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International Journal of Antennas and Propagation
Volume 2008, Article ID 636047, 6 pages
http://dx.doi.org/10.1155/2008/636047
Research Article

Wideband Flat Radomes Using Inhomogeneous Planar Layers

College of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran

Received 9 June 2008; Accepted 23 September 2008

Academic Editor: Joshua Li

Copyright © 2008 Mohammad Khalaj-Amirhosseini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Inhomogeneous planar layers (IPLs) are optimally designed as flat radomes in a desired frequency range. First, the electric permittivity function of the IPL is expanded in a truncated Fourier series. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. The performance of the proposed structure is verified using some examples.

1. Introduction

Radomes are sheltering structures to protect antennas against severe weather such as high winds, rain, icing, and/or temperature extremes [1]. In addition, radomes must not interfere with normal operation of the antennas. Therefore, the input reflection of radomes must be negligible at the usable frequency band of the protected antennas. Inhomogeneous planar layers (IPLs) are widely used in microwave and antenna engineering [24]. In this paper, we propose utilizing IPLs as flat radomes [5, 6] in a desired frequency range. To optimally design IPLs, the electric permittivity function of them is expanded in a truncated Fourier series, first. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. The identical procedure has been used to optimally design IPLs as impedance matchers between two different mediums previously [7]. Finally, the usefulness of the proposed structure is verified using some examples.

2. Analysis of IPLs

In this section, the frequency domain equations of the IPLs are reviewed. Figure 1 shows a typical IPL with thickness 𝑑, whose left and right mediums are the free space and whose electric permittivity function is 𝜀𝑟(𝑧). One way to fabricate the IPLs is to place several thin homogeneous dielectric layers beside each other. It is assumed that the incidence plane wave propagates obliquely toward positive x and z direction with an angle of incidence 𝜃𝑖 and electric filed strength 𝐸𝑖. Also, two different polarizations are possible, one is TM and the other is TE. Of course, we know that the wave radiated by an antenna can be decomposed to many plane waves with different angle of incidence.

fig1
Figure 1: A typical IPL as a flat radome (a) TE polarization mode, (b) TM polarization mode.

The differential equations describing IPLs have nonconstant coefficients and so, except for a few special cases, no analytical solution exists for them. There are some methods to analyze the IPLs such as finite difference [8], Taylor ’s series expansion [9], Fourier series expansion [10], the equivalent sources method [11], and the method of moments [12]. Of course, the most straightforward method to analyze IPLs is subdividing them into K thin homogeneous layers with thickness 𝑑Δ𝑧=𝐾𝜆min𝑐𝑓maxmax(𝜀𝑟(𝑧))(1) in which c is the velocity of the light and 𝑓max is the maximum analysis frequency. The ABCD parameters [13] of the IPL are obtained from those of the layers as follows: =𝐴𝐴𝐵𝐶𝐷1𝐵1𝐶1𝐷1𝐴𝑘𝐵𝑘𝐶𝑘𝐷𝑘𝐴𝐾𝐵𝐾𝐶𝐾𝐷𝐾,(2)where the ABCD parameters of the kth layer are as follows: 𝐴𝑘=𝐷𝑘=cosΔ𝜃𝑘,𝐵𝑘=𝑍2𝑐(𝑘0.5)Δ𝑧,𝜃𝑖𝐶𝑘=𝑗𝑍𝑐(𝑘0.5)Δ𝑧,𝜃𝑖sinΔ𝜃𝑘.(3) In (3), Δ𝜃𝑘=2𝜋𝑓𝑐𝜀𝑟𝑘0.5Δ𝑧sin2𝜃𝑖Δ𝑧(4)is the electrical length of the kth layer and 𝑍𝑐(𝑧,𝜃𝑖) is the characteristic impedance of the IPL, defined as the ratio of the transverse electric field to the transverse magnetic field, given by 𝑍𝑐(𝑧,𝜃𝑖)=𝜇0𝜀01𝜀𝑟𝑧sin2𝜃𝑖,TE,𝜇0𝜀01𝜀𝑟𝑧𝜀𝑟𝑧sin2𝜃𝑖,TM.(5)Finally, the input impedance and reflection coefficient of the radome are determined as follows: 𝑍in(𝑓,𝜃𝑖)=𝐴𝑍𝐿(𝜃𝑖)+𝐵𝐶𝑍𝐿(𝜃𝑖,Γ)+𝐷in(𝑓,𝜃𝑖𝑍)=in(𝑓,𝜃𝑖)𝑍𝑆(𝜃𝑖)𝑍in(𝑓,𝜃𝑖)+𝑍𝑆(𝜃𝑖),(6) where are the equivalent load and source impedances, respectively.

3. Synthesis of Radomes

In this section, a general method is proposed to optimally design the IPLs as radomes. First, we consider the following truncated Fourier series expansion for the electric permittivity function The reason to use logarithm function at the left of (8) is to keep 𝜀𝑟(𝑧)1. An optimum designed radome has to have the input reflection coefficient as small as possible in a desired frequency and incidence angle range. Therefore, the optimum values of the coefficients 𝐶𝑛 in (8) can be obtained through minimizing the following error function corresponding to M frequencies, J incidence angles, and two possible polarizations TE and TM: The defined error function should be restricted by some constraints such as not having a significant reflection at all incidence angles [0𝜃𝑖,max], and easy fabrication, respectively, as follows: |||Γin𝑓𝑚,𝜃𝑖,𝑗|||𝜌max𝜀,𝑚=1,2,,𝑀,𝑗=1,2,,𝐽,POL.=TEandTM,(10)𝑟𝜀(𝑧)𝑟max,(11)𝜀𝑟=1𝑑𝑑0𝜀𝑟𝜀(𝑧)𝑑𝑧𝑟min,(12) where (𝜀𝑟)max is the maximum value of 𝜀𝑟(𝑧), in the fabrication step. It is noticeable that the constraint (12) is necessary to avoid obtaining the wrong solution 𝜀𝑟(𝑧)=1 (the free apace) in the optimization process. Also, to enforce the designed radomes to be symmetric, we have to use the following truncated Fourier series instead of (8) for the electric permittivity function To solve the above constrained minimization problem, we can use the fmincon. m file in the MATLAB program. fmincon uses a sequential quadratic programming (SQP) method, in which a quadratic programming (QP) subproblem is solved at each of its iteration.

4. Examples and Results

We would like to design an IPL with thickness 𝑑=2.5 cm (a practical and feasible chosen) as a radome in a frequency range DC to 8.0 GHz, considering 𝜃𝑖,max=60,𝜌max=0.1=20 dB, and (𝜀𝑟)max=10. Using the proposed optimization approach, considering 𝑁=10 spatial harmonics, 𝑀=80 frequencies, 𝐽=2 incidence angles (0 and 60), and (𝜀𝑟)min=1.05 or 1.10, two radomes were synthesized. The unknown coefficients of the truncated Fourier series related to the synthesized radomes are written in Table 1. Figures 2 and 3 illustrate the obtained electric permittivity function 𝜀𝑟(𝑧), respectively, considering (𝜀𝑟)min=1.05 or 1.10. Figures 4, 5, 6, and 7 illustrate the magnitude of the input reflection coefficient |Γin(𝑓)| for TE and TM polarizations. It is observed that the designed radomes have a good performance in both desired frequency band and the incidence angle range. Meanwhile, the reflection coefficient degrades with increasing the angle of incidence. To show the effect of the thickness of IPLs, we increase d from 2.5 cm to 5, 7.5, and 10 cm. The unknown coefficients of the truncated Fourier series, the electric permittivity function 𝜀𝑟(𝑧), and the magnitude of the input reflection coefficient |Γin(𝑓)| corresponding to 𝜃𝑖=60 and considering (𝜀𝑟)min=1.05 are shown in Table 2 and Figures 8, 9, and 10, respectively. It is observed that as the thickness of the IPL is chosen larger, the obtained electric permittivity function tends to a continuous function, whose property is matching between the air mediums and an intermediate medium. Also, it is seen that the efficiency for TM polarization is better than that for the TE polarization.

tab1
Table 1: The unknown coefficients of the truncated Fourier series for 𝑑=2.5 cm.
tab2
Table 2: The unknown coefficients of the truncated Fourier series for (𝜀𝑟)min=1.05.
636047.fig.002
Figure 2: The electric permittivity function 𝜀𝑟(𝑧), considering (𝜀𝑟)min=1.05.
636047.fig.003
Figure 3: The electric permittivity function 𝜀𝑟(𝑧), considering (𝜀𝑟)min=1.10.
636047.fig.004
Figure 4: The magnitude of |Γin(𝑓)| for TE polarization, considering (𝜀𝑟)min=1.05.
636047.fig.005
Figure 5: The magnitude of |Γin(𝑓)| for TM polarization, considering (𝜀𝑟)min=1.05.
636047.fig.006
Figure 6: The magnitude of |Γin(𝑓)| for TE polarization, considering (𝜀𝑟)min=1.10.
636047.fig.007
Figure 7: The magnitude of |Γin(𝑓)| for TM polarization, considering (𝜀𝑟)min=1.10.
fig8
Figure 8: The electric permittivity function 𝜀𝑟(𝑧), considering (𝜀𝑟)min=1.05 (a) 𝑑=2.5 cm, (b) 𝑑=5.0 cm, (c) 𝑑=7.5 cm, (d) 𝑑=10 cm.
636047.fig.009
Figure 9: The magnitude of |Γin(𝑓)| for TE polarization with 𝜃𝑖=60, considering (𝜀𝑟)min=1.05.
636047.fig.010
Figure 10: The magnitude of |Γin(𝑓)| for TM polarization with 𝜃𝑖=60, considering (𝜀𝑟)min=1.05.

5. Conclusion

Inhomogeneous planar layers (IPLs) were optimally designed as flat radomes in a desired frequency. First, the electric permittivity function of the IPL is expanded in a truncated Fourier series. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. The performance of the proposed structure is verified using some examples. It was observed that the designed radomes have a good performance in both desired frequency band and the incidence angle range, where the efficiency for TM polarization is better than that for the TE polarization. Also, as the thickness of the IPL is chosen larger, the obtained electric permittivity function tends to a continuous function, whose property is matching between the air mediums and an intermediate medium. The proposed method can be extended for IPLs, whose magnetic permeability is inhomogeneous solely or along with their electric permittivity. Also, we can consider IPLs for spherical wavefronts instead of planar ones in the future.

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