ANALYSIS OF MICROSTRIP PATCH ANTENNAS ON SPHERICAL STRUCTURES Niksa Burum1, Zvonimir Sipus2 and Juraj Bartolic2 1
ITI-Computers Dubrovnik, Ćira Carića 3, HR-20000 Dubrovnik, Croatia e-mail:
[email protected] 2 University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, HR-10000 Zagreb, Croatia; e-mail:
[email protected],
[email protected]
ABSTRACT: The paper presents the program for calculating input impedance and radiation pattern of spherical patch antennas. The patches can be of rectangular or circular shape, and they are fed by a coaxial line. The multilayer spherical structure is rigorously taken into account using proper Green’s functions, and the electric field integral equation is numerically solved by applying the method of moments. This problem is solved in spectral domain, i.e. for each value of spectral variables only one-dimensional problem has to be solved. In order to avoid numerical difficulties a modified vector-Legendre transformation and normalized Legendre polynomials are defined and applied to the solution procedure. The influence of the antenna radius on radiation pattern and input impedance, as well as the possibility of designing a spherical array with hemispherical beam scanning, are studied. 1. INTRODUCTION Microstrip patch antennas are often used because of their thin profile, light weight and low cost. Furthermore, they can be made to conform the structure. Spherical arrays have possibility of directing single or multiple beams through complete hemisphere. Therefore, spherical array is an attractive solution for satellite tracking, telemetry and command application. The purpose of this paper is to describe a program for calculating input impedance and radiation pattern of rectangular or circular patches on the spherical structure. 2. METHOD OF ANALYSIS The geometry of the problem is shown in Figure 1. The rectangular or circular patches are placed on or embedded in a multilayer spherical structure. The radius of the grounded sphere and of the patches are rgnd and rpatch, respectively. The dimensions of each quasi-rectangular patch are Wθ and Wφ (Wθ = 2 θp rpatch , Wφ = 2φprpatch). The patches are fed by a coaxial line, and the distances of the feeding point from the center of the patch in the θ and φ directions are θfeed and φfeed , respectively. To determine the current on the patches, we consider the integral equation for electric field components tangential to the patches. The unknown currents are expanded into a sum of entire domain basis functions, and the unknown coefficients αi are determined by applying the moment method Galerkin's approach. The elements of the impedance matrix [Zji] and voltage vector [Vj] inside the moment method ( Z ji ⋅ [α i ] = V j ) are calculated in the spectral
[ ]
Figure 1. Geometry of the problem
[ ]
domain. Since the problem is defined in the spherical coordinate system we apply the vector - Legendre transformation to the patch current. [1], [2]. This
technique transforms the three-dimensional problem into spectrum of one-dimensional problems. The algorithm G1DMULT is used for calculating Green's function of general multilayer spherical structures [2]. We notice numerical problems in calculating Legendre polynomials and S(n,m) for large n, specially if m ≈ n when both both Legendre polynomials and S(n,m) become very large, and the solution is unstable (both terms are needed for performing vector Legendre transformation; S(n,m) is defined as S(n,m) = [2n(n+1)(n+|m|)!]/[(2n+1)(n-|m|!]). Notice that the number of used modes depends on the structure radius. To find the numerically stable solution first we have introduced the alternative definition of vector-Legendre transformation: ∞ ∞ ~ 1 J (r , θ , φ ) = L (n, m, θ ) J(r, n, m)e jmφ (1a) ∑ ∑ 2πS (n, m) m=−∞ n= m ~
J ( r , n, m ) =
1
π π
∫ ∫ L (n, m,θ )J (r ,θ , φ ) sin θe
2πS (n, m) −π 0
− jmφ
dθdφ
P m (cos θ ) n(n +1) 0 0 n m m L (n, m,θ ) = 0 ∂P n (cos θ ) ∂θ − jmPn (cos θ ) sinθ m m 0 jmPn (cos θ ) sinθ ∂P n (cos θ ) ∂θ
(1b)
(1c)
km
Here Pkn (cos θ ) are the associated Legendre functions of the first kind. Furthermore, we have normalized Legendre polynomials and their derivatives as m
m
P n = Pn
(n − m )! (n + m )!
m
,
m ∂ P n (cos θ ) ∂Pn (cos θ ) (n − m )! = ∂θ ∂θ (n + m )!
(2)
The recursive equations for normalized Legendre polynomials are: m
P n +1 ( z ) (n + 1 + m)(n + m) =
1 n − m +1
m m (2n + 1) z P n ( z ) (n + m)(n − m + 1) − (n + m) P n −1 ( z ) ( n − m)(n − m + 1)
( z 2 − 1)
(3a)
m
m m ∂ P n ( z) n + m = nz P n ( z ) n + m − (n + m) P n −1 ( z ) n − m ∂z
(3b)
Notice that the product L / S (n, m) in equations (9) and (10) enable us to calculate normalized Legendre polynomials instead of Legendre polynomials, and 2n(n+1)/(2n+1) instead of S(n,m), which is numerically stable. Thus, divisions of very large numbers are avoided. The far field radiation pattern is obtained as follows. If we consider e.g. the φ-component of the electric field in the outermost region with the r-coordinate larger than the r-coordinate of the patch, we have only i Hˆ n( 2) (k 0 r ) . Therefore, in the outermost region we can connect the outward-traveling waves described by a nm φ-component of the electric field with different r-coordinates r1 and r2 as
H ( 2) (k r ) ~ j n +1 e − jk0 r1 ~ ~ Eφ (r1 , n, m) = Eφ (r2 , n, m) n( 2) 0 1 = Eφ (r2 , n, m) ( 2) H n (k 0 r2 ) H n ( k 0 r2 )
(4)
Here r1 represents the r-component of the far field pattern. The final solution is obtained by superposing the spectral solutions, see eq. (1a). When calculating far field of spherical array it has been found convenient to introduce a local coordinate system with the origin located at the position of each antenna element. The far field is then calculated as a superposition of calculated patterns of each patch element calculated in local coordinate system. The transformation of coordinates between local and global coordinate system is obtained using approach given in [3]. 3. NUMERICAL RESULTS
First we have illustrated the effects of the sphere size. The square patch of dimension 4.8 × 4.8 cm is printed on the single-layer dielectric substrate with dielectric constant εr = 2.32 and thickness h = 1.58 mm. The working frequency is 2 GHz. The effect of the radius of the ground plane (grounded shell) is given in Fig. 2. For comparison, the radiation pattern of planar patch antenna of same dimensions is also given. It can be seen that with enlarging the radius of the sphere the main lobe of the spherical patch antenna approaches the main lobe of the planar counterpart, and that the back radiation is smaller for structures with larger radius. In order to steer the beam throughout the whole hemisphere the arrays should have a grid suitable for spherical structure, such as icosahedron structure (see [3] for details). In Fig. 3 radiation pattern of spherical patch array is given as a function of number of excited elements (α is the angle of activated area). The array has the icosahedron grid structure, the grounded shell radius is 35 cm, and the other antenna parameters are the same as in the previous example. Angles α = 15, 30 and 45 degrees correspond to 6, 16 and 31 activated elements, respectively. The patch excitations are phase corrected in order to compensate the propagation path difference. As expected, with enlarging the number of activated elements the main beam is narrower. However, the back radiation is larger for more activated elements. The influence of the sphere radius on input impedance of the spherical patch is shown in Fig. 4. The θpolarized rectangular patch of dimension 2.5 × 4.0 cm is printed on the single-layer dielectric substrate with dielectric constant ε r = 2.52 and thickness h = 1.576 mm. The measured results of a planar patch are shown for comparison. It can be seen that the sphere radius mostly influences the resonant frequency, while the resonant resistance is almost constant. ACKNOWLEDGEMENT
This material is based upon work supported by the European Office of Aerospace Research and Development, Air Force Office of Scientific Research, Airforce Research Laboratory, under Contract No. F61775-01-WE024. REFERENCES
[1] [2] [3]
W. Y. Tam and K. M. Luk, “Resonances in spherical-circular microstrip structures of cylindricalrectangular and wraparound microstrip antennas,” IEEE Trans. Microwave Theory Tech.., Vol. 39, pp. 700-704, 1991. Z. Sipus, P.-S. Kildal, R. Leijon, and M. Johansson, ”An algorithm for calculating Green’s functions for planar, circular cylindrical and spherical multilayer substrates,” Applied Computational Electromagnetics Society Journal, Vol. 13, pp. 243-254, 1998. D. L. Sengupta, T. M. Smith, and R. W. Larson, “Radiation Characteristics of Spherical Array ofPolarized Elements”, IEEE Trans. on Antennas and Propagat., Vol. 16, pp. 2-7, 1968.
Fig. 2: Radiation pattern of a spherical patch antenna for different sphere radii; (a) H-plane, (b) E-plane.
Fig. 3: Radiation pattern of a spherical patch array with elements placed uniformly by utilizing the symmetry properties of the icosahedron. (a) H-plane, (b) E-plane.
Figure 4. Input impedance of a θ-polarized rectangular patch printed on a spherical structure for different sphere radii; .(a) real part, (b) imaginary part.
