Millman's Theorem¶
A one-junction circuit rule that divides the sum of admittance-weighted known branch voltages and signed current injections by total admittance to obtain the unknown node voltage.
Core Idea¶
Millman's theorem calculates one unknown circuit-junction voltage from known potentials at the far ends of linear branch admittances. If \(Y_i\) is a branch admittance, \(E_i\) its remote potential relative to a common reference and \(V\) the junction potential, Kirchhoff's current law gives \(\sum_iY_i(E_i-V)=0\) and hence \(V=(\sum_iY_iE_i)/(\sum_iY_i)\), provided the total admittance is nonzero. An independent current injection toward the junction adds a signed current term to the numerator.[ref-ad22bcb3b702][ref-d010a86cfd42]
With positive DC conductances the voltage is a conductance-weighted mean of the known driving potentials. With complex AC admittances it is a phasor quotient, not an ordinary real-valued average. This is a special one-unknown case of nodal analysis, not a universal formula for any circuit with sources.[ref-d010a86cfd42][ref-eb6abbab4ce3]
Scope of Application¶
In a DC network from Harvey Mudd's circuit lecture, 32 V through 2 Ω, 20 V through 4 Ω and a passive 8 Ω branch to reference give \(V=(32/2+20/4)/(1/2+1/4+1/8)=24\) V. The passive branch changes the denominator although it contributes no source voltage.[^ref-d010a86cfd42]
Millman's original paper also lists an unbalanced three-phase Y-connected network. In a floating-neutral phasor model, with source-phase potentials \(E_a,E_b,E_c\) and load admittances \(Y_a,Y_b,Y_c\), the neutral displacement is \(V_n=(Y_aE_a+Y_bE_b+Y_cE_c)/(Y_a+Y_b+Y_c)\) under one reference and nonzero denominator. That is the same junction-current rule in a genuinely different electrical setting, not a claim that complex weights form a convex average.[ref-ad22bcb3b702][ref-eb6abbab4ce3]
Clarity¶
The rule makes every source polarity, conductance and passive load visible. A higher-conductance branch weighs more heavily in a resistive case; a zero-admittance open branch contributes nothing. An ideal zero-impedance voltage source is a constraint requiring separate treatment, not a finite term to be inserted as \(1/0\). If total complex admittance vanishes, the quotient cannot by itself determine a unique finite voltage.[^ref-d010a86cfd42]
Manages Complexity¶
The branch-current equations collapse into two sums: total signed driving-current contribution and total admittance. Dividing them replaces a simultaneous solve only if every remote-end potential is already known. A second unknown node or nonlinear branch law invalidates that shortcut and calls for a more general nodal equation.[ref-ad22bcb3b702][ref-d010a86cfd42]
Abstract Reasoning¶
Choose the reference and current-positive direction; mark the single unknown voltage; write each linear branch current as \(Y_i(E_i-V)\) and each independent injection with its signed direction; then apply current balance. Before dividing, test that the total admittance is nonzero. Check the result by substituting it into the original branch-current equation. In a positive-resistance DC case, a result outside the range of all driving potentials, absent extra current injection, warns of a sign or modeling error; no such interval test applies to complex phasors.[ref-d010a86cfd42][ref-eb6abbab4ce3]
Knowledge Transfer¶
The theorem transfers literally between DC practical-generator networks and sinusoidal steady-state three-phase junctions because the one-node KCL and linear admittance roles persist. Generic weighted averaging is only an algebraic relative; it does not preserve current conservation, voltage reference and branch-law conditions. Live Superposition is a different named pattern, and no strict DAG parent is asserted. A wider normalized-balance skeleton remains a future-prime question rather than a reason to call this domain-specific theorem a prime.
[^ref-ad22bcb3b702]: Jacob Millman, “A Useful Network Theorem,” Proceedings of the IRE 28, no. 9 (1940): 413–417, DOI 10.1109/JRPROC.1940.225885; original publisher abstract, full article not directly accessed. https://ieeexplore.ieee.org/document/1687226 [^ref-d010a86cfd42]: Ruye Wang, “Solving Circuits with Kirchhoff Laws,” Harvey Mudd College E84 lecture, Millman's theorem, equations (9)–(10). https://pages.hmc.edu/ruye/e84/lectures/ch2/node2.html [^ref-eb6abbab4ce3]: University of Moratuwa, EE101 Network Theorems (2001/02), §2.3.5, pp. 17–18 of course notes. https://uom.lk/sites/default/files/elect/files/EE101_2_Network_Theorems.pdf
Neighborhood in Abstraction Space¶
Millman's Theorem sits in a sparse region of the domain-specific corpus (76th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Magnetic Circuit — 0.85
- Bridging Fault — 0.85
- Pre-Charge — 0.83
- Image impedance — 0.82
- Network Synthesis — 0.82
Computed from structural-signature embeddings · 2026-10-08