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Resonant Frequencies in the Open Microstrip Structures Placed on Curved Surfaces Rafal Lech and Adam Kusiek Gdansk University of Technology Faculty of Electronics, Telecommunications and Informatics 80-233 Gdansk, Poland Abstract—The paper presents the research on open microstrip structures placed on curved surfaces such as cylindrical, elliptical or spherical. The numerical analysis of investigated structures is based on expansion of electric and magnetic fields into suitable function series. Utilizing the continuity conditions the boundary problem is formulated which is solved with the use of method of moments. The investigated structures find application in antenna technique for the construction of conformal antennas. Index Terms—Microstrip structure, conformal antennas, resonant structure, method of moments. I. I NTRODUCTION Conformal structure, i.e. the structures which shape is matched to the surface of the object on which they are embedded or integrated, are ideally suited due to the aerodynamic reasons for the use on the wings of the aircraft, missiles, highspeed trains or boats. The most popular application of these structures are conformal antenna. The ability to place these antennas on curved surfaces means that they can also find applications in communication systems located in the historic buildings, towers or columns without disturbing their aesthetics and local architecture. In addition to the high integrity of these structures, the undisputed advantages and benefits that can be gained by using these antennas (relative to the planar antennas) are e.g. larger angular coverage, possibility to control the beam without distortion of characteristics, wider operation band and lower RCS. However, there are various reasons for which the conformal antennas are not widespread. For example, the realization technology of conformal antenna is much more difficult than the planar antennas and is not yet fully mature. In addition, the market lacks universal design tools of this type of antennas. Due to the size of the designed antenna and its analysis method, the antennas are commonly divided into the electrically small and electrically large antennas. Electrically large antennas are those whose surface dimensions are many times larger than the wavelength, and the surfaces of the electrically small antennas are comparable to the wavelength. For both these categories, the time domain and frequency domain methods are utilized. The most commonly used and most effective approach is to use hybrid methods. Electrically small antenna are often analyzed using the mode matching method [1], method of moments [2]–[5], the finite element method often combined with other approaches [6], or the finite-difference time-domain method [7]. Electrically large 978-1-5090-2214-4/16/$31.00 ©2016 IEEE Fig. 1: Investigated structures: cylindrical, elliptical and spherical. antennas are mainly analyzed using high frequency approaches that are based on the asymptotic techniques allowing for finding the approximate solution [8]. In this article the problem of determining the resonant frequency of the microstrip structures, located on cylindrical, spherical and elliptical surfaces is considered. The curved surfaces are made of metal and coated with a dielectric material. For the case of elliptical surfaces both the structure made of confocal ellipses (for which the thickness of the dielectric layer is irregular) and the structure made of nonconfocal ellipses (where the dielectric layer is of constant thickness) are considered. In order to solve the problem the fullwave approach based on method of moments is utilized. Depending on the geometry of the investigated structure the expansion of electric and magnetic fields into suitable function series is used. II. D ESCRIPTION OF THE STRUCTURE AND ANALYSIS METHOD Schematic views of the structures under consideration are shown in Fig. 1. In each case, the structure can be divided into two regions: region 1 as a dielectric layer with a relative permittivity εr covering a metallic object (cylinder, ellipse or sphere) and region 2 located outside of the structure (open space). Microstrip patch is located on the outer surface of the dielectric - at the interface between region 1 and 2. For a cylindrical structure the z components of electric and magnetic fields take the form: Fz(κ) = Z∞ 1 2π dkz ejkz z m=−M −∞ M X ejmφ × (1) F Hm (kκρ ρ)AF mκ + Jm (kκρ ρ)Bmκ (1) F where F = {E, H}, AF mκ i Bmκ are unknown coefficients, (1) Jm (·) and Hm (·) are Bessel p and Hankel functions, respectively, of order m, kκρ = ω 2 µ0 ε0 εrκ − kz2 , for κ = {1, 2} denoting the region. For an elliptical structure the z components of electric and magnetic fields take the form: Fz(κ) Z∞ 1 = 2π X ν L X (ν) (ν),F Mcl (u, qκ,ξ )Al,κ,ξ cel (v, qκ,ξ )+ l=0 ! (ν) (ν),F Msl (u, qκ,ξ )Bl,κ,ξ sel (v, qκ,ξ ) l=1 (ν),F (ν),F (2) (ν) where Al,κ,ξ and Bl,κ,ξ are unknown coefficients, Mcl (·) (ν) and Msl (·) are even and odd Mathieu functions, respectively, of order l and kind ν = {1, 4}. cel (·) and sel (·) are even and odd angular Mathieu functions, respectively, of order l. √ Parameter qκ,ξ = (kκ2 − kz2 )d2ξ /4, where kκ = ω µ0 ε0 εrκ for region κ = {1, 2} and interface ξ = {1, 2}. It should be noted that for nonconfocal ellipses each ellipse has a different focal length dξ (ξ = 1 for the ellipse describing the metal surfaces of the core, and ξ = 2 for the ellipse describing the interface between the dielectric layer and the region outside), therefore the parameter q in region 1 would have different values in the equations describing the boundary conditions on these ellipses (interfaces). In the analysis for this case the addition theorems for Mathieu functions [9] need to be utilized. For a spherical structure the tangential components of electric and magnetic fields take the form: (κ) Ft = 2 M X X M X zn(i) (kκ r)Mtmn (θ, φ)AE,κ imn + i=1 m=−M n=m ′ zn(i) (kκ r)Ntmn (θ, φ)AH,κ imn between the regions. J is an unknown current distribution on the surface of microstrip patch and ia denotes a normal versor to the patch surface. In order to solve the above problem the patch current should be expanded in a series of current basic functions: P X J= ap Jp (5) p=1 dkz ejkz z × −∞ L X where Pnm is an associate Legandre polynomial. In every case of the above-described structures in order to determine the resonance frequency the boundary conditions must be satisfied for the tangential field components at both interfaces: (1) Et = 0 on the surface of metallic core and ( (2) (1) Et − Et = 0 (4) (2) (1) ia × (Ht − Ht ) = J (3) H,κ i where AE,κ imn and Aimn are unknown coefficients, zn (·) is t spherical Hankel function of kind i and order n, Mmn (θ, φ) and Ntmn (θ, φ) are vector functions defined as follows: ∂Pnm (cos θ) jm m Mtmn = Pn (cos θ)iθ − iφ ejmφ sin θ ∂θ m ∂P jm n (cos θ) Pnm (cos θ)iφ − iθ ejmφ Ntmn = sin θ ∂θ where Jp are the appropriate basis functions and ap are unknown coefficients of the expansion. In the proposed approach, the current basis functions are defined using the cavity model [4], [5]. In the case of complex shape of the patch, in order to determine the field distribution in the cavity the FDFD method is used. Substituting (5) to (4) and using the orthogonality properties of the respective functions (in different coordinate systems), we can determine the relation between the unknown coefficients in the external region and unknown coefficients of the current. Here, we can utilize the condition that the inner (2) product of the tangential electric field Et and current on the surface of the patch J is equal to zero: ZZ (2) Et J∗ ds = 0 (6) Sp where ∗ denotes complex conjugate and Sp is the surface of the patch. Then, using the method of moments, and assuming the current basis functions as testing functions (Galerkin procedure), we obtain a homogeneous system of equations. A nontrivial solutions are the resonant frequencies of a structure under investigation and current coefficients that allow for the determination of the current distribution on the microstrip patch. The procedure described above can be developed for the analysis of multilayer structures having any number of layers constituting the substrate (located under the patch), and any number of layers constituting the superstrate (located above the patch) as presented in [4]. III. N UMERICAL RESULTS This section presents the examples of analyzed conformal structures with calculated resonant frequency and radiation characteristics. A. Cylindrical structures The first example is a cylindrical structure with a metal core of radius r = 5 cm with a single dielectric substrate with relative permittivity εr1 = 2.32 and thickness h = 0.795 mm and with a single superstrate made of the same dielectric material with thickness t. Between the substrate and the superstrate the rectangular patch of dimensions L = 3 cm and W = 4 cm is placed. Fig. 2 shows the geometry of the structure and the calculated resonance frequencies of the TM01 mode as a function of thickness t of the supestrate. As one can see from the results the presence of a dielectric supestrate lowers the resonant frequency of the structure. The results are consistent with those presented in [2]. and the effect of the thickness of the whole substrate begins to dominate, which results with the decrease of resonance frequency. The results are consistent with the results presented in [3]. B. Elliptical structures An example of the elliptic structure is a microstrip rectangular patch of dimensions L = 3 cm and W = 4 cm located on a single dielectric substrate with a relative permittivity εr1 = 2.32 and thickness h = 0.795 mm covering the elliptical metal core of dimensions amax = 5 cm and amin ∈ (0.3, 0.999)amax. This example assumes two types of elliptical structures. The first structure is made of confocal ellipses. In this case, the two ellipses (one representing the outline of the metal core and the other is the interface between the substrate and air layer) have the same ellipticity which simplifies the numerical analysis. However, the thickness of the dielectric layer is irregular, causing a problem in the manufacturing process. The second structure has a dielectric layer with a constant thickness, thus it is made with the use of non-confocal ellipses. Fig. 4 shows the geometries of the structures and the calculated complex resonant frequencies as a function of the ellipticity of the structure. Fig. 2: The geometry of the structure with single substrate and single superstrate and resonant frequencies in function of substrate thickness. Solid line - this method; Circles - results form [2]. Fig. 4: Resonant frequencies of TM10 mode in function of structure ellipticity and the geometry of the elliptical structures (confocal and non-confocal) with single substrate. Solid and dashed lines - this method; Circles - HFSS. Fig. 3: The geometry of the structure with double substrate and resonant frequencies in function of air layer thickness. Solid line - this method; Circles - results form [3]. Another example is a rectangular patch with dimensions of L = 8 cm and W = 16.8 cm placed on a double substrate with a relative permittivity εr1 = 1 and εr2 = 2.32 (substrate with air gap) covering the metal core of radius r = 20 cm. Fig. 3 shows the geometry of the structure and the calculated resonance frequencies for the TM01 mode as a function of the thickness of the air gap at h = 0.795 mm. The obtained results shows that the increase of the air gap increases the resonant frequency of the structure which is caused by a decrease in the effective permittivity of the area under the patch. For thicker air gap the effective permeability changes slightly In the literature, often it is assumed that for small thicknesses of the dielectric layer one can approximate constant thickness of the substrate by confocal ellipses. The resonant frequency results shows that such an assumption, even for thin layers of the substrate, is incorrect, since with the increasing ellipticity the resonant frequencies vary considerably. C. Spherical structures The first example of a spherical structure is a circular patch placed on a single dielectric substrate with a relative permittivity εr2 = 2.32, covering a metallic sphere of radius r = 50 mm. Characteristics of the resonance frequency as a function of thickness of the substrate are shown in Fig. 5. As can be seen with increasing thickness of the substrate the resonant frequency decreases. The last example is a rectangular patch with four cut slits placed on a dielectric substrate with a relative permittivity Fig. 5: The geometry of the spherical structure and resonant frequencies of TM11 mode in function of substrate thickness. Solid line - this method; Circles - HFSS. εr1 = 3.5 and thickness h = 2 mm, covering a metallic sphere of radius r = 50 mm. The structure geometry and the obtained resonant frequencies for the first two modes (TM10 and TM01 ) as a function of the length of the slots ∆θs are shown in Fig. 6. From the presented results it can be seen that the increase of the length of the slots causes a decrease in the resonant frequency of the structure. Fig. 6: The geometry of the spherical structure and resonant frequencies of TM10 and TM01 modes in function of slot lengths. Solid line - this method; Circles - HFSS. IV. C ONCLUSION A method for determining the resonant frequency of microstrip open structures placed on curved surfaces was described. Several examples of multilayer cylindrical structures, confocal and non-confocal elliptical structures and spherical structures were presented. The obtained results are consistent with those found in the literature and the results of simulation using a fullwave electromagnetic simulator HFSS. ACKNOWLEDGMENT This work was supported from sources of National Science Center under grant decision no. DEC-2011/01/D/ST7/06639. R EFERENCES [1] P. Persson and R. G. Rojas, ”Efficient technique for mutual coupling calculations between apertures on a PEC circular cylinder covered with a dielectric layer,” Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2001, vol. 3, pp. 252255 [2] S.-Y. Ke and K.-L. Wong, ”Input impedance of a probe-fed superstrateloaded cylindrical-rectangular microstrip antenna,” Microw. Opt. Technol. Lett., vol. 7, pp. 232236, 1994 [3] K.-L. Wong et al., ”Analysis of a cylindrical-rectangular microstrip structure with an airgap,” IEEE Trans. Microw. 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