Electric Circuit Analysis: Chapter - 2: Network Theorems - DC

Millman's Theorem

Statement, Proof, Circuit Diagram, Formula, Solved Example Problems

Millman's Theorem states that 'n' voltage sources (V1, V2, ... Vn) are in parallel having their internal resistances (R1, R2, ... Rn) respectively. The arrangement can be replaced by a single voltage source 'V' and resistance 'R' as given below figure (a) (b).

MILLMAN'S THEOREM

Statement

Millman's Theorem states that 'n' voltage sources (V1, V2, ... Vn) are in parallel having their internal resistances (R1, R2, ... Rn) respectively. The arrangement can be replaced by a single voltage source 'V' and resistance 'R' as given below figure (a) (b).


(a) A number of parallel voltage sources feeding power to a load resistance.

(b) Equivalent voltage (V) and resistance (R) of the source network following Millman's Theorem.

As per Millman's Theorem,


Proof

The voltage sources in figure are replaced by equivalent current sources as shown in figure.


Here all the current sources are parallel. Therefore this circuit may be replaced by a single equivalent current source (I).

I = I1 + I2 + ... In

I = V1 G1 + V2 G2 + ... + Vn Gn

All the conductances are in parallel in this circuit diagram of figure can be replaced by a single equivalent conductance (G).

G = G1 + G2 +...+ Gn

R = 1/ G

Figure shows equivalent current source and conductance


This equivalent current source converted into an equivalent voltage source. It is shown in figure.


Here Millman's theorem is proved.

 

EXAMPLE 40: Using millman's theorem find the current through RL in the circuit shown in figure and find the voltage across R.

Solution:


The equivalent circuit of Millman's theorem is


Current through RL load resistor

IL = V / R + RL = 1.33 / 1.33 + 10

IL = 0.117 A

Voltage cross load resistor

VL = IL RL = 0.117 × 10 = 1.17 V

 

EXAMPLE 41: Find the current through the 12 resistor using millman's theorem of figure given below.


Solution :


Now this circuit converted into equivalent voltage source.


 

Electric Circuit Analysis: Chapter - 2: Network Theorems - DC : Tag: : Statement, Proof, Circuit Diagram, Formula, Solved Example Problems - Millman's Theorem