In practice, resonant circuits can never be ideal series or parallel resonant circuits. In some situations conversion of series to parallel, or parallel to series circuits makes the design calculations simpler. The following transformations are valid in narrow band of frequencies around resonance.
The input impedance of series R-X circuit shown in Figure 1 is
and the input impedance of parallel R-X circuit is,
One way of defining Q-factor of a circuit, with input impedance(), is .
Using Eq.(1) and this Q-definition, .
Using Eq.(2) and this Q definition, .
and are frequency dependent terms. Therefore and are also frequency dependent.
The condition under which series R-X and parallel R-X are equivalent is obtained by comparing Eq.(1) and Eq.(2).
After some arithmetic manipulations from Eq.(3) and using the definitions of and , we get .
At frequencies where , .
These impedance transformations are very helpful in designing narrow-band impedance matching networks.