Question

The frequency of oscillation for LC parallel resonant circuit at resonance is 'f. If the circuit is modified as shown in the circuit diagram, the frequency of oscillation at resonance is \[\mathrm{f}_1=\mathrm{xf},\] The value of 'x' is ______.

Options

• 3

• \[\frac{1}{3}\]

• \[\frac{1}{6}\]

• 6

Answer 1

The frequency of oscillation for LC parallel resonant circuit at resonance is 'f. If the circuit is modified as shown in the circuit diagram, the frequency of oscillation at resonance is \[\mathrm{f}_1=\mathrm{xf},\] The value of 'x' is \[\frac{1}{3}\].

Explanation:

Inductors L and 2L are in series

\[\therefore\] Equivalent Inductance \[\mathrm{L_{eq}=L+2L=3L}\] Capacitors C and 2C are in paralled

\[\therefore\] Equivalent Capacitance \[\mathrm{C_{eq}=C+2C=3C}\] Resonant Frequency for LC circuit.

\[\mathbf{f}=\frac{1}{2\pi\sqrt{\mathrm{LC}}}\]

Resonant frequency for modified LC circuit

\[\mathrm{f}=\frac{1}{2\pi\sqrt{3\mathrm{L}\cdot3\mathrm{C}}}=\frac{1}{2\pi3\sqrt{\mathrm{LC}}}\]

  \[=\frac{1}{3}\frac{1}{2\pi\sqrt{\mathrm{LC}}}\]

\[\mathrm{f}_1=\frac{1}{3}\mathrm{f}\]

\[\therefore\] Comparing with \[\mathrm{f}_{1}=\mathrm{xf}\]

\[\mathbf{x}=\frac{1}{3}\]