# VERY SIMPLE AND ACCURATE COMPUTER-AIDED-DESIGN (CAD) MODELS DEVELOPED BY GENETIC PROGRAMMING FOR THE QUASISTATIC ANALYSIS OF UNSHIELDED SUSPENDED AND INVERTED MICROSTRIP LINES.

Abstract: Very simple computer-aided-design models are introduced to determine the characteristic parameters such as effective permittivities and characteristic impedances of unshielded suspended and inverted microstrip lines. Computer-aided-design models are determined with the use of the genetic programming. The results of computer-aided-design models are compared with the results of quasi-static analysis, experimental works available in the literature and a commercial electromagnetic simulator. The comparison results clearly show that computer-aided-design models proposed in this work are in very good agreement with the simulation, theoretical and experimental results for the suspended and inverted microstrip lines. The design parameter ranges in this work are 2 [less than or equal to] [[epsilon].sub.r2] [less than or equal to] 20, 0.5 [less than or equal to] w/b [less than or equal to] 10, 0.1 [less than or equal to] a/b [less than or equal to] 1.5, and the respective characteristic impedances of unshielded suspended and inverted microstrip lines are 28[OMEGA] [less than or equal to] [Z.sub.0] [less than or equal to] 185[OMEGA], and 24 [OMEGA] [less than or equal to] [Z.sub.0] [less than or equal to] 159 [OMEGA], respectively. It is observed that the accuracies of computer-aided-design models proposed in this paper are good enough for the most practical cases.Keywords: Suspended and inverted microstrip, Computer-aided-design models, Genetic programming, Characteristic impedance, Effective permittivity

1. Introduction

Suspended and inverted microstrip (S&IM) lines are very useful transmission lines and they have an air gap between dielectric substrate and ground plane. They have many advantages such as lower dispersion, lower propagation loss and easy connections to conventional microstrip lines. S&IM lines are used to manufacture most of the microstrip components, such as amplifiers, mixers, power dividers, frequency multipliers, and directional couplers. Due to the symmetrical shielding and the wide range of characteristic impedance values achievable make these transmission lines particular suitable for filters [1, 2].

Many researches dealing with S&IM lines have been realized in the literature and they can be classified in three groups [3]. The first group consists of analytical works related with rigorous electromagnetic analysis of S&IM lines and their discontinuities. The works about the applications of S&IM lines such as filters, couplers, phase shifters, power combiners, mixers and antennas can be added in the second group. The third group contains many research papers about computer-aided-design (CAD) models for analyzing of different S&IM lines.

Four CAD models have been proposed for the quasistatic analysis of unshielded S&IM lines in the literature [4-9]. The first CAD model was reported in 1985 [4] and the accuracy of this model is better than [+ or -]1 % according to the exact theoretical data reproduced by Tomar etal [3]. This model contains an empirical expression depended on the geometrical dimensions of unshielded S&IM lines for the effective permittivity and it is valid for [[epsilon].sub.r2] [less than or equal to] 6. The most important disadvantage of this work is the restriction of the dielectric constant [[epsilon].sub.r2] [less than or equal to] 6 for substrate material, because substrate materials with [[epsilon].sub.r2] [greater than or equal to] 6 are often used in practice for example alumina ([[epsilon].sub.r2] = 9.6) and GaAs ([[epsilon].sub.r2] = 12.9). The second quasi-static CAD model is a generalized of the first CAD model [5]. This model is valid for a wider range of design parameters and the accuracy of this model according to the exact theoretical data reproduced in [3] is better than [+ or -]0.6 % for both unshielded suspended and inverted microstrip lines. This model contains a polynomial expression depending on geometrical dimensions of the lines for the effective permittivity. The third CAD model was first reported by Svacania in 1992 [6, 7] and later Schellenberg has modified the Svacania's formulas, uses a conformal mapping approach, coupled with curve-fitting to obtained theoretical data, and proposed two closed-form equations for the effective permittivities of unshielded S&IM lines [8]. These models are valid for [[epsilon].sub.r2] [less than or equal to] 12.9. The claimed accuracies of these models are better than [+ or -]0.65 % and [+ or -]1 % for unshielded S&IM lines, respectively [8]. The last CAD model was proposed by Yildiz and Saracoglu for the effective permittivities of unshielded S&IM lines [9]. This model is based on artificial neural network and does not contain closed-form equations.

The analysis CAD models proposed in the literature except last model contain long mathematical expressions and require extensive calculations to compute the effective permittivities of unshielded S&IM lines. Beside, these models do not have good accuracy except the second model [5]. More significantly, these CAD models do not contain closed-form equations to directly compute the characteristic impedances of unshielded S&IM lines and they are valid for only the effective permittivities of these lines. In the literature, the characteristic impedances of unshielded S&IM lines are calculated by using a formula depending on the effective permittivities of unshielded S&IM lines and characteristic impedance of a conventional microstrip line. For this reason, this formula becomes very complex, contains very long mathematical expressions and requires extensive calculations to compute the characteristic impedances of unshielded S&IM lines. Consequently, CAD models proposed in the literature are not very suitable to compute the propagation characteristics of S&IM lines. The aim of this paper is to present accurate and very simple CAD models developed by genetic programming (GP) for both effective permittivities and characteristic impedances of unshielded S&IM lines.

GP is an automated method for creating a working computer program from a high-level problem statement of the problem and it was applied to many engineering problems [10-15]. According to our researches, GP was also successfully applied to the design problems of microwave planar transmission lines for the first time by Kisioglu and Yildiz [16]. In this application, new and accurate synthesis models developed by GP were proposed to calculate the geometrical dimensions of coplanar waveguide with a finite width ground plane. New synthesis models consist of two closed-form design equations. One of them calculates the strip width of the coplanar waveguide and the other calculates the sloth width of the coplanar waveguide having a desired characteristic impedance for a given dielectric substrate material. In order to show the validities of proposed synthesis models for the coplanar waveguide with a finite width ground plane, obtained results are compared with the results of quasi-static analysis and very good agreements between them were observed [16].

In this paper, CAD models were developed by using GP to directly calculate both the effective permittivities and characteristic impedances of unshielded S&IM lines. CAD models can be used to easily and accurately calculate the propagation characteristics of unshielded S&IM lines. The ranges of validties of these CAD models valid very wide region and they are 2 [less than or equal to] [[epsilon].sub.r2] [less than or equal to] 20, 0.5 [less than or equal to] w/b [less than or equal to] 10, 0.1 [less than or equal to] a/b [less than or equal to] 1.5. In order to show the accuracies and validities of CAD models produced by GP, their results are compared with the results of theoretical [2, 5, 8], experimental works [17-19] available in the literature, and a commercial electromagnetic simulator HFSS [3].

2. Genetic Programming

Soft computing can be defined to be a collection of some computational techniques that try to imitate human intelligence for creating solution of complex problems, which attempt to overcome the impreciseness, uncertainty in the problem by some human-like capabilities such as reasoning, learning, decision making,. Components of soft computing include neural networks, fuzzy systems, evolutionary algorithms such as genetic algorithms, GP and swarm intelligence based algorithms.

GP was developed by Koza and adopted from the process of natural evolution and genetics [20]. GP belongs to the class of evolutionary algorithms which is based on the "principle of survival of the fittest". GP is however a relatively new addition to the group of other evolutionary algorithm techniques such as genetic algorithms, evolutionary programming and evolution strategies [21]. GP is similar to genetic algorithm [22]. The main difference between GP and genetic algorithm is that GP represents a solution as a tree instead of a binary string used in genetic algorithm.

GP differs from other data-driven models such as fuzzy rule-based systems and artificial neural networks [23]. It has found application mostly in the area of symbolic regression. The aim is to find a functional relationship between input and output variables in the symbolic regression problems. GP is a computer program which is often given as a tree model. There are terminals in the tree model where the internal nodes correspond to a set of functions used in the program and the external nodes indicate variables and constants used as the input to functions [24]. GP has been used as an optimization technique on the variety of problems and applications [25, 26].

In applying genetic programming to a problem, there are five major preparatory steps. These five steps involve determining terminal set, function set, fitness function, control parameter, and termination criteria. The first step is the set of terminals which may consist of the program's external inputs, functions with no arguments and constants. The second step is the set of primitive functions that are to be used to generate the mathematical expression that attempts to fit the given finite sample of data. The third step is the design of a fitness measure function. The fourth step includes determining the values of control parameters and in the last step a termination criterion for the algorithm is defined [27, 28].

3. CAD Models Developed by Genetic Programming for Unshielded S&IM Lines

The configurations of unshielded S&IM lines are shown in Figure 1. It is assumed that all conductors used in transmission lines are infinitely thin and perfectly conducting. GP finds a functional relationship between input and output variables for analyzing unshielded S&IM lines. In the analysis models, the inputs are relative permittivity ([[epsilon].sub.r2]), geometrical dimensions of the transmission lines (a/b, w/b) and output is effective permittivity ([[epsilon].sub.eff]) or characteristic impedance ([Z.sub.0]) of unshielded S&IM lines.

CAD models to be produced by GP for the quasistatic analysis of unshielded S&IM lines are essentially derived from the data set. The data set used in this paper has been obtained from the quasi-static analysis results [5]. The data set includes 3600 samples for each unshielded S&IM lines, separately. The ranges of the parameters in the data set are 2 [less than or equal to] [[epsilon].sub.r2] [less than or equal to] 20, 0.5 [less than or equal to] w/b [less than or equal to] 10, 0.1 [less than or equal to] a/b [less than or equal to] 1.5, and the respective characteristic impedances of unshielded S&IM lines are 28[OMEGA] [less than or equal to] [Z.sub.0] [less than or equal to] 185[OMEGA], and 24 [OMEGA] [less than or equal to] [Z.sub.0] [less than or equal to] 159 [OMEGA], respectively.

In order to find the suitable CAD models produced by GP for the quasi-static analysis of unshielded S&IM lines, many experiments were carried out. In these experiments, the most suitable values of population size, mutation rate, reproduction rate and crossover rate are chosen as 50, 0.1, 0.05 and 0.85, respectively. After many trials, accurate and very simple analysis CAD models were determined to compute the characteristic parameters of unshielded S&IM lines for a given relative dielectric constant [[epsilon].sub.r2] and physical dimensions (w/b and a/b) of substrate material. The analysis CAD models determined by GP for calculating the characteristic impedance and effective permittivity of unshielded SM line, which produce very good results, are given below;

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

The analysis CAD models produced by GP for computing the characteristic impedance and effective permittivity of unshielded IM line are given below;

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

with

u = [w/b] and x =[a/b] (5)

where [[eta].sub.0]=120[pi] [OMEGA] is the intrinsic impedance of free space and ([[alpha].sub.1], [[alpha].sub.2],... ., [[alpha].sub.11]) are unknown coefficients.

The values of unknown coefficients in the analysis CAD models are optimally found by GP and the results are listed in Table 1. The analysis CAD models are obtained by substituting the values of the coefficients in Equations 1, 2, 3 and 4 for the calculation of characteristic parameters of unshielded S&IM lines.

4. Numerical Results and Discussion

In this paper, analysis CAD models developed by GP for unshielded S&IM lines are presented. Many extensive comparisons have been made to confirm the accuracies and validities of CAD models proposed in this work for both unshielded S&IM lines. The results of CAD models are compared with the results of quasistatic analysis, experimental works available in the literature and a commercial electromagnetic simulator HFSS.

The first comparison is made for the effective permittivities and characteristic impedances of unshielded S&IM lines according to the ratio of normalized strip width (w/b) for different a/b ratios ([[epsilon].sub.r2] = 3.78). The obtained results are given in Figures 2 and 3. In these figures, the results of CAD models are compared with the results of Tomar and Bhartia's work [5] known as the best accurate model in the literature. As it can be clearly seen from these figures, there are very good agreement between our CAD model results and Tomar and Bhartia's results.

The results of second comparison are shown in Tables 2 and 3 for characteristic impedances and effective permittivities of unshielded S&IM lines with different geometrical dimensions and relative permittivities, respectively. In these tables, calculated results obtained from CAD models generated by GP are compared with the results of variational method in Fourier transform domain [2] and CAD models [5, 8] available in the literature, and a commercial electromagnetic simulator HFSS [3].

Finally, to consolidate the validities of CAD models proposed in this paper, their effective permittivity results are also compared with the results of experimental works [17-19] for both unshielded S&IM lines with different geometrical dimensions and relative permittivities. The calculated results of CAD models generated by GP in this work are given in Table 4. The results of variational method [2] and analysis CAD models [5, 8] available in the literature and HFSS results are also added in Table 4. The calculated values for effective permittivity obviously show that there are very good agreement again among CAD models developed by GP, theoretical [2, 5, 8], experimental results [17-19] available in the literature and HFSS [3].

Similar good results are also observed for all unshielded S&IM lines to be analyzed. The average percentage errors of CAD models for the characteristic impedance and effective permittivity are calculated to be 0.79% and 0.74%, respectively, for 3600 unshielded SM line samples. The average percentage errors of the CAD models for the characteristic impedance and effective permittivity are calculated to be 0.33% and 0.68%, respectively, for 3600 unshielded IM line samples, as compared with the results of quasi-static analysis [5].

Consequently, the agreement among the results of CAD models developed by GP, analysis CAD models, the results of variational method in Fourier transform domain, experimental works available in the literature, and HFSS obviously confirm the validity of CAD models proposed in this work for analyzing unshielded S&IM lines. The results also illustrate the performance of GP in obtaining high quality solutions.

5. Conclusions

In this work, accurate and very simple analysis CAD models for unshielded S&IM lines were developed by using GP. The proposed CAD models allow the designers to directly and easily calculate the characteristic parameters such as characteristic impedances and effective permittivities of unshielded S&IM lines. The results of CAD models proposed in this work are in good agreement with the result of theoretical, experimental works available in the literature and HFSS. It was observed that the accuracies of CAD models for unshielded S&IM lines were good enough for the most practical cases. The proposed CAD models are also very simple and useful for many microwave engineering applications. Finally, the procedure used in this paper could be also useful for developing simple and accurate analysis and design equations for other microwave transmission lines having different geometrical structure.

6. References

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Celal Yildiz was born in Kayseri, Turkey, 1958. He received the BS degree from Erciyes University, Kayseri, in 1983, the MS degree from Istanbul Technical University, in 1988, and the PhD degree from Erciyes University, in 1992, all in electronic engineering. He is now a professor at the Department of Electrical and Electronics Engineering, Erciyes University. He is working in the areas of neural networks, fuzzy inference systems, optimization techniques and their applications to electromagnetics and microwave engineering problems, the optical fibers, and the diffraction of electromagnetic waves.

Hakan Kisioglu was born in Kahramanmaras, Turkey, 1983. He received BS degree and M.S degree from Erciyes University in the Electrical and Electronics Engineering, Kayseri, Turkey in 2009 and 2011, respectively. He is now a PhD degree student in Erciyes University and he is working as a lecturer at the Department of Electrical and Electronics Engineering, Bozok University. His current research activities include optimization techniquies and their applications to electromagnetic and microwave engineering problems.

Celal YILDIZ (1) and Hakan KISIOGLU (2)

(1) Faculty of Engineering, Department of Electrical and Electronics Engineering, Erciyes University, Kayseri, Turkey.

(2) Faculty of Engineering and Architecture, Department of Electrical and Electronics Engineering, Bozok University, Yozgat, Turkey. yildizc@erciyes.edu.tr, hakan.kisioglu@bozok.edu.tr

Received on: 01.03.2017

Accepted on: 10.05.2017

Table 1. The values of the coefficients used in CAD models for the unshielded S&IM lines [[alpha].sub.1] [[alpha].sub.2] Suspended ([Z.sub.0]) 3.6450 0.9952 microstrip ([[epsilon].sub.eff]) 1.0501 0.7993 Inverted ([Z.sub.0]) 2.0540 1.0569 microstrip ([[epsilon].sub.eff]) 1.0068 0.0249 [[alpha].sub.3] [[alpha].sub.4] [[alpha].sub.5] Suspended 0.0533 1.2319 0.9952 microstrip 1.7814 1.0508 0.4085 Inverted 0.7599 3.4134 0.7029 microstrip 0.4235 0.0063 0.0249 [[alpha].sub.6] [[alpha].sub.7] [[alpha].sub.8] Suspended 5.7193 5.6497 0.9952 microstrip 0.5734 0.0331 1.5063 Inverted 3.7305 3.7305 1.4177 microstrip 0.5449 0.0152 0.0030 [[alpha].sub.9] [[alpha].sub.10] [[alpha].sub.11] Suspended 1.2505 3.3701 0.2066 microstrip 1.0508 Inverted 0.3105 microstrip Table 2. Results of CAD model developed by GP, analysis models available in the literature and HFSS for the characteristic impedance and effective permittivity of unshielded SM line Analysis Model [[epsilon].sub.r2] a/b w/b [2] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 0.5 1.1018 161.84 1 1.0830 127.11 2 1.0681 92.83 3 1.0613 74.22 4 1.0574 62.18 2.22 0.2 5 1.0548 53.65 6 1.0530 47.65 7 1.0517 42.26 8 1.0508 38.24 9 1.0500 34.94 10 1.0494 32.17 0.5 1.9184 108.80 1 1.8220 92.07 2 1.7096 74.58 3 1.6417 63.98 4 1.5957 56.40 12.9 1 5 1.5622 50.60 6 1.5367 45.97 7 1.5164 42.17 8 1.4999 38.99 9 1.4862 36.27 10 1.4746 33.93 Analysis Model [[epsilon].sub.r2] [5] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.0955 162.14 1.0816 126.61 1.0664 92.35 1.0587 73.85 1.0547 61.86 2.22 1.0526 53.36 1.0516 46.98 1.0512 42.00 1.0508 38.01 1.0502 34.75 1.0489 32.04 1.9121 108.81 1.8296 91.18 1.7149 73.77 1.6401 63.46 1.5883 56.09 12.9 1.5510 50.41 1.5232 45.84 1.5019 42.07 1.4849 38.90 1.4707 36.20 1.4582 33.88 Analysis Model [[epsilon].sub.r2] [8] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1146 159.35 1.0987 124.64 1.0825 90.98 1.0740 72.80 1.0688 61.04 2.22 1.0653 52.72 1.0628 46.48 1.0608 41.62 1.0593 37.70 1.0581 34.48 1.0572 31.79 1.9202 108.4 1.8257 91.37 1.7072 74.10 1.6373 63.57 1.5912 55.99 12.9 1.5583 50.17 1.5336 45.53 1.5142 41.73 1.4986 38.54 1.4856 35.84 1.4747 33.50 HFSS (1 GHz) [[epsilon].sub.r2] [3] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1027 162.95 1.0826 129.38 1.0668 96.16 1.0611 75.43 1.0574 62.80 2.22 1.0550 53.37 1.0540 48.50 1.0531 42.51 1.0515 38.71 1.0512 34.33 1.0502 31.46 1.9162 109.66 1.8237 92.14 1.7026 75.05 1.6377 64.60 1.5903 56.59 12.9 1.5560 50.57 1.5310 46.57 1.5076 42.12 1.4926 38.15 1.4826 35.43 1.4741 33.52 CAD model [[epsilon].sub.r2] (This work) [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1345 167.35 1.1038 128.69 1.0809 92.93 1.0716 74.31 1.0665 62.35 2.22 1.0633 53.87 1.0611 47.49 1.0595 42.50 1.0583 38.48 1.0573 35.16 1.0566 32.38 1.9109 108.54 1.8228 92.67 1.7075 75.08 1.6351 64.33 1.5853 56.64 12.9 1.5489 50.75 1.5211 46.03 1.4991 42.16 1.4814 38.90 1.4667 36.13 1.4544 33.73 Table 3. Results of CAD model developed by GP, analysis models available in the literature and HFSS for the characteristic impedance and effective permittivity of unshielded IM line Analysis Model [[epsilon].sub.r2] a/b w/b [2] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 0.5 1.1608 144.59 1 1.1348 112.44 2 1.1015 81.80 3 1.0810 65.43 4 1.0673 54.87 2.22 1 5 1.0575 47.39 6 1.0502 41.77 7 1.0444 37.39 8 1.0398 33.86 9 1.0360 30.96 10 1.0328 28.52 0.5 1.5434 108.75 1 1.4349 88.92 2 1.3174 68.39 3 1.2521 56.49 4 1.2097 48.41 9.8 0.6 5 1.1795 42.48 6 1.1568 37.92 7 1.1390 34.28 8 1.1246 31.31 9 1.1128 28.82 10 1.1028 26.71 Analysis Model [[epsilon].sub.r2] [5] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1636 143.36 1.1359 111.37 1.1006 80.95 1.0792 64.70 1.0651 54.23 2.22 1.0552 46.82 1.0479 41.26 1.0425 36.91 1.0383 33.42 1.0350 30.54 1.0325 28.13 1.5485 107.73 1.4430 87.67 1.3177 67.61 1.2462 56.03 1.2004 48.12 9.8 1.1689 42.27 1.1459 37.73 1.1287 34.10 1.1154 31.11 1.1048 28.61 1.0964 26.49 Analysis Model [[epsilon].sub.r2] [8] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1618 143.59 1.1711 108.03 1.0885 81.85 1.0701 65.25 1.0583 54.58 2.22 1.0501 47.05 1.0440 41.41 1.0393 37.03 1.0355 33.51 1.0325 30.62 1.0299 28.20 1.4966 111.47 1.5396 82.17 1.3286 67.06 1.2613 55.36 1.2177 47.44 9.8 1.1869 41.62 1.1640 37.14 1.1463 33.57 1.1321 30.65 1.1205 28.21 1.1108 26.15 HFSS (1 GHz) [[epsilon].sub.r2] [3] [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1601 148.64 1.1345 113.41 1.1036 80.72 1.0826 65.73 1.0686 56.34 2.22 1.0578 47.79 1.0502 41.43 1.0474 37.27 1.0416 33.86 1.0392 30.83 1.0300 29.48 1.5440 110.28 1.4279 89.76 1.3161 69.44 1.2329 57.15 1.1996 48.01 9.8 1.1790 42.21 1.1615 37.04 1.1446 32.69 1.1269 29.99 1.1050 27.62 1.1050 25.76 CAD model [[epsilon].sub.r2] (This work) [square root of [Z.sub.0] [[epsilon].sub.eff]] ([OMEGA]) 1.1956 142.02 1.1586 109.90 1.1142 79.79 1.0886 64.06 1.0719 53.89 2.22 1.0601 46.64 1.0514 41.16 1.0447 36.85 1.0393 33.38 1.0350 30.50 1.0313 28.09 1.5423 107.92 1.4336 88.34 1.3102 67.79 1.2416 56.09 1.1977 48.14 9.8 1.1627 42.27 1.1447 37.72 1.1274 34.07 1.1137 31.08 1.1026 28.57 1.0934 26.45 Table 4. Results of CAD models developed by GP, experimental works, analysis models published in the literature and HFSS for the effective permittivities of unshielded S&IM lines Effective Permittivity ([epsilon]eff) [[epsilon].sub.r2] a/b w/b Experimental Data [18] [17] [19] 1.6069 0.6541 1.7566 0.8034 0.4727 1.5986 2.55 0.4017 0.3041 1.4775 -- -- Suspended 0.2008 0.1775 1.3841 Microstrip 1 1.5269 2 1.3763 1.333 3 -- 1.2735 -- 4 1.2137 5 1.1709 3.78 0.5 1.3689 Inverted 1.0 1.2544 Microstrip 2.0 1.1664 0.333 3.0 -- -- 1.1342 4.0 1.1025 5.0 1.0712 Effective Permittivity ([epsilon]eff) Analysis Analysis Analysis HFSS CAD Model Model Model (1 GHz) model [2] [5] [8] [3] (This work) 1.5976 1.6002 1.5847 1.6703 1.5996 1.4879 1.4879 1.4751 1.5587 1.5609 1.3959 1.3806 1.3853 1.4568 1.5098 Suspended 1.3273 1.2760 1.3149 1.3884 1.4554 Microstrip 1.6276 1.6340 1.6215 1.6240 1.6340 1.4701 1.4689 1.4577 1.4660 1.4600 1.3740 1.3675 1.3684 1.3590 1.3574 1.3098 1.3000 1.3092 1.2929 1.2898 1.2642 1.2524 1.2670 1.2450 1.2418 1.4551 1.4682 1.3964 1.4299 1.4297 Inverted 1.3375 1.3391 1.3581 1.3165 1.3163 Microstrip 1.2285 1.2181 1.2301 1.2078 1.2077 1.1737 1.1608 1.1744 1.1552 1.1552 1.1340 1.1278 1.1408 1.1242 1.1241 1.1168 1.1063 1.1185 1.1038 1.1037

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Author: | Yildiz, Celal; Kisioglu, Hakan |
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Publication: | Istanbul University - Journal of Electrical & Electronics Engineering |

Article Type: | Report |

Date: | Jul 1, 2017 |

Words: | 5382 |

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