This article proposes that genetic algorithms

(GAs) are viable candidates as tools for design automation of

analog circuits (DAAC). The principles, strengths, and weaknesses

of GAs applied to sizing circuit parameters to meet performance

specifications are discussed and a software implementation is

introduced. Designs of BJT and CMOS operational amplifiers, a CMOS

operational transconductor, and a matching network are presented as

examples.

**Introduction**

The design of the analog portion of a mixed-signal integrated circuit often requires a large fraction of the overall design time, despite the fact that the analog circuit is often a relatively small portion of the overall circuit. Interestingly, much of the analog design time occurs after a candidate topology, having the

potential to meet required specifications, has been selected as a

building block for the system. This is due to the need to manually

iterate or “tweak” circuit parameters (such as, component values

and transistor sizes) to meet specifications. This iteration

process is particularly challenging when specifications must

include temperature and process corners. Moreover, designs using

short-channel MOSFET's further complicate the design process due to

a lack of manageable equations relating circuit parameters to

specifications that result. If design automation of analog circuits

(DAAC) is to be achieved, a solution to automating this phase of

the design process must be developed. The problem may be classified

as a global optimization problem.

While there have been efforts to develop optimization tools for

analog circuit design, they have generally been based on gradient

search methods. Unfortunately, these methods require initial

guesses, tend to get stuck in local minima, and have difficulty

with discrete variables and non-linear constraints. This paper

asserts genetic algorithms (GAs) may be a better alternative for global

optimization tools for DAAC.

GAs are based on the Darwinian principle of natural selection

and the concepts of natural genetics. GAs have many desirable

characteristics and offer significant advantages over traditional

methods. They are inherently robust and have been shown to

efficiently search large solution spaces containing discrete or

discontinuous parameters and non-linear constraints, without being

trapped in local minima. GAs do not require initial guesses or

derivative information and have often found non-intuitive solutions

to engineering problems.

While GAs have been applied extensively to many areas of

engineering, there are relatively few references in the analog

circuit literature that report the application of genetic

algorithms. In most instances only linear circuits were considered

(a relatively easy problem compared to non-linear circuits) or the

optimizations were based on derived circuit equations. Koza has performed non-linear circuit designs using

genetic programming, but emphasized topology generation. While his

results are very exciting, he has not yet demonstrated their

effectiveness in generating circuits suitable for production in an

industrial environment.

This article illustrates the application of GAs to select

component values and transistor sizes for a given circuit topology

to provide practical design solutions for a given set of

performance specifications. Since this phase of analog circuit

design is time intensive, significant time savings can be achieved

if a software tool can perform this task. The software discussed in

this article represents an important step in that direction.

**The Genetic Algorithm**

This section introduces some of the concepts of genetic

algorithms and their implementation in GA-Spice. It is provided

only as a guide and is not intended as a substitute for more

detailed expositions on GAs available in the literature.

**Chromosomes**

Central to all genetic algorithms is the concept of the

chromosome. The chromosome contains all information necessary to

describe an individual. In nature, chromosomes are composed of DNA.

In a computer, a long binary or character string is used.

Chromosomes are composed of genes for the various characteristics

to be optimized and can be any length depending on the number of

parameters to be optimized.

**Encoding**

Encoding defines the way genes are stored in the chromosome and

translated to actual problem parameters. A possible encoding scheme

for a hypothetical circuit using a 16-bit binary chromosome is

shown below:

**Figure 1:** Â Encoding Example

**Fitness**

Fitness is a single numerical quantity describing how well an

individual meets predefined design objectives and constraints.

Fitness can be computed based on the outputs of multiple analyses

using a weighted sum. The definition of good fitness functions is

highly problem dependent. The GA-SPICE program has several basic

fitness metrics that can be applied to any Spice output variable.

The basic metrics are described in terms of the output variables

*c* i , user specified

response *c* i , and user

specified point-by-point weighting values *s* i . All of the basic metrics are

defined such that the superior individuals have the lowest fitness

values. Using these definitions, the raw and standard fitness

defined by Koza are identical.

A raw fitness metric for minimizing an output variable ci

computed at N points can be defined as

Other metrics can also be defined to maximize an output variable

or to measure of the quality of a match of the calculated responses

to a specified response on either a relative or absolute basis.

Constraints are implemented by imposing a large penalty whenever

the constraint is violated. You can combine metrics for various

functions to yield a combined fitness for different output

variables, analysis types or circuit configurations. The total raw

fitness F is then calculated using

where *W* m is the

weighting applied to each of the basic fitness metrics. The

adjusted fitness, A, is defined by Koza as

and is used to rescale the fitness to the range 0 to 1 with

better individuals having the larger values. Subsequently, the

adjusted fitness of each individual is normalized by the total

adjusted fitness of the M members of the population.

The normalized fitness is used in determining the probability of

selecting a particular individual for mating.

**Crossover**

Crossover is a method of exchanging genetic material between two

parents to produce offspring. The operation is simple, a crossover

point is chosen at random and the genetic information to the left

of the cross-over point in parent A is combined with the genetic

information to the right of the cross-over point in parent B to

produce new offspring. A second offspring can be created by using

the information from the right of parent A and the left of parent

B. The operation is illustrated with the example below:

**Figure 2:** Â A single point crossover is widely used

to simulate the sharing of genetic information that occurs during

mating

A simple GA is described in this section. While more

sophisticated forms of the GA exist, they are all extensions of

this basic algorithm. The simple GA provides excellent performance

for a wide range of problems.

The simple GA begins with the creation of an initial random

population of individuals. The size of the population depends on

the problem size. Population size ranges from as few as 20 to tens

of thousands. Values of several hundred are commonly used.

The next step is to evaluate the fitness of each individual

according to the predefined criteria. After the evaluation, the

individuals are ranked according to fitness. Individuals are

selected for mating based on fitness. Fitter individuals have a

higher probability of mating and passing on genetic information to

subsequent generations while less fit individuals have a non-zero

probability of mating to preserve diversity.

Mating is simulated by applying the crossover operation to the

chromosomes of two individuals selected based their fitness.

Mutation is simulated by randomly changing a few bits in the

chromosome of the offspring. Mutation provides a mechanism for

exploring new regions of the solution space and prevents premature

convergence to local minima. Finally, the fitness of the new

generation is evaluated and the process is repeated for a specified

number of generations or until a desired fitness is attained. A

block diagram of the simple GA is shown below:

**Figure 3:** Â . The simple GA is adequate for many

engineering problems.

**The GA-Spice Program**

As part of this research, the GA-Spice program was developed.

The program allows the encoding of any parameter in a Spice netlist

as an optimization variable. This includes component values, node

numbers, model types, process parameters and other Spice

parameters. Fitness functions can be constructed using any output

variable available from Spice. Analysis types can include AC, DC,

transient, and noise. The designer can construct optimization

variables and fitness functions easily via a point & click

interface. The program is compatible with existing Spice tools for

schematic capture and output visualization. The general nature of

GA-Spice means the simulator is not limited to specific circuit

topologies like equation-based methods.

GA-Spice runs under the Microsoft Windows 9x/NT operating

systems. Since the GA is inherently a coarse-grain parallel

operation it is possible to reduce optimization time by employing

parallel computer architectures. The current version of GA-Spice

can use up to 4 processors in an symmetric multiprocessing system.

A more powerful multi-processor version based on the Parallel

Virtual Machine is under development.

**Practical Applications of GA-Spice**

The application of GA-Spice for the solution of several

practical analog circuit applications is presented in this

section.

**BJT Operational Amplifier**

The specifications for the BJT op-amp depicted in **Figure 5** were

a differential gain of at least 600V/V, an input offset voltage of

0V, and an output signal swing of at least Â±13V. Several

circuit designs produced by GA-Spice met the specifications. One of

the circuits was built and tested in the laboratory to confirm that

the design was practical and met specifications.

Interestingly, this circuit was also given as a design problem

to junior electronics students. It required at least two weeks

(three weeks or more for many) for the students to obtain a design

that met specifications. Using GA-Spice, the design be performed in

under 30 minutes with a 200 MHz Pentium notebook computer.

**Figure 5:** Â BJT Operational Amplifier

**CMOS Operational Amplifier**

The specifications for the CMOS Op-amp depicted in **Figure 6** were

to maximize gain and minimize the input offset voltage. The results

from GA-Spice gave designs that had gains in excess of 2kV/V and

input offset voltages of 0V. Transistor widths were fixed at 5

microns (5Âµm) except for M 6

and M 7 which were fixed at

2um. GA-Spice required one hour on a 200 MHz Pentium notebook

computer to complete the design.

**Figure 6:** Â CMOS Operational Amplifier

**Operational Transconductance Amplifier (OTA) Design**

The topology of the OTA is shown in **Figure 7** . GA-Spice was used

to determine the transistor widths and resistor value for R 1 required to properly bias the

circuit and provide a specified transconductance. For this problem,

the transistor widths were fixed at 2 um. The chromosome length was

76 bits, yielding 7.5×10^{22} possible designs. Desipite

this enormous solution space, the program located an acceptable

solution in approximately 15 minutes on 200 MHz Pentium notebook

computer. The following table summarizes the design specifications

and the resulting circuit performance.

**Figure 7:** Â Schematic diagram of OTA optimized by

GA-Spice

W1=W2=6.96um, W3=W4=101.5um, W5=W6=4.6um, W7=W8=27.4um,

W9=112.6um, W10=3.23um, W11=375.8um, W12=7.56um, and R_{1}

= 90.374k. The spice model for the NMOS used VTO=0.7V,

Kp=156uA/V, and lambda=30m/V and for the PMOS was VTO=-0.95V,

Kp=48uA/V, and lambda=4.2/V

**Input Impedance Matching Network for an LNA**

**Figure 8** shows the schematic for a model of an input stage of a

low-noise amplifier. C1 and

C2 tune the input impedance, Z

in , to 50 at 2.45GHz. L BOND models bond-wire inductance, C

pad models pad and trace

capacitance, C par models trace

capacitance, R 3 and C 3 model the input impedance of a

biasing circuit, R in and C

in model the impedance looking

into the gate of a MOSFET, R S1 , R S2 , C S1 , C S2 , and R lin models parasitics of an

integrated square spiral inductor.

**Figure 8:** Â Schematic for a model for the input stage

of a low-noise amplifier.

The results from GA-Spice provided Z in = 50.75 – j 3.5 at

2.45GHz. GA-Spice provided a solution after running about 15

minutes on a notebook computer with a 200MHz Pentium processor.

The values for the components are L BOND = 3nH, C pad = 500fF, C par = 12.56fF, R 3 = 1.796k, C 3 =

72.22fF, R in = 107.5, C in =

265.7fF, R S1 = 52.7, R S2 =

79.6, C S1

=113fF, C S2 =93.8fF, and R

lin = 21.9. C 1 and C

2 were selected by GA-Spice to

be 306fF and 20fF.

**Discussion**

Section 4 described four circuits with component values selected

using a genetic algorithm. Each of the these circuits met a set of

simple specifications in a very reasonable amount of computer

time.

The input matching circuit was an actual design problem in an

industrial setting. Realistic values for the components were used.

Linear circuits appear to be very easy for GA-Spice to

optimize.

For the BJT and CMOS operational amplifiers and the CMOS

operational transconductor amplifier, initial designs provided by

GA-Spice proved unrealistic. For example, we found that GA-Spice

had biased several of the Gate-Source voltages (VGS's) of several

MOSFET's at levels of mere millivolts above the threshold voltage.

This is very impractical. Thus, the fitness functions for the

transistor circuits must be defined to prevent this. For example,

we specified the overdrive voltages of all MOSFET's had to be

between 0.2V and 0.5V for threshold voltages of 0.7V. The DC

biasing for the BJT op-amp required similar constraints.

The user also must recognize when transistors in a design must

be matched and properly link the parameters of the second

transistor to the first using a symbolic link option provided by

the program. For example, in the CMOS OTA, M3 and M4, M5 and M6,

and M7 and M8, are MOSFET pairs using this symbolic-link encoding

scheme to provide proper OTA operation. An attempt to use a genetic

algorithm without proper linkage produced a design that met

specifications, however it resulted in a poor overall design due to

the circuit symetry being destroyed.

The CMOS op-amp and OTA used a basic square-law model and not

the current industry-standard BSIM3v3 model. This helped speed up

simulation times and still demonstrates the ability of the program

to design analog circuits where equations are not available.

One measurement used in this paper is the simulation time of the

GA-Spice. It has been stated that for simple circuits, this time is

relatively small even on a slow notebook computer and could be

comparable to the design time of an engineer. However, as GA-Spice

is applied to larger circuits, perhaps the measurement will be the

amount of time saved by the engineer. Even if a GA cannot design a

circuit as quickly as an engineer, if it saves an engineer hundreds

of man-hours, it is still an effective tool.

In an industrial environment, specifications and the MOSFET

models will not be so simple. An effort to show the successful

application of GAs to industrial-strength analog circuits is

presently underway. Future research also includes determining the

types of genetic algorithms most suitable for analog circuit

design, determining the best types of encoding schemes, and

developing optimal fitness functions for the different circuit

types.

**Conclusions**

This article demonstrates that the applications of GAs are viable

candidates for use in DAAC by successfully designing several

circuits. The methods used are general and can be applied to all

analog circuits. It has been observed that GA-Spice provides a

number of different solutions to a problem in a much shorter time

than trial and error. GA-Spice can help provide new insights into

circuit operation and produce novel designs. It appears GAs can

play a role for future tools for DAAC. At a minimum, they can speed

up or eliminate a considerable amount of the trial and error

engineering being done by many organizations today.

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