Synthesis of reciprocal piecewise-linear n-port resistors

Every reciprocal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port resistor represented by a continuous <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional piecewise-linear function as described in (12) is shown to be realizable by a circuit containing only 2-terminal piecewise-linear resistors and a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(p + q)</tex> -port transformer. An explicit circuit realization is given along with illustrative examples. The necessary and sufficient conditions under which this realization contains only passive elements are also given.