Sharpness of Resonance: Q Factor

13.9 Sharpness of Resonance — Q Factor

The amplitude of the current in a series LCR circuit is

$$i_0 = \dfrac{e_0}{\sqrt{R^2+\left(\omega L - \dfrac{1}{\omega C}\right)^2}}$$

If $\omega$ is varied, then at a particular frequency $\omega = \omega_r$ we have $X_L = X_C$, i.e. $\omega_r L = \dfrac{1}{\omega_r C}$. For a given resistance $R$ the amplitude of the current is maximum when

$$\omega_r L - \dfrac{1}{\omega_r C} = 0 \qquad \Rightarrow \qquad \omega_r = \dfrac{1}{\sqrt{LC}}$$

For values of $\omega$ other than $\omega_r$ the amplitude of the current is less than this maximum value.

Half-Power Frequencies and Bandwidth

Suppose we choose a value of $\omega$ for which the current amplitude is $\dfrac{1}{\sqrt2}$ times its maximum value. The power dissipated by the circuit is then half its maximum, so such a frequency is called a half-power frequency.

From the resonance curve there are two such values of $\omega$, say $\omega_1$ and $\omega_2$, one greater and the other smaller than $\omega_r$ and symmetrical about it:

$$\omega_1 = \omega_r + \Delta\omega, \qquad \omega_2 = \omega_r - \Delta\omega$$

The difference $\omega_1 - \omega_2 = 2\Delta\omega$ is called the bandwidth of the circuit. The quantity $\dfrac{\omega_r}{2\Delta\omega}$ is regarded as the measure of the sharpness of resonance.

Q Factor

The sharpness of resonance is measured by a coefficient called the quality factor or Q factor of the circuit.

Definition: the Q factor of a series resonant circuit is defined as the ratio of the resonant frequency to the difference between the two frequencies, taken on either side of the resonant frequency, at which the current amplitude becomes $\dfrac{1}{\sqrt2}$ times its value at the resonant frequency.

$$Q = \dfrac{\omega_r}{\omega_1-\omega_2} = \dfrac{\omega_r}{2\Delta\omega} = \dfrac{\text{Resonant frequency}}{\text{Bandwidth}}$$

• The Q factor is a dimensionless quantity.
• The larger the value of the Q factor, the smaller the value of $2\Delta\omega$, i.e. the smaller the bandwidth, and the sharper is the peak in the current — the series resonant circuit is then more selective.

Reading the Resonance Curve

Draw the current amplitude against $\omega$. The curve rises to a peak at $\omega_r$ and falls away on both sides; the horizontal line at $1/\sqrt2$ of the peak cuts it at $\omega_2$ and $\omega_1$, and the width between those cuts is the bandwidth.

The lower angular frequency side of the resonance curve is dominated by the capacitor's reactance (there $X_C > X_L$), the high angular frequency side is dominated by the inductor's reactance (there $X_L > X_C$), and resonance occurs in the middle where they are equal.

Application — Tuning a Radio or TV

The tuning circuit of a radio or a TV is an LCR resonant circuit. Signals are transmitted by different stations at different frequencies and are all picked up by the antenna, so a number of voltages appear across the series LCR circuit. But maximum current flows through the circuit for that AC voltage whose frequency equals

$$f_r = \dfrac{1}{2\pi\sqrt{LC}}$$

If the Q value of the circuit is large, the signals of the other stations will be very weak. By changing the value of the adjustable capacitor $C$, the signal from the desired station can be tuned in.

Standard Results

• $\omega_r = \dfrac{1}{\sqrt{LC}}$, and the current amplitude is maximum there.
• Half-power frequencies: current amplitude $= \dfrac{i_{0,max}}{\sqrt2}$, power $=$ half the maximum.
• Bandwidth $= \omega_1 - \omega_2 = 2\Delta\omega$.
• $Q = \dfrac{\omega_r}{2\Delta\omega}$, dimensionless; large $Q$ means narrow bandwidth and high selectivity.