KVL with Capacitor and inductor

 

Kirchhoff’s Voltage Law (KVL) with a Capacitor or inductor

Kirchhoff’s Voltage Law (KVL) is still fully applicable even if a mesh contains a capacitor or inductor.

Why it still works

KVL is based on the conservation of energy—the sum of voltages around any closed loop must be zero. This principle doesn’t depend on the type of component in the loop (resistor, capacitor, inductor, etc.).

What changes with a capacitor

The only difference is how you express the voltage across the capacitor:

V_C = (1 / C) ∫ i(t) dt

So when applying KVL in a mesh with a capacitor, you include this voltage term.

Example

In a loop with a source, resistor, and capacitor:

V_source - V_R - V_C = 0

Where:

• V_R = iR
• V_C = (1 / C) ∫ i(t) dt
• V_L = L* (di/dt)

Special cases

• DC steady state: Capacitor behaves like an open circuit (no current), but KVL still holds.
• AC analysis: Capacitor is handled using impedance:

Z_C = 1 / (jωC)

KVL is always valid—even with capacitors and inductoes in the mesh. You just need to use the correct voltage-current relationship for the capacitor.