Self-Induction — Question 10

Question 10

A nonlinear flux-linkage relation creates a third harmonic in voltage

An ideal nonlinear coil obeys $\Lambda=ai+bi^3$, with a and b positive. Impose $i=I_0\cos\omega t$. Use $\cos^3x=(3\cos x+\cos3x)/4$.

1. Find passive voltage as a sum of harmonics.

2. Find the ratio of third-harmonic amplitude to fundamental amplitude and its small-nonlinearity limit.

Question 10 - Solution

A nonlinear flux-linkage relation creates a third harmonic in voltage

(a) Method and calculation

$$\Lambda=(aI_0+3bI_0^3/4)\cos\omega t+(bI_0^3/4)\cos3\omega t.$$$$\boxed{v=-\omega(aI_0+3bI_0^3/4)\sin\omega t-3\omega bI_0^3\sin3\omega t/4}.$$

(b) Interpretation and further calculation

$$\boxed{V_3/V_1=\frac{3bI_0^2}{4a+3bI_0^2}}.$$$$bI_0^2\ll a:\quad\boxed{V_3/V_1\simeq3bI_0^2/(4a)}.$$

Physical check

A sinusoidal current need not produce sinusoidal voltage when linkage is nonlinear; the linear limit removes the third harmonic.