High Precision Synthesis of a Richards Immittance Via Parametric Approach

Abstract

A Richards immitance is a positive real function expressed in terms of the Richards variable <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\lambda = \tanh (p\tau) = \Sigma +j\Omega$</tex></formula> where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$p=\sigma +j\omega$</tex></formula> is the classical complex frequency. A Richards immittance can be synthesized as a lossless two port terminated in a resistance as in Darlington's synthesis such that the two-port consists of commensurate transmission lines. In this paper, a high precision method is presented to synthesize a Richards immittance as a lossless two-port constructed with cascade connections of equal length transmission lines, as well as short and open stubs. The new method of synthesis utilizes Bode procedure (or Parametric Method) to correct an immitance function specified in the complex Richards variable <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\lambda$</tex> </formula> at each step of the synthesis. It is verified that new technique can synthesize a randomly generated Richards immitance function yielding 25 commensurate lines with the accumulated numerical error less than <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$10^{-3}$</tex></formula> . A complete synthesis package is developed in MatLab and successfully integrated with the Real Frequency Technique to design broadband matching networks. Examples are presented to show the merits of the new Richards synthesis tool.