exploration of saw duplexer design space by modified differential evolution

7/24/2019 Exploration of SAW Duplexer Design Space by Modified Differential Evolution

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Exploration of SAW Duplexer Design Space

By Modied Differential Evolution

KIYOHARU TAGAWASchool of Science and Engineering

Kinki University3-4-1 Kowakae, Higashi-Osaka, 577-8502

[email protected]

Abstract: This paper proposes a new computer-aided design method to decide the electrode conguration of aSurface Acoustic Wave (SAW) duplexer used by mobile communication systems. First of all, in order to evaluate

the performance of the SAW duplexer through the computer simulation, a numerical model based on mixed-modescattering parameters is derived. Then the structural design of the SAW duplexer is formulated as a constraintsatisfaction problem instead of an optimization problem. In order to obtain various feasible solutions for theconstraint satisfaction problem, a new evolutionary algorithm, i.e., a modied Differential Evolution (DE), isproposed. Finally, from a set of feasible solutions, features of the SAW duplexers design space are discussed.

KeyWords: Computer-aided design, constraint satisfaction problem, SAW device, evolutionary algorithm

1 IntroductionA Surface Acoustic Wave (SAW) is an acoustic wavetravelling along the surface of a material exhibitingelasticity, with amplitude that typically decays expo-nentially with depth into the substrate. SAW devicesuse the acoustic wave in electronic components to pro-vide a number of different functions including delaylines, resonators, lters, convertors and so on [1, 2].SAW devices have one or more specic electrodes,which are called interdigital transducers (IDTs), toconvert acoustic waves to electrical signals and viceversa by exploiting the piezoelectric effect of certainmaterials. Recently, SAW lters are used widely inthe radio circuits of mobile communication systemssuch as cellular phones and Personal Digital Assis-tants (PDAs). That is because SAW lters providesmall, rugged and cost competitive mechanical band-pass lters with outstanding frequency response char-acteristics. Besides, SAW lters are also used as partsof a SAW duplexer [3, 4, 5]. The SAW duplexer isa key device of mobile communication systems thatisolates a receiver (Rx) from a transmitter (Tx) whilepermitting them to share a common antenna (ANT).

The frequency response characteristics of SAWdevices are governed primarily by their geometricalstructures, namely electrode congurations. As statedabove, a SAW device consists of one or more IDTsfabricated on a piezoelectric substrate. By decidingon an appropriate structure of the SAW device, we cancarry out its desirable function. In order to avoid the

repetition of the design by trial and error, the struc-tural design of the SAW device is usually formulatedas an optimization problem. Then an optimization al-gorithm combined with a simulator of the SAW deviceis used to solve the optimization problem [6, 7, 8].Author [9] has also reported such an optimum de-sign method of a balanced SAW lter using the latestevolutionary algorithm, namely Differential Evolution(DE) [10]. DE is arguably one of the most powerfuloptimization algorithms in current use [11, 12].

Conventional optimum design methods of SAWdevices reduce the cost and time of their products.However, we cannot occasionally use the optimalstructures of the SAW devices obtained by the opti-mum design methods because of the inuence of un-

certain factors, namely processing accuracy of elec-trodes, quality of materials, temperature uctuations,interference signals from other devices and so on.

In this paper, a new method is proposed and ap-plied to the structural design of a SAW duplexer. TheSAW duplexer has a complex structure consisting of two Double Mode SAW (DMS) lters and four SAWresonators. Therefore, in the structural design of theSAW duplexer, a lot of design parameters have to beconsidered. Besides, the SAW duplexer must be de-signed for operation in the frequency bands used byRx and Tx, and must provide the frequency separa-tion between Rx and Tx. In the proposed method,the structural design of the SAW duplexer is formu-lated as a constraint satisfaction problem instead of

Advances in Mathematical and Computational Methods

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an optimization problem. A solution of the problemthat satises all constraints is called a feasible solu-tion. For the constraint satisfaction problem, we try tond as many feasible solutions as possible. In order

to obtain various feasible solutions, we also proposea modied DE which has an archive to store all feasi-ble solutions found in search. From the set of feasiblesolutions obtained by the modied DE, we can esti-mate the feasible region in the SAW duplexers designspace. Furthermore, by choosing an appropriate fea-sible solution from the feasible region, it is expectedthat we can obtain a robust structure of the SAW du-plexer against the inuence of uncertain factors.

The rest of this paper is organized as follows.Section 2 provides the structure of a SAW duplexer.A numerical model of the SAW duplexer based onmixed-mode scattering parameters is also derivedfrom the equivalent circuit model of IDT. Section 3formulates the structural design of the SAW duplexeras a constraint satisfaction problem. Section 4 pro-poses a modied DE for nding various feasible solu-tions of the constraint satisfaction problem. In Section5, from a set of feasible solutions, the feasible regionin the SAW duplexers design space is analyzed anddiscussed. Finally, Section 6 concludes the paper.

2 SAW Duplexer

2.1 Basic Structure of SAW DuplexerFigure 1 shows a basic structure of a SAW duplexerthat consists of two DMS lters and four SAW res-onators. A comb-shaped IDT is composed of somefacing electrodes, which are called ngers, and usedto excite and detect acoustic waves. A grating reec-tor is provided by Short Metal Strip Array (SMSA).A DMS lter consists of ve components: three IDTssandwiched between two SMSAs. A SAW resonatorconsists of three components: one IDT sandwichedbetween two SMSAs. All the components shown inFig. 1 are fabricated on a piezoelectric substrate.

The two DMS lters in Fig. 1 provide differentpass- and stop-bands for Tx and Rx, respectively. Apair of port-1 and port-2 of the SAW duplexer is con-nected to a common ANT and works as a balancedport for operating a differential mode signal. On theother hand, the port-3 and the port-4 are connected toTx and Rx, respectively. The port-3 and the port-4 areunbalanced ports for operating single mode signals.

2.2 Numerical Model of SAW DuplexerIn order to evaluate the frequency response character-istics of the SAW duplexer through the computer sim-ulation, a numerical model of the SAW duplexer is

port-1 port-2

port-3 port-4

ANT

Tx Rx

Figure 1: Basic structure of SAW duplexer

derived. The behavior of IDT is analyzed by using athree-port equivalent circuit model shown in Fig. 2:port-a and port-b are acoustic ports, while port-c isan electric port. Impedances Z 1 and Z 2 , admittanceY m , and capacitance C T have been detailed in the lit-erature [13]. A two-port equivalent circuit model of SMSA can be also obtained by shorting the electricport (port-c) in the equivalent circuit model of IDT.

The SAW duplexer shown in Fig. 1 is composedof some elemental components: IDTs and SMSAs.Because all the components of the SAW duplexer are

connected acoustically in cascade on a piezoelectricsubstrate, a whole circuit model of the SAW duplexercan be made up from the equivalent circuit mod-els of its components in the same way with the bal-anced SAW lter [9]. Furthermore, considering theimpedances of the respective ports of the SAW du-plexer, the circuit model of the SAW duplexer can betransformed into a four-port network model.

By using the scattering matrix S = [snm ], thefour-port network model of the SAW duplexer can bedescribed as shown in (1). In the four-port network model, a n and bn (n = 1 ,

, 4) denote, respectively,

the input- and output-signals of the port- n in Fig. 1.

b1b2b3b4

=

s 11 s12 s13 s14s 21 s22 s23 s24s 31 s32 s33 s34s 41 s42 s43 s44

a 1a 2a 3a 4

(1)

In order to evaluate the frequency response char-acteristics of the SAW duplexer strictly, differentialmode signals need to be segregated from commonmode signals in the four-port network model in (1).According to the balanced network theory [14], theinput signals an and the output signals bn (n = 1 , 2)of the balanced port are reorganized as follows:


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port-a

port-c

port-bZ1

1: A 10 1: A 20

CT

Z2

mY

Z1

Figure 2: Equivalent circuit model of IDT

a d = 1

2 (a 1 a 2 )bd =

1 2 (b1 b2 )

(2)

a c = 1 2 (a 1 + a 2 )

bc = 1 2 (b1 + b2 )

(3)

where, a d and bd in (2) are differential mode signals,whereas ac and bc in (3) are common mode signals.

From (2) and (3), conventional s-parameters in(1) are converted into mix-mode s-parameters as

S mix = T S T 1 =

s dd sdc sd3 sd4s cd scc sc3 sc4s 3 d s3 c s33 s34s 4 d s4 c s43 s44

(4)

where, matrix T is given as follows:

T = 1 2

1 1 0 0

1 1 0 00 0 2 00 0 0 2

. (5)

By using the mix-mode s -parameter sd3 of signalsbd and a3 shown in (4), the transmission characteristicof Tx to ANT in the SAW duplexer is dened as

T = 20 log 10 (|s d3 |). (6)Similarly, the transmission characteristic of ANT

to Rx in the SAW duplexer is dened as

R = 20 log10 (|s 4 d |). (7)Furthermore, the transmission characteristic of

Tx to Rx in the SAW duplexer is dened as

I = 20 log10 (|s 43 |). (8)

3 Problem FormulationIn the structural design of the SAW duplexer, de-sirable frequency response characteristics can be de-

scribed by a set of constraints for them. Therefore,the structural design of the SAW duplexer is formu-lated as a constraint satisfaction problem. First of all,in order to describe an electrode conguration of theSAW duplexer, we choose D = 18 design parametersx = ( x 1 , , x j , , xD ) such as the number andlength of IDTs ngers, the thickness of electrode andso on. The values of respective design parameters x jare limited by the lower x j and upper x j bounds as

x j x j x j , j = 1 , , D. (9)Secondly, we dene some constraints for the char-

acteristics of the SAW duplexer which depend on thedesign parameters x and vary with the frequency. Let1 be a set of frequency points 1 sampled fromthe pass-band of Tx. Furthermore, let 2 be a set of frequency points 2 sampled from the stop-bandof Tx. Thereby, we specify constraints for the trans-mission characteristic of Tx to ANT dened in (6) byits upper U T () and lower LT () bounds as

f 1 (x, ) = LT () T (x, )< 0, 1 ,

f 2 (x, ) = T (x, ) U T ()< 0, 2 .(10)

Similarly, let 3 and 4 be the sets of frequencypoints sampled, respectively, from the pass- and stop-bands of Rx. Thereby, we specify constraints for thetransmission characteristic of ANT to Rx dened in(7) by its upper U R () and lower LR () bounds as

f 3 (x, ) = LR () R (x, )< 0, 3 ,

f 4 (x, ) = R (x, )

U R ()

< 0, 4 .(11)

Furthermore, let 5 be a set of frequency points 5 sampled from the range including both bandsof Tx and Rx. Thereby, we specify constraints for thetransmission characteristic of Tx to Rx dened in (8),i.e., the isolation characteristic of the SAW duplexerbetween Tx and Rx, by its upper U I () bounds as

f 5 (x, ) = I (x, ) U I ()< 0, 5 .

(12)

Let X be a set of solutions x = ( x1 , , xD )that satisfy the boundary constraints shown in (9). Theset X is called the design space of the SAW duplexer.


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Besides, let F (F X ) be a set of feasible solutionsx F that satises all inequality constraints in (10),(11) and (12). The set F is called the feasible region.Furthermore, in order to evaluate the goodness of aninfeasible solution x X \F , we dene the penaltyfunction f (x) to be minimized as follows:

f (x) =5

p=1 p

max{f p( x, ), 0 } (13)

where, f (x) = 0 holds for feasible solutions x F .

4 Modied Differential Evolution

4.1 Representation of SolutionAs well as the conventional DE, the modied DE hasa set of real vectors vi = ( v1 ,i , , v j,i , , vD,i )(i = 1 , , N p), which are called individuals. Eachof the individuals vi represents a solution x X of the constraint satisfaction problem. However, everyelement v j,i of an individual vi is restricted within therange [0, 1] such as 0 v j,i 1 ( j = 1 , , D ).A design parameter x j ( j = 1 , , D ) in (9)takes either a continuous value x j or a discretevalue x j Q apart by an interval e j , where Q de-notes a set of design parameters taking discrete val-ues. Therefore, an individual vi

[0, 1]D needs to be

converted into a corresponding solution x X as

x = c(vi ) (14)

where, round( r ) rounds r to the nearest integerand the conversion c : [0, 1]D X is dened as

x j =

(x j x j ) v j,i + x j , if x j ,round

(x j x j )e j

v j,i e j + x j , if x j Q.

4.2 Strategy of Modied DEDE uses an operator called strategy to generate a newindividual. Even though various strategies have beenproposed for DE [11], we employ a strategy namedDE/current/1/bin. That is because the strategy cansearch an interval between two feasible solutions.

A set of current individuals is called the popula-tion P . In the above strategy of DE, for each indi-vidual vi P (i = 1 , , N p), which is called thetarget vector, three different individuals vi 1 , vi 2 andvi 3 (i = i1 = i2 = i3) are selected randomly from

P . Then a new individual u, which is called the trial

vector, is generated from those four individuals as

u = K v i + (1 K ) vi 1 + F (vi 2 vi 3 ) (15)

where, K and F are parameters given by the user.If an element u j of the trial vector u falls outside

the range [0, 1], it is returned to the range as

u j = v j,i + (0 v j,i ), if u j < 0,v j,i + (1 v j,i ), if u j > 1.(16)

where, [0, 1] is a uniformly distributed randomnumber within the range between 0 and 1 [11].4.3 Procedure of Modied DEThe modied DE is an extended version of DispersiveDE (DDE) proposed by authors [15]. Therefore, themodied DE is based on the steady-state model thatuses only one population P . The initial population P is generated randomly within the region [0, 1]D . Themodied DE generates more than one trial vector ufrom one target vector vi P as well as ConstraintAdaptation by DE (CADE) contrived to solve con-straint satisfaction problems [16]. If a newborn trialvector u is not worse than the target vector vi P ,vi is replaced by u immediately. Besides, in order toobtain various feasible solutions, the modied DE hasan archive A to store all feasible solutions found sofar. If a newborn trial vector u corresponds to a feasi-ble solution x F , it is added to the archive A. Thecontrol parameters of the modied DE are the maxi-mum number of the penalty function evaluation N g ,the population size N p , the maximum number of thetrial vectors generated from one target vector N t , andtwo parameters used in (15). The procedure of theproposed modied DE is provided as follows:

[Modied DE]Initialize such as A = ;Randomly generate vi P (i = 1 , , N p);Evaluate f (c(vi )) for every vi P ; g = N p;while (g N g ) {

for (i = 1; i N p; i + +) {k = 1; Flag = TRUE ;while (k N t Flag ) {

Randomly select vi 1 , vi 2 , vi3 P ;Generate u from vi by (15) ; k + +;Evaluate f (c(u)); g + +;if (f (c(u)) f (c(vi ))) {

vi = u; f (c(vi )) = f (c(u));if (f (c( u )) = 0) A= A{c(u)};Flag = FALSE ;

}}

}}Output all solutions x A F ;


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Table 1: Design space of SAW duplexer

x i [x i , x i ] x i [x i , x i ]

x1

[250, 350] x10

[45, 65]x 2 [3900, 4030] x11 [2.20, 2.40]x 3 [16.5, 24.5] x12 [16.5, 24.5]x 4 [12, 20] x13 [12, 20]x 5 [35, 65] x14 [45, 65]x 6 [2.26, 2.46] x15 [2.22, 2.32]x 7 [250, 350] x16 [55, 95]x 8 [3900, 4030] x17 [35, 65]x 9 [55, 95] x18 [2.00, 2.20]

5 Computational Experiment

5.1 Setup of ExperimentThe simulator of the SAW duplexer based on the four-port network model in (1) was made by MATLAB andconverted to the library used by the Java language.The specications for the SAW duplexer, namely theupper and lower bounds of transmission characteris-tics, were specied at 301 frequency points within therange: 700 1000 [MHz]. The design space X of the SAW duplexer in (9) was given by Table 1.

The modied DE was also coded by the Java lan-guage. From a prior outcome of an experiment, thecontrol parameters of the modied DE were chosenas N g = 1000 , N p = 50 , N t = 5 , K = 0 .5 andF = 0 .4. Then the modied DE was applied to theconstraint satisfaction problem, namely the structuraldesign of the SAW duplexer shown in Fig. 1.

5.2 Results of ExperimentBy only one trial of the modied DE, we could obtain

|A| = 398 feasible solutions of the constraint satis-faction problem. Figure 3 summarizes the values of the design parameters x j ( j = 1 ,

, 18) included in

the feasible solutions x A F , where the respec-tive values are normalized between 0 and 1. The max-imum and minimum values of the design parametersare plotted by broken lines in Fig. 3. The medium val-ues of them are also plotted by a solid line. From theset of feasible solutions obtained by the modied DE,we can get appearances of the feasible region F X .For example, the values of sensitive design parametersare restricted within narrow intervals in Fig. 3.

The frequency response characteristics of theSAW duplexer achieved by one of the feasible solu-tions x

A are shown in Figs. 4 and 5. Figure 4

plots the transmission characteristics of Tx to ANTand ANT to Rx, which are dened in (6) and (7). Theupper and lower bounds for them are also described

by broken lines. Figure 5 plots the transmission char-acteristic of Tx to Rx, which is the isolation character-istic of the SAW duplexer dened in (8). From Figs. 4and 5, we can conrm that the solution x

Asatises

all constraints shown in (10), (11) and (12).Under the assumption that the feasible region F consists of only one island that is simply connected

and convex, a solution z = ( z1 , , zD ) X com-posed by the medium values z j of the design param-eters x j in Fig. 3 exists at the center of the feasibleregion. The literature [16] asserts that the solution zis also feasible and gives a robust structure of the SAWduplexer against the inuence of uncertain factors.

Figure 6 plots the transmission characteristicsachieved by the center solution z X . As we can seein Fig. 6, the solution z is infeasible. Consequently,the feasible region F may not be convex. Otherwise,there is an interaction between design parameters.

6 ConclusionThe structural design of a SAW duplexer was formu-lated as a constraint satisfaction problem. In orderto obtain various feasible solutions for the problem,a modied DE was proposed. Through the analysis of feasible solutions, it was found that the feasible region

F X had a complex topology in the design space.In our future work, we intend to analyze more in-depth the design space of the SAW duplexer.

Acknowledgements: The research was supported inpart by the Grant-in-Aid for Scientic Research (C)(Project No. 24560503) from Japan Society for thePromotion of Science (JSPS).

References:

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0 5 10 15 200

0.2

0.4

0.6

0.8

1

Design parameters

N o r m a l

i z e d v a

l u e

Figure 3: Design parameters of feasible solutions

700 750 800 850 900 950 100080

70

60

50

40

30

20

10

0

Frequency [MHz]

L o s s

[ d B ]

Figure 4: Transmission characteristics

700 750 800 850 900 950 1000140

120

100

80

60

40

20

Frequency [MHz]

L o s s

[ d B ]

Figure 5: Isolation characteristic

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70

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30

20

10

0

Frequency [MHz]

L o s s

[ d B ]

Figure 6: Transmission characteristics

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exploration of saw duplexer design space by modified differential evolution

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