Three Simple Circuits — Question 3

Question 3

An ideal inductor retains a DC current set by its initial condition

An ideal inductor L has applied voltage $v=V_p\cos\omega t$ and initial current Idc at t=0. No resistance is present.

1. Integrate the voltage law and identify the sinusoidal current component.

2. Find total rms current and mean magnetic energy.

Question 3 - Solution

An ideal inductor retains a DC current set by its initial condition

(a) Method and calculation

$$L\dot i=V_p\cos\omega t\Rightarrow\boxed{i=I_{\mathrm{dc}}+\frac{V_p}{\omega L}\sin\omega t}.$$$$I_{{\mathrm{ac}},p}=V_p/(\omega L),\quad X_L=\omega L.$$

(b) Interpretation and further calculation

$$\boxed{I_{\mathrm{rms}}^2=I_{\mathrm{dc}}^2+\frac12\left(\frac{V_p}{\omega L}\right)^2}.$$$$\boxed{\langle U_L\rangle=\tfrac12LI_{\mathrm{dc}}^2+\frac{V_p^2}{4\omega^2L}}.$$

Physical check

The ideal inductor has no mechanism to remove its integration constant; a sinusoidal-only assumption must be stated.