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Modelling and discretization of circuit problems. (English) Zbl 1207.78002
Schilders, W. H. A. (ed.) et al., Handbook of numerical analysis. Vol XIII. Special volume: Numerical methods in electromagnetics. Amsterdam: Elsevier/North Holland (ISBN 0-444-51375-2/hbk). Handbook of Numerical Analysis 13, 523-629 (2005).
Preface: Microelectronics is the core technology for numerous industrial innovations. Progress in microelectronics is highlighted by milestones in chip technology, i.e., microprocessor and memory chips. This ongoing increase in performance and memory density – accompanied with decreasing prices - would not have been possible without extensive use of computer simulation techniques, especially circuit simulation.
An important analysis type in circuit simulators is time domain analysis, which calculates the time-dependent (transient) behaviour of electrical signals in a circuit responding to time-varying input signals. A network description of the circuit is generated automatically in computer-aided electronics-design systems from designer’s drafts or fabrication data files. An input processor translates this network description into a data format reflecting the mathematical model of the system. The mathematical network equations are based on the application of basic physical laws like energy or charge conservation onto network topology and characteristic equations for the network elements. This automatic modeling approach preserves the topological structure of the network and does not aim at systems with a minimal set of unknowns. Hence an initial-value problem of differential-algebraic equations (DAEs) is generated which covers characteristic time constants of several orders of magnitude (stiff equations) and suffers from poor smoothness properties of modern transistor model equations.
In the first part of this article (Chapters I–III) we aim at filtering out the numerical analysis aspects time domain analysis is based on: The numerical integration of the very special differential-algebraic network equations. This task comprises the simulation core of all simulation packages. Although modelling, discretization and numerical integration can be clearly distinguished as different steps, all these levels are strongly interwoven (and therefore also somehow hidden) in commercial packages.
In Chapter I we discuss how these mathematical models are generated on the basis of a network approach with compact (lumped) models. The structural properties of these DAE models can be described by the DAE-index concept. We learn in Chapter II that these properties are fixed by the topological structure of the network model in most cases. However, if more general models for the network elements are incorporated, or refined models are used to include second order and parasitic effects then special circuit configurations may be built, which render ill-conditioned problems. These investigations form the basis for constructing numerical integration schemes that are tailored to the respective properties of the network equations. In Chapter III we describe the direct integration approach based on multi-step schemes, which is used in the extremely wide spread simulator SPICE [W. Nagel, “SPICE 2 – a computer program to simulate semiconductor circuits”, Berkeley, Berkeley: University of California (1975)] and has become a standard since almost 30 years. We include in our discussion a comparison with one-step methods, since recent developments have revealed an interesting potential for such schemes.
The second part (Chapters IV and V) deals with two challenges circuit simulation is faced actually in industry: The simulation of very large circuits with up to millions of transistors such as memory chips on the one hand, and oscillatory circuits with eventually widely separated time constants, appearing in radio frequency (RF) design on the other hand. For the reason of efficiency and robustness, and to make numerical simulation feasible at all, the time domain approach discussed in the first part has to be adapted in both cases. These fields are very much driven by actual industrial needs, and hence are rapidly evolving. So we can in the second part only describe the state of the art, rather than present an established mathematical theory, as is meanwhile available for numerical integration of DAE systems. Nevertheless we hope that the second part as well as the first one helps to get some feeling about the nature of the underlying problems and the attempts to solve them, and may be useful for both mathematical researchers and the users of the codes.
For the entire collection see [Zbl 1064.65001].

##### MSC:
 78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory 78A55 Technical applications of optics and electromagnetic theory 65L80 Numerical methods for differential-algebraic equations 65Y05 Parallel numerical computation
##### Software:
CHORAL; SPECTRE; SPICE; TR-BDF2