Star-mesh transform
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:
- [math]\displaystyle{ z_\text{AB} = z_\text{A} z_\text{B} \sum\frac{1}{z}, }[/math]
where [math]\displaystyle{ z_\text{A} }[/math] is the impedance between node A and the central node being removed.
The transform replaces N resistors with [math]\displaystyle{ \frac{1}{2}N(N - 1) }[/math] resistors. For [math]\displaystyle{ N \gt 3 }[/math], 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.
Special cases
When N is:
- For a single dangling resistor, the transform eliminates the resistor.
- For two resistors, the "star" is simply the two resistors in series, and the transform yields a single equivalent resistor.
- 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.
See also
References
- van Lier, M.; Otten, R. (March 1973). "Planarization by transformation". IEEE Transactions on Circuit Theory 20 (2): 169–171. doi:10.1109/TCT.1973.1083633.
- Bedrosian, S. (December 1961). "Converse of the Star-Mesh Transformation". IRE Transactions on Circuit Theory 8 (4): 491–493. doi:10.1109/TCT.1961.1086832.
- E.B. Curtis, D. Ingerman, J.A. Morrow. Circular planar graphs and resistor networks. Linear Algebra and its Applications. Volume 283, Issues 1–3, 1 November 1998, pp. 115–150| doi = https://doi.org/10.1016/S0024-3795(98)10087-3.
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