Initial (boundary) condition for capacitors

INITIAL (BOUNDARY) CONDITION FOR CAPACITORS

We know that, i_c (t) = C dV_c (t) / dt … (5)

To determine the capacitor voltage in terms of capacitor current, we write.

The above expression states that the voltage at any time t, is proportional to the integral of the current through the capacitor.

At  t = 0+

The interval from 0 to 0+ is almost zero. If ic is finite and does not contain impulses or derivatives.

We conclude that the voltage and hence the charge across the capacitor can not change instantaneously. But the capacitor current has no such restriction.

i_c (0+) ≠  i_c (0-)

When the capacitor has no initial charge or voltage, Q(0-) = 0 and Vc(0-) = 0. Therefore, from equations (9) and (10), V_c (0+) = q(0+) = 0.

Thus, the capacitor with no initial voltage or charge behaves like a short circuit at t = 0+.

The capacitor with an initial charge q_c (0-) = Qo is equivalent to a voltage source of value V_co = Q₀ / C at t = 0+.