Research Article Analog Circuit Design Optimization...

Research ArticleAnalog Circuit Design Optimization Based onEvolutionary Algorithms

Mansour Barari1 Hamid Reza Karimi2 and Farhad Razaghian1

1 Electrical Engineering Department Islamic Azad University South Tehran Branch Tehran 113654435 Iran2Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Hamid Reza Karimi hamidrkarimiuiano

Received 3 January 2014 Accepted 28 February 2014 Published 7 April 2014

Academic Editor Yang Tang

Copyright copy 2014 Mansour Barari et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates an evolutionary-based designing system for automated sizing of analog integrated circuits (ICs) Twoevolutionary algorithms genetic algorithm and PSO (Parswal particle swarm optimization) algorithm are proposed to designanalog ICs with practical user-defined specifications On the basis of the combination of HSPICE and MATLAB the system linkscircuit performances evaluated through specific electrical simulation to the optimization system in theMATLAB environment forthe selected topologyThe system has been tested by typical and hard-to-design cases such as complex analog blocks with stringentdesign requirements The results show that the design specifications are closely met Comparisons with available methods likegenetic algorithms show that the proposed algorithm offers important advantages in terms of optimization quality and robustnessMoreover the algorithm is shown to be efficient

1 Introduction

Today with the development of VLSI technology in order tointegrate digital and analogue circuits as a complete systemon a chip the issue of optimal electronic circuits design dueto its abundant advantages in contrast with manual designis very important In this regard because of their designcomplexity and difficulty analogue circuits have attractedgreater attention in optimization In generalthe analoguecircuits design is undertaken in two stages The first stageis related to the selection of the circuit structure so that itwould have the necessary abilities to fulfill our expectationsfrom the circuit The second stage which is often more time-consuming is the optimization of the circuit parametersfor example the size of the transistors and the value ofthe elements so that it could satisfy output parametersfor example consuming power and gain Moreover issuesdepending on the nature of themain function and limitations(eg whether the fitness function is convex or not limitsare linear) can be solved by KKT (Karush-Kuhn-Tucker)condition theories Lagrange coefficients nonlinearlinear

complementary problem (NLCPLCP) and nonaffineaffinevariational inequality (NAVIAVI) However due to its com-plexity being time-consuming and case-dependent therewas a need for newer andmore comprehensive solutions Forthese reasons and in order to respond to these problems theuse of evolutionary optimization methods such as geneticalgorithm particle swarm optimization ant colony andharmonic search algorithm was proposed

All these methods obey the following issues

min119909

119891 (119909)

subject to ℎ (119909) = 0

119892 (119909) le 0

119883119871le 119909 le 119883

119867

(1)

In this optimization 119891(119909) is the function and must beminimized and ℎ(119909) 119892(119909) are boundary conditions Theequality in (1) is our fitness function and refers to Kirchhofflaws (KVL and KCL)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 593684 12 pageshttpdxdoiorg1011552014593684

2 Mathematical Problems in Engineering

Vector 119909 is the design variable and 119883119871and 119883

119867are

their lower and upper bounds respectively Analog integratedcircuits use (1) as a constrained optimization problem and canbe solved by evolutionary algorithms

The selection depends on the conditions of the problemand the application Our circuits follow the same equation foroptimizationThen they are optimized by evolutionary algo-rithms In the past genetic algorithms were applied in orderto optimize analogue circuits but here we try to optimizeanalogue circuits by genetic algorithms PSO algorithm andafter that a new structure is introduced in order to make useof both of the algorithms simultaneously and in combinationwith an appropriate fitness function Nowadays with thedevelopment of electronic circuits due to the need to optimizeand speed limits there will be more Evolutionary algorithmsare implemented by means of MATLABThe fitness functionand the limitations are examined byHSPICE simulation soft-ware and there is a link between the twomentioned softwareprograms for analyzing and optimizationThe technology weapplied is based on stimulation and it is done as followsfirst the parameters related to the evolutionary algorithmsare produced by MATLAB and they are sent to the circuitsimulator HSPICE for optimization and examination Aftersimulation these parameters are sent back toMATLAB againso that they are examined by the evolutionary algorithm andthen they are optimized These stages are repeated until wereach the stop point of the algorithm In Section 2 relatedwork which has been done is listed Section 3 briefly reviewsthe genetic algorithm and its flowchart PSO algorithm andits flow chart are briefly reviewed in Section 4 Section 5presents stages of the proposed work In this section wetry to introduce the fitness functions and procedures whichhave been performed In Section 6 two circuits and theirparameters are listed which will be used in optimizationThis section is the most important part because the proposedmethod is discussed in this section that combines the geneticalgorithms and PSO and the simultaneous use of these twomethods Here we analyze the circuits with evolutionaryalgorithms and then compare them with each other to obtainthe best method for designing the analog circuit based onevolutionary algorithms

2 Related Works

The traditional design of analogue circuits [1 2] does nothave the precision robustness and efficacy of todayrsquos systemsTherefore designs based on optimization are applied Inmethods that are based on optimization techniques incomparison with traditional methods more precision androbustness are achieved In these methods the problem isstimulated as a function which is minimized in numericalissues and then it is located in an optimization loop whichis probably repeated several times It is difficult to choose thisfunction and it must be done with a lot of care

On the other hand the function must be written in theclosed form and this may cause the loss of precision andspeed If it could be written in the closed form this is for onecircuit and it is not generalizable

Byte code Netlist

Decode Spice circuitsimulation

Fitnesscalculation

Figure 1 An overview of circuit evaluation process starting withbyte coded representation with fitness score

One of the major limitations in the design of analogcircuits is the limited number of parameters that can be inves-tigated It may be more important when circuit designerswant to design circuits with more and more complexity andhigher performances and limitations In many ways a penaltyfunction is used to satisfy these constraints In thesemethodsthe constrained optimization problem is transformed into anunconstrained one by minimizing the following function [3]

1198911015840119909= 119891119909+119899

sum119894=1

119908119894(119892119894(119909)) (2)

where 119908119894are the penalty coefficients and 119892

119894(119909) gt 0

The results of thesemethods are depending on the penaltycoefficients and may not be very ideal to the designersThe values of penalty coefficients should have logical valuesbecause when they have illogical value the results willproduce infeasible solutions For finding good solution thedesigner usually needs to change penalty coefficients formany times

Researchers now try to find the ability to handle large-scale and multiobject design problems Most of the availablemethods can deal with about 10ndash20 variables simultaneouslybut analog circuits have many unknown variables

An appropriate location must be selected as the startingpoint These stages are shown in Figure 1 However theprocess is simpler and precise In a stage it is attempted toselect a good starting point [4] and a better convergence [5]

In recent years however evolutionary algorithms havereceived lots of attention see for instance [6ndash8] and thereferences therein At first genetic algorithm (GA) wasapplied but sometimes this method did not have a favorableoutcome For instance in GA [9] it was not convergent in theglobal minimum point

Ioana Saracut presents an optimization tool for the designof analog circuits with a topology able to select the best setof passive components values and active devices form withinwell-defined sets of available values and options Unlike mostof the existing circuit optimization tools it can search withindiscrete sets of possible solutions such as the standard seriesof values available commercially for passive components andlists of active devices given by the designer [10]

Xuesong Yan says that a major bottleneck in the evolu-tionary design of electronic circuits is the problem of scaleThis refers to the very fast growth of the number of gates usedin the target circuit as the number of inputs of the evolved

Mathematical Problems in Engineering 3

logic function increases This results in a huge search spacethat is difficult to explore even with evolutionary techniquesAnother related obstacle is the time required to calculatethe fitness value of a circuit Use of the traditional geneticalgorithm for electronic circuit optimization design is beingtrapped easily into a local optimum and the convergencespeed is slow They use PSO algorithm to overcome theshortcomings of GA By analyzing the testing results offour optimization Benchmarks they reach the followingconclusion in the optimization speed the PSO algorithm ismore efficient than the GA [11]

Fakhfakh et al [12] proposed particle swarm optimiza-tion technique for the optimal design of analog circuits tosolve optimization problems [13] Tlelo-Cuautle et al [14]presented how we can apply evolutionary algorithms for thesynthesis and sizing of analog integrated circuits Researchstudies on the examination of analogue circuits with a lot ofparameters continue [3]

In order to accommodate a larger design space with ahigher number of variables we may use techniques such asneural networks [15ndash17] But here we decide to investigategenetic algorithms and PSO

3 Genetic Algorithms

These algorithmsmake use ofDarwinrsquos natural selection prin-ciples in order to predict or adapt the pattern Genetic algo-rithms are often good options for prediction techniques basedon randomness It is briefly stated that genetic algorithm(GA) is a programming technique which uses genetic evo-lution as a problem-solving pattern In GA each unknownparameter is called a gene and the vector of parameters iscalled chromosome [12] Initially the chromosomes are gen-erated randomly from the population size Each chromosomestring ismade up of integer values to represent a set of param-eters to be optimized The problem to be solved is the inputand the solutions are coded according to a pattern which iscalled fitness function This function evaluates each selectedsolution that most of them are selected randomly First aprimary population (randomly or selectively) is initializedin the scope of problemrsquos limitations and chromosomes areranked according to their qualifications The rotating wheelof parentsrsquo selection is manufactured according to the quali-fications of the chromosomes and the parents are selectedThe new generation generates via mutation and crossoverand the stop condition (limited repetition combination andconvergence examination) and then the examined conditionaremet the algorithm is stopped and the result is announcedIf the condition is notmet the stop of chromosomes is rankedagain and the above stages are repeated again These stagesare shown in Figure 2 In Figure 3 we show how we cancalculate fitness and its process

4 PSO Algorithm

The PSO algorithm is a very important optimization algo-rithmwhich is inspired by the groupmovement of birds (andother animals which live in groups) The response of each

Populationinitialization

Fitnesscalculation

Fitnesscalculation

Selectioncrossover

Roundsolution

Finish

Yes

No

Figure 2 Genetic algorithm flowchart

Decodechromosome

Spicesimulation

Fitnessmeasurement

Figure 3 Fitness calculation flowchart

particle is stated as the location of a particle which has avelocity to approach the location of its food

Each particle generally adjusts its velocity toward its foodby means of its own location history the location history ofbirds in its specific vicinity the location history of all thebirds in search of the food and the predetermined inertiaThe movement of each particle depends on four factors (1)the present situation of the particle (2) the best situation theparticle has ever had (119875best) (3) the best situation which isobtained in the specific vicinity of the particle (119871best) and(4) the best situation which all the particles have ever had(119866best) each particlersquos update follows these relations

Present [] = present [] + V [] (3)

Here the last statement specifies the particlersquos velocity (theamount of location changes for each particle)This statementis calculated in this way

V119899= 119908 times V

119899+ 1198881times rand () times (119901best

119899

minus 119909119899)

+ 1198882times rand () times (119892best

119899

minus 119909119899)

(4)


Rand () produces the random number in the range of [0 1]1198881and 119888

2are called ldquoacceleration coefficientsrdquo Experience

shows that if the number of particles is approximately 10ndash50 it is enough for convergence in such problems 119888

1is the

significance related to the best situation of each particleand 1198882is the significance related to the best situation of the

vicinities It is usually considered that 1198881+ 1198882= 4 which is

only chosen because of experiential reasonsThe simultaneous application of the evolutionary algo-

rithms can be a way to cover the deficiencies of each algo-rithm and develop the analysis result This application can bedone in different ways for instance a combination of geneticalgorithm and PSO is used here These stages are shown inFigure 4

5 The Design Optimization of an Op-Amp bymeans of Evolutionary Algorithms

Circuit parameters are the parameters of the optimizationproblem and make the chromosomes include the transistorssize (length and width) and the amount of compensationcapacitor Four key features of the circuit whose optimizedvalues are more influential in the circuit performance areconsidered as the fitness functions including DC gain (AV)pw (consuming power) bandwidth and SR (Slew rate)Our objective is to obtain the appropriate values for circuitparameters maximizing values related to dc gain bandwidthGB (Gain Bandwidth) and SR and minimizing consumingpower simultaneously

The implementation of the evolutionary algorithms isdone in MATLAB and the calculations of the functions aredone by HSPICE In fact in this part fitness function is cal-culated and the inputs applied in evolutionary algorithmsare provided The calculation of the functions for parentsand children population which is a step of evolutionary algo-rithms is done as follows

First variables are read from MATLAB and they arewritten in the input file of HSPICE (sp) Then HSPICE isrun and finally the values of the fitness function are readfrom the output file of HSPICE (lis) and they are sent toMATLAB The first circuit has 900 input parameters and 4fitness functions (eight transistors and a capacitor 119888

119897to be

adjusted) and the second circuit has 1600 input parametersand 4 fitness functions (fifteen transistors and a capacitor 119888

119897

to be adjusted) When only genetic algorithm is applied thepopulation of each swarm is 100 and the number of swarmsis 100

Here we use the three fitness functions and compare theresponses obtained and then the best fitness function isintroduced The applied fitness functions are

119891119894= sum119908

119894119898119894 (5)

119891119894= prod119898 =

(1198981lowast 1198982lowast 1198983)

1198984

(6)

119891119894=(1198981205721

1lowast 1198981205722

2lowast 1198981205723

3)

1198984

(7)

Initializeswarm

Evaluatefitness

Gbest

Gbest

Replace

PbestReplace

Updateposition

Updatevelocity

Stopcondition

No Yes= best

Figure 4 PSO algorithm flowchart

These functions have been tested in algorithms and theresults are listed in the tables for each section separately Inthese functions 119898 is the circuit parameters such as GB andAV and 120572 is coefficient obtained from the optimization

6 Proposed Method

Several steps are considered in this section First the GAand PSO are selected for optimization The next step is todesign a procedure to link them HSPICE is selected forsimulation of electronic circuits and MATLAB is selected forimplementation of evolutionary algorithmsThefirst stepwasoptimization by GA At each step the GA is implementedand will attempt to run HSPICE HSPICE primary inputs aredefined in a GA as the starting point HSPICE processes theinput values and generates the output HSPICE outputs areconsidered as MATLAB inputs and MATLAB re-optimizedthe resultsThe optimization is determined by the parametersspecified in the program code For example the starting pointof the analysis of an analog circuit is obtained manuallyAfter the first step a series of analysis again is considered asHSPICE input These steps also were taken for PSO

The most important decision part is choosing a newtechnique to optimize the circuits Each of these methods hasalready been used separately but each has its own advantagesand disadvantages Here we decided to adopt an approachthat combines the advantages of themethods In other wordsthe disadvantage of this method is covered by benefits onthe other methods For example the GA has more accuracyand less speed and PSO has more speed and less accuracyin convergence So the proper use of these two methodscan provide the best resultsSelected fitness function is amilestone in this project Fitness functions that have beenused are given in (5)ndash(7) At first it was decided that 120572

1to 120572119899

in (7) should be swept The best reason for using this fitnessfunction is comprehensive equation But after the reviewsdue to the problems such as fluctuation it was decided to


swarm

Evaluate fitness

Update position

Update velocity

Stop No

Yes

No Yes

Finish

Fitness calculationSelection

crossover mutation

Round = best solutioncondition


Initialization

Gbest

Replace Gbest

Replace Pbest

Figure 5 Simultaneous PSO and genetic algorithm flow chart

use a combination of algorithms Thus 1205721to 120572119899with GA and

the circuit parameters are optimized by PSO This procedureis the main point of our work and is shown in Figure 5After many tests it is found that using this method wouldobtain much better results compared with when we applythe algorithms individually Starting points of HSPICE areproduced byMATLABThe output is called byMATLAB andthe first step of optimization is completed In this section firstGA optimizes 120572 parameters and then PSO runs 100 times tooptimize the circuit parametersThenGA is executed and theprocess is repeated Analysis of the results obtained from thismethod compared with the results that were obtained by GAand PSO methods is shown in the tables

When genetic and PSO algorithms are used simultane-ously the population of each swarm is 100 and the number ofthe swarms is 50 In this state algorithm precision is 2minus8 andmutation probability pm = 0005 and crossover probabilitypc = 09 Genetic algorithm works in the way that the valuesof input parameters are proposed by optimization algorithmand then the circuit is stimulated by those values andthe results are derived After 100 swarms the response isspecifiedThis is done by 06 120583m and 018 120583m technology andthe results are compared with each other All the examplesare carried out on a computer with the specifications of25 GHz and 4G RAM and the running time includes thetime consumed byMATLAB and HSPICE and making a linkbetween these two software programsThese stages are shownin Figure 5

Using various technologies enables us to display theadvantages of each method by the software separately andthen is considered to be the best choice If we want to use newtechnologies in this way it is not needed tomake fundamentalchanges in the program

Vin

Vdd = 25

IDC

Vss = minus25

CL

C1

M1

M2

M3

M4

M5

M6M7M8

minus

+30120583A

10 p

I1

3p

Figure 6 Two-stage amplifier

61 Example 1 Two-Stage Op-Amp Design A two-stage op-amp is shown in Figure 6 and the selected technologies are06 120583m and 018 120583m The design parameters are the lengthof transistors and the amount of compensation capacitor Allthese transistorsmustwork in saturation region In that119881DS gt(119881GS minus119881TH)must be set for NMOS transistors In every stagethe results of optimization by the algorithms are comparedtogether via different fitness functions

For both circuits in Figures 6 and 7 the genetic algorithmis implemented with functions equations (5) and (6) of thePSO with the same equations runs

The main problem after algorithm selection is selectingfitness function Fitness function mentioned in (7) was firstly


0 5 10 15 202

Iteration

Mean GB

25

3

35

4

45

5

55

6times106

(Hz)

Figure 7 GB value in each stage that PSO algorithm coefficients areswept

investigated by PSO algorithm in the way that 120572 parametersare swept at first in this stage sweep steps with accuracy of01 are considered and responses are recorded in Tables 1 to4 in the next stage 120572 parameters are optimized by geneticalgorithm and the circuit parameters are optimized by PSOalgorithm Best results are obtained by selecting the functionthat is mentioned in (7) This fitness function is actually aseries of multiply-and-accumulate operations and it is thecomplete set Each of these experiments is undertaken 10times and the best of them are presented in Tables 1 2 3 and4 Consider

FOM = AV lowast SR lowast GBpw

(8)

In the first stage if we compare 06 120583mand 018 120583m technolo-gies as we expect the results obtained from 018 120583m have farbetter results For example when only genetic algorithm isused and fitness function is in sum format FOM responsesof 018 120583m compared to 06120583m are approximately 20 better

When only PSO algorithm is used and fitness function isin sum format FOM responses of 018 120583m are 225 times of06 120583m technology If fitness function is in multiply formatFOM responses obtained from 018 120583m are 1699 times of06 120583m technology

When PSO algorithm coefficients are swept FOMresponses obtained from 018 120583m are 282 times of 06120583mThe stages of convergence are drawn in Figures 7 8 and 9

When genetic and PSO algorithms are used simultane-ously FOM responses of 018 120583m are approximately 13 timesof 06 120583m technologyThe stages of convergence are drawn inFigures 10 11 12 and 13

In 06 120583m technology when PSO and genetic algorithmsare used simultaneously FOMresponses are 14 times ofwhenonly PSO algorithm and fitness function in sum format areused FOM is also 20 times of when only genetic algorithm isused

0 5 10 15 203

4

5

6

7

8

9

10

Mean SR

Iteration

times106

(Vs

)

Figure 8 SR value in each stage that PSO algorithm coefficients areswapped

0 5 10 15 20

Mean PW

Iteration

minus138

minus136

minus134

minus132

minus13

minus128

minus126

minus124

times10minus3

(W)

Figure 9 PW value in each stage that PSO algorithm coefficientsare swapped

In 018 120583m technology when PSO and genetic algorithmsare used simultaneously FOM responses are 1008 times ofwhen only PSO algorithm and fitness function in multiplyformat are used FOM is also 277 times of when only geneticalgorithm is used The application of fitness function inmultiply format is better than its application in sum format

Obviously when fitness function coefficients are swa-pped the response has variation but when evolutionaryalgorithms are used in a shared manner better results areobtained but they are time-consuming

62 Example 2 The Design of Folded Cascade AmplifierA folded cascade amplifier is shown in Figure 14 and the


Table 1 The results of fitness function stimulation in the sum format and 06 120583m technology

Specification Constraint GA PSO PSO(120572) GA + PSODC gain 5000 51198905 3541198905 18321198905 6981198904

GB (MHz) 5 82 257 1598 675119878119877 (V120583119904) 10 144 207 1867 564POWER (mw) 2 0997 117 12 202FOM(1198905) 125 59 16096 455 13154




Table 3 The results of fitness function stimulation in the multiply format and 06 120583m technology



















1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration

Figure 10 GB value in one stage when PSO algorithm coefficientsare selected by genetic algorithm

applied technologies are 06120583m and 018 120583m The designparameters are the length of transistors and the amount ofcompensation capacitor All these transistors must work insaturation region That is 119881DS gt (119881GS minus 119881TH)must be set forNMOS transistors In each stage the results of optimizationof algorithms are compared by means of different fitnessfunctions Fitness functions are the same functions used inthe previous example

Here also fitness function (5) was firstly investigated byPSO algorithm in the ways that 120572 parameters are swept atfirst and in the next stage 120572 parameters are optimized bygenetic algorithm and circuit parameters are optimized byPSO algorithm Each of these experiments is undertaken 10times and the best of them are presented in Tables 5 6 7 and

1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)

Figure 11 SR value in one stage when PSO algorithm coefficientsare selected by genetic algorithm

8 The stages of convergence are drawn in Figures 15 16 1718 19 and 20

In the first stage if we compare 06 120583m and 018 120583mtechnologies as we expect the results obtained from 018120583mhave far better specifications For example when only geneticalgorithm is used and fitness function is in sum format FOMresponses of 018 120583m compared to 06 120583m are approximately41 better

When only PSO algorithm is used and fitness function isin sum format FOM responses of 018 120583m are 5000 times of06 120583m technology If fitness function is in multiply formatFOM responses obtained from 018 120583m are 1199 times of06 120583m technology When PSO and genetic algorithms areused simultaneously FOM responses of 018 120583m are 1032times of 06 120583m technology


1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration

Figure 12 Gain value in one stage when PSO algorithm coefficientsare selected by genetic algorithm

1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)

Figure 13 PW value in one stage when PSO algorithm coefficientsare selected by genetic algorithm

In 06 120583m technology when PSO and genetic algorithmsare used simultaneously FOM responses are 10 times ofwhen only PSO algorithm and fitness function in sum formatare used In 018120583m technology when PSO and geneticalgorithms are used simultaneously FOM responses are 21times of when only genetic algorithm and fitness function inmultiply format are used The application of fitness functioninmultiply format is better than its application in sum format

As it is clear from the figures when PSO and geneticalgorithms are applied simultaneously the results are approx-imately convergent after 7 stages which indicate the appro-priate selection of fitness function but when 120572s are swappedthere may be variation and it is not a good result Checking

Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A

Figure 14 Folded cascade op-amp

1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)


the results indicates that using a combination of optimizationalgorithms with selecting the appropriate fitness function inthe optimization process highly required being effective Eachof these algorithms has a prominent feature for exampledifferent speed or accuracy that better results can be achievedby combining them

Using a combination of genetic algorithms and PSOsimultaneously gives us the best results compared with usingthese algorithms separately

7 Conclusion

This method is a new look to design and optimize analogelectronic circuits that enables users by taking advantage of


1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration


software to design hardware and reduce the design complex-ity

The circuit design has been performed by SPICE simu-lator This paper investigated two methods of evolutionaryalgorithms namely genetic algorithm (GA) and PSO algo-rithm and their combination in order to optimize analoguecircuits

It was shown that combination of both algorithms hasbetter results Coefficients are based on GA and parametersare optimized by PSO algorithmThe reason for this choice isthat the PSO algorithm can converge faster than GA becausethe number of its operators is less than that of GA The

1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46

Figure 19 AV value in one stage when PSO algorithm coefficientsare selected by genetic algorithm

application of a combination of these two algorithms has thefollowing advantages

(1) More precision in search space and not being trappedin the local minimum

(2) Satisfaction of the designerrsquos desire specifications andits limitations

(3) Being appropriate for circuits with lots of elements(4) Using this method can meet the designerrsquos specifica-

tions for highly constrained problems(5) It can be suitable for large-scale problems


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)


Evolutionary algorithm includes a wide class of algo-rithms inspired from nature and function (such as harmonicsearch algorithm ant colony worms greedy and gravity)More diverse set of algorithms can be used for furtherresearch in order to select the appropriate method for thedesign of analog circuits is proposed Other interestingaspects of this work are (1) the use of real time methodsand considering the delay effects [18ndash20] (2) the use ofany electronic CAD software which improves the process ofdesign and fabrication In future also we can use other costfunctions and software such as cadence

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research leading to these results has received fundingfrom the Polish-Norwegian Research Programme operatedby the National Centre for Research and 24 Developmentunder the Norwegian FinancialMechanism 2009ndash2014 in theframe of Project Contract no Pol-Nor200957472013

References

[1] M G R Degrauwe O Nys E Dijkstra et al ldquoIDAC an inter-active design tool for analog CMOS circuitsrdquo IEEE Journal ofSolid-State Circuits vol 22 no 6 pp 1106ndash1116 1987

[2] C A Makris and C Toumazou ldquoAnalog IC design automationPart II Automated circuit correction by qualitative reasoningrdquoIEEE Transactions on Computer-Aided Design of IntegratedCircuits and Systems vol 14 no 2 pp 239ndash254 1995

[3] B Liu YWang Z Yu et al ldquoAnalog circuit optimization systembased on hybrid evolutionary algorithmsrdquo IntegrationTheVLSIJournal vol 42 pp 137ndash148 2009

[4] G Stehr M Pronath F Schenkel H Graeb and K AntreichldquoInitial sizing of analog integrated circuits by centering withintopology-given implicit specificationsrdquo in Proceedings of theIEEEACM International Conference on Computer Aided Design(ICCAD rsquo03) pp 241ndash246 November 2003

[5] M del Mar Hershenson S P Boyd and T H Lee ldquoOptimaldesign of a CMOS op-amp via geometric programmingrdquo IEEETransactions on Computer-Aided Design of Integrated Circuitsand Systems vol 20 no 1 pp 1ndash21 2001

[6] D Nam Y D Seo L J Park C H Park and B Kim ldquoParameteroptimization of an on-chip voltage reference circuit usingevolutionary programmingrdquo IEEETransactions on EvolutionaryComputation vol 5 no 4 pp 414ndash421 2001

[7] S Balkir G Dundar andG Alpaydm ldquoEvolution based synthe-sis of analog integrated circuits and systemsrdquo in Proceedings ofthe NASADOD Conference on Evolvable Hardware pp 26ndash29June 2004

[8] W Kruiskamp and D Leenaerts ldquoDARWIN CMOS opampsynthesis by means of a genetic algorithmrdquo in Proceedings of the32nd Design Automation Conference pp 433ndash438 June 1995

[9] G Rudolph ldquoConvergence analysis of canonical genetic algo-rithmsrdquo IEEE Transactions on Neural Networks vol 5 no 1 pp96ndash101 1994

[10] I Saracut M Neag R Onet and I Kovacs ldquoDesign optimiza-tionof analog active circuits using the genetic algorithmrdquoActat-echnicanapocensis Electronics and Telecommunications vol 53no 4 2012

[11] Y A N Xuesong H U Chengyu Y A O Hong F A NYuanyuan Qingzhong LIANG and L I U Chao Electronic Cir-cuit Optimization Design Algorithm Based on Particle SwarmOptimization PRZEGLAD ELEKTROTECHNICZNY 2013

[12] M Fakhfakh Y Cooren A Sallem M Loulou and P SiarryldquoAnalog circuit design optimization through the particle swarmoptimization techniquerdquo Analog Integrated Circuits and SignalProcessing vol 63 no 1 pp 71ndash82 2010

[13] P Prem Kumar K Duraiswamy and A Jose Anand ldquoAn opti-mized device sizing of analog circuits using genetic algorithmrdquoEuropean Journal of Scientific Research vol 69 no 3 pp 441ndash448 2012

[14] E Tlelo-Cuautle I Guerra-Gomez M A Duarte-Villasenoret al ldquoApplications of evolutionary algorithms in the designautomation of analog integrated circuitsrdquo Journal of AppliedSciences vol 10 no 17 pp 1859ndash1872 2010

[15] O Garitselov S P Mohanty and E Kougianos ldquoFast-accuratenon-polynomial metamodeling for nano-CMOS PLL designoptimizationrdquo in Proceedings of the 25th International Confer-ence on VLSI Design (VLSID rsquo12) pp 316ndash321 January 2012

[16] O Okobiah S P Mohanty E Kougianos and O GaritselovldquoKriging-assisted ultra-fast simulated-annealing optimizationof a clamped bitline sense amplifierrdquo in Proceedings of the 25thInternational Conference on VLSI Design (VLSID rsquo12) pp 310ndash315 January 2012

[17] H R Karimi and H Gao ldquoNew delay-dependent exponentialHinfin synchronization for uncertain neural networks withmixedtime delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[18] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I Regular Papersvol 58 no 8 pp 1910ndash1923 2011


[19] S Huang Z Xiang and H R Karimi ldquoRobust 1198972-gain control

for 2D nonlinear stochastic systems with time-varying delaysand actuator saturationrdquo Journal of the Franklin InstituteEngineering and Applied Mathematics vol 350 no 7 pp 1865ndash1885 2013

[20] Z Li H Gao and H R Karimi ldquoStability analysis and 119867infin

controller synthesis of discrete-time switched systemswith timedelayrdquo Systems amp Control Letters vol 66 pp 85ndash93 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Vector 119909 is the design variable and 119883119871and 119883

119867are

their lower and upper bounds respectively Analog integratedcircuits use (1) as a constrained optimization problem and canbe solved by evolutionary algorithms

The selection depends on the conditions of the problemand the application Our circuits follow the same equation foroptimizationThen they are optimized by evolutionary algo-rithms In the past genetic algorithms were applied in orderto optimize analogue circuits but here we try to optimizeanalogue circuits by genetic algorithms PSO algorithm andafter that a new structure is introduced in order to make useof both of the algorithms simultaneously and in combinationwith an appropriate fitness function Nowadays with thedevelopment of electronic circuits due to the need to optimizeand speed limits there will be more Evolutionary algorithmsare implemented by means of MATLABThe fitness functionand the limitations are examined byHSPICE simulation soft-ware and there is a link between the twomentioned softwareprograms for analyzing and optimizationThe technology weapplied is based on stimulation and it is done as followsfirst the parameters related to the evolutionary algorithmsare produced by MATLAB and they are sent to the circuitsimulator HSPICE for optimization and examination Aftersimulation these parameters are sent back toMATLAB againso that they are examined by the evolutionary algorithm andthen they are optimized These stages are repeated until wereach the stop point of the algorithm In Section 2 relatedwork which has been done is listed Section 3 briefly reviewsthe genetic algorithm and its flowchart PSO algorithm andits flow chart are briefly reviewed in Section 4 Section 5presents stages of the proposed work In this section wetry to introduce the fitness functions and procedures whichhave been performed In Section 6 two circuits and theirparameters are listed which will be used in optimizationThis section is the most important part because the proposedmethod is discussed in this section that combines the geneticalgorithms and PSO and the simultaneous use of these twomethods Here we analyze the circuits with evolutionaryalgorithms and then compare them with each other to obtainthe best method for designing the analog circuit based onevolutionary algorithms

2 Related Works

The traditional design of analogue circuits [1 2] does nothave the precision robustness and efficacy of todayrsquos systemsTherefore designs based on optimization are applied Inmethods that are based on optimization techniques incomparison with traditional methods more precision androbustness are achieved In these methods the problem isstimulated as a function which is minimized in numericalissues and then it is located in an optimization loop whichis probably repeated several times It is difficult to choose thisfunction and it must be done with a lot of care

On the other hand the function must be written in theclosed form and this may cause the loss of precision andspeed If it could be written in the closed form this is for onecircuit and it is not generalizable

Byte code Netlist

Decode Spice circuitsimulation

Fitnesscalculation

Figure 1 An overview of circuit evaluation process starting withbyte coded representation with fitness score

One of the major limitations in the design of analogcircuits is the limited number of parameters that can be inves-tigated It may be more important when circuit designerswant to design circuits with more and more complexity andhigher performances and limitations In many ways a penaltyfunction is used to satisfy these constraints In thesemethodsthe constrained optimization problem is transformed into anunconstrained one by minimizing the following function [3]

1198911015840119909= 119891119909+119899

sum119894=1

119908119894(119892119894(119909)) (2)

where 119908119894are the penalty coefficients and 119892

119894(119909) gt 0

The results of thesemethods are depending on the penaltycoefficients and may not be very ideal to the designersThe values of penalty coefficients should have logical valuesbecause when they have illogical value the results willproduce infeasible solutions For finding good solution thedesigner usually needs to change penalty coefficients formany times

Researchers now try to find the ability to handle large-scale and multiobject design problems Most of the availablemethods can deal with about 10ndash20 variables simultaneouslybut analog circuits have many unknown variables

An appropriate location must be selected as the startingpoint These stages are shown in Figure 1 However theprocess is simpler and precise In a stage it is attempted toselect a good starting point [4] and a better convergence [5]

In recent years however evolutionary algorithms havereceived lots of attention see for instance [6ndash8] and thereferences therein At first genetic algorithm (GA) wasapplied but sometimes this method did not have a favorableoutcome For instance in GA [9] it was not convergent in theglobal minimum point

Ioana Saracut presents an optimization tool for the designof analog circuits with a topology able to select the best setof passive components values and active devices form withinwell-defined sets of available values and options Unlike mostof the existing circuit optimization tools it can search withindiscrete sets of possible solutions such as the standard seriesof values available commercially for passive components andlists of active devices given by the designer [10]

Xuesong Yan says that a major bottleneck in the evolu-tionary design of electronic circuits is the problem of scaleThis refers to the very fast growth of the number of gates usedin the target circuit as the number of inputs of the evolved







4 PSO Algorithm



Fitnesscalculation

Fitnesscalculation

Selectioncrossover

Roundsolution

Finish

Yes

No


Decodechromosome

Spicesimulation

Fitnessmeasurement






V119899= 119908 times V


119899

minus 119909119899)


119899

minus 119909119899)

(4)





1is the









119897to be


119897



119891119894= sum119908

119894119898119894 (5)

119891119894= prod119898 =

(1198981lowast 1198982lowast 1198983)

1198984

(6)

119891119894=(1198981205721

1lowast 1198981205722

2lowast 1198981205723

3)

1198984

(7)

Initializeswarm

Evaluatefitness

Gbest

Gbest

Replace

PbestReplace

Updateposition

Updatevelocity

Stopcondition

No Yes= best



6 Proposed Method



1to 120572119899



swarm

Evaluate fitness

Update position

Update velocity

Stop No

Yes

No Yes

Finish


crossover mutation



Initialization

Gbest

Replace Gbest

Replace Pbest






Vin

Vdd = 25

IDC

Vss = minus25

CL

C1

M1

M2

M3

M4

M5

M6M7M8

minus

+30120583A

10 p

I1

3p






0 5 10 15 202

Iteration

Mean GB

25

3

35

4

45

5

55

6times106

(Hz)




(8)






0 5 10 15 203

4

5

6

7

8

9

10

Mean SR

Iteration

times106

(Vs

)


0 5 10 15 20

Mean PW

Iteration

minus138

minus136

minus134

minus132

minus13

minus128

minus126

minus124

times10minus3

(W)































1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration




1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)






1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















4 PSO Algorithm



Fitnesscalculation

Fitnesscalculation

Selectioncrossover

Roundsolution

Finish

Yes

No


Decodechromosome

Spicesimulation

Fitnessmeasurement






V119899= 119908 times V


119899

minus 119909119899)


119899

minus 119909119899)

(4)





1is the









119897to be


119897



119891119894= sum119908

119894119898119894 (5)

119891119894= prod119898 =

(1198981lowast 1198982lowast 1198983)

1198984

(6)

119891119894=(1198981205721

1lowast 1198981205722

2lowast 1198981205723

3)

1198984

(7)

Initializeswarm

Evaluatefitness

Gbest

Gbest

Replace

PbestReplace

Updateposition

Updatevelocity

Stopcondition

No Yes= best



6 Proposed Method



1to 120572119899



swarm

Evaluate fitness

Update position

Update velocity

Stop No

Yes

No Yes

Finish


crossover mutation



Initialization

Gbest

Replace Gbest

Replace Pbest






Vin

Vdd = 25

IDC

Vss = minus25

CL

C1

M1

M2

M3

M4

M5

M6M7M8

minus

+30120583A

10 p

I1

3p






0 5 10 15 202

Iteration

Mean GB

25

3

35

4

45

5

55

6times106

(Hz)




(8)






0 5 10 15 203

4

5

6

7

8

9

10

Mean SR

Iteration

times106

(Vs

)


0 5 10 15 20

Mean PW

Iteration

minus138

minus136

minus134

minus132

minus13

minus128

minus126

minus124

times10minus3

(W)































1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration




1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)






1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1is the









119897to be


119897



119891119894= sum119908

119894119898119894 (5)

119891119894= prod119898 =

(1198981lowast 1198982lowast 1198983)

1198984

(6)

119891119894=(1198981205721

1lowast 1198981205722

2lowast 1198981205723

3)

1198984

(7)

Initializeswarm

Evaluatefitness

Gbest

Gbest

Replace

PbestReplace

Updateposition

Updatevelocity

Stopcondition

No Yes= best



6 Proposed Method



1to 120572119899



swarm

Evaluate fitness

Update position

Update velocity

Stop No

Yes

No Yes

Finish


crossover mutation



Initialization

Gbest

Replace Gbest

Replace Pbest






Vin

Vdd = 25

IDC

Vss = minus25

CL

C1

M1

M2

M3

M4

M5

M6M7M8

minus

+30120583A

10 p

I1

3p






0 5 10 15 202

Iteration

Mean GB

25

3

35

4

45

5

55

6times106

(Hz)




(8)






0 5 10 15 203

4

5

6

7

8

9

10

Mean SR

Iteration

times106

(Vs

)


0 5 10 15 20

Mean PW

Iteration

minus138

minus136

minus134

minus132

minus13

minus128

minus126

minus124

times10minus3

(W)































1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration




1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)






1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










swarm

Evaluate fitness

Update position

Update velocity

Stop No

Yes

No Yes

Finish


crossover mutation



Initialization

Gbest

Replace Gbest

Replace Pbest






Vin

Vdd = 25

IDC

Vss = minus25

CL

C1

M1

M2

M3

M4

M5

M6M7M8

minus

+30120583A

10 p

I1

3p






0 5 10 15 202

Iteration

Mean GB

25

3

35

4

45

5

55

6times106

(Hz)




(8)






0 5 10 15 203

4

5

6

7

8

9

10

Mean SR

Iteration

times106

(Vs

)


0 5 10 15 20

Mean PW

Iteration

minus138

minus136

minus134

minus132

minus13

minus128

minus126

minus124

times10minus3

(W)































1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration




1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)






1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 202

Iteration

Mean GB

25

3

35

4

45

5

55

6times106

(Hz)




(8)






0 5 10 15 203

4

5

6

7

8

9

10

Mean SR

Iteration

times106

(Vs

)


0 5 10 15 20

Mean PW

Iteration

minus138

minus136

minus134

minus132

minus13

minus128

minus126

minus124

times10minus3

(W)































1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration




1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)






1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration




1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107(V

s)






1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 10

Mean AV

32

34

36

38

4

42

44

46

times108

Iteration


1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)




Vin

Vss

Vdd

IDC

I1

minus

+

10 p1 k1 k CL

M1M2

M3

M4M5

M6M7

M8 M9

M10 M11

M12

M14

M13

M15

R1 R2

0A


1 2 3 4 5 6 7 8 9 10

Mean SR

Iteration

82

825

83

835

84

845

times107

(Vs

)




7 Conclusion



1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 100089

009

0091

0092

0093

0094

0095

Mean PW

Iteration

(W)


1 2 3 4 5 6 7 8 9 109917

9918

9919

992

9921

9922

9923

9924

9925

Mean GB

times106

(Hz)

Iteration





1 2 3 4 5 6 7 8 9 10436

438

44

442

444

446

448

45

Mean SR

Iteration

times107

(Vs

)


1 2 3 4 5 6 7 8 9 10

Mean AV

Iteration

times108

32

34

36

38

4

42

44

46








1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 100031

00311

00312

00313

00314

00315

00316

00317

Mean PW

Iteration

(W)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Analog Circuit Design Optimization...

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