Three Simple Circuits — Question 10

Question 10

A balanced square voltage produces triangular current in an ideal inductor

An ideal inductor L receives +Vp for T/2 and -Vp for T/2 repeatedly. Choose the periodic current solution with zero mean.

1. Find the peak current by integrating over one half-cycle.

2. Find rms current and maximum magnetic energy.

Question 10 - Solution

A balanced square voltage produces triangular current in an ideal inductor

(a) Method and calculation

$$\Delta i_{T/2}=V_pT/(2L)=2I_p.$$$$\boxed{I_p=V_pT/(4L)}.$$

(b) Interpretation and further calculation

$$\langle i^2\rangle=\frac1{2I_p}\int_{-I_p}^{I_p}i^2di=I_p^2/3.$$$$\boxed{I_{\mathrm{rms}}=V_pT/(4\sqrt3L),\quad U_{\max}=V_p^2T^2/(32L)}.$$

Physical check

Zero mean voltage guarantees a periodic solution; zero mean current selects its otherwise arbitrary DC offset.