Superposition Preliminaries

The questions below are due on Tuesday February 18, 2025; 10:00:00 PM.
 
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In this exercise, we'll use what we've learned over the past week to examine a critical property of linear circuits. Over the course of the next week, we'll develop a strategy for simplifying the process of solving linear circuits by taking advantage of the property described below.

Let's start by looking at an example circuit, comprised of two independent sources and four resistors:

Here, we've started solving this circuit for you by choosing the bottom node in the circuit as our reference point. Go ahead and solve for the node potentials e_1 and e_2 in terms of the labeled values using the node method, and use the result to solve for the current i_1.

It turns out that the node potential e_1 can be written as an expression of the form e_1 = A_1 V_o + B_1 I_o for some values of A_1 and B_1 expressed in terms of the resistances in the circuit. What are A_1 and B_1?
 
Enter your answers as expressions (using Python syntax, so * for multiplication and / for division and () for grouping) in terms of R_1, R_2, R_3, and R_4.
 
  

  

It turns out that the node potential e_2 can also be written as an expression of the form e_2 = A_2 V_o + B_2 I_o for some values of A_2 and B_2 expressed in terms of the resistances in the circuit. What are A_2 and B_2?
 
Enter your answers as expressions (using Python syntax, so * for multiplication and / for division and () for grouping) in terms of R_1, R_2, R_3, and R_4.
 
  

  

What's more, we can also write i_1 as an expression of the form i_1 = A_3 V_o + B_3 I_o for some values of A_3 and B_3 expressed in terms of the resistances in the circuit. What are A_3 and B_3?
 
Enter your answers as expressions (using Python syntax, so * for multiplication and / for division and () for grouping) in terms of R_1, R_2, R_3, and R_4.
 
  

  

Beyond those simple examples, if we continued solving for voltages and currents throughout the circuit, every one of those values would end up being expressible as AV_o + BI_o for some values of A and B, which is kind of a cool result.

And this is not just a property of this circuit; for any network consisting of the components we've seen so far (independent current sources, independent voltage sources, and constant resistors), the solution for any branch current or branch voltage will be a linear combination of the strengths of the independent sources in the circuit.

For example, in the circuit below, every branch current and branch voltage would be expressible in the form AV_1 + BI_1 + CV_2 for some values of A, B, and C that depend only on the resistances in the circuit.

For now, this may just seem like an interesting coincidence/curiosity. But over the course of the next week, we'll see that this property is something that we can exploit, to simplify the process of solving a large category of circuits, using a method called superposition.