# Millman Theorem

Published on 24/2/2012 & updated on 9/8/2018HOME / CIRCUIT THEORY / CIRCUIT THEOREM

**was named after famous electrical engineering professor JACOB MILLMAN who proposed the idea of this theorem. Millman’s theorem acts as a very strong tool in case of simplifying the special type of complex electrical circuit. This theorem is nothing but a combination of Thevenin's Theorem and Norton's Theorem. It is very useful theorem to find out voltage across the load and current through the load.This theorem is also called as**

*Millman’s theorem***PARALLEL GENERATOR THEOREM**.

**Millman's theorem**is applicable to a circuit which may contain only voltage sources in parallel or a mixture of voltage and current sources connected in parallel. Let’s discuss these one by one.

### Circuit consisting only Voltage Sources

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

Here V_{1}, V_{2} and V_{3} are voltages of respectively 1^{st}, 2^{nd} and 3^{rd} branch and R_{1}, R_{2} and R_{3} 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 : We can go through any solving method to solve this problem but the most effecting and time saving method will be none another than **Millman’s theorem**. Given circuit can be reduced to a circuit shown in fig-d where equivalent voltage V_{E} can be obtained by millman’s theorem and that is
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,

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