Forced Oscillations — Question 1

Question 1

A sinusoidal drive determines two quadrature coefficients

A driven series circuit obeys $L\ddot q+R\dot q+q/C=V_p\cos\omega t$, with R positive. Seek the steady state $q=A\cos\omega t+B\sin\omega t$. Define $D=1/C-L\omega^2$.

1. Solve for A and B by matching sine and cosine terms.

2. Find the charge amplitude and phase lag delta, with $q=Q\cos(\omega t-\delta)$.

Question 1 - Solution

A sinusoidal drive determines two quadrature coefficients

(a) Method and calculation

$$DA+R\omega B=V_p,\quad DB-R\omega A=0.$$$$\boxed{A=\frac{V_pD}{D^2+(R\omega)^2},\quad B=\frac{V_pR\omega}{D^2+(R\omega)^2}}.$$

(b) Interpretation and further calculation

$$Q=\sqrt{A^2+B^2},\quad\cos\delta=A/Q,\quad\sin\delta=B/Q.$$$$\boxed{Q=\frac{V_p}{\sqrt{D^2+(R\omega)^2}},\quad\delta=\operatorname{atan2}(R\omega,D)}.$$

Physical check

The two-argument phase selects the correct quadrant above resonance.