Energy balance of the RLC circuit

The energy stored in the circuit is the sum of the electrical energy in the capacitor and the magnetic energy in the inductor:

$\ev{E(t) = \frac{Q^2}{2C} + \frac{L\,i^2}{2}}$

Differentiating with respect to time using $i = -dQ/dt$:

$\begin{aligned} \frac{dE}{dt} &= \frac{Q}{C}\,\dot{Q} + L\,i\,\dot{\imath} \\ &= -\frac{Q}{C}\,i + L\,i\,\dot{\imath} \end{aligned}$

From Kirchhoff’s equation, isolating $L\,di/dt$: $L\,\dot{\imath} = \Delta V_\text{gen} - R\,i + Q/C$. Substituting:

$\frac{dE}{dt} = -\frac{Q}{C}\,i + i\left(\Delta V_\text{gen} - R\,i + \frac{Q}{C}\right) = \Delta V_\text{gen}\cdot i - R\,i^2$

Principle — Energy balance of the RLC circuit

$\ev{\frac{dE}{dt} = P_\text{gen} - P_R}$ with $P_\text{gen} = \Delta V_\text{gen}\cdot i$ (power delivered by the generator, positive when it feeds energy into the circuit) and $P_R = R\,i^2 \geq 0$ (power dissipated by Joule heating in the resistor). If $dE/dt > 0$ the circuit is storing energy; if $dE/dt < 0$ it is releasing it.

Key formula

$E = \frac{Q^2}{2C} + \frac{L\,i^2}{2}, \qquad \frac{dE}{dt} = \Delta V_\text{gen}\cdot i - R\,i^2$

The balance is transparent: the only source of energy is the generator, the only sink is the resistor. The capacitor and inductor dissipate nothing, they merely exchange energy back and forth; it is the resistance that settles the irreversible bill in heat.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Alternating current · Joule heating · Field energy density

Esercizi collegati: Worked exercise — Forced RLC, find the generator’s EMF · Bicycle dynamo · Worked exercise — loop entering and leaving a field