In the given RLC circuit below, resonance will never occur, if

SOLUTION

The impedance of given RLC circuit is

Z = (R₁ + jωL) // (R₂ - j/ωL)

\(= \frac{{\left( {R + j\omega L} \right)\left( {{R_2} - \frac{j}{{\omega C}}} \right)}}{{{R_1} + j\omega L + {R_2} - \frac{j}{{\omega C}}}}\)

\(= \frac{{{R_1}{R_2} - j\frac{{{R_1}}}{{\omega C\;}} + j\omega L{R_2} + \frac{L}{C}}}{{\left( {{R_1} + {R_2}} \right) + j\left( {\omega L - \frac{1}{{\omega C}}} \right)}}\)

\(= \frac{{\left[ {\left( {{R_1}{R_2} + \frac{L}{C}} \right) + j\left( {\omega L{R_2} - \frac{{{R_1}}}{{\omega C}}} \right)} \right]\left[ {\left( {{R_1} + {R_2}} \right) - j\left( {\omega L - \frac{1}{{\omega C}}} \right)} \right]}}{{{{\left( {{R_1} + {R_2}} \right)}^2} + {{\left( {\omega L - \frac{1}{{\omega C}}} \right)}^2}}}\)

Equating imaginary part to zero

\(- j\left( {{R_1}{R_2} + \frac{L}{C}} \right)\left( {\omega L - \frac{1}{{\omega C}}} \right) + j\left( {\omega L{R_2} - \frac{{{R_1}}}{{\omega C}}} \right)\left( {{R_1} + {R_2}} \right) = 0\)

\(\omega L{R_2}\left( {{R_1} + {R_2}} \right) - \frac{{{R_1}}}{{\omega C}}\left( {{R_1} + {R_2}} \right) - {R_1}{R_2}\omega L + \frac{{{R_1}{R_2}}}{{\omega C}} - \frac{{\omega {L^2}}}{C} + \frac{L}{{\omega {c^2}}} = 0\)

\(\Rightarrow \omega LR_2^2 - \frac{{R_1^2}}{{\omega c}} - \frac{{\omega {L^2}}}{c} + \frac{L}{{\omega {c^2}}} = 0\)

\(\omega L\left( {R_2^2 - \frac{L}{c}} \right) = \frac{1}{{\omega c}}\left( {R_1^2 - \frac{L}{C}} \right)\)

\(\Rightarrow \omega = \frac{1}{{\sqrt {LC} }}\sqrt {\frac{{\left( {R_1^2 - \frac{L}{C}} \right)}}{{\left( {R_2^2 - \frac{L}{C}} \right)}}} \)

For \(R_1^2 < \frac{L}{C}\;and\;R_2^2 > \frac{L}{C}\) the resonant frequency is negative, i.e. no resonance occurs.