(A) Write a Mesh Equation for Mesh 1 (Left Side)

Essay

$$\text { In the circuit shown below, } i \text { is the current through } \mathrm { R } _ { 2 } \text { and } v \text { is the voltage across } \mathrm { R } _ { 2 } \text {. }$$
(a) Write a mesh equation for mesh 1 (left side) by summing the voltage drops around mesh 1 in a clockwise direction.
(b) Write a mesh equation for mesh 2 (right side) by summing the voltage drops around mesh 2 in a clockwise direction.
(c) Solve the mesh equations to find $\mathrm { I } _ { 2 }$ .
(d) Find the Thévenin equivalent voltage $V _ { \text {th, } }$ , which is the voltage across $R _ { 3 }$ .
(e) Deactivate the voltage source $\mathrm { V } _ { \mathrm { s } }$ and apply a test voltage of $1 \mathrm {~V}$ between $a$ and $b$ to find the Thévenin resistance $R _ { \text {th. } }$ .