Damped Oscillations in an RLC Circuit — Question 4

Question 4

Large resistance creates a slow capacitive tail as well as a fast inductive transient

An overdamped series RLC circuit has $R^2\gg4L/C$. Its charge modes are proportional to $e^{s_\pm t}$.

1. Find the exact roots and their large-R approximations.

2. Identify the two time constants and explain why increasing R does not make the slow discharge faster.

Question 4 - Solution

Large resistance creates a slow capacitive tail as well as a fast inductive transient

(a) Method and calculation

$$Ls^2+Rs+1/C=0\Rightarrow s_\pm=\frac{-R\pm\sqrt{R^2-4L/C}}{2L}.$$$$\boxed{s_+\simeq-1/(RC),\quad s_-\simeq-R/L}.$$

(b) Interpretation and further calculation

$$\tau_{\mathrm{slow}}=-1/s_+\simeq RC,\quad\tau_{\mathrm{fast}}=-1/s_-\simeq L/R.$$$$\boxed{R\uparrow\Rightarrow\tau_{\mathrm{slow}}\uparrow,\quad\tau_{\mathrm{fast}}\downarrow}.$$

Physical check

Two distinct relaxation processes coexist; the late-time behavior is governed by the slower root.