Chapter 5, Problem 5.1HP

Write the differential equations fort $t>0$ for $i_L$ and $v_3$ Figure P5.21. How are they related?

To determine

The differential equation for $t>0$ for $i_L$ and $v_3$. The relation between them.

Answer to Problem 5.1HP

The differential equation for $v_3$ is $\frac{d{v}_3}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)v_3=0$ and for $i_L$ is $\frac{d{i}_L}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)i_L=0$, also they are related as both the equations are identical except for the variables.

Explanation of Solution

Calculation:

The given diagram is shown in Figure 1

  

The circuit diagram for $t>0$ is shown below.

The required diagram is shown in Figure 2

  

Apply KCL at node $v_3$ for $t>0$ .

  $\begin{array}{l}\frac{v_3-0}{R_1+R_2}+\frac{1}{L}{\displaystyle {\int }_0^t{v}_{3}dt}+\frac{v_3-0}{R_3}=0\\ v_3\left(\frac{1}{R_1+R_2+R_3}\right)\frac{d{v}_3}{dt}+\frac{1}{L}{v}_3=0\\ \frac{d{v}_3}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)v_3=0\end{array}$ ...... (1)

The differential equation for t $t>0$ is $\frac{d{v}_3}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)v_3=0$ .

Apply KCL at node $v_3$ for $t>0$ for $i_L$ .

  $\frac{v_3-0}{R_1+R_2}+i_L+\frac{v_3-0}{R_3}=0$

Substitute $L\frac{d{i}_L}{dt}$ for $v_3$ in the above equation.

  $\begin{array}{l}\frac{L\frac{d{i}_L}{dt}}{R_1+R_2}+i_L+\frac{L\frac{d{i}_L}{dt}}{R_3}=0\\ L\frac{d{i}_L}{dt}\left(\frac{1}{R_1+R_2}+\frac{1}{R_3}\right)+i_L=0\\ \frac{d{i}_L}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)i_L=0\end{array}$

From above and from equation (1) it is clear that both the equations are identical except for the variable.

Conclusion:

Therefore, the differential equation for $v_3$ is $\frac{d{v}_3}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)v_3=0$ and for $i_L$ is $\frac{d{i}_L}{dt}+\left(\frac{\left(R_1+R_2\right)R_3}{L\left(R_1+R_2+R_3\right)}\right)i_L=0$, also the they are related as both the equations are identical expect for the variables.