Forced Oscillations — Question 5

Question 5

Transient suppression takes a calculable number of decay times

For an underdamped driven RLC circuit, the difference between its full charge and its sinusoidal steady state has envelope $A_h e^{-Rt/(2L)}$. Let L=.300 H and R=12.0 ohms.

1. Find the time required to reduce this envelope to one percent of its initial value.

2. How does the time change if R is doubled while the response remains underdamped?

Question 5 - Solution

Transient suppression takes a calculable number of decay times

(a) Method and calculation

$$e^{-Rt/(2L)}=.0100\Rightarrow t=(2L/R)\ln100.$$$$\boxed{t=.0500\ln100=.230259\,s}.$$

(b) Interpretation and further calculation

$$t\propto1/R\quad\text{for a specified envelope ratio}.$$$$\boxed{t_{2R}=t_R/2=.115129\,s}.$$

Physical check

The criterion concerns the transient envelope, avoiding misleading ratios at zero crossings.