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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.
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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 =
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For two resistors, the "star" is simply the two resistors in series, and the transform yields a single equivalent resistor.
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into an equivalent network with one less node. The equivalence follows from the
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Bedrosian, S. (December 1961). "Converse of the Star-Mesh
Transformation".
111:{\displaystyle z_{\text{AB}}=z_{\text{A}}z_{\text{B}}\sum {\frac {1}{z}},}
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van Lier, M.; Otten, R. (March 1973). "Planarization by transformation".
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For a single dangling resistor, the transform eliminates the resistor.
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is the impedance between node A and the central node being removed.
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The equivalent impedance betweens nodes A and B is given by:
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The special case of three resistors is better known as the
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identity applied to the
Kirchhoff matrix of the network.
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193:{\textstyle {\frac {1}{2}}N(N-1)}
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