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 α = π2 −δ (6) mn,

l ⎛⎞ Y mn ⎛⎞1 l ⎜⎟ omn Y −−rrmnnmncosϕϕ cos－ sin ϕ⎜⎟ sin ϕ n mn, ⎜⎟sinϕ ⎝⎠n βmn, = ar cos⎜⎟ (7) a l 2 XXmn, ⎜⎟ p 2 ⎛⎞1 ⎜⎟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

Z ll 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. Thus for an arbitrary sidelobe area dmm =−⎢⎥ +() −1cosδ . (12) ⎣sinδ 2 ⎦ S111(θ ,ϕ ) and mainbeam area S000(θ ,ϕ ) in (2), the fitness function of the conformal conical array can be given Based on the prevalent simulated means of conformal by array[18]-[20], a multifaceted surface can be a very good approximation to the curved counterpart. Here, the conical

fw(mn,1 )=−μθ max⎣⎦⎡⎤ SLL co() ,ϕ SLVL1+array is created by using 36 segment multifaceted surfaces. ()(θϕ,,∈S θ ϕ ) 111 (10) On each multifaceted surface, the patch radiating element ⎪⎪⎧⎫can be designed exactly, as shown in Fig. 3. μθb − max⎡⎤ SLL ,ϕ− SLVL 2x⎨⎬⎣⎦p() 2 For each element of the mth stacked subarray in ⎩⎭⎪⎪()θϕ,,∈+SS000() θ ϕ 111() θ , ϕ the ϕ -direction, radiating elements are arranged with an where wmn, is the excited weight and can be found in (1), equal angle-spacing, and here the angle-spacing

μ1 is the tuning weight of co-polarization pattern, μ2 is the is Δ=ϕ 0.075 rad . A microstrip patch with a length of tuning weight of cross-polarization pattern, SLVL1 is a L = 1.8 mm and a width of W = 1.1 mm is selected as the desired co-polarization sidelobe level, SLVL2 is a desired radiating element. The ground plane dimensions of the patch 2 cross-polarization level, SLLco is the co-polarization antennas are chosen to be GG×=×44 mm. Taconic RF-60 with a thickness h = 0.508 mm and a relative sidelobe level, SLLxp is the cross-polarization level, and b is the maximum of the co-polarization pattern and given by permittivity ε r = 6.15 is used as the substrate. A 50 Ω probe feed, which is located at the b = Ε θϕ, . (11) ∑ co ( ) point P ()0.25 mm,0.25 mm away from the mnth patch ()θϕ,,∈S000() θ ϕ mn, center, is used to excite the patch. The first part of (10) indicates letting co-polarization sidelobe levels reach the desired low sidelobe level and the second part letting cross-polarization levels reach the Z W Ymn desired low cross-polarization in the whole space domain. P • mn X mn L O ⎡⎤ mn max⎣ SLLxp ()θ ,ϕ ⎦is the max level of the ()θϕ,,∈+SS000() θ ϕ 111() θ , ϕ

G cross-polarization pattern and b − max⎡⎤ SLLxp ()θ ,ϕ is the ⎣⎦ Array Element ()θϕ,,∈+SS000() θ ϕ 111() θ , ϕ normalization level of the cross-polarization pattern referring to the b. In implementing the GA process, each individual of each generation can be expressed as a group of the excited weight. For the co-polarization pattern, b can Y be calculated based on the wmn, , and then b is seen as the benchmark for optimizing the cross-polarization pattern in Fig. 3. Model of the conformal array and its radiating element. 74 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 1, MARCH 2011

Especial position element plays a significant role for the sidelobe and cross-polari- zation levels of the conformal conical array patterns. The first sidelobe starts to merge with the main beam and a high cross-polarization level is produced. In Fig. 6, the “Co” denotes a co-polarization pattern and the “cross” denotes a Interior elements cross-polarization pattern.

Adjacent edge elements 0

Edge elements

B) B) −5 Fig. 4. Sketch of the edge, interior, adjacent, and special position elements. Isolated element pattern Here, an active element pattern technique[11] is used to −10 Interior element pattern Edge element pattern obtain array element patterns. The active element pattern (d Amplitude technique, which uses the simulated or measured patterns of −15 individual elements in the array environment to calculate the −80 −40 0 40 80 pattern of the fully excited array, can often be employed ϕ (°) when classical analysis and numerical techniques cannot. (a)

Active element patterns are either measured on a test range 0 or simulated in some manner. Generally, the array elements can be divided into edge elements, interior elements, and B) B) −5 adjacent edge elements as shown in Fig. 4, where the adjacent edge elements locate between edge elements and Isolated element pattern interior elements. Of course, for certain especial position −10 Interior element pattern such as cone tip in Fig. 4, the effect of mutual coupling is (d Amplitude Edge element pattern possibly different, and the mutual coupling analysis should −15 be specially considered. Here, there is not especial position 0 40 80 120 160 element in the proposed example. Through careful analysis, θ (°) it is found that the traits of adjacent edge elements are very (b) similar to that of interior elements. Thus, the array elements Fig. 5. Isolated element pattern and active element patterns may be divided into edge elements and interior elements. (including the effects of mutual coupling and mounted platform): (a) xoy-plane and (b) xoz-plane. The edge elements typically taken are the first and final rows and columns of the active elements on the sides of the 0 conformal conical array. Corresponding active element patterns about the edge elements and interior elements are

B) B) −15 obtained with the HFSS (high frequency structure simulator) simulations. Fig. 5 shows the co-polarization patterns for the −30 Simulated Co Calculated Co isolated element and the active elements. It is clear that (d Amplitude Simulated cross Calculated cross these patterns are different because of the effects of element −45 mutual coupling and mounted platform. Similarly, the −80 −40 0 40 80 ϕ (°) cross-polarization patterns for the isolated element and the (a) active elements possess the same case. 0 Fig. 6 plots the far field patterns for a 88× conformal Simulated Co Calculated Co conical array by using the proposed method and HFSS Simulated cross

B) simulation, respectively. Here xoy60-plane denotes a surface −15 Calculated cross where θ is 60° and ϕ varies from −90° to +90°, and

° ° xoz-plane denotes a plane where ϕ is 0 andθ varies from 0 −30 to 180°. Obviously, it is seen that th e results obtained by the (d Amplitude proposed method are consistent with those of HFSS −45 0 40 80 120 160 simulation for co-polarization patterns. In the xoz-plane, θ (°) because of the modeling approximations between (b) multifaceted surface and true curve-surface, the cross- Fig. 6. Far-field patterns for the 8×8 conical array based on the polarization patterns have been different in the range from proposed method and HFSS simulation: (a) xoy60-plane and (b) 110° to 180°. Furthermore, Fig. 6 reveals that the curvature xoz-plane. HE et al.: Pattern Synthesis and Polarization Optimization of a Conical Array 75

) 0.8 V

) 0.8 V 0.4 0.4

Amplitude ( Amplitude 0

8 ( Amplitude 6 8 0 6 4 4 8 N 2 2 6 8 M 4 6 (a) N 4 2 2 M

(a) 3

2 3

1 2 Phase (rad) 0 8 1 8 6 Phase (rad) 6 4 4 N 2 2 0 M 8 (b) 6 8 Fig. 7. Optimized co-polarization weights: (a) amplitudes and (b) 4 6 N 4 phases. 2 2 M

(b ) In order to obtain good sidelobe and low cross-polari- zation, the excited weights w of (10) need to be optimized. Fig. 8. Optimized cross-polarization weights: (a) amplitudes and mn, (b) phases. For resolving the optimized pr oblem of a large amount of 0 parameters, a GA complying with the adaptive rule[13] has merit. Based on the geometrical symmetry about xoz-plane, −10 the optimized parameters can be reduced half in all space domains. The number of the excited amplitudes and phases −20 for the co-polarization quantity are 64 and the number of the excited amplitudes and phases for the cross-polarization −30 quantity are also 64. All excited amplitudes vary from 0 to 1, Amplitude (dB) (dB) Amplitude and all excited phases vary from 0 to 2π . The GA is −40 encoded using a population size of approximately 180 members. The unrepeatable strategies are used for crossover −50 and mutation. After performing the crossover and mutation −80 −40 0 40 80 manipulations, the elites are maintained in order to prevent ϕ (°) the losses of the optimized results and accelerate the (a) convergence of this algorithm. For evolutional each −20 generation, the crossover and mutation probabilities comply with the adaptive rule. The target function is defined by the sidelobe level (−20 dB) and cross-polarization level (−30 dB). −30 After about 210 iterations, the optimized co-polarization weights are shown in Fig. 7 and the optimized cross- polarization weights are shown in Fig. 8. −40 The corresponding patterns with the proposed method (dB) Amplitude are shown in Fig. 9 and Fig. 10, and good results are obtained. Through further calculation and analysis, it is −50 found that the far-field behavior of conformal array is −80 −40 0 40 80 different from that of classical planar array. With the ϕ ( °) curvature effects, it is very difficult to define the radiating (b) characteristic plane for a conformal array. Thus the pattern Fig. 9. View of the radiating patt erns for conical array in the optimization for a conformal array often needs to be xoy-plane: (a) co-polarization radiated pattern and (b) cross- performed in the whole space domain. polarization radiated pattern. 76 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 1, MARCH 2011

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Bing-Zhong Wang was born in Sichuan, Qing-Qiang He was born in Chongqing, China, China, in 1962. He received the Ph.D. degree in 1977. He received the M.Sc. degree in in electrical engineering from UESTC in 1988. electromagnetic and microwave technology He joined UESTC in 1984 and is currently a and Ph.D. degree in radio physics from professor there. He has been a visiting scholar University of Electronic Science and at the University of Wisconsin- Milwaukee, a Technology of China (UESTC), Chengdu, research fellow at the City University of Hong China, in 2005 and 2008, respectively. Now, he Kong, and a visiting professor in the Electromagnetic works with the Southwest China Institute of Communication Laboratory, Pennsylvania State University. His Electronic Technology and is a senior engineer. His current current research interests are in the areas of computational research interests are in the areas of antenna theory and technique, electromagnetics, antenna theory and technique, electromagnetic electromagnetic compatibility analysis, and computer-aided compatibility analysis, and computer-aided design for passive design for passive microwave integrated circuits. microwave integrated circuits.