Millman's Theorem

Quick Answer

Millman's theorem gives the voltage at a node where several branches meet, each branch being a voltage source behind a resistance. The answer is the average of the source voltages weighted by branch conductance. It is nodal analysis for the one-unknown case, collapsed into a single expression.

Take each branch, divide its source voltage by its resistance, add those up, then divide by the sum of the reciprocal resistances. What comes out is the common voltage the whole arrangement settles at. A branch with no source is not an exception: its source voltage is zero, and it still contributes its conductance to the denominator, which is how a load resistor pulls the answer down.

The theorem is not a new result. It is Kirchhoff's current law written at a single node and rearranged, which is what nodal analysis does before it starts building arrays. The value is in the shape of the answer: a stiff branch, meaning a low resistance, drags the node toward its own source, and a weak one barely votes at all.

That makes it the quick tool for anything that parallels sources. Cells or supplies of slightly different voltage tied to one bus settle at a weighted average, and the difference between each source and that average, divided by its own resistance, is the current that branch contributes: a cell below the average is being charged by the rest. A resistive summing junction is the same calculation. The limit is the topology: every branch has to run between the same two nodes, and nothing else may be attached.