IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 11, NOVEMBER 2015
Wide-Angle Scanning Phased Array Using an Efficient Decoupling Network Run-Liang Xia, Shi-Wei Qu, Peng-Fa Li, De-Qiang Yang, Shiwen Yang, and Zai-Ping Nie
Abstract—The active impedance of each element in a phased array changes substantially with scan angle due to mutual coupling. Therefore, a key challenge in the design of wide-angle scanning phased array is to achieve wide-angle impedance matching (WAIM). In this communication, an efficient decoupling network is developed to reduce the mutual coupling between adjacent elements for this purpose. A complete scattering matrix analysis of the decoupling network is given and the design procedure is summarized. A 1 × 16 linear microstrip phased array has been designed and fabricated for experimental verification. Good agreement is obtained between measured and simulated results. The measured mutual coupling between adjacent elements is reduced to lower than −35 dB at the center frequency. Due to the reduced mutual coupling, the array can experimentally scan to 66◦ with a gain reduction of only 3.04 dB. Index Terms—Decoupling network, phased array, wide-angle scanning.
I. I NTRODUCTION Phased array antennas with wide-angle scanning capability are always pursued for a variety of applications. However, unavoidable mutual coupling among array elements causes difficulties in impedance matching of the system over a wide scan range. The severe mismatching at large angles can result in a sharp drop of the array realized gain. Therefore, a key issue in the design of wide-angle scanning phased array is to achieve wide-angle impedance matching (WAIM). In the past decades, many research works have been carried out to achieve WAIM. Reducing or compensating the mutual coupling between adjacent elements is a very intuitive and natural way. In 1960, Edelberg and Oliner [1] investigated the use of conducting baffles between dipole elements to compensate the mutual coupling. However, these baffles can also cause the scan blindness [2]. In 1966, Magil and Wheeler [3] proposed a method of placing a dielectric sheet in front of a waveguide array to remove some of the susceptance variation as the array scans. It has been successfully applied to improve the scan range in tightly coupled dipole array (TCDA) [4]. However, in practice, the permittivity, thickness, location, and number of the dielectric layers should be carefully chosen and optimized. Meanwhile, the additional dielectric layers often introduce surface waves and increase the weight, profile, and cost of the system. Instead of traditional dielectric layer, using a metamaterial layer with anisotropic properties to achieve WAIM was reported in [5]. It can provide more degrees of freedom than traditional isotropic dielectric layer. However, the material properties specified in [5] are very difficult to be achieved in practice. In recent years, periodic structures such as electromagnetic band gap Manuscript received Januvary 23, 2015; revised May 12, 2015; accepted August 29, 2015. Date of publication September 02, 2015; date of current version October 28, 2015. This work was supported in part by the Natural Science Foundation of China (NSFC) Projects under Grant 61371051 and Grant 61101036, and in part by the General Financial Grant from the China Postdoctoral Science Foundation under Grant 2012M521682 and Grant 2013T60847. The authors are with the School of Electrical Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China (e-mail:
[email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2015.2476342
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(EBG) structures [6], ground defected structures (DGS) [7], are often used to reduce the mutual coupling in microstrip antenna arrays by suppressing surface waves. However, some difficulties may be encountered in practical applications. For example, EBG structures require enough units to keep the periodic property, which will occupy much space between adjacent elements, then impose restrictions in array applications. For DGS, various patterns printed on the ground may lead to strong backward radiation. In summary, the aforementioned methods are all based on the concept of reducing or compensating the mutual coupling by adding extra structures. Another kind of scheme to achieve WAIM is to design a coupling compensation network connected to the antenna array. In [8], by interconnecting the feed lines through lumped capacitors, the mutual coupling could be compensated, and thereby the impedance matching over wide scan angles was attained. Meanwhile, the authors also mentioned that the lumped capacitors may be replaced by inductors, transmission lines, or more complex networks. Recently, Bhattacharyya studied the scan performance of phased arrays with overlapped subarrays [9]. In his work, the overlapped subarrays are realized by overlapping the two-way power dividers in the feeding network. Consequently the feed lines of all array elements are actually connected to each other, which is similar to the connected circuits in [8]. This method was experimentally investigated in our previous work [10] which has been proven useful in achieving WAIM but in a very narrow band. In [11], Cook and Pecina proposed an idea by adding a Bethe coupling hole in the common wall of adjacent waveguide elements. If the hole is properly designed, an indirect coupling wave can be transferred from one waveguide to the other. The deliberately introduced indirect coupling has a phase and amplitude that can cancel the direct aperture-coupled wave propagating toward the generator. Unfortunately, no experimental results can be found therein. Furthermore, the heavy and bulky waveguide elements are not welcome in many applications. A similar work was reported in [12] in which the mutual coupling between microstrip patch elements is compensated by using directional couplers to introduce the indirect coupling. However, it needs lumped resistors to terminate the isolated ports of the couplers, which will introduce obvious loss in the network, especially in large arrays. Briefly, the aforementioned methods are all based on the concept of connecting the feed lines in a direct or indirect way. In our previous work [13], a decoupling network using connected couplers but without any lumped components was proposed. In this communication, we significantly extend the work reported in [13] by giving a complete scattering matrix analysis and a summarized design procedure. Then the decoupling network is applied to a 16element linear microstrip antenna array. Consequently, the array can experimentally scan to 66◦ with gain reduction of 3.04 dB.
II. S CATTERING M ATRIX A NALYSIS To begin with the simplest case, a two-element array in Fig. 1(a) is first assumed, and each element is matched to an impedance of 50 Ω. The scattering matrix at the reference plane can thus be expressed as [SA ] =
0 |SA12 | ejϕA jϕA |SA12 | e 0
(1)
where |SA12 | and ϕA are the magnitude and phase of the coupling coefficient between the two elements, respectively. A 90◦ coupled line coupler is adopted in this case, as shown in Fig. 1(b). Each port is assumed to match to an impedance of 50 Ω. An
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Then (3a)–(3c) can be reduced as SD12 = −Co2 e−jθ1 /(1 + T r 2 ) 2
SD13 = −2T r/(1 + T r ) 2 −jθ2
SD34 = −Co e
(5a) (5b)
2
/(1 + T r ).
(5c)
Finally, the output ports (ports 3d and 4d) of the decoupling network are connected to the array feed ports (ports 1a and 2a). The total scattering matrix, including the array and the decoupling network, can be derived as 0 S12 (6a) [S] = S12 0 Fig. 1. Schematics of (a) two-element array. (b) Coupled line coupler. (c) Decoupling network.
ideal isolation is considered between the input and the isolated ports. Thus, the scattering matrix of the coupler can be expressed as ⎡
0
⎢ SC12 [SC ] = ⎢ ⎣ 0 SC14
⎤ SC12 0 SC14 0 SC14 0 ⎥ ⎥ SC14 0 SC12 ⎦ 0 SC12 0
(2a)
where SC12 = T rejϕC12 SC14 = jCoe
jϕC14
(2b) .
(2c)
Because the coupler is assumed to be lossless, the parameters Co and Tr satisfy the relation Co2 + T r 2 = 1.
(2d)
Fig. 1(c) shows the schematic of the proposed decoupling network. It consists of two coupled-line couplers and two sections of transmission lines with a characteristic impedance of 50 Ω and electrical lengths of θ1 and θ2 . After a simple derivation, the scattering matrix of the decoupling network can be written as ⎡
0
⎢ SD12 [SD ] = ⎢ ⎣ SD13 0
⎤ SD12 SD13 0 0 0 SD13 ⎥ ⎥ 0 0 SD34 ⎦ SD13 SD34 0
where S12 =
SA12 − Co2 e−jθ1 /(2 − Co2 ) . 1 + SA12 Co2 e−jθ2 /(2 − Co2 )
(6b)
A zero value of S12 means an ideal cancellation of the mutual coupling. Then, from (6b), the required parameters Co and θ1 can be worked out (7a) Co = 2 |SA12 | /(1 + |SA12 |) θ1 = −ϕA .
(7b)
From the above analysis, it is quite clear that the parameters Co and θ1 represent the amplitude and phase of the indirect coupling introduced by the network, respectively. If designed properly, the indirect coupling can completely cancel out the direct coupling completely between the array elements. Based on the derivation process, a simple design procedure for the decoupling network can be summarized as follows. 1) Calculate the array by using full-wave simulation software to obtain the values of |SA12 | and ϕA . Then determine the initial values of Co, θ1 , and θ2 using (7). 2) Design a directional coupler according to the parameter Co and make sure that ϕC12 = π. Convert the electric lengths θ1 and θ2 to their physical dimensions. Then, connect two couplers by the two transmission lines to form the decoupling network. 3) Connect the decoupling network to the array. Adjust the joint location, i.e., the reference plane, and the lengths of the two transmission lines to achieve the best decoupling performance.
III. A PPLICATION TO A 16-E LEMENT A RRAY (3a)
where
2 2 (3b) e−jθ1 / 1 − SC12 e−j(θ1 +θ2 ) SD12 = SC14 2 2 SD13 = SC12 (1 + SC14 e−j(θ1 +θ2 ) /(1 − SC12 e−j(θ1 +θ2 ) )) (3c) 2 2 e−jθ2 /(1 − SC12 e−j(θ1 +θ2 ) ). (3d) SD34 = SC14
Now if θ1 + θ2 = π, 3π, 5π . . .
(4a)
ϕC12 = π.
(4b)
A. Preparing Work For an element in a linear array, the mutual coupling from two adjacent elements on each side is strong, which should be reduced. Following the design procedure, a 16-element array without the decoupling network was first simulated to evaluate the mutual coupling between adjacent elements. As depicted in Fig. 2, the patch element is etched on the upper substrate with a relative permittivity of 2.2 and thickness of 3 mm. The center-to-center distance between the two patches is 0.5 λ0 (λ0 is the free space wavelength at the center frequency of 7.5 GHz). The lower substrate has an identical relative permittivity and thickness of 0.5 mm, which is separated from the upper substrate by an aluminum board 5 mm thick. Each patch element is matched to a 50 microstrip feed line by a 0.8-mm diameter probe passing through a 2-mm diameter hole in the aluminum ground. The specific size of an array unit cell is given in Fig. 3.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 11, NOVEMBER 2015
Fig. 2. Array configurations. (a) Top view. (b) Cross-sectional view. (c) Bottom view.
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Fig. 4. Simulated active reflection coefficient and realized gain.
Though the design equations may be different from one kind of coupler to another due to different scattering matrices, the decoupling mechanism is consistent. B. Array Unit Cell Design
Fig. 3. Array unit cell. (a) Top view. (b) Bottom view.
The simulated S9,8 and S9,10 are −22.5 dB which means that a coupler with −8.6 dB coupling is required according to (7a). Although the mutual coupling in this case is not very strong, it still greatly deteriorates the impedance matching at large scan angles. The simulated ϕA at the reference plane [d = 4.3 mm, Fig. 2(c)] is −137.2◦ . Then the initial values of the electric lengths of the two transmission lines can be obtained: θ1 = 137.2◦ and θ2 = 402.8◦ . The corresponding physical lengths of θ1 and θ2 are 11.2 and 32.7 mm, respectively. A six-port coupler has been designed to form the decoupling network as shown in Fig. 3(b). The simulated magnitudes of the coupling coefficients from the input port to the two coupled ports are −9.3 dB, which is a little weaker than the required value of −8.6 dB. But the mutual coupling still can be theoretically reduced to as low as −38 dB according to (6b). In general, traditional microstrip quarter-wavelength coupled-line coupler can provide a maximum coupling of about −8 dB [14]. If the mutual coupling is stronger than the given example, e.g., −15 dB, the decoupling network can only reduce the mutual coupling to −20.6 dB according to (6b). Therefore, the main limitation of this technique is the maximum coupling coefficient a coupler can provide. To reduce the mutual coupling to a lower level, e.g., below −30 dB, it is necessary to adopt other kinds of couplers with coupling of at least −5.8 dB.
To avoid spending a lot of time in full-wave simulations of the whole array, an array unit cell has been built for parameter optimization, as plotted in Fig. 3. Periodic boundary condition is established at right- and left-hand sides of the unit cell to simulate the array environment. Note that to be consistent with the number of the finite array elements, the number of the unit cell is set to be 16. Although the reduced mutual coupling cannot be shown up directly in the simulated results of the unit cell, it is reflected by the improved WAIM. To make the active reflection coefficient be lower than −15 dB over the entire scan range, the parameters d, θ1 , and θ2 were optimized. After optimization, θ2 is fixed to be 446.3◦ and the joint is moved to a new location, i.e., d = 4.5 mm in Fig. 3. The optimized active reflection coefficient is depicted by black curve with rectangular symbols in Fig. 4, which remains lower than −15 dB from 0◦ to 70◦ . For comparison, the results obtained by using conventional feed technique (the array element is fed separately and matched well near boresight scanning) are also depicted by black curve with up triangle symbols in Fig. 4. It increases as the array scans away from boresight. The variations of the realized gain versus scan angle are also shown by red curves in Fig. 4. The gain reduction in the decoupling network case is only 2.64 dB when the array scans from boresight to 70◦ , while the gain reduction in conventional feed case is 3.88 dB. The gain improvement is impressive when using the proposed decoupling network especially in a large array. In our original simulation model, an ideal wave-port was employed to feed the microstrip line. In practice, the microstrip line is fed by an SMA connector. Thus, to take the SMA connector into account, the SMA connector was also modeled in the new model. Meanwhile, to eliminate the effect on impedance matching brought by the discontinuity from SMA connector to the microstrip line, a transition section is designed as plotted in Fig. 3(b). The simulated active reflection coefficient of the new model still remains lower than −15 dB from 0◦ to 70◦ while the gain reduction is increased to 4.05 dB, as shown in Fig. 4. Then, it was found that this unexpected phenomenon is caused by the destructive combinations of the scattering from the periodic SMA connectors. To solve this problem, all the SMA connectors were fixed on a 2-mm thick aluminum board, as shown in Fig. 6(b). The parameter
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 11, NOVEMBER 2015
Fig. 5. Active reflection coefficient varies with frequency at different scan angles.
Fig. 7. Simulated and measured scattering parameters.
Fig. 8. Radiation pattern measurement setup.
Fig. 6. Antenna prototype. (a) Top view. (b) Bottom view.
θ2 was then optimized to be 483.3◦ (39.2 mm) while other parameters remain unchanged. As depicted in Fig. 4, the active reflection coefficient of the improved model still remains lower than −15 dB over the entire scan range. Importantly, the gain reduction at 70◦ is only 2.28 dB. The variation of active reflection coefficient versus frequency in the scan range (0◦ –70◦ ) is presented in Fig. 5. A weak blindness appears at around 7.7 GHz when the array scans to 70◦ due to the surface wave propagation [15]. And note that the grating lobe starts to enter into the real space at 7.73 GHz when the array scans to 70◦ . The −10 dB impedance bandwidth over the scan range is 3.3% (7.35–7.6 GHz). The active reflection coefficient of a center element in a finite array is also depicted in Fig. 5 for comparison. The infinite and finite array results show a reasonable agreement, which confirms the effectiveness of unit cell in evaluating and optimizing the array performance. C. 1 × 16 Finite Array Experimental Results Based on the design described in previous section, a 1 × 16 array prototype was fabricated for measurements as shown in Fig. 6.
To provide a uniform neighboring environment for the elements near the array edge, another two dummy elements terminated by matched loads are put at each end of the array. The finite array prototype was calculated for comparison by full-wave simulation method. A comparison plot of the simulated and measured embedded reflection coefficient (S99 ) of the center element (no.: 9) is presented in Fig. 7. The main difference appears at frequencies below 7.4 GHz primarily due to the assembly errors. Coupling coefficients S9,7 , S9,8 , S9,10 , and S9,11 are also depicted in Fig. 7 and a reasonable agreement is obtained between the measured and the simulated results. The measured S9,8 and S9,10 is lower than −35 dB at the center frequency, which again validates the theoretical analysis. Fig. 8 shows the radiation pattern measurement setup. Five fourway power dividers are used to provide a uniform excitation to each element. The array is scanned by adjusting the lengths of the feed cables. Two sets of cables for boresight and 70◦ scanning were handmade by the authors so that the average phase error of all cables is relatively large, about ±7◦ . The measured radiation patterns normalized to the maximum gain at 0◦ are shown in Fig. 9. The array scans to 66◦ instead of 70◦ mainly due to the phase errors brought by the feed cables. The simulated radiation patterns are also depicted for comparison. The measured and simulated patterns agree well at boresight. When the array scans to 66◦ , the measured gain reduction is 3.04 dB while the simulated is 2.12 dB, which is primarily due to the large phase error. The simulated radiation patterns of the finite array without the decoupling network are also depicted in Fig. 9. The simulated gain reduction is 3.6 dB for 66◦ scanning, which is about 1.5 dB worse than that in the array with the decoupling network.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 11, NOVEMBER 2015
Fig. 9. Finite array E-plane radiation patterns scanned to 0◦ and 66◦ .
IV. C ONCLUSION A complete scattering matrix analysis and a detailed design procedure of the proposed decoupling network have been presented in this communication. The network is then used to achieve WAIM in a linear microstrip array. Simulated and measured results validate the theoretical analysis and confirm the effectiveness of the decoupling network for WAIM. The array can experimentally scan to 66◦ with gain reduction of only 3.04 dB. However, the simulated −10 dB bandwidth over the scan range (0◦ −70◦ ) is only 3.3% primarily due to the inherently narrow decoupling bandwidth of the network and the narrow band properties of the probe fed microstrip patch element. The reason of narrow decoupling bandwidth is that the network is designed at the center frequency so that (7a) and (7b) cannot maintain over a wide bandwidth. A possible way to extend the decoupling bandwidth is to cascade several decoupling networks designed at different frequencies and those arrays with wider bandwidth are always helpful to extent the bandwidth. Further investigations will focus on improving the bandwidth over a wide scan range. ACKNOWLEDGMENT The authors would like to thank L.-W. Zhang in UESTC for helping with the measurement data entry work. R EFERENCES [1] S. Edelberg and A. Oliner, “Mutual coupling effects in large antenna arrays II: Compensation effects,” IRE Trans. Antennas Propag., vol. 8, no. 4, pp. 360–367, Jul. 1960. [2] S. Mailloux, “Surface waves and anomalous wave radiation nulls in phased arrays of TEM waveguides with fences,” IEEE Trans. Antennas Propag., vol. 16, no. 2, pp. 160–166, Jan. 1972. [3] E. G. Magil and H. A. Wheeler, “Wide-angle impedance matching of a planar array antenna by a dielectric sheet,” IEEE Trans. Antennas Propag., vol. 14, no. 1, pp. 49–53, Jan. 1966. [4] B. Munk et al., “A low-profile broadband phased array antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 2003, vol. 2, pp. 448–451. [5] S. Sajuyigbe, M. Ross, P. Geren, S. A. Cummer, M. H. Tanielian, and D. R. Smith, “Wide angle impedance metamaterials for waveguide-fed phased-array antennas,” IET Microw. Antennas Propag., vol. 4, no. 8, pp. 1063–1072, Aug. 2010. [6] F. Yang and Y. R. Samii, “Microstrip antennas integrated with electromagnetic band-gap EBG structures: A low mutual coupling design for array applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2936–2946, Oct. 2003.
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[7] C. Y. Chiu, C. H. Cheng, R. D. Murch, and C. R. Rowell, “Reduction of mutual coupling between closely-packed antenna element,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1732–1738, Jun. 2007. [8] P. Hannan, D. Lerner, and G. Knittel, “Impedance matching a phased array antenna over wide scan angles by connecting circuits,” IEEE Trans. Antennas Propag., vol. 13, no. 1, pp. 28–34, Jan. 1965. [9] A. K. Bhattacharyya, “Floquet modal based analysis of overlapped and interlaced subarrays,” IEEE Trans. Antennas Propag., vol. 60, no. 4, pp. 1814–1820, Apr. 2012. [10] R.-L. Xia, S.-W. Qu, X. Bai, Q. Jiang, S. Yang, and Z.-P. Nie, “Experimental investigation of wide-angle impedance matching of phased array using overlapped feeding network,” IEEE Antennas Wireless Propag., vol. 13, pp. 1284–1287, Jul. 2014. [11] J. Cook and R. Pecina, “Compensation coupling between elements in array antennas,” in Proc. Antennas Propag. Soc. Int. Symp., Jul. 1963, pp. 234–236. [12] S. M. Duffy, G. A. Brigham, and J. S. Herd, “Integrated compensation network for low mutual coupling of planar microstrip antenna arrays,” in Proc. IEEE Antennas Propag. Int. Symp., Jun. 2007, pp. 1389–1392. [13] R.-L. Xia, S.-W. Qu, P.-F. Li, Q. Jiang, and Z.-P. Nie, “An efficient decoupling feeding network for microstrip antenna array,” IEEE Antennas Wireless Propag., vol. 14, pp. 871–874, Apr. 2015. [14] R. Mongia, I. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA, USA: Artech House, 1999, pp. 1–21. [15] D. M. Pozar and D. Schaubert, “Scan blindness in infinite phased arrays of printed dipoles,” IEEE Trans. Antennas Propag., vol. 32, no. 6, pp. 602–610, Jun. 1984.
Ray Tracing Theory in a Radially Uniaxial Sphere Mohsen Yazdani, Jay Kyoon Lee, Joseph Mautz, Ercument Arvas, and Kepei Sun
Abstract—Ray tracing theory for a radially uniaxial dielectric sphere is presented in this communication. Reflection and transmission of the plane waves obliquely incident on the spherical interface between air and uniaxial are briefly examined and the general formulations for phase velocities as well as the refractive indices of the ordinary and extraordinary waves propagating in a radially uniaxial medium are investigated. The ray tracing method is implemented for both ordinary and extraordinary rays and the results are applied to evaluate the propagation paths and arrival times of the shortcut wave returns appearing in the transient backscattering response of a uniaxial dielectric sphere. The effect of the variation in relative permittivity on the arrival times of shortcut waves is studied. The results of the ray tracing method are compared with those computed using the Mie and Debye series solutions with good agreement. Index Terms—Debye series, extraordinary wave, Mie series, ray tracing method, transient scattering, uniaxial sphere.
I. I NTRODUCTION The process of electromagnetic scattering by isotropic and nonisotropic dielectric spheres has been an attractive subject over the past few decades. A rigorous EM scattering solution by an isotropic dielectric sphere is introduced in [1]. Wong and Chen [2] developed a generalized Mie series scattering solution for a radially uniaxial Manuscript received October 25, 2014; revised June 22, 2015; accepted June 07, 2015. Date of publication September 07, 2015; date of current version October 28, 2015. The authors are with the Syracuse University, Syracuse, NY 13244 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2015.2477100
0018-926X © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
