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Star-mesh transform

Mathematical circuit analysis technique

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated December 11, 2025
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The star-mesh transform, or star-polygon transform, is a mathematical circuit analysis technique to transform a resistive network into an equivalent network with one less node. The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network.

The equivalent impedance betweens nodes A and B is given by:

z_{\text{AB}}=z_{\text{A}}z_{\text{B}}\sum {\frac {1}{z}},

where z_{\text{A}} is the impedance between node A and the central node being removed.

The transform replaces N resistors with {\textstyle {\frac {1}{2}}N(N-1) resistors. For {\textstyle N>3, the result is an increase in the number of resistors, so the transform has no general inverse without additional constraints.

It is possible, though not necessarily efficient, to transform an arbitrarily complex two-terminal resistive network into a single equivalent resistor by repeatedly applying the star-mesh transform to eliminate each non-terminal node.

01Special cases

When N is:

  1. For a single dangling resistor, the transform eliminates the resistor.
  2. For two resistors, the "star" is simply the two resistors in series, and the transform yields a single equivalent resistor.
  3. The special case of three resistors is better known as the Y-Δ transform. Since the result also has three resistors, this transform has an inverse Δ-Y transform.
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Sources and credits

This article is adapted from the Wikipedia article “Star-mesh transform”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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