**Parallel Resonance:**

Given figure represents a parallel resonating circuit where a coil is connected in parallel with a capacitor C and the combination is connected across an AC voltage source of variable frequency. Figure represents the vector diagram of the given circuit.

Vector Diagram of the Parallel AC Circuit

Let I_{C} = The current through the capacitor,

I_{L} = The current through the coil,

I = Vector sum of I_{L} and I_{C}, that is the source current,

V = Supply voltage, V_{R} = Drop across R, V_{L} = Drop across L, V_{C} = Drop across C,

φ = p.f. angle of the coil (i.e. the angle of lag of IL with respect to V),

Z_{L} = Coil impedance, and

X_{C} = Capacitor impedance (or simple capacitor reactance).

Here I_{C} = V/ X_{C}

And

cos φ= R/ Z

At resonance the capacitive current should be equal to the inductive part of the coil current, that means the imaginary components of I_{L} and I_{C} should cancel each other at resonance.

i.e. I_{C} = I_{ L }sin φ

or V / X_{ C} = ( V / Z _{L}) × (X _{L}/ Z _{L})

where sin φ= X_{ L} / Z _{L} ,

or (Z _{L} )^{2} = X_{C }X_{L}

Also

[ω_{0} represents resonance frequency]

or

(ω_{o })^{2}L^{2} = (L/C)-R^{2}

i.e. (ω_{o })^{2}L^{2} = (1/ LC)-R^{2 }/L^{2}

Again, at resonance, as the reactive components of I_{L} and L_{C} balance each other, the only remaining part of the current is I_{L} cos φ (= I)

I = I_{ L} cos φ

or,

V / Z_{Ω} = V/ Z _{L} . (R / Z _{L}) [Z_{Ω} = equivalent impedance of parallel circuit.]

or,

Z _{Ω} =( Z _{L})^{2 } /R = (L/C )/R = L/(CR) [? Z_{L }= √(L/C)]

Then, the equivalent impedance of the parallel resonating circuit is L/CR at resonance. This impedance is called as dynamic resistance of the parallel circuit. In general R being loss, this impedance is extremely high at resonance and then the current is much lower in the parallel circuit. Then, this circuit is also called as rejector circuit.