A semi‐analytic and cellular approach to rational system characterization through equivalent circuits

Abstract

The partial fraction form of linear time-invariant system transfer function is characterized through a cellular perspective, where each pole/residue fraction term is transformed into an equivalent circuit branch via an exact transformation. Minimal expressions for transformation of partial fraction form to/from equivalent circuit form are provided. The time-domain and frequency-domain impedance and admittance transfer function for resistor (R)inductor (L), and resistance (R), inductance (L), capacitance (C), and conductance (G) equivalent circuit branches are presented in a form that is amenable to expedient inspection of cellular causality and stability and used to derive explicit expressions for the average power amenable to inspection of cellular passivity. The characteristics of passivity, causality, and stability at the cellular level are discussed to gain insight into the macro-level network characteristics. Numerical examples are given to elucidate the aforementioned concepts and to provide insight into the behavior of linear time-invariant systems. A. ZADEHGOL R 1 ı.t/dt D 1. The time-derivative of the Dirac Delta distribution is ı 0 .t/ D d dt ı.t/. The exponential function exp.x/ D e x , where e 2:71828 is the base of the natural logarithm function. The

Passivity verification and enforcement—A review paper

“…() Pordanjani et al proposed a method that transforms the frequency domain response into an equivalent circuit consisting of passive components only, whereas Yamin and Zadehgol describes a method that adds resistance to the equivalent circuit of the frequency response so that the system becomes passive yet the response being minimally changed. The frequency response is transformed into equivalent circuit using the conversions stated in and . …”

Pole residue equivalent system solver (PRESS)

“…If higher accuracy is required, then additional iterations may be performed which adds more partial fraction terms to H sum . We compare PRESS against method of relaxed VF [7], by fitting Zu and synthesizing an equivalent circuit [5] from each method, then excite each circuit by a 1 V step source of IOns rise-time in the ADS circuit simulator [8]; there is excellent agreement in responses from the circuit based on PRESS and the circuit based on relaxed VF, as shown in Figure 7. …”

“…In other applications where time domain SPICE based simulation is necessary, the frequency domain models are processed further [2]; for example, performance of package interconnects [3] can be assessed only in time domain by running a long sequence of Is and Os to evaluate the actual degradation caused to eye diagrams [4]. In such cases, the frequency domain model needs to be converted to a pole/residue representation before being synthesized [5] as lumped element network. Many pole-zero fitting methods exist; für example, the well-known method of vector fitting (VF) [6].…”

A novel iterative algorithm for approximating equivalent circuits of numerical transfer functions

“…The above rational TF H(s) may be written as a sum of partial fractions and realized as an equivalent circuit. 6 If circuit synthesis is required, then a convenient choice is to have H(s) represent the admittance transfer function, because of the ease in adding the admittance of parallel networks. This can be done without loss of generality, given that converting between network parameters (e.g., impedance, admittance, and scattering) is a straightforward process.…”

“…Recently, we presented a novel algorithm pole residue equivalent system solver (PRESS) 4,5 that uses the given frequency response of a known system, to interpolate the poles and residues of an equivalent rational transfer function that approximates the given response; subsequently, the poles/residues may be used to synthesize the equivalent circuit model. 6 The method iterates until a specified target error is reached while offering the unique advantage of being amenable to parallel and simultaneous computation of operations across all frequency sample points within each iteration, thus, enabling solution of arbitrarily large multiport systems 7 and reducing time to solution. The proposed algorithm directly handles sampled data based on arbitrary sampling schemes including linear, logarithmic, and adaptive.…”