Rusty bolt effect

The transfer characteristic of an object can be represented as a power series:

${\displaystyle E_{\text{out}}=\sum _{n=1}^{\infty }{K_{n}E_{\text{in}}^{n}}}$

Or, taking only the first few terms (which are most relevant),

${\displaystyle E_{\text{out}}=K_{1}E_{\text{in}}+K_{2}E_{\text{in}}^{2}+K_{3}E_{\text{in}}^{3}+K_{4}E_{\text{in}}^{4}+K_{5}E_{\text{in}}^{5}+...}$

For an ideal perfect linear object K₂, K₃, K₄, K₅, etc. are all zero. A good connection approximates this ideal case with sufficiently small values.

For a 'rusty bolt' (or an intentionally designed frequency mixer stage), K₂, K₃, K₄, K₅, etc. are not all zero. These higher-order terms result in generation of harmonics.

The following analysis applies the power series representation to an input sine-wave.

If the incoming signal is a sine wave {E_in sin(ωt)}, (and taking only first-order terms), then the output can be written:

${\displaystyle {\begin{aligned}E_{\text{out}}&=\sum _{i=1}^{\infty }{K_{i}E_{\text{in}}^{i}\sin(i\omega t)}\\&=K_{1}E_{\text{in}}\sin(\omega t)+K_{2}E_{\text{in}}^{2}\sin(2\omega t)+K_{3}E_{\text{in}}^{3}\sin(3\omega t)+K_{4}E_{\text{in}}^{4}\sin(4\omega t)+K_{5}E_{\text{in}}^{5}\sin(5\omega t)+\cdots \end{aligned}}}$

Clearly, the harmonic terms will be worse at high input signal amplitudes, as they increase exponentially with the amplitude of E_in.

To understand the generation of nonharmonic terms (frequency mixing), a more complete formulation must be used, including higher-order terms. These terms, if significant, give rise to intermodulation distortion.

${\displaystyle {\begin{aligned}E_{f_{1}+f_{2}}&=kE_{f_{1}}E_{f_{2}}\\E_{f_{1}-f_{2}}&=kE_{f_{1}}E_{f_{2}}\end{aligned}}}$

${\displaystyle {\begin{aligned}E_{f_{1}+f_{2}+f_{3}}&=kE_{f_{1}}E_{f_{2}}E_{f_{3}}\\E_{f_{1}-f_{2}+f_{3}}&=kE_{f_{1}}E_{f_{2}}E_{f_{3}}\\E_{f_{1}+f_{2}-f_{3}}&=kE_{f_{1}}E_{f_{2}}E_{f_{3}}\\E_{f_{1}-f_{2}-f_{3}}&=kE_{f_{1}}E_{f_{2}}E_{f_{3}}\end{aligned}}}$

Hence the second-order, third-order, and higher-order mixing products can be greatly reduced by lowering the intensity of the original signals (f₁, f₂, f₃, f₄, …, f_n)