RL Circuits — Question 7

Question 7

The resistance seen by an inductor sets its time constant in a branched circuit

A source V feeds series resistance Rs into a node connected to ground through resistor R and ideal inductor L in parallel. The source steps on at t=0 with zero inductor current.

1. Find the Thevenin voltage and resistance seen by L, then its current response.

2. Find initial current slope and final inductor current.

Question 7 - Solution

The resistance seen by an inductor sets its time constant in a branched circuit

(a) Method and calculation

$$V_{\mathrm{th}}=\frac{VR}{R_s+R},\quad R_{\mathrm{th}}=R_s\parallel R=\frac{R_sR}{R_s+R}.$$$$\boxed{i_L(t)=\frac{V}{R_s}(1-e^{-t/\tau}),\quad\tau=\frac{L(R_s+R)}{R_sR}}.$$

(b) Interpretation and further calculation

$$\boxed{\dot i_L(0^+)=V_{\mathrm{th}}/L=\frac{VR}{L(R_s+R)}}.$$$$\boxed{i_L(\infty)=V/R_s};\quad v_{\mathrm{node}}(\infty)=0.$$

Physical check

At steady state the ideal inductor shorts the parallel resistor; the transient resistance is nevertheless Rs in parallel with R.