Millman's Theorem for Parallel Voltage Sources

When several voltage sources, each with its own series (or output) impedance, feed the same node in parallel — think redundant power rails, paralleled batteries, multiple sensor outputs summed at a node, or a resistive DAC ladder — solving the node voltage by full nodal analysis is more work than it needs to be. Millman's theorem collapses the whole network into a single equivalent voltage source in one algebraic step, which is exactly the kind of shortcut worth having memorized for bench work and quick hand-checks of SPICE results.

Statement of the Theorem

For n branches, each consisting of an ideal voltage source Vᵢ in series with an impedance Zᵢ (or conductance Gᵢ = 1/Zᵢ), all connected in parallel between a common node A and a common reference node (ground), the voltage at node A is:

V_A = (V₁/Z₁ + V₂/Z₂ + ... + Vₙ/Zₙ) / (1/Z₁ + 1/Z₂ + ... + 1/Zₙ)

V_A = ΣGᵢVᵢ / ΣGᵢ      (Gᵢ = 1/Zᵢ)

This is nothing exotic — it is Kirchhoff's Current Law at node A, written after converting each source-plus-series-impedance branch into its Norton equivalent (a current source Iᵢ = Vᵢ/Zᵢ in parallel with Zᵢ), summing the currents, and summing the conductances. Millman's theorem is a named shortcut for a Norton-equivalent KCL sum, not a separate physical law.

If a branch is a bare resistor to ground with no source (Vᵢ = 0), it still contributes its conductance to the denominator — it just adds zero to the numerator. If a branch has zero series impedance (an ideal source direct to the node), Zᵢ → 0 makes that branch's Vᵢ dominate completely, forcing V_A = Vᵢ, which is the physically correct result (an ideal source pins the node).

Derivation from KCL

Convert every branch to its Norton equivalent and sum currents into node A, defining all currents as flowing into the node:

I₁ + I₂ + ... + Iₙ = 0   (with node A also loaded by ΣGᵢ back to ground)

Each Norton current source is Iᵢ = Vᵢ/Zᵢ = Vᵢ·Gᵢ, and each Norton resistance Zᵢ sits between node A and ground. KCL at node A:

ΣIᵢ = V_A · ΣGᵢ
ΣGᵢVᵢ = V_A · ΣGᵢ
V_A = ΣGᵢVᵢ / ΣGᵢ

This confirms the formula is just one weighted average: each source votes for the node voltage, weighted by how strongly it's coupled (its conductance). A low-impedance source dominates; a high-impedance source barely moves the result.

Worked Example

Three branches feed node A, referenced to ground:

(Branch 3 is a plain 50 Ω resistor to ground — no source, but it still loads the node.)

Numerator: ΣGᵢVᵢ = (0.1000×5) + (0.0500×3) + (0.0200×0) = 0.5 + 0.15 + 0 = 0.65

Denominator: ΣGᵢ = 0.1000 + 0.0500 + 0.0200 = 0.1700 S

V_A = 0.65 / 0.1700 = 3.8235 V

Check by direct KCL at node A (currents into node A must sum to zero):

(5 − 3.8235)/10 + (3 − 3.8235)/20 + (0 − 3.8235)/50 = ?
= 0.11765 + (−0.04118) + (−0.07647)
= 0.11765 − 0.04118 − 0.07647 = 0.00000  ✓

The three branch currents sum to zero, confirming V_A = 3.8235 V is correct. Branch 1 (lowest impedance, highest source voltage) pulls current into the node; branches 2 and 3 pull current out, exactly balancing.

Equivalent Thevenin Source

Millman's result is precisely the Thevenin equivalent of the parallel combination, seen from node A:

V_TH = V_A = ΣGᵢVᵢ / ΣGᵢ
R_TH = 1 / ΣGᵢ = 1 / (0.1700) = 5.882 Ω

Any external load placed at node A now sees a single source (3.8235 V) with a single series resistance (5.882 Ω) — exactly the simplification Thevenin's theorem promises, arrived at faster because all sources already share the same two nodes.

Where the Theorem Applies — and Where It Doesn't

• Applies directly: any set of branches that are truly in parallel between the same pair of nodes — one being the node of interest, the other a common reference (ground). This includes resistor ladders with reference voltages (as in a flash ADC or resistive DAC), paralleled supply rails through current-limiting/balancing resistors, and multiple sensor outputs summed through fixed source impedances.
• Does not apply directly if any branch contains a dependent source whose controlling variable is the node voltage itself (needs re-derivation, though the same KCL approach still works), or if branches are coupled through mutual impedance (transformers, coupled inductors). It also does not apply if the sources connect to different pairs of nodes — that's a general nodal-analysis problem, not a Millman one.
• Ideal sources only in the strict statement — but "ideal source + series impedance" is exactly the Thevenin model of any real source, so real batteries, regulated rails with output impedance, and driven op-amp outputs all fit the form directly.

Practical Design Implications

• Paralleling supplies or batteries: if two rails at slightly different voltages are tied together through balancing resistors, Millman's formula gives the shared-node voltage instantly — useful for redundant power architectures and for sizing balancing resistors to control cross-current.
• Resistor-ladder references (DACs, R-2R networks): each ladder tap can be modeled as a Millman node summing weighted contributions from reference voltages, letting you compute an intermediate node voltage without solving the full matrix.
• Sensor/signal summing nodes: when multiple low-impedance sources (e.g., op-amp outputs) drive a shared node through resistors — a resistive summing junction — Millman's theorem directly gives the summed/weighted output before any active buffering.
• Sanity-checking simulations: because the formula is a single line of arithmetic, it's a fast way to hand-verify a SPICE .op result at a summing node without re-deriving the whole netlist.

Key Takeaways

• Millman's theorem gives the voltage at a common node fed by parallel voltage sources (each with series impedance) as a conductance-weighted average: V_A = ΣGᵢVᵢ / ΣGᵢ.
• It is derived directly from KCL after converting each branch to its Norton equivalent — not an independent law, but a fast shortcut for that specific topology.
• The equivalent Thevenin resistance seen at the node is R_TH = 1/ΣGᵢ, so the result doubles as a Thevenin-equivalent simplification.
• It applies only when all branches share the same two nodes; dependent sources tied to the node voltage or mutual coupling between branches break the direct formula.
• Typical uses: paralleled supply/battery rails, resistor-ladder reference networks, and resistive summing junctions — anywhere several source-plus-impedance branches land on one shared node.