RLC circuit: damped oscillations

If we also add a resistor $R$ to the LC circuit, things become more complicated: the energy, besides oscillating between electrical and magnetic form, is also dissipated as heat. Kirchhoff’s loop equation becomes:

$\text{FEM}_\text{gen} - R\,i - \frac{Q}{C} - L\,\frac{di}{dt} = 0$

With $i = dQ/dt$ (choosing the convention in which the current enters the capacitor) and $di/dt = d^2Q/dt^2$, we obtain a differential equation in $Q(t)$:

$L\,\frac{d^2Q}{dt^2} + R\,\frac{dQ}{dt} + \frac{Q}{C} = \text{FEM}_\text{gen}$

Example — An RLC circuit with concrete coefficients

Consider $L = 2$ H, $R = 1\;\Omega$, $C = 3$ F, with constant EMF $\Delta V_\text{gen} = 4$ V and initial condition $Q(0) = 0$. We look for a solution of the form $Q(t) = a + b\,e^{-\gamma t}$: an exponential “transient tail” superposed on the steady-state value.

The steady-state value ($t\to\infty$) is $a = C\cdot\Delta V_\text{gen} = 12$ C (the capacitor charges up until it matches the generator’s voltage). The exponential part $b\,e^{-\gamma t}$ must satisfy the associated homogeneous equation: $L\gamma^2 - R\gamma + \frac{1}{C} = 0 \quad\Rightarrow\quad 2\gamma^2 - \gamma + \frac{1}{3} = 0$ The discriminant is $1 - 8/3 < 0$: the roots are complex! $\gamma = \frac{1\pm i\sqrt{5/3}}{4} = \frac{1}{4} \pm i\,\frac{\sqrt{15}}{12}$ Interpreting $e^{-\gamma t}$ with real and imaginary parts as a damped oscillation: $\ev{Q(t) = 12\left(1 - e^{-t/4}\cos\!\left(\frac{\sqrt{15}}{12}\,t\right)\right)}$ A function that starts at $Q = 0$, oscillates a couple of times around $12$, and finally settles at $12$ C. This is the regime of damped oscillations.

The complex root of the characteristic equation is the mathematical signature of the damped oscillation: the real part ($1/4$) governs the exponential decay of the amplitude, and the imaginary part ($\sqrt{15}/12$) governs the angular frequency of the residual oscillation.

Collegamenti

Argomenti: Electromagnetic induction Concetti: LC and RLC oscillations · Damped oscillations

Esercizi collegati: Problem — Natural frequency of an LC circuit · Problem — Capacitor of an AM radio receiver · Worked exercise — Forced RLC, find the generator’s EMF