Shooting, finite difference, or harmonic balance techniques in conjunction with a damped Newton method are widely employed for the numerical calculation of limit cycles of (free-running, autonomous) oscillators. In some cases, however, nonconvergence occurs when the initial estimate of the solution is not close enough to the exact one. Generally, the higher the quality factor of the oscillator the tighter are the constraints for the initial estimate. A 2-D homotopy method is presented in this paper that overcomes this problem. The resulting linear set of equations is underdetermined, leading to a nullspace of rank two. This underdetermined system is solved in a least squares sense for which a rigorous mathematical basis can be derived. An efficient algorithm for solving the least squares problem is derived where sparse matrix techniques can be used. As continuation methods are only employed for obtaining a sufficient initial guess of the limit cycle, a coarse grid discretization is sufficient to make the method runtime efficient.

Homotopy/continuation methods are attractive for finding dc operating points of circuits because they offer theoretical guarantees of global convergence. Existing homotopy approaches for circuits are, however, often ineffective for large mixed-signal applications. In this paper, we describe a robust homotopy technique that is effective for solving large metal-oxide-semiconductor (MOS)-based mixed-signal circuits. We demonstrate how certain common circuit structures involving turning-point nesting can lead to extreme inefficiency, or failure, of conventional probability-one homotopy methods. We also find that such situations can lead to numerical ill-conditioning and homotopy paths that fold back upon themselves, leading to algorithm failure. Our new homotopy model for MOS devices, dubbed Arc-tangent Schichman-Hodges (ATANSH), features decoupled continuation parameters that are instrumental in avoiding these problems. ATANSH-based homotopy methods in production use have led to the routine solution of large previously hard-to-solve industrial circuits, several examples of which are presented.

Shooting, finite difference or Harmonic Balance techniques in conjunction with Newton's method are widely employed for the numerical calculation of limit cycles of oscillators. The resulting set of nonlinear equations are normally solved by damped Newton's 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 Newton's method is under-determined and is solved in a least squares sense for which a rigorous mathematical basis can be derived.

We describe powerful new techniques for the analysis of RF circuits. Next-generation CAD tools based on such techniques should enable RF designers to obtain a more accurate picture of how their circuits will operate. These new simulation capabilities will be essential in order to reduce the number of design iterations needed to produce complex RF ICs. 1 Introduction Design methodology and superior computer-aided design tools are key to success in the integrated circuit (IC) business. They are particularly important in the case of radio-frequency (RF) IC applications, where the digital IC divide-and-conquerdesign style, based on partitioning by functional blocks and abstraction levels, does not apply. The goal of an RF designer is to get a manufacturable design that meets the specifications with minimum cost, under severe time-to-market constraints. Unlike traditional discretecomponent RF design, prototyping is practically impossible, and the validation of a design can only be done b...

HOMPACK90 is a Fortran 90 version of the Fortran 77 package HOMPACK (Algorithm 652), a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probability-one homotopy algorithms.Three qualitatively different algorithmsordinary differential equation based, normal flow, quasi-Newton augmented Jacobian matrix-are provided for tracking homotopy zero curves, as well as separate routines for dense and sparse Jacobian matrices.A high level driver for the special case of polynomial systems is also provided.Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.

A new technique for obtaining a DC operating point of large, hard-to-solve MOS circuits is reported in this paper. Based on homotopy, the technique relies on the provable global convergence of arc length continuation and uses a novel method for embedding the continuation parameter into MOS devices. The new embedding circumvents inefficiencies and numerical failures that limit the practical applicability of previous simpler embeddings. Use of the technique in a production environment has led to the routine solution of large, previously hard-to-solve circuits.

A sufficient condition is given for which a continuation method is guaranteed to find at least three operating points possessed by a multistable circuit. A practical realization of this condition in a homotopy using passive components is also given and demonstrated in circuit examples.

Efficient and robust computation of one or more of the operating points of a nonlinear circuit is a necessary first step in a circuit simulator. The application of globally convergent probability-one homotopy methods to various systems of nonlinear equations that arise in circuit simulation is discussed. The coercivity conditions required for such methods are established using concepts from circuit theory. The theoretical claims of global convergence for such methods are substantiated by experiments with a collection of examples that have proved difficult for commercial simulation packages that do not use homotopy methods. Moreover, by careful design of the homotopy equations, the performance of the homotopy methods can be made quite reasonable. An extension to the steady-state problem in the time domain is also discussed.>

Automatic differentiation as a technique for accurate and reliable computation of partial derivatives in device models used by circuit simulation is introduced. The requirements for derivative computations in several simulation algorithms are reviewed, and two automatic differentiation methods are discussed. The application of automatic differentiation to circuit simulation and device modeling is demonstrated through several circuit analysis examples.>