Damped Oscillations in an RLC Circuit — Question 1

Question 1

A released charged RLC circuit needs a sine term even when initial current is zero

A source-free series RLC loop is underdamped. Define $\alpha=R/(2L)$, $\omega_0=1/\sqrt{LC}$, and $\omega_d=\sqrt{\omega_0^2-\alpha^2}$. Initially $q=Q_0$ and $i=0$.

1. Determine the constants in $q=e^{-\alpha t}(A\cos\omega_dt+B\sin\omega_dt)$.

2. Differentiate to obtain the exact current.

Question 1 - Solution

A released charged RLC circuit needs a sine term even when initial current is zero

(a) Method and calculation

$$q(0)=A=Q_0,\quad i(0)=-\alpha A+\omega_dB=0.$$$$\boxed{q=Q_0e^{-\alpha t}[\cos\omega_dt+(\alpha/\omega_d)\sin\omega_dt]}.$$

(b) Interpretation and further calculation

$$i=Q_0e^{-\alpha t}[-\omega_d-\alpha^2/\omega_d]\sin\omega_dt.$$$$\boxed{i=-\frac{Q_0\omega_0^2}{\omega_d}e^{-\alpha t}\sin\omega_dt}.$$

Physical check

A decaying cosine alone has nonzero initial current and would violate the stated release condition.