Modelling and Discretization of Circuit Problems

doi:10.1016/S1570-8659(04)13006-8

Handbook of Numerical Analysis

Volume 13, 2005, Pages 523-659

https://doi.org/10.1016/S1570-8659(04)13006-8 Get rights and content

Publisher Summary

This chapter discusses the modeling aspect of differential-algebraic equations (DAEs). In computational engineering, the network modeling approach forms the basis for computer-aided analysis of time-dependent processes in multibody dynamics, process simulation, or circuit design. Its principle is to connect compact elements via ideal nodes, and to apply some kind of conservation rules for setting up equations. The mathematical model, a set of so-called network equations, is generated automatically by combining network topology with characteristic equations describing the physical behavior of network elements under some simplifying assumptions. Interconnects and semiconductor devices (i.e., transistors) are modeled by multi-terminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities. Interconnects and semiconductor devices (i.e., transistors) are modeled by multiterminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities.

Section snippets

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

DAE-Systems – the Modelling Aspect

In computational engineering the network modelling approach forms the basis for computer-aided analysis of time-dependent processes in multibody dynamics, process simulation or circuit design. Its principle is to connect compact elements via ideal nodes, and to apply some kind of conservation rules for setting up equations. The mathematical model, a set of so-called network equations, is generated automatically by combining network topology with characteristic equations describing the physical

DAE-index – the Structural Aspect

So we are faced with network equations of differential-algebraic type when simulating electrical circuits. Before attacking them numerically, we have to reveal the analytical properties of DAEs. In a first step we inspect linear systems and apply, in a second step, the results to nonlinear systems. We will see that for a rather general class of circuits the network topology determines the structural properties of the DAE network equations. However, if more general models for the network

Numerical Integration Schemes

The numerical integration of the network equations defines (at least from a mathematical point of view) the kernel of simulation packages in circuit design. This chapter does not aim at an introduction into numerical integration schemes for DAE systems: Neither in theory (convergence and stability) nor in general aspects of implementation (adaptivity, solution of nonlinear and linear systems). For this, the reader may consult a bunch of excellent textbooks Ascher and Petzold, 1998, Brenan,

Numerical Treatment of Large Problems

Due to their reliability and robustness software codes employing the standard algorithms are established as workhorses, which are inevitable when designing electronic circuits. Especially for integrated circuit design one can distinguish two different steps in the design flow, where these tools are used:

•
The electrical design stage comprises standard applications for characterization and optimization of functional building blocks, such as gates, operational amplifiers, oscillators etc. These

Periodic Steady-State Problems

Periodic Steady-State (PSS) Problems have received special attention for simulating analog circuits. The aim was to efficiently study solutions of problems where a highly oscillating signal (carrier) was modulated by another signal. Due to nonlinear components the response to a single tone may give rise to higher harmonics, which in general is considered as (harmonic) distortion. When two tones are considered, intermodulation distortion may arise. Then an IC-designer is interested in detecting

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A second order multirate scheme for the evolutionary coupled Stokes–Darcy problem is proposed and analyzed. The scheme decouples the problem into two smaller subproblems at each time step by exploiting the mismatch in flow activity between the domains. Specifically, the slower Darcy subproblem is integrated with a larger time step compared to the more active Stokes subproblem. This avoids expending unnecessary computational effort on sampling the slow changing Darcy solution. The slow varying Darcy solution is extrapolated/interpolated onto the fine grid for the integration of the Stokes subproblem. The long-time stability and convergence of the method is proved under a time step restriction that depends on physical parameters and the ratio of the time steps in each subdomain. The scheme includes a stabilization term that relaxes the time step restriction. Numerical experiments confirm convergence of the method and the effectiveness of the stabilization technique at relaxing the stability time step restriction.
Sensitivity analysis of random linear differential–algebraic equations using system norms
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We consider linear dynamical systems composed of differential–algebraic equations (DAEs), where a quantity of interest (QoI) is assigned as output. Physical parameters of a system are modelled as random variables to quantify uncertainty, and we investigate a variance-based sensitivity analysis of the random QoI. Based on expansions via generalised polynomial chaos, the stochastic Galerkin method yields a new deterministic system of DAEs of high dimension. We define sensitivity measures by system norms, i.e., the $H_{\infty}$ -norm of the transfer function associated with the Galerkin system for different combinations of outputs. To ameliorate the enormous computational effort required to compute norms of high-dimensional systems, we apply balanced truncation, a particular method of model order reduction (MOR), to obtain a low-dimensional linear dynamical system that produces approximations of system norms. MOR of DAEs is more sophisticated in comparison to systems of ordinary differential equations. We show an a priori error bound for the sensitivity measures satisfied by the MOR method. Numerical results are presented for two stochastic models given by DAEs.
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We consider the efficient numerical solution of coupled dynamical systems, consisting of a low dimensional nonlinear part and a high dimensional linear time invariant part, e.g., stemming from spatial discretization of an underlying partial differential equation. The linear subsystem can be eliminated in frequency domain and for the numerical solution of the resulting integro-differential algebraic equations, we propose a combination of Runge–Kutta or multistep time stepping methods with appropriate convolution quadrature to handle the integral terms. The resulting methods are shown to be algebraically equivalent to a Runge–Kutta or multistep solution of the coupled system and thus automatically inherit the corresponding stability and accuracy properties. After a computationally expensive pre-processing step, the online simulation can, however, be performed at essentially the same cost as solving only the low dimensional nonlinear subsystem. The proposed method is, therefore, particularly attractive, if repeated simulation of the coupled dynamical system is required.
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Therefore different scheme were tested. For the integration of the electric part CHORHAL [16], MEBDF [10] as well as Adaptive Runge Kutta (Implicit Explicit) (ARKIMEX L2,2e,3 to 5) [14,21,31] were suitable and yielded good integration properties. The MEBDF and ARKIMEX schemes yielded better results than CHORHAL12 for the coupled system (charge transport, mass transport and crystal growth).
Modeling the performance of electrochemical storage systems divides mostly into two approaches; (extended) impedance bases models, which can model the fast dynamics very good but fail for processes with long time constants, and physical-chemical models which usually neglects the fast dynamics and focus on the slower time constants. This works aims to introduce the faster dynamics into a physical-chemical model for lead-acid batteries, which can help to understand the processes inside the battery during e.g. a primary control events. This parts covers the basic model on which future works will build on.
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Algebraic stiffness, which directly stems from large differences in the timescales of the underlying equations. Some examples are discretized convection–diffusion–reaction equations [3], molecular dynamics [4] or electrical circuits [5]. Grid-induced stiffness, which occurs in the spatial discretization of partial differential equations (PDEs) with non-uniform meshes.
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Modelling and Discretization of Circuit Problems

Publisher Summary

Section snippets

Preface

DAE-Systems – the Modelling Aspect

DAE-index – the Structural Aspect

Numerical Integration Schemes

Numerical Treatment of Large Problems

Periodic Steady-State Problems

Comp. Math. Appl.

J. Comput. Appl. Math.

Appl. Numer. Math.

Appl. Numer. Math.

Comp. Appl. Math.

Zur Theorie und zur numerischen Lösung von Anfangswertproblemen für differentiell-algebraische Systeme von höherem Index

Preconditioned dynamic iteration for coupled differential-algebraic systems

BIT

Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations

Transient simulation of silicon devices and circuits

IEEE Trans. Comp. Aided Des. CAD

XPSim: A MOS VLSI simulator

A parallelizable GMRES-type method for p-cyclic matrices with applications in circuit simulation

SCEPTRE: A Computer Program for Circuit and System Analysis

A simulation tool for the analysis and verification of the steady state of circuit designs

Int. J. Circ. Theory Appl.

Numerical steady state analysis of electronic circuits driven by multi-tone signals

Electr. Eng.

A time-frequency algorithm for the simulation of the initial transient response of oscillators

Analyse des transienten Verhaltens von Oszillatoren durch eine inverse Charakteristikenmethode

On consistent initial conditions for circuit's DAEs with higher index

IEEE Trans. Circ. Syst. CAS I

Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations

Simulation of autonomous circuits in the time domain

A stable, accurate method of numerical integration for nonlinear systems

Proc. IEEE

Computer Aided Network Design

MOTIS – An MOS timing simulator

IEEE Trans. Circ. Syst. CAS

Finding zeros of maps: Homotopy methods that are constructive with probability one

Math. Comput.

Computer-Aided Analysis of Electronic Circuits: Algorithms & Computational Techniques

Nonlinear circuits

IEEE Trans. Circ. Syst. CAS

New implicit integration method for efficient latency exploitation in circuit simulation

IEEE Trans. Comp. Aided Des. CAD

Direct circuit simulation algorithms for parallel processing

IEEE Trans. Comp. Aided Des. CAD

Waveform techniques

Mixed mode circuit simulation techniques and their implementation in DIANA

Decomposition techniques for large scale circuit analysis and simulation

Phase noise in oscillators: DAEs and coloured noise sources

Computing phase noise eigenfunctions directly from Harmonic Balance/shooting matrices

Phase noise in oscillators: A unifying theory and numerical methods for characterization

IEEE Trans. Circ. Syst. CAS I

An improved numerical integration method in the circuit simulator SPICE2-S

Mathematical models in electric circuit simulation and their numerical treatment

Accurate simulation of phase noise in oscillators

Numerische Mathematik II

Adaptively controlled explicit simulation

IEEE Trans. Comp. Aided Des. CAD

Parallel algorithms for solving linear equations in VLSI circuit simulation