Millman Theorem

Millman’s 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 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.
 Here V₁, V₂ and V₃ are voltages of respectively 1^st, 2^nd and 3^rd branch and R₁, R₂ & 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 2ohm 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 VE 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,