# Quick Step Kirchhoff’s Laws for AC Circuits

Kirchhoff’s laws can also be used in ac electric circuit analysis. With the basic KVL and KCL from dc circuit, we can modify those two to be used for a sinusoidal electric circuit.

Make sure to read what is ac circuit first.

We cannot do circuit analysis in the frequency domain without Kirchhoff’s current and voltage laws. Hence, we need to express them in the frequency domain.

## Kirchhoff’s Laws for AC Circuit

For KVL, let v1v2, …, vn be the voltages around a close loop. Then

 (1)

In the sinusoidal steady-state, each voltage may be written in cosine form, so that Equation.(1) becomes

 (2)

This can be written as

 (3)

If we let Vk = Vmkejθk, then

 (4)

Since ejωt ≠ 0,

 (5)

indicating that Kirchhoff’s voltage law holds for phasors.

By following a similar procedure, we can show that Kirchhoff’s current law holds for phasors. If we let i1i2, …,in be the current leaving or entering a closed surface in a network at time t, then

 (6)

If I1I2, …., In are the phasor forms of the sinusoids i1i2, …,in, then

 (7)

which is Kirchhoff’s current law in the frequency domain.

Once we have shown that both KVL and KCL hold in the frequency domain, it is easy to do many things, such as impedance combination, nodal and mesh analyses, superposition, and source transformation.