1. Computer tools for the analysis and design of electrically large antennas
Iván González*, Felipe Cátedra*, Mª Jesús Algar** , Lorena Lozano*, Álvaro Somolinos**, Gustavo Romero*, Javier Moreno**
*Universidad de Alcalá (Spain)
** newFASANT S.L. (Spain)

2. INDEX
Introduction
Method of Moment (MoM) formulation for multilayer planar structures
MoM for arbitrary shaped large structures
Design of reflectarrays antennas
Lens antennas
Design of Dicroical Reflector antennas
Radomes
Phased arrays

3. INTRODUCTION (I)
Two EM formulations will be presented using MoM. The first one is for the analysis of single/multiple layers of periodic structures using Green Functions for layered media.
The second EM formulation is based on solving full 3D Integral Equations.
A Graphical User Interface (GUI) has been developed for helping users to analyse and design large antennas.
The GUI includes full developed design tools, wizard projects, friendly creation of scripts and custom functions, standards and custom post-processing and visualization.

4. Method of Moment (MoM) formulation for multilayer plannar structures(I)
An approach for planar structures based on MoM is used to analyse problems with single/multiple layers.
The Green Function for dielectric multilayer media is computed in the real domain from its spectral form using the Hankel transform for both infinite and finite sized structures by:

5. Method of Moment (MoM) formulation for multilayer plannar structures(II)
For infinite structures the Poisson formula is used for obtain the Green Function in the real domain for solving the MoM Integral Equation in the Period Unit Cell under Floquets Boundary conditions.
For finite structures the Hankel transform is computed using the WA ( Weighted Average Technique) for the addition of infinite series and the DE (Double Exponential Technique) for the calculation of the singularities. That transform can be applied for periodical and non periodic structures.

6. Method of Moment (MoM) formulation for multilayer planar structures(III)
A tool has been included in the GUI for setting the parameters to analyse and design multilayer periodic structures.
The tool permits to define the geometry of the unit cell between several primitives or create new ones. In both cases the geometries elements can be parametrized.
This tool provides reflection coefficient matrices for lineal
circular polarization for arbitrary incidence angles and
the cell data base for reflectarrays designs.

7. Method of Moment (MoM) formulation for arbitrary shaped large structures (I)
A full 3D Integral Equation (IE) is solved. This approach permits the analysis and design of antennas, analysis of antennas on board platforms, electromagnetic compatibility studies, analysis and design of devices based on periodic structures, near and far fields scattering studies, impulse response, antenna radiation maps and antenna diagnoses, etc.
The IE is solved using Moment Method (MoM) with MLFMM (Multilevel Fast Multipole Method). We consider the Electric Field Integral Equation (EFIE), the Magnetic Field Integral Equation (MFIE) and the Combined Field Integral Equation (CFIE) for perfect conductor material and/or material bodies.

8. Method of Moment (MoM) formulation for arbitrary shaped large structures (II)
Anisotropic material can be treated.
The MoM is based on curved rooftop functions that are fitted to every surface. The computer approach provides excellent results in accuracy and efficiency.

9. Reflectarray applications overview
Reflectarrays
Reflectarray applications overview

10. 1. Features of REFLECTARRAY Tool
Planar and curved surfaces
Periodical structures module to design the cell database
Custom functions to place the cells for obtaining a desired radiation pattern
PO analysis for quick design
Linear and circular polarizations
Full MoM analysis with MLFMM for accurate results

11. 2. Test case:Multi-Spot Beam Reflectarray for Satellite Telecommunication in Ka-Band
The Objective is to design a full dual-band (transmit/receive) multiple spot beam coverage in Ka- band using only two reflectarrays.
Example of multiple spot beam coverage in Ka-band reusing four kinds of beam spots: A (pol1, freq1), B(pol2,freq1), C(pol1, freq2) and D (pol2, freq2)

12. A solution based on reflectors needs four apertures (four reflectors)
A solution based on reflectors needs four apertures (four reflectors). Each reflectors is feed by a set of horns to obtain the beam of the same type. For instance the reflector in the right and upper corner can give all the beam spots A. Each beam spot is due only to one horn.

13. The need of four reflectors is due to the fact that we need two allocated one horn for each beam in a different position in the focal plane. As the horns have a high gain their apertures are large and to avoid collision between contiguous horns we need lot of space for allocated all the horns.

14. We could reduce the number of reflectors if we are able to obtain two different bean spots using the same horn as indicated in the figure.
That can be achieved by substituting the regular metallic surface of the reflector by a properly designed parabolic reflector and using dual polarized horns.

15. 2. Reflectarray design objetive
Full dual-band (transmit/receive) multiple spot beam coverage in Ka-band using only two main apertures while maintaining single-feed-per-beam operation.
Combination of the capabilities of the reflectarrays and parabolic surfaces.
We will show here a design for 19.7 Ghz

16. 2.2. Reflector Meshing at 10 divisions per wavelength
Parameters
unit
Value
Frequency
GHz
19,7
Main reflector diameter m
1,812
Main reflector f/D -
1,5
Focal length
2,718
Clearance
0,35
Half sub-tended angle
deg. 18
Taper dB
19,7 GHz
29,5 GHz
Beam type
Gaussian
Meshing at 10 divisions per wavelength
Feed by pattern file with LHCP
Simulated with MoM
elements

17. 2.1. Feed
The feed is modeled by a cosq(q) distribution, with q = 24 at 19.7 GHz
LHCP and RHCP polarizations

18. Reflector Results
Cut phi=0º, theta from -20º to 20º

19. 2.3. Reflectarray cell design
Thereflectarray design is based on the variable rotation technique (VRT)
Using VRT the reflected field has the same sense of circular polarization as the incident field. The matrix of reflection
coefficients for circular components shall be:
𝑅 𝑟𝑟 = 𝑒 𝑗𝜑 𝑅 𝑙𝑟 =0
𝑅 𝑟𝑙 =0 𝑅 𝑙𝑙 = 𝑒 −𝑗𝜑
That can be achieved when the reflection
coefficients of the linear components are of
magnitude 1 and a phase difference of 180 degrees
A element that can give these ref. coefficients is
a cross with its arms of different lengths as
shown in the figure.

20. 2.3. Reflectarray cell design
Using VRT the angle of the period element is rotated. The phase of the reflection coefficients for circular components are approximately twice the rotation angle 𝛼:
𝜑= 𝜑 𝑜 +2∗𝛼
and we have for the reflection matrix
𝑅 𝑟𝑟 = 𝑒 𝑗( 𝜑 𝑜 +2∗𝛼) 𝑅 𝑙𝑟 =0
𝑅 𝑟𝑙 =0 𝑅 𝑙𝑙 = 𝑒 −𝑗( 𝜑 𝑜 +2∗𝛼)

21. 2.3.2. Cell design (PS – Module) -> Cross
L1: Step 20 ( º) 6.9mm
L2: Step 5 (49.339º) to 8 (33.53º) 5.4mm – 5.7mm
VV – HH Phase variation
Step 11: 5.5mm -> Difference ≈ 180º

22. 2.3.3. Cell design (PS – Module) -> Rotated Cross
L1: 6.9mm
L2: 5.5mm
180 steps: 0º to 179º
RR and LL Phase variation ≈ 2*rotation angle

23. 2.4. Reflectarray Surface: Reflector Database: Rotated cross
Layout: 42,566 crosses

24. 2.4.1. Reflectarray Simulation
Simulated with MoM (LHCP and RHCP): unknowns
Intel Xeon with 12 cores: 8 hours and 50 GB of RAM

25. Reflectarray Results
Cut phi=0º, theta from -10º to 10º

26. 2.4.3. Reflectarray -> Multiple Feed
Cut phi=0º, theta from -10º to 10º

27. Results

28. Lens applications overview
Lens Antennas
Lens applications overview

29. 1. Introduction
A dielectric lens serves to transform an incident wave from the antenna feeder.
Modify the amplitude and the phase.
The most common transformation is from spherical wave to plane wave, that increases the directivity and the gain of the antenna.
Other utilities are decrease the secondary lobes, steering the main lobe or focus the EM field to a near point.

30. 2. Design of a dielectric lens
It has developed a tool that allows modeling curves and surfaces using algorithms defined by the user.
The algorithm is designed by obtain a geometric model of lens, which applies the desired effect on the antenna.
The most frequent case is that the lens has symmetry of revolution. In this case the user obtains a curve, which revolutionizes and the base is closed.

31. 2.2. Design of an arbitrary lens
The surfaceFunction command is similar to the above.
Two ranges are defined, u and v.
The functions receive two parameters.

32. 2.3. Transformation of spherical wave to plane wave
The function, using the following parameters:
Range of values in the ‘x’ coordinate, where the lens is located.
Distance of the source and observation plane.
Frequency
Dielectric material.
Will have to calculate the dielectric thickness in the ‘x’ coordinate.

33. 2.3. Transformation of spherical wave to plane wave
The problem can be solved analytically equalizing the phases:

34. 3. Analysis with dielectric volumes using MoM with full volumetric approach
The volumetric mesh is obtained cubing the original volume. The result are regular hexaedrons of planar surfaces.
It is a simple and quick method which produces a regular grid.
The mesh is analyzed with MoM using the VFIE formulation (Volumetric Field Integral Equation).

35. 4. Results Using the above techniques we have designed a primary lens
Frequency: 220 GHz
Substrate and hemisphere: 1,08 mm y εr 11,48
Antireflection layer: 0,272 mm y εr 3,39

36. Coplanar line antenna, feeding of 1V
4. Results
Coplanar line antenna, feeding of 1V

37. Radiation pattern without secondary lens (17,27 dBi)
4. Results
Radiation pattern without secondary lens (17,27 dBi)

38. 4. Results
To the above primary antenna is superimposed a secondary dielectric lens.
They have been made the parametric studios modifying the radius of the lens and the distance from the source.
Radius of 10 mm, distance from the source of 25 mm and εr 2,77.
After performing both simulations we obtained the radiation patterns.

39. 4. Results Radiation pattern without secondary lens (17,27 dBi)
Radiation pattern with secondary lens (26,66 dBi)

40. Cut Phi=0 of the radiation pattern
4. Results
Cut Phi=0 of the radiation pattern

41. Cut Phi=90 of the radiation pattern
4. Results
Cut Phi=90 of the radiation pattern

42. Radomes and FSS applications
Radomes and FSS applications overview

43. 2. FSS Applications
The FSS (Frequency Selective Surfaces) are placed for achieving a frequency selective performance.
The most common distributions are placed on planar structures.
Developed methods for FSS distribution on arbitrary (curved) structures:
The planar FSS cells are body-fitted to the target surfaces.
Parametric
Global
Projected

44. 2.1. Jerusalem cross Metallic Cross
It is the reference FSS cell to be used in next examples:
Transmission performance at 9.5 GHz.
Transparent at 6 and 15 GHz.

45. 2.1. Jerusalem cross
1 dielectric layer: analysis of performance by varying the thickness
Metallic cross on top of material.

46. 2.1. Jerusalem cross
1 dielectric layer: S parameters for dielectric thickness = 0,25 cm

47. 2.1. Jerusalem cross
1 dielectric layer: S parameters for dielectric thickness = 1 cm

48. 2.3. Cassegrain dichroic
Main reflector feed by a rectangular horn at 9.5 GHz
Subreflector feed by a rectangular horn at 15 GHz
4 different set-up:
Isolated main reflector with its horn at 9.5 GHz.
Cassegrain by using the equivalent subreflector with Jerusalem crosses. Feeding the main reflector with its horn at 9.5 GHz.
Cassegrain feeding the subreflector at 15 GHz.
Cassegrain by using the equivalent subreflector with Jerusalem crosses. Feeding the subreflector at 15 GHz.

49. 2.3. Cassegrain dichroic
a) 9,5 GHz
b) 9,5 GHz
c) 15 GHz
d) 15 GHz

50. 2.3. Cassegrain dichroic
a) 9,5 GHz
b) 9,5 GHz
c) 15 GHz
d) 15 GHz

51. 2.3. Cassegrain dichroic
a) 9,5 GHz
b) 9,5 GHz

52. 2.3. Cassegrain dichroic
c) 15 GHz
d) 15 GHz

53. 2.3. Dichroic system
Two equivalent reflectors with Jerusalem crosses and no materials
Upper reflector feed by a circular horn at 5 GHz
Lower reflector feed by a circular horn at 8 GHz
The Jerusalem cross has been scaled to reflect at 5 GHz and to be transparent a 8 GHz

54. 2.3. Dichroic system Dichroic Reflectors system FSS-reflectors system

55. FSS-reflector fed at 5 GHz
2.3. Dichroic system
a) Upper
FSS-reflector fed at 5 GHz

56. FSS-reflector fed at 8 GHz
2.3. Dichroic system
b) Lower
FSS-reflector fed at 8 GHz

57. 3. Radome Applications
A radome is a structure designed for covering antennas
It is made of dielectric material that should be transparent at the design frequency
¿Main goals?
Physical protection
Aerodynamic performance
Environmental isolation
Hiding the antennas
¿Disadvantages?
Insertion losses
Boresight error
Distortion of transmission and reflection coefficients
Proposed analysis technique: MOM
Dielectric analysis by a thin layer approach
Dielectric analysis by a parallelepipeds approach

58. 3. Radome Applications Thin layer modelling
Dielectric of homogeneous thickness (single layer, sandwich,…)
Field incidence normal to radomes (polarization parallel to surfaces)
Only the boundary interfaces are required

59. 3. Radome Applications Volumetric modelling
Volumes are approximated by parallelepipeds
VFIE (Volumetric Field Integral Equation) optimized using Toeplitz symmetry

60. 3.1. Helmet-pilot in cockpit
Metallic structure with 2 monopoles at 2 GHz
Metallic structured covered with dielectric
εr1=1.5 ; thickness = 2.2 cm
Full structure centred in a aircraft cockpit
εr2=1.35 ; thickness = 2 cm

61. 3.1. Helmet-pilot in cockpit
Electric field components

62. 3.1. Helmet-pilot in cockpit
Radiated Power (Watts)
Insertion Losses (dB)
Boresight Error (°)
Antenna
0.0033 -
Helmet
0.0039
0.182 4
Cockpit
0.0032
0.2381 1
Parameters resume with and without radomes

63. 3.2. Reflector in ogive radome
Currents in the isolated reflector
Currents in the ogive radome
R = 50 cm ; F = 40 cm ; offset = 20 cm ; f = 6 GHz
RO = 75 cm ; HO = 150 cm ; RS = 10 cm
εr=3 ; tanδ= ; thickness = 2.5 cm

64. 3.2. Reflector in ogive radome
Electric field components

65. 3.2. Reflector in ogive radome
Radiated Power (Watts)
Insertion Loss(dB)
Boresight Error (°)
Antenna
0.0021 -
Radome
0.0022
3.431
0.5
Parameters resume with and without radome

66. 3.3. Parametric Radomes 3-layers sandwich radome f = 5 GHz
Ri = 250 mm ; thickness = 0.25 mm ; t ∈ [3 5] mm
εr1=4 ; tanδ=0.02
εr2=1.1 ; tanδ=0.005

67. Parametric gain with and without FSS
3.3. Parametric Radomes
Parametric gain with and without FSS

68. Radiated Power (Watts)
3.3. Parametric Radomes
Radiated Power (Watts)
Insertion Loss (dB)
Boresight Error (°)
Antenna -
3 mm
3 mm FSS
1.5
3.5 mm
3.5 mm FSS
4 mm
4 mm FSS
0.0237 3
4.5 mm
4.5 mm FSS
3.5
5 mm
0.0026 1
5 mm FSS
Parameters resume with and without radome

69. 3.3. Parametric Radomes Custom nose-airplane shape radome
3 dielectric layers with parametric thickness
Circular horn at 5 GHz.

70. 3.3. Parametric Radomes Custom nose-airplane shape radome
3 dielectric layers with parametric thickness
Circular horn at 5 GHz.

71. 3.3. Parametric Radomes Custom nose-airplane shape radome
3 dielectric layers with parametric thickness
Circular horn at 5 GHz.
Intermediate Thickness (mm) 3
3,5 4
4,5 5
Insertion Losses
(dB)
Radiated Power – Antenna (RPA)
(Watts)
Radiated Power with radome
(RPR)
RPA/RPR (dB)

72. 3.4. Electrically-Large Radome
Cassegrain system fed with a circular horn at 60 GHz
PVC radome – thickness of 1 mm
unknowns, analysed with:
OpenMP Architecture – 16 cores
EFIE
GMRES Solver
SAI preconditioner
Simulation time: 22’17’’
RAM memory: 11.1 GHz

73. 3.4. Electrically-Large Radome

74. 3.4. Electrically-Large Radome

75. Arrays Antennas Applications Overview
Phased Arrays
Arrays Antennas Applications Overview

76. Design of antennas array
We start by building the unit antenna element of the array
We obtain the array by replicating the element in an uniform rectangular distribution.
We can modify, if necessary, the position and orientation of each element
The first element has been created using the feature of "coaxial feed“
To create the array we use the command line "array" command

77. Design of the array of antennas. Considering an uniform feeding
The array of antennas is analyzed with MoM module and considering an uniform feeding

78. Design of the array of antennas. How can we modify the sidelobe level?
We can modify the feeding (amplitude and phase) of each antenna in order to change the direction of the main beam and the side lobe levels using an algorithm for the synthesis of a desired radiation pattern.
As example we will apply the Dolph- Chebychev algorithm.

79. Design of the array of antennas. How can we modify the sidelobe level?
In order to obtain the amplitude and phase of each array element, we have a repertory of “functions” with different algorithms for the array radiation pattern synthesis. Users can introduce custom functions to satisfy his needs.
In our example we select the Bidirectional_Dolph function.
bidirectional_dolph(d1, N1, R1, d2, N2, R2, theta, phi)
We feed each array element with the weight obtained from this function.

80. Design of the array of antennas. Considering a sidelobe level of -50 dB
The Dolph-Chebychev algorithm has been used to compute the module and phase to feed each element.

81. 5. Design of the array of antennas.
Get the radiation on the desired orientation
Once, we have designed the array antenna and the antenna feeding, we can rotate it, in order to get the radiation on the desired orientation.

82. Radiation pattern combining MoM and
GTD (II)
We can analyze easily a large antenna on-board a platform combining MoM and GTD.
A large antenna mounted on the platform cannot be analyzed using its far field radiation pattern emanating from a single phase center point.
In this case:
First, the antenna alone is analyzed using MoM-MLFMM.
Then, we obtain the multipoles (set of radiation patterns) that model the antenna currents.
We analyze the antenna on-board the platform using GTD for modelling the radiation of each multipole.

83. Radiation pattern combining MoM and GTD (II)
We analyze the radiation pattern obtained mounted on the platform

84. Thank you