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Millman’s Theorem to Voltage and Current Sources

We can express the Millman’s Theorem in terms of voltage sources. Also, we can express in terms of current sources.

Millman’s Theorem Applicable to Voltage Sources

This theorem combines both Thevenin’s theorem and Norton’s theorem. In order to find a common voltage across a network, we use Millman’s Theorem. The network contains a number of parallel voltage sources.

Suppose there are n numbers of parallel voltage sources.

millmans theorem with voltage source

Here, in the above figures, R1, R2, R3, and Rn are the internal resistance of the voltage sources. On the other hand, the voltage V1, V2, V3 … and Vn are the emf of the voltage sources.

The overall equivalent voltage of the above circuit across A and B is,

This voltage is nothing but the Thvenin’s voltage across terminals A and B.

Proof of Millman’s Theorem

Millman’s theorem tells the open circuit voltage of an active circuit across two terminals of the circuit. Also, Thevenin’s theorem tells the same.

Proof of Millman's Theorem

Let us consider the voltage at node A is VA and voltage at node B is 0.

So, as per nodal analysis, we can write,

 

As the voltage at B is 0, we can write,

Example of Millman’s Theorem

Let us take the circuit as shown below.Example of Millman's Theorem

As per Millman’s theorem, the voltage across A and B

The Thevenin equivalent resistance of the circuit across node A and B is

Now, the rule of current through the 25Ω resistance is,

Millman’s Theorem  Applicable to Current Sources. Suppose there are n numbers of current sources connected in parallel.

Example of Millman's Theorem

We can consider this circuit as a single current source. As per Millman’s theorem, the current of the source is the sum of individual source currents.

The equivalent resistance of the circuit is the internal resistance of the current source.

The equivalent resistance or Norton resistance of the network against terminals  A and B, is the parallel combination of individual source resistances. That is

So, the equivalent current source of the above parallel combinations of n number of current sources is, as shown below.

Example of Millman’s Theorem Applicable to Connect Sources

Let us consider the circuit below.

Example of Millman's Theorem Applicable to Connect Sources

As per Millman’s theorem, the Norton equivalent current of the circuit against terminal A and B is,

The Norton equivalent resistance of the circuit is the parallel combination of 12Ω, 4Ω and 6Ω. That is

Therefore, the Norton equivalent source of the above circuit is as below.

Hence, the current through, the resistance of 8Ω resistance, connected between A and B is,

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