Worked exercise — Driven RLC, find the generator EMF

Problem

In a series circuit with $R=3\ \Omega$, $C=2\ \text{F}$, $L=4\ \text{H}$, the charge on the capacitor is $Q(t)=3\sin(2t)$. Find the generator voltage $\Delta V_{\text{gen}}(t)$ and check the energy balance.

Solution

Derivatives of the charge. $\dot Q=6\cos(2t),\qquad \ddot Q=-12\sin(2t)$

Kirchhoff’s law (series loop). With $i=-\dot Q$, summing the drops across $L$, $R$, $C$: $\Delta V_{\text{gen}}=-L\,\ddot Q-R\,\dot Q-\frac{Q}{C}=-4(-12\sin 2t)-3\cdot 6\cos 2t-\frac{3\sin 2t}{2}$ $\Delta V_{\text{gen}}(t)=48\sin 2t-18\cos 2t-\tfrac{3}{2}\sin 2t=-18\cos(2t)+\tfrac{93}{2}\sin(2t)$ $\ev{\Delta V_{\text{gen}}(t)=-18\cos(2t)+\tfrac{93}{2}\sin(2t)}$

Energy balance. The current is $i=-\dot Q=-6\cos(2t)$. Stored energy: $E(t)=\frac{Q^2}{2C}+\frac{L\,i^2}{2}=\frac{9\sin^2 2t}{4}+\frac{4\cdot 36\cos^2 2t}{2}=\tfrac{9}{4}\sin^2 2t+72\cos^2 2t$ $\ddt{E}=\tfrac{9}{4}\cdot 4\sin 2t\cos 2t-72\cdot 4\sin 2t\cos 2t=(9-288)\sin 2t\cos 2t=-279\sin 2t\cos 2t$

Power delivered by the generator and dissipated by $R$: $P_{\text{gen}}=\Delta V_{\text{gen}}\cdot i=\left(-18\cos 2t+\tfrac{93}{2}\sin 2t\right)(-6\cos 2t)=108\cos^2 2t-279\sin 2t\cos 2t$ $P_R=R\,i^2=3\cdot 36\cos^2 2t=108\cos^2 2t$ $P_{\text{gen}}-P_R=-279\sin 2t\cos 2t=\ddt{E}\quad\checkmark$ The balance checks out: the net power supplied minus that dissipated in $R$ equals the rate of change of the energy stored in $L$ and $C$.

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Topics: Induzione elettromagnetica Concepts: Corrente alternata · Oscillazioni LC e RLC · Effetto Joule