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:
where is the impedance between node A and the central node being removed.
The transform replaces N resistors with resistors. For , 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 casesedit
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.
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.
January 01, 1970
star, mesh, transform, star, mesh, transform, star, polygon, transform, mathematical, circuit, analysis, technique, transform, resistive, network, into, equivalent, network, with, less, node, equivalence, follows, from, schur, complement, identity, applied, ki. 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 AB z A z B 1 z displaystyle z text AB z text A z text B sum frac 1 z where z A displaystyle z text A is the impedance between node A and the central node being removed The transform replaces N resistors with 1 2 N N 1 textstyle frac 1 2 N N 1 resistors For N gt 3 textstyle N gt 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 Special cases editWhen 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 D transform Since the result also has three resistors this transform has an inverse D Y transform See also editTopology of electrical circuits Network analysis electrical circuits References editvan 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 Retrieved from https en wikipedia org w index php title Star mesh transform amp oldid 1081464152, wikipedia, wiki, book, books, library,
