Jacob Millman

Jacob Millman was born in Russia in 1911. He received a Ph.D. from MIT in 1935. In 1951 he became professor at Columbia University. He went on to write eight textbooks on electronics between 1941 and 1987. Millman's Theorem is named after him.

Millman's Theorem

In electrotechnics , Millman's theorem is a useful method to simplify the solution of a circuit. It has been proved by Millman, but a similar method, known as Tank's method, was already commonly used before Millman's proof.

Millman's theorem gives a very quick method to compute the tension at the ends of a circuit made up of only branches in parallel.

Let e_k be the tension generators and a_m the current generators.

Let R_i be the resistances on the branches with no generator.

Let R_k be the resistances on the branches with tension generators.

Let R_m be the resistances on the branches with current generators.

Then Millman states that the tension at the ends of the circuit is given by:

$v=\frac{\sum_{}\frac{\pm e_{k}}{R_{k}}+\sum_{}\pm a_{m}}{\sum_{}\frac{1}{R_{k}}+\sum_{}\frac{1}{R_{i}}}$

It can be proved by considering the circuit as a single supernode . Then, according to Ohm and Kirchhoff, the tension between the ends of the circuit is equal to the total current entering the supernode divided by the total equivalent conductance of the supernode.

The total current is the sum of the currents flowing in each branch.

The total equivalent conductance of the supernode is the sum of the conductance of each branch, since all the branches are in parallel. When computing the equivalent conductance all the generators have to be switched off, so all tension generators become short circuits and all current generators become open circuits. That's why the resistances on the branches with current generators do not appear in the expression of the total equivalent conductance.

Jacob Millman

Millman's Theorem

See also