Alternating Current

Electrical Resonance

A series LCR circuit is said to be in electrical resonance when the circuit allows maximum current for a given frequency (called resonant frequency) of the source of alternating supply. This occurs when the inductive reactance $(X_L)$ becomes equal to the capacitive reactance $(X_C)$ so that the impedance of the circuit becomes minimum and the circuit allows maximum current.

The impedance $(Z)$ of a series LCR circuit is given by \[Z=\sqrt{R^2+(X_L-X_C)^2}\] where, \[X_L=2πfL\;\text{ and }\;X_C=\frac{1}{2πfC}\] Hence, at low frequency, $X_L$ is very small and $X_C$ is very large and vice versa. Also, resistance $R$ does not depend upon frequency. Since $X_L$ and $X_C$ depends upon frequency $f$ of the source, $Z$ also varies with frequency. The variation of $Z$ with frequency is shown in the following graph.

Variation of impedance with frequency: Minimum impedance at resonant frequency

At certain frequency $f_0$, $X_L$ becomes equal to $X_C$ so \[Z=\sqrt{R^2}=R\] Hence, the circuit becomes purely resistive and maximum current passes through it.

Electrical Resonance: Maximum current at resonant frequency

Also, the phase angle $(\phi)$ between $I$ and $E$ given by \[\tan\phi=\frac{X_L-X_C}{R}=0\] Hence, the applied alternating emf and alternating current are in same phase. This frequency $f_0$ is called resonant frequency and the phenomenon is called electrical resonance.

Determination of Resonant Frequency

For electrical resonance, \[X_L=X_C\] \[\omega_0L=\frac{1}{\omega_0C}\] \[2πf_0L=\frac{1}{2πf_0C}\] \[f_0^2=\frac{1}{4π^2LC}\] \[\therefore f_0=\frac{1}{2π\sqrt{LC}}\] This gives the value of resonant frequency. The resonant frequency depends only on the value of inductance and capacitance but independent of the value of resistance. However, the sharpness of resonance decreases with increase in resistance.

Due to electrical resonance, LCR circuits can be used to tune the desired frequency or filter the unwanted frequencies. So they are used in transmitters and receivers of radio, television, telephone, etc.

Quality Factor (Q-Factor)

The sharpness of tuning of resonance is determined by Q-factor of the circuit. It is defined as the ratio of either the inductive reactance or capacitive reactance at the resonant frequency to the total resistance of the circuit. Q stands for Quality.

Mathematically, \[Q=\frac{X_L}{R}=\frac{\omega_0L}{R}=\frac{2πf_0L}{R}\text{ __(1)}\] \[\text{or, }Q=\frac{X_C}{R}=\frac{1}{\omega_0CR}=\frac{1}{2πf_0CR}\text{ __(2)}\] At resonance, \[X_L=X_C\] \[\omega_0L=\frac{1}{\omega_0C}\] \[\therefore\omega_0=\frac{1}{\sqrt{LC}}\text{ __(3)}\]

From $\text{(1)}$ and $\text{(3)}$, the Q-factor for inductor is given by \[Q=\frac{1}{\sqrt{LC}}.\frac{L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}\] From $\text{(2)}$ and $\text{(3)}$, the Q-factor for capacitor is given by \[Q=\frac{1}{\frac{1}{\sqrt{LC}}×CR}=\frac{1}{R}\sqrt{\frac{L}{C}}\] Hence, quality factor is same for both inductor and capacitor. As $R$ increases, $Q$ decreases.

$Q$ has no unit and dimension. It is a pure number. Generally, its value lies between $10$ to $200$ but with a well designed circuit with very high frequency, $Q$ can be more than $200$. Higher the value of $Q$, the sharper is the resonance.