Setting up the equation for the series RLC circuit

The series RLC circuit

A series RLC circuit consists of a resistance R, a coil of inductance L, and a capacitor of capacitance C, connected in series with no generator (free circuit). The capacitor is initially charged and then the switch is closed: the system evolves on its own, with no external energy input.

Setting up the equation

Kirchhoff's voltage law gives, with q(t) the charge of the capacitor and i = dq/dt:

Ld^2q/dt^2 + Rdq/dt + q/C = 0

This is a second-order linear differential equation with constant coefficients and no right-hand side.

Canonical form

Two characteristic parameters are introduced:

• omega0 = 1/sqrt(L*C): natural angular frequency of the circuit
• Q = (1/R)*sqrt(L/C): quality factor

The equation is then rewritten in canonical form:

d^2q/dt^2 + (omega0/Q)dq/dt + omega0^2q = 0

Characteristic equation

Looking for solutions of the form exp(r*t), we obtain the characteristic equation:

r^2 + (omega0/Q)*r + omega0^2 = 0

The sign of the discriminant Delta = (omega0/Q)^2 - 4*omega0^2 determines the nature of the transient regime, studied in the next lesson.

Common pitfall

Do not confuse the charge q(t) with the current i(t) = dq/dt: writing Kirchhoff's voltage law directly in terms of i without going back through q is a frequent source of sign errors on the resistive term.