Three Simple Circuits — Question 7

Question 7

A nonzero mean voltage prevents periodic current in a lossless inductor

An ideal inductor is driven by $v=V_0+V_p\cos\omega t$ with nonzero V0. Its initial current is zero.

1. Find i(t).

2. Find the current change per source period and the condition required for any periodic inductor current.

Question 7 - Solution

A nonzero mean voltage prevents periodic current in a lossless inductor

(a) Method and calculation

$$i(t)=\frac1L\int_0^t(V_0+V_p\cos\omega u)du.$$$$\boxed{i(t)=V_0t/L+(V_p/\omega L)\sin\omega t}.$$

(b) Interpretation and further calculation

$$i(t+T)-i(t)=\frac1L\int_t^{t+T}v\,dt=V_0T/L.$$$$\boxed{\Delta i_T\ne0;\quad\text{periodicity requires}\ \int_t^{t+T}v\,dt=0}.$$

Physical check

A zero-mean AC component cannot cancel cumulative current growth caused by a DC voltage.