336 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 11, NO. 3, JUNE 2007 An Evolutionary Algorithm-Based Approach to Automated Design of Analog and RF Circuits Using Adaptive Normalized Cost Functions Abhishek Somani, Partha P. Chakrabarti, Senior Member, IEEE, and Amit Patra Abstract—Typical analog and radio frequency (RF) circuit sizing optimization problems are computationally hard and require the handling of several conflicting cost criteria. Many researchers have used sequential stochastic refinement methods to solve them, where the different cost criteria can either be combined into a single-objective function to find a unique solution, or they can be handled by multiobjective optimization methods to produce tradeoff solutions on the Pareto front. This paper presents a method for solving the problem by the former approach. We pro- pose a systematic method for incorporating the tradeoff wisdom inspired by the circuit domain knowledge in the formulation of the composite cost function. Key issues have been identified and the problem has been divided into two parts: a) normalization of objective functions and b) assignment of weights to objectives in the cost function. A nonlinear, parameterized normalization strategy has been proposed and has been shown to be better than traditional linear normalization functions. Further, the designers’ problem specific knowledge is assembled in the form of a partially ordered set, which is used to construct a hierarchical cost graph for the problem. The scalar cost function is calculated based on this graph. Adaptive mechanisms have been introduced to dynam- ically change the structure of the graph to improve the chances of reaching the near-optimal solution. A correlated double sampling offset-compensated switched capacitor analog integrator circuit and an RF low-noise amplifier in an industry-standard 0.18 m CMOS technology have been chosen for experimental study. Optimization results have been shown for both the traditional and the proposed methods. The results show significant improvement in both the chosen design problems. Index Terms—Analog, circuits, genetic algorithms (GAs), opti- mization, sizing. NOMENCLATURE Normalization Function: Normalization function. Objective function, normalized objective function. Minimum possible value of and . Maximum possible value of and . Target optimal value of and . . . Set of integers. Manuscript received February 21, 2005; revised April 26, 2006. The authors are with the Indian Institute of Technology, Kharagpur 721302, West Bengal, India (e-mail: somani_iitkgp@yahoo.com). Digital Object Identifier 10.1109/TEVC.2006.882434 Set of integers modulo x. Normalization sensitivity. Optimization Formulation: Linear normalization flat cost function. Nonlinear normalization flat cost function. Hierarchical approach (nonlinear normalization hierarchical cost function). HA with Hierarchical cost graph (HCG) mutation. Design Objectives: Dynamic range. Settling time. Settling error. Output voltage range. Noise figure. Third intercept point (measure of linearity). I. INTRODUCTION A UTOMATING the design of analog and radio frequency (RF) circuits has generally been considered a difficult problem. Quite a number of attempts have been made in the past to build generic analog synthesis systems. However, the stiff generality-complexity tradeoffs inherent to the problem have resisted wide acceptance of any particular synthesis method- ology. On a parallel note, the advent of the deep-submicron CMOS era has made manual optimization of the performance of analog and RF circuits exceedingly difficult. Few practical optimization problems bring forth the issue of effectively han- dling multiple, usually noncommensurate and often conflicting objectives as prominently as the problem of optimization of analog and RF circuits. A typical analog circuit like an opera- tional amplifier has objectives like Gain, Bandwidth, Slew Rate, Phase Margin, CMRR, PSRR, Dynamic Range, Output Range, Power, and Area. The order of the numerical values of these objectives, as well as their individual ranges can be diverse. One of the earliest analog circuit synthesis methods was presented in OPASYN [1] which is a design-equation-based opAmp generator that uses decision tree and heuristic pruning for topology selection and steepest descent method for device sizing. This problem has also been formulated as a con- strained optimization problem [2]. Notable among various 1089-778X/$20.00 © 2006 IEEE