave.
Which is to say that we can get an even better representation of the diode’s output by **adding some signal with twice the frequency of the input:**
https://www.wolframalpha.com/input?i=plot%28max%28sin%282+pi+t%29%2C0%29%2C1%2Fpi+%2B+%281%2F2%29+sin%282+pi+t%29%2C+1%2Fpi+%2B+%281%2F2%29+sin%282+pi+t%29+-+%282%2F%283+pi%29%29+cos%284+pi+t%29%29+with+0%3Ct%3C2
… which is better, but still not perfect: it’s too high, too low, 4 times per repetition of the diode’s output signal. So we can fix it by **adding some signal with *four times* the frequency of the input:**
https://www.wolframalpha.com/input?i=plot%28max%28sin%282+pi+t%29%2C0%29%2C1%2Fpi+%2B+%281%2F2%29+sin%282+pi+t%29%2C+1%2Fpi+%2B+%281%2F2%29+sin%282+pi+t%29+-+%282%2F%283+pi%29%29+cos%284+pi+t%29+-+%282%2F%2815+pi%29%29+cos%288+pi+t%29%29+with+0%3Ct%3C2
… and so on. The diode output can be represented as — no, the diode output *is* — a signal composed of the base frequency, plus twice the base frequency, four times that, and so on.
The important thing, is that the output of a non-linear device depends only on the current input. That means, that if the input repeats, so does the output.
If the input is periodic with the period T, then the output must also be periodic with the period T. But that means, that it can only be the fundamental and it’s harmonics (they all repeat with a period T).
You may also study, how simple powers (x^(n)) affect the frequencies: they create scaled copies at sums and differences of every two frequencies of the input signal (including the negative ones). Now remember, that many nonlinear functions can be decomposed into power series (called Taylor series) – so they are just a scaled sum of powers.
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