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In https://en.wikipedia.org/wiki/Frequency_mixer#Diode, it talks about diode's current-input voltage relationship, and using Taylor approximation of e^x demonstrates frequency mixer abilities.

But this works only if input voltage is small. But I also heard that frequency mixers can operate for large-voltage signals. How is math being done there?

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There's two easy ways to answer your question:

  1. Add another term to the Taylor series and evaluate it in either a computer algebra system (Mathematica or Maple or Maxima) to find when the higher-order terms become significant.

  2. Simulate your circuit under transient analysis in a SPICE-like simulator, and look for higher-order terms.

Keep in mind that when "unwanted" terms fall well outside the desired output band (for example, \$3\omega_1\$, \$2\omega_1+\omega_2\$), they'll be relatively easy to filter out. The terms that cause practical problems are the ones that fall near the desired output band (for example \$3\omega_1-\omega_2\$, assuming the input frequencies aren't widely separated)

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