# Millman's Theorem

Millman's Theorem, aka Parallel Generator Theorem, is a theorem which can be used to simplify circuits consisting of multiple sources connected in parallel branches.

It allows us to calculate the voltage across all branches in terms of the voltage sources and resistances.

The node voltage A shown in the circuit structure shown below can be solved with Millman's Theorem:

It will be shown that the node voltage is given by:

## Derivation:

We can derive the above by working backwards from the result. We can apply Ohm's Law and recall the definition of conductance G to simplify to the following:

Which is equal to I*R by Ohm's Law, i.e.

In other words, the node voltage is given by the sum of the currents divided by the conductances which is in agreement with Ohm's Law and that currents in branches sum together at the node.

Another way to look at it is from the perspective of Thevenin and Norton Equivalent circuits, take the above example where we effectively have 4 Thevenin sub-circuits connected together.

We can redraw each of these as Norton Equivalent circuits where the Norton currents are given by (Vth/R) add the currents together and combine the parallel resistances into a single resistor and apply Ohm's law.

From here we can then combine all the current sources and resistances in parallel:

With Ohm's law we then arrive at the above result :)

## Who was Jacob Millman?

Jacob (Jack) Millman was born in Russia in 1911, and he came to Lawrence, Mass. at the age of 1 1/2 years. He obtained the B.S. in Physics from MIT in 1932. A Fellowship allowed him to spend the next year at the University of Munich studying under the great teacher A. Sommerfield. He returned to M.I.T. to complete the requirements for the Ph.D. in Theoretical Physics in 1935.

He joined Columbia University in 1951, and retired in 1975. From 1941 to 1987, Millman wrote eight textbooks on electronics. His obituary was printed in the New York Times newspaper on 24 May 1991.

Millman's Theorem is named after him.

He received the IEEE Education Medal in 1970.

## Examples:

1. Calculate node voltage Va for the circuit below using Millman's Theorem:

We can apply Millman's Theorem directly to this circuit:

2. Calculate node voltage Va for the circuit below using Millman's Theorem:

This looks remarkably similar to the above circuit! However notice that R1 is not connected to a voltage source this time, but what does this mean in terms of Millman's Theorem? It simply means that V1 is 0V as we could consider the resistor to be connected to a 0V source. We can then apply Millman's Theorem:

## Questions:

1. Calculate node voltage Va for the circuit below using Millman's Theorem: