Millman’s Theorem (3 Branches)

V = Σ(Ei/Ri) / Σ(1/Ri)

Quick answer

This calculator solves V = (E1/R1 + E2/R2 + E3/R3) / (1/R1 + 1/R2 + 1/R3) for Node Voltage. With Source 1 26.4 V, Resistance 1 0.47 kΩ, Source 2 56.4 V, Resistance 2 1.03 kΩ, Source 3 5.64 V, Resistance 3 2.21 kΩ, the result is 31.96 V.

Calculator

Node Voltage (V)
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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 — Branch 1 source voltage (V)
  • R1 — Branch 1 series resistance (Ω)
  • E2 — Branch 2 source voltage (V)
  • R2 — Branch 2 series resistance (Ω)
  • E3 — Branch 3 source voltage (V)
  • R3 — Branch 3 series resistance (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.

Symbols and units

Symbol Meaning Default unit Accepted units
V Node Voltage V pV, nV, µV, mV, V, kV
E1 Source 1 V pV, nV, µV, mV, V, kV
R1 Resistance 1 Ω mΩ, Ω, kΩ, MΩ
E2 Source 2 V pV, nV, µV, mV, V, kV
R2 Resistance 2 Ω mΩ, Ω, kΩ, MΩ
E3 Source 3 V pV, nV, µV, mV, V, kV
R3 Resistance 3 Ω mΩ, Ω, kΩ, MΩ

Step by step

  1. 1. What you know: Source 1 26.4 V, Resistance 1 0.47 kΩ, Source 2 56.4 V, Resistance 2 1.03 kΩ, Source 3 5.64 V, Resistance 3 2.21 kΩ
  2. 2. The relation: V = (E1/R1 + E2/R2 + E3/R3) / (1/R1 + 1/R2 + 1/R3)
  3. 3. Substitute: E1 = 26.4 V, R1 = 0.47 kΩ, E2 = 56.4 V, R2 = 1.03 kΩ, E3 = 5.64 V, R3 = 2.21 kΩ
  4. 4. Result: Node Voltage = 31.96 V

Related quantities from the same inputs

  • P = V² / R = 1.483 W
  • I = V / R = 56.17 mA

Worked examples

Minimum range

Given: Source 1 0.726 V, Resistance 1 1 Ω, Source 2 1.55 V, Resistance 2 2.2 Ω, Source 3 155 mV, Resistance 3 4.7 Ω

Result: Node Voltage = 0.8778 V

Low range

Given: Source 1 7.26 V, Resistance 1 10 Ω, Source 2 15.5 V, Resistance 2 22 Ω, Source 3 1.55 V, Resistance 3 47 Ω

Result: Node Voltage = 8.778 V

Typical

Given: Source 1 26.4 V, Resistance 1 0.47 kΩ, Source 2 56.4 V, Resistance 2 1.03 kΩ, Source 3 5.64 V, Resistance 3 2.21 kΩ

Result: Node Voltage = 31.96 V

High range

Given: Source 1 0.506 kV, Resistance 1 4.7 kΩ, Source 2 1.08 kV, Resistance 2 10.3 kΩ, Source 3 108 V, Resistance 3 22.1 kΩ

Result: Node Voltage = 0.6122 kV

Maximum range

Given: Source 1 5.06 kV, Resistance 1 47 kΩ, Source 2 10.8 kV, Resistance 2 103 kΩ, Source 3 1.08 kV, Resistance 3 0.221 MΩ

Result: Node Voltage = 6.122 kV

How Node Voltage changes with Source 1

Source 1 (input) Node Voltage (result)
2.64 V 17.72 V
6.6 V 20.09 V
13.2 V 24.05 V
19.8 V 28 V
26.4 V 31.96 V
39.6 V 39.87 V
52.8 V 47.77 V
105.6 V 79.41 V
0.1848 kV 126.9 V
0.264 kV 0.1743 kV

The same result in other units

Unit Value
mV 31960
V 31.96
kV 0.03196

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Frequently asked questions

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

Use V = (E1/R1 + E2/R2 + E3/R3) / (1/R1 + 1/R2 + 1/R3). With Source 1 26.4 V, Resistance 1 0.47 kΩ, Source 2 56.4 V, Resistance 2 1.03 kΩ, Source 3 5.64 V, Resistance 3 2.21 kΩ, Node Voltage works out to 31.96 V.

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

The formula is 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.

What units does Millman’s Theorem (3 Branches) use?

Node Voltage is shown in mV, V, kV. Values are held in SI base units internally, so you can mix prefixes without converting by hand.

Related Concepts

All Basic Electronics formulas →
Ohm's Law Electrical Power Voltage Divider Current Divider Thévenin Equivalent Norton Equivalent Temperature Coefficient of Resistance Millman’s Theorem (3 Branches)

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