For a parallel RLC resonant circuit with resistance in series with inductor, what will be the effect on resonant frequency, if we increase the value resistance?

Concept:

\({Y_{eq}} = \frac{1}{{R + j\omega L}} + j\omega C\)

\( = \frac{{R - j\omega L}}{{{R^2} + {\omega ^2}{L^2}}} + j\omega C\)

\(= \frac{R}{{{R^2} + {\omega ^2}{L^2}}} + \left[ {\omega C - \frac{{\omega L}}{{{R^2} + {\omega ^2}{L^2}}}} \right]\)

At resonant frequency, imaginary part of equivalent admittance is zero.

\(\Rightarrow \omega c = \frac{{\omega L}}{{R{\;^2} + {\omega ^2}{L^2}}}\)

\(\Rightarrow {R^2} + {\omega ^2}{L^2} = \frac{L}{C}\)

\(\Rightarrow {L^2}{\omega ^2} = \frac{L}{C} - {R^2}\)

\(\Rightarrow {\omega ^2} = \frac{1}{{LC}} - {\left( {\frac{R}{L}} \right)^2}\;\)

\(\omega = \sqrt {\frac{1}{{LC}} - {{\left( {\frac{R}{L}} \right)}^2}}\)

Application:

From the above expression, the resonant frequency decreases with an increase in resistance.