Parallel Resonant Circuit

A parallel resonant circuit consisting of an inductance L and a capacitance C connected in parallel to the alternating e.m.f.
Let the applied emf be
        E = E₀ sin ω_t


Current through the inductance lags behind the applied emf by π/2 is given by,
        IL = E₀ /ωL sin (ωt – π/2)
Here, ωL is the reactance of the inductance.
The current through the capacitor leads the applied emf by π/2 and is given by
        Ic = E₀ /(1/ωC) sin (ωt + π/2)
Here, (1/ωC) is the reactance of the capacitance.
The total current through the circuit is given by
        I = IL + IC
        = E₀ /ωL sin(ωt – π/2) + E₀ /1/ωC sin(ωt + π/2)
        = - E₀ cos ωt + E₀ /(1/ωC) cos ωt
Or        1 = E₀ (ωC – 1/ωL) cos ωt                                … (1)

If for a particular frequency v₀, ωC = 1/ωL, then from Eq.(1), the current becomes zero. The circuit now offers infinite impedance. Such a circuit which offers an infinite impedance to the A.C. is called a parallel resonant circuit. The frequency is called the resonant frequency.

Thus at v₀, we have, ωC = 1/ωL
Or    ω = 1/√LC
Or 2πv₀ = 1/√LC
. :    v₀ = 1/2π√LC                                        … (2)

The variation of the current with frequency.

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