Two-step Structural Design of Mesh Antennas for High Beam Pointing Accuracy
© The Author(s) 2017
Received: 9 May 2016
Accepted: 2 April 2017
Published: 21 April 2017
A well-designed reflector surface with high beam pointing accuracy in electromagnetic performance is of practical significance to the space application of cable mesh reflector antennas. As for space requirements, circular polarizations are widely used in spaceborne antennas, which usually lead to a beam shift for offset reflectors and influence the beam pointing accuracy. A two-step structural design procedure is proposed to overcome the beam squint phenomenon for high beam pointing accuracy design of circularly polarized offset cable mesh reflectors. A simple structural optimal design and an integrated structural electromagnetic optimization are combined to alleviate the beam squint effect of circular polarizations. It is implemented by cable pretension design and adjustment to shape the offset cable mesh surface. Besides, in order to increase the efficiency of integrated optimization, an update Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian matrix is employed in the optimization iteration with sequential quadratic programming. A circularly polarized offset cable mesh reflector is utilized to show the feasibility and effectiveness of the proposed procedure. A high beam pointing accuracy in order of 0.0001º of electromagnetic performance is achieved.
In recent years, the stringent requirements on large space reflectors become demanding for high electromagnetic performance . As for space applications, circular polarizations are usually used in spaceborne antennas. With circularly polarized feeds, there exists a beam squint phenomenon in offset reflector antennas . The squint angle, which is manifested by a small beam shift of the radiation pattern in the plane perpendicular to the principal offset plane, can significantly affect the beam pointing accuracy. As one of the most widely used space antennas, cable mesh reflector antenna has attracted much attention due to its advantages of large diameter, light weight, and reasonable cost . Similarly with the smooth solid reflectors, the beam squint phenomenon can also be observed in offset cable mesh reflector antennas with reflecting mesh leakage [4, 5]. The beam squint angle should be taken into account for space applications such as satellite communications, deep-space telemetry, and radio astronomy , which concentrate more on beam pointing accuracy. With the stringent requirements on space reflector antennas, the compensation technology to overcome the antenna pattern degradation including beam squint to achieve high pointing accuracy becomes more demanding .
Since the simple formula which accurately predicts the squint angle in circularly polarized offset reflectors was proposed by ADATIA and RUDGE , the beam squint phenomenon and its compensation method have attracted many authors’ interests. A squint compensation method by properly tilting the feed to make the interpreted angle between the incident beam and the radiated beam zero is a natural choice for symmetrical reflectors with off-focus feeds . A squint free approach for symmetrical dual reflector antennas is also proposed by properly choosing geometrical parameters . Furthermore, XU and RAHMAT-SAMII  summarized the beam squint compensation methods, and proposed a compensation technology by optimally displacing circularly polarized feeds in the perpendicular plane to obtain high beam pointing accuracy. However, these methods in Refs. [2, 7–9] are presented from the simple electromagnetic disciplinary, and they are just practical for undistorted reflectors in the nominal state for preliminary design. In actual engineering, space reflectors including cable mesh antennas are easily susceptible to surface distortion under thermal load and other impacts, which enlarge the beam squint angle and seriously affect the beam pointing accuracy. Simply displacing and tilting the antenna feed cannot thoroughly compensate the distorted electromagnetic performance in actual engineering. Another consideration should be taken into account is that feed remains on focus with a satisfactory reflector surface is preferred due to the limited size in satellites. How to produce a cable mesh reflector with high beam pointing accuracy in electromagnetic performance is an urgent problem for space applications.
As for structural design of cable mesh reflectors, pretension design of cable nets is an important process to obtain the required reflector surface. Recently, there are several methods which investigate the form-finding analysis for cable mesh reflectors, such as the method presented by TANAKA, et al , optimal design method of initial surface in Ref. , simple technique in Ref. , numerical form-finding method proposed by MORTEROLLE, et al  to ensure uniform tension, form-finding analysis with PZT actuators  and pretension design under multi-uncertainty . These methods aim to design a surface profile with minimum or zero root-mean-square (rms) error to ensure its surface accuracy. Although the reflector shape can be obtained with high surface accuracy by these methods, its beam pointing accuracy cannot be easily guaranteed, even for circularly polarized feeds. Thus, there rises a problem that is it possible to provide a pretension structural design considering electromagnetic performance to obtain high beam pointing accuracy for circularly polarized feeds? The integrated structural electromagnetic design concept [16, 17] inspires us with a combined procedure, which makes a pretension design from multidisciplinary viewpoint of structure and electromagnetism.
The main purpose of this paper is to present a two-step structural design technology for circularly polarized offset cable mesh reflectors with high beam pointing accuracy. The two-step pretension design combines a simple structural design and an integrated structural electromagnetic optimization. With this technology, high electromagnetic performance especially high beam pointing accuracy can be achieved in the antenna structural design. This technology not only can compensate the beam squint angle of circular polarizations, but also can produce a well-designed cable mesh reflector with on-focus feeds. Comparing with the compensation methods proposed by electromagnetism designers, the limited weakness of aforementioned methods can be overcame.
This paper is organized as follows. Section 2 of this study outlines the procedure of the two-step structural design technology. In this technology, an update Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian matrix is employed to increase the efficiency of optimization iteration. In section 3, a circularly polarized offset cable mesh reflector is utilized to show the feasibility and effectiveness of this procedure with an on-focus feed to achieve high beam pointing accuracy in electromagnetic performance. The major achievements are summarized in section 4.
2 Two-step Structural Design Procedure
After determining the surface cable length l, the number division in radius can be obtained in preliminary design. Thus, with the required parabolic surface equation, the predesigned surface nodal positions in front and rear cable nets can be calculated.
In the following structural design, in order to obtain a circularly polarized cable mesh reflector with high beam pointing accuracy, the predesigned nodes are firstly assumed in the nominal undistorted state. As mentioned before, there exists a beam squint phenomenon for this offset antenna. Then, with the integrated structural electromagnetic optimization, the beam squint phenomenon will be compensated.
To determine the cable tension, the Singular Value Decomposition (SVD) is performed on the equilibrium matrix A, and the cable tensions can be expressed as the linear combination of the independent states of self-stress . With optimizing the combination coefficients of multiple states of self-stress, the cable tensions can be obtained in this nominal state.
The derivation of G is based on two sensitivities - one is the electromagnetic sensitivity of boresight directivity with respect to surface nodal displacements, and the other is the structural sensitivity of surface nodal displacements with respect to cable dimensions. Its expression is illustrated in Refs. [17, 23]. The constraint gradient matrix G t is based on structural sensitivity analysis of cable tensions with respect to cable dimensions . By using the nonlinear optimization function—quadprog in MATLAB, this optimization model can be solved.
- Step 1:
Provide the initial parameters of cable mesh reflector, including the diameter, focal length, offset height, mesh tension, working frequency, and feed polarization;
- Step 2:
Perform the preliminary design by the relationship between surface rms error and cable length;
- Step 3:
Obtain the equilibrium equation in the nominal state;
- Step 4:
Perform SVD operation to obtain the independent states of self-stress;
- Step 5:
Solve the pretension optimization model in (5);
- Step 6:
Perform structural and electromagnetic (EM) sensitivity analysis;
- Step 7:
Approximate Hessian matrix using BFGS update formula;
- Step 8:
Update cable dimensions;
- Step 9:
Obtain the structural and EM performance in the present state;
- Step 10:
Does the EM performance satisfy the convergence criterion? If no, go to Step 6, otherwise, export the optimum design.
It should be mentioned that the implementation from Step 3 to Step 5 belongs to the structural pretension design, and the procedure from Step 6 to Step 10 is a typical integrated structural electromagnetic optimization design. With this two-step structural design, an offset cable mesh reflector with high beam pointing accuracy under circular polarization will be obtained.
3 Simulation and Application
Cable mesh reflector specifications
Value or character
Single offset parabola
Aperture diameter d/m
Focal length f/m
Offset height h/m
Minimum distance between the front and rear cable nets h e/m
Young’s modulus of cables E/GPa
Cable cross-sectional area A/mm2
Mesh tension N m /(N m−1)
Cosine-Q feed Q x Q y
Feed tilt angle ψ 0/(°)
With the structural pretension design, a surface with uniform tie cable tension distribution can be obtained and all of the surface nodes are located at their nominal states. In the next, the electromagnetic performance is examined for this circular polarization. Beam squint occurs in this circularly polarized offset cable mesh reflector antenna. In the nominal state for RCP feed illumination, there exists a linear phase shift across the reflector aperture and the phase in the left side aperture region is lagging compared with the phase in the right side aperture region. The radiated left-hand circularly polarized (LCP) beam, which is launched from the RCP feed and reflected by this reflector, squints toward the right in xz plane and produces a negative squint angle.
Optimal results with different convergence criterions
Major parameters of far field patterns with RCP illumination
Left sidelobe Level/dB
Right Sidelobe level/dB
Considering the far field patterns in Fig. 4 and the optimization model in Eq. (7), it can be seen that although the constraints of the other electromagnetic performance such as sidelobe levels and cross-polarization are not added in the optimization model, the simulation result in Fig. 4 shows very satisfactory far field patterns in sidelobes and cross-polarization pattern. This can be explained that the effects of surface error on boresight directivity and the other performance are harmonious; the nonuniform phase distribution will produce a lower directivity and higher sidelobe levels; as the phase distribution across the reflector aperture becomes more uniform, the antenna electromagnetic performance including boresight directivity, sidelobe levels and cross-polarization will become better.
From the application, it can be concluded that a well-designed cable mesh reflector with high beam pointing accuracy in electromagnetic performance is obtained by a two-step structural design. This beam squint free technology is accomplished by structural design to shape surface with a uniform phase distribution in the aperture plane, and the linear phase shift caused on the polarized components of the incident field is thus reduced. This procedure benefits the radiation pattern with no need to displace feed position and orientation.
From the above comparative simulation between BFGS update Hessian matrix and exact Hessian matrix by second-order derivative, it can be seen that the BFGS approximation matrix can provide less iteration time and a little worse electromagnetic performance with maximum directivity in accuracy of 0.000 1 dB than the exact one in the cable mesh reflector antenna design.
Compared with the previous compensation methods, this technology can not only compensate the beam squint angle with an on-focus circular polarized feed, but also provide a well-designed surface with high beam pointing accuracy considering actual engineering. The mentioned pretension structural design can also be improved with considering electromagnetic performance. A statement should be addressed that the drawback of this method is that the procedure cannot handle both two circular polarizations simultaneously, which is also the drawback of other previous compensation methods.
Less iteration time and a little worse electromagnetic performance with maximum directivity in accuracy of 0.000 1 dB than the exact one are provided by BFGS approximation Hessian matrix in the two-step structural design. A helpful guideline for the cable mesh reflector antennas design can be presented.
A tilt-like surface deformation to achieve a uniform phase distribution in reflector aperture for circularly polarized offset cable mesh reflector antennas is provided in the optimal structural design, and the electromagnetic performance including boresight directivity, beam squint angle, sidelobe levels and cross-polarization approaches better as the phase distribution becomes uniform.
Even though the other antenna electromagnetic performance besides boresight directivity is not added in the multidisciplinary optimization model, once the boresight directivity is optimized as its extremely maximum value with sufficiently small convergence criterion, the other performance will also be made as an acceptable value due to the electromagnetism property.
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