RL Circuits — Question 3

Question 3

Winding resistance changes both the final current and the rise time

A coil with inductance L=.240 H and winding resistance r=2.00 ohms is placed in series with external resistance R=4.00 ohms and a 9.00-V step source. Initial current is zero.

1. Find current as a function of time.

2. Find the steady terminal voltage across the complete coil and the error in the time constant if r is ignored.

Question 3 - Solution

Winding resistance changes both the final current and the rise time

(a) Method and calculation

$$L\dot i+(R+r)i=V\Rightarrow\boxed{i=1.50(1-e^{-t/.0400})\,A}.$$$$\tau=L/(R+r)=.0400\,s,\quad i_\infty=1.50\,A.$$

(b) Interpretation and further calculation

$$v_{\mathrm{coil,\infty}}=ri_\infty=\boxed{3.00\,V}.$$$$\tau_{\mathrm{naive}}=L/R=.0600\,s\Rightarrow\boxed{(\tau_{\mathrm{naive}}-\tau)/\tau=50.0\%}.$$

Physical check

An ideal inductor has zero steady voltage, but a real coil retains the voltage drop across its winding resistance.