RL Circuits — Question 5

Question 5

A finite charging pulse followed by a different discharge resistance has two time constants

An initially unenergized inductor L is charged from source V through Rc for time T. It is then disconnected from the source and closed through Rd. Switching keeps a continuous current path and introduces no impulse voltage.

1. Find current at the end of charging and during the subsequent discharge.

2. Find the total charge through Rd after switching.

Question 5 - Solution

A finite charging pulse followed by a different discharge resistance has two time constants

(a) Method and calculation

$$\tau_c=L/R_c,\quad\boxed{i_T=\frac V{R_c}(1-e^{-T/\tau_c})}.$$$$\boxed{i(T+s)=i_Te^{-s/\tau_d},\quad\tau_d=L/R_d,\quad s\ge0}.$$

(b) Interpretation and further calculation

$$Q_d=\int_0^\infty i_Te^{-s/\tau_d}ds=i_T\tau_d.$$$$\boxed{Q_d=\frac{LV}{R_cR_d}(1-e^{-TR_c/L})}.$$

Physical check

Current is continuous at the switching time, but its derivative generally changes because the driving circuit changes.