# Voltage across Capacitor and Inductor with DC Voltage and Current source

As we know, capacitors and inductors are most important components in circuit design.

I am just curious about how they will behave when connected across voltage or current source as per below image and voltage and current wave-forms in each case.

PS: This can be tutorial question for most of the Electronics learner like us.

• Should there be a switch in each circuit that closes at t=0, or should it be assumed the circuits have been closed for a long period?
– Chu
Commented Apr 13, 2016 at 6:58
• Yes. Wave-forms will be add on for understanding. :) Commented Apr 13, 2016 at 7:03
• Any half-decent simulator could plot those for you. Commented Apr 13, 2016 at 7:54

There are only two formula that fully encapsulate the theory of inductors and capacitors in AC circuits and transient circuits. Understanding these formulas is more important than trying to ring-fence capacitor and inductor behavior by four trivial scenarios.

For an inductor $V = L \dfrac{di}{dt}$ or put another way...

If voltage is constant across an inductor, then $\dfrac{di}{dt}$ is constant.

This means current will ramp up to infinite amps unless current is limited.

For a capacitor $I = C \dfrac{dv}{dt}$ or put another way...

If current is constant through the capacitor, then $\dfrac{dv}{dt}$ is constant.

This means voltage will ramp up to infinity unless voltage is limited.

Case 1: voltage across C is V, current is 0. Case 2: voltage across L is 0, current is I.

With the assumption of ideal sources.

Derivation using the impedance: Z_C=1/(jwC), Z_L=jwL, where w in DC is 0 -> replace C with open circuit and L with short circuit.

1a: At $\small t=0$, I is an impulse of strength VC that charges the capacitor instantaneously to V. Thereafter, $\small I=0$.

1b: A constant current, I, flows from $\small t=0$; the voltage across C increases linearly with time (until the dielectric breaks-down).

2a: Current increases linearly with time (until something melts), satisfying $V=L\frac{di}{dt}$ or $\frac{di}{dt}=\frac{V}{L}$

2b: At $\small t=0$ a voltage impulse of strength $\small IL$ is generated (by induction), thereafter $\small V=0$ and a steady current, I, flows. Zero voltage is required from the current generator to support this current.