JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 1, MARCH 2011 71 Pattern Synthesis and Polarization Optimization of a Conical Array Qing-Qiang He, Yu-Hong Liang, and Bing-Zhong Wang Abstract⎯A feasible method for synthesizing milli- is fixed with respect to the surface of the platform[9]. In [10], meter-wave conical array and optimizing low cross- an adaptive array theory is applied to a conformal antenna polarization is proposed. Starting from the far-field array to synthesize a mainbeam with optimized polarization superposition principle, an efficient approach including employing dual polarized patch antennas as radiators. In element mutual coupling and mounted platform effects [11], the low cross-polarization of an arc-array is optimized is used to calculate the far-field patterns. A coordinate by using the alternating projection method. However, these transform is applied to create polarization quantities, synthesis techniques mentioned above mostly neglect the and a general process for the element polarized pattern element mutual couplings, and the effect of the mounted transformation is performed. Corresponding numerical platform is also not rigorously taken into account. example is given and the desired sidelobe level and low In the paper, our research presents a new method to cross-polarization are optimized. The numerical results synthesize millimeter-wave conical array including the indicate the proposed method is valid. effects of element mutual couplings and mounted platform. [12] Index Terms⎯Conical array, coordinate transform- Based on the active element pattern technique , a conical ation, low cross-polarization, genetic algorithm, pattern array model is firstly created and polarization quantities synthesis. with Euler rotation[13] for each element are obtained. Different from the axis rotations in [11], plane rotations are proposed to realize the element polarization transformation. 1. Introduction In order to obtain the good cross-polarization and sidelobe [14],[15] Conformal arrays have good potential for application in levels, a genetic algorithm (GA) , including a large aerospace vehicles with excellent aerodynamic characteris- amount of optimized parameters, is implemented. Different tics. Because conformal arrays generally are curved, new from [16] and [17], the two-dimension curved-face conical far-field behaviors emerge and most of the traditional linear array is investigated and GA combining with the active and planar array synthesis methods are not valid. A variety element pattern technique is used to calculate the of techniques have been used to solve the synthesis complicated radiation pattern function. The numerical problem for conformal array. These techniques include results show the proposed method is feasible and valid. such as space mapping technique[1], non-linear optimization method[2], intersection approach[3], least squares methods[4], 2. Theories simulated annealing technique[5], adaptive array theory[6], As shown in Fig. 1, by applying the principle of and particle swarm optimization[7],[8]. Furthermore, in the superposition, the total radiated field of an arbitrary array design of conformal arrays, problems arise, which are can be expressed as different from those appearing in arrays on planar surfaces. ˆ The polarization in the far field changes as a function of jkrR⋅ mn, EF(,θϕ )= ∑∑wemn,, mn(,θϕ ) (1) directional angle for radiating elements whose polarization mn where mM= 1, 2, ", , n= 1, 2, ", N, wmn, is the Manuscript received October 21, 2010; revised November 28, 2010. excited complex current applied to the mnth element, M is This work was supported by the Emphases Foundation of Southwest China the number of stacked arc subarrays and N is the number Institute of Electronic Technology under Grant No. H090024. Q.-Q. He and Y.-H. Liang are with the Southwest China Institute of of radiators in the mth arc subarray, Fmn, (,θϕ )is the active Electronic Technology, Chengdu 610036, China. (email: heqingqiang518 jkrRˆ⋅ mn, @126.com). element pattern of the mnth element, e is the spatial B.-Z. Wang is with the Institute of Applied Physics, University of phase term, rˆ is the unit radial vector from the coordinate Electronic Science and Technology of China, Chengdu 610054, China. origin to the observation direction (,θ ϕ ), Rmn, is a position Color versions of one or more of the figures in this paper are available online at http://www.intl-jest.com/. vector from the origin to the center of the mnth element, Digital Object Identifier: 10.3969/j.issn.1674-862X.2011.01.013 and rRˆ ⋅ is dot product. Usually, F (,θϕ ) in (1) is mn, mn, 72 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 1, MARCH 2011 different from the isolated element pattern because other Obviously, the unusual geometry of the conformal elements will radiate some power due to the effects of conical array requires a special consideration of the element mutual coupling and mounted platform. coordinate system used to represent the patterns as shown Furthermore, Fmn, (,θϕ ) also depends on the polarization in Fig. 1. In order to easily obtain the far field pattern of the of the array element and the position of the array element. conformal conical array, the coordinate transformation for Generally, the whole array radiating field in (1) can be element pattern between the element local coordinate divided into a co-polarization quantity and cross-polariza- system and the global coordinate system needs to be carried tion quantity. Thus the total field may be rewritten as out to count the contribution of each element to the conformal conical array radiation. The corresponding EE(,θ ϕθϕθ )=+co ( ,) Exp ( ,ϕ) (2) transformation relation from the local coordinate system to the global coordinate system is shown in Fig. 2. where Eco (θ ,ϕ ) is the co-polarization quantity and In Fig. 2, the local coordinate system is defined by lll OXYZmn,,,− mn mn mn,, and the superscript l represents the Exp (θ,ϕ ) is the cross-polarization quantity. l For the conical surface as shown in Fig. 1, its local coordinate system. Zmn, -axis of local system is curved-surface equation is oriented along the direction of the local normal vector of curved surface where the mnth element is placed. The c z =−xy2 + 2 (3) directions of X l - and Y l -axis are determined by the a mn, mn, particular position of the mnth element. Thus the where c is the cone height, a is the bottom radius, cone transformation from the local coordinate system to the angle δ is determined by arctan (ac) . Thus, for the mth global coordinate system can be obtained as arc subarray, the arc radius is l ⎡⎤Xr⎡⎤− mncosϕ ⎡⎤Xmn, ⎢⎥ ⎢⎥ ⎢l ⎥ Yr=−msinϕβαγ n +R ()mn,,,,, mn mn⎢Y mn ,⎥ (5) rdmm= tanδ (4) ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥l ⎣⎦Zd ⎣mm ⎦ ⎣⎦Z,n where dm is the distance from origin of the global where ϕ is the mnth element position in the mth arc coordinate system to the mth arc subarray plane. n subarray, R(βαγmn,,,,, mn mn) is a 33× Euler rotation matrix. Z Based on the created rotation relation in Fig. 2, αmn, , o β , and γ can be obtained through mathematical Y mn, mn, manipulation: X δδ l ddmm Z l Zmn, c αmn, = π2 −δ (6) l ⎛⎞ Y mn ⎛⎞1 l ⎜⎟ omn Y −−rrmnnmncosϕϕ cos－ sin ϕ⎜⎟ sin ϕ n mn, ⎜⎟sinϕ ⎝⎠n βmn, = ar cos⎜⎟ (7) a l 2 XX ⎜⎟ mn, 2 ⎛⎞1 p ⎜⎟Rmn, ()cosϕϕ n +−⎜⎟ sin n ⎜⎟sinϕ Fig. 1. Geometry of conformal conical array. ⎝⎠⎝⎠n ⎛⎞sinϕϕ cos Z nn γ mn, = arctan ⎜⎟. (8) sinϕϕ sin− 1 Y ⎝⎠nn ll Z o Zmnmn, p In (7), Rmn, is the distance from the origin of the global X αα mn, mn coordinate system to the mnth element on the radial direction of the conical surface, and Yll Ymnmn, Oomnmn, Rrp =++cosϕϕ22 r sin d2. (9) ββ M mn, ()()( m n m n m) γ mn, mn mnmn, Applying (5), (6), (7), and (8), the element pattern X ll Xmn, polarization transformation can be achieved, and the total Fig. 2. Corresponding coordinate system for element radiators. radiated field can be computed by (2). HE et al.: Pattern Synthesis and Polarization Optimization of a Conical Array 73 For certain applications, the conformal array patterns the second part of (10). If the first part of (10) cannot reach are usually designed to have low sidelobe and low cross the optimized target in the tuning process, μ1 is increased, polarization. But conformal array factor often cannot be and vice versa, μ is increased. Here, the tun ing factors μ separated because of the effects of structure curvature, and 2 1 and μ are in [0. 5, 1.5] and [0.4, 1], respectively. the traditional linear/planar array synthesis methods often 2 do not suitable for curved arrays. Thus the pattern synthesis and low cross polarization optimization for conformal 3. Example and Results conical array need powerful and attractive methods. To validat i ng the proposed method, a numerical Furthermore, the placement of an element will be greatly experiment is firstly implemented. Assuming the spacing influenced by the shape of the mounted platform. Instead, if between each element is 0.5λ along the radial direction of individual element patterns, which include the effects of the 0 shape of the platform, mutual coupling between elements, the conical surface, where λ0 is the free space wavelength and polarization, are used, a more general superposition corresponding to an operatin g frequency of 32 GHz, we computation must be performed in the whole area. One obtain that the bottom radius of the cone is R = 8λ0 , the half possible approach for solving this generalized problem is D through the use of GA combining with the active element conical angle is δ = 30 , thus the coordinate of t he mth technique to select the excitation magnitudes and phases stacked arc subarray is that can produce the closest possible match to the desired ⎡⎤R λ0 array radiating patterns.