Using Fig. 7.92, design a problem to help other students better understand source-free RL...

For , the switch in the circuit is closed and inductor acts as short circuit. Draw the modified circuit diagram:

Figure 2

Calculate the initial current through the inductor.

For , the switch in the circuit is open. Draw the modified circuit diagram:

Figure 3

The circuit shown in Figure 2 is a source free circuit. Write an expression for \(R L\) source free circuit.

$$ i(t)=i(0) e^{\frac{1}{r}} \cdots \ldots(1) $$

Calculate the time constant of \(R L\) circuit at \(t>0\).

$$ \begin{aligned} \tau &=\frac{l}{R_{\mathrm{cq}}} \\ &=\frac{L}{R_{2}} \end{aligned} $$

Substitute \(\frac{v}{R_{1}}\) for \(i(0)\) and \(\frac{L}{R_{2}}\) for \(\tau\) in the equation (1).

$$ \begin{aligned} i(t) &=\left(\frac{v}{R_{1}}\right) e^{\frac{1}{\left(\frac{L}{R_{2}}\right)}} \\ &=\frac{v}{R_{1}} e^{\frac{R_{2} t}{L}} \end{aligned} $$

Hence, the current through inductor \(i(t)\) for \(t>0\) is \(\frac{v}{R_{1}} e^{-\frac{R_{t}}{L}}\).