High Precision Synthesis of a Richards Immittance Via Parametric Approach

A Richards immitance is a positive real function expressed in terms of the Richards variable λ = tanh(pτ) = Σ+jΩ where p=σ+jω 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 λ 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 10-3. 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.

[1]  Raafat R. Mansour,et al.  Microwave Filters for Communication Systems: Fundamentals, Design and Applications , 2007 .

[2]  Malcolm C. Smith,et al.  A Note on Tests for Positive-Real Functions , 2009, IEEE Transactions on Automatic Control.

[3]  Malcolm C. Smith,et al.  Restricted Complexity Network Realizations for Passive Mechanical Control , 2009, IEEE Transactions on Automatic Control.

[4]  D. Rhodes,et al.  A reactance theorem , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[5]  Kai Wang,et al.  Generalized Series–Parallel $RLC$ Synthesis Without Minimization for Biquadratic Impedances , 2012, IEEE Transactions on Circuits and Systems II: Express Briefs.

[6]  B. Siddik Yarman,et al.  High Precision LC Ladder Synthesis Part I: Lowpass Ladder Synthesis via Parametric Approach , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[7]  Alfred Fettweis,et al.  Computer aided double matching via parametric representation of Brune functions , 1988, 1988., IEEE International Symposium on Circuits and Systems.

[8]  P.I. Richards,et al.  Resistor-Transmission-Line Circuits , 1948, Proceedings of the IRE.

[9]  Binboga Siddik Yarman,et al.  Design of Ultra Wideband Antenna Matching Networks: Via Simplified Real Frequency Technique , 2008 .

[10]  H. J. Carlin,et al.  Distributed circuit design with transmisssion line elements , 1971 .

[11]  B. Siddik Yarman,et al.  High Precision LC Ladder Synthesis Part II: Immittance Synthesis With Transmission Zeros at DC and Infinity , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[12]  D. Pozar.  Microwave Engineering , 1990 .

[13]  H. J. Carlin,et al.  Improved Accuracy for Commensurate-Line Synthesis (Short Papers) , 1976 .

[14]  J S Seeley,et al.  New algorithms for synthesized design of optical filters. , 1985, Applied optics.

[15]  Binboga Siddik Yarman,et al.  Design of Ultra Wideband Power Transfer Networks , 2010 .

[16]  S. Ramo,et al.  Fields and Waves in Communication Electronics , 1966 .

[17]  Kai Wang,et al.  Supplementary Material to: Realizations of a Special Class of Admittances with Strictly Lower Complexity than Canonical Forms , 2015, 1501.04488.

[18]  R. Levy.  Tables of Element Values for the Distributed Low-Pass Prototype Filter , 1965 .

[19]  H. Carlin.  A new approach to gain-bandwidth problems , 1977 .

[20]  Gabor C. Temes,et al.  Filter Design Using Transformed Variables , 1968 .

[21]  J. Bingham.  A New Method of Solving the Accuracy Problem in Filter Design , 1964 .

[22]  Binboga Siddik Yarman,et al.  The double matching problem: Analytic and real frequency solutions , 1983 .

[23]  R. Saal,et al.  On the Design of Filters by Synthesis , 1958 .