Resonance In Parallel Circuits

Parallel branches containing inductance and capacitance are in resonance when the reactive current in the inductive branch is equal to the reactive current in the capacitive branch.

The resultant reactive current for the circuit is zero.

In the vector diagram shown here

IC = IL sin θ

Thus, net reactive component of current

=  IC - IL sin θ = 0

Hence the resultant current IL cos θ flowing is in phase with the applied voltage, and the power factor of the whole circuit is 1.  This is sometimes called unity - power – factor resonance.

Using the equations IL = V/Z, sin θ = XL /Z and IC = V/XC, it can be derived that

V/ XC = V/Z * XL /Z for resonance in parallel circuits

i.e.Z2 = XC * XL = 2πfL. 1/2πfC = L/C

Therefore, L/C = R2 + XL 2 = R2 + (2πfoL)2

Resonant frequency fo = 1/2πLC

Thus, any of the quantities L, C, f, RL or RC may be varied to obtain resonance.