Markus Brunk

Dynamic Iteration for Coupled Problems of Electric Circuits and Distributed Devices

Coupled systems of differential-algebraic equations (DAEs) may suffer from instabilities during a dynamic iteration. We extend the existing analysis on recursion estimates, error propagation, and stability to (semiexplicit) index-1 DAEs. In this context, we discuss the influence of certain coupling structures and the computational sequence of the subsystems on the rate of convergence. Furthermore, we investigate in detail convergence and divergence for two coupled problems stemming from refined electric circuit simulation. These are the semiconductor-circuit and field-circuit couplings. We quantify the convergence rate and behavior also using Lipschitz constants and suggest an enhanced modeling of the coupling interface in order to improve convergence.

Numerical Coupling of Electric Circuit Equations and Energy-Transport Models for Semiconductors

A coupled semiconductor-circuit model including thermal effects is proposed. The charged particle flow in the semiconductor devices is described by the energy-transport equations for the electrons and the drift-diffusion equations for the holes. The electric circuit is modeled by the network equations from modified nodal analysis. The coupling is realized by the node potentials providing the voltages applied to the semiconductor devices and the output device currents for the network model. The resulting partial differential-algebraic system is discretized in time by the 2-stage backward difference formula and in space by a mixed-hybrid finite-element method using Marini-Pietra elements. A static condensation procedure is applied to the coupled system reducing the number of unknowns. Numerical simulations of a one-dimensional

Progress in industrial mathematics at ECMI 2010

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On the convergence rate of dynamic iteration for coupled problems with multiple subsystems

-junction diode with time-dependent voltage and of a rectifier circuit show the heating of the electrons which allows one to identify hot spots in the devices. Moreover, the choice of the boundary conditions for the electron density and energy density is numerically discussed.

A Convergent Iteration Scheme for Semiconductor/Circuit Coupled Problems

ECMI is the brand associated with European mathematics for industry and organizes successful biannual conferences. In this series, the 16th conference was held in the Historical City Hall of Wuppertal (Germany). It covered mathematics in a wide range of applications and methods, from Circuit and Electromagnetic Device Simulation, Model Order Reduction for Chip Design, Uncertainties and Stochastics, Production, Fluids, Life and Environmental Sciences to Dedicated and Versatile Methods. These proceedings of ECMI 2010 emphasize mathematics as an innovation enabler for industry and business, and as an absolutely essential pre-requiste for Europe on its way to becoming the leading knowledge-based economy in the world.We present a new adaptive circuit simulation algorithm base d on spline wavelets. The unknown voltages and currents are expanded in to a wavelet representation, which is determined as solution of nonlinear equ ations derived from the circuit equations by a Galerkin discretization. The spline wavelet representation is adaptively refined during the Newton iteration. The resulti ng approximation requires an almost minimal number of degrees of freedom, and in additi on the grid refinement approach enables very efficient numerical computation s. Initial numerical tests on various types of electronic circuits show promising resu lts when compared to the standard transient analysis.

SIMULATION OF THERMAL EFFECTS IN OPTOELECTRONIC DEVICES USING COUPLED ENERGY-TRANSPORT AND CIRCUIT MODELS

In multiphysical modeling coupled problems naturally occur. Each subproblem is commonly represented by a system of partial differential-algebraic equations. Applying the method of lines, this results in coupled differential-algebraic equations (DAEs). Dynamic iteration with windowing is a standard technique for the transient simulation of such systems. In contrast to the dynamic iteration of systems of ordinary differential equations, convergence for DAEs cannot be generally guaranteed unless some contraction condition is fulfilled. In the case of convergence, it is a linear one. In this paper, we quantify the convergence rate, i.e., the slope of the contraction, in terms of the window size. We investigate the convergence rate with respect to the coupling structure for DAE and ODE systems and also for two and more subsystems. We find higher rates (for certain coupling structures) than known before (that is, linear in the window size) and give sharp estimates for the rate. Furthermore it is revealed how the rate depends on the number of subsystems.

Heating of Semiconductor Devices in Electric Circuits

A dynamic iteration scheme is proposed for a coupled system of electric circuit and distributed semiconductor (pn-diode) model equations. The device is modelled by the drift-diffusion (DD) equations and the circuit by MNA-equations. The dynamic iteration scheme is investigated on the basis of discrete models and coupling via sources and compact models. The analytic divergence and analytic convergence results are verified numerically.

Numerical Simulation of Thermal Effects in Coupled Optoelectronic Device-circuit Systems

A coupled model with optoelectronic semiconductor devices in electric circuits is proposed. The circuit is modeled by differential-algebraic equations derived from modified nodal analysis. The transport of charge carriers in the semiconductor devices (laser diode and photo diode) is described by the energy-transport equations for the electron density and temperature, the drift-diffusion equations for the hole density, and the Poisson equation for the electric potential. The generation of photons in the laser diode is modeled by spontaneous and stimulated recombination terms appearing in the transport equations. The devices are coupled to the circuit by the semiconductor current entering the circuit and by the applied voltage at the device contacts, coming from the circuit. The resulting time-dependent model is a system of nonlinear partial differential-algebraic equations. The one-dimensional transient transport equations are numerically discretized in time by the backward Euler method and in space by a hybridized mixed finite-element method. Numerical results for a circuit consisting of a single-mode heterostructure laser diode, a silicon photo diode, and a high-pass filter are presented.

Self-heating in a coupled thermo-electric circuit-device model

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.

Positivity preserving discretization of time dependent semiconductor drift-diffusion equations

The control of thermal effects becomes more and more important in modern semiconductor circuits like in the simplified CMOS transceiver representation described by U. Feldmann in the above article Numerical simulation of multiscale models for radio frequency circuits in the time domain. The standard approach for modeling integrated circuits is to replace the semiconductor devices by equivalent circuits consisting of basic elements and resulting in so-called compact models. Parasitic thermal effects, however, require a very large number of basic elements and a careful adjustment of the resulting large number of parameters in order to achieve the needed accuracy.