#### The Application of Single-step High Order Integration Methods for Periodic Steady-state Analysis of Integrated Circuits

Gourary M.M., Zharov M.M., Rusakov S.G., Ulyanov S.L. (IPPM RAS)

Abstract - The periodic steady state analysis is one of the most important tool in design of analog and RF integrated circuits. The application of conventional transient analysis to find a periodic steady state solution often results in a long simulation time and hence special purpose means are needed. The method of shooting-Newton is used in periodic steady state analysis to solve a periodic boundary value problem. The shooting-Newton method transforms the solution of the periodic boundary value problem to the solution of sequence of initial value problems on the one period of input signal. The initial value problem is solved using transient analysis. The efficiency of the method depends on both the computation of sensitivity matrix and the solution of linear system with dense Jacobian matrix. The efficient numerical techniques have been proposed. Another factor that determines the computational cost of the method is a numerical technique used to integrate differential equations on the period of input signal. To perform numerical integration of ordinary differential or differential-algebraic equations the comprehensive variable order and variable time step integration algorithms based on implicit multistep integration methods are used. The most commonly used methods are backward Euler, trapezoidal and backward-differentiation formulas (BDF). The common drawback of methods is the lack of A-stability for order higher than 2. As a result high order methods are not implemented in modern simulators. However the usage of high order and A-stable methods allows to improve accuracy and speed up time-domain transient analysis. In this paper the method of periodic steady-state analysis in analog integrated circuits is proposed. The periodic boundary value problem is solved using shooting-Newton method, in which to solve the initial value problem the single step integration methods based on the Obreshkov formula are suggested to apply. The formulation of methods of order 1 up to 4 are obtained for charge oriented circuit equations. The formulas for computing sensitivity matrix are presented. The numerical examples of steady-state analysis are given which demonstrate the numerical accuracy and efficiency of the proposed method.

Keywords - circuit simulation, periodic steady-state, boundary value problem, shooting method, integration methods

#### Применение одношаговых методов интегрирования высокого порядка точности для анализа установившихся периодических режимов в интегральных схемах

Гурарий М.М., Жаров М.М., Русаков С.Г., Ульянов С.Л. (Институт проблем проектирования в микроэлектронике РАН, г. Москва)

Аннотация - В работе предлагается вычислительный метод анализа установившегося периодического режима аналоговых интегральных схем. Решение периодической краевой задачи выполняется методом пристрелки-Ньютона, в котором для решения задачи с начальными значениями предлагается применить одношаговые методы численного интегрирования ОДУ, полученные на основе формулы Обрешкова. Приведены примеры анализа периодического установившегося режима.

Ключевые слова - схемотехническое моделирование, установившийся периодический режим, краевая задача, метод пристрелки, методы интегрирования.