Robust Parameter Design Methodology for Microwave Circuits Considering the Manufacturing Variations

Transcription

1 IAENG Internatonal Journal of Computer Scence, 39:, IJCS_39 9 Robust Parameter Desgn Methodology for Mcrowave Crcuts Consderng the Manufacturng Varatons Takafum Nakagawa and Tasuku Krkosh Abstract Our target s to acheve a hgher frst run rate n the quantty producton of mcrowave crcuts. For ths purpose, we propose a useful robust parameter desgn methodology n whch the mult-obectve problem s treated as a sngle optmzaton problem under lmtng condtons. A set of controllable factors, whch provde an acceptable producton, s calculated by consderng such nose factors as manufacturng varatons. We used the teratve technque wth the Monte Carlo method to search for these values. The nose factors are assgned to Taguch's orthogonal array to reduce the CPU tme. Our proposed method s appled to the desgn of a mcrowave amplfer. Ths method s performance s compared wth four optmzaton methods n the mcrowave crcut smulaton, and ts effectveness s expermentally confrmed. The calculated controllable factors are not unque among these optmzaton methods to mnmze the varatons of the gan n manufactures. Our method s more effcent to fnd many canddates than the other optmzaton methods. The produced amplfers have acheved a frst run rate of 97% n ts manufacture. Index Terms robust desgn, mcrowave crcut, mult-obectve optmzaton, SN rato R I. INTRODUCTION obust desgn s an mportant technology that provdes an acceptable product for varablty n a frst run and upgrades product qualty at low cost. Computer aded engneerng (CAE) can be used as an alternatve to assst product desgn n many cases of mcrowave crcut desgns. Conventonal technques usng statstcal or worst-case modelng have been usually used by many desgners [1], [], [3], []. In these works, they ascertan the degree of the performance varablty by the Monte Carlo approach or an expermental desgn method after decdng the parameters. The conventonal desgn tres to fnd the values of the controllable factors for the allowance of manufacturng varatons. But t s unknown whether t gves smaller varablty untl the manufacturng s completed. Moreover, Takafum Nakagawa s wth Advanced Technology R&D Center, Mtsubsh Electrc Corporaton, -1-1, Tsukaguch-Honmach, Amagasak, Hyogo, JAPAN (correspondng author to provde phone: ;fax: ;e-mal:Nakagawa.Takafum@d r.mtsubshelectrc.co.p). Tasuku Krkosh s wth Communcaton Systems Center, Mtsubsh Electrc Corporaton, -1-1, Tsukaguch-Honmach, Amagasak, Hyogo, JAPAN (correspondng author to provde phone: ; fax: ;e-mal:krkosh.tasuku@ap.mtsubshelectrc.co.p). consderng the tradeoff among frequency response, gan, nose fgure, power consumpton, VSWR, and cost, the desgn leads to a mult-obectve problem. At present, the smulated annealng algorthm (SA) and a stochastc algorthm based on evoluton theory such as genetc algorthms (GA) are usually used to solve the mult-obectve problem [5], [], [7]. When usng these tradtonal methods, t generally takes much CPU tme to determne the optmal values. Therefore, many approaches have reduced the CPU tme usng optmzaton methods based on orthogonal desgn [], [9] or technques fndng the pareto front of tradeoff functons [1], [11], [1]. However, these works dd not consder the effects of noses, whch are an mportant part of the phlosophy of robust desgn. The conventonal approach by Taguch s well known as the qualty control to mprove the performance of products at low cost [13], [1], [15]. Taguch employed an orthogonal array (OA) to arrange the experments and used sgnal-to-nose ratos (SN rato) to evaluate the varablty of response n an expermental run. But Taguch s method has a lmtaton because t s an addtve lnear model and s ncompatble wth the mult-obectve problem. Several approaches have been appled to multple-obectve problems [], [17], [1], [19], []. However, these works cannot prevent trappng n a local mnmum wthout reachng global optmzaton. Other technques such as response-surface methodology have been studed for the desgns of mcrowave crcuts [1], [], [3]. In these works, regresson technques are used to ft the recorded response values to a user-defned model. As a result, computng tme s greatly requred to decde the fttng functon when the number of desgnable factors and obectves becomes large. A method usng GA combned wth Taguch s method was also proposed to consder the effects of noses [], [5]. In these works, the qualty loss functon s mnmzed wth OA assgned nose factors. Another effectve nteractve technque for solvng mult-obectve problems has been proposed [], [7], where the tradeoff between obectve functons s analyzed wth a newly defned tradeoff matrx, and the nteractve mult-obectve desgn optmzaton based on the Satsfcng Trade-Off Method s used. In ths paper, we propose a useful robust desgn methodology for mcrowave crcut desgn and apply t to the desgn of a mcrowave amplfer. The valdty of ths method s studed wth computer smulatons and experments. (Advance onlne publcaton: May 1)

2 IAENG Internatonal Journal of Computer Scence, 39:, IJCS_39 9 II. ROBUST PARAMETER DESIGN METHODOLOGY We treat the mult-obectve problem as a sngle optmzaton problem []. We calculate the sets of desgn values under the lmtng condtons based on the specfcatons. The Taguch s SN rato s used to evaluate the robustness of a crcut s deal performance. The mult-obectve problem s evaluated by the followng formulaton (1): Maxmze η (xˆ ) m Subect to xˆ X {ˆ x R g ( xˆ),( 1, m)}, (1) where (xˆ ) s the SN rato, whch descrbes the varablty of the performance. The performance s descrbed by a functon of f ( xˆ, M ). xˆ ( x1, xs ) s a set of controllable factors, and M s the nput sgnal. Suffx s refers to the number of controllable factors. R m s the feasble regon, and g (xˆ ) denotes the lmtng condton. m s the number of obectve functons whch refer to the specfcatons. s calculated by Eq. () []: η 1 log( β / σ ). () Slope β s determned by the least-squares method of y : β / q (3) y β M e, 1, p; 1, q () M y q, (5) / where M s the average of calculatons for all nose. and e s the regresson error. refers to the expermental runs n OA. refers to the number of nput sgnals. The total square error from regresson lne σ s gven by σ () k n 1 1 When each ( pq 1) q (( β β) M ) ( y β M k 1 1 M y concdes wth n ) M, β equals one. The obectve functons are calculated aganst the controllable factors decded wth the Monte Carlo method. Taguch's OA s used to consder the nose factors to reduce the CPU tme. The proposed method searches for the optmal values n the drecton that ncreases the SN rato. A bgger SN rato gves smaller varablty from Eq. (). The procedure runs n the followng steps, and ts detals are shown n Fg. 1. Step 1: Assgn the nose factors to Taguch's OA. Step : In the frst step, set of controllable factors x s randomly searched wth the Monte Carlo method n the range of [ x, hgh, x, low] : x = rand (,1) ( x x )., hgh, low. After the second step, x s randomly selected n the range of [-Δ, Δ]: x = w rand (,1) [ -Δ, Δ], where w s the prevous values of x and Δ s a search strp wdth. Step 3: Calculate obectve functons f ( xˆ, M ) for each expermental number of OA and set of controllable factors xˆ. Step : When max [ (xˆ ) ] s larger than the prevous one under the satsfacton of the lmtng condtons, w s replaced by x. If there was no desrable result, the wdth of Δ% s reduced by half of the prevous one. Step 5: A set of controllable parameters, whch gves the maxmum SN rato, s selected among the calculated results. 1: ( assgn nose factors to OA) : do k= 1,n; ( n s the number of teratons.) 3: do =1,p ; ( p s a number to search x n each step k.) : If k=1 then 5: do r=1,s ; x = rand (,1) ( x r, hgh x r, low) ;end do; r : else do r=1,s; x = r w r rand (-Δ k, +Δk ) ; end do; 7: end f; : xˆ = ( x 1,, x s ) ; 9: do =1, q; ( q s the expermental number n OA ) 1: calculate target functon f (xˆ,m) 11: end do; 1 : end do; 13: calculate η (xˆ ) 1: If ( η= k (max( k η ) ( k )( g ( xˆ) k ( 1, m )} Then 15: set w x ; η =η; : else 17: Δ k =.5 Δk-1 ; 1 : end If ; 19 : end do; : ( end of calculaton ) Fg. 1 Algorthm of proposed method III. APPLICATION TO MICROWAVE AMPLIFIER We desgned an nput matchng crcut for a mcrowave amplfer wth our proposed method. Fg. shows the layout of an amplfer wth FETs. The schematc drawng of a CAE model s llustrated n Fg. 3. The nput sgnal s dvded nto four crcuts through the nput matchng crcut and amplfed by four FETs n parallel. All sgnals are combned wth the output matchng crcut. It s mportant to reduce the varance of gan for stable performance. Commercal CAE code [9] s used to calculate the performance of the mcrowave crcut. FET Fg. Layout archtecture of mcrowave amplfer (Advance onlne publcaton: May 1)

3 IAENG Internatonal Journal of Computer Scence, 39:, IJCS_39 9 W 3,L 3 C 1, L 3 C 3 R L W,L R 1 W 1, L 1 Input matchng crcut L 5 L FET C 3 L 1 L 5 C, L Output matchng crcut Fg. 3 Schematc drawng of CAE model. Ten knds of nose factors and nne knds of controllable factors are llustrated. Controllable factors are expressed wth symbols enclosed n squares. TABLE II OA OF L1 ( 11 ) No Ten nose factors are assgned to OA. Number n matrx denotes nose levels descrbed n Table I. A. Identfyng Nose and Controllable Factors Ten knds of nose factors and nne knds of controllable factors are llustrated n Fg. 3. The nose factors are tabulated n Table I, where A and B are the manufacturng varatons of the thckness and the permttvty of the base plate. C and D are the manufacturng tolerances about the nductance of the lnes connected to FETs. E and F are related to the varatons of nductance on the DC cut-off crcut, and H and I are the manufacturng tolerances of the capactors of both nput and output DC cut-off crcuts. G s the varaton of the nductance of the mcrostrp lne connectng the capactor, and J s the manufacturng tolerance of the bypass capactors. Ten knds of nose factors are assgned to OA of L1 ( 11 ) n Table II, where the number denotes the nose levels descrbed n Table I. In Fg,3, nne knds of controllable factors are descrbed as symbols enclosed n the squares. The controllable factors are lengths L 1, L, and L 3 and wdths W 1, W, and W 3 on the mcrostrp lnes and gate wre nductance L 5 connected to each FET. In addton, two knds of resstance, R 1 and R, are optmzed. The FET s modeled by measured S parameters. The calculaton s done by a lnear computaton. TABLE I NOISE FACTORS Nose factors Level 1 Level A thckness -1% 1% B permttvty -1% 1% C nductance L 1-1% 1% D Inductance L -1% 1% E nductance L 3 -. nh. nh F Inductance L -. nh. nh G Inductance L 5 -. nh. nh H capactance C 1 -% % I capactance C -% % J capactance C 3 -% % B. Calculatons We appled the proposed method to a desgn of mcrowave amplfer. A performance example and ts target specfcaton are shown n Fg.. and ndcate the low and hgh frequency wthn the range of use. The dotted lne shows the lower lmt of a target value. The devaton caused by the nose factors s shown n Fg. 5. The vertcal lne denotes the normalzed gan, whch s expressed by Eq. (): Gan 1 ( gan T arg et ) /1 where gan s the averaged gan at each frequency. In Fg. 5, the sold lne ndcates the averaged value and the vertcal dotted symbols show the devaton produced by the nose factors. When the gan s equal to the target value, the normalzed gan agrees wth one. The SN rato s calculated wth a lnearzed functon. Fg. shows an example of an evaluaton. The calculated data for the expermental run of No. 5 n Table II are plotted by dotted crcles, whch are expressed wth regresson lne y 5 β 5 M. A straght dotted lne refers to the averaged gan calculated wth the expermental runs. If data of No. 5 agree wth the averaged values, slopeβ 5 equals 1.. These procedures are done for all expermental runs n Table II, and the SN rato s calculated from (). In ths desgn, RF stablzaton coeffcent k of a power amplfer s also consdered as a lmtng condton. Gan (db) Desgn regon () Target Fg. Example of performance and ts target n a mcrowave amplfer. Dotted lne s a mnmum of a target value. (Advance onlne publcaton: May 1)

4 IAENG Internatonal Journal of Computer Scence, 39:, IJCS_39 9 Gan Target devaton Averaged gan mn(gan) A SA Target B C D.5 Gan y M 5 β 5 Fg. 5. The example of the performance n the amplfer, and ts targeted value. The vertcal lne denotes the normalzed gan. y β M Averaged gan M IV. RESULTS AND DISCUSSION β 1 Fgure 7 shows the calculated results by the teratve technque wth the Monte Carlo method. The horzontal lne gves SN Ratoη, and the vertcal lne shows the mnmum value of Gan. 1 ponts are plotted for each of ten teratons. Plotted symbols are the calculated results wth SA. At the ntal step, search strp wdth Δs set as half of the nomnal values. In the followng steps, Δ decreases as shown n Fg., whch shows the relatonshp between SN ratoη and Δfor the each of teratons. From Fg. 7, the calculated results gradually converge to the pareto front calculated by SA. Ths shows the effectveness of our method. In Fg., ηncreases from 35. to 3.1. Ths means that the coeffcent of varaton decreased to 3% of the ntal one. The calculaton s almost converged by ten teratons, and ts CPU tme s 3 sec wth Intel Core 5-5 processor n a Wndows PC. We also compared the calculated results wth gradent search (GR), SA and GA n the commercal CAE code [9], [3]. Fg. 9 compares the optmal values, whch are tabled n Table III. Intal desgn shows the values wthout consderng the effects of the nose factors. The optmal values are dfferent among the optmzaton methods; the calculated result s not unque. Ths means that many combnatons of parameters can reduce the effect of manufacturng varatons. Therefore, we must fnd many canddates for the change of specfcatons, and for ths purpose the random search method wth the Monte Carlo method s more effcent than the other optmzaton algorthms. e Fg. Example of evaluaton for SN rato usng a lnearzed functon Fg. 7 Calculated results by teratve technque wth Monte Carlo method. 1 ponts are plotted for each of ten teratons. In the frst step, ntal value s decded by random search n the entre desgn space. In followng steps, search ranges are decreased by half of prevous one. SN rato η D A η mn(gan) η.. A D Δ SN rato Search strp wdth Δ (%) Number of teratons Fg. Relatonshp between SN rato η and search strp wdth Δ correspondng to number of teratons Values (mm) Proposed Method GR Intal Desgn GA SA W1 L1 W L W3 L3 L R1 R Knds of controllable factors Fg. 9 Comparson of optmal values of controllable factors among three optmzaton methods. Gradent method s a gradent search and SA method s a smulated annealng algorthm. GA s a genetc algorthm. Intal desgn shows desgn values wthout consderng effects of nose factors. Theses controllable factors are descrbed n Fg.3. TABLE III COMPARISONS OF OPTIMAL VALUES Controllable Factors SN W 1 L 1 W L W 3 L 3 L R 1 R Intal Desgn Proposed Method GR SA GA CONTROLLABLE FACTORS ARE DESCRIBED IN FIG.3. (Advance onlne publcaton: May 1)

5 IAENG Internatonal Journal of Computer Scence, 39:, IJCS_39 9 Fgure 1 shows the frequency response of gan at desgn ponts A D n Fg. 7. A and D gve the mnmum and the maxmum of the SN rato. B gves the maxmum gan, and C shows the fnal desgn. From Fg. 1, the larger SN rato gves smaller varance of gan. Desgn C s selected to acheve the target value even f the worst case producton occurred. Gan (db) Gan (db) (a) desgn ponts A (C) desgn ponts C Gan (db) Gan (db) The fnal confguraton on the basal plate was decded for under the lmtaton of the substrate sze. The crcut response was confrmed by electromagnetc feld computaton, and t was tuned to avod the undesrable oscllatons n the CAE model. We manufactured the prototype amplfer based on these results. The calculated results are compared wth the experments n Fg. 11. The bold lne shows the measurement, and the flux of thn brown lnes are the calculatons that nclude the varance caused by the nose factors. The calculated result qualtatvely agrees wth the measurement, and the gan satsfes the specfcaton. The measured gan n the mass producton s plotted n Fg. 1. The maxmum and mnmum values correspond to the mnmum and maxmum gan n all measurements, respectvely. The varablty of average gan was wthn ±. db, and the standard devaton was. db. These mcrowave amplfers have acheved a frst run rate of 97% n the manufactures. (b) desgn ponts B (d) desgn ponts D Fg. 1 response of gan for desgn ponts on Fg. 7 Gan (db) measure calculatons average target Fg. 11 Comparson between calculaton and experment. Bold lne s measurement and flux of brown thn lnes are calculatons Fg.1 Normalzed gan obtaned by experments n manufacturng V. CONCLUSION We proposed a robust parameter desgn methodology for mcrowave crcuts consderng manufacturng varatons. We calculated a set of controllable factors, whch provde acceptable producton, by consderng such nose factors as manufacturng varatons. The mult-obectve problem s treated as a sngle optmzaton problem under the lmtng condtons based on the specfcatons. We appled our proposed method to the desgn of a mcrowave amplfer and studed ts effectveness wth CAE smulatons and experments. The mcrowave amplfers desgned by our proposed method have acheved a frst run rate of 97% n the manufactures. ACKNOWLEDGMENT The authors are grateful to Dr. M. Shmozawa and Dr. M. Myazak, and the frst author also wshes to thank Dr. Haruna for ther many valuable comments and suggestons. REFERENCES [1] Shuch. Ota, Relablty Engneerng Assocaton of Japan (REAJ), Vol. 5, No., 3, pp. 5-, (n Japanese) [] N. Shgyo, H. Tanmoto, T. Morshta, K. Sugawara, N. Wakta, and Y. Asah, Statstcal smulaton of MOSFETs usng TCAD: meshng nose problem and selecton of factors, 3rd Internatonal Workshop on Statstcal Metrology, 199, pp [3] Stephen W., Peter Feldmann, and Kannan Krshna, Statstcal Integrated Crcut Desgn, IEEE Journal of Sold-State Crcuts, Vol.. No. 3, 1993, pp [] Norman J. Elas, Acceptance Samplng: An Effcent, Accurate Method for Estmatng and Optmzng Parametrc Yeld, IEEE Journal of Sold-Stats, Vol. 9. No. 3, 199, pp [5] Rcardo S. Zebulum, Marco Aurélo Pacheco, and Marley Vellasco, A Mult-Obectve Optmzaton Methodology Appled to the Synthess of Low-Power Operatonal Amplfers, Proceedngs of the XIII Internatonal Conference n Mcroelectroncs and Packagng, Vol. 1.,199, pp. -71 [] Leonard C. Brto and Paul H. P. de Carvalho, An evolutonary approach for mult-obectve optmzaton of nonlnear mcrowave crcuts, IEEE MTT-S Internatonal Mcrowave Symposum Dgest, Vol.,, pp [7] Eckart Ztzler and Lothar Thele, Multobectve Evolutonary Algorthms: A Comparatve Case Study and the Strength Pareto Approach, IEEE Transactons on evolutonary Computaton, Vol. 3, No., 199, pp [] Lnglng Sun, Zh Zhou, Xungen L and Wenmng Zhao, A new optmzaton method for mcrowave broad band amplfer based on orthogonal desgn, Proceedngs. th Internatonal Conference on Sold-State and Integrated-Crcut Technology, 1, pp [9] Hroyuk KAWAGISHI and Kazuhko KUDO, Development of Grobal Optmzaton Method by Orthogonal Array (Applcaton to Mechancal Desgn Problem), The Japan Socety of Mechancal Engneerng, 7, pp [1] Fatemeh Kashf, Safar Hatam, and Massoud Pedram, Mult-obectve optmzaton technques for VLSI crcuts, IEEE (Advance onlne publcaton: May 1)

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