The Bott-Duffin synthesis of electrical circuits. (English) Zbl 1201.94146

Hubbard, John H.

Kotiuga, P. Robert (ed.), A celebration of the mathematical legacy of Raoul Bott. Based on the conference, CRM, Montreal, Canada, June 9–13, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4777-0/pbk). CRM Proceedings and Lecture Notes 50, 33-40 (2010).

R. Bott and R. J. Duffin [J. Appl. Phys. 20, 816 (1949)] proved:
Theorem: A rational function is the impedance of a passive circuit if and only if it has a real coefficients and maps the right half-plane to itself.
In the paper under review, the author not only recalls the ingredients of the proof of that so-called synthesis theorem, but he also reads it in terminology understandable to mathematicians. Moreover, he derives it here from P.I. Richards’ theorem [Duke Math. J. 14, 777–786 (1947; Zbl 0029.25705)]. In addition the author shows that Richard’s theorem is in fact Schwarz’s lemma in “light disguise” (quoted from the author’s paper).
This very nice paper gives nice connections between electronic circuits theory and rational functions with real coefficients that map the right halve plane to itself.
For the entire collection see [Zbl 1186.00040].

Reviewer: R. W. van der Waall (Huizen)

MSC:

94C05	Analytic circuit theory
01A70	Biographies, obituaries, personalia, bibliographies
94-03	History of information and communication theory
30C30	Schwarz-Christoffel-type mappings

Keywords:

rational function; Schwarz’s lemma; impedance of a passive circuit; laws of Kirchhoff; Ohm and Lenz

Citations:

Zbl 0029.25705