Forced Oscillations — Question 6

Question 6

Switching on a resonant source requires a natural transient to enforce zero initial current

An underdamped series RLC circuit starts at q=i=0 when $v_s=V_p\cos\omega_0t$ is applied. Let $\alpha=R/(2L)$ and $\omega_d=\sqrt{\omega_0^2-\alpha^2}$.

1. Combine the resonant steady charge with a homogeneous term to meet both initial conditions.

2. Check the initial current derivative against the circuit equation.

Question 6 - Solution

Switching on a resonant source requires a natural transient to enforce zero initial current

(a) Method and calculation

$$q_s=\frac{V_p}{R\omega_0}\sin\omega_0t,\quad q_h=e^{-\alpha t}(A\cos\omega_dt+B\sin\omega_dt).$$$$\boxed{q=\frac{V_p}{R\omega_0}\sin\omega_0t-\frac{V_p}{R\omega_d}e^{-\alpha t}\sin\omega_dt}.$$

(b) Interpretation and further calculation

$$q(0)=0,\quad\dot q(0)=V_p/R-V_p/R=0.$$$$\boxed{\ddot q(0)=2\alpha V_p/R=V_p/L}.$$

Physical check

Using the steady sinusoid alone would incorrectly create an instantaneous current jump.