Millman's Theorem
The fastest way to solve parallel source branches. When several branches — each a source with a resistance — share the same two terminals, Millman's theorem gives the terminal voltage in one line: V = ΣVG / ΣG. Learn the formula, its derivation, a step-by-step method, a fully solved example and how it compares with Thevenin, Norton and superposition.
Complete Learning Path — Millman's Theorem
From the statement and the V = ΣVG/ΣG formula, to conductance, a KCL derivation, the step-by-step method, a solved example, current sources, sign rules and comparisons
What is Millman's Theorem?
Millman's theorem (named after Jacob Millman) is a fast circuit-analysis shortcut. It says that when several branches — each containing a source in series with a resistance — are connected in parallel across the same two terminals, you can find the voltage across those terminals with a single formula, without writing full mesh or nodal equations.
It is really nodal analysis at one node, packaged as a ready-made formula — perfect for circuits with many parallel batteries or supplies feeding a common load.
Consider three branches with sources V1, V2, V3 and resistances R1, R2, R3, all tied between terminals A and B. Millman's theorem finds the voltage VAB in one step.
When Millman shines
Whenever you see many parallel branches each with its own source and resistance across a common pair of nodes — parallel batteries, multiple supplies, or a mix — Millman gives the answer far faster than mesh or superposition.
The Formula & Conductance
The heart of the theorem is one compact equation:
Everything hinges on conductance, the reciprocal of resistance:
Derivation from Kirchhoff's Current Law
Millman's theorem is not magic — it drops straight out of Kirchhoff's current law (KCL) applied at the single common node.
Let the common node sit at voltage V. The current flowing into the node from branch k is (Vk − V)/Rk. KCL requires the branch currents to sum to zero:
Expanding and grouping the V terms gives ΣVk/Rk = V · Σ(1/Rk), so:
The Step-by-Step Method
Applying Millman's theorem is a short, repeatable recipe.
The procedure in words
- Check the layout — all branches must be in parallel across the same two terminals.
- Find each conductance, Gk = 1/Rk (in siemens).
- Weight each source by its conductance: form each VkGk (with the correct sign).
- Apply the formula, V = (ΣVkGk) / (ΣGk), to get the terminal voltage.
- Find the currents with Ik = (Vk − V)/Rk and Ohm's law for any load.
Solved Example
Find the terminal voltage for three parallel branches: V1 = 12 V, R1 = 4 Ω; V2 = 6 V, R2 = 2 Ω; V3 = 4 V, R3 = 4 Ω.
Conductances
G1 = 1/4 = 0.25 S, G2 = 1/2 = 0.50 S, G3 = 1/4 = 0.25 S.
Weighted sources (VkGk)
12×0.25 = 3, 6×0.50 = 3, 4×0.25 = 1 → ΣVG = 7; ΣG = 0.25+0.50+0.25 = 1.0 S.
Branch currents (a quick check)
I1 = (12−7)/4 = +1.25 A, I2 = (6−7)/2 = −0.5 A, I3 = (4−7)/4 = −0.75 A. With no external load they sum to 0 A — exactly what KCL demands.
The Millman Equivalent Source & a Load
Millman's result is really a single equivalent source: an EMF Veq = ΣVG/ΣG in series with a resistance Req = 1/ΣG. That equivalent then drives whatever load you connect.
From the solved example, Veq = 7 V and Req = 1/1.0 = 1 Ω. Connect a load RL = 6 Ω across the terminals and the load current and voltage follow at once:
Millman with Current Sources
Millman's theorem is not limited to voltage sources. Parallel current sources with conductances fit the same pattern.
A voltage source V in series with R is equivalent to a current source I = V/R in parallel with R (source transformation). So any Millman branch can be written either way, and the numerator becomes the sum of the branch short-circuit currents.
Sign Convention — Watch the Polarity
The single most common Millman mistake is a wrong sign. Each source's contribution VkGk carries a sign set by its polarity relative to the chosen terminal.
Rule of thumb
Pick a reference terminal (say A). For every branch, if its source drives current toward A, take VkGk as positive; if it opposes, take it negative. A branch that is just a resistor (no source) contributes only to ΣG, not to the numerator.
Millman vs Thevenin, Norton & Superposition
Millman is one of several network theorems. Knowing when to reach for each saves time.
| Theorem | Best for | What it gives |
|---|---|---|
| Millman | Parallel source branches on two common terminals | The node voltage in one formula |
| Thevenin | Any two-terminal linear network | Veq in series with Req |
| Norton | Any two-terminal linear network | Ieq in parallel with Req |
| Superposition | Circuits with several sources | Sum of each source's contribution |
Advantages & Limitations
Advantages
- Very fast — one formula, no simultaneous equations.
- Ideal for many parallel sources on a common load.
- Gives a clean equivalent source (Veq, Req).
- Handles voltage and current sources via source transformation.
Limitations
- Only for branches all in parallel across the same two nodes.
- Applies to linear circuits.
- Not a fit for general multi-node networks.
- Dependent sources need extra care.
Applications
Millman's theorem is a working shortcut wherever parallel sources meet a common node.
Parallel batteries
Combine cells or supplies of different EMF and internal resistance.
Multi-source DC
Solve DC networks with several parallel sources fast.
Op-amp inputs
Find the node voltage of a resistive summing/averaging network.
Voltage averaging
Weighted-average circuits and sensor summing nodes.
Power distribution
Feeders from several sources onto a common bus.
Exams & homework
A huge time-saver on parallel-source problems.
Key Terms — Glossary
| Term | Meaning |
|---|---|
| Millman's theorem | V = (ΣVG)/(ΣG) for parallel source branches on two common terminals. |
| Conductance (G) | Reciprocal of resistance, G = 1/R, in siemens (S). |
| Siemens (S) | The SI unit of conductance (1 S = 1 A/V = 1/Ω). |
| Common node / terminals | The two points all branches share; Millman finds the voltage between them. |
| Branch | One parallel path: a source in series with (or a current source in parallel with) a resistance. |
| Equivalent EMF (Veq) | The Millman result ΣVG/ΣG, the open-terminal voltage. |
| Equivalent resistance (Req) | 1/ΣG — the resistance seen by a load. |
| Source transformation | Converting a voltage source + series R into a current source I = V/R + parallel R. |
| KCL | Kirchhoff's current law — the sum of currents at a node is zero; the basis of the proof. |
Frequently Asked Questions
Quick, expert answers to the questions people ask most about Millman's theorem.
What is Millman's theorem in simple words?
It is a shortcut for circuits where several branches, each with a voltage source and a resistance, are connected in parallel across the same two points. It gives the voltage across those points directly as V = (ΣVG)/(ΣG).
What is the formula for Millman's theorem?
V = (V₁G₁ + V₂G₂ + V₃G₃ + …) / (G₁ + G₂ + G₃ + …), where each G = 1/R is the conductance of that branch. Equivalently, V = (ΣVₖ/Rₖ)/(Σ1/Rₖ).
What is conductance in Millman's theorem?
Conductance G is the reciprocal of resistance, G = 1/R, measured in siemens (S). Each source is weighted by its branch conductance, so lower-resistance branches influence the result more.
When can Millman's theorem be used?
For linear circuits where the branches are all in parallel between the same two terminals, each branch containing a source with a resistance. It is ideal for many parallel voltage sources feeding a common load.
How is Millman's theorem derived?
From KCL at the single common node: write each branch current as (Vₖ − V)/Rₖ, set their sum to zero, and solve for V. This gives V = (ΣVₖ/Rₖ)/(Σ1/Rₖ).
Can Millman's theorem be used with current sources?
Yes. Convert a voltage source with a series resistance to a current source I = V/R in parallel with that resistance. With parallel current sources the formula becomes V = (ΣIₖ)/(ΣGₖ).
What are the limitations of Millman's theorem?
It works only when the branches are all in parallel across the same two terminals, applies to linear circuits, and does not directly handle general multi-node networks or dependent sources without extra work.
How is Millman's theorem different from Thevenin's theorem?
Both reduce a network to an equivalent source, but Millman is a fast single-formula method for parallel source branches sharing two terminals, while Thevenin applies to any two-terminal linear network and gives a voltage source in series with a resistance.
Conclusion & Key Takeaways
Millman's theorem turns a cluster of parallel source branches into a single equivalent source with one elegant formula — the quickest route to the terminal voltage.
Parallel branches
Same two terminals.
V = ΣVG/ΣG
Conductance-weighted average.
G = 1/R
Weights in siemens.
From KCL
One-node derivation.
Veq, Req
A clean equivalent source.
Mind signs
Polarity sets each term's sign.