Substitution Theorem

In mathematics all theorems are easy to find their demonstration or full statement meanwhile in circuit theory I have been looking for several days and did not find anything. Only Thevenin, Norton and Superposition.

Not looking for the demonstration of the theorem but for the full statement of it which would answer me this questions:

Could be applied in non-linear circuits?
Could be used if there are dependant sources?

Maybe I was searching with the wrong words.

can you explain about what you are searching. Your statement is not clear.

CataM, the key word "substitution theorem" is correct. However, most of the textbooks do not "know" this theorem - although it is applied very often.
As an example, it is given in "Desoer, C.A. and Kuh, E.S. : Basic Circuit Theory, Mc Graw-Hill, 1969".

First, to answer your questions
* Yes - it can be applied to non-linear circuits
* No - dependent sources are not allowed.

Here is the basic content of the theorem:
"In an electrical network (time-invariant) any arbitrary branch Z with the branch voltage Vz and the branch current Iz can be replaced by an independent source (Vz or Iz) without influencing all node voltages and branch currents - provided the network matrix has only one single solution."

This theorem is applied very often - however, mostly without knowing it.
Two simple examples:
1.) Two transistor gain stages in cascade. We calculate the gain of the first stage (taking the input impedance of the second stage into account) and consider the signal output voltage of this stage as an ideal input voltage source for the second stage. This is only allowed because of the Substituition Theorem.
2.) For calculating the quiescent DC conditionc (Ic and Vce) for a simple transistor-based gain stage we can substitute the base emitter voltage Vbe (assumed to be app. 0.7V) by a DC source of Vz=0.7V.
This simplifies the calculkation.

The time-invariance of the network is not required. Is required, instead that the network has a unique solution: this is always true for linear networks, while attention has to be paid for non-linear networks

In the textbook provided by LvW ("Desoer, C.A. and Kuh, E.S. : Basic Circuit Theory, Mc Graw-Hill, 1969".) I don't understand what the author want to say. I have highlighted it.


That reference after "... to other branches of the network." is referenced at the bottom of the page like this "That is, branch k may not be a branch of a coupled inductor nor a branch of a dependent source."

Why first he say that it has not to be coupled to other branches and then say that is still true when coupled inductors are present ?

It refers that you can apply Substitution Theorem when coupled inductors are present in a different branch but do not apply it on the branch that has the coupled inductor?

For example, in the circuit of this thread: https://www.edaboard.com/threads/350293/
It is correct to replace the Graetz bridge but not correct to replace the graetz bridge and the coupled inductor ?