What is the formula for Millman’s Theorem (3 Branches)?

The Millman’s Theorem (3 Branches) formula is V = (E1/R1 + E2/R2 + E3/R3) / (1/R1 + 1/R2 + 1/R3), where V—Common node voltage (V); E1, E2, E3—Branch source voltages (V); R1, R2, R3—Branch series resistances (Ω).

Millman’s Theorem (3 Branches)

Q: How do you calculate Millman’s Theorem (3 Branches)?

Use the formula V = (E1/R1 + E2/R2 + E3/R3) / (1/R1 + 1/R2 + 1/R3). Millman’s theorem gives the voltage at a common node fed by several voltage-source-plus-resistance branches in a single step, without writing node equations. Enter the values you know in the calculator above and it solves for the remaining variable.

V = Σ(Ei/Ri) / Σ(1/Ri)

Calculator

Node Voltage

Source 1

Resistance 1

Source 2

Resistance 2

Source 3

Resistance 3

Result

—

Formula

V = (E1/R1 + E2/R2 + E3/R3) / (1/R1 + 1/R2 + 1/R3)

Description

Millman’s theorem gives the voltage at a common node fed by several voltage-source-plus-resistance branches in a single step, without writing node equations. It is the weighted average of the branch source voltages, each weighted by its conductance. It is the fastest way to find the output of paralleled sources—for example summing amplifiers, battery banks, or biasing networks.

Variables

V — Common node voltage (V)
E1, E2, E3 — Branch source voltages (V)
R1, R2, R3 — Branch series resistances (Ω)

Practical Notes

Set a branch’s source to 0 to model a plain resistor to ground. The theorem extends to any number of branches as the conductance-weighted average of the source voltages.

Related Concepts

All Basic Electronics formulas →

Ohm's Law Electrical Power Voltage Divider Current Divider Thévenin Equivalent Norton Equivalent Temperature Coefficient of Resistance Kirchhoff's Voltage Law (KVL)

Need more features?

Save calculations, import telemetry data, simulate battery discharge, and collaborate with your team.

Try the App