On the Minimality and Uniqueness of the Bott–Duffin Realization Procedure

This paper establishes some new results on the Bott-Duffin procedure-a fundamental result in both passive network synthesis and linear systems theory. Since its appearance in 1949, this procedure has puzzled circuit and systems researchers as the networks produced contain an apparently extravagant number of elements. In this paper, we prove that the networks obtained by the Bott-Duffin procedure contain the least number of reactive elements (six) and the least number of resistors (two) among all series-parallel networks realizing a certain type of impedance function (a biquadratic minimum function). We further show that the Bott-Duffin procedure, in combination with some simple network transformations, can be used to produce all of the series-parallel networks containing exactly six reactive elements and two resistors which realize any specified biquadratic minimum function. In the course of the argument, we prove several additional interesting results pertaining to series-parallel realizations of minimum functions. The results are motivated by the application of the inerter device to passive mechanical control.