Chapter 13, Problem 1P

To determine

Derive the inductor current in s-domain as shown in given equivalent circuit.

Answer to Problem 1P

The inductor current in s-domain for the given equivalent circuit is derived as $\frac{V\left(s\right)}{sL}+\frac{I_0}{s}$.

Explanation of Solution

Given data:

Refer to the given circuit in textbook.

The inductor current is a function of terminal voltage.

Calculation:

Write a general expression to calculate the inductor current in time domain.

$i=\frac{1}{L}{\displaystyle \underset{0}{\overset{t}{\int }}vdt}+i_0$        (1)

Here,

$L$ is the value of inductance,

$v$ is the terminal voltage, and

$i_0$ is the initial current.

Apply Laplace transform for equation (1) to find $I\left(s\right)$.

$\begin{array}{c}I\left(s\right)=\frac{1}{L}\left(\frac{V\left(s\right)}{s}\right)+\frac{I_0}{s}\left\{\begin{array}{l}\because \mathcal{L}\left({\displaystyle \underset{0}{\overset{t}{\int }}f\left(t\right)dt}\right)=\frac{1}{s}F\left(s\right)\\ \mathcal{L}\left(1\right)=\frac{1}{s}\end{array}\right\}\\ =\frac{V\left(s\right)}{sL}+\frac{I_0}{s}\end{array}$

Conclusion:

Thus, the inductor current in s-domain for the given equivalent circuit is derived as $\frac{V\left(s\right)}{sL}+\frac{I_0}{s}$.