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There are plenty of topics about this question, but I did not find a precise answer.

Let's consider two points A and B of a generic circuit and suppose you want to find the equivalent resistance between them. It is possible to evaluate it by inserting an ideal voltage source Vx and by evaluating the ratio Vx/Ix.

I have some doubts about the following steps that are usually performed:

1) Independent Voltage Sources are short - circuited. I read on the web that this is due to the fact that a resistance is the ratio ∆V/∆I, that means that if V changes, then I changes proportionally. So, since ∆V = 0 (voltage does not change in an independent voltage source), we get that its resistance is 0, that means that it is equivalent to a short (for the purpose of evaluating a resistance).

But it is a qualitative answer. What I am looking for is a proof that if we replace it with a short, we get a circuit that is equivalent for the calculation of the resistance between A and B. It does not seem true to me if for instance I consider the following circuit:

enter image description here

Let's put a voltage source Vx and let's call Ix the current provided by it.

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By the KVL we get:

V1 + R1×Ix - Vx = 0

that means Rab = Vx / Ix = R1 + V1/Ix. In this case I do not know how to proceed.

If instead we short V1, we ger Rab = Vx/Ix = R1. But how can we be sure that the operation of replacing V1 with a short is correct? If it is, we should get the same result of that of the circuit with V1, but we get different equations.

2) Independent Current Source are replaced by open circuits. Also in this case: where is the precise proof that this substitution produces a result which is exactly the same of the original circuit?

3) Dependent Sources are not turned off. Why? Can you give me an example in which it is easy to understand that if we turn off them, we make a mistake?

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