Download Millmans Theorem - Wintec Learning

DE5410 Class Note – Millman’s Theorem
In electrical engineering, Millman's theorem (or the parallel generator theorem) is a
method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute
the voltage at the ends of a circuit made up of only branches in parallel.
It is named after Jacob Millman, who proved the theorem.
It permits any number of parallel branches consisting of voltage sources and impedances to
be reduced to a single equivalent voltage source and equivalent impedance. Such multibranch circuits are frequently encountered in both electronics and power applications
Figure 1
ek is the voltage generators and am the current generators.
Ri be the resistances on the branches with no generator.
Rk be the resistances on the branches with voltage generators.
Rm be the resistances on the branches with current generators.
Millman’s theorem states that the voltage at the ends of the circuit is given by:
For a familiar three phase system this may be stated as follows:
It can be proved by considering the circuit as a single supernode. Then, according to Ohm
and Kirchhoff, "the voltage between the ends of the circuit is equal to the total current
entering the supernode divided by the total equivalent conductance of the supernode".
The total current is the sum of the currents flowing in each branch.
The total equivalent conductance of the supernode is the sum of the conductance of each
branch, since all the branches are in parallel. When computing the equivalent conductance
all the generators have to be switched off, so all voltage generators become short circuits
and all current generators become open circuits. That's why the resistances on the branches
with current generators do not appear in the expression of the total equivalent conductance.
V
V
V
V
V
V
V
V