Spectral methods for circuit analysis

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1999.Includes bibliographical references (p. 119-124).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Harmonic balance (HB) methods are frequency-domain algorithms used for high accuracy computation of the periodic steady-state of circuits. Matrix-implicit Krylov-subspace techniques have made it possible for these methods to simulate large circuits more efficiently. However, the harmonic balance methods are not so efficient in computing steady-state solutions of strongly nonlinear circuits with rapid transitions. While the time-domain shooting-Newton methods can handle these problems, the low-order integration methods typically used with shooting-Newton methods are inefficient when high solution accuracy is required. We first examine possible enhancements to the standard state-of-the-art preconditioned matrix-implicit Krylovsubspace HB method. We formulate the BDF time-domain preconditioners and show that they can be quite effective for strongly nonlinear circuits, speeding up the HB runtimes by several times compared to using the frequency-domain block-diagonal preconditioner. Also, an approximate Galerkin HB formulation is derived, yielding a small improvement in accuracy over the standard pseudospectral HB formulation, and about a factor of 1.5 runtime speedup in runs reaching identical solution error. Next, we introduce and develop the Time-Mapped Harmonic Balance method (TMHB) as a fast Krylov-subspace spectral method that overcomes the inefficiency of standard harmonic balance for circuits with rapid transitions. TMHB features a non-uniform grid and a time-map function to resolve the sharp features in the signals. At the core of the TMHB method is the notion of pseudo Fourier approximations. The rapid transitions in the solution waveforms are well approximated with pseudo Fourier interpolants, whose building blocks are complex exponential basis functions with smoothly varying frequencies. The TMHB features a matrix-implicit Krylov-subspace solution approach of same complexity as the standard harmonic balance method. As the TMHB solution is computed in a pseudo domain, we give a procedure for computing the real Fourier coefficients of the solution, and we also detail the construction of the time-map function. The convergence properties of TMHB are analyzed and demonstrated on analytic waveforms. The success of TMHB is critically dependent on the selection of a non-uniform grid. Two grid selection strategies, direct and iterative, are introduced and studied. Both strategies are a priori schemes, and are designed to obey accuracy and stability requirements. Practical issues associated with their use are also addressed. Results of applying the TMHB method on several circuit examples demonstrate that the TMHB method achieves up to five orders of magnitude improvement in accuracy compared to the standard harmonic balance method. The solution error in TMHB decays exponentially faster than the standard HB method when the size of the Fourier basis increases linearly. The TMHB method is also up to six times faster than the standard harmonic balance method in reaching identical solution accuracy, and uses up to five times less computer memory. The TMHB runtime speedup factor and storage savings favorably increase for stricter accuracy requirements, making TMHB well suited for high accuracy simulations of large strongly nonlinear circuits with rapid transitions.by Ognen J. Nastov.Ph.D

Frequency domain steady-state simulation of oscillators

The focus of this work is on developing algorithms for frequency domain steady-state analysis of oscillators. Convergence problems associated with the frequency domain harmonic balance simulation of oscillators have been examined. Globally convergent homotopy methods have been combined with the harmonic balance method for robust high-Q oscillator simulation. Various homotopy options are evaluated leading to an algorithm that is applicable to a wide variety of oscillator circuits. Two new approaches have also been developed for the simulation of ring oscillators using the harmonic balance method. These include a single-delay cell method and a multiple-probe method. The new methods that have been proposed are robust compared to traditional methods and readily converge for a wide range of single-ended and differential oscillators. They enable harmonic balance simulation of “difficult-to-converge” oscillator circuits

Inverter design and analysis using multiple reference frame theory

Multiple reference frame theory allows for periodically time varying signals to be represented as a set of dc signals. In other words, every periodic signal can be expanded into a Fourier series representation. By modeling an inverter connected to a boost maximum power point tracker (MPPT) in this manner, frequency transfer properties can be preserved and harmonics throughout the system can be predicted. A state space model taking into account the dc and fundamental grid frequency is presented and used to optimize the controller gains of the system. Using information from the dq-axis values of the measured grid current and voltage, the double frequency dc-link voltage component is predicted. The double frequency component is removed from the controller input using feedforward. As a result, there is a reduction in output harmonics in the grid current. The same method is applied to the MPPT, where the double frequency component is predicted and removed from the controller input. This allows for a MPPT with reduced oscillations in the input power waveform. Next, a method is presented to generate a large-signal model of a H-bridge inverter. A set of algorithms are presented, which take a standard set of large-signal (user generated) dynamic equations and performs a Fourier series expansion on the inputs and states of the equations. These algorithms work for an arbitrary finite set of harmonics and preserve the frequency transfer properties between harmonics. The solution to the generated equations is the steady state output of the inverter. Lastly, a set of algorithms are presented which take a user generated netlist in and automatically outputs a truncated harmonic transfer function (THTF) --Abstract, page iii

Computationally Efficient Steady--State Simulation Algorithms for Finite-Element Models of Electric Machines.

The finite element method is a powerful tool for analyzing the magnetic characteristics of electric machines, taking account of both complex geometry and nonlinear material properties. When efficiency is the main quantity of interest, loss calculations can be affected significantly due to the development of eddy currents as a result of Faraday’s law. These effects are captured by the periodic steady-state solution of the magnetic diffusion equation. A typical strategy for calculating this solution is to analyze an initial value problem over a time window of sufficient length so that the transient part of the solution becomes negligible. Unfortunately, because the time constants of electric machines are much smaller than their excitation period at peak power, the transient analysis strategy requires simulating the device over many periods to obtain an accurate steady-state solution. Two other categories of algorithms exist for directly calculating the steady-state solution of the magnetic diffusion equation; shooting methods and the harmonic balance method. Shooting methods search for the steady-state solution by solving a periodic boundary value problem. These methods have only been investigated using first order numerical integration techniques. The harmonic balance method is a Fourier spectral method applied in the time dimension. The standard iterative procedures used for the harmonic balance method do not work well for electric machine simulations due to the rotational motion of the rotor. This dissertation proposes several modifications of these steady-state algorithms which improve their overall performance. First, we demonstrate how shooting methods may be implemented efficiently using Runge-Kutta numerical integration methods with mild coefficient restrictions. Second, we develop a preconditioning strategy for the harmonic balance equations which is robust against large time constants, strong nonlinearities, and rotational motion. Third, we present an adaptive framework for refining the solutions based on a local error criterion which further reduces simulation time. Finally, we compare the performance of the algorithms on a practical model problem. This comparison demonstrates the superiority of the improved steady-state analysis methods, and the harmonic balance method in particular, over transient analysis.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113322/1/pries_1.pd

Multilevel harmonic balance analysis of large-scale nonlinear RF circuits via Newton-Krylov and tensor-Krylov methods

Orientador: Hugo Enrique Hernandez FigueroaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Este trabalho, tem como objetivo o desenvolvimento de novas técnicas, para análise de regime permanente não-autonoma de circuitos de alta-velocidade não-lineares em grande-escala. Para tal, é proposto um novo método do balanço harmônico (BH) fundamentado em uma eficiente metodologia de decomposição multi-níveis, que subdivide um circuito não-linear em grande escala em uma estrutura hierarquica de super-redes (SuRs) esparsamente interconectadas. Mais precisamente, em cada nível de hierarquia, o circuito é composto por SuRs intermediárias, SuRs de fundo, e redes de conexão (RCs). As SuRs de fundo são decompostas em um aglomerado de subredes não-lineares (SRNs) correspondendo a dispositivos semicondutores, que por sua vez, estão envolvidos por uma sub-rede linear (SRL). A equação de estado e de sonda das SuRs de fundo foram obtidas utilizando uma nova metodologia que combina a formulação de espaço de estado (FEE) para as SRNs com a formulação nodal modificada (FNM) para a SRL. Esta metodologia FEE/FNM produz um sistema quadrado de equações com menor tamanho possível. Para realização das conversões do sinal entre os domínios do tempo e da frequência, foram discutidas e implementadas diferentes transformadas de Fourier discreta (TFDs), para operação em regime multi-tons, incluindo sinais com modulação digital. A equação determinante do BH multi-níveis do circuito assume uma estrutura hierarquica do tipo bloco diagonal com borda , que pode ser eficientemente resolvida utilizando técnicas de processamento paralelo. A matriz jacobiana de cada SuR de fundo é processada utilizando eficientes técnicas de matrizes esparsas, junto com o conceito de espectro de derivada. Para a solução da equação determinante, foram utilizados os métodos de Newton e do tensor para problemas de pequena- e média-escala, e os métodos de Newton inexato e do tensor inexato para problemas em grande-escala. A globalização via pesquisa-em-linha com retrocedimento, foi adotada para nestes solucionadores não-lineares. Entretanto, para o método do tensor e do tensor inexato, também foi adotada a técnica de pesquisa-em-linha curvilinear. Nos métodos inexatos, técnicas de pré-condicionamento foram utilizadas, para aumentar a eficiência e a robustez do solucionador linear iterativo em subespaço de Krylov (GMRES, GMRES-Bt e TGMRES-Bt). Finalmente, a formulação proposta foi validada e a eficiência do método do tensor e do tensor inexato comparada com o método de Newton e de Newton inexato, para diferentes topologias de circuitos utilizando diodos, FETs e HBTs, e operando sob diferentes regimes de excitação multi-tons.Abstract: This work deals with the development of new techniques for nonautonomous nonlinear steady-state analysis of high-speed large-scale integrated circuits. To this end, it is proposed a novel harmonic balance (HB) method fundamented on a efficient multi-level decomposition methodology, that divides a large-scale circuit into hierarchical structure of sparsely interconnected supernetworks (SuNs). More precisely, the circuit is composed by intermediary SuRs, bottom SuRs and connection networks (CNs). The bottom SuNs are decomposed into a cluster of nonlinear subnetworks (NSNs) corresponding to the opto-electronic semiconductor devices, which in turn, are embedded by a linear subnetwork (LSN). Multi-port elements can be included in the LSN, in order to use measured data or results from electromagnetic analysis of structures with complex geometries. The formulation of the bottom SuN state and probe equations uses an improved table-oriented statespace formulation (SSF), that produces a square system with the lowest possible size, which is equal to the number of nonlinear state-variables (branch voltages and currents) that act as argument of the fuctions representing the semiconductor devices nonlinearities. The SSF is compared with the classical modified nodal formulation (MNF). For dealing with signal timefrequency conversions, discrete Fourier transform (DFT) techniques for different multi-tone regimes are discussed, including complex digitally modulated signals. The multi-level HB determining equation of the circuit assumes a hierarchical block bordered structure that can be efficiently tackled by parallel processing techniques. The HB jacobian matrix is handled using efficient sparse matrix techniques with a proper definition of the derivatives spectra. For the solution of a large-size HB problem, we investigated the applications of inexact tensor method based on Krylov-subspace techniques. Preconditioning are used to improve the robustness of the iterative tensor solver. To determine the circuit DC regime, we employ the tensor method. We adopted the backtracking linesearch technique as a globalisation strategy. However, for the tensor method, in particular, a curvilinear linesearch was also implemented. Finally, the formulation was validated and, the tensor and inexact tensor method efficiency was compared with the Newton and inexact Newton method, respectively, for several different circuits using diodos, FETs and HBTs, and operating under different multi-tone regimes.DoutoradoEngenharia de TelecomunicaçõesDoutor em Engenharia Elétric

Time-Mapped Harmonic Balance

Matrix-implicit Krylov-subspace methods have made it possible to efficiently compute the periodic steady-state of large circuits using either the time-domain shooting-Newton method or the frequencydomain harmonic balance method. However, the harmonic balance methods are not so efficient at computing steady-state solutions with rapid transitions, and the low-order integration methods typically used with shooting-Newton methods are not so efficient when high accuracy is required. In this paper we describe a Time-Mapped Harmonic Balance method (TMHB), a fast Krylovsubspace spectral method that overcomes the inefficiency of standard harmonic balance in the case of rapid transitions. TMHB features a non-uniform grid to resolve the sharp features in the signals. Results on several examples demonstrate that the TMHB method achieves several orders of magnitude improvement in accuracy compared to the standard harmonic balance method. The TMHB method is also several times faster than the standard harmonic balance method in reaching identical solution accuracy