My book states that two circuits can be considered equivalent if they produce the same voltage and current in any resistor connected to the two ends of the circuit. The book then goes about proving how various different circuits are interchangeable because they produce the same voltage and current in a resistor with arbitrary resistance connected to the two ends of said circuit. However, what is bothering me is that this strategy seems to assume that the rest of the circuit can be reduced to a single resistor. But clearly this strategy is used when the rest of the circuit cannot be reduced to a single resistor (such as when the rest of the circuit has a voltage or current source). Could someone please explain to me why this strategy is still valid in said situation?
Electronic – I do not completely understand the intuition behind equivalent sources used in source transformations
current-sourcenortonsourcetheveninvoltage divider
Related Solutions
I'm not going to just give you the answer to your homework problem.
However, consider what it really means to be a voltage source. Let's say I have a 6V source that is good to 1 A. What voltage is it when nothing is connected? When I put a 60 Ω resistor accross it? A 30 Ω resistor accross it?
If I put the 6 V source and resistor inside a black box and brought out only the two leads, what difference could someone observe on the outside to distinguish the no load, 60 Ω and 30 Ω cases? Now consider this is the case you have. Your +6V and R are inside the box and the rest of the circuit on the outside.
Redraw the circuit as follows to see why \$V_X\$ is across the 60 ohm resistor, the controlled current source and the output terminals:
It is often helpful to remember that two points connected by an ideal wire are the same circuit node which can be seen by 'shrinking' the wire to zero length as I've done above.
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Best Answer
There is an unstated assumption that the circuit is linear. It's easy to devise a nonlinear circuit that would look the same to any real value of resistor but fail when certain voltage sources were connected. Think of a series battery + resistor with a reverse-biased diode across the terminals.
Since the circuit is linear, the superposition theorem applies. That (and the uniqueness theorem) is adequate to prove Thevenin's theorem. It also applies more generally with resistors replaced by impedances.
Since it's linear it only has to behave the same with two values of resistor attached to be identical for all possible applied voltages to the two terminals (ignoring the degenerate case where the voltage is zero for both resistor values).