A fuzzy logic based approach for parametric optimization of MOS operational amplifiers

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Abstract

In this work, we have used the concept of fuzzy logic to build a CAD tool for the parametric optimization of MOS operational amplifiers (op-amps). In order to capture human intentions to express the requirements for a particular application, e.g. minimize power, maximize gain, etc., each of the performance specifications of a given topology is assigned a membership function to measure the degree of fulfillment of the objectives and the constraints. A number of objectives are optimized simultaneously by assigning weights to each of them representing their relative importance, and then by clustering them to form the objective function, which is solved by Powell's direct search algorithm. After optimization, the program creates a SPICE netlist of the circuit topology for the verification of the design. Initially, this approach was used for parametric optimization of simple bipolar and MOS circuits, e.g. current mirrors, gain stages, differential amplifiers, etc. Encouraged by these results, it was applied to much more complex blocks, such as op-amps. The design results obtained from our optimization program showed an excellent agreement with those obtained from SPICE simulation for the op-amp topologies considered in this work.

Introduction

A study of the progress in the area of analog design automation (DA) can be found in the literature [1], [2], [3]. Among the many analog DA approaches, the optimization-based design approach is becoming very popular of late. It uses recent advances in the optimization theory and algorithms, and relates these to the parametric optimization of the analog ICs. A survey of the various optimization techniques for IC designs can be found in the literature [4], [5]. The synthesis problem is formulated as one of mathematical programming. The circuit performances are considered to be the objective functions, which are to be minimized or maximized, subject to a set of specification constraints.

Historically, the very first attempt towards analog DA were numerical optimization-based. Systems such as DELIGHT.SPICE [6], ECSTACY [7], and the more recent ADOPT [8], consider the sizing of the individual transistors in a given circuit topology as an optimization problem. Typically, these systems employ optimization algorithms, which iteratively adjust the individual transistor sizes in order to meet the constraints and objectives specified by the user. A simulator is used within the optimization loop to assess the performance of the circuit during each iteration. These design approaches are referred to as the simulation-based optimization. ASTRX/OBLX [9] developed at CMU also uses this technique. Systems based on numerical optimization techniques are independent of the actual circuit used, the technology, and the fabrication process.

Several attempts have been made to avoid the time consuming and expensive simulator inside the optimization loop. One approach is to adopt simplified, but sufficiently accurate, analytical models that predict circuit performances, and these are used inside the optimization loop. This approach is referred to as the analytical equation based optimization. A number of prototypes that use this technique, have evolved out in recent times, e.g. OPASYN [10], STAIC [11], FPAD [12], FASY [13], and those reported by Mandal and Viswanathan [14], Maulik et al. [15], and Taumazou and Markis [16].

Among the various sub-domains of analog design, i.e. topology selection, parametric optimization, and layout generation, each one plays an important role to achieve higher performance for a particular application. Even if the topology of the circuit to be designed is fixed, finding the optimal set of circuit parameter values is not an easy and obvious task, since all the circuit performances are highly dependent on these choices. The parametric optimization is complicated because of the conflicting design objectives and performance constraints, which are generally implicit nonlinear functions of the circuit parameters.

In all techniques used for parameter optimization, it is very crucial to formulate the problem in the right manner, and use an optimization algorithm that reflects the users' intentions as accurately as possible. The focal point of this work is aimed towards the development of a tool in order to find a set of circuit parameters (i.e. design variables), such that all the design objectives are optimized while satisfying the performance constraints, assuming that the topology of the circuit to be designed is known. The approach is based on a constrained multi-objective algorithm, avoiding the limitations of both unconstrained and single objective optimization. Also, it combines the advantages of both the optimization-based and the knowledge-based approaches and tries to overcome the limitations of both. Analytic circuit equations and device models are used to avoid a simulator in the optimization loop, and, therefore, the cost of repeated simulation is avoided.

Also, an attempt has been made to formulate the optimization problem in a more realistic manner by capturing the human intentions in the inherently imprecise terms used by the user, e.g. minimize power, small output resistance, etc. The fuzzy set theory [17], as applied in FPAD [12], is used to formulate the design objectives and constraints. Depending on the objectivity associated with a particular performance or constraint, it is transformed into either a fuzzy objective or a fuzzy constraint. The trade-offs among a number of specifications, depending on the users' interest, are handled by means of membership functions (MFs). Each of the performance specifications (PSs) is assigned a weight, which reflects its relative importance. To allow trade-offs and limit repeated modifications of the input specifications, they are not assigned precise target values; instead, each specification is formulated as a fuzzy set, i.e. a range of possible values are assigned to each of them with varying degrees of acceptability.

The performance functions of the circuit, as required by the optimization routine, are evaluated by using analytical circuit equations, which describe the performance objectives in terms of the design variables. The technology parameters needed during the performance function computations are directly read from a technology file,2 making the synthesis routine technology independent. The time-consuming circuit simulator is avoided inside the optimization loop. The objective function (OF) is formulated by clubbing all the MFs associated with the specifications in proportion to their assigned weights. The formulation takes into account performance tolerances and allows varying degrees of a particular solution. Finally, the solution to the formulated problem is carried out using Powell's direct search algorithm [18], [19], which is a very simple optimization algorithm and is found to work effectively for the cases considered in this work.

The development of the synthesis tool involved the creation of an optimization module in order to optimize an n-variable objective function. To provide the user a number of MFs for transforming the PSs into fuzzy objectives or constraints, a library is created. In order to develop the optimization modules for each of the topologies currently supported by our program, the following steps are followed: (i) mathematical conceptualization of the analog circuit topology, identification of the PSs and the design variables, (ii) elimination of the unimportant and dependent design variables by using heuristics and circuit knowledge, thus filtering out the independent design variables, and expressing the performance in terms of these variables, (iii) development of the program to read the PSs, type of constraints or objectives (e.g. maximize, minimize, etc.), weight of each of the performances, and, thereby, formulating the OF for optimization, (iv) development of the modules to create output files for storing the output results, operating point information, and an input file containing the netlist of the circuit topology for subsequent verification using the SPICE simulation, and (v) verification of the module by running it for a number of sets of PSs, obtaining the output, and their comparison with the SPICE simulation.

The methodology adopted in this work can be interpreted as the development of a part of a complete analog DA tool. In other words, our attempt is to provide reliable and useful parametric optimization procedures for cell level analog circuit blocks. An attempt has been made to look at the synthesis problem in a more systematic and organized manner. The tool developed here can be used for evaluating the circuit parameters, considering the fact that almost all the important performances are considered for the design. In order to explore the viability of the analog circuit design methodology as discussed above, a CAD tool is constructed in this work in the C-language (having ANSI features) and applied for the synthesis of analog building blocks in both bipolar and MOS technologies. A variety of such modules (both bipolar and MOS), e.g. current sources and biasing stages, gain stages, differential amplifiers, output stages, and various op-amp topologies (e.g. two-stage OTA and folded cascode) have been optimized in this work, however, for brevity, only the results for the Miller compensated op-amp are presented here. The design results, as obtained from our program, match reasonably well with those obtained from SPICE simulations.

Section snippets

The fuzzy optimization approach

The general structure of the synthesis procedure adopted in this work can be represented by means of a flow chart, as shown in Fig. 1. First, the program asks the user to decide the circuit that he wants to optimize among the various topologies available, after which the program starts with a set of input PSs, and deduces the optimal set of design parameter values (e.g. lengths and widths of transistors, bias voltages, bias currents, compensation capacitor values, etc.). Each of the PSs is

Optimization of analog modules

As a first step towards parametric optimization of analog modules, some simple bipolar and MOS analog blocks were considered in this work, e.g. current mirrors, gain stages, differential amplifiers, output stages, etc. The design results obtained from our optimization program showed an excellent match with those obtained from SPICE simulation for each of the topologies considered. Encouraged by the results, we extended our work to parametric optimization of a more complex analog block, i.e. the

Procedure and results

In order to determine the initial values of the design variables, an algorithm is developed and implemented in this work, which deduces the values of the design variables from the PSs specified by the user, and the circuit knowledge using the simple square-law current equation for the MOS transistors. The algorithm works as described below.

  • 1.

    The overall gain of the circuit is obtained by taking the product of the gains of the first stage and that of the second stage. Assuming that both these

Summary and conclusions

A fuzzy logic based optimization approach for analog circuit DA is presented in this work. In order to model the real world terms, e.g. high, maximize, minimize, etc., used by the user while specifying the performance objectives, we have applied the concept of fuzzy MFs [17]. A library is developed for different MFs, which are subsequently used by the modules for synthesis of different analog circuits. Among a set of PSs, some are considered as objectives and others as constraints, and using

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    Current address: Visiting Associate Professor. Department of Electrical Engineering, Louisiana State University, Baton Rouge, la 70803, USA. Tel. +1-225-578-5532, Fax: +1-225-578-5200.

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