Scientific Computing in Electrical Engineering SCEE 2008

Simulations are indispensable to industry nowadays. They allow one to perform virtual experiments which are faster and cheaper than the physical ones. Simulation environments are created or improved on the basis of numerical methods applied to solve specific problems. Electrical and electronics engineering need at least the computation of the electromagnetic (EM) field. For instance, in electronics, the increase of the operating frequency makes that effects specific to the EM field, and neglected until now, be relevant. Consequently, on the one hand, the development of a new approach based on the EM field computation is needed, since the old techniques, based mainly on circuits, do not correspond to the necessities any more. On the other hand, solely the numerical approach of obtaining the field cannot be effective, due to the enormous computational resources needed. Thus, in spite of the commercial advertisements, software tools’ functionality is low at high frequencies reaching 60 GHz. Even if the general software EM field packages solve many of the designers’ problems, they need appropriate solutions that are not offered yet by the present available software commercial tools.

This paper focuses on the following key tasks in numerical modeling of high frequency EMC (electromagnetic compatibility) problems: overcoming computational difficulties caused by multi-scale features and obtaining absolute (as opposed to normalized) values of the magnetic field emissions radiated from a printed circuit board. The first task is solved by using an iterative procedure combining codes for 2.5D and 3D EM field computations. The second task is considered using the technique to approximate (“tune”) parameters of the EM source in numerical models for 3D EM computations using measured data. Examples of applications of both techniques are included.

This paper addresses the problem of building a low complexity macromodel of an electromagnetic device based on measurements of its scattering parameters. For devices with a large number of ports, currently available techniques are very expensive. The approach we propose is based on a system-theoretic tool, the Loewner matrix pencil constructed in the context of tangential interpolation. Several implementations are possible. They are fast, accurate and robust; they construct models of low order and are especially designed for devices with a large number of terminals. Moreover, they allow to identify the underlying system, rather than merely fitting the measurements. We compare our algorithms to industry standard vector fitting method on two examples. This paper is a summary of [1].

Further downscaling of the integrated circuits pushes the limits of lithographic technologies and certain variability effects previously considered negligible now should be taken into account. This paper proposes an efficient approach that addresses the problem of interconnect process variations. New models for line parameters parameterized with respect to the geometric transversal dimensions, subject to small or large variations are proposed. The parametric models are solely based on the computation of first order sensitivities. In the multiparametric case the use of multiplicative models can be a better choice than the use of traditional models based on first order Taylor Series truncation.

This paper shows how to obtain models for passive integrated components that take into consideration the variability inherent to their design. To achieve this, the computational domain is split into sub-domains in which the electromagnetic circuit element (EMCE) formulation is used. The variability is described by using first order Taylor Series representation for the semi-state space matrices. The novelty of the paper is that it describes how the EMCE based parametric models can be obtained. The parametric sub-models can be interconnected afterwards to obtain a global parametric model that can be simulated or reduced. The advantage of this approach is that it bears an inherent parallelism. The sub-models can be treated independently both from the point of view of the variability, and from the point of view of electromagnetic field formulation. Both aspects are illustrated with a simple test case as well as a real benchmark designed and characterized at austriamicrosystems.

Continued device scaling into the nanometer region has given rise to new effects that previously had negligible impact but now present greater challenges and unprecedented complexity to designing successful mixed-signal silicon. This paper presents a novel graphical tool for semi automatic extraction of magnetic reluctances between on-chip current loops. The novel graphical tool seamlessly integrates within the workflow of the CHAMELEON-RF software prototype developed.

A robust technique for modelling linear and nonlinear lumped elements spanning multiple cells in an FDTD-based electromagnetic field simulator is presented. The nonlinear models require iteration as part of the model. The technique is applied to produce a highly stable LE-FDTD diode model that works well far beyond normal operational voltage ranges. Simulation results are in good agreement with those obtained with the circuit simulator APLAC and those in the literature [1].

In the infrared spectrum noble metals do not act like perfect conductors, but have to be described by dispersive material models. Eigenvalue problems including such frequency-dependent material properties occur for instance, when the dispersion relations of periodic structures such as photonic crystals and metamaterials are analyzed by electromagnetic field simulations of the corresponding unit cells. We show that the commonly used Drude dispersion model leads to a polynomial eigenvalue problem which can be solved by a modified Jacobi-Davidson method.

The paper presents a concept of the region-oriented 3D formulation for the boundary element method (BEM) applied to computation of electric fields. Differences between the region-oriented and the traditional BEM are explained. Numerical tests performed for simple arrangements with high permittivity components show that the new approach leads to better accuracy than the traditional BEM.

This paper describes how the Surface Integrated Field Equations method (SIFE) is implemented to solve 3D Electromagnetic (EM) problems on substrates in which high contrast materials occur. It gives an account of the promising results that are obtained with it when compared to traditional approaches. Advantages of the method are the highly improved flexibility and accuracy for a given discretization level, at the cost of higher computational complexity.

We explain the reduced basis (RB) method applied to electromagnetic field computations with the finite element method. Rigorous numerical simulations for practical applications often become very time consuming. The RB method allows to split up the solution process of an e.g. geometrically parameterized problem into an expensive offline and a fast online part. For an actual simulation only the fast online part is evoked.

We apply the RB method to the rigorous simulation of light scattering from a parameterized phase shift mask.

In this paper we explore the viability of using nodal discontinuous Galerkin methods as implemented in the software library Nudg++ [1] to compute the space charge field of an electron or positron bunch. We will use a benchmark problem to evaluate this method by comparing results from this scheme with solutions obtained analytically.

In IC failure analysis the detection of currents is often used to indicate the presence of a defective device. One method used for this analysis is the Magnetic Force Microscopy (MFM). Employing this technique measurement errors often occur as for instance due to heterogeneous magnetic tip coatings, fabrication/abrasion errors of the MFM tips and vibrations during a MFM scanning process. Hence, in this work a theoretical model of the MFM was developed to verify and improve the results of laboratory MFM measurements. Therefore a scanning process is simulated and different force calculation methods are implemented and compared with each other in order to obtain the total magnetic force acting on the cantilever as well as the local magnetic force densities.

A recently developed high order field solver for the complete Maxwell equations provides all information needed by a new relativistic particle push method based on a truncated Taylor series expansion up to the desired order of convergence. The property and capability of this approach is demonstrated for different numerical experiments.

A statistical characterization of random electromagnetic interactions affected by resonances is presented. It hinges on the analysis of the variance and the kurtosis to assess the intensity of the resonances. The method is illustrated by the study of a randomly varying thin wire modeled by a Pocklington integral equation.

This introduction gives some background to

circuit simulation

in general and provides a short overview of the 15 papers that follow. Before proceeding further, it is pointed out that Section 1 of the first paper, the invited paper by Dautbegovic, gives a nice two-page introduction to circuit simulation. Thus, the reader may first want to read that section before returning to the text at hand, which completes the introduction to circuit simulation.

Wavelet theory is a relatively recent area of scientific research, with a very successful application in a broad range of problems such as image, audio and signal processing, numerical analysis, electromagnetic scattering, data compression and denoising, stohastics, mathematics and physics, (bio)medicine, astronomy and many more. The key wavelet property contributing to its success in such a variety of disciplines is the capability of a simultaneous time and frequency representation of a signal embedded within a multi-resolution analysis (MRA) framework. The potential exploitation of this property for next-generation, wavelet-based techniques for analog circuit simulation is discussed in this paper.

The burden of solving inner equations in compact models of semiconductor devices (such as transistors) is often shifted to the host circuit simulator. Schur complement techniques for local handling of these equations may help to reduce the size of the model stamp, which – depending on the host simulator – may have a positive impact on CPU time and memory needs. Some practical aspects of applying these concepts in compact modeling are discussed. A formulation is presented which accounts for the specific way of model evaluation in circuit simulation. It can be realized in a standard code for flat model evaluation by adding a software shell around the model core function itself.

First tests with an advanced high voltage MOS model demonstrate the feasibility of this approach in terms of accuracy, iterations and runtimes.

Commercial packages for transient circuit simulation are often based on the modified nodal analysis (MNA) which allows an automatic setup of model equations and requires a nearly minimal number of variables. However, it may lead to differential-algebraic equations (DAEs) with higher index. Here, we present a hybrid analysis for nonlinear time-varying circuits leading to DAEs with index at most one. This hybrid analysis is based merely on the network topology, which possibly leads to an automatic setup of the hybrid equations from netlists. Moreover, we prove that the minimum index of the DAE arising from the hybrid analysis never exceeds the index from MNA. As a positive side effect, the number of equations from the hybrid analysis is always no greater than that one from MNA. This suggests that the hybrid analysis is superior to MNA in numerical accuracy and computational effort.

A new approach for transient analysis of nonlinear circuits is presented. The circuit equations are formulated as functions of incident and reflected waves at the device ports. Only one large matrix decomposition is necessary if time step is constant. The proposed method is parallelizable, allows straightforward inclusion of complex nonlinear device models and has better convergence properties compared to existing methods. Simulation results are provided to demonstrate the approach.

Noise in electronic components is a random phenomenon that can adversely affect the desired operation of a circuit. Transient noise analysis is designed to consider noise effects in circuit simulation. Taking noise into account by means of Gaussian white noise currents, mathematical modelling leads to stochastic differential algebraic equations (SDAEs) with a large number of small noise sources. Their simulation requires an efficient numerical time integration by mean-square convergent numerical methods. As efficient approaches for their integration we discuss adaptive linear multi-step methods, together with a new step-size and path selection control strategy. Numerical experiments on industrial real-life applications illustrate the theoretical findings.

This paper illustrates the use of term-wise Volterra analysis tool that can plot both IM3 tone and relevant 2nd order tones as vector sums of all important contributions. As an example the nonlinear distortion behaviour in a fully differential amplifier is studied, when driven either with single-ended or balanced input signal. It is shown that with single-ended drive a small tail-impedance of the differential pair generates 2nd-order distortion into the output of the first stage, and this mixes further to 3rd-order distortion in the 2nd-degree nonlinearity of the second stage.

A general expression for the phase transfer functions of an oscillator for frequencies close to the harmonics of the oscillator fundamental is derived. Numerical testing and comparison with some known results are performed.

Numerical simulation of electric circuits uses systems of differential algebraic equations (DAEs) in general. We examine forced oscillators, where the DAE models involve periodic solutions. Uncertainties in physical parameters can be described by random variables. We apply the strategy of the generalised polynomial chaos (gPC) to resolve the stochastic model. In particular, failure probabilities are determined using the approximation from gPC. We present results of numerical simulations for a system of DAEs modelling a Schmitt trigger.

Multitone Harmonic Balance (HB) is widely used for the simulation of the quasiperiodic steady-state of RF circuits. HB is based on a Fourier expansion of the waveforms. Unfortunately, trigonometric polynomials often exhibit poor convergence properties when the signals are not quasi-sinusoidal, which leads to a prohibitive run-time even for small circuits. Moreover, the approximation of sharp transients leads to the well-known Gibbs phenomenon, which cannot be removed by an increase of the number of Fourier coefficients, because convergence is only guaranteed in the

L

2

norm. In this paper we present alternative approaches based on cubic or exponential splines for a periodic or quasiperiodic steady state analysis. Furthermore, it is shown below that the amount of coding effort is negligible if an implementation of HB exists.

The Devil’s Staircase of an Injection-Locked Frequency Divider (ILFD) is simulated in a novel and efficient manner in this contribution. In particular, the Multiple-Phase-Condition Envelope Following (MPCENV) method is employed. The locking range of the ILFD is then determined from the Devil’s Staircase. The proposed method is applied to an LC oscillator based ILFD and the results are validated by comparison with experimental results.

This paper provides a tutorial overview of artificial neural network/ dynamic neural network (ANN/DNN) for radio frequency (RF) and microwave modeling and design. We will describe neural network structures suitable for representing high-speed/high-frequency behaviors in components and circuits, ANN training exploiting RF/microwave device and circuit data, formulation of ANN/DNN for microwave component and circuit behavioral modeling, and use of ANN/DNN models for high-level RF/microwave simulation and design optimization.

Although the behavior of several RF circuit blocks can be accurately evaluated via transistor-level simulations, the design space exploration is limited by the high computational cost of such simulations. Therefore, cheap-to-evaluate surrogate models of the circuit simulator are introduced. This paper presents some results of a feasibility study concerning the development of surrogate models of low noise amplifiers.

In this paper, we practically implement a systematical method for thin-film transistor liquid-crystal display (TFT-LCD) design optimization and sensitivity analysis. Based upon a three-dimensional (3D) field solver and a Design of Experiments, we construct a second-order response surface model (RSM) to examine the capacitances’ effect on the performance of an interested TFT-LCD pixel. The constructed RSMs are reduced using a step-wise regression. We verify the accuracy using the normal residual plots and their residual of squares. According to the models, we then analyze the sensitivity of the capacitances by considering the design parameters as changing factors (i.e., the size variation and the position shift) under an assumption of Gaussian distribution. Consequently, we further apply the models to optimize the designed circuit. The designing parameters of these models are selected and optimized to fit the designing target of the examined circuit by the genetic algorithm in our unified optimization framework. This computational statistics method predicts the capacitances’ effects on the gate delay time and compares with full 3D simulation approaches, it shows the engineering practicability in display panel industry.

This work presents a data-based behavioral modeling scheme for switched-capacitor integrator settling error. In a typical SIMULINK behavioral model, settling behavior is implemented as a conditional, equation-based block. Here, the amplifier model is first characterized by a full range of amplifier’s initial input and output voltages. The resulting settling errors are tabulated and finally, the settling error table is used directly as a lookup-table in behavioral simulations. One- or two-dimensional lookup-tables are standard library blocks in SIMULINK. This means that the actual settling error model is independent of the modeled amplifier topology, which is clearly a welcome feature in behavioral modeling.

Many combined analog-digital (mixed-signal) systems involve quite a lot of Digital Signal Processing (DSP) functions. These circuits are simulated either by using sample-based behavioral models (often with fixed time-step), or by combining digital event-driven simulation with traditional transient analysis. The above approaches are expensive in some applications, and this paper presents ways of speeding up behavioural simulations of mixed-signal systems. As a system design example, linear state-space models are applied to study the effect of small timing errors in a time-interleaved digital-to-analog converter system.

This part addresses the challenging topic of solving coupled problems. The increasing necessity to solve complex problems in the science and engineering community, accounting for all the coupling occurring at the different scales of the problem, requires the development of new ideas and methods which can effectively provide accurate numerical solutions in affordable computation times. The state of the art is discussed here as well as mathematical, numerical, and computational methods for solving coupling problems of multidisciplinary character, with an emphasis on coupling with electromagnetic (EM) and/or circuit simulation. Special attention is paid to showing the potential of new computational methods for solving practical multidisciplinary problems of industrial interest.

Thermal effects in a coupled circuit-device system are modeled and numerically simulated. The circuit equations arise from modified nodal analysis. The transport in the semiconductor devices is modeled by the energy-transport equations for the electrons and the drift-diffusion equations for the holes, coupled to the Poisson equation for the electric potential. The lattice temperature is described by a heat equation with a heat source including energy relaxation heat, recombination heat, hole Joule heating, and radiation. The circuit-device model is coupled to a thermal network. The resulting system of nonlinear partial differential-algebraic equations is discretized in time using backward difference formulas and in space using (mixed) finite elements. Heating effects from numerical simulations in a

pn

-junction diode and a clipper circuit are presented.

In this work we address the well-posedness of the steady-state and transient problems stemming from the coupling of a network of lumped electric elements and a PDE model of heat diffusion in the chip substrate. In particular we consider the thermal element model presented in [1] and we prove that it can be controlled by any combination of voltage sources (imposing the average current in a region of the chip) and current sources (imposing the Joule power per unit area produced in a region) connected to its temperature nodes. This result justifies the implementation of the element as a linear n-port conductance as carried out in [2].

We present a new strategy to perform chip-level electro-thermal simulation. In our approach electrical behaviour of each circuit element is modeled by standard compact models with an added temperature node (1; 2). Mutual heating is accounted for by a 2-D or 3-D diffusion reaction PDE, which is coupled to the electrical network by enforcing instantaneous energy conservation. To cope with the multiscale nature of heat diffusion in VLSI circuit a suitable spatial discretization scheme is adopted which allows for efficient meshing of large domains with details at a much smaller scale. Preliminary numerical results on a realistic test case are included as a validation of the model and of the numerical method.

Recently an energy-transport models has been formulated based on the maximum entropy principle for the coupled phonon-electron system in silicon in order to cope with the effects of heating of the crystal lattice. Here the numerical simulations of some benchmark devices are presented in order to assess the validity of the model.

For a coupled circuit device simulation in the time domain, consistent initial values have to be calculated. We study the structure and properties of the differential-algebraic equations (DAEs) that arise after space discretization of the partial differential equation part coming from the device modelling. Exploiting the special DAE structure, we show that a consistent initial value can be computed within two steps. Firstly, one determines an operation point. Secondly, a linear system is solved for correcting the operation point such that the hidden constraints are also satisfied. Finally, an algorithm for the calculation of such values is proposed.

We address the problem of coupling a system of network equations corresponding to an electric circuit with a detailed model for a device connected to the circuit. The device is modeled by an hydrodynamic model based on the maximum entropy principle, which results in a hyperbolic system of partial differential equations. We perform a numerical simulation with an oscillator coupled with an

n

-

n

-

n

channel.

The increasing characteristics variability in nano-CMOS devices becomes a major challenge to scaling and integration. In this work, a large-scale statistically sound “atomistic” circuit-device coupled simulation methodology is presented to explore the discrete-dopant-induced characteristic fluctuations in nano-CMOS digital circuits. According to the simulation scenario, the discrete-dopant-induced characteristic fluctuations are examined for a 16-nm-gate MOSFET and inverter circuit. The fluctuations of the intrinsic current-voltage and capacitance-voltage characteristics, and timing behaviors for the explored device and circuit are estimated. The timing fluctuation may result in a significant signal delay in the digital circuit. Consequently, links should be established between circuit design and fundamental device technology to allow circuits and systems to accommodate the individual behavior of every transistor on a silicon chip. The proposed simulation approach could be extended to outlook the fluctuations in various digital and analog circuits.

Electromagnetic coupling between devices in an microelectronic layout can become a serious design concern. In this paper, the problem of electromagnetic coupling is addressed from field computational point of view. Approximation schemes are justified by evaluating dimensionless parameters in the set up of the field equations and scale considerations of devices. The discretization scheme is reviewed and a simulation method is presented to compute the S-matrix directly by imposing boundary conditions that map directly to the experimental set up. An example demonstrates the validity of the scheme.

Continued device scaling into the nanometer region has given rise to new effects that previously had a negligible impact but now present greater challenges to successful design of mixed-signal silicon. This paper evaluates Domain Decomposition (DD) strategies for compact simulation of on-chip coupled problems from a computational perspective, using the recently completed CHAMELEON-RF software prototype on several standard benchmark structures.

In this paper the field/circuit coupling is reconsidered for (non-linear) lumped electric circuits refined by 3-D magnetoquasistatic conductor models, where the circuit is described by modified nodal analysis and the field is discretized in terms of the finite integration technique. This leads to the coupling of systems of differential-algebraic equations, for which two numerical approaches are proposed, the weak coupling (co-simulation) and strong coupling (monolithic). The DAE-index of the subproblems and of the full problem are analyzed, then convergence properties of the co-simulation are studied. Finally computational results of a simple half rectifier circuit are exemplarily given to prove the applicability of the concepts.

This part is concerned with

mathematical and computational methods

in electrical engineering, including also multiobjective optimization and space-mapping methods. Theoretical results, novel approaches, and simulations cover some important issues mainly arising in the field of computational electromagnetics, including finite-element/volume discretization and the differential/integral formulation of Maxwell’s equations, large interconnect structures, uncertainty quantification, and electron devices. Both the mathematical aspects and the applicative importance are outlined in this part, and as such may appeal to both engineers and theoreticallyoriented readers.

Under certain conditions, electromagnetic time-domain modeling can be performed using the regimes of quasistatic approximations. The corresponding mathematical models represent then systems of first order ordinary differential equations or index 1 differential-algebraic equations. To resolve the time dependencies of the transient processes described by these equations, numerous time integration schemes can be employed. In this work, we give an overview of the mostly used time integration algorithms and discuss the main features, peculiarities and typical numerical difficulties associated with them. The materials presented in the paper are illustrated with corresponding numerical examples.

In this work a novel, staggered finite volume time domain method for Cartesian grids is presented, analyzed and validated. An important characteristic of the method is the use of a rather unorthodox staggering of the degrees of freedom.

EM scattering from PEC surfaces are mostly calculated through the induced surface current

J

. In this paper, we consider PEC surfaces homeomorphic to the sphere, apply Hodge decomposition theorem to a slightly rewritten surface current, and show how this enables us to replace the unknown current with two scalar functions which serve as potentials for the current. Implications of this decomposition are pointed out, and numerical results are demonstrated.

A new family of source integral equations is presented, dedicated to the solution of time-harmonic Maxwell scattering problems. Regardless of the composition of the obstacle – metallic, full dielectric or coated with an impedance layer – we show that a general methodology is able to guide the construction of some special equations whose the foremost feature is to be well-conditioned. Indeed, all of them are free of spurious modes and appear as some compact perturbations of positive operators (when it is not the identity), leading therefore to fast iterative solutions without the help of any preconditioner. These intrinsically well-conditioned equations open the way for interesting new developments in the field of boundary equation methods for Maxwell applications.

For a fast simulation of interconnect structures we consider preconditioned iterative solution methods for large complex valued linear systems. In many applications the discretized equations result in ill-conditioned matrices, and efficient preconditioners are indispensable to solve the linear systems accurately. We apply the dual threshold incomplete LU (ILUT) factorization as preconditioners for the BICGSTAB iterative solver. On complicated problems with a different range of frequencies we show that the BICGSTAB method with the ILUT preconditioner provides a very efficient solution for the linear systems.

We discuss the basics of discontinuous Galerkin methods (DG) for CEM as an alternative of emerging importance to the widely used FDTD. The benefits of DG methods include geometric flexibility, high-order accuracy, explicit time-advancement, and very high parallel performance for large scale applications. The performance of the scheme shall be illustrated by several examples. As an example of particular interest, we further explore efficient probabilistic ways of dealing with uncertainty and uncertainty quantification in electromagnetics applications. Whereas the discussion often draws on scattering applications, the techniques are applicable to general problems in CEM.

Tensors are the natural mathematical objects to describe physical quantities that evolve over multiple independent variables. This paper considers the computation of empirical projection spaces by decomposing a tensor that can be associated with measured data. We show how these projection spaces can be used to derive reduced order models. The procedure is applied to a two-dimensional heat diffusion problem and a problem in fluid flow dynamics.

Finite element tearing and interconnecting (FETI) methods are efficient parallel domain decomposition solvers for large-scale finite element equations. In this work we investigate the robustness of FETI methods in case of highly heterogeneous (multiscale) coefficients. Our main application are magnetic field computations where both large jumps and large variation in the reluctivity coefficient may arise. We give theoretical condition number bounds which are confirmed in numerical tests.

In this paper we present exact closures of the 8-moment and the 9-moment models for the charge transport in silicon semiconductors based on the maximum entropy principle. The validity of these models is assessed by numerical simulations of an n-n-n device. The results are compared with those obtained from the numerical solution of the Boltzmann Transport Equation both by Monte Carlo method and directly by a finite difference scheme.

In this paper multiobjective optimization is applied to antenna design. The optimization algorithm is a novel response surface method based on approximation with radial basis functions. It is combined with CAD and mesh generation software, and electromagnetic solvers. To demonstrate the procedure we optimize the geometric design and feed position of a PIFA antenna located on a ground plane.

In this paper we solve an optimal doping profile control problem for semiconductors using the manifold mapping technique. As coarse and fine approximation we employ the drift diffusion and energy transport model, respectively. In this work the manifold mapping technique is applied for the first time to a problem in which the number of design variables varies with the finite element mesh points employed. The advantage of our approach is that it allows to optimize the energy transport model without having to implement an adjoint code while at the same preserving computational efficiency. Numerical results giving evidence of this claim for different values of the applied voltage will be shown.

The surrogate methods have been used to ease the computational burden in various disciplines. In this work, a surrogate method based on space mapping is proposed to solve inverse problems. Even though the efficiency of space mapping and its variants has been demonstrated in numerous work, using it for inverse problems is addressed for the first time in this work. The efficiency of the proposed method is demonstrated solving the shape reconstruction of a conducting cylinder.

Over the years, model-order reduction (MOR) always was greatly inspired by problems from the electronics industry. Especially problems from interconnect and from parasitics extraction offered a nice class of large, linear problems. MOR aims to compress large systems but requires that its input-output behavior is preserved (within tolerances). There are several techniques available. For good general references on MOR the reader is referred to [1–3].

We discuss algorithms for balanced truncation (BT) based model reduction of linear systems. BT is known to have good global approximation properties and to preserve important system properties. A computable error bound allows to choose the order of the reduced-order model adaptively. We will emphasize those aspects that makes the application of BT to models arising in circuit simulation a non-straightforward task. In recent years, these issues have been addressed by several authors. We will survey some of these developments and demonstrate that BT is now suitable for linear descriptor systems encountered in circuit simulation.

We consider passivity-preserving model reduction of circuit equations using the bounded real balanced truncation method applied to a Moebius-transformed system. This method is based on balancing the solutions of the projected Lur’e or Riccati matrix equations. We also discuss their numerical solution exploiting the underlying structure of circuit equations. A numerical example is given.

A new model reduction method for circuit simulation is presented, which preserves passivity by interpolating dominant spectral zeros. These are computed as poles of an associated Hamiltonian system, using an iterative solver: the subspace accelerated dominant pole algorithm (SADPA). Based on a dominance criterion, SADPA finds relevant spectral zeros and the associated invariant subspaces, which are used to construct the passivity preserving projection. RLC netlist equivalents for the reduced models are provided.

We suggest a simple and an efficient iterative method based on both the Gerschgorin eigenvalue inclusion theorem and the deflation methods to compute a Reduced Order Model (ROM) to lower greatly the order of a given state space system. This method is especially efficient in symmetric state-space systems but it works for the other cases with some modifications.

This paper proposes a new approach for the Model-Order Reduction (MOR) of RLC circuits: Global-Approximation-Based Order Reduction (GABOR). GABOR preserves passivity and reciprocity, and matches the ‘moments’ of the underlying global approximation. However, GABOR has some problematic features, too. First, many matrices must be recursively precomputed into the memory space. Second, it is difficult to circumvent the singularity of the conductance matrix by any conventional frequency shifting. On the other hand, some tryouts for solving the second problem lead to finding interesting links between GABOR and other MOR methods. The correct operation of GABOR is verified with a simulation example.

We consider model order reduction of a system described by a non-rational transfer function. The systems under consideration result from the discretization of electromagnetic systems with surface losses (1). In this problem, the frequency parameter

s

appears nonlinearly. We interpret the nonlinear functions containing

s

as parameters of the systems and apply parametric model order reduction (PMOR) to the system. Since the parameters are functions of the frequency

s

, they are coupled to each other. Nevertheless, PMOR treats them as individual parameters. We review existing PMOR methods, and discuss their applicability to the problem considered here. Based on our findings, we propose an optimized method for the parametric system considered in this paper. We report on numerical experiments performed with the optimized method applied to real-life data.

We discuss methods for model order reduction (MOR) of linear systems with many input and output variables, arising in the modeling of linear (sub) circuits with a huge number of nodes and a large number of terminals, like power grids. Our work is based on the approaches SVDMOR and ESVDMOR proposed in recent publications (1; 2; 3; 4; 5). In particular, we discuss efficient numerical algorithms for their implementation. Only by using efficient tools from numerical linear algebra, these methods become applicable for truly large-scale problems.

The super node algorithm performs model order reduction based on physical principles. Although the algorithm provides us with compact models, its stability and passivity have not thoroughly been studied yet. The loss of passivity is a serious problem because simulations of the reduced network may encounter artificial behavior which render the simulations useless. In this paper we explain why the algorithm delivers not passive reduced order models and present a way in order to overcome this problem.

This paper presents a hierarchical model-order reduction (HMOR) flow, where the linear parts of a hierarchically defined circuits are divided into independently reducable subcircuits. The impact of the hierarchical structure and circuit partitioning on two MOR methods is discussed and some simulation results are presented.

This paper proposes a passive, stable, netlist-in–netlist-out-type Model-Order Reduction (MOR) method suitable for the reduction of very large RL circuit blocks. The method relies on partitioning the circuit into subcircuits that can be efficiently approximated with low-order macromodels. The efficiency of the method is demonstrated with several simulations and comparison to the PRIMA method.

Electro Static Discharge (ESD) analysis is of vital importance during the design of large-scale integrated circuits, since it gives insight in how well the interconnect can handle unintended peak charges. Due to the increasing amount of interconnect and metal layers, ESD analysis may become very time consuming or even unfeasible. We propose an algorithm for the reduction of large resistor networks, that typically arise during ESD, to much smaller equivalent networks. Experiments show reduction and speed-ups up to a factor 10.

Refined models for MOS-devices and increasing complexity of circuit designs cause the need for Model Order Reduction (MOR) techniques that are capable of treating nonlinear problems. In time-domain simulation the Trajectory PieceWise Linear (TPWL) approach is promising as it is designed to use MOR methodologies for linear problems as the core of the reduction process. We compare different linear approaches with respect to their performance when used as kernel for TPWL.

Due to refined modelling of semiconductor devices and increasing packing densities, reduced order modelling of large nonlinear systems is of great importance in the design of integrated circuits (ICs). Despite the linear case, methodologies for nonlinear problems are only beginning to develop. The most practical approaches rely either on linearisation, making techniques from linear model order reduction applicable, or on proper orthogonal decomposition (POD), preserving the nonlinear characteristic. In this paper we focus on POD. We demonstrate the missing point estimation and propose a new adaption of POD to reduce both dimension of the problem under consideration and cost for evaluating the full nonlinear system.

The paper addresses the dynamical properties of large-scale perturbed nonlinear systems of the Hopfield type with feedback. In particular, it focuses on the hyperstability of the equilibria of the system. It proceeds to examine the effect of the empirical balanced truncation model reduction technique on the hyperstability properties. Finally, estimates of the additional conditions for preserving hyperstability when perturbations are present are derived.