# Can inductor voltage and capacitor current change abruptly?

I understand that inductor current and capacitor voltage cannot change abruptly, but can inductor voltage and capacitor current change abruptly?

I have a feeling the answer is no but I cannot explain why.

In an ideal world, where a capacitor has no series inductance and an inductor has no parallel capacitance, and voltage and current sources can provide voltages and currents with a step-shaped profile, the current into a capacitor and the voltage over an inductor can change abruptly.

Not that this ideal world is an mathematical abstraction, you can't buy such components.

• Thank you. I am only doing mathematical abstractions at this point of my education, so that's completely fine. – user53172 Sep 13 '14 at 17:46

but can inductor voltage and capacitor current change abruptly?

Yes, in the context of ideal circuit theory. Indeed, this is often the case when there is a switch in the circuit.

Mathematically, if the slope of inductor current (capacitor voltage) changes abruptly, the inductor voltage (capacitor current) is discontinuous.

So, for example, consider the case that a charged capacitor, an open switch, and a resistor are in series (as in problem 2 here)

At the instant the switch is closed, the voltage across the resistor instantaneously changes from zero to the initial capacitor voltage $V_0$. Thus, the capacitor current discontinuously changes from zero to non-zero and is given by

$$i_C(t) = \frac{V_0}{R}e^{-\frac{t}{RC}} \cdot u(t)$$

where $u(t)$ is the unit step function

The dual of this is an inductor, with non-zero current $I_0$, in parallel with a closed switch and a resistor. At the instant the switch is opened, the current through the resistor changes instantly from zero to the initial inductor current. Thus, the inductor voltage discontinuously changes from zero to non-zero and is given by

$$v_L(t) = I_0R\;e^{-\frac{tR}{L}} \cdot u(t)$$

In physical circuits, voltages and currents cannot instantaneously change but depending on the characteristic time scale, they can effectively change instantaneously.

I = C*dv/dt. So a step of voltage on a capacitor (infinite dv/dt) leads to infinite current. The response to a step of current is 1/C times the integral of the current. The integral of a step is a ramp, so the capacitor voltage will ramp linearly in response to a step of current. (From whatever initial condition voltage is on it.)

An inductor's response is analogous with current and voltage switched.