A practical regularization technique for modified nodal analysis in large-scale time-domain circuit simulation

Fast full-chip time-domain simulation calls for advanced numerical integration techniques with capability to handle the systems with (tens of) millions of variables resulting from the modified nodal analysis (MNA). General MNA formulation, however, leads to a differential algebraic equation (DAE) system with singular coefficient matrix, for which most of explicit methods, which usually offer better scalability than implicit methods, are not readily available. In this paper, we develop a practical two-stage strategy to remove the singularity in MNA equations of large-scale circuit networks. A topological index reduction is first applied to reduce the DAE index of the MNA equation to one. The index-1 system is then fed into a systematic process to eliminate excess variables in one run, which leads to a nonsingular system. The whole regularization process is devised with emphasis on exact equivalence, low complexity, and sparsity preservation, and is thus well suited to handle extremely large circuits. © 2012 IEEE.published_or_final_versio

Globally stable, highly parallelizable fast transient circuit simulation via faber series

Time-domain circuit simulation based on matrix exponential has attracted renewed interested, owing to its explicit nature and global stability that enable millionth-order circuit simulation. The matrix exponential is commonly computed by Krylov subspace methods, which become inefficient when the circuit is stiff, namely, when the time constants of the circuit differ by several orders. In this paper, we utilize the truncated Faber Series for accurate evaluation of the matrix exponential even under a highly stiff system matrix arising from practical circuits. Experiments have shown that the proposed approach is globally stable, highly accurate and parallelizable, and avoids excessive memory storage demanded by Krylov subspace methods. © 2012 IEEE.published_or_final_versio

An Algorithmic Framework for Efficient Large-Scale Circuit Simulation Using Exponential Integrators

We propose an efficient algorithmic framework for time domain circuit simulation using exponential integrator. This work addresses several critical issues exposed by previous matrix exponential based circuit simulation research, and makes it capable of simulating stiff nonlinear circuit system at a large scale. In this framework, the system's nonlinearity is treated with exponential Rosenbrock-Euler formulation. The matrix exponential and vector product is computed using invert Krylov subspace method. Our proposed method has several distinguished advantages over conventional formulations (e.g., the well-known backward Euler with Newton-Raphson method). The matrix factorization is performed only for the conductance/resistance matrix G, without being performed for the combinations of the capacitance/inductance matrix C and matrix G, which are used in traditional implicit formulations. Furthermore, due to the explicit nature of our formulation, we do not need to repeat LU decompositions when adjusting the length of time steps for error controls. Our algorithm is better suited to solving tightly coupled post-layout circuits in the pursuit for full-chip simulation. Our experimental results validate the advantages of our framework.Comment: 6 pages; ACM/IEEE DAC 201

MATEX: A Distributed Framework for Transient Simulation of Power Distribution Networks

We proposed MATEX, a distributed framework for transient simulation of power distribution networks (PDNs). MATEX utilizes matrix exponential kernel with Krylov subspace approximations to solve differential equations of linear circuit. First, the whole simulation task is divided into subtasks based on decompositions of current sources, in order to reduce the computational overheads. Then these subtasks are distributed to different computing nodes and processed in parallel. Within each node, after the matrix factorization at the beginning of simulation, the adaptive time stepping solver is performed without extra matrix re-factorizations. MATEX overcomes the stiff-ness hinder of previous matrix exponential-based circuit simulator by rational Krylov subspace method, which leads to larger step sizes with smaller dimensions of Krylov subspace bases and highly accelerates the whole computation. MATEX outperforms both traditional fixed and adaptive time stepping methods, e.g., achieving around 13X over the trapezoidal framework with fixed time step for the IBM power grid benchmarks.Comment: ACM/IEEE DAC 2014. arXiv admin note: substantial text overlap with arXiv:1505.0669

Analog and Mixed Signal Verification using Satisfiability Solver on Discretized Models

With increasing demand of performance constraints and the ever reducing size of the IC chips, analog and mixed-signal designs have become indispensable and increasingly complex in modern CMOS technologies. This has resulted in the rise of stochastic behavior in circuits, making it important to detect all the corner cases and verify the correct functionality of the design under all circumstances during the earlier stages of the design process. It can be achieved by functional or formal verification methods, which are still widely unexplored for Analog and Mixed-Signal (AMS) designs. Design Verification is a process to validate the performance of the system in accordance with desired specifications. Functional verification relies on simulating different combinations of inputs for maximum state space coverage. With the exponential increase in the complexity of circuits, traditional functional verification techniques are getting more and more inadequate in terms of exhaustiveness of the solution. Formal verification attempts to provide a mathematical proof for the correctness of the design regardless of the circumstances. Thus, it is possible to get 100% coverage using formal verification. However, it requires advanced mathematics knowledge and thus is not feasible for all applications. In this thesis, we present a technique for analog and mixed-signal verification targeting DC verification using Berkeley Short-channel Igfet Models (BSIM) for approximation. The verification problem is first defined using the state space equations for the given circuit and applying Satisfiability Modulo Theories (SMT) solver to determine a region that encloses complete DC equilibrium of the circuit. The technique is applied to an example circuit and the results are analyzed in turns of runtime effectiveness

Time-domain analysis of large-scale circuits by matrix exponential method with adaptive control

We propose an explicit numerical integration method based on matrix exponential operator for transient analysis of large-scale circuits. Solving the differential equation analytically, the limiting factor of maximum time step changes largely from the stability and Taylor truncation error to the error in computing the matrix exponential operator. We utilize Krylov subspace projection to reduce the computation complexity of matrix exponential operator. We also devise a prediction-correction scheme tailored for the matrix exponential approach to dynamically adjust the step size and the order of Krylov subspace approximation. Numerical experiments show the advantages of the proposed method compared with the implicit trapezoidal method. © 1982-2012 IEEE.published_or_final_versio

computer-aided transient simulation of switched power electronic circuits

Die Leistungselektronik zählt zu den sogenannten Schlüsseltechnologien. Sie findet überall dort Verwendung, wo elektrische Energie umgeformt werden muss. Sei es die Stromversorgung eines Prozessors, die Motorelektronik einer Waschmaschine, das Vorschaltgerät einer LED-Lampe, unzählige Steuergeräte im Automobil oder die Anbindung regenerativer Energiequellen an das Versorgungsnetz. Die Anwendungsbereiche sind äußerst vielfältig. Fortschritte im Gebiet der Leistungselektronik wirken sich dadurch direkt auf die Einsatzgebiete aus und erlauben noch effizientere, kleinere und kostengünstigere Gesamtlösungen. In den letzten Jahren prägt ein hoher Anteil an Modellbildung und Simulation zunehmend die Entwicklung moderner Leistungselektronik. Der große wirtschaftliche Druck auf die Hersteller sowie die zunehmende Komplexität der Schaltungen stellen den Schaltungsentwickler vor Herausforderungen, die er nur durch eine simulationsbasierte Vorgehensweise beherrschen kann. So helfen Simulationsergebnisse die Funktionsweise und Zusammenhänge besser und schneller zu verstehen sowie die Schaltungsparameter optimal an die Entwicklungsziele anzupassen. Ausgehend von einem Überblick über aktuelle Lösungsansätze zur Schaltungssimulation, beschäftigt sich die vorliegende Arbeit mit der Entwicklung eines neuen Programms zur transienten Simulation getakteter, leistungselektronischer Schaltungen. Die rechnerinterne Beschreibung geschieht, ausgehend von einer SPICE-Netzliste, mit Hilfe stückweise linearer Netzwerke im Zustandsraum. Durch eine Eingangsgrößenmodellierung, d. h. dem Ersetzen der unabhängigen Strom- und Spannungsquellen durch Ersatznetzwerke, gelingt die Reduktion auf ein homogenes Differentialgleichungssystem. Virtuelle Widerstände helfen unterbestimmte Netzwerke, wie sie im Zusammenhang mit idealen Schaltern auftreten können, zu beheben. Ebenfalls eine mögliche Folge der Verwendung idealer Schaltelemente sind inkonsistente Anfangswerte der dynamischen Schaltungselemente, die das Programm selbstständig mit Hilfe der Gesetze zur Ladungs- bzw. Flusserhaltung löst. Die dabei auftretenden Spannungs- und Stromimpulse werden mittels ihres Gewichts quantitativ erfasst und ermöglichen die automatische Auffindung des korrekten Zustands aller als stückweise linear modellierten, nichtlinearen Schaltungselemente. Maßgeblich für die benötigte Simulationsdauer ist unter anderem die Bestimmung des Zeitpunkts intern gesteuerter Unstetigkeiten, bspw. dem Ein- bzw. Ausschaltzeitpunkt idealer Dioden. Dieser wird mit Methoden der Intervallarithmetik abgeschätzt und durch iteratives Auswerten durch immer kleinere Zeitintervalle zuverlässig eingegrenzt. Eine Blockdiagonalisierung der Systemmatrix mit anschließender Eigenwertverschiebung liefert die für die Abschätzung nötige, analytische Lösung der Matrixexponentialfunktion. Die Kombination all dieser Methoden erlaubt eine hochgradige Ausnutzung des Potenzials der stückweise linearen Modellierung. Das im Rahmen dieser Arbeit entwickelte Simulationswerkzeug ermöglicht es dem Schaltungsentwickler, einzig auf Basis einer SPICE-Netzliste, zuverlässige und hochgenaue Ergebnisse mit geringem Rechenaufwand zu erhalten.Power electronics is a key enabling technology which can be found wherever electric power has to be controlled and converted. Its extremely wide application area ranges from power supplies for CPUs to motor electronics of washing machines to LED lamp ballasts to countless car control units and to grid integration of regenerative energy sources. Progress in power electronics, thus, has large implications on many other areas and enables more efficient, smaller and less expensive solutions. The design process of modern power electronics is characterised by a large amount of modeling and simulation. A simulation-based approach helps the circuit designer to master the conflict between the demand for shorter time to market and an ever increasing circuit complexity. Simulation results allow to understand a circuit's basic operation and let the designer optimize parameter values to reach specified design constraints. Based on an overview of the state of the art of circuit simulation this thesis develops a new computer simulation software aimed at switched power electronic circuits. Within computer memory, state-space matrices, derived from piece-wise linear networks, represent the circuit, which itself is defined by a SPICE-netlist. Replacing the networks' independent sources by equivalent circuits allows a formulation as an homogeneous differential equation system. Under-determined networks, which can occur with ideal switches, are fixed using virtual resistors. Networks with ideal switches may as well exhibit inconsistent initial conditions of energy storing circuit elements. The simulator uses the laws of charge and flux conservation to solve this issue. Accompanying impulses in voltage or current are quantified by their weight and allow the automatic state detection of all piece-wise linear elements. During the transient analysis some circuit components, e. g. diodes, cause breakpoints controlled by internal quantities. The detection of these events exhibits a large impact on simulation time. The corresponding time instants are bound iteratively by increasingly narrow intervals employing methods from interval analysis. Block diagonalisation of the system matrix in combination with eigenvalue shifting enables the analytical expressions required for the upper and lower bounds of the matrix exponential function. The combination of all these methods provides access to the full potential of piece-wise linearly modeled circuits. The proposed simulation tool allows circuit designers to get reliable and highly accurate simulation results in short time on the basis of a SPICE-netlist without the need for any further user input