Energy Stored in a Magnetic Field — Question 9

Question 9

Constant-power energy extraction makes the required inductor voltage grow as current falls

An ideal inductor L initially carries I0 and supplies constant positive power P to a controlled load. Assume no other loss and fixed geometry. The load is stopped before the current reaches zero.

1. Find i(t) and the magnitude of the inductor terminal voltage.

2. If that voltage must not exceed Vmax, find the smallest allowed current and maximum extractable energy, assuming the initial current is sufficient.

Question 9 - Solution

Constant-power energy extraction makes the required inductor voltage grow as current falls

(a) Method and calculation

$$\frac d{dt}(\tfrac12Li^2)=-P\Rightarrow\boxed{i(t)=\sqrt{I_0^2-2Pt/L}}.$$$$v i=-P\Rightarrow\boxed{|v|=P/i}.$$

(b) Interpretation and further calculation

$$P/i\le V_{\max}\Rightarrow\boxed{i_{\min}=P/V_{\max}}.$$$$\boxed{W_{\mathrm{extracted,max}}=\tfrac12L[I_0^2-(P/V_{\max})^2]}.$$

Physical check

Constant power is incompatible with finite terminal voltage all the way to zero current; the formula states the permitted endpoint explicitly.