ACCURATE SYNTHESIS FORMULAS OBTAINED BY USING A DIFFERENTIAL EVOLUTION ALGORITHM FOR CONDUCTOR-BACKED COPLANAR WAVEG- UIDES

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1 Progress In Electromagnetics Researc M, Vol. 10, 71 81, 2009 ACCURATE SYNTHESIS FORMULAS OBTAINED BY USING A DIFFERENTIAL EVOLUTION ALGORITHM FOR CONDUCTOR-BACKED COPLANAR WAVEG- UIDES S. Kaya, K. Guney, C. Yildiz, and M. Turkmen Department of Electrical and Electronics Engineering Faculty of Engineering Erciyes University Kayseri 38039, Turkey Abstract In tis paper, accurate syntesis formulas obtained by using a differential evolution DE algoritm for conductor-backed coplanar waveguides CBCPWs are presented. Te syntesis formulas are useful to microwave engineers for accurately calculating te pysical dimensions of CBCPWs. Te results of te syntesis formulas are compared wit te teoretical and experimental results available in te literature. A full-wave electromagnetic simulator IE3D and experimental results are obtained in tis work. Te average percentage error of te syntesis formulas obtained by using DE algoritm is computed as 0.67% for 1086 CBCPW samples aving different electrical parameters and pysical dimensions, as compared wit te results of quasi-static analysis. 1. INTRODUCTION Te conductor-backed coplanar waveguide CBCPW as te advantage of mecanical strengt, eat sinking ability, and lower caracteristic impedance compared wit conventional coplanar waveguide in designing microwave integrated circuits MICs. Among tese advantages, CBCPWs allow easy implementation of mixed coplanar/microstrip circuits, reduce radiation effects, and raise effective permittivity. Tese and several oter advantages make CBCPWs ideally suit for MIC as well as monolitic MIC MMIC applications [1 12]. Corresponding autor: S. Kaya sabrikaya@erciyes.edu.tr.

2 72 Kaya et al. Various types of CBCPW structures ave been presented and analyzed in te literature [4 11]. Te CBCPWs were analyzed by using te spectral domain approac SDA [4], conformal mapping metod CMM [5, 6], and quasi-static SDA [7]. Te dispersion caracteristics of CBCPWs ave been obtained by using te rigorous mode matcing metod [8], alternative formulation of te transverse resonance tecnique [9], full-wave SDA and Galerkin s metod [10], and a ybrid two-dimensional finite-difference time-domain/marquardt curve-fitting tecnique [11]. Te syntesis formulas were also proposed in [12]. In tis paper, accurate syntesis formulas obtained by using te differential evolution DE [13, 14] algoritm are presented for CBCPWs. DE algoritm, one of te evolutionary algoritms, as been proven to be a igly efficient tecnique for solving numerical optimization problems. It is recognized tat DE algoritm is practical and powerful optimization tools for a variety of microwave engineering problems [15 23]. Te syntesis formulas proposed ere are used to successfully compute te pysical dimensions of CBCPWs. Te validity and accuracy of te proposed syntesis formulas are verified by comparing teir results wit tose of experimental works [3], CMM [5], syntesis formulas [12], a full-wave electromagnetic simulator IE3D [24], and experimental works realized in tis study. It was sown tat te syntesis formulas proposed in tis paper provide more accurate results tan te syntesis formulas proposed in [12]. 2. DIFFERENTIAL EVOLUTION DE ALGORITHM DE algoritm is an exceptionally simple, fast and robust computation metod for solving optimization problems [13, 14]. DE algoritm uses only a few control parameters, and tese remain fixed trougout te entire optimization procedure. DE algoritm operates on a population wit N P OP cromosomes, and eac cromosome is a symbolic representation of te vector consisting of te N PAR optimization parameters. To establis a starting point for optimum seeking, te population must be initialized. Te initial population is created wit random values selected from witin te given boundaries: v P i,j = v min j + M j v max j vj min, j = 1, 2,..., NPAR 1 were M j is a random number, uniformly distributed between 0 and 1, and vj min and vj max represent te minimum and maximum permissible values of te jt parameter, respectively.

3 Progress In Electromagnetics Researc M, Vol. 10, After te initialization, te algoritm goes into genetic evolution, and tree genetic operations, namely, mutation, crossover and selection are executed in sequence. In tis work, te mutation operation, wic generates a mating partner for eac individual by producing a difference vector called te mutant vector, is used. Mutant vector v i,g+1 is produced by v i,g+1 = t r1,g + F t r2,g t r3,g 2 were r 1, r 2, and r 3 belong to te set {1, 2,..., N P OP }, and t r1,g, t r2,g, and t r3,g represent tree random individuals cosen in te current generation, G, to produce te mutant vector for te next generation, v i,g+1. Te random numbers r 1, r 2, and r 3 sould be different from te running index i. F is te real scaling factor wic controls te amplification of te differential variation between two random vectors. Following mutation operation, in order to control te amount of diversity of te mutant vectors, te crossover is used. Te crossover may partially suppress te effect of mutation by forming a trial vector p ji,g+1 { vji,g+1, R p ji,g+1 = j P CR 3 t ji,g, oterwise were R j is a real random number in te range of [0, 1], and P CR is te probability of a real-valued crossover factor. Finally, te selection operation is used to produce better offspring. Eac cild competes wit its parent, and survives only if its fitness value is better. Following tis, te next round of genetic evolution ten begins. Te above procedure continues until a termination criterion is attained or a predetermined generation number is reaced. 3. SYNTHESIS FORMULAS OBTAINED FROM DE ALGORITHM Te cross-section of a CBCPW is illustrated in Fig. 1. In tis figure, s is te central strip widt, w is te slot widt, and is tickness of te dielectric substrate wit relative permittivity ε r. In tis paper, accurate syntesis formulas for computing te pysical dimensions of CBCPWs are obtained wit te use of DE algoritm. Tese syntesis formulas are essentially derived from te data set. Te data set used in tis work as been obtained from te respective quasi-static analysis results [5] and contains 1086 samples. Te design parameter ranges of CBCPWs used in tese samples are 2 ε r 50, 0.1 s/ 5.5, 0.1 w/ 1.90,

4 74 Kaya et al. 20 µm 3000 µm, and te respective caracteristic impedance is 10 Ω Z Ω. Figure 1. Cross-section of a CBCPW. In tis paper, first, te CBCPW parameters related to te pysical dimensions are determined by using te results available in te literature. After tis determination, models for te pysical dimensions are cosen, ten te unknown coefficient values of te models are determined by te DE algoritm. It is clear from te literature [5] tat four parameters ε r,, s, and w are needed to determine te caracteristic impedance of a CBCPW. Te first design step is te selection of a suitable dielectric substrate ε r, for a CBCPW aving a required caracteristic impedance Z 0. Ten, te pysical dimensions w and s are determined. In tis paper, two syntesis formulas for w and s are presented. Te first syntesis formula calculates te slot widt wz 0, s, ε r, for a given dielectric substrate ε r, and a required caracteristic impedance Z 0 by coosing an appropriate strip widt s. Te second syntesis formula computes te strip widt sz 0, w, ε r, for a given dielectric substrate ε r, and a required caracteristic impedance Z 0 by coosing an appropriate slot widt w. To find proper computer-aided design CAD models for te pysical dimensions of CBCPWs, many experiments were carried out in tis study. After many trials, te following syntesis models, wic produce very good results, were cosen to determine te slot and strip widts, respectively, α 1 x α 2 + α 3 y α 4 + α 5 s α6 w = α7 x α8 s α9 + α 10 s α11 y + α 12 α 13 α 14 4 x α 15 y + α 16 α 17 + α 18 s α19 and β 1 x β2 y β3 w β4 w +β 5 x+β 6 y+β 7 +β8 x 2 β10 +β 9 s= β 11 x β 12 y β 13 w β14 + β 15 x + β 16 y + β 17 w 5

5 Progress In Electromagnetics Researc M, Vol. 10, wit x = Z 0 η 0 6 y = ε r 7 were η 0 = 120π Ω is te intrinsic impedance of free space, α 1, α 2,..., α 19 and β 1, β 2,..., β 17 are te unknown coefficients. In tis paper, te coefficient values are optimally found by te DE algoritm. Te following slot and strip widt formulas are ten obtained by substituting tese optimum coefficient values into Eqs. 4 and 5: w = x y s x s s x y s y and x y w x y w x s= x 3.76 y w x y w 9 Te proposed syntesis formulas are valid for te ranges of s/ 10/1 + lnε r, w/ 10/[31 + lnε r ], 2 ε r 50, and 10 Ω Z Ω. In te DE algoritm optimization process, te values of population size, mutation rate, and crossover rate are taken as 30, 0.8, and 0.8, respectively. 4. NUMERICAL RESULTS AND DISCUSSION In order to sow te validity and accuracy of te proposed syntesis formulas given in 8 and 9, te results of te syntesis formulas obtained by using DE algoritm are compared wit te results of respective quasi-static analysis [5] in Figs. 2 and 3. Te results of syntesis formulas proposed by Yildiz and Turkmen [12] are also given in tese figures for comparison. Figs. 2 and 3, respectively, illustrate te quasi-static analysis [5] contours, te slot widt wz 0, s, ε r, results obtained by first syntesis formula and te strip widt sz 0, w, ε r, results obtained by second syntesis formula for CBCPWs wit ε r = 12.9 and = 200 µm and a required caracteristic impedance. It

6 76 Kaya et al. Normalized Slot Widt w/ Normalized Slot Widt w/ Ω Ω Normalized Strip Widt s/ Normalized Strip Widt s/ Figure 2. Comparison of te wz 0, s, ε r, results obtained by using te first syntesis formula proposed in tis paper, te syntesis formula proposed in [12] and te quasi-static analysis [5] contours for CBCPWs ε r = 12.9 and = 200 µm. Analysis [5], Yildiz and Turkmen [12], First Syntesis Formula. Figure 3. Comparison of te sz 0, w, ε r, results obtained by using te second syntesis formula proposed in tis paper, te syntesis formula proposed in [12] and te quasi-static analysis [5] contours for CBCPWs ε r = 12.9 and = 200 µm. Analysis [5], Yildiz and Turkmen [12], Second Syntesis Formula. is clear from Figs. 2 and 3 tat tere is a very good agreement between te results of quasi-static analysis [5] and te syntesis formulas proposed in tis paper. Tis good agreement supports te validity of te syntesis formulas proposed ere. Te average percentage errors of te syntesis formulas obtained by using DE algoritm and te syntesis formulas proposed Yildiz and Turkmen [12] are computed as 0.67% and 0.98%, respectively, for 1086 CBCPW samples, as compared wit te results of quasi-static analysis [5]. Terefore, better accuracy wit respect to te previous syntesis formulas [12] is obtained. Te caracteristic impedances calculated by using te results of syntesis Formulas 8 and 9 for a given s and w, respectively, are compared wit tose of quasi-static analysis [5] for CBCPWs wit ε r = 10.2 and = 635 µm in Fig. 4. In tis figure, te caracteristic impedance results are plotted wit respect to te sape ratio s+w/ for four different s/ values. It can be seen from Fig. 4 tat te results of te syntesis formulas are in very good agreement wit te results of quasi-static analysis. It is also apparent from tis figure tat tere is a very good self-consistent agreement between te first and second

7 Progress In Electromagnetics Researc M, Vol. 10, syntesis formulas. In tis paper, tree different CBCPWs are fabricated on RT/duroid laminates ε r = 6.15 and = 1270 µm by using te printed circuit board PCB excavation tecnique. Te caracteristic impedances of tese CBCPWs are calculated from te measured S-parameters for 1.75 GHz. We also calculated te caracteristic impedances by using a full-wave electromagnetic simulator IE3D [24]. In Table 1, te results of te syntesis formulas obtained by using DE algoritm are compared wit te results of measured [3], CMM [5], te formulas presented by Yildiz and Turkmen [12], IE3D [24], and experimental works realized in tis study. In tis table, Z 0 is te measured caracteristic impedance values, w and s represent te measured geometrical dimensions of slot and strip widts of CBCPWs, respectively. Also, Z 0 represents te caracteristic impedance values obtained from CMM and IE3D by using w and s. Z 0w and Z 0s are te final-ceck quasi-static analysis results calculated by using te w and s values obtained from te first and second syntesis formulas, respectively. As it can be seen from Table 1, a close agreement is obtained between te teoretical and experimental results. It is also clear tat te best results are obtained from Eq. 8 wit respect to te experimental results. In tis paper, te syntesis models, wic are simpler and more complicated tan te models given by Eqs. 4 and 5, were also tired. It was observed tat te results of simpler models are not in good Caracteristic impedance Om s/ = 0.5 s/ = 0.75 s/ = 1.0 s/ = s + w/ Figure 4. Comparisons of te caracteristic impedances calculated by using te results of te first syntesis Formula 8 for a given s; te results of te second syntesis Formula 9 for a given w; and te quasi-static analysis [5] for CBCPWs ε r = 10.2 and = 635 µm. Analysis [5], First Syntesis Formula, Second Syntesis Formula.

8 78 Kaya et al. Table 1. Comparisons of te caracteristic impedance results of te present syntesis formulas, experimental works [3], CMM [5], syntesis formulas proposed by Yildiz and Turkmen [12], IE3D [24], and experimental works realized in tis study. Measured CMM [5] IE3D [24] Yildiz and Turkmen [12] Present Results w µm s µm Z 0 Ω Z 0 Ω Z 0 Ω Z 0w Ω Z 0s Ω Z 0w Ω Z 0s Ω * * * *Measured in tis paper for CBCPWs wit ε r = 6.15 and = 1270 µm and te remainder measured in [3] for CBCPWs wit ε r = 12.9 and = 100 µm. agreement wit te teoretical and experimental results available in te literature, and tat te more complicated models provide only a little improvement in te results, at te expense of te simplicity of te syntesis formulas. Te advantages of te proposed syntesis formulas are simplicity and accuracy. Similar good results are obtained for all CBCPW samples to be designed wit different electrical parameters and pysical dimensions. Te results obtained from te proposed syntesis formulas clearly illustrate te performance of DE algoritm in obtaining ig quality solutions. DE algoritm can be applied to obtain syntesis formulas for oter transmission lines by coosing proper models. 5. CONCLUSION In tis paper, syntesis formulas obtained by using DE algoritm are presented for computing accurately te pysical dimensions of CBCPWs. Te caracteristic impedance values calculated by using te results of tese syntesis formulas are in good agreement wit te teoretical and experimental results available in te literature and experimental results obtained in tis study. Tis good agreement supports te validity and accuracy of te syntesis formulas proposed ere. Te syntesis formulas allow te designers to determine te pysical dimensions of CBCPWs for te required design specifications in a very simple and convenient way, rater tan by te iteration approac of applying te analysis tecnique.

9 Progress In Electromagnetics Researc M, Vol. 10, ACKNOWLEDGMENT Te work described in tis paper is supported by Te Erciyes University Scientific Researc Project Center, 2009, Project No: FBA REFERENCES 1. Nguyen, C., Analysis Metods for RF, Microwave, and Millimeter-wave Planar Transmission Line Structures, Jon Wiley and Sons, Simons, R. N., Coplanar Waveguide Circuits, Components and Systems, Jon Wiley and Sons, Si, Y. C., Broadband caracterization of conductor-backed coplanar waveguide using accurate on-wafer measurement tecniques, Microwave Journal, Vol. 34, , Si, Y. C. and T. Ito, Analysis of conductor-backed coplanar waveguide, Electronic Letters, Vol. 18, , Gione, G. and C. U. Naldi, Parameters of coplanar waveguides wit lower ground plane, Electronic Letters, Vol. 19, , Gione, G. and C. U. Naldi, Coplanar waveguides for MMIC applications: Effect of upper sielding, conductor backing, finite-extent ground planes, and line-to-line coupling, IEEE Transactions on Microwave Teory and Tecniques, Vol. 35, , Ceng, K. K. M. and J. K. A. Everard, A new tecnique for te quasi-tem analysis of conductor-backed coplanar waveguide structures, IEEE Transactions on Microwave Teory and Tecniques, Vol. 41, , Tien, C. C., C. K. C. Tzuang, S. T. Peng, and C. C. Cang, Transmission caracteristics of finite-widt conductor-backed coplanar waveguide, IEEE Transactions on Microwave Teory and Tecniques, Vol. 41, No. 9, , Neto, A. G., C. S. D. Roca, D. Bajon, and H. Baudrand, Analysis of te conductor-backed coplanar waveguide by an alternative formulation of te transverse resonance tecnique, SBMO/IEEE MTT-S Int., , Huang, J. F. and C. W. Kuo, More investigations of leakage and nonleakage conductor-backed coplanar waveguide, IEEE Transactions on Electromagnetic Compatibility, Vol. 40, No. 3, , 1998.

10 80 Kaya et al. 11. Hotta, M., Y. Qian, and T. Ito, Efficient FDTD analysis of conductor-backed CPW s wit reduced leakage loss, IEEE Transactions on Microwave Teory and Tecniques, Vol. 47, , Yildiz, C. and M. Turkmen, Syntesis formulas for conductorbacked coplanar waveguide, Microwave and Optical Tecnology Letters, Vol. 50, No. 4, , Price, K., Differential evolution: A fast and simple numerical optimizer, IEEE Nort American Fuzzy Info. Process. Conf., , Berkeley, CA, Storn, R. and K. Price, Differential evolution: A simple and efficient euristic for global optimization over continuous spaces, Journal of Global Optimization, Vol. 11, , Micalski, K. A., Electromagnetic imaging of circular cylindrical conductors and tunnels using a differential evolution algoritm, Microwave and Optical Tecnology Letters, Vol. 27, , Qing, A., Electromagnetic inverse scattering of multiple twodimensional perfectly conducting objects by te differential evolution strategy, IEEE Transactions on Antennas and Propagation, Vol. 51, , Luo, X. F., P. T. Teo, A. Qing, and C. K. Lee, Design of doublesquare-loop frequency-selective surfaces using differential evolution strategy coupled wit equivalent-circuit model, Microwave and Optical Tecnology Letters, Vol. 44, , Luo, X. F., A. Qing, and C. K. Lee, Application of te differential-evolution strategy to te design of frequency-selective surfaces, Int. J. RF and Microwave CAE, Vol. 15, , Yildiz, C., A. Akdagli, and M. Turkmen, Simple and accurate syntesis formulas obtained by using a differential evolution algoritm for coplanar strip lines, Microwave and Optical Tecnology Letters, Vol. 48, , Guney, K., C. Yildiz, S. Kaya, and M. Turkmen, New and accurate syntesis formulas for multilayer omogeneous coupling structure, Microwave and Optical Tecnology Letters, Vol. 49, , Guney, K., C. Yildiz, S. Kaya, and M. Turkmen, Syntesis formulas for multilayer omogeneous coupling structure wit ground sielding, Journal of Electromagnetic Waves and Applications, Vol. 21, No. 14, , Guney, K., C. Yildiz, S. Kaya, and M. Turkmen, Syntesis formulas for microcoplanar striplines, Microwave and Optical

11 Progress In Electromagnetics Researc M, Vol. 10, Tecnology Letters, Vol. 50, , Guney, K., C. Yildiz, S. Kaya, and M. Turkmen, New and accurate syntesis formulas for asymmetric coplanar stripline wit an infinitely wide strip, Journal of Infrared, Millimeter and Teraertz Waves, Vol. 50, , Zeland Software Inc., IE3D, Version 12.12,