Hans Georg Brachtendorf

Numerical steady state analysis of electronic circuits driven by multi-tone signals

ContentsCharacteristics of analogue circuits such as intermodulation distortion and transfer characteristics can often be received from the steady state behavior. This paper presents a unified approach for the simulation of non-autonomous circuits with multi-tone excitation. The steady state is here regarded as the solution of a partial differential-algebraic equation. A suitable numerical method for its solution is a variational method with trigonometric basis functions. The Harmonic Balance technique based either on the multi-dimensional Fourier transformation or the Artificial Frequency Map technique can be interpreted as a special variant of this method.ÜbersichtDie Eigenschaften analoger Schaltungen, die etwa Intermodulationsverzerrungen und Übertragungscharakteristiken beschreiben, lassen sich häufig im eingeschwungenen Zustands ermitteln. Dieser Beitrag stellt ein vereinheitlichtes Verfahren zur Simulation von nicht-autonomen Schaltungen bei einer Mehrton-Erregung vor. Der eingeschwungene Zustand wird als spezielle Lösung einer partiellen Algebro-Differentialgleichung formuliert. Zur numerischen Berechnung eignen sich Variationsverfahren mit trigonometrischen Ansatzfunktionen. Interessant ist, daß das Verfahren der Harmonischen Balance, sowohl basierend auf einer mehrdimensionalen Fouriertransformation als auch basierend auf einer Transformation auf ein künstliches Hilfsspektrum, als spezielle Variante dieses Ansatzes angesehen werden kann.

A novel time-frequency method for the simulation of the steady state of circuits driven by multi-tone signals

In this paper we introduce a novel algorithm for efficient computing of the steady state of circuits driven by multi-tone excitation. The efficiency is the result of combining finite difference and the harmonic balance techniques. Unlike existing techniques the algorithm shown here is based on reformulating the underlying system of ordinary DAEs by a system of partial DAEs. The steady state leads to simple boundary conditions for these partial DAEs.

Fast simulation of the steady-state of circuits by the harmonic balance technique

When simulating analog and microwave circuits, the steady-state behavior is of primary interest. One method for simulating the steady-state is the harmonic balance technique (HB). HB is characterized by the use of trigonometric basis functions. The resulting nonlinear equations are solved by Newtons method (NR). The linear systems arising from NR are very large, indefinite but sparse. They can be solved by direct, stationary or Krylov subspace methods. This paper deals with the solution of linear systems arising from NE using preconditioned Krylov subspace methods (CGS, BiCGSTAB, BiCGSTAB(2), TFQMR).

Adaptive Multi-Rate Wavelet Method for Circuit Simulation

In this paper a new adaptive algorithm for multi-rate circuit simulation encountered in the design of RF circuits based on spline wavelets is presented. The circuit ordinary differential equations are first rewritten by a system of (multi-rate) partial differential equations (MPDEs) in order to decouple the different time scales. Second, a semi-discretization by Rothes method of the MPDEs results in a system of differential algebraic equations (DAEs) with periodic boundary conditions. These boundary value problems are solved by a Galerkin discretization using spline functions. An adaptive spline grid is generated, using spline wavelets for non-uniform grids. Moreover the instantaneous frequency is chosen adaptively to guarantee a smooth envelope resulting in large time steps and therefore high run time efficiency. Numerical tests on circuits exhibiting multi-rate behavior including mixers and PLL conclude the paper.

A simulation tool for the analysis and verification of the steady state of circuit designs

Analogue and microwave design requires accurate and reliable simulation tools and methods to meet the design specifications. System properties are often measured in the steady state. Well-suited algorithms for calculating the steady state can be classified into shooting methods, finite difference methods and the harmonic balance (HB) technique. Harmonic balance is a frequency domain method which approaches the problem of finding the steady state by a trigonometric polynomial. Depending on the size of the circuit and the number of Fourier coefficients of the polynomial, the resulting system of non-linear equations can become very large. These non-linear equations are solved by using Newtons method. The sparse linear system arising from Newtons method can be solved by direct, stationary or non-stationary iterative solvers. Iterative methods are normally easy to parallelize or vectorize. In this paper a tool for the simulation of the steady state of electronic circuits is presented. the steady state is calculated using the harmonic balance technique. Non-linear equations are solved by Newtons method and linear equations by preconditioned non-stationary iterative solvers (CGS, Bi-CGSTAB, BiCGSTAB(2), TFQMR). the run time is reduced dramatically, by up to an order of magnitude.

Optimal frequency sweep method in multi-rate circuit simulation

Purpose – Radio-frequency circuits often possess a multi-rate behavior. Slow changing baseband signals and fast oscillating carrier signals often occur in the same circuit. Frequency modulated signals pose a particular challenge. The paper aims to discuss these issues. Design/methodology/approach – The ordinary circuit differential equations are first rewritten by a system of (multi-rate) partial differential equations in order to decouple the different time scales. For an efficient simulation the paper needs an optimal choice of a frequency-dependent parameter. This is achieved by an additional smoothness condition. Findings – By incorporating the smoothness condition into the discretization, the paper obtains a non-linear system of equations complemented by a minimization constraint. This problem is solved by a modified Newton method, which needs only little extra computational effort. The method is tested on a phase locked loop with a frequency modulated input signal. Originality/value – A new optimal ...

Grid Size Adapted Multistep Methods for High

The numerical calculation of the limit cycle of oscillators with resonators exhibiting a high-quality factor Q such as quartz crystals is a difficult task in the time domain. Time domain integration formulas, when not carefully selected, introduce numerical damping that leads to erroneous limit cycles or spurious oscillations. A novel class of adaptive multistep integration formulas based on finite difference (FD) schemes is derived, which circumvent the aforementioned problems. The results are compared with the well-known harmonic balance (HB) technique. Moreover, the range of absolute stability is derived for these methods. The resulting discretized system by FD methods is sparser than that of HB and, therefore, easier to solve and easier to implement.

Q

New algorithms for fast wavelet transforms with biorthogonal spline wavelets on nonuniform grids are presented. In contrary to classical wavelet transforms, the algorithms are not based on filter coefficients, but on algorithms for B-spline expansions (differentiation, Oslo algorithm, etc.). Due to inherent properties of the spline wavelets, the algorithm can be modified for spline grid refinement or coarsening. The performance of the algorithms is demonstrated by numerical tests of the adaptive spline methods in circuit simulation.

Oscillators

Shooting, finite difference or Harmonic Balance techniques in conjunction with Newtons method are widely employed for the numerical calculation of limit cycles of oscillators. The resulting set of nonlinear equations are normally solved by damped Newtons method. In some cases however divergence occurs when the initial estimate of the solution is not close enough to the exact one. A two-dimensional homotopy method is presented in this paper which overcomes this problem. The resulting linear set of equations employing Newtons method is under-determined and is solved in a least squares sense for which a rigorous mathematical basis can be derived.

Fast Algorithms for Adaptive Free Knot Spline Approximation Using Nonuniform Biorthogonal Spline Wavelets

In this paper we study efficient numerical methods for obtaining consistent initial conditions for systems of differential-algebraic equations (DAEs) with higher index arising, for example, from electronic circuits. We show that the class of Gears backward differentiation formulas, unlike other multi-step techniques, are a useful means for obtaining consistent initial conditions when carefully implemented. Because the method employs sparse matrix techniques, it is efficient even for large circuits, The numerical experiments suggest that the method works reliably even for index-3 DAEs.