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Analysis and design of polygonal resistors by conformal mapping

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Abstract

To compute the electrical resistance (≈ conformal modulus) of a polygonally shaped resistor cut from a sheet of uniform resistivity, it suffices to find a conformal map of the polygon onto a rectangle. Constructing such a map requires the solution of a Schwarz-Christoffel parameter problem. First we show by examples that this is practical numerically. Then we consider an inverse “resistor trimming” problem in which the aim is to cut a slit in a given polygon just long enough to increase its resistance to a prescribed value. We show that here the solution can be obtained by solving a “generalized parameter problem.” The idea of a generalized parameter problem is applicable also in many other Schwarz-Christoffel computations.

Zusammenfassung

Um den elektrischen Widerstand eines polygonalen Resistors aus einem Material homogener Leitfähigkeit zu berechnen, genügt es, eine konforme Abbildung des Polygons auf ein Rechteck zu finden. Die Konstruktion einer solchen Abbildung erfordert die Lösung eines Schwarz-Christof-felschen Parameterproblems. Wir zeigen zunächst anhand von Beispielen, daß dies numerisch durchführbar ist. Dann betrachten wir ein inverses Problem: Die Aufgabe besteht hier darin, einen Schlitz in ein gegebenes Polygon zu schneiden, dessen Länge gerade so gewählt ist, daß der Widerstand auf einen vorgegebenen Wert erhöht wird. Wir zeigen, daß dieses Problem auf ein „verallgemeinertes Parameterproblem“ zurückgeführt werden kann. Die Idee des verallgemeinerten Parameterproblems ist auch auf viele weitere Schwarz-Christoffel-Probleme anwendbar.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Lloyd N. Trefethen

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  1. Lloyd N. Trefethen
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Additional information

Supported by NSF Mathematical Sciences Postdoctoral Fellowship, and by the U.S. Dept. of Energy under contract DE-AC02-76-ER03077-V. This work was performed at the Courant Institute of Mathematical Sciences, New York University.

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Trefethen, L.N. Analysis and design of polygonal resistors by conformal mapping. Z. angew. Math. Phys. 35, 692–704 (1984). https://doi.org/10.1007/BF00952114

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  • DOI: https://doi.org/10.1007/BF00952114

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