Two Circuits Two Ways

In this problem, we'll analyze two circuits using two methods each: superposition, as well as the direct application of the node method. You should compare for yourself the work required to analyze each circuit by these two methods. It is very useful to have multiple approaches to analyzing a circuit; not only does it open multiple avenues for solving a given circuit, but it can also give us a useful independent check of our results.

1) Circuit 1

Consider the following circuit, with a reference node and two node potentials labeled:

Using Superposition:
Solve the circuit using superposition. First, find expressions for each of the node potentials (e_1 and e_2) when only the voltage source is active (i.e., when I_s = 0{\rm A}). Then find expressions for each of the node potentials when only the current source is active (i.e., when V_s = 0{\rm V}). Finally, find expressions for each of the node potentials when both sources are active (how should these relate to your previous answers?).

Using Direct Application of the Node Method:
Solve the circuit using the node method (without superposition) by writing KCL equations at each of e_1 and e_2 in terms of node potentials. Show that solving these equations (solve for one variable and plug into the other equation) produces the same results as the superposition method.

2) Circuit 2

Now repeat that process for the following circuit, which again has the node potentials labeled for you:

Using Superposition:
Solve the circuit using superposition. First, find expressions for each of the node potentials (e_1, e_2, and e_3) when only one of the current sources is active (e.g., when I_2 = 0{\rm A}). Then find expressions for each of the node potentials when only the other current source is active (e.g., when I_1 = 0{\rm A}). Finally, find expressions for each of the node potentials when both sources are active (how should these relate to your previous answers?).

Using Direct Application of the Node Method:
Solve the circuit using the node method (without using superposition) by writing KCL equations at each of e_1, e_2, and e_3 in terms of node potentials. Show that solving these equations produces the same results as the superposition method.

 
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