Scientific Computing in Electrical Engineering

Nowadays, after more than a century of inconsiderate divergence between electromagnetic and mechanical field theories, we find it hard to bring them together. This can be best exemplified by the problematic status of the electrodynamics of deformable media. The blame can be laid mainly on the limitations of the underlying theoretical frameworks and on the practitioners’ education, too narrow to bridge the gap between them. I would like to concentrate here on the first problem—even though I am convinced that the second one carries more weight.

In this article, we describe two types of conservative variational techniques that aim at improving the use of FDTD methods for the treatment of complex geometries with time dependent Maxwell’s equations.

RF circuits and systems are gaining importance because we are moving further into a society where information is very important and should be available any time and anywhere. In this paper we give an overview of RF circuit simulation with an emphasis on noise simulation which is important functionality for RF designers. Due to the high frequency signals, the standard circuit formulation using Kirchhoff and lumped elements is not sufficient anymore to accurately predict the behaviour of a design and Maxwell’s equations should be used. We give several approximations of Maxwell’s equations and scenarios how the results can be incorporated in RF circuit simulation.

We review some recent extensions of the Finite Integration Technique (FIT), which is known to be a generalization of the Finite Difference Time Domain (FDTD) method. Some shortcomings of the standard formulation are discussed which limit the applicability or at least the efficiency of the time domain variant of FIT. The novel developments which are proposed in this paper cover both the basic geometrical modeling in space and time and advanced methods to solve the algebraic problems in time and frequency domain. A numerical application is presented to demonstrate the performance of the algorithms in the high frequency regime.

The paper explores a class of “Finite Element Difference” (FED) schemes with Finite Difference-type data structures but based on Finite Element — variational principles. Curved material boundaries are approximated algebraically on relatively coarse regular rectangular or hexahedral grids by a judicious choice of local approximating functions, rather than geometrically on conforming meshes. The grids do not have to resolve small geometric details. The proposed approach combines the ideas of the Generalized Finite Element — Partition of Unity methods, Discontinuous Galerkin Methods and Finite Difference / Finite Volume / Finite Integration Techniques.

In refined network analysis, a compact network model is combined with drift-diffusion models for the semiconductor devices which are part of the network, in a multiphysics approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that combines the differentialalgebraic network equations of the circuit with elliptic boundary value problems modelling the diodes. For this mixed initial-boundary value problem of partial differential-algebraic equations a first existence result is given, based on a nonstandard application of Schauder’s fixed point theorem.

Electric circuits designers are frequently interested in the transient behaviour of the designed circuit. A common method for time integration of the Differential Algebraic circuit Equations (DAE) is the Backward Differentiation Formula (BDF) method. In 1983, J. Cash proposed the Modified Extended BDF (MEBDF) method, which combines better stability properties and higher order of convergence than BDF, but requires more computations per step. We prove reduction of convergence order for MEBDF when applied to DAE’s with higher DAE-index. However, because in practice, in circuit analysis, the DAE-index does not exceed 2, the reduction is quite moderate and it equals the BDF-order in that case. One gains better, or even unconditional, stability. One also obtains consistent solutions.

In this paper we present a theoretical foundation for tail electron hydrodynamical models (TEHM) in semiconductors with application to bulk silicon.

Thermal effects influence the electrical behaviour of circuits more and more. Therefore it is necessary to take power dissipation and temperature evolution into account. In order to analyize large systems of integrated circuits, this has to be realized very efficiently. Thus we introduce a thermal network model consisting of 0D and 1D thermal elements approximating the full heat aspect, but keeping the system relatively small. After semi-discretization, this approach yields a coupled DAE system. According to the largely differing time scales, we outline the basics of a multirate co-simulation algorithm, which bases on an averaging technique. Its potential and feasibility is demonstrated on a simple, however, instructive testcircuit. As an outlook we discuss the application to thermal models of SOI-chips.

For solving sparse linear systems from circuit simulation whose coefficient matrices include a few dense rows and columns, a parallel Bi-CGSTAB algorithm with distributed Schur complement (DSC) preconditioning is presented. The parallel efficiency of the solver is increased by transforming the equation system into a problem without dense rows and columns as well as by exploitation of parallel graph partitioning methods. The costs of local, incomplete LU decompositions are decreased by fill-in reducing reordering methods of the matrix and a threshold strategy for the factorization. The efficiency of the parallel solver is demonstrated with real circuit simulation problems on a PC cluster.

This paper focuses on the development of a model to obtain qualitative insight in the behaviour of large, but finite, phased arrays of microstrip antennas. This model concerns a finite array of simple elements, namely perfectly conducting, infinitely thin, narrow rings, excited by voltage gaps and positioned in free or half space. The currents on the rings, and from that the electromagnetic field, are calculated by a moment method. Dimension analysis is carried out to reduce numerical effort and to acquire insight in the behaviour of the array. The qualitative analysis shows promising results and although numerically a brute force method has been applied, CPU times are still acceptable.

The idea of modelling space as two interacting equivalent networks, one for currents, one for magnetic fluxes, pervades computational electromagnetics since its beginnings. The Yee scheme, the TLM method, can thus be interpreted. But this is also true of finite element- or finite volume-inspired more recent proposals, as we show, so the idea is not incompatible with “unstructured” meshes. Yet, meshes with some rotational and translational symmetry (locally, at least) are desirable on many accounts. The tetrahedral Sommerville mesh we describe here, able to fit curved boundaries and yet regular, looks like an interesting compromise.

A novel substrate coupling simulation tool named SubCALM is presented. It is well suited to floorplanning of large mixed-signal designs since it exploits the boundary element method and contains a Poisson solver based on a hierarchical O(n) conjugate gradient algorithm. Sophisticated preconditioners are applied, which further increase the computation speed by a factor of about 10. The approach is verified by experimental results in a 0.25 μm BiCMOS technology.

Widely seperated time scales appear in many electronic circuits, making analysis with the usual numerical methods very difficult and costly. In this article we present a quasilinear system of partial differential equations (PDE) of first order, where the time scales are treated seperately. The PDE corresponds to the system of differential-algebraic equations (DAE) describing the electronic circuit in the sense that the solution of the PDE restricted to one of its characteristics is the solution of the DAE. This embedding method is described in a general setting. Hence it can be used for various applications in circuit simulation.Since generalized quasiperiodic functions, which are presented here, conceptualize physical properties, they have a basic significance for the embedding method.Theoretical investigations are presented as well as new approaches for numerical methods based on the connection between the PDE and the DAE.

In this work we deal with the numerical simulation of thermal oxidation in silicon device technology. This application is a complex coupled phenomen, involving the solution of a diffusion-reaction problem and of a fluid-structure interaction problem. Suitable iterative procedures are devised for handling nonlinearities and strong coupling between the sub-problems to be solved. In particular, we propose a unified dual-mixed hybrid formulation that allows for the simultaneous solution of the compressible/incompressible Navier equations in both solid and fluid domains. The accuracy and the flexibility of the proposed approach are demonstrated on benchmark test problems.

Nonlinear transient eddy current simulations require the solution of nonlinear differential-algebraic systems of equations of index 1, for which linearimplicit time marching methods of Rosenbrock-type are proposed. These methods avoid the iterative solution of nonlinear systems within each time step due to their built-in Newton procedures. Embedded lower order schemes allow an error-controlled adaptive time step selection to take into account the nonlinear dynamics of the underlying process. Extrapolation methods used for start value generation include a new subspace projection method to improve the numerical performance of the simulations.

The finite element model of a superconductive dipole magnet is equipped with a specialised conductor model which accounts for the inter-strand currents caused by the ramping of the magnet without explicitly meshing the individual strands.

New modeling technology is developed that allows engineers to define the frequency range, layout parameters, material properties and desired accuracy for automatic generation of simulation models of general passive electrical structures. It combines electromagnetic (EM) accuracy of parameterized passive models with the simulation speed of analytical models. The adaptive algorithm does not require any a priori knowledge of the dynamics of the system to select an appropriate sample distribution and an appropriate model complexity. With this technology, designers no longer must put up with legacy modeling techniques or invest resources in examining new ones.

Magnetic circuits can be represented with a topological dual circuit. In the dual circuit, flux paths are modelled by hysteretic permeances instead of reluctances. Hysteresis effect is taken into account by using the Jiles-Atherton (JA) approach. In addition, iron losses due to eddy current are also included to the model. Comparison of simulated results with the experimental results from a core type transformer demonstrates the capability of the proposed method.

The field distribution at the ports of the transmission line structure is computed by applying Maxwell’s equations to the structure. Assuming longitudinal homogeneity an eigenvalue problem can be derived, whose solutions correspond to the propagation constants of the modes. The nonsymmetric sparse system matrix is complex in the presence of losses and Perfectly Matched Layer. The propagation constants are found solving a sequence of eigenvalue problems of modified matrices with the aid of the invert mode of the Arnoldi method. Using coarse and fine grids, and a new parallel sparse linear solver, the method, first developed for microwave structures, can be applied also to high dimensional problems of optoelectronics.

A method to optimize delay and power dissipation in on-chip interconnect is reported. Propagation delay can be represented by the dominant time constant of the corresponding RC circuit or as a p-q% delay [1]. The optimization problem is formulated as a sequence of semi-definite programming problems. The method is applied to interconnect with inclusion of the fringing capacitance and capacitive coupling between wires. Shapes of single wires and models of real-life bus designs are optimized. It is shown that the optimal wire shape depends on the chosen delay metric and that it can be described accurately with a linear model. The differences between wire sizing and wire tapering are discussed. The importance of capacitive coupling in the optimization of multi-wire buses is demonstrated. Future extensions of the approach are discussed.

The paper introduces a systematic procedure to derive the expression of the Maxwell stress tensor associated with a given expression of the electromagnetic energy density.

With interconnect increasingly contributing to the electrical behaviour of integrated circuits, both by higher frequencies and smaller dimensions, it becomes increasingly important to incorporate its behaviour into simulations of ICs. This can be done rather elegantly by summarizing interconnect behaviour into a compact or reduced order model which is then co-simulated with the circuit. A similar approach can be used in the case of more conventional printed circuit boards. The SVD-Laguerre algorithm proposed by Knockaert and De Zutter [4] can be used for this purpose. In this paper, we describe an efficient implementation of the algorithm for multiple inputs, and show how the mathematical reduced order models can be translated into realizable circuit elements.

In this paper we show how the existing numerical time-integration methods of the Maxwell equations can be handled in the framework of the theory of operator splitting methods. We consider the classical Yee-method, the Namiki-Zheng-Chen-Zhang alternating direction implicit method (NZCZ) and the Kole-Figge-De Raedt method (KFR). The unconditional stability of the NZCZ-method has been proven only by means of extensive use of computer algebraic tools. We give a pure mathematical proof. We compare the methods from the point of view of accuracy and computational speed.

A new method for the simulation of circuits with widely-varying time scales is given. The method makes a splitting of the behaviour of the circuit in a fast-varying and a slowly-varying component. The method is attractive because it can handle frequency modulated (FM) circuits, unlike existing methods. Numerical results are given.

The paper presents an efficient numerical method to extract accurate R, L, C parameters of passive on-chip structures. This method is based on two main original ideas. First, the accuracy is controlled by using two complementary approaches based on scalar and vector potential, which provide lower and upper bounds for the extracted parameter. The convergence is accelerated by using the Richardson extrapolation of the average value of the two complementary bounds. Second, the field equations are solved by multigrid finite element method with local adaptive mesh subgriding. The refining process is stopped as soon as the desired accuracy is reached.

We are investigating inductive coupling optimization schemes and quantization effects for microscopic metal rings as a possible basis for a quantum bit (qbit). Faraday induction is proposed to provide electromagnetic coupling between the rings, therefore acting as an information carrier. Quantizing this information will produce distinguishable ring states that can be denoted by |0〉 and |1〉, representing the logic states of the qbit. We have set up simulation case studies with the aim of reducing signal loss between the rings. Further, different quantization mechanisms are investigated analytically. A combination of the two concepts can in theory be used to design qbits, consisting of metal rings with I/O facilities.

The self-inductance of the operating coil of a magnetizing device is calculated using different methods. The winding of the coil under investigation basically consists of copper sheets with rectangular concentric inner and outer contours. These plates form the turns of a Bitter coil. They are stacked together with an electric insulation between them and connected in series to form a helix-like winding. Analytical formulae for cylindrical coils can only be applied as a coarse approximation due to the rectangular cross-section and because of exact geometric measures of current paths such as for arrangements of filamentary wires not being available. More reliable results are obtained, if first self- and mutual inductances of all turns are determined according to Neumann’s formula and the resulting self-inductance is determined afterwards with respect to the connection for all turns in series. A 3D-FEM analysis is carried out in order to verify the method described above and to judge the influence of eddy-current phenomena, i.e. the skin-effect, which might become important in the usual transient operation mode.

In this paper we will discuss the EXOR function synthesized with a modulo function. As an implementation of the modulo function we will use the single-electron tunneling (SET) electron-box as a basic structure. A SET electronbox consists of one SET junction in series with a normal capacitor. This basic electron-box structure is extended with a number (equal to the number of inputs) of normal capacitors.Simulations are carried out with the commonly used simulator in SET-electronics: SIMON [1], indeed showing the expected outcome.

For the coupling of the magnetic field and the electric circuit equations, there are different approaches. In any case, the flux linkage has to be taken into account by augmenting the finite element system by additional equations. Recently, it has been proposed to eliminate the circuit part by taking the Schur complement, which results in symmetric and positive definite matrices. The rank of the circuit’s contribution to the Schur complement equals the number of linear independent coupling variables. Field-circuit coupling therefore introduces a low rank correction into the equations of the field problem. The consequences of this key observation are discussed in the paper. If a direct solution of the finite element system is considered, the circuit coupling can be treated elegantly by using the Woodbury formula. The Woodbury formula gives an explicit expression for the inverse of a matrix with low rank correction in terms of the inverse of the original matrix. In the framework of a preconditioned conjugate gradient solver it turns out that it is sufficient to include the circuit equations into the matrix-by-vector product, while the finite element preconditioner can be retained. These considerations will be illustrated by numerical results that have been obtained from a simple model problem.

In practice the shape of the electrodes and the spacers in high voltage equipments are so designed that the electric stress on the electrodes, mainly on the live electrode, and on the spacers are well within the limits. This necessitates that the practical contours of the insulators have complex geometries. The complexities are often increased by the constraints imposed by the mechanical considerations in the Gas Insulated Systems (GIS). Suitable techniques are required for accurate yet efficient simulation of such complex geometries. This paper highlights a modified algorithm for computation of electric field distribution by indirect boundary element method (indirect BEM) around a complex electrode-spacer configuration used in practise in high voltage system arrangements.

This work deals with the calculation of touch voltages and leakage current density distribution in direct-current subway grounding systems.A new approach for direct-current subway grounding systems calculations was developed. In this approach, all system components are modelled as multi-input, multi-output blocks, which are interconnected using appropriate electrical equation.Results obtained using the proposed approach show good agreement with results reported in previous works.

Recently a new approach was presented to determine the high-frequency electromagnetic behavior of on-chip passives and interconnects. The method solves the electric scalar and magnetic vector potentials in a prescribed gauge. The latter one is included by introducing an additional independent scalar field, whose field equation needs to be solved. This additional field is a mathematical aid that allows for the construction of a gauge-conditioned, regular matrix representation of the curl-curl operator acting on edge elements. This paper reports on the convergence properties of the new method and shows the first results of this new calculation scheme for VLSI-based structures at high frequencies. The high-frequent behavior of the substrate current, the skin effect and current crowding is evaluated.

We deal with a Quantum-Drift-Diffusion (QDD) model for the description of transport in semiconductors which generalizes the standard Drift-Diffusion model (DD) through extra terms that take into account some quantum dispersive corrections. We also study numerically the influence on the I-V curve of the electron effective mass, the barrier height and width, and of the ambient temperature. The performance of several linearization algorithms, i.e. a two Gummel-type iterations and the fully-coupled Newton method are also compared.

We present applications of homotopy methods, which make it possible to compute multiple dc-operating points of transistor circuits with standard network analysis programs. It is possible to capture all dc-operating points at least of smaller transistor networks with the help of one- and two-parametric homotopies. Uniqueness criteria of network theory can help to find a parameterization of the homotopy path. As an example for appropriate uniqueness criteria a well known theorem of Nielsen and Willson is applied. Bounds for the parameter space can be found by the no-gain property of transistor circuits.

Numerical prediction of charged particle dynamics in accelerators is essential for the design and understanding of these machines. The calculation of space charge forces influencing the behaviour of a particle bunch is still a bottleneck of existing tracking codes.We report on our development of a new 3D space-charge routine in the General Particle Tracer (GPT) code. It scales linearly with the number of particles in terms of CPU time, allowing over a million particles to be tracked on a normal PC. The model is based on a non-equidistant multigrid Poisson solver that is used to solve the electrostatic fields in the rest frame of the bunch.A reliable multigrid scheme for the tracking of particles should be very fast, stable and show good convergence for a great variety of meshes. Numerical results demonstrate the effect of the choice of the multigrid components. Further, the values of physical quantities show good agreement compared to the values calculated by a well-tested 2D routine in the GPT code.

In electric circuits, signals often include widely separated frequencies. Thus numerical simulation demands a large amount of computational work, since the fastest rate restricts the integration step size. A multidimensional signal model yields an alternative approach, where each time scale is given its own variable. Consequently, underlying differential algebraic equations (DAEs) change into a PDAE model, the multirate partial differential algebraic equations (MPDAEs). A time domain method to determine multiperiodic MPDAE solutions is presented. According discretisations rest upon the specific information transport in the MPDAE system along characteristic curves. In contrast, general time domain methods produce unphysical couplings. Hence enormous savings in computational time and memory arise in the linear algebra part. This technique is applied to driven oscillators including two periodic time scales as well as to oscillators, where one periodic rate is forced and the other is autonomous.

The application of Jacobi-Davidson style methods in electric circuit simulation will be discussed in comparison with other iterative methods (Arnoldi) and direct methods (QR, QZ). Numerical results show that the use of a preconditioner to solve the correction equation may improve the Jacobi-Davidson process, but may also cause computational and stability problems when solving the correction equation. Furthermore, some techniques to improve the stability and accuracy of the process will be given.

An electromagnetic field can be considered as slowly varying if the wavelength is large compared to the problem region. In the electro-quasistatic case it then may be assumed that the time-derivative of the magnetic flux is negligible, whereas the displacement currents have to be taken into account. Under these assumptions Maxwell’s equations for time harmonic fields reduce to a complex Poisson’s equation and discretization yields a complex symmetric system of equations. Krylov-subspace methods with an algebraic multigrid (AMG) preconditioner are used for fast solution. The electro-quasistatic model is applicable in many different constellations. This paper deals with applications from three different fields: high-voltage engineering, neural sensor-actor systems and the influence of slowly varying fields on human tissue.

In present-day IC’s, substrate noise can have a significant impact on performance. Thus, modeling the noise-propagation characteristics of the substrate is becoming ever more important. Two ways of obtaining such a model are the Finite Element Method (FEM) and the Boundary Element Method (BEM). The FEM makes a full 3D discretization of the entire substrate and is very accurate and flexible, but, in general, it is also slow. The BEM only discretizes contact areas on the substrate-boundary, and is usually faster, but less flexible, because it assumes the substrate to consist of uniform layers. Sometimes, layout-dependent doping patterns near the top of the substrate may also play a significant role in noise-propagation. The FEM would easily be able to model such patterns, but it can often be too slow. The BEM, on the other hand, might not always be accurate enough. This paper describes a combination between BEM and FEM, which results in a method that is faster than FEM but more accurate than BEM. Through a number of experiments, the method is validated and successfully verified against 2 commercially available tools.

The numerical simulation of the induction heating of three-dimensional metal bodies by moving inductors can be problematic from several points of view when using standard FE or BE schemes. In this paper we describe the main difficulties and propose an alternative modelling approach based on the reformulation of the Maxwell’s equations into a system of second-kind Fredholm integral equations. These equations for the eddy currents are coupled with the heat transfer equation with non-linear temperature-dependent material parameters. Mathematical analysis of the existence and uniqueness of solution to the continuous as well as discrete problem is provided and convergence of the numerical scheme is shown. An illustrative numerical example is presented.

The application of multigrid (MG) methods for the solution of electromagnetic problems has attracted attention in recent years (e.g. [8]). These problems are related to bilinear forms (curl·, curl·)L_Ω2 +α (·, ·)L_Ω2, α ∈ ℝ, which require special smoothers as presented in [1,7]. This paper shows by numerical experiments that these ideas also work for the time-harmonic eddy currents, i.e. for complex bilinear forms (curl·, curl·)L_Ω2 +iα (·, ·)L_Ω2. Furthermore, an approximate projection procedure is presented that allows the application of multigrid to an un-gauged electric formulation even if there are regions with zero conductivity. Numerical results are shown for the TEAM Workshop problem 7.

This article introduces a method for modeling and analysis of perturbed oscillator behavior, i.e. the behavior of “ideal” oscillators subjected to weak interactions with the outside world. These interactions can e.g. involve an amplituderegulating mechanism, as in harmonic oscillators, couplings to other oscillators, as in quadrature-type oscillators, and noise, both white and colored. The method is grounded on perturbation theory and averaging. Perturbation techniques allow us to separate the analysis of the unperturbed, ideal, oscillator from the analysis of the perturbed one. Averaging is used to separate the fast-varying and the slow-varying components of the oscillator’s behavior. Applications of this method include oscillator phase noise analysis and the construction of compact behavioral models for harmonic oscillators.

This paper presents an analytical approach for calculating fringing permeances in gapped inductors. For most of the gapped inductors, the permeance of other field paths out of the air gap (the fringing paths) is not negligible. Existing three-dimensional modelling techniques using finite element analysis for magnetic components are accurate, but require prohibitive amount of simulation time. Twodimensional models are often used, but the accuracy is low as a 2D simulation fails taking into account important 3D effects. We present analytical approximations for fringing permeance calculation for the most usual field patterns, denoted as basic cases. The proposed fringing coefficients can be used to present all symmetrical cases and cases with multiple air gaps. The derived equations are sufficient for a normal engineering accuracy.

In this paper we describe how stochastic differential-algebraic equations (SDAEs) arise as a mathematical model for network equations that are influenced by additional sources of Gaussian white noise. We give the necessary analytical theory for the existence and uniqueness of strong solutions, provided that the systems have noise-free constraints and are uniformly of DAE-index 1. We express these conditions in terms of the network-topology for reasons of use within a circuit simulator. In the second part we analyze discretization methods. Due to the differential-algebraic structure, implicit methods will be necessary. By the examples of the drift-implicit Euler and Milstein schemes we show how drift-implicit schemes for SDEs can be adapted to become directly applicable to stochastic DAEs and prove that the convergence properties of these methods known for SDEs are preserved. For illustration we apply the drift-implicit Euler scheme to an oscillator circuit.