Electrical Resonance in Series LCR Circuit and Quality Factor

Electrical Resonance in Series LCR Circuit

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 f₀, 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 F₀ is called electrical resonance.

Determination of Resonant Frequency

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*}

Application of LCR Circuit

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.

Quality Factor of Resonance Circuit

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.