Parallel Resonant Circuit

Here, capacitor C is connected in parallel to the series combination of resistance R and inductance L. The combination is connected across the AC source. The applied voltage is sinusoidal, represented by
    E = E₀ e^jωt
Complex impedance of L – branch
    Z₁ = R + jLω
Complex impedance of C – branch
    Z₂ = 1/jCω

Z₁ and Z₂ are in parallel.
    1/Z = 1/R + jωL + 1/I/ jωC = 1/R + jωL + jωC
     = R - jωL /(R + jωL) x (R - jωL) + jωC
    = R /R² + (ωL)² + j[Cω - Lω/ R² - (Lω)² ]

The current I = E/Z = E X 1/Z
. :    I = E[R /(R₂ + (Lω)₂ + j(Cω - Lω/R² + (Lω)² ]

Let A cos Φ = R /R² + (Lω)² ; A sin Φ = Cω - Lω/R² + (Lω)²

. :    I = E (A cos Φ + j A sin Φ = Cω - Lω/R² + (Lω)²

Where ∅= tan^-1 Cω- (Lω)/R²+ (Lω)² )/R/(R²+ (Lω)²  

A² = R²/(R²+ ω² L² )² + (Cω- Lω/R²+ ω² L² )²

The magnitude of the admittance

Y= 1/z  √[R²+ (ωCR²+ ω³ L² C- ωL)² ]/(R²+ ω² L² )

The admittance will be minimum, when
    ωCR² + ω³L²C - ωL = 0    ,

or    ω = ω₀ = √[1/LC-R²/L²]

or         V₀= 1/2π √1/LC-R²/L²

This is the resonant frequency of the circuit.

If R is very small so that R2/L2 is negligible compared to 1/LC.
   
    V₀  = 1/2π√(LC)

At such a minimum admittance, i.e., maximum impedance, the circuit current is minimum.
Impedance at Resonance

At resonance, Z = R² + (Lω)² /R

But R² + (Lω)² = L/C at resonance
. :    Z = L/RC

The smaller the resistance R, larger is the impedance. If R is negligible, the impedance is infinite at resonance.

Rejector Circuit. The parallel resonant circuit does not allow the current of the same frequency as the natural frequency of the circuit. Thus it can be used to suppress the current of this particular frequency out of currents of many other frequencies. Hence the circuit is known as a ‘rejector’ or ‘filter’ circuit.