For KVL, let v1, v2, …, vn be the voltages around a close loop. Then
v1 + v2 + ... + vn = 0 .....(1)
In the sinusoidal steady-state, each voltage may be written in cosine form, so that Equation.(1) becomes
Vm1 cos(ωt + θ1) + Vm2 cos (ωt + θ2) + ... + Vmn cos(ωt + θn) = 0 .....(2)
This can be written as
or
.....(3)
If we let Vk = Vmkejθk, then
Re[(V1 + V2 + ... + Vn) ejωt] = 0 .....(4)
Since ejωt ≠ 0,
V1 + V2 + .... + Vn = 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, …,in, then
I1 + I2 + ... + In = 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.
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