Millman Theorem

Millman’s theorem is named after Jacob Millman, a well-known electrical engineering professor. This theorem simplifies complex electrical circuit, especially those with parallel voltage or current sources. It combines elements of Thevenin’s Theorem and Norton’s Theorem, making it a powerful tool to find the voltage across a load and the current through it. This theorem is also known as the PARALLEL GENERATOR THEOREM.
Millman’s theorem applies to circuits with either only parallel voltage sources or a mix of voltage and current sources in parallel. Let’s discuss these cases one by one.

Circuit consisting only Voltage Sources

Let us have a circuit as shown in below figure a.

Here V₁, V₂ and V₃ are voltages of respectively 1^st, 2^nd and 3^rd branch and R₁, R₂ and R₃ are their respective resistances. I_L, R_L and V_T are load current, load resistance and terminal voltage respectively.
Now this complex circuit can be reduced easily to a single equivalent voltage source with a series resistance with the help of Millman’s Theorem as shown in figure b.

The value of equivalent voltage V_E is specified as per Millman’s theorem will be –

This V_E is nothing but Thevenin voltage and Thevenin resistance R_TH can be determined as per convention by shorting the voltage source. So R_TH will be obtained as

Now load current and terminal voltage can be easily found by

Let’s try to understand whole concept of Millman’s Theorem with the help of a example.

Example – 1
A circuit is given as shown in fig-c. Find out the voltage across 2 Ohm resistance and current through the 2 ohm resistance.

Answer: While various methods can solve this problem, the most effective and time-saving approach is using Millman’s theorem. The given circuit can be reduced to the one shown in figure d, where the equivalent voltage (V_E) is found using Millman’s theorem.


Equivalent resistance or Thevenin resistance can be found by shorting the voltage sources as shown in fig – e.


Now we can easily found the required current through 2 Ohm load resistance by Ohm’s law.

Voltage across load is,

Circuit is Consisting Mixture of Voltage and Current Source

Millman’s Theorem is also helpful to reduce a mixture of voltage and current source connected in parallel to a single equivalent voltage or current source. Let’s have a circuit as shown in below figure – f.

Here all letters are implying their conventional representation.This circuit can be reduced to a circuit as shown in figure – g.

Here V_E which is nothing but thevenin voltage which will be obtained as per Millman’s theorem and that is

And R_TH will be obtained by replacing current sources with open circuits and voltage sources with short circuits.

Now we can easily find out load current I_L and terminal voltage V_T by Ohm’s law.

Let’s have a example to understand this concept more properly.

Example 2 :

A circuit is given as shown in fig-h. Find out the current through load resistance where R_L = 8 Ω.

Answer : This problem may seem to be difficult to solve and time consuming but it can easily be solved in a very less time with the help of Millman’s Theorem. The given circuit can be reduced in a circuit as shown in fig – i. Where, V_E can be obtained with the help of Millman’s theorem,


Therefore, current through load resistance 8 Ω is,