Yield-driven em optimization using space mapping-based neuromodels

Transcription

1 Institut Tecnlógic y de Estudis Superires de Occidente Repsitri Institucinal del ITESO rei.ites.mx Departament de Electrónica, Sistemas e Inrmática DESI - Artículs y pnencias cn arbitraje 2-9 Yield-driven em ptimizatin using space mapping-based neurmdels Rayas-Sánchez, Jsé E.; Bandler, Jhn W.; Zhang, Qi J. J. W. Bandler, J. E. Rayas-Sánchez and Q.J. Zhang, Yield-driven EM ptimizatin using space mapping-based neurmdels, in 3st Eurpean Micrwave Cn., vl. 2, Lndn, England, Sep. 2, pp (.9/EUMA ) Enlace direct al dcument: Este dcument btenid del Repsitri Institucinal del Institut Tecnlógic y de Estudis Superires de Occidente se pne a dispsición general baj ls términs y cndicines de la siguiente licencia: (El dcument empieza en la siguiente página)

2 3st Eurpean Micrwave Cnerence, Lndn, England, September 2 YIELD-DRIVEN EM OTIMIZATION USING SACE MAING-BASED NEUROMODELS J.W. Bandler,,2 J.E. Rayas-Sánchez 2 and Q.J. Zhang 3 Bandler Crpratin,.O. Bx 883, Dundas, Canada L9H 5E7, j.bandler@ieee.rg 2 McMaster University, 28 Main St. West, Hamiltn, Canada L8S 4K, bandler@mcmaster.ca 3 Carletn University, 25 Clnel By Drive, Ottawa, Canada KS 5B6, qjz@de.carletn.ca ABSTRACT In this wrk, an eicient prcedure t realize electrmagnetics-based yield ptimizatin and statistical analysis micrwave structures using space mapping-based neurmdels is prpsed. A generalized relatinship between the ine and carse mdel sensitivities thrugh the Jacbian the neurmapping is prpsed. Our technique is illustrated by the EM-based statistical analysis and yield ptimizatin an HTS micrstrip ilter. INTRODUCTION With the increasing availability cmmercial EM simulatrs, it is very desirable t include them in statistical analysis and yield-driven design micrwave circuits. Given the high cst in cmputatinal ert impsed by the EM simulatrs, creative prcedures must be develped t eiciently use them r statistical analysis and design. We prpse the use space mapping-based neurmdels r eicient and accurate EM-based statistical analysis and yield ptimizatin micrwave structures. A general equatin t express the relatinship between the ine and carse mdel sensitivities thrugh a nnlinear, requencysensitive neurmapping is presented. We illustrate ur technique by the yield analysis and ptimizatin a high-temperature supercnducting (HTS) quarter-wave parallel cupled-line micrstrip ilter. YIELD ANALYSIS AND OTIMIZATION VIA SACE MAING BASED NEUROMODELS Let the vectrs x c, x R n represent the design parameters the carse and ine mdels, respectively. The perating requency ω, used by the ine mdel, can be dierent t that used by the carse mdel ω c. Let R c (x c,ω c ), R (x,ω) R r represent the carse and ine mdel respnses at ω c and ω, respectively. We dente the crrespnding SM-based neurmdel respnses at requency ω as R SMBN (x,ω), given by with R x, ω) = R ( x, ω ) () SMBN ( c c c This wrk was supprted in part by the Natural Sciences and Engineering Research Cuncil Canada (NSERC) under Grants OG7239, STR and thrugh the Micrnet Netwrk Centres Excellence. J.E. Rayas-Sánchez is supprted by an Ontari Graduate Schlarship, by McMaster University and by ITESO (Institut Tecnlógic y de Estudis Superires de Occidente, Mexic).

3 xc = ( x, ω) ω c (2) where the mapping unctin is implemented by a neural netwrk llwing any the 5 neurmapping variatins (SM, FDSM, FSM, FM r FSM) described in the wrk by Bandler et al. []. We assume that a suitable mapping unctin has already been und (i.e., a neural netwrk with suitable cmplexity has already been trained). I the SM-based neurmdel is prperly develped, r all x and ω in the training regin. R ( x, ω) R ( x, ω) (3) SMBN Let the Jacbian the ine mdel respnses w.r.t. the ine mdel parameters be J R r n ; let the Jacbian the carse mdel respnses w.r.t. the carse mdel parameters and mapped requency be J c R r (n+) and let the Jacbian the mapping w.r.t. the ine mdel parameters be J R (n+) n. Then the sensitivities the ine mdel respnses can be apprximated using J J J (4) c The accuracy the apprximatin J using (4) will depend n hw well the SM-based neurmdel reprduces the behavir the ine mdel in the training regin. I the mapping is implemented with a 3-layer perceptrn with h hidden neurns, (2) is given by ( x, ω ) = W Φ( x, ω) + b (5) Φ ( x, ω ) = [ ϕ( ) ϕ( s2) ϕ( s h s )] x ω h s = W + b h T (6) (7) where W R (n+) h is the matrix utput weighting actrs, b R n+ is the vectr utput bias elements, Φ R h is the vectr hidden signals, s R h is the vectr activatin ptentials, W h R h (n+) is the matrix hidden weighting actrs, b h R h is the vectr hidden bias elements and h is the number hidden neurns. A typical chice r the nnlinear activatin unctins is hyperblic tangents, i.e., ϕ( ) = tanh( ). All the internal parameters the neural netwrk, b, b h, W and W h are cnstant since the SM-based neurmdel has been already develped. The Jacbian J is btained rm (5-7) as J h = W JΦW (8) where J Φ R h h is a diagnal matrix given by J Φ = diag(ϕ ' (s j )), with j = h. I the SM-based neurmdel uses a 2-layer perceptrn, the Jacbian J is simply J = W (9) 2

4 which crrespnds t the case a requency-sensitive linear mapping. Ntice that by substituting (9) in (4) and assuming a requency-insensitive neurmapping we btain the lemma und by Bakr et al. [2], since in the case a 2-layer perceptrn with n requency dependence, W R n n. YIELD OTIMIZATION OF AN HTS FILTER (SYMMETRIC CASE) Cnsider a high-temperature supercnducting (HTS) parallel cupled-line micrstrip ilter [, 3] illustrated in Fig.. OSA9/hpe built-in linear elements cnnected by circuit thery rm the carse mdel. Snnet s em driven by Empipe rms the ine mdel, using a high-reslutin grid. The SM-based neurmdel the HTS ilter [3] is used. The crrespnding SM-based neurmdel is illustrated in Fig. 2, which implements a requency partial-space mapped neurmapping with 7 hidden neurns, mapping nly, S and the requency (3L:7-7-3). Applying direct minimax ptimizatin t the carse mdel, we btain the ptimal carse slutin x c *. We apply direct minimax ptimizatin t the SM-based neurmdel, starting at x c *, t btain the ptimal SM-based neurmdel nminal slutin x SMBN *. Fr yield analysis, we cnsider.2% variatin r the dielectric cnstant and r the lss tangent, as well as 75 micrn variatin r the physical dimensins, with unirm statistical distributins. We perrm Mnte Carl yield analysis the SM-based neurmdel arund x SMBN * with 5 utcmes. This takes a ew tens secnds n a C (AMD 64MHz, 256M RAM, Windws NT 4.). A single utcme calculatin r the same circuit using an EM simulatin takes abut 5 hurs. The respnses r 5 utcmes are shwn in Fig. 3. The yield calculatin is shwn in Fig. 4. A yield nly 8.4% is btained at x SMBN *. We then apply yield ptimizatin t the SM-based neurmdel with 5 utcmes using the Yield-Huber ptimizer available in OSA9/hpe, btaining the ptimal yield slutin: x SMBN Y*. The crrespnding respnses r 5 utcmes are shwn in Fig. 5. The yield is increased rm 8.4% t 66%, as shwn in Fig. 6. Excellent agreement between the EM respnses and the SM-based neurmdel respnses was und at bth the ptimal nminal slutin and the ptimal yield slutin. CONCLUSIONS An eicient prcedure t realize EM-based statistical analysis and yield ptimizatin micrwave structures is prpsed. A general equatin relates the ine and carse mdel sensitivities thrugh the Jacbian the neurmapping. The yield-driven design an HTS ilter is illustrated. ACKNOWLEDGEMENT The authrs thank Dr. J.C. Rauti, resident Snnet Stware, Inc., Liverpl, NY, r making em available. REFERENCES [] M.H. Bakr, J.W. Bandler, M.A. Ismail, J.E. Rayas-Sánchez and Q.J. Zhang, Neural space mapping ptimizatin r EM-based design, IEEE Trans. Micrwave Thery Tech., vl. 48, 2, pp [2] M.H. Bakr, J.W. Bandler, N. Gergieva and K. Madsen, A hybrid aggressive space-mapping algrithm r EM ptimizatin, IEEE Trans. Micrwave Thery Tech., vl. 47, 999, pp [3] J.W. Bandler, M.A. Ismail, J.E. Rayas-Sánchez and Q.J. Zhang, Neurmdeling micrwave circuits expliting space mapping technlgy, IEEE Trans. Micrwave Thery Tech., vl. 47, 999, pp

5 L S S 2 SM-based neurmdel L L 3 W S 3 H H ε r W L L 3 S 2 S 3 carse mdel Re{S Im{S Re{S 2 ε r S ω ANN c S c ω c Im{S 2 S 2 S Fig.. HTS quarter-wave parallel cupled-line micrstrip ilter. Fig. 2. SM-based neurmdel the HTS ilter r yield analysis assuming symmetry (c and S c crrespnd t and S as used by the carse mdel). 5 S number utcmes 5 yield = 8.4% requency (GHz) max errr Fig. 3. Mnte Carl yield analysis the SM-based neurmdel respnses arund the ptimal nminal slutin x SMBN * with 5 utcmes. Fig. 4. Histgram the yield analysis the SM-based neurmdel arund the ptimal nminal slutin x SMBN * with 5 utcmes. 35 S requency (GHz) number utcmes yield = 66% max errr Fig. 5. Mnte Carl yield analysis the SM-based neurmdel respnses arund the ptimal yield slutin x SMBN Y* with 5 utcmes. Fig. 6. Histgram the yield analysis the SM-based neurmdel arund the ptimal yield slutin x SMBN Y* with 5 utcmes (cnsidering symmetry). 4