International Journal of Antennas and Propagation

Volume 2008 (2008), Article ID 636047, 6 pages

http://dx.doi.org/10.1155/2008/636047

## 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 [2–4]. 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 , whose left
and right mediums are the free space and whose electric permittivity function
is .
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
and electric filed
strength . 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.

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 in which *c* is the velocity of the light and is the maximum analysis frequency. The *ABCD* parameters [13] of the IPL
are obtained from those of the layers as
follows: where the *ABCD* parameters of the *k*th layer are
as follows: In (3), is the electrical length of the
*k*th layer and
is the characteristic impedance of the IPL, defined as the ratio of the
transverse electric field to the transverse magnetic field, given by
Finally, the input impedance and
reflection coefficient of the radome are determined as follows:
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 . 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
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
, and easy
fabrication, respectively, as follows:
where is the
maximum value of , in the
fabrication step. It is noticeable that the constraint (12) is necessary to
avoid obtaining the wrong solution (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 cm (a practical and feasible chosen) as a radome in a frequency range DC to 8.0 GHz, considering dB, and . Using the proposed optimization approach, considering
spatial harmonics, frequencies, incidence angles ( and ), and
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 ,
respectively, considering or 1.10.
Figures 4, 5,
6, and 7 illustrate the
magnitude of the input reflection coefficient
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 , and the
magnitude of the input reflection coefficient corresponding to and considering 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.

#### 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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