Inductors and Inductance — Question 5

Question 5

Unequal turn fluxes require summing linkages before defining inductance

A linear N-turn coil has per-turn flux $\Phi_k=cI[1-\epsilon k/(N-1)]$ for $k=0,1,\ldots,N-1$, with N>1 and $0<\epsilon<1$. All turn senses agree.

1. Find total linkage and inductance.

2. For N=10 and epsilon=.200, compare the true L with the estimate Nc based on the first turn.

Question 5 - Solution

Unequal turn fluxes require summing linkages before defining inductance

(a) Method and calculation

$$\Lambda=\sum_{k=0}^{N-1}\Phi_k=cI\left[N-\frac\epsilon{N-1}\frac{N(N-1)}2\right].$$$$\boxed{L=Nc(1-\epsilon/2)}.$$

(b) Interpretation and further calculation

$$N=10,\ \epsilon=.200:\quad\boxed{L=9.00c}.$$$$\frac{Nc-L}{L}=\frac{10c-9c}{9c}=\boxed{1/9=11.1111\%}.$$

Physical check

Using one representative flux is valid only if that flux is the turn-average or all turns link equally.