LC and RLC Oscillations

An LC circuit is simply a capacitor and an inductor wired in a loop, and an RLC circuit adds a resistor; the letters are just the standard symbols for the three parts ($C$, $L$ and $R$). Charge one up and it swings back and forth on its own, the way a mass on a spring bounces. That's what makes these circuits the tuning heart of every radio and the timekeeper in many oscillators. The equation $Lq'' + Rq' + q/C = 0$ looks heavy, but it's three pieces: $Lq''$ (the inductor resisting changes in current, like mass), $Rq'$ (the resistor bleeding off energy, like friction) and $q/C$ (the capacitor pushing the charge back, like a spring). It's the spring equation from differential equations in different clothes; bear with it.

The scene. Charge a capacitor, disconnect the battery and connect the capacitor across a coil. It starts to discharge through the coil, but the inductor won't let the current change suddenly, so the current builds gradually. By the time the capacitor is empty the current is at its largest, and the inductor keeps it flowing, piling charge onto the other plate. Now the capacitor is charged backward, and the whole thing runs in reverse. Charge sloshes back and forth like a mass on a spring: the capacitor is the spring and the inductor is the mass.

The equation. Let $q$ be the charge on one plate and $i = dq/dt$ the current flowing onto it. Going around the loop, the voltages across the inductor, resistor and capacitor add to zero (Kirchhoff's voltage law):

Put it next to the spring of mechanical vibrations, $mu'' + \gamma u' + ku = 0$:

Same equation, same solutions: everything you know about springs is now a fact about circuits.

LC: pure oscillation. With $R = 0$ the equation is $q'' = -\frac{1}{LC}q$, so

Energy confirms the picture. With $q = Q_0\cos\omega_0t$ and $i = -\omega_0Q_0\sin\omega_0t$, the capacitor holds $\frac{Q_0^2}{2C}\cos^2\omega_0t$ and the inductor holds $\frac12L\omega_0^2Q_0^2\sin^2\omega_0t = \frac{Q_0^2}{2C}\sin^2\omega_0t$, using $L\omega_0^2 = 1/C$. They add to the constant $\frac{Q_0^2}{2C}$. The energy crosses from one field to the other twice per cycle, and when the capacitor is empty the current peaks at $I_{\max} = \omega_0Q_0$.

Resistance drains it. Differentiate the total energy and use the circuit equation:

Think about what that says. The resistor takes energy only while charge is flowing, so the total falls in steps, pausing each time the current passes through zero. There's an animation further down that shows those steps.

Three kinds of damping. Try $q = e^{rt}$: $Lr^2 + Rr + \frac1C = 0$, so

• $R < 2\sqrt{L/C}$: underdamped. The roots are complex, $-\alpha \pm j\omega_d$ (circuits write $j$ for $\sqrt{-1}$ and save $i$ for current), so $q = e^{-\alpha t}(c_1\cos\omega_dt + c_2\sin\omega_dt)$: ringing inside a shrinking envelope.
• $R = 2\sqrt{L/C}$: critically damped. A repeated root gives $q = (c_1 + c_2t)e^{-\alpha t}$, the fastest return to zero without overshooting.
• $R > 2\sqrt{L/C}$: overdamped. Two negative real roots: a slow creep back with no oscillation.

For light damping the envelope falls like $e^{-\alpha t} = e^{-Rt/2L}$ and the energy like $e^{-Rt/L}$, while the ringing frequency $\omega_d$ sits just below $\omega_0$.

Initial conditions come from continuity. A capacitor's voltage and an inductor's current can't jump, so $q(0)$ is the charge the capacitor already had and $q'(0) = i(0)$ is the current the inductor already carried. A capacitor charged to $V_0$ and switched onto an idle coil starts with $q(0) = CV_0$ and $q'(0) = 0$.

As a system. Writing $q' = i$ and $i' = -\frac{1}{LC}q - \frac{R}{L}i$ makes it a two-dimensional linear system. In the $(q, i)$ plane an LC circuit's state runs around an ellipse of constant energy, and with resistance it spirals inward, governed by the complex eigenvalues $-\alpha \pm j\omega_d$. Drive the circuit with an AC source instead of leaving it alone and you get resonance when the drive matches $\omega_0$: that's Resonance and Filters, in the AC circuits unit coming up next.

A common mistake. It's tempting to use the capacitor's starting voltage as the second initial condition, since that's the number you're usually given. It isn't: the first condition is the starting charge, $q(0) = CV_0$, and the second is the starting current, $q'(0) = i(0)$, which the inductor carries over from before the switch. Two other slips: $f_0 = \frac{1}{2\pi\sqrt{LC}}$ is in hertz while $\omega_0 = \frac{1}{\sqrt{LC}}$ is in rad/s, and the critical resistance is $2\sqrt{L/C}$, not $\sqrt{L/C}$.

Every radio, television and phone tunes with LC circuits: the antenna delivers every station at once, and the tank circuit responds only near its own $\omega_0$. LC and crystal oscillators set the clock frequencies of processors and Wi-Fi chips. Power engineers fight the unwanted version, since stray inductance and capacitance in a fast-switching converter ring like an RLC circuit, and they add a damping resistor (a snubber) sized near critical to kill the ringing. Defibrillators shape their pulse with a deliberately damped RLC discharge, and a car's shock absorbers solve the same equation in steel and oil.