What is the difference between the substitution principle and Thevenin theorem?

Question

The substitution principle (as seen in this book; in italian, sorry)

Let A and B be any two part of an electrical network with voltage and current sources and only resistances, connected by ideal conductors. Suppose the voltage between the conductors is v. Then, in order to study A, B can be replaced with an ideal generator of voltage v.

Thévenin's theorem

Any linear electrical network with voltage and current sources and only resistances can be replaced at terminals A-B by an equivalent voltage source Vth in series connection with an equivalent resistance Rth.

There must be something I do not understand.

It seems to me that these statements cannot be both true (or at least, the first one would imply the equivalent resistance in Thevenin's theorem to be zero, and so the theorem loses any meaning.)

Thank you all in advance for the answers!

Answer 1

SUBSTITUTION THEOREM:

If the voltage across and the current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.

As long as terminal voltage and current is same, accordance with substitution theorem, you can substitute whatever in the branch. Here is an example that demonstrate how it works.

THÉVENIN’S THEOREM:

Any two-terminal dc network can be replaced by an equivalent circuit consisting solely of a voltage source and a series resistor.

You have asked an important question indeed. Thévenin’s equivalent circuit has a series resistor but in the second circuit diagram I have only used a source and in the third I have used both (more possible). Both are accordance with substitution theorem.It means one can replace a branch with any combination of elements which is not true for Thévenin’s theorem.For the marked branch in the main circuit if you use Thévenin’s theorem you will get Vth or Eth = 0V and Rth =3 Ohm. This is because Thévenin’s theorem doesn’t care about rest of the network or the load resistance but substitution theorem does. Without the whole circuit substitution theorem is not applicable but in Thévenin equivalent circuit the load resistance may vary.

Answer 2

Hello Ant: To answer your question, I think that both theorems have no direct connection.

The "substitution principle" as quoted by you is a special form of the "General Substitution Theorem" of network theory. This one of the less known network theorems which, however, is applied very often - mostly without knowing.

Here is the contents of the theorem: Any arbitrary branch Z of a time-invariant network can be replaced by an indpendent voltage or current source of the same value (branch voltage resp. current) without influencing other node voltages or branch currents - provided the network matrix has one singular solution only.

Example: Series connection of two transistor-based gain stages. The output voltage of the first stage (taking the input impedance of the 2nd stage into account) can be seen as a voltage source driving the 2nd stage. Thus, the output two-pole of the 1st stage is replaced by an ideal voltage source having the same voltage. This is allowed only because of the existence of the "Substitution Theorem".

This approach has nothing to do with Thevenin´s theorem.

EDIT: Of course, the substitution theorem applies for DC as well as AC circuits.