Bandwidth and Q-factor

Derives the bandwidth and Q-factor of a RLC resonator. Visualizes the Bode magnitude for different zeta values. Part of the article RLC resonator.\(\)

Bandwidth and Q-factor

Oscillators with a high quality factor oscillate with a smaller range of frequencies and are therefore more stable. The quality factor is defined as the natural frequency \(\omega_n\) multiplied with the ratio of the maximum energy stored and the power loss. The maximum energy stored can be calculated from the maximum energy in the inductor or capacitor. The equation below uses the maximum energy in the inductor \(LI_{rms}^2\). At the natural frequency \(\omega_n\), the impedance of the capacitor and inductor cancel each other and power is only dissipated in the resistor \(RI_{rms}^2\).

$$ \begin{align} Q &= \omega_n\frac{L\, I_{rms}^2}{R\,I_{rms}^2}=\omega_n\frac{L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \nonumber \\ &= \frac{1}{2\zeta} \end{align} \label{eq:qfactor1} $$

The Q factor also relates the frequencies \(\omega_1\) and \(\omega_2\) where the dissipated power equals half the power stored. Consequently, the transfer function \(H(s)\) equals \(\frac{1}{\sqrt{2}}\) (-3dB) as shown in the illustration below

own work

The half power bandwidth BW follows from solving the equation \(H(s)=\frac{1}{\sqrt{2}}\)

$$ \begin{align} |H(\omega)| &= \frac{R}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}\equiv\frac{1}{\sqrt{2}} \nonumber \\ \Rightarrow\ R^2&+\left(\omega L-\frac{1}{\omega C}\right)^2=2R^2 \nonumber \\ \Rightarrow\ \omega^2 &\pm \frac{R}{L}\omega – \frac{1}{LC}=0 \nonumber \\ \Rightarrow\ \omega_{1,2}\ &= \pm\frac{R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}},\ \omega_{1,2}>0 \nonumber \\ &= \pm\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}} \nonumber \\ \Rightarrow\ BW &= \omega_1-\omega_2=\frac{R}{L} \end{align} $$

The Q factor equals the ratio of resonant frequency \(\omega_n\) to half power bandwidth \(\omega_2-\omega_1\). $$ Q = \frac{\omega_n}{\omega_2-\omega_1}=\frac{1}{R}\sqrt{\frac{L}{C}} = \frac{1}{2\zeta} \label{eq:qfactor2} $$

High quality factor \(Q>1\) results in a sharp resonance peak.

own work
Bode magnitude for different \(\zeta\)

Note that the frequency-dependent definition can be uses to describe circuits with a single capacitor or inductor, opposed to the frequency-to-bandwidth ratio definition.

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