Millman Theorem

The Millman theorem is expressed this way:

Any network made of voltage or current sources connected in parallel may be replaced by only one equivalent source and only one serial resistor for the voltage source and a parallel resistor for the current source.

In realty, the Millman theorem is only a specific application of Norton and Thevenin theorems.  These theorems tell us that it is always possible to replace a network, as complex as possible, by either a current source with a parallel resistor or a voltage source with a serial resistor.

Fig. 1

It is like replacing the Fig. 1 right portion by the lest portion.  This is done in three steps.

Fig. 2

We simply transform all voltage sources into their equivalent current sources with a current equal to the one supplied by each voltage source.  Giving the fact that we are dealing with a parallel circuit, we convert all resistors into conductances; this is to simplify the calculations because currents and conductances simply added each other in parallel circuits.

Step 2:
We add all the currents and all the conductances to do only one source with one conductance and we get the Fig. 3 schematic.

Fig. 3

In which:  I_eq = I₁ + I₂ + I₃ and G_eq = G₁ + G₂ + G₃

Step 3:
We then transpose the Fig. 3 circuit to obtain a voltage source with a serial resistor and we get the fig. 4 schematic.

Fig. 4

Exercise:
Back to the Fig. 1 and suppose that the three voltage sources give 50 V respectively and resistor are 10 Kohms each.  Calculate source voltage and serial resistor values equivalent to the Fig. 4.

Your answers: (Do all your calculations with two significant figures.)