Formula & Calculator
Millman's Theorem
Gives the voltage at a common node of several parallel branches, each with its own source and conductance.
Interpretation
Millman's theorem: multiple parallel voltage sources (with series resistances) can be combined into a single equivalent voltage source.
The equivalent voltage is the sum of each branch's current (V/R) divided by the sum of conductances (1/R).
Example: Branches with currents 2A,3A,1A and conductances 0.1S,0.2S,0.05S → V = (2+3+1)/(0.1+0.2+0.05) ≈ 17.14V.
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
Frequently Asked Questions
Millman's theorem gives the voltage at a common node of several parallel branches, each with its own source and conductance: V = (Σ I_k) / (Σ G_k), where I_k are equivalent current sources and G_k are conductances.
I_k = equivalent current source in each branch (voltage source in series with resistance converted to current source), G_k = conductance of each branch (1/R_k).
A voltage source V in series with resistance R becomes a current source I = V/R in parallel with conductance G = 1/R.
It simplifies the analysis of parallel branches to find the voltage at a common node without solving multiple equations.
Yes, using phasor notation and admittances (complex conductances).
If ΣG_k = 0 (no conductance), the voltage is infinite, which is not physically possible; it indicates an open circuit condition.
It is essentially a shortcut for the node voltage method when multiple branches connect to a single node with no series resistance between sources.
It applies only to circuits with independent sources and linear elements, and only for the voltage at a common node.
Analysis of power distribution networks, resistor ladder networks, and circuits with multiple parallel branches.
Common errors include: 1) forgetting to convert voltage sources to current sources, 2) using the wrong sign for sources, 3) not summing conductances correctly, 4) applying the theorem to nodes with series resistance, and 5) using it for non-linear circuits.