Differential equation of the forced RLC circuit

Let us connect a sinusoidal generator, a resistor $R$, a capacitor $C$ and an inductor $L$ in series. Going round the circuit in the direction of the current $i$ and applying Kirchhoff’s law: the sum of the potential differences must be zero.

We agree that $Q(t)$ is the charge accumulated on the capacitor and that the current leaves the capacitor, so $i = -\,dQ/dt$. Going round the circuit in the direction of $i$:

• generator: $+\Delta V_\text{gen}(t)$ (potential gain);
• resistor: $-R\,i$ (drop);
• capacitor: $+Q/C$ (the accumulated charge pushes the current in the direction traversed);
• inductor: $-L\,di/dt$ (drop due to self-induction).

Substituting $i = -dQ/dt$ and $di/dt = -d^2Q/dt^2$:

$\Delta V_\text{gen}(t) + R\,\frac{dQ}{dt} + \frac{Q}{C} + L\,\frac{d^2Q}{dt^2} = 0$

Differential equation of the forced RLC circuit

$\ev{L\,\ddot{Q} + R\,\dot{Q} + \frac{Q}{C} = -\Delta V_\text{gen}(t)}$ where the dot denotes the derivative with respect to time. The forcing term is the generator’s voltage; when $\Delta V_\text{gen} = 0$ this reduces to the free damped oscillation.

The equation is formally identical to that of a driven, damped pendulum, with the dictionary: $L \leftrightarrow m$ (mass), $R \leftrightarrow b$ (damping), $1/C \leftrightarrow k$ (stiffness), $-\Delta V_\text{gen} \leftrightarrow F_\text{esterna}$ (driving force).

As with any driven oscillator, the amplitude of the response is greatest when the generator oscillates at the natural frequency of the LC circuit. This resonance frequency is:

$\ev{\omega_0 = \frac{1}{\sqrt{LC}}}$

Collegamenti

Argomenti: Electromagnetic induction Concetti: Alternating current · LC and RLC oscillations · Resonance

Esercizi collegati: Series RLC resonance and quality factor · Problem — Natural frequency of an LC circuit · Problem — Reactances of an RLC as omega varies