- Millman’s Theorem Definition: Millman’s theorem is defined as a method to simplify circuits with multiple parallel voltage or current sources into a single equivalent source.
- Application to Voltage Source Circuits: Millman’s theorem simplifies circuits with only voltage sources in parallel into an equivalent voltage source with a series resistance.
- Equivalent Voltage Calculation: The equivalent voltage (VE) is calculated using the theorem, representing the Thevenin voltage.
- Application to Mixed Source Circuits: The theorem also applies to circuits with both voltage and current sources in parallel, simplifying them into an equivalent source.
- Example Applications: Example problems demonstrate how Millman’s theorem simplifies complex circuits, making it easier to find voltage and current across specific components.
Millman’s theorem reduces parallel voltage or current sources to one equivalent source. Jacob Millman, an electrical engineering teacher, is the usual namesake. It applies to a parallel electrical circuit. The equivalent matches Thevenin’s Theorem and Norton’s Theorem on those branches, and it gives the voltage across a load and the current through it. The same result is also called the PARALLEL GENERATOR THEOREM.
Millman’s theorem covers only parallel voltage sources, or a mix of voltage and current sources in parallel. The two cases follow.
Circuit consisting only Voltage Sources
Take the circuit in figure a.
Here V1, V2 and V3 are the voltages of the 1st, 2nd and 3rd branch and R1, R2 and R3 are the resistances of those branches. IL, RL and VT are load current, load resistance and terminal voltage.
That network reduces to one equivalent voltage source and a series resistance by Millman’s Theorem, as in figure b.
Millman’s theorem gives the equivalent voltage VE as
VE is the Thevenin voltage. The Thevenin resistance RTH is found in the usual way by shorting the independent voltage sources. So RTH is
Load current and terminal voltage then follow as
The next numerical example uses the same steps.
Example – 1
The circuit in fig-c asks for the voltage across the 2 Ohm resistor and the current through it.
Answer: Several methods work. Millman’s theorem is the short path. Reduce the circuit to figure d and find the equivalent voltage (VE) from Millman’s theorem.
The equivalent (Thevenin) resistance is found by shorting the independent voltage sources, as in fig – e.
The current through the 2 Ohm load resistance then follows from Ohm’s law.
Voltage across the load is
Circuit is Consisting Mixture of Voltage and Current Source
Millman’s Theorem also reduces a parallel mix of voltage and current sources to one equivalent voltage or current source. Take figure – f.
The symbols have their usual meaning. That network reduces to figure – g.
VE is the Thevenin voltage from Millman’s theorem:
RTH is found by replacing independent current sources with open circuits and independent voltage sources with short circuits.
Load current IL and terminal voltage VT then follow from Ohm’s law.
A second numerical example follows.
Example 2 :
Find the current through the load in fig-h, where RL = 8 Ω.
Answer : Millman’s Theorem shortens the work. Reduce the circuit to fig – i and find VE from Millman’s theorem,
The current through the load resistance of 8 Ω is
