12.11 PARALLEL RESONANCE

A parallel resonant circuit consists of an inductor L in parallel with a capacitor
C as shown in Figure 12.17. R is a small resistance associated with the coil.
The capacitor C is assumed to be lossless. The tuned circuit is driven by a
voltage source V. Such a parallel-tuned circuit is commonly used in tuned
amplifiers, oscillators, etc.

Analysis of a parallel-tuned circuit may be done more conveniently
in terms of admittances instead of impedances. Thus, admittance of the

FIGURE 12.17   Parallel resonant circuit.

inductive branch is given by

Admittance of capacitor C is given by

Total admittance

At resonance, the imaginary part should be equal to zero,

or

Hence,

or

If coil resistance (R) is very small (R 0), then f_r =

Equation (12.57) may also be put as

Considering the three elements, L, C, and R in series, the Q_r is given by
, where ω_rs is the series resonant frequency in radians/sec.

Then

Substituting Q_rs for , Equation (12.59) yields

But the frequency of series resonance is given by

Hence,

From Equation (12.62) we find that the frequency of parallel resonance
f_r differs from the frequency of series resonance frs. However, if Q_rs exceeds
10, then the factor