Millman's Theorem Calculator
Combine any number of parallel branches (voltage sources with series resistance, current sources, or plain resistors) into one node voltage. See the circuit, branch currents, weighting and a KCL check update live.
Parallel branches top node → ground
Green = branch delivers current into the node. Orange = branch absorbs current. Totals balance (KCL).
Equivalent circuit & KCL check
Conductance weighting Gk / ΣG
Branch table
| Branch | Ik (A) | Gk (S) | i into node (A) | P (W) |
|---|
The theorem in one minute
Write KCL at the top node: Σ(Vk − V)·Gk = 0. Solving gives V = ΣVkGk / ΣGk, which equals ΣIk / ΣGk once each source is written as its Norton current Ik = Vk/Rk.
A weighted average
V is the average of the source voltages weighted by conductance: the lowest-resistance branch pulls the node hardest.
Thevenin / Norton result
The whole network equals one source VTH = V with RTH = 1/ΣG, or IN = ΣIk in parallel with G = ΣG.
Where it applies
Parallel branches with Rk > 0. Ideal voltage sources in parallel (R = 0) conflict, and ideal current sources add G = 0.
Worked example
ΣI = 10/2 + 5/5 − 3/10 = 5 + 1 − 0.3 = 5.7 A
ΣG = 0.5 + 0.2 + 0.1 = 0.8 S
V = 5.7 / 0.8 = 7.125 V and RTH = 1/0.8 = 1.25 ohm
FAQ
A zero-ohm branch is an ideal voltage source straight across the node: G is infinite and V simply equals that source. Parallel ideal sources of different voltages contradict each other.
Add a branch of type "R only" with the load resistance. V is then the load voltage and its current appears in the branch table.
Open Advanced mode: the KCL residual (the sum of branch currents into the node) should be zero, and each branch current is i = Ik − Gk·V.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| V | Voltage at the common node | V |
| I_k | Branch current source | A |
| G_k | Branch conductance | S (siemens) |
What it means
Millman’s theorem provides a method to find the voltage at a common node in a network of parallel voltage sources, each with a series resistance. It states that the equivalent voltage V is the sum of each branch current (V_i / R_i) divided by the sum of branch conductances (1/R_i). The formula is V = (Σ (V_i/R_i)) / (Σ (1/R_i)). This is useful for simplifying circuits with multiple sources feeding a common node. It is particularly convenient for circuits with many parallel branches, reducing them to a single equivalent source. Millman’s theorem is also known as the parallel‑source theorem and is often applied in electronics for summing voltages in op‑amp circuits. It can be derived from both KCL and Ohm’s law. Example: Branches with voltages 2V,3V,1V and resistances 10Ω,5Ω,20Ω give currents 0.2A,0.6A,0.05A and conductances 0.1,0.2,0.05 S. Sum currents = 0.85A, sum conductances = 0.35S, so V = 0.85/0.35 = 2.43V.
Worked example
Millman's Theorem – Practical Example
Real‑World| Parameter | Value |
|---|---|
| V₁, R₁ | 5 V, 1 Ω |
| V₂, R₂ | 10 V, 2 Ω |
| V₃, R₃ | 15 V, 3 Ω |
| Formula | V = (Σ Vk/Rk) / (Σ 1/Rk) |
Common mistakes
- I_k: The current contribution of each branch – V_k/R_k, where V_k is the branch source voltage.
- G_k: Conductance of each branch (1/R_k) – not resistance.
- Summation: The numerator is the sum of currents, denominator is the sum of conductances.
- Sign: Use the correct sign for current directions (sources aiding the output are positive).
- Applicability: Only for circuits with multiple parallel voltage sources and series resistances.
Applications
Millman's theorem provides a method for combining multiple parallel voltage sources (each with its series resistance) into a single equivalent voltage source. It is particularly useful in circuits with several branches, such as in power distribution networks or multi‑source analog circuits. Engineers use it to simplify analysis, to determine the voltage at a common node, and to design circuits with redundant supplies. By applying Millman's theorem, they can quickly calculate the equivalent voltage and resistance without solving simultaneous equations. This theorem is a practical tool in circuit analysis, especially when dealing with non‑ideal voltage sources. Understanding Millman's theorem enhances analytical capabilities in complex circuits.
- Analysis of multi‑source circuits and power supplies
- Redundant supply systems and load sharing
- Analog circuit design with multiple reference voltages
- Simplification of network for simulation
- Educational insight into source transformations
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