Question 2.10: Write an independent set of equations for the node voltages ...
Write an independent set of equations for the node voltages shown in Figure 2.28.
First, we write KCL equations at each node, including the current of the controlled source just as if it were an ordinary current source:
\frac{v_1-v_2}{R_1} = i_s + 2i_x (2.39)
\frac{v_2-v_1}{R_1} + \frac{v_2}{R_2}+ \frac{v_2-v_3}{R_3} = 0 (2.40)
\frac{v_3-v_2}{R_3} + \frac{v_3}{R_4} + 2i_x= 0 (2.41)
Next, we find an expression for the controlling variable i_x in terms of the node voltages. Notice that i_x is the current flowing away from node 3 through R_3. Thus, we can write
i_x=\frac{v_3-v_2}{R_3} (2.42)
Finally, we use Equation 2.42 to substitute into Equations 2.39, 2.40, and 2.41.
Thus, we obtain the required equation set:
\frac{v_1-v_2}{R_1} = i_s + 2\frac{v_3-v_2}{R_3} (2.43)
\frac{v_2-v_1}{R_1} + \frac{v_2}{R_2}+ \frac{v_2-v_3}{R_3} = 0 (2.44)
\frac{v_3-v_2}{R_3} + \frac{v_3}{R_4} + 2\frac{v_3-v_2}{R_3}= 0 (2.45)
Assuming that the value of i_s and the resistances are known, we could put this set of equations into standard form and solve for v_1, v_2, and v_3.