KCL & KVL in AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

For KVL, let v₁, v₂, …, v_n be the voltages around a close loop. Then

v₁ + v₂ + ... + v_n = 0    .....(1)

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

V_m1 cos(ωt + θ₁) + Vm2 cos (ωt + θ₂) + ... + V_mn cos(ωt + θ_n) = 0     .....(2)

This can be written as

or
     .....(3)

If we let V_k = V_mke^jθk, then

Re[(V₁ + V₂ + ... + V_n) e^jωt] = 0    .....(4)

Since e^jωt ≠ 0,

V₁ + V₂ + .... + V_n = 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 i₁, i₂, …,in be the current leaving or entering a closed surface in a network at time t, then

i₁ + i₂ + ... + in = 0     .....(6)

If I₁, I₂, …., In are the phasor forms of the sinusoids i₁, i₂, …,i_n, then

I₁ + I₂ + ... + I_n = 0     .....(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.