Inductors and Inductance — Question 9

Question 9

A nonlinear coil has distinct secant and incremental inductances

A coil has single-valued quasistatic linkage $\Lambda(I)=aI+bI^3$, where a=.0200 H and b=.00500 H/A squared. There is no hysteresis or winding resistance.

1. Find the secant inductance Lambda/I and incremental inductance dLambda/dI at I=2.00 A.

2. Find the terminal voltage when the current at that instant increases at 3.00 A/s.

Question 9 - Solution

A nonlinear coil has distinct secant and incremental inductances

(a) Method and calculation

$$L_{\mathrm{sec}}=a+bI^2=.0200+.00500(2.00)^2=\boxed{.0400\,H}.$$$$L_{\mathrm{inc}}=a+3bI^2=.0200+3(.00500)(2.00)^2=\boxed{.0800\,H}.$$

(b) Interpretation and further calculation

$$v=\frac{d\Lambda}{dt}=\frac{d\Lambda}{dI}\dot I=L_{\mathrm{inc}}\dot I.$$$$\boxed{v=(.0800)(3.00)=.240\,V}.$$

Physical check

For a nonlinear coil, the slope of the linkage curve governs voltage; the ratio Lambda/I generally differs.