Consider the parallel resonant circuit shown in adjacent figure. One branch contains an inductor of inductance
L and a small ohmic resistance
R, whereas the other branch contains a capacitor of capacitance
C. The circuit is fed by a source of alternating emf
E=E0eiωt=E0sinωtThe impedance of inductor branch,
Z1=R+jωLThe impedance of capacitor branch,
Z2=(1/jωC)Net impedance
Z of the two parallel branches is given by
Z1=Z11+Z21=R+jωL1+jωC=R2+L2ω 2R+jω[C−R2+L2ω 2L ]The current flowing in the circuit
I=ZE=E[R2+L2ω 2R+jω[C−R2+L2ω 2L ] ]For resonance to occur, the current must be in phase with the applied emf. For this, the reactive component of current should be zero, ie
ω(C−R2+L2ω 2L )=0This gives resonant angular frequency
ω r=LC1−L2R2 At parallel circuit resonance, the impedance is maximum and current is minimum. Parallel resonant circuit is sometimes called the anti-resonance in order to distinguish from series resonance.