Equivalent Circuits Primer
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Over the past week, we have been looking at equivalent circuits of various kinds. In particular, we found that a sequence of resistors connected in series (or in parallel) could be replaced by a single equivalent resistance in such a way that the constraint imposed by the combination on the current flowing through it and the voltage developing across it would be the same for the single resistor as it would for the combination.
For example, the following two circuits are equivalent in the sense that they both impose the constraint that v = i(R_1 + R_2):
Because they impose the same constraints on their respective currents and voltages, any external circuit hooking up to either of these configurations would not be able to tell them apart (the same currents and voltages would develop throughout that external circuit regardless of which of these boxes it was hooked up to).
We saw a similar relationship for resistors connected in parallel, and we could use those two in combinations to simplify down many (but not all!) networks of resistors to a single resistor.
In this exercise, we'll start to take another step, exploring the idea of other circuits that are equivalent to each other in this same sense (but that are not comprised only of resistors).
For example, let's consider the following two circuits, each of which has a voltage and a current labeled (on a single pair of exposed terminals):
Here, we'll take a look at connecting each of these circuits up to a few different components and see how v_1, i_1, v_2, and i_2 relate to each other.
Let's first consider hooking each circuit up to a single 5{\rm k}\Omega resistor:
Now let's instead hook each circuit up to a 7.5{\rm V} voltage source instead:
Finally, let's instead hook each circuit up to a 3{\rm mA} current source instead:
This is really interesting! For everything that we connected up, both circuits had precisely the same relationship between v and i: these circuits are equivalent in exactly the same sense that our resistive circuits from the top of this writeup are: they both exert precisely the same constraint on v and i.
This constraint can be expressed as v_1 = A + Bi_1 (or, for the same values of A and B), v_2 = A + Bi_2. What are the values and units of A and B?
What are the units of A in the formula above? What are the units of B in the formula above?
In this problem, we've seen an example of applying the same ideas we saw with resistors last week (in particular, the notion of equivalence) to circuits consisting of more than just resistors! This is a really powerful idea that will further help us simplify the process of solving circuits.
In particular, next week, we'll develop a systematic way of finding simpler equivalents for arbitrary linear circuits, and we'll see some ways in which we can use that strategy to simplify solving bigger circuits.