Coupled systems of differential-algebraic equations (DAEs) may suffer from instabilities during a dynamic iteration. For a general dynamic iteration, we extend the existing analysis on recursion estimates, error transport and stability to a general DAE setting. 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 convergence and divergence for two coupled problems stemming from refined electric circuit simulation in detail. These are the semiconductorcircuit and field-circuit coupling. As a result of our analysis, 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.
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 pn-junction diode with timedependent 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.
Introduction.In industrial applications, complex semiconductor device models are usually substituted by circuits of basic network elements (resistors, capacitors, inductors, voltage, and current sources) resulting in simpler so-called compact models. Such a strategy was advantageous up to now since integrated circuit simulations were possible without computationally expensive device simulations. Parasitic effects and high frequencies in the circuits, however, require one to take into account a very large number of basic elements and to adjust carefully a large number of parameters in order to achieve the required accuracy. Moreover, device heating and hot spots cannot be easily modeled by this approach.Therefore, it is preferable to model those semiconductor devices which are critical for the parasitic effects by semiconductor transport equations. Since structural information about the resulting device-circuit equations was missing for a long time, the first approaches to couple circuits and devices were based on an extension of existing device simulators by more complex boundary conditions [36,40] or the combination of device simulators with circuit simulators as a "black box" solver [15]. Both approaches, however, are not suitable for complex circuits in the high-frequency domain. In this work, we follow an approach that includes the device model into the network equations directly. As the device model is described by partial differential equations and the network equations are given by differential-algebraic equations
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.
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