A connected planar circuit occupies two adjacent rectangular windows whose shared…

A connected planar circuit occupies two adjacent rectangular windows whose shared vertical branch is a voltage-controlled voltage source. The bottom rail is ground. Label the top-left node \(A\), the top of the shared source \(B\), and the top-right node \(C\). The branches are as follows: - an independent \(12\,\mathrm{V}\) voltage source from ground to \(A\), with the positive terminal at \(A\); - a \(3\,\Omega\) resistor from \(A\) to \(B\); - a voltage-controlled voltage source occupying the entire shared vertical from ground to \(B\), with the positive terminal at \(B\) and value \(0.5 v_x\), where \(v_x\) is the voltage across the \(3\,\Omega\) resistor with positive polarity at \(A\) (so \(v_x=v_A-v_B\)). The voltage gain is the dimensionless factor \(0.5\); - a \(6\,\Omega\) resistor from \(B\) to \(C\); - a \(2\,\Omega\) resistor from \(C\) to ground. Take clockwise mesh current \(I_1\) in the left window and clockwise mesh current \(I_2\) in the right window. The current through the controlled source, directed downward, is \(I_1-I_2\). (a) Express \(v_x\) in terms of mesh currents, and write the two mesh-current KVL equations. (b) Solve for \(I_1\) and \(I_2\). (c) Determine \(v_B\) and the current leaving the positive terminal of the controlled source.

A shared voltage source — here a VCVS — does not create a supermesh; write KVL in each window.