Microstrip Antennas Part 4 doc

Particle-Swarm-Optimization-Based Selective Neural Network Ensemble and Its Application to Modeling Resonant Frequency of Microstrip Antenna 79 No f ME [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] 1 7740 7804 7697 7750 7791 7635 7737 7763 7720 7717 412 7765 2 8450 8496 8369 8431 8478 8298 8417 8446 8396 8389 488 8451 3 3970 4027 3898 3949 3983 3838 3951 3950 3917 3887 510 3977 4 7730 7940 7442 7605 7733 7322 7763 7639 7551 7376 1610 7730 5 4600 4697 4254 4407 4641 4455 4979 4759 4614 4430 113 4618 6 5060 5283 4865 4989 5070 4741 5101 4958 4924 4797 1621 5077 7 4805 5014 4635 4749 4824 4520 4846 4724 4688 4573 1460 4830 8 6560 6958 6220 6421 6566 6067 6729 6382 6357 6114 2550 6563 9 5600 5795 5270 5424 5535 5158 5625 5414 5374 5194 1769 5535 10 6200 6653 5845 6053 6201 5682 6413 5987 5988 5735 2860 6193 11 7050 7828 6566 6867 7052 6320 7504 6682 6769 6433 4792 7030 12 5800 6325 5435 5653 5801 5259 6078 5552 5586 5326 3259 5787 13 5270 5820 4943 5155 5287 4762 5572 5030 5081 4842 3383 5273 14 7990 9319 7334 7813 7981 6917 8885 7339 7570 6822 8674 8101 15 6570 7412 6070 6390 6550 5794 7076 6135 6264 5951 5486 6543 16 5100 5945 4667 4993 5092 4407 5693 4678 4830 4338 5437 5193 17 8000 8698 6845 7546 7519 6464 8447 6889 7160 6367 8067 7948 18 7134 7485 5870 6601 6484 5525 7342 5904 6179 5452 7242 7169 19 6070 6478 5092 5660 5606 4803 6317 5125 5341 4735 6103 6026 20 5820 6180 4855 5423 5352 4576 6042 4886 5100 4513 5875 5817 21 6380 6523 5101 5823 5660 4784 6453 5122 5396 4729 6546 6515 22 5990 5798 4539 5264 5063 4239 5804 4550 4830 4196 5976 6064 23 4660 4768 3746 4227 4141 3526 4689 3770 3949 3479 4600 4613 24 4600 4084 3201 3824 3615 2938 4209 3168 3446 2921 4603 4550 25 3580 3408 2668 3115 2983 2485 3430 2670 2845 2461 3574 3628 26 3980 3585 2808 3335 3162 2590 3668 2790 3015 2572 3955 3956 27 3900 3558 2785 3299 3133 2573 3629 2771 2987 2555 3895 3907 28 3980 3510 2753 3294 3112 2522 3626 2721 2966 2509 3982 3922 29 3900 3313 2608 3147 2964 2364 3473 2554 2823 2356 3903 3747 30 3470 3001 2358 2838 2675 2146 3129 2317 2549 2137 3493 3381 31 3200 2779 2183 2623 2474 1992 2889 2151 2357 1983 3197 3123 32 2980 2684 2102 2502 2370 1936 2752 2086 2259 1924 2982 2972 33 3150 2763 2168 2600 2453 1982 2863 2139 2338 1972 3160 3096 Errors 13136 24097 11539 12322 30669 8468 22572 18148 30504 56698 1393 Resonant frequencies and errors are in MHz. Table 6. Resonant frequency obtained from traditional methods for rectangular MSAs and sum of absolute errors between experimental results and theoretical results Microstrip Antennas 80 conventional methods [3]-[13] are listed in table 6. The sum of absolute errors between experimental and theoretical results for every method is also listed in the last row of table 6. It is clear from table 5 and table 6 that the computing results of the chaos BiPSO-based selective NNE are better than these of previously proposed methods, which proves the validity of the algorithm further. 7. Conclusion Selective neural network ensemble (NNE) methods based on decimal particle swarm optimization (DePSO) algorithm and binary particle swarm optimization (BiPSO) algorithm are proposed in this study. In these algorithms, optimally select neural networks (NNs) to construct NNE with the aid of particle swarm optimization (PSO) algorithm, which can maintain the diversity of NNs. In the process of ensemble, the performance of NNE may be improved by appropriate restriction on combination weights based on BiPSO algorithm. And this may avoid calculating the matrix inversion and decrease the multi-dimensional collinearity and the over-fitting problem of noise. In order to effectively ensure the particles diversity of PSO algorithm, chaos mutation is adopted during the iteration process according to randomicity, ergodicity and regularity in chaos theory. Experimental results show that the chaos BiPSO algorithm can improve the generalization ability of NNE. By using the chaos BiPSO-based selective NNE, resonant frequency of rectangular microstrip antenna (MSA) is modeled, and the computing results are better than available ones, which mean that the proposed NNE in this study is effective. The method of NNE proposed in this study may be conveniently extended to other microwave engineering and designs. 8. Acknowledge This work is supported by Pre-research foundation of shipping industry of China under grant No. 10J3.5.2, and Natural Science Foundation of Higher Education of Jiangsu Province of China under grant No. 07KJB510032. 9. Reference [1] Wong K L, “Compact and broadband microstrip antennas”, New York: John Wiley & Sons Inc., 2002. [2] Kumar G, and Ray K P, “Broadband microstrip antennas”, MA: Artech House, 2003. [3] Howell J Q, “Microstrip antennas”, IEEE Transactions on Antennas and Propagation, 1975, 23(1): 90-93. [4] Hammerstad E O, “Equations for microstrip circuits design”, Proc. 5th Eur. Microw. Conf., Hamburg, Germany, Sep. 1975, pp. 268–272. [5] Carver K R, “Practical analytical techniques for the microstrip antenna”, Proc. Workshop Printed Circuit Antenna Tech., New Mexico State Univ., Las Cruces, NM, Oct. 1979, pp. 7.1–7.20. [6] Bahl I J, and Bhartia P, “Microstrip Antennas”, MA: Artech House, 1980. [7] James J R, Hall P S, and Wood C, “Microstrip antennas-theory and design”, London: Peregrinus, 1981. [8] Sengupta D L, “Approximate expression for the resonant frequency of a rectangular patch antenna”, Electronics Letters, 1983, 19: 834-835. Particle-Swarm-Optimization-Based Selective Neural Network Ensemble and Its Application to Modeling Resonant Frequency of Microstrip Antenna 81 [9] Garg R, and Long S A, “Resonant frequency of electrically thick rectangular microstrip antennas”, Electronics Letters, 1987, 23: 1149-1151. [10] Chew W C, and Liu Q, “Resonance frequency of a rectangular microstrip patch”, IEEE Transactions on Antennas Propagation, 1988, 36: 1045-1056. [11] Guney K, “A new edge extension expression for the resonant frequency of electrically thick rectangular microstrip antennas”, Int. J. Electron., 1993, 75: 767-770. [12] Kara M, “Closed-form expressions for the resonant frequency of rectangular microstrip antenna elements with thick substrates”, Microwave and Optical Technology Letters,1996, 12: 131-136. [13] Guney K, “A new edge extension expression for the resonant frequency of rectangular microstrip antennas with thin and thick substrates”, J. Commun. Tech. Electron., 2004, 49: 49-53. [14] Zhang Q J, and Gupta K C, “Neural networks for RF and microwave design”, MA: Artech House, 2000. [15] Christodoulou C, and Georgiopoulos M, “Applications of Neural Networks in Electromagnetic”, MA: Artech House, 2001. [16] Guney K, Sagiroglu S, and Erler M, “Generalized neural method to determine resonant frequencies of various microstrip antennas”, International Journal of RF and Microwave Computer-Aided Engineering, 2002, 12(1): 131-139. [17] Sagiroglu S, and Kalinli A, “Determining resonant frequencies of various microstrip antennas within a single neural model trained using parallel tabu search algorithm”, Electromagnetics, 2005, 25(6): 551-565. [18] Kara M, “The resonant frequency of rectangular microstrip antenna elements with various substrate thicknesses”, Microwave and Optical Technology Letters, 1996, 11: 55-59. [19] Hansen L K, and Salamon P, “Neural network ensembles”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(10): 993-1001. [20] Kennedy J, and Eberhart R C, “Particle Swarm Optimization”, IEEE International Conference on Neural Networks, Piscataway, NJ: IEEE Press, 1995, 1942-1948. [21] Zeng J C, Jie J, and Cui Z H, “Particle swarm optimization”, Beijing: Science Press, 2004. [22] Clerc M, “Particle Swarm Optimization”, ISTE Publishing Company, 2006. [23] R. Poli. Analysis of the publications on the applications of particle swarm optimization. Journal of Artificial Evolution and Applications, 2008, Article No. 4. [24] Robinson J, and Rahmat-Samii Y, “Particle swarm optimization in electromagnetics”, IEEE Transactions on Antennas and Propagation, 2004, 52(2): 397-407. [25] Mussetta M, Selleri S, Pirinoli P, et al., “Improved Particle Swarm Optimization algorithms for electromagnetic optimization”, Journal of Intelligent and Fuzzy Systems, 2008, 19(1): 75-84. [26] M. T. Hagan, H. B. Demuth, and M. H. Beale, Neural Network Design, Boston: PWS Pub. Co., 1995. [27] S. Haykin. Neural Networks: A Comprehensive Foundation (2nd Edition), Prentice Hall, 1999. [28] Y. B. Tian, Hybrid neural network techniques, Beijing: Science Press, 2010. [29] Merz C J, and Pazzani M J, “A principal components approach to combining regression estimates”, Machine Learning, 1999, 36 (1-2): 9-32. Microstrip Antennas 82 [30] Hashem S, “Treating harmful collinearity in neural network ensembles”, In: Sharkey A J C, ed. Combining artificial neural nets: Ensemble and modular multi2net systems, Great Britain: Springer-Verlag London Limited, 1999. 101-123. [31] Zhou Zhihua, Wu Jianxin, and Tang Wei, “Ensembling neural networks: Many could be better than all”, Artificial Intelligence, 2002, 137 (1-2): 239-263. [32] Dietterich T G, “An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting, and randomization”, Machine Learning, 2000, 40: 139-157. [33] Kennedy J, and Eberhart R, “A discrete binary version of the particle swarm optimization”, Proceedings IEEE International Conference on Computational Cybernetics and Simulation. Piscataway, NJ: IEEE, 1997: 4104-4108. [34] Huang R S, and Huang H, “Chaos and its applications”, China Wuhan: Wuhan University press, 2005. [35] Zhang L P, “Theory and applications of particle swarm optimization”, Ph. D. dissertation, Zhejiang University, China, 2005. [36] Shi Y, and Eberhart R, “Empirical study of particle swarm optimization”, Proceedings of the 1999 Congress on Evolutionary Computation, 1999: 1945-1950. 5 Microstrip Antennas Conformed onto Spherical Surfaces Daniel B. Ferreira and J. C. da S. Lacava Technological Institute of Aeronautics Brazil 1. Introduction Microstrip antennas are customary components in modern communications systems, since they are low-profile, low-weight, low-cost, and well suited for integration with microwave circuits. Antennas printed on planar surfaces or conformed onto cylindrical bodies have been discussed in many publications, being the subject of a variety of analytical and numerical methods developed for their investigation (Josefsson & Persson, 2006; Garg et al., 2001; Wong, 1999). However, such is not the case of spherical microstrip antennas and arrays composed of these radiators. Even commercial electromagnetic software, like HFSS ® and CST ® , do not provide a tool to assist the design of spherical antennas and arrays, i.e., electromagnetic simulators do not have an estimator tool for establishing the initial dimensions of a spherical microstrip antenna for further numerical analysis, as available for planar geometries. Moreover, this software is time-consuming when utilized to simulate spherical radiators, hence it is desirable that the antenna geometry to be analyzed is not too far off from the final optimized one, otherwise the project cost will likely be affected. Nonetheless, spherical microstrip antenna arrays have a great practical interest because they can direct a beam in an arbitrary direction throughout the space, i.e., without limiting the scan angles, differently from the planar antenna behaviour. This characteristic makes them feasible for use in communication satellites and telemetry (Sipus et al., 2006), for example. Rigorous analysis of spherical microstrip antennas and their respective arrays has been conducted through the Method of Moments (MoM) (Tam et al., 1995; Wong, 1999; Sipus et al., 2006). But the MoM involves highly complex and time-consuming calculations. On the other hand, whenever the objective is the analysis of spherical thin radiators, the cavity model (Lima et al., 1991) can be applied, instead of the MoM. However, for both MoM and cavity model, the behaviour of the antenna input impedance and radiated electric field is described by the associated Legendre functions, hence efficient numerical routines for their evaluation are required, otherwise the scope of the antennas analyzed is restricted. In order to overcome the drawbacks described above, a Mathematica ® -based CAD software capable of performing the analysis and synthesis of spherical-annular and -circular thin microstrip antennas and their respective arrays with high computational efficiency is presented in this chapter. It is worth mentioning that the theoretical model utilized in the CAD can be extended to other canonical spherical patch geometries such as rectangular or triangular ones. The Mathematica ® package, an integrated scientifical computing software with a vast collection of built-in functions, was chosen mainly due to its powerful Microstrip Antennas 84 algorithms for calculating cylindrical and spherical harmonics functions what makes it suitable for the analysis of conformed antennas. Mathematica ® permits the analysis and synthesis of various spherical microstrip radiators, thus avoiding the use of the normalized Legendre functions that are sometimes employed to overcome numerical difficulties (Sipus et al., 2006). Furthermore, it is important to point out that the developed CAD does not require a powerful computer to run on, working well and quickly in a regular classroom PC, since its code does not utilize complex numerical techniques, like MoM or finite element method (FEM). In Section 2, the theoretical model implemented in the developed CAD to evaluate the antenna input impedance, quality factor, radiation pattern and directivity is discussed. Furthermore, comparisons between the CAD results and the HFSS ® full wave solver data are presented in order to validate the accuracy of the utilized technique. An effective procedure, based on the global coordinate system technique (Sengupta, 1968), to determine the radiation patterns of thin spherical meridian and circumferential arrays is utilized in the special-purpose CAD, as addressed in Section 3. The array radiation patterns so obtained with the CAD are also compared to those from the HFSS ® software. Section 4 is devoted to present an alternative strategy for fabricating a low-cost spherical-circular microstrip antenna along with the respective experimental results supporting the proposed antenna fabrication approach. 2. Analysis and synthesis of spherical thin microstrip antennas The geometry of a probe-fed spherical-annular microstrip antenna embedded in free space (electric permittivity ε 0 and magnetic permeability μ 0 ) is shown in Fig. 1. It is composed of a metallic sphere of radius a, called ground sphere, covered by a dielectric layer (ε and μ 0 ) of thickness h = b – a. z a x y Metallic sphere Dielectric layer Probe position h Annular patch 1 θ p θ 2 θ b z a x y Metallic sphere Dielectric layer Probe position h Annular patch 1 θ p θ 2 θ b Fig. 1. Geometry of a probe-fed spherical-annular microstrip antenna. A symmetrical annular metallic patch, defined by the angles θ 1 and θ 2 (θ 2 > θ 1 > 0), is fed by a coaxial probe positioned at (θ p , φ p ). The radiators treated in this chapter are electrically thin, i.e., h << λ (λ is the wavelength in the dielectric layer), so the cavity model (Lo et al., Microstrip Antennas Conformed onto Spherical Surfaces 85 1979) is well suited for the analysis of such antennas. Based on this model it is possible to develop expressions for computing the antenna input impedance and for estimating the electric surface current density on the patch without employing any complex numerical method such as the MoM. Before starting the input impedance calculation, the expression for computing the resonant frequencies of the modes established in a lossless equivalent cavity is determined. In the case of electrically thin radiators, the electric field within the cavity can be considered to have a radial component only, which is r-independent. Therefore, applying Maxwell’s equations to the dielectric layer region, and disregarding the feeder presence, the following equation for the r-component of the electric field is obtained 2 2 2222 11 sin 0 sin sin rr r EE kE aa ∂∂∂ ⎛⎞ θ ++= ⎜⎟ ∂θ ∂θ θθ∂φ ⎝⎠ , (1) where k 2 = ω 2 μ 0 ε and ω denotes the angular frequency. Consequently, only TM r modes can be established in the equivalent cavity. Solving the wave equation (1) via the method of separation of variables (Balanis, 1989), results in the electric field ( , ) [ P (cos ) Q (cos )][ cos( ) sin( )] θ φ= θ+ θ φ+ φ AA mm r EA B CmDm, (2) where P(.) A m and Q(.) A m are the associated Legendre functions of ℓ-th degree and m-th order of the first and the second kinds, respectively, ℓ(ℓ +1) = k 2 a 2 and A, B, C and D are constants dependent on the boundary conditions. Enforcing the boundary conditions related to the equivalent cavity of annular geometry and taking into account that it is symmetrical in relation to the z-axis, the electric field (2) reduces to (,) R(cos)cos( ) θ φ= θ φ AA m r m EE m, (3) where 11 1 R (cos ) sin [P (cos )Q (cos ) Q (cos )P (cos )] mmmmm cc c ′′ θ =θ θ θ− θ θ AAAAA , (4) m = 0, 1, 2,… and the index ℓ must satisfy the transcendental equation 12 12 P (cos )Q (cos ) Q (cos )P (cos ) 0 mm mm cc cc ′′ ′′ θ θ− θ θ= AA AA , (5) with the angles θ 1c and θ 2c (θ 2c > θ 1c ) indicating the equivalent cavity borders in the θ direction, i.e., θ 1c ≤ θ ≤ θ 2c , 0 ≤ φ < 2π, E ℓm are the coefficients of the natural modes and the prime denotes a derivative. Hence, once the indexes ℓ and m are determined it is possible to evaluate the TM ℓm mode resonant frequency from the following expression 0 (1) 2 m f a + = π με A AA . (6) Before solving the transcendental equation (5) it is necessary to determine the equivalent cavity dimensions θ 1c and θ 2c , which correspond to the actual patch dimensions with the Microstrip Antennas 86 addition of the fringe field extension. However, differently from planar microstrip antennas, the literature does not currently present expressions for estimating the dimensions of spherical equivalent cavities based on the physical antenna dimensions and the dielectric substrate characteristics. Therefore, in this chapter, the expressions used for estimating the equivalent cavity dimensions of a planar-annular microstrip antenna are extended to the spherical-annular case (Kishk, 1993), i.e., the spherical-annular equivalent cavity arc lengths were considered equal to the respective linear dimensions of the planar-annular equivalent cavity. The proposed expressions are given below; equations (7.a) and (7.b), 1 11 1 2F( ) 1 c r h b θ θ=θ − π θε , (7.a) 2 22 2 2F( ) 1 c r h b θ θ=θ + π θε , (7.b) where F( ) n( /2 ) 1.41 1.77 (0.268 1.65) / rr bh h bθ= θ + ε+ + ε+ θA and ε r is the relative electric permittivity of the dielectric substrate. 2.1 Input impedance The input impedance of the spherical-annular microstrip antenna illustrated in Fig. 1 fed by a coaxial probe can be evaluated following the procedure proposed in (Richards et al., 1981), i.e., the coaxial probe is modelled by a strip of current whose electric current density is given by, () 2 1 ˆ ,()() sin p f p JJr a θφ = φδθ−θ θ G , (8) where δ(.) indicates the Dirac’s delta function and 0 ,if /2 /2 () 0, otherwise pp J J φ −Δφ ≤φ≤φ +Δφ ⎧ ⎪ φ= ⎨ ⎪ ⎩ (9) with Δφ denoting the strip angular length relative to the φ−direction. In our analysis, also following the procedure established in (Richards et al., 1981) for planar microstrip antennas, the strip arc length has been assumed to be five times the coaxial probe diameter d, expressed as 5/sin p da Δ φ= θ . (10) It is important to point out that the electric current density (8) is an r-independent function since the antenna under analysis is electrically thin. Thus, to take into account the current strip, the wave equation (1) for the electric field is modified to () 2 2 0 2222 11 ˆ sin , sin sin rr rf EE kE j J r aa ∂∂∂ ⎛⎞ θ ++=ωμθφ⋅ ⎜⎟ ∂θ ∂θ θθ∂φ ⎝⎠ G . (11) Expanding the r-component of the electric field into its eigenmodes (3), the solution for equation (11) is given by Microstrip Antennas Conformed onto Spherical Surfaces 87 1 00 2cos 22 2 cos R (cos )sinc( /2)cos( ) ( , ) R (cos )cos( ) [][R()] c c m p p m r m m m m mm J Ej m a kk d 2 θ ν= θ θΔφ φ ωμ Δφ θφ = θ φ π ξ− νν ∑∑ ∫ A A A AA , (12) where 2, if 0 1, otherwise m m = ⎧ ξ= ⎨ ⎩ , (1)/ m ka=+ A AA and sinc(x)=sin(x)/x. Since the procedure just described has been developed for ideal cavities, equation (12) is purely imaginary. So, for incorporating the radiated power and the dielectric and metallic losses into the cavity model, the concept of effective loss tangent, tan δ eff , (Richards et al., 1979) is employed. Based on this approach, the wave number k is replaced by an effective wave number 1tan eff eff kk j = −δ. (13) Once the electric field inside the equivalent cavity has been determined, the antenna input voltage V in can be computed from the expression, in r VEh=− , (14) where r E denotes the average value of (,) p r E θ φ over the strip of current. Consequently, the input impedance Z in of the spherical-annular microstrip antenna is given by 1 22 2 0 2cos 22 2 0 cos [R (cos )] sinc ( /2)cos ( ) [ (1 tan )] [R ( )] c c m p p in in m m meff m mm Vh Zj J a kk j d 2 θ ν= θ θΔφφ ωμ == Δφ π ξ −−δ νν ∑∑ ∫ A A AA . (15) An alternative representation for frequencies close to the TM LM resonant mode but sufficiently apart from the other modes can be obtained by rewriting the antenna input impedance as 2222 (, )(, ) (1 tan ) LM m in LM eff m m mLM j Zj j ≠ α ωα ≅+ω ω −− δω ω−ω ∑ ∑ A A A A , (16) where 1 cos 22 2 2 2 cos [R (cos )] sinc ( / 2)cos ( ) / [R ( )] . c c m m p pm m hmma d 2 θ ν= θ α= θ Δφ φ πεξ ν ν ∫ AA A The expression (16) corresponds to the equivalent circuit shown in Fig. 2, i.e., a parallel RLC circuit with a series inductance L p . In this case, the series inductance represents the probe effect and its value is that of the double sum in (16). However, as this is a slowly convergent series, the developed CAD utilizes, alternatively, the equation due to (Damiano & Papiernik, 1994) for calculating the probe reactance X p , given by 0 60 n( /2) p Xjkhkd = − A , (17) Microstrip Antennas 88 where 000 k =ω μ ε and provided 0 0.2kd < < . in Z p L R L C in Z p L R L C p L R L C Fig. 2. Simplified equivalent circuit for thin microstrip antennas. The previous expressions developed for computing the resonant frequencies and the input impedance of spherical-annular microstrip antennas can also be used for analysing wraparound radiators. However, in the limit case when θ 1 → 0, i.e., the antenna patch corresponding to a circular one (Fig. 3), the associated Legendre function of the second kind becomes unbounded for θ → 0, so it is no longer part of the function that describes the electromagnetic field within the equivalent cavity. So, to obtain the expressions for spherical-circular microstrip antennas it is enough to eliminate the Legendre function of the second kind from the previously developed solution for spherical-annular antennas. These expressions are presented in Table 1. In Section 2.3 examples are given for spherical-annular and -circular microstrip antennas. Resonant frequency 0 (1) 2 m f a + = π με A AA , the index ℓ is obtained from P 2 (cos ) 0 m c ′ θ= A Input impedance 1 22 2 0 2cos 22 2 cos [P (cos )] sinc ( / 2)cos ( ) [ (1 tan )] [P ( )] c c m p p in m m meff m mm h Zj a kk j d 2 θ ν= θ θΔφφ ωμ = π ξ −−δ νν ∑∑ ∫ A A AA Table 1. Spherical-circular microstrip antenna expressions. z a Metallic sphere Dielectric layer Probe position h Circular patch 2 θ b x y p θ z a Metallic sphere Dielectric layer Probe position h Circular patch 2 θ b x y p θ Fig. 3. Geometry of a probe-fed spherical-circular microstrip antenna. [...]... WileyInterscience, ISBN: 0 -47 1 -46 5 84- 4, New Jersey Kishk, A A (1993) Analysis of spherical annular microstrip antennas IEEE Transactions on Antennas and Propagation, Vol 41 , No 3, pp 338- 343 , ISSN: 0018-926X Lima, A C C.; Descardeci, J R & Giarola, A J (1991) Microstrip antenna on a spherical surface, Proceedings of Antennas and Propagation Society International Symposium, pp 820-823, ISBN: 0-7803-0 144 -7, London,... manufacture spherical conformal microstrip antennas, Proceedings of Antennas and Propagation Society International Symposium, pp 3525-3528, ISBN: 0-7803-8302-8, Monterey, June 20 04, IEEE, New York 108 Microstrip Antennas Richards, W F.; Lo, Y T & Harrison, D D (1979) Improved theory for microstrip antennas Electronics Letters, Vol 15, No 2, pp 42 -44 , ISSN: 0013-51 94 Richards, W F.; Lo, Y T & Harrison,... Wiley & Sons, ISBN: 0 -47 1621 94- 3, New York Balanis, C A (2005) Antenna theory: analysis and design, Wiley-Interscience, ISBN: 0 -47 166782-X, New York Damiano, J P & Papiernik, A (19 94) Survey of analytical and numerical models for probefed microstrip antennas IEE Proceedings Microwaves, Antennas and Propagation, Vol 141 , No 1, pp 15-22, ISSN: 1350- 241 7 Ferreira, D B (2009) Microstrip antennas conformed... spherical arrays of microstrip antennas using moment method in spectral domain IEE Proceedings Microwaves, Antennas and Propagation, Vol 153, No 6, pp 533- 543 , ISSN: 1350- 241 7 Sipus, Z.; Skokic, S.; Bosiljevac, M & Burum, N (2008) Study of mutual coupling between circular stacked-patch antennas on a sphere IEEE Transactions on Antennas and Propagation, Vol 56, No 7, pp 18 34- 1 844 , ISSN: 0018-926X Tam,... spherical-circular microstrip antennas by electric surface current models IEE Proceedings H Microwaves, Antennas and Propagation, Vol 138, No 1, pp 98-102, ISSN: 0950-107X Tam, W Y.; Lai, A K Y & Luk, K M (1995) Input impedance of spherical microstrip antenna IEE Proceedings Microwaves, Antennas and Propagation, Vol 142 , No 3, pp 285-288, ISSN: 1350- 241 7 Wong, K –L (1999) Design of Nonplanar Microstrip Antennas. .. Ittipiboon, A (2001) Microstrip Antenna Design Handbook, Artech House, ISBN: 0-89006-513-6, Massachusetts Giang, T V B.; Thiel, M & Dreher, A (2005) A unified approach to the analysis of radial waveguides, dielectric resonators, and microstrip antennas on spherical multilayer structures IEEE Transactions on Microwave Theory and Techniques, Vol 53, No 1, pp 40 4 -40 9, ISSN: 0018- 948 0 Josefsson, L & Persson,... of the far electric field radiated by spherical microstrip antenna arrays z α α N 1 2 βN β1 x Fig 20 Circumferential-spherical array y 103 Microstrip Antennas Conformed onto Spherical Surfaces z 3 4 2 α 1 h a y x Fig 21 Four-element circumferential array 0° [dB] 0 330° CAD HFSS 30° -10 60° CAD HFSS 30° 60° 300° -20 -20 270° 90° -30 90° 270° -20 -20 240 ° 240 ° 120° 120° -10 -10 0 330° -10 300° -30 0° [dB]... experiment on microstrip antennas IEEE Transactions on Antennas and Propagation, Vol 27, No 2, pp 137- 145 , ISSN: 0018-926X Olver, F W J (1972) Bessel functions of integer order, In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz, M & Stegun, I A., pp 355-389, Dover Publications, ISBN: 978-0 -48 6-61272-0, New York Piper, B R & Bialkowski, M E (20 04) Modelling... 1.01 1.02 Normalized frequency Fig 24 Input impedance 0° 0° [dB] 0 330° CAD HFSS 30° [dB] 300° CAD HFSS 30° 300° 60° 60° -20 -20 270° -30 90° -20 270° 90° -20 240 ° 240 ° 120° -10 0 330° -10 -10 -30 0 120° -10 210° 150° 210° 0 180° 150° 180° Eφ radiation pattern: yz plane Eθ radiation pattern: xz plane Fig 25 Radiation patterns rF rp rd 17.20 4. 68 20 .47 1.3% 6.8% 0% Table 4 Final dimensions in mm and percent... the TM11 mode of a spherical-circular cavity which is employed by the CAD is: 11 ( θ2 c ) 3 4 = 54. 46 − 11.06 θ2 c + 1.13 θ2 c − 6.21 × 10 −2 θ2 c + 1.67 × 10 −3 θ2 c 2 5 − 5.36 × 10 −6 θ2 c − 8.71 × 10 −7 θ6 c + 2.07 × 10 −8 θ7 c − 1.53 × 10 −10 θ8 c , 2 2 2 where θ2c is given in degrees (46 ) 96 Microstrip Antennas To illustrate the CAD synthesis procedure, a spherical-circular antenna conformed onto . 7 940 744 2 7605 7733 7322 7763 7639 7551 7376 1610 7730 5 46 00 46 97 42 54 440 7 46 41 44 55 49 79 47 59 46 14 443 0 113 46 18 6 5060 5283 48 65 49 89 5070 47 41 5101 49 58 49 24 4797 1621 5077 7 48 05 50 14. 52 64 5063 42 39 58 04 4550 48 30 41 96 5976 60 64 23 46 60 47 68 3 746 42 27 41 41 3526 46 89 3770 3 949 347 9 46 00 46 13 24 4600 40 84 3201 38 24 3615 2938 42 09 3168 344 6 2921 46 03 45 50 25 3580 340 8 2668. 46 67 49 93 5092 44 07 5693 46 78 48 30 43 38 543 7 5193 17 8000 8698 6 845 7 546 7519 646 4 844 7 6889 7160 6367 8067 7 948 18 71 34 748 5 5870 6601 648 4 5525 7 342 59 04 6179 545 2 7 242 7169 19 6070 647 8