Theory
A pure sinewave at frequency f has no harmonics. If it goes through a linear amplifier, the result continues to be pure (but may acquire a phase shift).
If the sinewave is run through a stateless nonlinear circuit (transcribing function), the resulting distortion creates harmonics. The distorted signal can be described by a Fourier series in f.
The nonzero ck represent the generated harmonics. The Fourier coefficients are given by integrating over the fundamental period T:
These harmonics can be selected by a bandpass filter.
The power in the distorted signal is spread across all the resulting harmonics. An ideal halfwave rectifier, for example, has all nonzero coefficients. An approximate circuit could use a diode.
From a conversion efficiency standpoint, the nonlinear circuit should maximize the coefficient for the desired harmonic and minimize the others. Consequently, the transcribing function is often specially chosen. Easy choices are to use an even function to generate even harmonics or an odd function to for odd harmonics. See Even and odd functions#Harmonics. A full wave rectifier, for example, is good for making a doubler. To produce a times-3 multiplier, the original signal may be input to an amplifier that is over driven to produce nearly a square wave. This signal is high in 3rd order harmonics and can be filtered to produce the desired x3 outcome.
YIG multipliers often want to select an arbitrary harmonic, so they use a stateful distortion circuit that converts the input sine wave into an approximate impulse train. The ideal (but impractical) impulse train generates an infinite number of (weak) harmonics. In practice, an impulse train generated by a monostable circuit will have many usable harmonics. YIG multipliers using step recovery diodes may, for example, take an input frequency of 1 to 2 GHz and produce outputs up to 18 GHz. Sometimes the frequency multiplier circuit will adjust the width of the impulses to improve conversion efficiency for a specific harmonic.
Read more about this topic: Frequency Multiplier
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