Subject: Physics

Electrical resonance is said to take place in a series LCR circuit when the circuit allows maximum current for a given frequency of the source of alternating supply for which capacitive reactance becomes equal to the inductive reactance.

The current (I) in a series LCR circuit is given by

\begin{align*} I &= \frac EZ = \frac {E}{\sqrt {R^2 + \left (l\omega - \frac {1}{C\omega }\right )^2}} \end{align*}

At high frequency, \(X_L = L\omega \) is very large and \(X_C = \frac {1}{C\omega } \) is very small.

At low frequency, \( X_L = L\omega \) is very small and \(X_C = \frac {1}{C\omega} \) is very large.

If \(X_C = X_L \) for a particular frequency f_{0}, then the impedance of LCR circuit is given by

$$ Z = \sqrt {R^2 + 0} = R $$

Impedance of LCR circuit is minimum and hence current is maximum. This frequency F_{0} is called electrical resonance.

For electrical resonance, we have

\begin{align*} X_L &= X_C \\ \text{or,} \: L\omega &= \frac {1}{C\omega } \\ \text {or,} \: \omega ^2 &=\frac {1}{LC} \\ \text {or,} \: \omega &= \frac {1}{\sqrt {LC}} \\ \text {or,} \: 2\pi f_o &= \frac {1}{\sqrt {LC}} \\ \therefore f_o &=\frac {1}{2\pi \sqrt {LC}} \\ \end{align*}

At resonance of LCR circuits admit maximum current at particular frequencies. LCR circuit is used in transmitters and receivers of radio, television and telephone carrier equipment etc.

The quality factor or Q-factor of a series resonant circuit is defined as the ratio of a voltage developed across the inductance or Capacitance at resonance to the impressed voltage, which is the voltage applied across R.

**Q-factor by inductor**

\begin{align*} \text {i.e.} Q &= \frac {\text {Voltage across L}}{\text {applied voltage}} \\ \text {or,} \: Q &= \frac {IX_L}{IR} \\ \text {or,} \: Q &= \frac {\omega L}{R} \\ \text {or,} \: Q &= \frac {1}{\sqrt {LC}}.\frac LR \\ \text {or,} \: Q &= \sqrt {\frac LC}. \frac 1R \\ \text {or,} \: Q &= \frac 1R \sqrt {\frac LC} \\ \end{align*}

**Q-factor by capacitor**

\begin{align*} \text {i.e.} Q &= \frac {\text {Voltage across C}}{\text {applied voltage}} \\ \text {or,} \: Q &= \frac {IX_C}{IR} \\ \text {or,} \: Q &= \frac {1 }{\omega C R} \\ \text {or,} \: Q &= \frac {1}{\frac {1}{\sqrt {LC}}C.R} \\ \text {or,} \: Q &=\frac{ \sqrt { LC}}{CR} \\ \text {or,} \: Q &= \frac 1R \sqrt {\frac LC} \\ \end{align*}

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al.*Principle of Physics*. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. *A text Book of Physics*. Kathmandu: Surya Publication, 2003.

Electrical resonance is said to take place in a series LCR circuit when the circuit allows maximum current for a given frequency of the source of alternating supply for which capacitive reactance becomes equal to the inductive reactance.

The impedance of LCR circuit is minimum and hence current is maximum. This frequency F_{0} is called electrical resonance.

At resonance of LCR circuits admit maximum current at particular frequencies. LCR circuit is used in transmitters and receivers of radio, television and telephone carrier equipment etc.

The quality factor or Q-factor of a series resonant circuit is defined as the ratio of a voltage developed across the inductance or Capacitance at resonance to the impressed voltage, which is the voltage applied across R.

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