Microstrip Antennas Part 2 pptx

Design of Low-Cost Probe-Fed Microstrip Antennas 19 and a 75-mm square ground plane was designed using the HFSS software for operation at 1.603 GHz. The optimized antenna dimensions are shown in Fig. 28(a), the simulated input impedance and axial ratio results are presented in Fig. 28(b) and the reflection coefficient magnitude in Fig. 29. As expected, the microstrip antenna with the new geometry exhibits very good AR (0.1 dB) and reflection coefficient magnitude (-48 dB) characteristics at 1.603 GHz, without the need for any external matching network. 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 -30 -25 -20 -15 -10 -5 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Axial ratio [dB] | Γ| [dB] Frequency [GHz] |Γ| Axial ratio Fig. 29. CP truncated-corner patch: axial ratio and reflection coefficient magnitude. 5.2.2 Globalstar Antenna Moderately thick microstrip antennas can also be used for bandwidth improvement. To exemplify such application a prototype of a Globalstar antenna was manufactured using a low-loss substrate ( ε r = 2.55, tan δ = 0.0022 and h = 4.572 mm). Left-handed CP Globalstar mobile-terminals require two radiators. The first is designed for uplink frequencies (Tx - 1.61073 to 1.62549 GHz) while the other receives the downlink ones (Rx - 2.48439 to 2.49915 GHz) (Nascimento et al., 2007a). The antenna geometry and a photo of the prototype are shown in Figs. 30(a) and (b), respectively. The optimized antenna dimensions (using the HFSS software) are presented on Table 2 for the radiators designed on finite ground plane and dielectric ( L = 140 mm; W = 85 mm). Tx Rx L T1 54.90 mm L R1 34.40 mm L T2 55.90 mm L R2 35.85 mm C T 7.55 mm C R 5.75 mm P T 15.85 mm P R 9.00 mm D T 71.00 mm D R 15.00 mm Table 2. Globalstar antenna dimensions. Microstrip Antennas 20 x y T D 2 T L 1T L 1 R L 2 R L R D T C T C R C R C X R R P 2 1 / R L X T T P 2 1 / T L L W (a) (b) Fig. 30. Globalstar antenna: (a) geometry - (b) photo of the prototype. The axial ratio and reflection coefficient magnitude are presented in Figs. 31 and 32 for the Tx and Rx antennas, respectively. 1.59 1.60 1.61 1.62 1.63 1.64 1.65 0 1 2 3 4 5 6 7 8 9 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 |Γ| [dB] Axial ratio [dB] Fre q uenc y [GHz] Axial ratio |Γ| Fig. 31. Globalstar antenna axial ratio and reflection coefficient magnitude: Tx radiator. Design of Low-Cost Probe-Fed Microstrip Antennas 21 2.42 2.44 2.46 2.48 2.50 2.52 2.54 2.56 0 1 2 3 4 5 6 7 8 9 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 |Γ| [dB] Axial ratio [dB] Frequency [GHz] Axial ratio |Γ| Fig. 32. Globalstar antenna axial ratio and reflection coefficient magnitude: Rx radiator. Results for the input impedance on the Smith chart are presented in Figs. 33 and 34 for the Tx and Rx antennas, respectively. These results indicate that the antenna meets the Globalstar specifications. 10 25 50 100 250 -10j 10j -25j 25j -50j 50j -100j 100j -250j 250j Prototype HFSS 1.540 GHz 1.696 GHz 1.618 GHz Fig. 33. Globalstar antenna input impedance: Tx radiator. Microstrip Antennas 22 10 25 50 100 250 -10j 10j -25j 25j -50j 50j -100j 100j -250j 250j Prototype HFSS 2.492 GHz 2.210 GHz 2.774 GHz Fig. 34. Globalstar antenna input impedance: Rx radiator. 5.3 CP antenna radiation efficiency measurements The radiation efficiency of a LP microstrip antenna can be efficiently measured using the Wheeler cap (Choo et al., 2005; Pozar & Kaufman, 1988; Sona & Rahmat-Samii 2006). According to Wheeler, the radiation resistance of an antenna can be separated from its loss resistance by enclosing the antenna with a radiation shield cap placed at a distance greater than λ /(2π) (Wheeler, 1959). Consequently, since a linearly-polarized microstrip antenna can be modeled as a parallel RLC circuit, its efficiency is calculated by out ca p out GG G η − = , (11) where G cap is the conductance of the admittance measured with the cap in place and G out is the conductance of the admittance measured with the cap removed. In the case of a CP microstrip antenna an innovative radiation efficiency analysis using the Wheeler cap method was presented in (Nascimento & Lacava, 2009). This procedure is discussed next, for the case of the Glonass antenna designed in Section 5.2.1. Differently from the standard design, the two orthogonal resonant modes in the new approach are now asymmetrically positioned in relation to the frequency for optimal axial ratio as presented in Fig. 28 (b). In addition, at the lower resonant frequency (1.468 GHz), its 15.45-dB axial ratio shows the antenna tends to be linearly polarized around this frequency. This result supports the use of the Wheeler cap method for measuring the antenna radiation efficiency at this frequency. The cap geometry is shown in Fig. 35 where the radiator is positioned inside a cubic cavity of electrically conducting walls of 270-mm internal dimension. Design of Low-Cost Probe-Fed Microstrip Antennas 23 Fig. 35. Geometry of the Wheeler cap simulation through the HFSS package. HFSS simulation results for the real part of the input impedance are presented in Fig. 36, both with and without the cubic cavity. Making use of equation (11) for the lower resonant mode ( G cap = 1.92 mS and G out = 7.43 mS), the radiation efficiency computed from the Wheeler method is 74.16%. The free-space radiation efficiency, computed with the HFSS package is 74.68% at 1.468 GHz, which is only 0.7% off. Consequently, the Wheeler cap method can be used for accurately determining the radiation efficiency of TCRP radiators. 1.43 1.48 1.53 1.58 1.63 1.68 0 60 120 180 240 300 360 420 480 540 Antenna in free space Antenna inside the cap Real part of Z in [Ω] Fre q uenc y [GHz] Fig. 36. Low-cost Glonass antenna: real part of its input impedance. 6. Conclusion In this chapter, new effective strategies for designing probe-fed moderately thick LP and CP microstrip antennas were presented. As the design procedures do not make use of external networks, the antenna construction process is considerably simplified. Besides, as the new Microstrip Antennas 24 methodologies are based on properties of the antenna equivalent circuit, they can be applied to the design of microstrip radiators of arbitrary patch shapes. Moreover, it is not restricted to low-cost substrate thus applying equally well to the design of LP or CP microstrip patches printed on any moderately thick commercial microwave laminates. Experimental results for LP and CP radiators validate the design strategies for both the LP and CP cases. Moreover, the Wheeler cap method is shown to be an effective means for simulating the radiation efficiency of CP microstrip antennas. The excellent practical results obtained when matching microstrip patch radiators to a 50- Ω SMA connector can be readily extended to the synthesis of inductive or capacitive input impedances, as for example in the case of optimization of the noise figure and stability of low-noise power amplifiers connected directly to the antenna. Another possible application is the design of low-cross-polarization probe-fed microstrip arrays (Marzall, et al., 2009; Marzall et al., 2010). 7. References Alexander, M. J. (1989). Capacitive matching of microstrip antennas. IEE Proceedings of Microwaves, Antennas and Propagation , Vol. 137, No. 2, (Apr. 1989) (172-174), ISSN: 0950-107X. Chang, F. S. & Wong, K. L. (2001), A broadband probe-fed patch antenna with a thickened probe pin, Proceedings of Asia-Pacific Microwave Conference, (1247-1250), ISBN: 0- 7803-7138-0, Taipei, China, Dec. 2001 Chen, H. M.; Lin, Y. F.; Cheng, P. S.; Lin, H. H.; Song, C. T. P. & Hall, P. S. (2005), Parametric study on the characteristics of planar inverted-F antenna. IEE Proceedings of Microwaves, Antennas and Propagation , (Dec. 2005) (534-538), ISSN: 1350-2417. Choo, H.; Rogers, R. & Ling, H. (2005), Comparison of three methods for the measurement of printed antennas efficiency, IEEE Transactions on Antennas and Propagation, Vol. 53, No. 7, (Jul. 2005) (2328-2332), ISSN: 0018-926X. Dahele, J. S.; Hall, P. S. & Haskins, P. M. (1989), Microstrip patch antennas on thick substrates, Proceedings of Antennas and Propagation Society International Symposium, pp. 458-462, San Jose, CA, USA, Jun. 1989. Engest, B. & Lo, Y. T. (1985), A study of circularly polarized rectangular microstrip antennas, Technical Report, Electromagnetics Laboratory, University of Illinois. Gardelli, R.; La Cono, G. & Albani, M. (2004), A low-cost suspended patch antenna for WLAN access points and point-to-point links, IEEE Antennas and Wireless Propagation Letters, Vol. 3, (2004) (90-93), ISSN: 1536-1225. Garg, R.; Bhartia, P.; Bahl, I. & Ittipiboon, A. (2001). Microstrip Antenna Design Handbook, Artech House, ISBN: 0-89006-513-6, Boston. Hall, P. S. (1987). Probe compensation in thick microstrip patches. Electronics Letters, Vol. 23, No. 11, (May 1987) (606-607), ISSN: 0013-5194. Haskins, P. M. & Dahele, J. S. (1998), Capacitive coupling to patch antenna by means of modified coaxial connectors, Electronics Letters, Vol. 34, No. 23, (Nov. 1998) (2187- 2188), ISSN: 0013-5194. HFSS (2010), Product overview, Available: http://www.ansoft.com/products/hf/hfss/, (Sept. 2010). Design of Low-Cost Probe-Fed Microstrip Antennas 25 IEEE Std 145 (1993). IEEE Standard Definitions of Terms for Antennas, ISBN: 1-55937-317-2, New York, USA. James, J. R. & Hall, P. S. (1989). Handbook of Microstrip Antennas, Peter Peregrinus, ISBN: 0- 86341-150-9, London. Lee , K. F. & Chen, W. (1997). Advances in Microstrip and Printed Antennas, John Wiley, ISBN: 0-471-04421-0, New York. Lumini, F.; Cividanes, L. & Lacava, J. C. S. (1999), Computer aided design algorithm for singly fed circularly polarized rectangular microstrip patch antennas, International Journal of RF and Microwave Computer-Aided Engineering, Vol. 9, No. 1, (Jan. 1999) (32-41), ISBN: 1096-4290. Marzall, L. F., Schildberg, R. & Lacava, J. C. S. (2009), High-performance, low-cross- polarization suspended patch array for WLAN applications, Proceedings of Antennas and Propagation Society International Symposium , pp. 1-4, ISBN: 978-1-4244-3647-7, Charleston, SC, USA, June 2009. Marzall, L. F., Nascimento D.C., Schildberg, R. & Lacava, J. C. S. (2010), An effective strategy for designing probe-fed linearly-polarized thick microstrip arrays with symmetrical return loss bandwidth, PIERS Online, Vol. 6, No. 8, (July 2010) (700-704), ISSN: 1931-7360. Nascimento, D. C.; Mores Jr., J.A.; Schildberg, R. & Lacava, J. C. S. (2006), Low-cost truncated corner microstrip antenna for GPS application, Proceedings of Antennas and Propagation Society International Symposium , pp. 1557-1560, ISBN: 1-4244-0123-2, Albuquerque, NM, USA, July 2006. Nascimento, D. C.; Bianchi, I.; Schildberg, R. & Lacava, J. C. S. (2007a), Design of probe-fed truncated corner microstrip antennas for Globalstar system, Proceedings of Antennas and Propagation Society International Symposium , pp. 3041-3044, ISBN: 978-1-4244- 0877-1, Honolulu, HI, USA, June 2007. Nascimento, D. C.; Schildberg, R. & Lacava, J. C. S. (2007b), New considerations in the design of low-cost probe-fed truncated corner microstrip antennas for GPS applications, Proceedings of Antennas and Propagation Society International Symposium, pp. 749-752, ISBN: 978-1-4244-0877-1, Honolulu, HI, USA, June 2007. Nascimento, D. C.; Schildberg, R. & Lacava, J. C. S. (2008). Design of low-cost microstrip antennas for Glonass applications. PIERS Online, Vol. 4, No. 7, (2008) (767-770), ISSN: 1931-7360. Nascimento, D. C. & Lacava, J. C. S. (2009), Circularly-polarized microstrip antenna radiation efficiency simulation based on the Wheeler cap method, Proceedings of Antennas and Propagation Society International Symposium , pp. 1-4, ISBN: 978-1-4244- 3647-7, Charleston, SC, USA, June 2009. Niroojazi, M. & Azarmanesh, M. N. (2004), Practical design of single feed truncated corner microstrip antenna, Proceedings of Second Annual Conference on Communication Networks and Services Research, 2004 , pp. 25-29, ISBN: 0-7695-2096-0, Fredericton, NB, Canada, May 2004. Pozar, D. M. & Kaufman, B. (1988), Comparison of three methods for the measurement of printed antennas efficiency, IEEE Transactions on Antennas and Propagation, Vol. 36, No. 1, (Jan. 1988) (136-139), ISSN: 0018-926X. Microstrip Antennas 26 Richards, W. F.; Lo, Y. T. & Harrison, D. D. (1981), An improved theory for microstrip antennas and applications, IEEE Transactions on Antennas and Propagation, Vol. 29, No 1, (Jan. 1981) (38-46), ISSN: 0018-926X. Sona, K. S. & Rahmat-Samii, Y. (2006), On the implementation of Wheeler cap method in FDTD, Proceedings of Antennas and Propagation Society International Symposium, pp. 1445-1448, ISBN: 1-4244-0123-2, Albuquerque, NM, USA, July 2006. Teng, P. L.; Tang, C. L. & Wong, K. L. (2001), A broadband planar patch antenna fed by a short probe feed, Proceedings of Asia-Pacific Microwave Conference, pp. 1243-1246, ISBN: 0-7803-7138-0, Taipei, China, Dec. 2001. Tinoco S., A. F.; Nascimento, D. C. & Lacava, J. C. S. (2008), Rectangular microstrip antenna design suitable for undergraduate courses, Proceedings of Antennas and Propagation Society International Symposium , pp. 1-4, ISBN: 978-1-4244-2041-4, San Diego, CA, USA, July 2008. Tzeng, Y. B.; Su, C. W. & Lee, C. H. (2005), Study of broadband CP patch antenna with its ground plane having an elevated portion, Proceedings of Asia-Pacific Microwave Conference , pp. 1-4, ISBN: 0-7803-9433-X, Suzhou, China, Dec. 2005 Vandenbosch, G. A. E. & Van de Capelle, A. R. (1994), Study of the capacitively fed microstrip antenna element, IEEE Transactions on Antennas and Propagation, Vol. 42, No. 12, (Dec. 1994) (1648-1652), ISSN: 0018-926X. Volakis, J. L. (2007). Antenna Engineering Handbook. 4th ed., McGraw-Hill, ISBN: 0-07-147574- 5, New York. Wheeler, H. A. (1959), The radiansphere around a small antenna, Proceedings of the IRE, Vol. 47, No. 8, (Aug. 1959) (1325-1331), ISSN: 0096-8390. 2 Analysis of a Rectangular Microstrip Antenna on a Uniaxial Substrate Amel Boufrioua Electronics Department University of Constantine, 25000 Constantine, Algeria 1. Introduction Over the past years microstrip resonators have been widely used in the range of microwave frequencies. In general these structures are poor radiators, but by proper design the radiation performance can be improved and these structures can be used as antenna elements (Damiano & Papiernik, 1994). In recent years microstrip patch antennas became one of the most popular antenna types for use in aerospace vehicles, telemetry and satellite communication. These antennas consist of a radiating metallic patch on one side of a thin, non conducting, supporting substrate panel with a ground plane on the other side of the panel. For the analysis and the design of microstrip antennas there have been several techniques developed (Damiano & Papiernik, 1994; Mirshekar-Syahkal, 1990). The spectral domain approach is extensively used in microstrip analysis and design (Mirshekar-Syahkal, 1990). In such an approach, the spectral dyadic Green’s function relates the tangential electric fields and currents at various conductor planes. It is found that the substrate permittivity is a very important factor to be determined in microstrip antenna designs. Moreover the study of anisotropic substrates is of interest, many practical substrates have a significant amount of anisotropy that can affect the performance of printed circuits and antennas, and thus accurate characterization and design must account for this effect (Bhartia et al. 1991). It is found that the use of such materials may have a beneficial effect on circuit or antenna (Bhartia et al. 1991; Pozar, 1987). For a rigorous solution to the problem of a rectangular microstrip antenna, which is the most widely used configuration because its shape readily allows theoretical analysis, Galerkin’s method is employed in the spectral domain with two sets of patch current expansions. One set is based on the complete set of orthogonal modes of the magnetic cavity, and the other employs Chebyshev polynomials with the proper edge condition for the patch currents (Tulintsef et al. 1991). This chapter describes spectral domain analyses of a rectangular microstrip patch antenna that contains isotropic or anisotropic substrates in which entire domain basis functions are used to model the patch current, we will present the effect of uniaxial anisotropy on the characterization of a rectangular microstrip patch antenna, also because there has been very little work on the scattering radar cross section of printed antennas in the literature, including the effect of a uniaxial anisotropic substrate, a number of results pertaining to this case will be presented in this chapter. Microstrip Antennas 28 2. Theory An accurate design of a rectangular patch antenna can be done by using the Galerkin procedure of the moment method (Pozar, 1987; Row & Wong, 1993; Wong et al., 1993). An integral equation can be formulated by using the Green’s function on a thick dielectric substrate to determine the electric field at any point. The patch is assumed to be located on a grounded dielectric slab of infinite extent, and the ground plane is assumed to be perfect electric conductor, the rectangular patch with length a and width b is embedded in a single substrate, which has a uniform thickness of h (see Fig. 1), all the dielectric materials are assumed to be nonmagnetic with permeability μ 0 . To simplify the analysis, the antenna feed will not be considered. The study of anisotropic substrates is of interest, however, the designers should, carefully check for the anisotropic effects in the substrate material with which they will work, and evaluate the effects of anisotropy. Fig. 1. Geometry of a rectangular microstrip antenna Anisotropy is defined as the substrate dielectric constant on the orientation of the applied electric field. Mathematically, the permittivity of an anisotropic substrate can be represented by a tensor or dyadic of this form (Bhartia et al., 1991) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = zzzyzx yzyyyx xzxyxx 0 εεε εεε εεε .εε (1) For a biaxially anisotropic substrate the permittivity is given by ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = z y x 0 ε00 0ε0 00ε .εε (2) h x y z zx ε,ε a b 0 a. Plan view y x z h zx ε,ε Radiatin g conductor 0 a b b. Cross sectional view Ground plan [...]... results 2. 32 2. 32 1 4. 123 4. 121 4.64 2. 32 2 4.0 42 4.041 2. 32 1.16 2 5.476 6.451 1.16 2. 32 0.5 4.174 4.171 2. 32 4.64 0.5 3.0 32 3. 028 Table 2 Dependence of resonant frequency on relative permittivity ( ε x , ε z ) The anisortopic ratio AR = εx εz 37 Analysis of a Rectangular Microstrip Antenna on a Uniaxial Substrate 5 0 Radar cross section, (dB) -5 -10 -15 -20 -25 -30 -35 -40 0 10 (a) ε z changed, 20 30... cos ⎢ 2 ⎜ y + ⎟⎥ 2 ⎠⎦ 2 ⎠⎦ ⎣ a ⎝ ⎣ b ⎝ (31a) ⎡m π ⎛ ⎡m π ⎛ a ⎞⎤ b ⎞⎤ J ym (rs ) = cos ⎢ 1 ⎜ x + ⎟⎥ sin ⎢ 2 ⎜ y + ⎟⎥ a ⎝ 2 ⎠⎦ b ⎝ 2 ⎠⎦ ⎣ ⎣ (31b) The Fourier transforms of Jxn and Jym are obtained from equation (27 ) and given by ~ Jxn (k s ) = ~ Jym (k s ) = b /2 a /2 ⎛n π⎛ ⎛n π⎛ b ⎞⎞ a ⎞⎞ − ik y dy e cos⎜ 2 ⎜ y + ⎟ ⎟ dx e −ik x sin⎜ 1 ⎜ x + ⎟ ⎟ × ⎜ a ⎟ ⎜ b 2 ⎠⎟ 2 ⎠ ⎠ −b /2 ⎝ ⎝ ⎝ ⎝ ⎠ −a /2 ∫ ∫ x a /2 y b /2 ⎛m... Resonant frequency Ghz Band width BW % Quality factor Q isotropic 2. 35 2. 35 8.6360194 9.0536891 11.04 522 13 isotropic 7.0 7.0 5 .22 53631 3.1806887 31.4397311 positive 1.88 2. 35 8. 724 1 626 9.1377564 10.9436053 negative 2. 82 2.35 8.5537694 8.9779555 11.1383933 negative 8.4 7.0 5.1688307 3.1550166 31.6955535 positive 5.6 7.0 5 .28 69433 3 .21 24019 31. 129 3545 Table 1 Resonant frequency, band width and quality factor... frequency From the above equations and after some algebraic manipulation, the wave equations for E z and Hz are respectively ∂ 2 Ez ∂ 2 Ez εx ∂ 2 Ez 2 + + + εz k 0 Ez = 0 ∂ x2 ∂ y2 εz ∂ z2 (9) 30 Microstrip Antennas ∂ 2 Hz ∂ 2 Hz εx ∂ 2 Hz 2 + + + εz k 0 Hz = 0 ∂ x2 ∂ y2 εz ∂ z2 (10) With k 0 propagation constant for free space, k 0 = ω ε 0 μ0 By assuming plane wave propagation of the form e ± i k x... 1 ⎡k e 0 ⎤ ⎛ ε 2 2 2 h 2 2 , k e = ⎜ ε x k 0 − x k s ⎟ and k z = (ε x k 0 − k s )2 kz = ⎢ z z h⎥ ⎟ ⎜ εz ⎠ ⎝ ⎣ 0 kz ⎦ e h k z and k z are respectively propagation constants for TM and TE waves in the uniaxial dielectric By eliminating the unknowns A and B, in the equations (21 ) and (22 ) we obtain the following matrix which combines the tangential field components on both sides z1 and z2 of the considered... π⎛ b ⎞⎞ a ⎞⎞ −ik y dx e −ik x cos⎜ 1 ⎜ x + ⎟ ⎟ × dy e sin⎜ 2 ⎜ y + ⎟ ⎟ ⎜ a ⎟ ⎜ b 2 ⎠ ⎠ −b /2 2 ⎠⎟ ⎝ ⎝ ⎝ ⎝ ⎠ −a /2 ∫ (32a) ∫ x y (32b) Since the chosen basis functions approximate the current on the patch very well for conventional microstrips, only one or two basis functions are used for each current component Using the equations ( 32. a) and ( 32. b), the integral equation describing the field E in the patch... Transactions on Antennas and Propagation, Vol 35, No 3, (March 1987), (24 5 -25 1) ISSN 0018- 926 X Pozar, D M (1987) Radiation and Scattering from a Microstrip Patch on a Uniaxial Substrate, IEEE Transactions on Antennas and Propagation, Vol 35, No 6, (June 1987), ( 613- 621 ), ISSN 0018- 926 X Pozar, D M & Voda, S M (1987) A Rigorous Analysis of a Microstripline Fed Patch Antenna, IEEE Transactions on Antennas and... discretized into the following matrix ⎡(Z 1 )N×N ⎢(Z ) ⎣ 3 M ×N (Z 2 )N×M ⎤ ⎡(a )N×1 ⎤ ⋅ =0 (Z 4 )M×M ⎥ ⎢(b)M×1 ⎥ ⎦ ⎦ ⎣ (33) ] (34a) Where the impedance matrix terms are ∞ (Z 1 )kn = ∫∫ dk s −∞ [ 1 2 e ~ ~ k x G + k 2 G h × Jxk (− k s ) Jxn (k s ) y 2 ks 34 Microstrip Antennas ∞ k xk y (Z 2 )km = ∫∫ dk s 2 ks −∞ ∞ (Z 3 )ln = ∫∫ dk s k xk y 2 ks −∞ ∞ (Z 4 )lm = ∫∫ dk s −∞ [G [G ] (34b) ] (34c) e ~ ~ − G... dyadic Green’s function (25 ) 32 Microstrip Antennas ⎡G e G=⎢ ⎣0 0 ⎤ ⎥ Gh ⎦ (26 ) G e , G h are given by Ge = 1 −k e k z sin (k z1 h ) z e iωε 0 ik z sin (k z1 h ) + ε x k z cos(k z1 h ) (26 a) 2 1 −k 0 sin (k z1 h ) h iωε 0 ik z sin (k z1 h ) + k z cos(k z1 h ) In the case of the isotroipc substrate Gh = cos(k z1 h ) μ0 ε 0 1 − i ε r k z cot(k z1 h ) k z1 Ge = Gh = ( (26 b) (26 c) ) μ0 1 ε 0 cos(k z1 h... resistive part of input impedance includes both the radiation and conduction loss The imaginary part is proportional to the difference of time averaged stored magnetic and electric energy outside the spherical volume of radius a This could be easily deduced from the following Poynting formulation, − 1 2 ∫ ( ExH S * 1 )•dA=j2ω ∫ ⎛⎜⎝ 4 μ H V 2 ( ) 1 1 1 2 - ε E ⎟ dV+ ∫ E•J * dV = I 2 Z(ω) 4 2V 2 ⎠ (1) . et al., 1999) Our results 2. 32 2. 32 1 4. 123 4. 121 4.64 2. 32 2 4.0 42 4.041 2. 32 1.16 2 5.476 6.451 1.16 2. 32 0.5 4.174 4.171 2. 32 4.64 0.5 3.0 32 3. 028 Table 2. Dependence of resonant frequency. z H are respectively 0kε zε ε yx 2 0z 2 2 z x 2 2 2 2 =+ ∂ ∂ + ∂ ∂ + ∂ ∂ z zzz E EEE (9) Microstrip Antennas 30 0kε zε ε yx 2 0z 2 2 z x 2 2 2 2 =+ ∂ ∂ + ∂ ∂ + ∂ ∂ z zzz H HHH (10). radiator. Design of Low-Cost Probe-Fed Microstrip Antennas 21 2. 42 2.44 2. 46 2. 48 2. 50 2. 52 2.54 2. 56 0 1 2 3 4 5 6 7 8 9 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 |Γ| [dB] Axial ratio [dB] Frequency