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Development of 2.5D Analytical Regularization Method for reflector antenna analysis
Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Regular paper
Development of 2.5D Analytical Regularization Method for reflector antenna analysis q Okan Mert Yücedag˘ a,⇑, Ahmet Serdar Türk b a b
TUBITAK Informatics and Information Security Research Center, Kocaeli, Turkey Yildiz Technical University, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 7 June 2016 Accepted 19 October 2016 Available online xxxx 2010 MSC: 00-01 99-00
a b s t r a c t A novel and efficient approach, which is called two-and-a-half dimensional Analytical Regularization Method (2.5D ARM), is proposed to analyse reflector antennas for microwave, millimeter wave or ultra-wide band applications. To present computational performance of the proposed method, its results are compared with well known Physical Optics (PO) and Method of Moments (MoM) solutions for a PEC hollow cylinder and an open-ended corner reflector. Moreover, radiation pattern of a ridged horn fed parabolic reflector antenna is measured to validate the method experimentally. It is observed that either the simulated or the measured data and the 2.5D ARM results are in close agreement with each other. Ó 2016 Elsevier GmbH. All rights reserved.
Keywords: Analytical Regularization Method Reflector antenna Radiation pattern
1. Introduction Reflector antennas, which usually consist a feed to radiate electromagnetic energy and a curved reflecting surface to collimate this energy over a large aperture, have applications in radar, radio astronomy, and microwave communication systems [1]. Therefore, a large number of analytical and numerical techniques have been developed to obtain accurate solutions for reflector antenna radiation problems. Many high frequency techniques, preferred for their relatively low computational memory and time requirements, are suitable for calculating radiation characteristics of electrically large reflector antennas. One of them is Geometrical Optics (GO) method which assumes that electromagnetic energy transports with rays having same character as a plane wave. But, the GO method is unable to calculate fields in shadowed regions and can not deal with diffraction effects. To overcome this failure, Keller extended the GO to include diffracted rays produced by scattering from sharp edges of boundary surfaces and called that Geometrical Theory of Diffraction (GTD) [2]. Afterward, GTD was applied to axially symmetric reflector antennas fed by point source [3], with general rim shapes [4] and illuminated by feeds that do not radiate an
q
Fully documented templates are available in the elsarticle package on CTAN.
⇑ Corresponding author.
exact spherical wave front [5]. However, the Keller’s GTD technique based on a diffraction coefficient which is determined by an asymptotic evaluation of Sommerfeld’s integral. Because of the singularity of the integral, the Keller’s GTD becomes invalid in transition regions adjacent to the shadow boundaries. This singularity can be eliminated by using Uniform Theory of Diffraction (UTD) [6] or Uniform Asymptotic Theory (UAT) [7]. Yazgan and Safak showed that the difference between the UTD and UAT results for focus-fed axially symmetric reflector antenna were insignificant [8]. But, either the GTD or its uniform versions can not calculate the fields at the caustics which occur whenever a group of ray intersect exterior of a reflector antenna. On the other hand, Physical Optics (PO) method, which is based on the integration of the induced currents predicted by the GO, can accurately predict the field in the main beam region and near-side lobes. But, edge diffraction of the reflector may become important for the far-side lobes or cross-polarized field predictions. In this situation, Ufimtsev’s Physical Theory of Diffraction (PTD) [9] should be included to the PO [10]. Also, note that the PO with its extension based on the PTD are useful for calculating the fields in shadow transition and caustic regions [11]. Thus, PO method makes possible to analyse single and dual reflector antennas efficiently [12]. Mentioned studies prove that considering the diffracted fields is necessary to obtain accurate radiation patterns for the reflector antennas. But, it should be emphasized that the diffracted fields
E-mail address: [email protected] (O.M. Yücedag˘). http://dx.doi.org/10.1016/j.aeue.2016.10.020 1434-8411/Ó 2016 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
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O.M. Yücedag˘, A.S. Türk / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
can not be easily calculated in the caustics regions of arbitrary shaped reflectors and the PO method becomes less accurate for antennas whose dimensions are not large in terms of wavelength. However, some computational electromagnetic methods usually deal with first-kind integral equations for determining the surface currents can be preferred to overcome these kind of difficulties. These equations are obtained from the boundary conditions of the antenna problem, and then, they are discretized by using entire or sub-domain basis functions in a classical numerical solution approach such as Method of Moments [13,14]. But, especially Fredholm integral equations of the first kind with continuous kernels cause ill-posed problems which have consequences for its numerical solutions. Specifically, these solutions show low accuracy and can yield numerical instability. Unfortunately, there are not any general theorems proving convergence, or even the existence of an exact solution, for such equations because of their logarithmic or higher-order singular kernels [15]. But, an approach exists to obtain second-kind integral equations of the Fredholm type, with a smoother kernel, from the first kind equations. This causes the condition number of the matrix equations to remain small while the matrix size gets larger. The mentioned approach is called Analytical Regularization Method (ARM). Hence, the ARM can be defined as a mathematical technique, which allows constructing two-sided regulator of initial diffraction problem of the first kind and then it reduces the problem to an equation of the secondkind [16]. In diffraction theory, the first example of the ARM is semiinversion procedure [17]. However, this procedure works only simplest diffraction problems which have an obstacle surface coincides with a part of a coordinate system [18]. The main reason of the ARM existence is to eliminate disadvantages of the semiinversion procedure. At the initial phase, the ARM was applied to the wave scattering by arbitrary profile unclosed cylindrical screen with Dirichlet and Neuman boundary condition by Tuchkin in 1985 [19,20]. Karacuha and Turk implemented ARM to solve two dimensional problem of E-polarized wave diffraction by arbitrary shaped, smooth and perfectly conductive cylindrical obstacle [21]. Depending on this study, in 2006, Turk used the ARM to analyse the aperture illumination and edge rolling effects of two dimensional (2D) parabolic antennas which are illuminated by plane wave [22]. This study showed that a waveguide or horn antenna model needs to be added as a suitable feeder to illuminate the parabolic reflector antennas. Then, Turk and Yucedag applied the ARM to investigate the main structural design parameters of an 2D H-plane sectoral horn antenna [23]. Moreover, it makes it possible to calculate the horn antenna near field distribution which can be used to illuminate parabolic reflector antennas. In 2012, the authors proposed a waveguide array fed on-set parabolic reflector antenna to obtain electronically switchable pencil beam and cosecant-squared patterns for air, naval and coastal surveillance radars [24]. In 2014, the ARM algorithm was upgraded to calculate radiation characteristics of horn fed 2D off-set parabolic reflector antennas that may have pencil beam or cosecant-squared patterns [25]. Since mentioned ARM solutions are valid for 2D geometries, they are not suitable for three dimensional (3D) problems. To solve the electromagnetic scattering or radiation problems for 3D perfect electrical conductor (PEC) obstacles, a new and more efficient solution approach called two-and-a-half dimensional Analytical Regularization Method (2.5D ARM) is introduced in this paper.
posed feeder model for a 2.5D problem is discussed and meshing technique which depends on separating an 3D reflector into vertical slices is explained. 2.1. Mathematical basis of the regularization procedure In order to explain the 2.5D ARM approach, let us take one of the infinitely thin and perfectly conductive cylindrical screens. Its xy-plane cross section forms a smooth and closed contour S where n is the normal vector directed outward as seen in Fig. 1. In the vicinity of the contour, Helmholtz equation must be satisfied for an E-polarized incident scalar wave (ui ) which is independent from z-axis. Therefore, the Dirichlet boundary value problem reduces to determine scattered field (us ) which respectively satisfies Helmholtz equation: 2
p 2 R2
ðM þ k Þus ðpÞ ¼ 0;
us ðpÞ 2 C 2 ðR2 n SÞ
ð1Þ
Dirichlet boundary condition:
u ðpÞ ¼ ui ðpÞ;
p2S
s
ð2Þ
and Sommerfeld radiation condition:
s 1 du ðpÞ s iku ðpÞ ¼ Oðjpj2 Þ; p!1 djpj
us ðpÞ ¼ lim
k>0
ð3Þ
Using the Green’s formula, solution of the boundary value problem can be written as
i 4
Z
S
ð1Þ
H0 ðkjq pjÞZðpÞdS ¼ ui ðqÞ;
q2S
ð4Þ
ð1Þ
where H0 is the 0th -order Hankel function of the first kind and ZðqÞ is unknown function. Parametrization of the S contour specified by a function gðhÞ ¼ ðxðhÞ; yðhÞÞ allows to rewrite the (4):
1 2p
Z p h s ln2sinð Þ þ Kðh; sÞ Z D ðsÞds ¼ gðhÞ 2 p
ð5Þ
The logarithmic part of (5) represents the main singularity and Kðh; sÞ is the rather smooth section of Green’s function. The functions can be represented by their Fourier series expansions with ks;m ; zm and g m coefficients:
K ðh; sÞ ¼
1 1 X X
ks;m eiðshþmsÞ
ð6Þ
s¼1m¼1
Z D ð sÞ ¼
1 X
zn einðhsÞ
ð7Þ
n¼1
and expansion of the singular part can be given by [21]
1 h s 1 X ¼ ln2sin jnj1 einðhsÞ 2 n¼1;n–0 2
ð8Þ
2. Theory of the 2.5D ARM approach for Dirichlet boundary value problem In this section, mathematical basis of the 2.5D-ARM for the Dirichlet boundary value problem is presented. Afterwards, pro-
Fig. 1. xy-plane cross section of the smooth and closed contour.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
O.M. Yücedag˘, A.S. Türk / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
3
where h; s 2 ½p; p. Thus, the boundary value problem is reduced to an infinite linear algebraic system of the second kind. Finally, the scattered field us ðqÞ for q 2 R2 is obtained any required accuracy by the truncation method [26]:
^zs þ
1 X
^ ^z ¼ g^ k s;m m s
s; m ¼ 1; 2; . . .
ð9Þ
m¼1
where
^s;m ¼ 2ss sm ks;m þ 1 ds;0 dm;0 ; k 2 ^zn ¼ s1 ^ n zn ; g ¼ 2ss g s ; sn ¼ max jnj1=2 n ¼ 0; 1; 2; . . .
ð10Þ
Here, ds;m is the Kronecker delta function. 2.2. Feeder model In case of the 2D problems, the incident field can be defined as a plane wave or a line source, which has a propagation vector in the cross sectional plane of the problem geometry. However, the propagation vector may have oblique direction to the obstacle in the 2.5D case and its mathematical definition can be given as:
ui ðq; pÞ ¼ ui0
eikjqpj ; 4pkjq pj
p 2 R3 n S;
q2S
ð11Þ
where ¼ Fðh; /Þ is field distribution of the feeder antenna in the spherical coordinate system. Every different value of p for same q causes different scattered field values in (11). On the other hand, using the symmetry feature of Green function, q and p values can be interchanged and different q values become the source of the total scattered field while the p values remain the same (Gð~ q; ~ pÞ ¼ Gð~ p; ~ qÞ). Speaking in Physics terms, it means the potential at p, maintained by a point source at q is equal to the potential at q maintained by a point source at p. This makes it possible to analyse the scattered fields of the 3D objects by dividing them into homogeneous 2D slices, unbounded along the z-axis in Cartesian coordinates and illuminated by a point source which is illustrated in Fig. 2. Then, total field can be calculate from the summation of the fields of every single scatterer and the solution of the boundary value problem becomes:
Fig. 3. RCS result of the point source approximation for the infinitely long circular cylinder.
and electric field values in the far field region can be calculated from:
Eh ¼
2 Sn X Ik eikr sinh 4 p i x e r j¼0
ð14Þ
E/ ¼
Sn X iIk eikr sinh 4p r j¼0
ð15Þ
ui0
N iX 4 j¼1
S
us ðqÞ ¼
Z Sj
ð1Þ
H0 ðkjq pjÞZ j ðpÞdSj ;
q 2 Sj
ð12Þ
Then the infinite system of the linear algebraic equations of the second kind has the following form Sn Sn X Sn 1 X X X ^ j ^z j ¼ ^zsj þ k g^sj s;m m j¼0
j¼0 m¼1
j¼0
s ¼ 1; 2; . . .
ð13Þ
where I ¼ ^z j and r ¼ jq pj. To justify the feeder model, the 2.5D ARM is applied to compute radar cross section (RCS) of an infinitely long PEC cylinder for TMz case. The point source is taken 106 k away from the origin to obtain a plane wave. Then, computed RCS results are compared with the analytical solution and presented in Fig. 3. 2.3. Meshing method for the reflector Very robust meshing methods, such as flat triangle patches or tetrahedrons, have been using for the EM simulations. However, these methods usually cause heavy computational burdens for electrically large problems. Thus, an 2.5D mesh can be proposed to solve these kind of problems instead of an 3D meshing approach. Nevertheless, dividing an 3D body into 2D slices needs some precautions. For instance, if the object of the problem has electrically small geometric details on the axis of slicing, the slicing method can not be model the problem geometry properly. On the other hand, if the object of the problem is relatively smooth then it can be modelled by the slices easily. Here, radiation problem of the
Fig. 2. Illustration of 2.5D approach.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
O.M. Yücedag˘, A.S. Türk / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
4
reflector antenna which has very smooth geometry is investigated. General structure of a reflector antenna consists of two components: a reflecting surface and a feeder antenna which illuminates that surface. The most popular and basic reflector type is parabolic reflector which can also be called dish antenna. A paraboloidal surface can be define as [27]
q0 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4F F zf ;
q0 6 a
ð16Þ
Here, a is the radius of the aperture circle, q0 is the distance from a point A to the focal point O where A is the projection of the point R on the reflector surface onto the aperture plane at the focal point which is presented in Fig. 4. For a given displacement q0 from the axis of the reflector, the point R on the reflector surface has a distance rf away from the focal point O. The position can be defined in polar coordinates by ðr f ; hf Þ. A relation between ðr f ; hf Þ and F is readily found from Eq. (17):
rf ¼
F cos2 hf =2
ð17Þ
where
hf ¼ 2tan1
q0
2F
Fig. 5. Parametric definitions of the 2D parabolic reflector antenna for ARM.
ð18Þ
Thus, Eq. (19) can be defined to calculate the axis shift, DF, of a 2D slice from its initial position towards the focal point for every displacement of the source.
DF ¼ F
2Fcoshf 1 þ coshf
ð19Þ
This axis shift produce geometrical shape of the paraboloidal reflector in horizontal plane. To determine the vertical geometric shape of the reflector, it should be modelled as a closed contour L which goes from point A to point M and back to A by means of the variable l 2 ½0; L as seen in Fig. 5 where a is distance of the inner side of the reflector from the origin, b is distance of the outer side of the reflector from the origin. Segment length and parametric definitions of a typical radar reflector, which may have a csc2 shaped radiation pattern for a surveillance radar, can be found in [25].
3. Numerical and experimental results It is known that, as a high frequency method, the PO is computationally much more faster than the MoM which provides more accurate results than the PO. Consequently, comparing the 2.5DARM with the PO and MoM gives an reasonable opinion about its computational performance. Thus, bistatic RCS of a PEC hollow cylinder and an open-ended corner reflector are computed by using the PO, MoM and 2.5D-ARM at 300 MHz and 5 GHz, respectively. They are illuminated by a plane wave which propagates x direction and has an electric field z direction. The cylinder has 1 m radius and 10 m length. It is modelled by two different type of mesh which are constructed by the 2D smooth and closed contours (slices) and triangles. The mesh for the 2.5D-ARM contains 64 slices as shown in Fig. 6 while the triangular mesh, which is assembled by 27,182 triangles, is used by the PO and MoM. Comparison of bistatic RCS computation results are presented for h ¼ 90 and / ¼ 90 in Figs. 7 and 8(c), respectively. Memory and time requirements of these methods to compute bistatic RCS of the cylinder are given in Table 1. The open-ended corner reflector has 0.6 m width, 0.6 m height and 0.3 m length. The corner reflector is also modelled by two different type of mesh which are constructed by the slices and triangles. The mesh for the 2.5D-ARM contains 42 slices as shown in
5
y(m)
1 0 0 −1 1 0 −1
x(m) Fig. 4. xz-plane cross section of the parabolic reflector and its definition parameters.
−5
z(m)
Fig. 6. Sliced mesh of the hollow cylinder.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
O.M. Yücedag˘, A.S. Türk / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
0
0 2.5D−ARM PO MoM
−2 −4 −6 −8 −10 −12 −14
2.5D−ARM PO MoM
−5
Normalized RCS (dBsm)
Normalized RCS (dBsm)
5
−10 −15 −20 −25 −30
−16 −35
−18 −20 0
30
60
90
120 150 180 210 240 270 300 330 360
φ (Degree)
Fig. 7. Bistatic RCS of the hollow cylinder for h ¼ 90 .
0
20
40
60
80
100
φ (Degree)
120
140
160
180
Fig. 10. Bistatic RCS of the open-ended corner reflector for h ¼ 90 .
2.5D−ARM PO MoM
−5
Normalized RCS (dBsm)
−40 0
−10 −15 −20 −25 −30 −35 −40 90
120
150
180
θ (Degree)
210
240
270
Fig. 8. Bistatic RCS of the hollow cylinder for / ¼ 90 . Table 1 Memory and time requirements for bistatic RCS computation of the hollow cylinder. Frequency
Method
Memory usage
Time
300 MHz
2.5D ARM 3D PO 3D MoM
4 MB 10.5 MB 12.39 GB
16.5 s 1.80 s 2 h 45 m
0.3 0.2
y(m)
0.1 0
Fig. 11. Bistatic RCS of the open-ended corner reflector for / ¼ 90 .
Fig. 9 while the triangular mesh, which is assembled by 23,530 triangles, is used by the PO and MoM. Comparison of the computation results are presented for h ¼ 90 and / ¼ 90 in Fig. 10 and Fig. 11, respectively. Memory and time requirements of these methods to compute bistatic RCS of the reflector are given in Table 2. It should be noted that these benchmark analyses are performed on a PC with the following technical specifications: 2.80 GHz Intel i7 930 CPU and 22 GB RAM. As mentioned previous section, the proposed method mathematically based on the 2D ARM. This is a drawback and enforces the 2.5D ARM to apply an obstacle which can be meshed by using the slices. Moreover, even if the obstacle is meshed properly, coupling between the slices can not be considered. Fortunately, reflector type structures have suitable geometrical shapes for the meshing procedure. Therefore, the numerical results show that the 2.5D-ARM complies with the 3D MoM in the vertical plane. However, due to neglecting the slice couplings, the horizontal
−0.1 −0.2 −0.3 0
−0.2 0 0.1
0.2
0.2
z(m)
Fig. 9. Sliced mesh of the open-ended corner reflector.
Table 2 Memory and time requirements for bistatic RCS computation of the open-ended corner reflector. Frequency
Method
Memory usage
Time
5 GHz
2.5D ARM 3D PO 3D MoM
1.50 MB 8.93 MB 9.19 GB
7.51 m 2.50 s 1 h 50 m
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
O.M. Yücedag˘, A.S. Türk / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
6
Fig. 12. Experimental reflector antenna structure.
minated by a ridged horn antenna [28]. Fabricated reflector antenna structure can be seen in Fig. 12. Dish antenna is modeled by 110 slices for the 2.5D-ARM analysis as seen in Fig. 13. Measurement and the 2.5D-ARM analysis results for the vertical radiation pattern of the dish antenna at 10 GHz are compared and presented in Fig. 14. Results demonstrate very good agreement even for side lobe levels. Measurement results prove that the 2.D ARM is valid for the parabolic reflector antenna which is illuminated by (not only a hypothetical but a practical source) a ridged horn antenna. Here, it should be noted that field distribution of the feeder antenna should be obtained mathematically, numerically or experimentally before applying to the 2.5D ARM, if it is necessary.
0.3
y−axis (m)
0.2 0.1 0 −0.1 −0.2 −0.3
0.2 0 0.2 0
x−axis (m)
4. Conclusion
−0.2
z−axis (m)
Fig. 13. Sliced mesh of the experimental dish antenna.
plane solutions may tend to the PO results. Taking into account the coupling effects is considered as future study. Briefly, the 2.5DARM is computationally much faster than the MoM and more accurate than the PO as expected. The 2.5D-ARM is applied to compute the vertical radiation pattern of an on-set symmetrical 3D parabolic reflector which is illu-
In this study, the 2.5D-ARM procedure is developed as an efficient analysis technique for reflector antenna problems. The proposed method is validated by comparing its results with simulation and measurement data. Obtained results are showed that the 2.5D-ARM can be considered as a suitable tool for reflector antenna analysis at both resonance and high frequency regions. Acknowledgements This work was supported by grant 114E241 of TUBITAK (The Scientific and Technological Research Council of Turkey) research fund. References
Fig. 14. Measurement and ARM result comparison for the dish antenna.
[1] Franceschetti G, Mohsen A. Recent developments in the analysis of reflector antennas: a review. IEEE Proc 1986;133:65–76. [2] Keller JB. Geometrical theory of diffraction. J Opt Soc Am 1962;52:116–30. [3] Rusch WVT, Sorensen O. The geometrical theory of diffraction for axially symmetric reflectors. IEEE Trans Antennas Propag 1975;23:414–9. [4] Rudduck RC, Lee SH, Burnside, WD. Gtd analysis of reflector antennas with general rim shapes. In: IEEE Antennas and Propagation Society International Symposium, vol. 17; 1979. p. 495–98. [5] Narasimhan MS, Govind KR. Front-to-back ratio of paraboloidal reflectors. EEE Trans Antennas Propag 1991;39:877–82. [6] Kouyoumjian RG. Asymptotic high frequency methods. Proc IEEE 1965;53:864–76. [7] Ahluwalia DS, Lewis RM, Boersma J. Uniform asymptotic theory of diffraction by a plane screen. SIAM J Appl Math 1968;16:783–807. [8] Yazgan E, Safak M. Comparison UTD and UAT in axially symmetric reflectors. IEEE Trans Antennas Propag 1987;35:113–6. [9] Ufimtsev PY. Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies. Soviet Phys 1957;27:1708–18. [10] Samii YR, Duan DW, Gri DV, Libelo LF. Canonical examples of reflector antennas for high-power microwave applications. IEEE Trans Electromagn Compat 1992;34:197–205.
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[21] Karacuha E, Turk AS. E polarized scalar wave diffraction by perfectly conductive arbitrary shaped cylindrical obstacles with finite thickness. Int J Infrared Milimeter Waves 2001;22:1531–46. [22] Turk AS. Analysis of aperture illumination and edge rolling effects for parabolic reflector antenna design. Int J Electron Commun 2006;60:257–66. [23] Turk AS, Yucedag OM. Parametric analysis of flare edge, rolling throat bending and asymmetric flare effects for h-plane horn radiator. Int J Electron Commun 2012;66:297–304. [24] Yucedag OM, Turk AS. Parametric design of open ended waveguide array feeder with reflector antenna for switchable cosecant-squared pattern. Appl Comput Electromagnet Soc 2012;27:668–75. [25] Yucedag OM, Turk AS. Design of horn fed offset parabolic reflector antennas with analytical regularization method. J Electromagn Waves Appl 2014;28:1502–11. [26] Thuchkin YA, Karacuha E, Turk, AS. Analytical regularization method for epolarized electromagnetic wave diffraction by arbitrary shaped cylindrical obstacles. In: International Conferance on Mathematical Methods Electromagnetic Theory; 1998. p. 733–35. [27] Nikolova NK. Modern antennas in wireless telecommunications. Canada: MacMaster University; 2010. [28] Keskin AK, Senturk MD, Caliskan A, Turk AS, Demirel S. Compact partially dielectric loaded ridged horn antenna. In: Applied Computational Electromagnetics Society Conference; 2014.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
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Regular paper
Development of 2.5D Analytical Regularization Method for reflector antenna analysis q Okan Mert Yücedag˘ a,⇑, Ahmet Serdar Türk b a b
TUBITAK Informatics and Information Security Research Center, Kocaeli, Turkey Yildiz Technical University, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 7 June 2016 Accepted 19 October 2016 Available online xxxx 2010 MSC: 00-01 99-00
a b s t r a c t A novel and efficient approach, which is called two-and-a-half dimensional Analytical Regularization Method (2.5D ARM), is proposed to analyse reflector antennas for microwave, millimeter wave or ultra-wide band applications. To present computational performance of the proposed method, its results are compared with well known Physical Optics (PO) and Method of Moments (MoM) solutions for a PEC hollow cylinder and an open-ended corner reflector. Moreover, radiation pattern of a ridged horn fed parabolic reflector antenna is measured to validate the method experimentally. It is observed that either the simulated or the measured data and the 2.5D ARM results are in close agreement with each other. Ó 2016 Elsevier GmbH. All rights reserved.
Keywords: Analytical Regularization Method Reflector antenna Radiation pattern
1. Introduction Reflector antennas, which usually consist a feed to radiate electromagnetic energy and a curved reflecting surface to collimate this energy over a large aperture, have applications in radar, radio astronomy, and microwave communication systems [1]. Therefore, a large number of analytical and numerical techniques have been developed to obtain accurate solutions for reflector antenna radiation problems. Many high frequency techniques, preferred for their relatively low computational memory and time requirements, are suitable for calculating radiation characteristics of electrically large reflector antennas. One of them is Geometrical Optics (GO) method which assumes that electromagnetic energy transports with rays having same character as a plane wave. But, the GO method is unable to calculate fields in shadowed regions and can not deal with diffraction effects. To overcome this failure, Keller extended the GO to include diffracted rays produced by scattering from sharp edges of boundary surfaces and called that Geometrical Theory of Diffraction (GTD) [2]. Afterward, GTD was applied to axially symmetric reflector antennas fed by point source [3], with general rim shapes [4] and illuminated by feeds that do not radiate an
q
Fully documented templates are available in the elsarticle package on CTAN.
⇑ Corresponding author.
exact spherical wave front [5]. However, the Keller’s GTD technique based on a diffraction coefficient which is determined by an asymptotic evaluation of Sommerfeld’s integral. Because of the singularity of the integral, the Keller’s GTD becomes invalid in transition regions adjacent to the shadow boundaries. This singularity can be eliminated by using Uniform Theory of Diffraction (UTD) [6] or Uniform Asymptotic Theory (UAT) [7]. Yazgan and Safak showed that the difference between the UTD and UAT results for focus-fed axially symmetric reflector antenna were insignificant [8]. But, either the GTD or its uniform versions can not calculate the fields at the caustics which occur whenever a group of ray intersect exterior of a reflector antenna. On the other hand, Physical Optics (PO) method, which is based on the integration of the induced currents predicted by the GO, can accurately predict the field in the main beam region and near-side lobes. But, edge diffraction of the reflector may become important for the far-side lobes or cross-polarized field predictions. In this situation, Ufimtsev’s Physical Theory of Diffraction (PTD) [9] should be included to the PO [10]. Also, note that the PO with its extension based on the PTD are useful for calculating the fields in shadow transition and caustic regions [11]. Thus, PO method makes possible to analyse single and dual reflector antennas efficiently [12]. Mentioned studies prove that considering the diffracted fields is necessary to obtain accurate radiation patterns for the reflector antennas. But, it should be emphasized that the diffracted fields
E-mail address: [email protected] (O.M. Yücedag˘). http://dx.doi.org/10.1016/j.aeue.2016.10.020 1434-8411/Ó 2016 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020
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can not be easily calculated in the caustics regions of arbitrary shaped reflectors and the PO method becomes less accurate for antennas whose dimensions are not large in terms of wavelength. However, some computational electromagnetic methods usually deal with first-kind integral equations for determining the surface currents can be preferred to overcome these kind of difficulties. These equations are obtained from the boundary conditions of the antenna problem, and then, they are discretized by using entire or sub-domain basis functions in a classical numerical solution approach such as Method of Moments [13,14]. But, especially Fredholm integral equations of the first kind with continuous kernels cause ill-posed problems which have consequences for its numerical solutions. Specifically, these solutions show low accuracy and can yield numerical instability. Unfortunately, there are not any general theorems proving convergence, or even the existence of an exact solution, for such equations because of their logarithmic or higher-order singular kernels [15]. But, an approach exists to obtain second-kind integral equations of the Fredholm type, with a smoother kernel, from the first kind equations. This causes the condition number of the matrix equations to remain small while the matrix size gets larger. The mentioned approach is called Analytical Regularization Method (ARM). Hence, the ARM can be defined as a mathematical technique, which allows constructing two-sided regulator of initial diffraction problem of the first kind and then it reduces the problem to an equation of the secondkind [16]. In diffraction theory, the first example of the ARM is semiinversion procedure [17]. However, this procedure works only simplest diffraction problems which have an obstacle surface coincides with a part of a coordinate system [18]. The main reason of the ARM existence is to eliminate disadvantages of the semiinversion procedure. At the initial phase, the ARM was applied to the wave scattering by arbitrary profile unclosed cylindrical screen with Dirichlet and Neuman boundary condition by Tuchkin in 1985 [19,20]. Karacuha and Turk implemented ARM to solve two dimensional problem of E-polarized wave diffraction by arbitrary shaped, smooth and perfectly conductive cylindrical obstacle [21]. Depending on this study, in 2006, Turk used the ARM to analyse the aperture illumination and edge rolling effects of two dimensional (2D) parabolic antennas which are illuminated by plane wave [22]. This study showed that a waveguide or horn antenna model needs to be added as a suitable feeder to illuminate the parabolic reflector antennas. Then, Turk and Yucedag applied the ARM to investigate the main structural design parameters of an 2D H-plane sectoral horn antenna [23]. Moreover, it makes it possible to calculate the horn antenna near field distribution which can be used to illuminate parabolic reflector antennas. In 2012, the authors proposed a waveguide array fed on-set parabolic reflector antenna to obtain electronically switchable pencil beam and cosecant-squared patterns for air, naval and coastal surveillance radars [24]. In 2014, the ARM algorithm was upgraded to calculate radiation characteristics of horn fed 2D off-set parabolic reflector antennas that may have pencil beam or cosecant-squared patterns [25]. Since mentioned ARM solutions are valid for 2D geometries, they are not suitable for three dimensional (3D) problems. To solve the electromagnetic scattering or radiation problems for 3D perfect electrical conductor (PEC) obstacles, a new and more efficient solution approach called two-and-a-half dimensional Analytical Regularization Method (2.5D ARM) is introduced in this paper.
posed feeder model for a 2.5D problem is discussed and meshing technique which depends on separating an 3D reflector into vertical slices is explained. 2.1. Mathematical basis of the regularization procedure In order to explain the 2.5D ARM approach, let us take one of the infinitely thin and perfectly conductive cylindrical screens. Its xy-plane cross section forms a smooth and closed contour S where n is the normal vector directed outward as seen in Fig. 1. In the vicinity of the contour, Helmholtz equation must be satisfied for an E-polarized incident scalar wave (ui ) which is independent from z-axis. Therefore, the Dirichlet boundary value problem reduces to determine scattered field (us ) which respectively satisfies Helmholtz equation: 2
p 2 R2
ðM þ k Þus ðpÞ ¼ 0;
us ðpÞ 2 C 2 ðR2 n SÞ
ð1Þ
Dirichlet boundary condition:
u ðpÞ ¼ ui ðpÞ;
p2S
s
ð2Þ
and Sommerfeld radiation condition:
s 1 du ðpÞ s iku ðpÞ ¼ Oðjpj2 Þ; p!1 djpj
us ðpÞ ¼ lim
k>0
ð3Þ
Using the Green’s formula, solution of the boundary value problem can be written as
i 4
Z
S
ð1Þ
H0 ðkjq pjÞZðpÞdS ¼ ui ðqÞ;
q2S
ð4Þ
ð1Þ
where H0 is the 0th -order Hankel function of the first kind and ZðqÞ is unknown function. Parametrization of the S contour specified by a function gðhÞ ¼ ðxðhÞ; yðhÞÞ allows to rewrite the (4):
1 2p
Z p h s ln2sinð Þ þ Kðh; sÞ Z D ðsÞds ¼ gðhÞ 2 p
ð5Þ
The logarithmic part of (5) represents the main singularity and Kðh; sÞ is the rather smooth section of Green’s function. The functions can be represented by their Fourier series expansions with ks;m ; zm and g m coefficients:
K ðh; sÞ ¼
1 1 X X
ks;m eiðshþmsÞ
ð6Þ
s¼1m¼1
Z D ð sÞ ¼
1 X
zn einðhsÞ
ð7Þ
n¼1
and expansion of the singular part can be given by [21]
1 h s 1 X ¼ ln2sin jnj1 einðhsÞ 2 n¼1;n–0 2
ð8Þ
2. Theory of the 2.5D ARM approach for Dirichlet boundary value problem In this section, mathematical basis of the 2.5D-ARM for the Dirichlet boundary value problem is presented. Afterwards, pro-
Fig. 1. xy-plane cross section of the smooth and closed contour.
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where h; s 2 ½p; p. Thus, the boundary value problem is reduced to an infinite linear algebraic system of the second kind. Finally, the scattered field us ðqÞ for q 2 R2 is obtained any required accuracy by the truncation method [26]:
^zs þ
1 X
^ ^z ¼ g^ k s;m m s
s; m ¼ 1; 2; . . .
ð9Þ
m¼1
where
^s;m ¼ 2ss sm ks;m þ 1 ds;0 dm;0 ; k 2 ^zn ¼ s1 ^ n zn ; g ¼ 2ss g s ; sn ¼ max jnj1=2 n ¼ 0; 1; 2; . . .
ð10Þ
Here, ds;m is the Kronecker delta function. 2.2. Feeder model In case of the 2D problems, the incident field can be defined as a plane wave or a line source, which has a propagation vector in the cross sectional plane of the problem geometry. However, the propagation vector may have oblique direction to the obstacle in the 2.5D case and its mathematical definition can be given as:
ui ðq; pÞ ¼ ui0
eikjqpj ; 4pkjq pj
p 2 R3 n S;
q2S
ð11Þ
where ¼ Fðh; /Þ is field distribution of the feeder antenna in the spherical coordinate system. Every different value of p for same q causes different scattered field values in (11). On the other hand, using the symmetry feature of Green function, q and p values can be interchanged and different q values become the source of the total scattered field while the p values remain the same (Gð~ q; ~ pÞ ¼ Gð~ p; ~ qÞ). Speaking in Physics terms, it means the potential at p, maintained by a point source at q is equal to the potential at q maintained by a point source at p. This makes it possible to analyse the scattered fields of the 3D objects by dividing them into homogeneous 2D slices, unbounded along the z-axis in Cartesian coordinates and illuminated by a point source which is illustrated in Fig. 2. Then, total field can be calculate from the summation of the fields of every single scatterer and the solution of the boundary value problem becomes:
Fig. 3. RCS result of the point source approximation for the infinitely long circular cylinder.
and electric field values in the far field region can be calculated from:
Eh ¼
2 Sn X Ik eikr sinh 4 p i x e r j¼0
ð14Þ
E/ ¼
Sn X iIk eikr sinh 4p r j¼0
ð15Þ
ui0
N iX 4 j¼1
S
us ðqÞ ¼
Z Sj
ð1Þ
H0 ðkjq pjÞZ j ðpÞdSj ;
q 2 Sj
ð12Þ
Then the infinite system of the linear algebraic equations of the second kind has the following form Sn Sn X Sn 1 X X X ^ j ^z j ¼ ^zsj þ k g^sj s;m m j¼0
j¼0 m¼1
j¼0
s ¼ 1; 2; . . .
ð13Þ
where I ¼ ^z j and r ¼ jq pj. To justify the feeder model, the 2.5D ARM is applied to compute radar cross section (RCS) of an infinitely long PEC cylinder for TMz case. The point source is taken 106 k away from the origin to obtain a plane wave. Then, computed RCS results are compared with the analytical solution and presented in Fig. 3. 2.3. Meshing method for the reflector Very robust meshing methods, such as flat triangle patches or tetrahedrons, have been using for the EM simulations. However, these methods usually cause heavy computational burdens for electrically large problems. Thus, an 2.5D mesh can be proposed to solve these kind of problems instead of an 3D meshing approach. Nevertheless, dividing an 3D body into 2D slices needs some precautions. For instance, if the object of the problem has electrically small geometric details on the axis of slicing, the slicing method can not be model the problem geometry properly. On the other hand, if the object of the problem is relatively smooth then it can be modelled by the slices easily. Here, radiation problem of the
Fig. 2. Illustration of 2.5D approach.
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reflector antenna which has very smooth geometry is investigated. General structure of a reflector antenna consists of two components: a reflecting surface and a feeder antenna which illuminates that surface. The most popular and basic reflector type is parabolic reflector which can also be called dish antenna. A paraboloidal surface can be define as [27]
q0 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4F F zf ;
q0 6 a
ð16Þ
Here, a is the radius of the aperture circle, q0 is the distance from a point A to the focal point O where A is the projection of the point R on the reflector surface onto the aperture plane at the focal point which is presented in Fig. 4. For a given displacement q0 from the axis of the reflector, the point R on the reflector surface has a distance rf away from the focal point O. The position can be defined in polar coordinates by ðr f ; hf Þ. A relation between ðr f ; hf Þ and F is readily found from Eq. (17):
rf ¼
F cos2 hf =2
ð17Þ
where
hf ¼ 2tan1
q0
2F
Fig. 5. Parametric definitions of the 2D parabolic reflector antenna for ARM.
ð18Þ
Thus, Eq. (19) can be defined to calculate the axis shift, DF, of a 2D slice from its initial position towards the focal point for every displacement of the source.
DF ¼ F
2Fcoshf 1 þ coshf
ð19Þ
This axis shift produce geometrical shape of the paraboloidal reflector in horizontal plane. To determine the vertical geometric shape of the reflector, it should be modelled as a closed contour L which goes from point A to point M and back to A by means of the variable l 2 ½0; L as seen in Fig. 5 where a is distance of the inner side of the reflector from the origin, b is distance of the outer side of the reflector from the origin. Segment length and parametric definitions of a typical radar reflector, which may have a csc2 shaped radiation pattern for a surveillance radar, can be found in [25].
3. Numerical and experimental results It is known that, as a high frequency method, the PO is computationally much more faster than the MoM which provides more accurate results than the PO. Consequently, comparing the 2.5DARM with the PO and MoM gives an reasonable opinion about its computational performance. Thus, bistatic RCS of a PEC hollow cylinder and an open-ended corner reflector are computed by using the PO, MoM and 2.5D-ARM at 300 MHz and 5 GHz, respectively. They are illuminated by a plane wave which propagates x direction and has an electric field z direction. The cylinder has 1 m radius and 10 m length. It is modelled by two different type of mesh which are constructed by the 2D smooth and closed contours (slices) and triangles. The mesh for the 2.5D-ARM contains 64 slices as shown in Fig. 6 while the triangular mesh, which is assembled by 27,182 triangles, is used by the PO and MoM. Comparison of bistatic RCS computation results are presented for h ¼ 90 and / ¼ 90 in Figs. 7 and 8(c), respectively. Memory and time requirements of these methods to compute bistatic RCS of the cylinder are given in Table 1. The open-ended corner reflector has 0.6 m width, 0.6 m height and 0.3 m length. The corner reflector is also modelled by two different type of mesh which are constructed by the slices and triangles. The mesh for the 2.5D-ARM contains 42 slices as shown in
5
y(m)
1 0 0 −1 1 0 −1
x(m) Fig. 4. xz-plane cross section of the parabolic reflector and its definition parameters.
−5
z(m)
Fig. 6. Sliced mesh of the hollow cylinder.
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0
0 2.5D−ARM PO MoM
−2 −4 −6 −8 −10 −12 −14
2.5D−ARM PO MoM
−5
Normalized RCS (dBsm)
Normalized RCS (dBsm)
5
−10 −15 −20 −25 −30
−16 −35
−18 −20 0
30
60
90
120 150 180 210 240 270 300 330 360
φ (Degree)
Fig. 7. Bistatic RCS of the hollow cylinder for h ¼ 90 .
0
20
40
60
80
100
φ (Degree)
120
140
160
180
Fig. 10. Bistatic RCS of the open-ended corner reflector for h ¼ 90 .
2.5D−ARM PO MoM
−5
Normalized RCS (dBsm)
−40 0
−10 −15 −20 −25 −30 −35 −40 90
120
150
180
θ (Degree)
210
240
270
Fig. 8. Bistatic RCS of the hollow cylinder for / ¼ 90 . Table 1 Memory and time requirements for bistatic RCS computation of the hollow cylinder. Frequency
Method
Memory usage
Time
300 MHz
2.5D ARM 3D PO 3D MoM
4 MB 10.5 MB 12.39 GB
16.5 s 1.80 s 2 h 45 m
0.3 0.2
y(m)
0.1 0
Fig. 11. Bistatic RCS of the open-ended corner reflector for / ¼ 90 .
Fig. 9 while the triangular mesh, which is assembled by 23,530 triangles, is used by the PO and MoM. Comparison of the computation results are presented for h ¼ 90 and / ¼ 90 in Fig. 10 and Fig. 11, respectively. Memory and time requirements of these methods to compute bistatic RCS of the reflector are given in Table 2. It should be noted that these benchmark analyses are performed on a PC with the following technical specifications: 2.80 GHz Intel i7 930 CPU and 22 GB RAM. As mentioned previous section, the proposed method mathematically based on the 2D ARM. This is a drawback and enforces the 2.5D ARM to apply an obstacle which can be meshed by using the slices. Moreover, even if the obstacle is meshed properly, coupling between the slices can not be considered. Fortunately, reflector type structures have suitable geometrical shapes for the meshing procedure. Therefore, the numerical results show that the 2.5D-ARM complies with the 3D MoM in the vertical plane. However, due to neglecting the slice couplings, the horizontal
−0.1 −0.2 −0.3 0
−0.2 0 0.1
0.2
0.2
z(m)
Fig. 9. Sliced mesh of the open-ended corner reflector.
Table 2 Memory and time requirements for bistatic RCS computation of the open-ended corner reflector. Frequency
Method
Memory usage
Time
5 GHz
2.5D ARM 3D PO 3D MoM
1.50 MB 8.93 MB 9.19 GB
7.51 m 2.50 s 1 h 50 m
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Fig. 12. Experimental reflector antenna structure.
minated by a ridged horn antenna [28]. Fabricated reflector antenna structure can be seen in Fig. 12. Dish antenna is modeled by 110 slices for the 2.5D-ARM analysis as seen in Fig. 13. Measurement and the 2.5D-ARM analysis results for the vertical radiation pattern of the dish antenna at 10 GHz are compared and presented in Fig. 14. Results demonstrate very good agreement even for side lobe levels. Measurement results prove that the 2.D ARM is valid for the parabolic reflector antenna which is illuminated by (not only a hypothetical but a practical source) a ridged horn antenna. Here, it should be noted that field distribution of the feeder antenna should be obtained mathematically, numerically or experimentally before applying to the 2.5D ARM, if it is necessary.
0.3
y−axis (m)
0.2 0.1 0 −0.1 −0.2 −0.3
0.2 0 0.2 0
x−axis (m)
4. Conclusion
−0.2
z−axis (m)
Fig. 13. Sliced mesh of the experimental dish antenna.
plane solutions may tend to the PO results. Taking into account the coupling effects is considered as future study. Briefly, the 2.5DARM is computationally much faster than the MoM and more accurate than the PO as expected. The 2.5D-ARM is applied to compute the vertical radiation pattern of an on-set symmetrical 3D parabolic reflector which is illu-
In this study, the 2.5D-ARM procedure is developed as an efficient analysis technique for reflector antenna problems. The proposed method is validated by comparing its results with simulation and measurement data. Obtained results are showed that the 2.5D-ARM can be considered as a suitable tool for reflector antenna analysis at both resonance and high frequency regions. Acknowledgements This work was supported by grant 114E241 of TUBITAK (The Scientific and Technological Research Council of Turkey) research fund. References
Fig. 14. Measurement and ARM result comparison for the dish antenna.
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[21] Karacuha E, Turk AS. E polarized scalar wave diffraction by perfectly conductive arbitrary shaped cylindrical obstacles with finite thickness. Int J Infrared Milimeter Waves 2001;22:1531–46. [22] Turk AS. Analysis of aperture illumination and edge rolling effects for parabolic reflector antenna design. Int J Electron Commun 2006;60:257–66. [23] Turk AS, Yucedag OM. Parametric analysis of flare edge, rolling throat bending and asymmetric flare effects for h-plane horn radiator. Int J Electron Commun 2012;66:297–304. [24] Yucedag OM, Turk AS. Parametric design of open ended waveguide array feeder with reflector antenna for switchable cosecant-squared pattern. Appl Comput Electromagnet Soc 2012;27:668–75. [25] Yucedag OM, Turk AS. Design of horn fed offset parabolic reflector antennas with analytical regularization method. J Electromagn Waves Appl 2014;28:1502–11. [26] Thuchkin YA, Karacuha E, Turk, AS. Analytical regularization method for epolarized electromagnetic wave diffraction by arbitrary shaped cylindrical obstacles. In: International Conferance on Mathematical Methods Electromagnetic Theory; 1998. p. 733–35. [27] Nikolova NK. Modern antennas in wireless telecommunications. Canada: MacMaster University; 2010. [28] Keskin AK, Senturk MD, Caliskan A, Turk AS, Demirel S. Compact partially dielectric loaded ridged horn antenna. In: Applied Computational Electromagnetics Society Conference; 2014.
Please cite this article in press as: Yücedag˘ OM, Türk AS. Development of 2.5D Analytical Regularization Method for reflector antenna analysis. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.10.020