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**Aperture Efficiency Study of Square Reflect Array Antennas** ()

*η*). Effects of quantization-phase and limited phase-range errors on radiation pattern, half-power beam width (HPBW) and

_{a}*η*for different feed locations are investigated. Results show an in-crease in side-lobe levels (SLLs) and a slightly reduction in

_{a}*η*with quantization-phase augmentation or element phase-range reduction, however, the effects on HPBW are negligible. Nevertheless, the degradation in

_{a}*η*is negligible when the quantization-phase is lower than 30° or phase-range is more than 300°. Parametric studies have been carried out to provide design guidelines to maximize

_{a}*η*. It is perceived that the offset-angle plays an important role to determine

_{a}*η*, especially for feed with narrow beam width.

_{a}Keywords

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*Wireless Engineering and Technology*,

**9**, 66-78. doi: 10.4236/wet.2018.93006.

1. Introduction

Reflect array (RA) antenna is comprised of a quasi-periodic set of unit elements mostly set in a regular lattice to emulate a specific phase-front transformation [1] [2] [3] . RAs have many technological benefits compared to parabolic reflectors [4] , like ameliorated cross-polarization performance due to the polarization sensitive elements, feed blockage reduction with center-fed offset-beam structure, simply folded mechanism for packaging and transportation by division into small segments, keeping the price low with easy manufacturing process for shaped-beam RAs.

Considering the electrically large size of the RAs, composed of many elements with small size lower than a wavelength, a full-wave simulation requires a considerably high computational time and huge resource. Similar to the look of the standard reflectors, η_{a} and radiation properties need to be predicted in initial RA design procedure to judge the antenna performances. A design parameter r_{f} (is the focal length to aperture side length ratio) should be correctly selected. Larger r_{f} value results in smaller phase sensitivity to frequency variation and typically enhances radiation performance in terms of cross-polarization level, gain bandwidth, and scan capability. On the other hand, it increases feed size and overall profile of RA antenna, therefore, it demands more mechanical effort to hold the antenna exactly in place [4] . A smaller r_{f} increases variation of spatial phase delays and causes large incident angles for edge elements.

The main objective of this paper is to study aperture efficiency of a square planar space-fed system. In practice, the phase of each RA element is chosen to resemble the nearest quantization phase. Besides, some phasing elements have a phase-range lower than 360˚. An investigation is presented to survey these errors on side-lobe levels (SLLs), half-power beam widths (HPBWs) and η_{a} of RAs. By plotting η_{a} versus configuration parameters, an economical and comparatively correct approximated design procedure ought to be introduced. Comparison between the center- and offset-fed square RAs for different feed locations is given. Square apertures are suitable for development of small spacecrafts based on the Cube Sat standard (3U, equal to 30 cm), which has grown considerably in recent years for low-cost space missions [5] [6] .

2. Aperture Efficiency Analysis

Figure 1 shows a typical configuration of a square RA system consist of an array of radiating elements and a feeding source. As demonstrated in Figure 1, four sets of coordinate systems are usually employed to analysis an RA. The subscript f points out the feed coordinates where
${r}_{f}$ is the location of feed phase center. In Figure 2 the feed beam direction (FBD) is marked by P_{o}(x_{o}, y_{o}, 0) where the maximum radiation of the feed horn is directed. The θ_{o} and θ_{e} are offset-angle of feed source and incidence-angle of mn^{th} element, respectively. The system configuration parameters are listed in Table 1. In Figure 3 Σ points to the spherical surface centered at feed phase center. A shows the RA aperture that is specified by the aperture boundary, and surface σ shares a similar angle with A and Σ [7] .

Table 1. The configuration parameters.

Figure 1. The coordinate systems of a typical reflect array antenna.

Figure 2. Configuration parameters of a typical reflect array antenna.

Figure 3. Spherical geometry of a typical reflect array antenna.

The η_{a} of the whole RA is presented by the product of several sub-efficiency factors [8] [9] :

${\eta}_{a}={\eta}_{spill}\times {\eta}_{ph}\times {\eta}_{t}\times {\eta}_{b}\times {\eta}_{x}\times {\eta}_{e}\times {\eta}_{et}\times {\eta}_{o}$ . (1)

where η_{spill}, η_{ph}, η_{t}, η_{b}, η_{x}, η_{e} and η_{et} are the efficiencies for spill-over, phase, taper, blockage, polarization, element and edge-taper, respectively. The η_{o} is the sum efficiencies of manufacturing accuracy, environmental factors, assembling errors and measurement mechanism. The η_{spill} is part of radiated power emanated at the feed on the RA aperture. The illumination efficiency (η_{ill}) is the product of η_{t} and η_{ph} as follows η_{ill} = η_{t} × η_{ph}. The η_{t} and η_{ph} are the uniformity of amplitude and phase distribution over the RA aperture. At design frequency, the phase error is almost zero once the element achieves complete phase range of 360˚. Therefore, some authors solely take into account η_{t} in η_{ill}. Among these sub-efficiencies, the product of η_{spill} and η_{ill} has the major effect on η_{a}. Finally, typical sub-efficiencies for RAs are tabulated in Table 2 [7] [8] [9] .

Element dimensions can be determined by the phase versus element change curve, when the necessary phase shift for each element is computed. However, element dimensions vary by a discrete value associated with the fabrication resolution, therefore, a sustained phase control is impossible. The discrepancy between desired element phase and quantized phase of the chosen element is classified as quantization-phase error. In this section, a study is undertaken to investigate the effects of quantization-phase errors on SLLs, HPBW and η_{a}. A broadside center-fed 30 cm square RA with sub-wavelength unit elements (lambda/3 at 10 GHz), r_{f} = 1, q_{f} = 8.2, and q_{e} = 0.85 is used in this study. The feed and element patterns are respectively modeled by the
${\mathrm{cos}}_{f}^{2q}\left({\theta}_{f}\right)$ and
${\mathrm{cos}}_{e}^{2q}\left({\theta}_{e}\right)$ due to its simplicity. Figure 4 shows the phase distribution of the RA aperture at the ideal and various quantization phase, 45˚, 90˚ and 180˚ equal to 3-, 2- and 1-bit(s), respectively. Figure 5 shows the radiation pattern which calculated by array-theory method [10] .

(a) (b) (c) (d)

Figure 4. Phase distribution for different quantization-phase values. (a) 0˚; (b) 45˚; (c) 90˚; (d) 180˚.

(a)(b)

Figure 5. Radiation patterns for different values of: (a) quantization-phase; (b) phase-range.

Table 2. Typical sub-efficiencies for reflect array antennas.

Sub-wavelength single resonance phasing elements have a phase-range below 360˚. So, some elements have unachievable phase shift. The radiation patterns for various element phase-ranges of 30 cm side-length square RA with θ_{o} = 0˚, θ_{b} = 0˚, r_{f} = 1, q_{f} = 8.2, and q_{e} = 0.85 are represented in Figure 5(b). As can be observed from Figure 5(a) and Figure 5(b), no grating lobe appears due to the pseudo-random distribution of phase errors [11] . These effects are compared in Table 3 and Table 4. It is noticed that these errors increase SLLs, but have no major effects on the HPBW. Even the 45˚ quantization-phase or 180˚ phase-range cases has similar HPBW as RA with ideal phases. It can be a very helpful to cut back to the system complexity and cost, when the HPBW is a major demand.

A parametric study has been performed for center- and offset-fed square aperture RA, with side length 30 cm, λ/3 element spacing at 10 GHz, q_{e} = 0.85, x˳ = 0 and y_{o} = 0, with different r_{f}, for each case q_{f} is considered to be maximum η_{a}. In this study, the offset-feed and main beam angles are equal. The η_{a} is derived from gain value [12] , which includes η_{spill}, η_{ph} and η_{t}. In Figure 6, the acceptable quantization-phase without η_{a} reduction is around 30 and the threshold for phase-range is around 300˚. It shows reduction of η_{a} depends on quantization-phase or phase-range values, however, it is independent of offset-angle and feed location. The η_{a} reduction occurs when the SLLs is increased which in turn causes gain loss.

(a)(b)

Figure 6. Reflect array antenna aperture efficiency versus. (a) Quantization-phase; (b) Phase-range.

Table 3. Effects of quantization-phase errors on reflect array antenna.

Table 4. Effects of limited phase-range errors on reflect array antenna.

3. Aperture Efficiency Study

Parameters of square aperture RA with a side length of 10λ_{0} (λ_{0} is lambda at 10 GHz) are studied. Two designs with a θ_{o} = 0˚, q_{f} = 8.2; and θ_{o} = 20˚, q_{f} = 2.8 are considered, other parameters are: x_{o} = 0, y_{o} = 0, and q_{e} = 0.85. η_{spill}, η_{ill} and η_{a} = η_{spill} × η_{ill} are plotted in Figure 7 and as shown the maximum accessible η_{a} value for center-fed (76.44%) is greater than offset-fed (62.58%). Also, in Figure 7 as r_{f} grows the η_{spill} decreases due to a bigger r_{f} reduced aperture angle of the RA plane with respect to the feed source. In addition, η_{ill} increases since it forms a more uniform field distribution on the array.

For a precise design, the effects of excitation angle (θ_{inc}, φ_{inc}) for each element ought to be considered. Figure 8 displays the range and distribution of plane wave excitation angles in the RA aperture. It can be observed that the upper part of the RA aperture has a maximum θ_{inc}. So, the element spacing should have selected small enough that no distributed grating lobe radiated [1] . Comparison of Figure 8(a) and Figure 8(b) indicates that the percentage of aperture area illuminated with incident angle larger than 30˚ in the central case is less than 5.6% whereas in offset-feed RA is about 66.7%. Therefore, the impact of incidence angle is anticipated to be more significant in offset fed RAs. So, the use of sub-wavelength element appears to be necessary [13] . In the RA design process, an excitation plane wave can always be decomposed into a combination of the zero TE- and TM-waves [1] . In [14] , it is indicated that magnitude of reflection components depends on both the θ_{inc} and φ_{inc}, and it ought to be considered for every element.

The feed position is determined by offset angle (θ_{o}) and the distance r_{f}. Figure 9(a) and Figure 9(b) show the η_{a} versus r_{f} and q_{f} with other parameters set as: x_{o} = 0, y_{o} = 0, and q_{e} = 0.85. In Figure 9(a), in center-fed case for every r_{f} it is possible to find a q_{f} that maximize η_{a}. However, in offset-fed RA, the maximum of η_{a} is obtained just for lower r_{f} and q_{f} values, as presented in Figure 9(b). In Figure 9(a), an η_{a} around 70% is realized for various mixtures of r_{f} and q_{f}, however for larger r_{f} and q_{f} a wider bandwidth can be achieved [12] . Since the feed is

(a)(b)

Figure 7. η_{ill}, η_{spill}, and η_{a} for square aperture. (a) θ_{o} = 0˚, q_{f} = 8.2; (b) θ_{o} = 20˚, q_{f} = 2.8.

(a) (b) (c) (d)

Figure 8. Excitation angles distribution on the RA aperture θ_{inc} for (a) θ_{o} = 0˚, r_{f} = 1; (b) θ_{o} = 20˚, r_{f} = 0.6; φ_{inc} with mapping into the 0˚ - 90˚ for (c) θ_{o} = 0˚, r_{f} = 1; (d) θ_{o} = 20˚, r_{f} = 0.6.

(a) (b)

Figure 9. The η_{a} for square aperture versus both q_{f} and r_{f}.(a) θ_{o} = 0˚; (b) θ_{o} = 20˚.

placed in y-z plane, the plane of incidence, it is proper to use the coordinate (y_{f}, z_{f}) for parametric study. The results are displayed in Figure 10(a) and Figure 10(b) for q_{f} = 2.8 and q_{f} = 8.2, respectively, other fixed parameters are: x_{o} = 0, y_{o} = 0, and q_{e} = 0.85. It is noted that once q_{f} = 2.8, η_{a} is varied from 60% to 70%, and when q_{f} = 8.2, the maximum η_{a} is achieved at higher feed position. Larger q_{f} value yields a narrower feeding beam width. So, a larger r_{f} ought to have an additional uniform field distribution on the aperture. Using this contour map, one might find a correct feed location. The maximum η_{a} is acquired close to z_{f} = 290 mm, as determined in Figure 10(b). The η_{a} keeps nearly constant until y_{f} is shifted to -50mm with z_{f} fixed in 297 mm.

The contoured η_{a} plot versus x_{o} and y_{o} are depicted in Figure 11, with the fixed parameters: r_{f} = 1.0, θ_{o} = 0˚, q_{f} = 8.2, and q_{e} = 0.85 for center-fed; r_{f} = 0.6, θ_{o} = 20˚, q_{f} = 2.8, and q_{e} = 0.85 for offset-fed RA. It is noted that for offset case maximum efficiency is obtained once the feeding beam is pointed at 13 mm away from aperture center, when η_{a} reaches 63%. Besides, the symmetry of the η_{a} with respect to x-axis is observed. In most RAs x_{o} = 0, and y_{o} = 0, therefore center elements have a stronger illumination and contribute more to total radiation. Accordingly, the useful information of an RA performance can be deduced without simulating whole structure. For example, a decent approximation of the RA gain bandwidth can be estimated by calculating the scattering from the middle row of a large RA enclosed by perfectly magnetic conductor (PMC) boundaries [15] . For gain bandwidth enhancement, one might place elements with smaller reflection loss at the central area of the aperture. This can be done by adding a phase constant to the phase distribution over the RA aperture. Considering both q_{f} and q_{e} results a contour plot of η_{a} in Figure 12 with constant parameters: r_{f} = 1.0, θ_{o} = 0˚, x_{o} = 0, and y_{o} = 0 for center-fed; r_{f} = 0.6, θ_{o} = 20˚, x_{o} = 0, and y_{o} = 0 for offset-fed. Note that the parameters q_{e} solely effects η_{ill}. Figure 12(a) and Figure 12(b) show the fact that the q_{e} plays a smaller role in the η_{a} than q_{f}.

Finally, effects of configuration parameters are studied separately. Figure 13(a)

(a) (b)

Figure 10. The η_{a} versus feed location in y-z plane. (a) q_{f} = 2.8; (b) q_{f} = 8.2.

(a) (b)

Figure 11. The η_{a} versus feed orientation P_{o}(x_{o}, y_{o}, 0). (a) Center-fed; (b) Offset-fed.

(a) (b)

Figure 12. The aperture efficiency versus both q_{f} and q_{e}. (a) Center-fed; (b) Offset-fed.

demonstrates the maximum η_{a} at r_{f} = 1.02 once the feed has θ_{o} = 0˚ and q_{f} = 8.2. Comparison of center- and offset-cases shows the importance of right selection of RA parameters. Selecting an incorrect q_{f} and r_{f} for a given RA leads to a considerably low η_{a}. Another curve, shown in Figure 13(b), provides η_{a} as a function of θ_{o}. The maximum η_{a} appears at the center feed position (θ_{o} = 0˚) and

(a) (b) (c) (d)

Figure 13. The aperture efficiency versus. (a) r_{f}; (b) θ_{o}; (c) q_{f}; (d) q_{e}.

maintains a certain offset angle depending on r_{f} and q_{f} values. For RAs with higher r_{f} and q_{f} this offset-angle is around 10˚ and experience shows a maximum of 15˚ offset-angle is allowed for acceptable η_{a}. The curve of feed pattern function is described in Figure 13(c). The optimum q_{f} for center-fed case with r_{f} = 0.6 is 3.2 and 6.3 for the 20˚ offset-fed RA with r_{f} = 1.0. The variation of q_{f} to obtain a maximum η_{a} between center- and offset-fed for r_{f} = 0.6 is smaller amount than r_{f} = 1.0. Likewise, the curve of the η_{a} versus q_{e} is depicted in Figure 13(d). RAs with smaller r_{f} and offset-fed location, the η_{a} is further attenuated by the element pattern.

4. Conclusion

The quantization-phase and limited phase-range errors reduce RA antenna efficiency and increase SLLs, however, the HPBW remains mostly constant. The maximum acceptable quantized phase with negligible η_{a} diminution is around 30˚ and this threshold for limited phase-range is near 300˚. The limitations are independent of offset angle and feed location. Based on conducted parametric studies for a square aperture RA with side length 30 cm, it was observed that the appropriate selection of r_{f} and q_{f} has a significant effect on η_{a} and a center-fed RA has the maximum η_{a}. However, the η_{a} preserves its behavior up to 15˚ for offset-fed with smaller r_{f} and q_{f}.

Conflicts of Interest

The authors declare no conflicts of interest.

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