Resonance and Filters

Resonance is simply a circuit responding far more strongly at one frequency than at any other, the way a swing goes highest when you push it in time with its own rhythm. (The word comes from the Latin for "resound".) A filter is a circuit that lets some frequencies through and blocks others, and between them they're how a radio picks out one station and a sensor gets rid of noise. The favorite frequency $\omega_0 = 1/\sqrt{LC}$ depends only on $L$ (the inductor, which fights fast changes) and $C$ (the capacitor, which blocks slow ones); there the two cancel, and the current gets as big as $R$ (the resistor) allows. A filter's $|H|$ is just a number between 0 and 1 (how much of each frequency makes it through). It looks like a pile of formulas, but it's one idea used two ways; bear with it.

Tuning a radio. An antenna picks up every station at once, hundreds of signals at different frequencies piled on top of each other. Turning the dial picks out one. Behind the dial is a series RLC circuit whose response is enormous at one frequency and tiny elsewhere, and the dial turns a variable capacitor that moves that frequency.

Resonance, derived. Drive a series RLC circuit with $V_0\cos\omega t$. From the phasors lesson, $Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$, so the current amplitude is

The numerator is fixed, and the denominator is smallest when the reactances cancel, $\omega L = \frac{1}{\omega C}$, at

Notice that's the same natural frequency the LC circuit rings at on its own. That's no accident. It's the swing again: push in time with its own rhythm and each push adds to the last. At resonance $Z = R$, so the current is in phase with the source and as large as it can be, $V_0/R$. Below $\omega_0$ the capacitor dominates and the current leads; above it the inductor dominates and the current lags.

Voltage magnification. At resonance the inductor's voltage amplitude is $I_0\omega_0L = \frac{V_0}{R}\omega_0L = QV_0$, and the capacitor's is the same. The two are opposite in phase, so they cancel around the loop, but each one on its own can be huge. A 1 V signal can put 100 V across the capacitor of a $Q = 100$ circuit. Radio receivers use that gain, and power engineers fear it, since a resonance at a harmonic of the grid frequency can overload capacitor banks.

Bandwidth and Q, derived. How wide is the peak? We need a fair way to measure it. The average power delivered is $P = \frac12I_0^2R$, which falls to half its peak where $I_0$ is down by $\sqrt2$, that is, where the reactance equals $\pm R$:

The positive roots are $\omega_\pm = \sqrt{\left(\frac{R}{2L}\right)^2 + \omega_0^2} \pm \frac{R}{2L}$, so the bandwidth is exactly

and the product is $\omega_+\omega_- = \omega_0^2$: the resonant frequency is the geometric mean of the half-power frequencies, not their average. The quality factor $Q = \frac{\omega_0}{\Delta\omega} = \frac{\omega_0L}{R}$ compares the peak's position with its width. The same $Q$ measures ringing: a free RLC circuit's envelope decays like $e^{-Rt/2L} = e^{-\omega_0t/2Q}$, so a struck high-Q circuit rings for about $Q/\pi$ cycles before falling by a factor of $e$.

Where the peak sits. The current, like a spring's velocity, peaks exactly at $\omega_0$. The capacitor's voltage follows the charge, like the spring's displacement, and peaks a little below $\omega_0$: the practical resonance of mechanical vibrations. For high $Q$ the difference is negligible.

Filters. Resonance picks out one frequency. More generally, we'd like to describe what a circuit does to every frequency. That's the job of the transfer function $H(\omega) = \tilde V_{\text{out}}/\tilde V_{\text{in}}$: $|H|$ scales the amplitude and $\arg H$ shifts the phase. Take an RC voltage divider with the output across the capacitor:

For $\omega \ll 1/RC$, $|H| \approx 1$: low frequencies pass. For $\omega \gg 1/RC$, $|H| \approx \frac{1}{\omega RC}$: ten times the frequency, a tenth of the amplitude. That's a low-pass filter, and its cutoff $\omega_c = 1/RC$ is where $|H| = 1/\sqrt2$ (half power) and the phase shift is $-45^\circ$. It's no coincidence that $1/RC$ is the reciprocal of the RC time constant: a circuit that needs time $\tau$ to respond can't follow wiggles much faster than $1/\tau$.

Swap the parts, taking the output across $R$, and $H = \frac{j\omega RC}{1 + j\omega RC}$: a high-pass filter with the same cutoff. A series RLC circuit with the output across $R$ has $H = R/Z$, which passes a band of width $R/L$ around $\omega_0$: a band-pass filter, which is what the radio tuner really is. The first picture further down lets you play with one. Engineers quote gains in decibels, $20\log_{10}|H|$: $-3$ dB at cutoff and $-20$ dB for a factor of 10 in amplitude.

A common mistake. Because the cutoff is called the "half-power" point, it's tempting to think $|H| = \frac12$ there. It isn't: $|H| = 1/\sqrt2$, and it's the power, which goes as $|H|^2$, that's halved. Similarly, it's tempting to put $\omega_0$ halfway between the two half-power frequencies, but it sits at their geometric mean, not their average. And keep $\omega$ and $f$ apart: the bandwidth in hertz is $R/(2\pi L)$.

Resonance and filtering are everywhere signals are. Radio and TV receivers, Wi-Fi chips and phones select channels with resonant circuits and their crystal and acoustic-wave cousins. Every analog-to-digital converter sits behind a low-pass anti-aliasing filter, audio equalizers are banks of band-pass filters, and the RC filter on a microcontroller's input smooths a noisy sensor before it's sampled. Wireless charging pads and MRI coils are tuned to resonance so that a small drive builds large currents, while power engineers hunt down unwanted resonances that let one harmonic of the grid frequency build up dangerous voltages.