# Direct Steady-State Determination in SPICE Using Krylov-Subspace Methods

## Abstract

Given today's increasing demand for communication devices, including cellular telephones and other cordless devices, more effective methods are needed to examine integrated circuits that make up these devices in the steady-state mode. Quantities such as power, distortion, and noise are evaluated in the steady-state and need to be studied in detail. Standard simulators, such as SPICE, can use transient analysis to determine the steady-state response by simulating until all the transients have died out. Unfortunately, for the type of circuits used in communication devices, this takes much too long for a detailed analysis to be made. Harmonic balance methods are widely used for steady-state determination but are not optimal for strongly nonlinear circuits. Direct time-domain methods are a good choice for steady-state determination. An improved and expanded approach has been developed to determine the steady-state response for analog and RF integrated circuits in the time-domain. The approach uses the standard shooting-Newton algorithm to directly determine the steady-state. Different Krylov-subspace algorithms have been coded and implemented that can be used to solve the resulting iterative equations generated by this method. The approach also incorporates distributed devices that are needed for many communication circuits. The purpose of the new and improved steady-state method is to efficiently determine the steady-state response for analog and microwave circuits that are difficult in standard methods. This class of circuits includes strongly nonlinear communication circuits. The shooting-Newton algorithm is a direct approach to determine the steady-state of integrated circuits. The calculation involves the solution of an iterative equation. Historically, this system is solved via methods such as Gaussian elimination. This limits the method to relatively small circuits. With the implementation of the Krylov-subspace methods, greater computational efficiency is gained and larger, more complex circuits can be simulated. Different Krylov- subspace algorithms are available to compute the solution so as to take advantage of the particular matrix being solved. Also, the distributed elements are included for analysis. Elements such as transmission lines are modeled and simulated in the time domain. The shooting-Newton method along with the other aspects of the approach is implemented and tested in the most popular analog circuit simulator, SPICE. Quantities computed during the transient analysis in SPICE are reused in the calculation of the direct determination of the steady-state response. Transient analysis and the shooting-Newton method are efficient for strongly nonlinear circuits that are encountered often in RF systems. Results from simulations are compared with standard transient simulation, standard shooting-Newton with Gaussian elimination, and a harmonic balance simulation. The convergence behavior of the different Krylov methods BiCG, CGS, BiCGSTAB, GMRES, QMR, and GMRES are presented along with matrix characteristics. This approach is shown to be much more efficient than transient analysis when finding the steady-state solution of certain classes of circuits, like self-biasing amplifiers, narrow-band filters, and circuits with high Qs, lightly damped circuits with long time constants. Because the approach is computationally efficient it allows for simulation of larger circuits, including transmission lines. It also shows greater efficiency than standard Gaussian elimination solution of the shooting-Newton iterate. Since distributed components are included in the approach, a much wider class of circuits can be simulated. Incorporated into SPICE this approach, is accurate and efficient for the steady-state response for nonlinear circuits that are found in communication devices.