Question 7.2: Develop the input–output differential equation relating eo t...

Here the loop method is used and the elemental equations are developed as needed for each loop. Like the use of the node method, the use of the loop method helps to eliminate quickly the unwanted variables. The two loops, I and II, are chosen as shown in Fig. 7.12, carrying the loop currents i_{I} and i_{II}. Loop I is needed only to note that i_{I} =i_{s}. Using Kirchhoff’s voltage law for loop II yields

R_{2} i_{II} +L\frac{di_{II} }{dt} +\left(\frac{1}{C} \right) \int{i_{II}dt } +R_{1} \left(i_{II}-i_{s} \right) =0.          (7.36)

Because i_{II} ={Cde_{o} }/{dt}, substituting for i_{II} in Eq. (7.36) yields 

\left(R_{1}+R_{2} \right) C\frac{de_{o} }{dt} +LC\frac{d^{2}e_{ o} }{dt^{2} } +e_{o} =R_{1 } i_{s} .          (7.37)

Rearranging, we have

LC\frac{d^{2}e_{ o} }{dt^{2} }+\left(R_{1}+R_{2} \right) C\frac{de_{o} }{dt} +e_{o} =R_{1 } i_{s} .          (7.38)