This paper provides a formalized basis for initialization in the fractional calculus The intent is to make the fractional calculus readily accessible to engineering and the sciences A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization Conceptualizations of fractional derivatives and integrals are shown Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions A new generalized form for the Laplace transform of the generalized differintegral is derived The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions

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Initialization, conceptualization, and application in the generalized (fractional) calculus.

In this paper, we introduce a general method of interconnect simulation based on distributed circuits. The algorithm is very efficient and consists of two main steps. In the first step, each distributed line is modeled by a finite order system with passivity preservation and multipoint moment matching of its input admittance/impedance matrix. In the second step, an Arnoldi-based congruence transform is applied to the network to form its reduced order model. The main feature of the algorithm is in its first step, where a passive multipoint moment matching model of a distributed line can be generated without any discretization of the line. We provide an integrated-congruence transform which can be directly applied to the partial differential equations of a distributed line and generate a passive finite order system as its model. We also provide an algorithm based on the L/sup 2/ Hilbert space theory so that exact moment matching at multiple points can be obtained, We demonstrate the accuracy of our method with examples and show the advantage of ours over conventional ones based on lumped circuit models.

Passive multipoint moment matching model order reduction algorithm on multiport distributed interconnect networks

This paper proposes a circuit model for lossy multiconductor transmission lines (MTLs) suitable for implementation in modern SPICE simulators, as well as in any simulator supporting differential operators. The model includes the effects of a uniform or nonuniform disturbing field illuminating the line and is especially devised for the transient simulation of electrically long wideband interconnects with frequency dependent per-unit-length parameters. The MTL is characterized by its transient matched scattering responses, which are computed including both dc and skin losses by means of a specific algorithm for the inversion of the Laplace transform. The line characteristics are then represented in terms of differential operators and ideal delays to improve the numerical efficiency and to simplify the coding of the model in existing simulators. The model can be successfully applied to many kinds of interconnects ranging from micrometric high-resistivity metallizations to low-loss PCBs and cables, and can be considered a practical extension of the widely appreciated lossless MTL SPICE model, which maintains the simplicity and efficiency.

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Analysis of crosstalk and field coupling to lossy MTLs in a SPICE environment

Electronic system design requires noise and delay analyses of lossy, low-loss, frequency dependent, and coupled transmission lines, along with lumped parasitic elements. Numerous models and techniques have been developed for simulating these interconnect circuits, however, no single method represents the optimal approach for all transmission line problems. Accuracy and efficiency considerations would suggest the use of different models and analyses for the various types of interconnects. Such an approach to simulating transmission lines requires a measure of goodness for selecting the appropriate model and analysis in terms of the circuit parameters. Toward this goal, this paper presents an efficient methodology for characterizing transmission line models into solution domains. This provides an efficient technique for automatically selecting the type of interconnect model and analysis. Specifically, for a specified acceptable modeling error, analysis domains are established for automatic model selection between the classical method of characteristics, a lumped modeling approach and the state-based convolution technique within a simulation and analysis tool. In general, however, this approach can be applied to any combination of models and analyses, if a measure of accuracy and efficiency can be established.

Domain characterization of transmission line models and analyses

This paper presents an on-chip, interconnect-aware methodology for high-speed analog and mixed signal (AMS) design which enables early incorporation of on-chip transmission line (T-line) components into AMS design flow. The proposed solution is based on a set of parameterized T-line structures, which include single and two coupled microstrip lines with optional side shielding, accompanied by compact true transient models. The models account for frequency dependent skin and proximity effects, while maintaining passivity requirements due to their pure RLC nature. The signal bandwidth supported by the models covers a range from DC to 100 GHz. The models are currently verified in terms of S-parameter data against hardware (up to 40 GHz) and against EM solver (up to 100 GHz). This methodology has already been used for several designs implemented in SiGe (silicon-germanium) BiCMOS technology.

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An Interconnect-Aware Methodology for Analog and Mixed Signal Design, Based on High Bandwidth (Over 40 Ghz) On-Chip Transmission Line Approach

We present circuit models of a length of transmission line with skin effect and frequency-dependent dielectric loss, treating the ordinary RLGC line as a special case. The line is modeled as a characteristic two-port in which the characteristic impedance is approximated as the input impedance of a lumped circuit and the propagation function is approximated as the transfer function of a second lumped circuit plus an ideal delay. A unified approach based on frequency matching is presented to form the model. In the approximation, the skin effect is represented as the impedance of an RL ladder, and the complex dielectric parameter as the admittance of an RC ladder. A table of the rms error of the approximation is derived from which the complexity of the model can be found for a desired accuracy. The model is guaranteed to be stable, and since it consists entirely of lumped circuit elements and ideal delay lines and since the model parameters can be easily computed, it can be incorporated as a subcircuit model in a general circuit simulator, and the solution can be obtained by integrating the circuit equations in the usual manner. No FFT or convolution is required. >

Qingjian Yu

Papers

Computational models of transmission lines with skin effects and dielectric loss