Performance of Planar Arrays for Microwave Power Transmission With ...

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 14, 2015

Performance of Planar Arrays for Microwave Power Transmission With Position Errors Xun Li, Jinzhu Zhou, Baoyan Duan, Yang Yang, Yiqun Zhang, and Jianyu Fan

Abstract—An analytical method based on Statistical Analysis (SA) is proposed to predict the influence of position errors of planar array elements on microwave power transmission (MPT) efficiency. The average power pattern of arrays with arbitrary shape is formulated according to the rules of probability method. Then a closed-form expression describing the relationship between position errors and the average beam collection efficiency (BCE) is derived. A set of representative numerical experiments dealing with planar arrays of different sizes and shapes is reported and discussed to point out the effect of the position errors on MPT. Index Terms—Beam collection efficiency, microwave power transmission, planar arrays, position errors, space solar power satellite, statistical analysis.

M

I. INTRODUCTION

ICROWAVE power transmission (MPT) is a promising technology of delivering energy from one point/area to another via microwave. It can be used to supply fixed and mobile systems without using electric cables [1]–[3], and thus is obtaining more and more attention from the applicative viewpoint. A comprehensive review of long distance MPT array techniques is reported in [4]. One of the potential application fields of MPT is the so-called space solar power satellite (SSPS) [5], which converts solar energy to electricity in the geostationary orbit (GEO) and transmits the electricity energy to the earth through microwave. Transmitting antenna and rectenna are the two key components of MPT systems. Transmitting antenna is used to concentrate the radiated power to the receiving region, and the rectenna is used to collect and convert the incoming microwave to DC. It must be stressed that MPT is a technology different from using microwave for communication, due to high efficiency requirement and large magnitude of the power handled at the receiving point, being in many cases over 90% of the power transmitted at the transmitting point [6]. The collection and conversion of impinging microwave to DC is a unique technology that has little relation to traditional methods of receiving and processing microwave in communication applications. The objective of MPT applications is high transmission efficiency and low

Manuscript received March 28, 2015; accepted April 14, 2015. Date of publication April 17, 2015; date of current version November 17, 2015. This work was supported by the National Natural Science Foundation of China under Grants 51305323, 51405361, and 51490660 and the Fundamental Research Funds for the Central Universities. (Corresponding author: Jinzhu Zhou.) The authors are with the Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi’an 710071, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2015.2424227

sidelobes, so it is to some degree similar to traditional communication applications, which mainly concerns high gain and low sidelobes. However, to maximize MPT efficiency is not equivalent to maximize the gain of the antenna [7]. Owing to the high requirement on MPT efficiency, specific studies have been carried out on the maximization of the ratio between the power radiated over a given region and the total transmitted power, called beam collection efficiency (BCE) [7], [8]. Stochastic algorithm for the minimization of sidelobes and grating lobes for MPT was investigated in [9]. However, previous works [7]–[9] were concerned with the ideal situation, and the array elements with position errors were not considered. In practice, position errors are inevitably introduced during antenna manufacturing and assembly processes, and they are characterized by uncertainties. The effects of position, amplitude, and phase errors described by random and interval methods on linear and planar arrays are studied in [10]–[13]. Nevertheless, to the best of the authors’ knowledge, few people have analyzed the effect of random position errors on BCE, which is of great importance to MPT applications. In this letter, the effect of position errors of planar arrays with different sizes and shapes on BCE is studied. Representative results concerned with square, circular, and hexagonal arrays are presented and discussed. II. MATHEMATICAL FORMULATION Consider a planar array of an arbitrary shape consisting of radiating elements and, without loss of generality, located in the XOY plane. The effect of mutual coupling among elements is ignored, and the element patterns in the array are assumed to be isotropic. The ideal array factor and the corresponding power pattern can be expressed as (1)

(2) and are, respectively, the complex excitation where coefficient and position coordinate of the th radiating element. The symbol denotes the wave number, “ ” stands for the com, , where and plex-conjugate, are the elevation angle and azimuth angle, respectively. Suppose that the target area is in the far-field region and no excitation or position errors exist, then BCE can be written as

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(3)

LI et al.: PERFORMANCE OF PLANAR ARRAYS FOR MICROWAVE POWER TRANSMISSION WITH POSITION ERRORS

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Assume a case where the standard deviation of the errors in , , and directions are equal, i.e., . Note that , then (8) can be simplified as

(9) Substituting (9) into (3), the impact of random position errors on the average BCE of an arbitrary planar array antenna can be determined as Fig. 1. Geometry of transmitting antenna and rectenna.

(10)

where and indicate the power radiated over the angular region and the total transmitted power over the visible region , respectively, as shown in Fig. 1. The power and can be formulated as (4) (5) By rewriting (1) as the product of weight vector and steering vector, and substituting first the two vectors into (4) and (5), and then (4) and (5) into (3), the BCE maximization problem can be reduced to the solution of a largest generalized eigenvalue problem as described in [7]. Through solving the generalized eigenvalue problem, the maximum BCE and the corresponding optimal tapering can be readily obtained. In practical application, position errors introduced in the manufacturing and assembly processes are inevitable. Moreover, in outer space application, environmental factors, such as large temperature difference, intensive radiation and vibration, also contribute to position errors. Taking into consideration the position errors, the array factor and the corresponding power pattern can be stated as

(6)

(7) where , , and are the random position errors of , the th element in , , and directions. Assume that , and are statistically independent and have a normal distribution with zero mean and standard deviations equal to , , and , respectively. Then, the average power pattern with position errors can be described as

(8)

where denotes the beam collection efficiency of arrays without position errors. Equation (10) shows that the average BCE with position errors depends not only on the operation frequency (the wave number ) and the scale of the standard deviation but also the angular region of the radiated area and the visible region , which is unique compared to the traditional communication antennas. Moreover, it is irrelevant to the direction angles and . III. RESULTS AND DISCUSSIONS The purpose of this section is to investigate the influence of the scale of the standard deviation of elements position errors on the BCE when dealing with different aperture sizes and shapes. For this purpose, a set of representative numerical simulations involving square, circular, and hexagonal arrays have been carried out. As a typical case, suppose that the arrays are half-wavelength spaced and illuminate a square collection area limited . The operation frequency is 5.8 GHz. by eleTo begin with, square arrays of , and elements having a diameter of are dealt with. and denote ments having a diameter of the row number and column number of the array, respectively. is the total number of the radiating elements. The element excitation coefficients are obtained by solving the generalized eigenvalue problem of the array without position errors. Here we consider the standard deviation of element position , , , , and , respectively; errors to be the corresponding 3D power patterns are obtained and compared to the ideal one. Due to space limits, only results of array are plotted and shown in Fig. 2, and the arising power patterns clearly show that more power is lost outside of the receiving region with the increasing errors, due to the appearance of the higher sidelobes. This is numerically confirmed by the BCE and CSL listed in Table I. eleNumerical results concerning square array of ments are also given in Table I. It is shown that the BCE of and arrays without position errors are 95.42% and 99.68%, respectively, which agrees well with the results 96.45% for array and 99.97% achieved in [7] ( array). However, the BCE decreases sharply when for , and is even less than 70% when the erthe errors exceed . Moreover, position errors lift the side lobe level rors reach outside of the collection region (CSL [7]), defined as

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Fig. 2. Square array–square collection area ( , ; (e) standard deviation (d) standard deviation TABLE I SQUARE ARRAY—SQUARE COLLECTION AREA

TABLE II CIRCULAR ARRAY—SQUARE COLLECTION AREA

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 14, 2015

). (a) Ideal array; (b) standard deviation (f) standard deviation .

; (c) standard deviation

;

TABLE III HEXAGONAL ARRAY—SQUARE COLLECTION AREA

Additionally, circular and hexagonal array antennas of the same dimension as the square arrays are studied as well, and similar results can be obtained. Owing to the limited space of this letter, plots like Fig. 2 are omitted, and only the numerical simulation data are listed in Table II and Table III. A comparison of square, circular, and hexagonal arrays with the same aperture dimension is made and shown in Fig. 3. Obviously, the sidelobes are uplifted and become flat with the increasing errors. The power pattern is always symmetrical and no beam pointing error exists as random position errors are considered and average power pattern is calculated. By further studying the relationship between the BCE and the position errors, comparison of the BCE of square, circular, and hexagonal arrays with errors is made, and the result is shown in , Fig. 4. These arrays have the same aperture size, i.e., and the elements needed for square, circular, and hexagonal arrays are 400, 316, and 290, respectively. It can be observed from

LI et al.: PERFORMANCE OF PLANAR ARRAYS FOR MICROWAVE POWER TRANSMISSION WITH POSITION ERRORS

Fig. 3. BCE of different shape of array antennas with the same dimension. (a) Square array—square collection area ( , ); (c) Hexagonal array—square collection area ( , array—square collection area (

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,

); (b) Circular ).

arrays having a dimension of , and for arrays . having a dimension of (2) Once the element spacing is given, the BCE depends mainly on the dimension of the antenna and has little relation to the shape of the antenna. (3) Compared with the rectangular array antenna, the circular or hexagonal array antennas have nearly the same BCE but much less elements, which greatly reduces the cost. Therefore, the circular or hexagonal array antennas are preferable to the rectangular one in engineering applications. REFERENCES

Fig. 4. Effect of position errors on the BCE of different shapes of array antenna.

Fig. 4 that the obtained BCE of the three arrays are nearly same, while circular array can save 21% of the elements, and hexagonal array can save 27.5% of array elements. A similar concluarray. sion can be drawn for IV. CONCLUSION An approach based on statistical analysis has been proposed for the analysis of the effect of array antenna position errors on the BCE, and formulas that precisely describe the relationship between the errors and the BCE have been developed. This is, to the best of the authors’ knowledge, the first time that the influence of the position errors on the BCE has been analyzed. In the numerical analysis, the effectiveness of the proposed approach in evaluating the impact of position errors on the BCE has been studied by considering various error scales on kinds of antennas with different shapes and dimensions. The following significant conclusions can be drawn. (1) The BCE decreases with the increasing position errors. To keep the BCE larger than 90%, the standard deviation for of elements position errors should be less than

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