# Parallel Resonant Circuit

A parallel resonant RLC circuit is shown in Figure 1.

Figure 1. Parallel RLC circuit excited with a current source

The input admittance of a parallel R-L-C circuit is given by,

(1)

By definition, at resonant frequnecy inductive and capacitive susceptances are equal in magnitude and opposite in phase. Therefore they cancel each other.

(2)

Quality factor(Q) of a parallel R-L-C circuit at resonance is

(3)

So the Q-factor at resonance is

(4)

From Eq-(1), Eq-(2) and Eq-(4),

(5)

The transsfer function is

(6)

At resonant frequency , which is real and minimum.

The input impedance of parallel RLC circuit value deceases by at half power frequencies .

From Eq-(6) the condition for is and for is

(7)

(8)

The bandwidth of the circuit is . Using Eq-(7) and Eq-(8), bandwidth is given as . The resonant frequency is geometric mean of upper and lower 3dB frequencies, .

Quality factor is

For small deviations in frequency from resonant frequency , , input impedance is given by,

(9)

or when the deviations are very small compared to , input impedance is approximated as

(10)

From the above equation, at resonant frequency, the input impedance is R and is maximum at that frequency. Below it is inductive and above it is capacitive.

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