The above answer is OK but needs some more details. (1) The answer seems to be valid for resistive networks containing only independent voltage and current sources. That is inclusion controlled sources have not been considered. (2) I quote from the above answer "The last current source is the external IM
, which produces this voltage,
vM=IMRM
where RM is the equivalent resistance looking into the port when all the internal sources are suppressed" I think this needs justification in a proof.
All these can be addressed if we use modified node equation technique to represent an electric network. The technique may be described as follows.
Modified Node Equation Technique
Assume that the circuit has n+1 nodes of which node #0 is labeled as reference node. The circuit contains LRCM passive elements, independent voltage and current sources and linear controlled sources. The following algorithm would augment the circuit so that node equation technique may be used for its analysis. The underlying concept is to replace all the sources other than the independent current sources by unknown current sources. With these replacements included one can easily obtain n node equation associated with n nodes. To solve the circuit, however, one would require n+m number of equations, where, m is the total number of active sources that are replaced by unknown current generators in the augmented circuit. The extra m equations are readily obtained from the specifications of the replaced sources.
Augmentation Algorithm:
• Replace all independent voltage sources, controlled current sources and controlled voltage sources by unknown current sources. • Obtain the set of n node equations corresponding to n nodes of the circuit. 628
• For an independent voltage source equate the output voltage with the difference between the terminal node voltages to get an extra equation. • For a controlled source express the control law in terms of the node voltages and the circuit parameters to get the necessary extra equation.
We may now state <PERSON>'s theorem as follows. Consider a one port network A that is terminated to a load network B.
Network A may contain linear passive elements and linear independent and controlled sources. On the other hand there is no restriction of linearity for network B. However, following constraint exist 1. There is no electro-magnetic coupling between network A and B. 2. There exists no source at network A that is controlled by any voltage or current in network B. Under this condition <PERSON>’s theorem states that the network A may be replaced by a voltage source VTH connected in series with an impedance ZTH where VTH is the open circuit voltage measured at the port of A and ZTH is the driving point impedance of A with all the independent sources in A reduced to zero.
Since there exists no coupling between the circuits A and B we can consider network A separately with an out put current I1 flowing away from its port. The port current I1 and the port voltage V1 will be the solution of the pair of equations relating V1 and I1 in A and in B respectively. <PERSON>’s theorem aims at deriving the relationship valid for the first network that is network A.
Following the modified node equation technique one can find the general representation scheme for an arbitrary A network. (for detail description you are referred to https://www.researchgate.net/publication/262764661_Thevenin%27s_Theorem_An_Easy_Proof_Suitable_for_Undergraduate_Teaching
Using this representation one can very easily express the voltage at the port of A similar to the statement of <PERSON>'s theorem. This is well explained in the above link. 844