RC Circuits — Question 9

Question 9

Periodic switching settles into two reproducible capacitor-voltage limits

An ideal input alternates between E and zero, spending duration T at each level. It drives a series R and shunt C with tau=RC. Consider the periodic steady state. Let L be the capacitor voltage just before each high interval and H just after it; set $a=e^{-T/\tau}$.

1. Derive H, L and the peak-to-peak ripple.

2. Find the time-averaged capacitor voltage over a complete cycle and the small-T limit of the ripple.

Question 9 - Solution

Periodic switching settles into two reproducible capacitor-voltage limits

(a) Method and calculation

$$H=E+(L-E)a,\quad L=aH\Rightarrow H=\frac E{1+a},\quad L=\frac{aE}{1+a}.$$$$\Delta V=H-L=E\frac{1-a}{1+a}.$$

(b) Interpretation and further calculation

$$\tau\langle\dot v_C\rangle+\langle v_C\rangle=\langle v_{\mathrm{in}}\rangle\Rightarrow\langle v_C\rangle=E/2.$$$$T/\tau\ll1:\quad a\simeq1-T/\tau,\quad\Delta V\simeq\frac{ET}{2\tau}.$$

Physical check

Periodic steady state requires zero net capacitor charge change per cycle, not zero instantaneous current.