Forced Oscillations — Question 8

Question 8

The energy stored each cycle returns to its initial value in steady forcing

A damped driven circuit obeys $L\dot i+Ri+q/C=v_s(t)$ with $i=\dot q$. Its steady state is periodic with the source period T.

1. Derive an exact instantaneous energy balance.

2. Integrate over a cycle and find the work supplied when $i=I_p\cos(\omega t-\phi)$.

Question 8 - Solution

The energy stored each cycle returns to its initial value in steady forcing

(a) Method and calculation

$$i(L\dot i+q/C)=v_si-Ri^2.$$$$\boxed{\dot U=v_si-Ri^2,\quad U=Li^2/2+q^2/(2C)}.$$

(b) Interpretation and further calculation

$$\Delta U_T=0\Rightarrow W_{s,T}=\int_0^TRi^2dt.$$$$\boxed{W_{s,T}=RI_p^2T/2}.$$

Physical check

The source replaces dissipation each cycle; it does not continually increase stored energy after transients vanish.