Thévenin’s theorem

Thévenin’s theorem states that if a part of a linear electrical circuit is comprised between two terminals A and B, this part in question can be replaced by an equivalent circuit that is constituted solely by a voltage generator in series with an impedance such that when connect an element between the two terminals A and B, the voltage that falls on it and the current that passes through it are the same both in the real circuit and in the equivalent. Thévenin’s theorem is the dual of Norton’s Theorem .

Summary

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• 1 Thévenin’s theorem
• 2 Statement of Theorem
• 1 Demonstration
• 3 Equivalence between Thevenin and Norton
• 4 Bibliography
• 5 Sources

Thévenin’s theorem

It is one of the most important and most widely applied. Let be a linear circuit, in which there can be everything, R, L, C, M, independent and dependent voltage and current sources. We distinguish two terminals A and B of that circuit and connect an external impedance Z

It is a matter of calculating the current flowing through this impedance, without solving the entire circuit. We make one more hypothesis: there is no mutual between Z and the rest of the circuit

1. Vacuum or Open Circuit Voltage: VAB It is the voltage that appears between A and B when there is no impedance Z It is what an “ideal” voltmeter would measure (ideal in the sense that when connected it does not modify the voltage that existed before between those points. We will already specify what this means). At Laplace , the vacuum voltage will be VAB (s).
2. View Impedance: ZAB To define it, let’s cancel all the sources. There remains a “passive” circuit (better said: without sources). What does “nullify the sources” mean? The voltage sources are short-circuited; the current ones open. Which? The independent and previous data; not so the dependents that are not generators but links. Once the sources are canceled, we apply a voltage source E between A and B.

A current circulates I. The E / I ratio, which does not depend on E, due to the linearity of the circuit and since E is the only source, is what is called impedance seen. ZAB (s) = E (s) / I (s) E (s) is any; we do not specify it. In simple cases, there is no need to calculate ZAB; it is enough with “looking” from A and B, and recognizing a combination (for example series and / or parallels) of simple impedances. There are, therefore, two methods to calculate ZAB: the definition or “look”.

Statement of the Theorem

“The current that passes through the impedance Z connected between terminals A and B is I = VAB /) ZAB + Z)”

In other words, regardless of what is inside the “black box”, if we know these two parameters VAB and ZAB, we are in a position to know what current is going to pass through any Z In particular, if we short-circuit A and B we have a current that we call short circuit: Icc = VAB / ZAB

Demonstration

It relies on the linearity of the circuit, which allows us to apply superposition. We will superimpose two states in order to obtain the original circuit.

By overlapping, the sources go away and the original circuit remains. The internal configuration of the black box is the same -except for the cancellation of the sources in (2) Then: I = I1 + I2 In circuit (1), remembering the definition of VAB, I say that I1 = 0 is a solution (since the black box point of view is “open”: compatible pair I1 = 0 VAB voltage and from the load point of view it is also compatible, because with I1 = 0, there is no drop in Z – if there is no mutual -) Accepting uniqueness of the solution I1 = 0 This as long as there is no mutual between Z and the interior of the circuit, because if there is, the voltage between A and B would change. In circuit (2), recalling the definition of ZAB, it is clear that I2 = VAB / (ZAB + Z)

So: for the purposes of what happens in Z, we can replace the black box with its Thévenin equivalent: VAB source and ZAB impedance

Why? Well, in this one too: I = VAB / (ZAB + Z)

For the voltmeter to measure VAB, that is, so that when connecting it the voltage is not altered , it should be Zv = ¥. Strictly speaking, Zv >> ZAB It is said that the voltmeter does not “load” the rest of the circuit. In common (analog) testers, the Zv is given in W / V, eg 10k / V. It means that on the 10V scale, Zv = 10×10 = 100k. If we connect it in a circuit with a Z view of 1k, the error made by the fact of measuring is 1%. If the Z seen is 100k, another type of voltmeter must be used (digital, for example) that has a higher own impedance.

Short circuit current

We have already seen that if we short-circuit A and B: Icc = VAB / ZAB This in particular suggests another method of calculating the impedance seen. So far we have seen two:

– Put an external source, which we call E, cancel the internal sources, and calculate E / I

– Simply “look” from A and B.

– If we know VAB and Icc, it is ZAB = VAB / Icc

Equivalence between Thevenin and Norton

Whatever the equivalent obtained, it is very easy to go to the other equivalent without applying the corresponding theorem, so for example, suppose we have calculated the Thévenin equivalent of a circuit and we have obtained the circuit on the left of the following figure: Applying the Norton’s theorem to the figure on the left, we will short-circuit the output and calculate the current that passes between them which will be the current: Ith = 10/20 = 0.5 A. and the Norton resistance is 20 W. so we will have the Norton equivalent circuit on the right