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 J Electromagn Eng Sci > Volume 18(3); 2018 > Article Kim, Song, Shin, and Park: Radiation from a Millimeter-Wave Rectangular Waveguide Slot Array Antenna Enclosed by a Von Karman Radome

### Abstract

In this paper, electromagnetic radiation from a slot array antenna enclosed by a Von Karman radome is analyzed by using the ray tracing method and HuygensŌĆÖs principle. We consider the rectangular slot array antenna and the Von Karman radome. The radiation patterns are calculated by using the surface currents of the radome to illustrate the electromagnetic behaviors of the radome-enclosed waveguide slot array antenna.

### II. Field Analysis

The waveguide slot array antenna enclosed by a Von Karman radome is shown in Fig. 1. The waveguide array antenna has 112 elements . The dielectric constant of the radome and tilt angle are ╔ør and ╬▒x. The shape of the Von Karman radome satisfies the following equations in the cylindrical coordinates.
t=arccos(1-2zLi)
##### (2)
Žü=RiŽĆt-sin(2t)2
where Li, Ri, and t are the ith radome surface length, radius, and intervening variable, respectively. Fig. 2 illustrates the analysis procedure of the radome-enclosed waveguide slot array antenna based on the ray tracing and HuygensŌĆÖs principle.
The waveguide slot array antenna based on an actual model  has 112 elements and the electric fields of the slots are obtained by the simulation of the ANSYS High Frequency Structure Simulator (HFSS) based on the 3D FEM (Fig. 3). Details on how to design the antenna can be found in . We assume that the point source is at the center of each slot. Using the surface equivalence theorem , the magnetic currents on the slot are given by
##### (3)
MŌåÆ=-2n^├ŚEŌåÆt
where n╠é is the normal vector of the slot and EtŌåÆ is the tangential electric field of the slot. On the radome surface, the meshes are created in the vertical direction (z╠é) and in the azimuthal direction (Žå╠é) (Fig. 4). The radome surface is divided into M (vertical direction) ├Ś N (azimuthal direction) meshes. We generate the rays from rectangular slots, the origins of which are the centers of the slots (x0, y0, z0) and the ray direction vector is ki,jŌåÆ. Note that (xi,j, yi,j, zi,j) is the center of the mesh (M = i and N = j). To apply the ray tracing technique inside the radome, the ray path is calculated . We find the intercept point of the ray incident on a ray radome surface based on the iterative method . At the intercept point (xj, yi, zi), the normal vector (n╠év) of the ith Von Karman radome surface can be expressed as
##### (4)
n^v=xixi2+yi2┬ĘtztŽü2+tz2x^+yixi2+yi2┬ĘtztŽü2+tz2y^-tŽütŽü2+tz2z^
##### (5)
tŽü=-RiŽĆ┬Ę(1-cos(2t))2t-sin(2t)2
##### (6)
tz=Li2ŌĆēsin(t)
where tŽü and tz are the horizontal and vertical components of the tangential vectors, respectively. At the intercept point, the incident fields are split by the perpendicular and parallel components. The reflected and transmitted waves are obtained by using the reflection and transmission coefficients of each polarization. On the outer radome surface, we enforce HuygensŌĆÖs principle to calculate the radiation pattern of the Von Karman radome-enclosed slot array antenna . We use the far-field approximation  to calculate the radiated fields from each radome mesh as
##### (7)
E╬Ė(m,n)=-j╬▓e-j╬▓r(m,n)4ŽĆr(m,n)(LŽå(m,n)+╬ĘN╬Ė(m,n))
##### (8)
EŽå(m,n)=j╬▓e-j╬▓r(m,n)4ŽĆr(m,n)(L╬Ė(m,n)-╬ĘNŽå(m,n))
##### (9)
H╬Ė(m,n)=j╬▓e-j╬▓r(m,n)4ŽĆr(m,n)(NŽå(m,n)-L╬Ė(m,n)╬Ę)
##### (10)
HŽå(m,n)=j╬▓e-j╬▓r(m,n)4ŽĆr(m,n)(N╬Ė(m,n)-LŽå(m,n)╬Ę)
##### (11)
N╬Ė(m,n)=A(m,n)┬Ę(Jx(m,n)ŌĆēcosŌĆē╬ĖŌĆēcosŌĆēŽå+Jy(m,n)ŌĆēcosŌĆē╬ĖŌĆēsinŌĆēŽå-Jz(m,n)ŌĆēsinŌĆē╬Ė)
##### (12)
NŽå(m,n)=A(m,n)┬Ę(-Jx(m,n)ŌĆēsinŌĆēŽå+Jy(m,n)ŌĆēcosŌĆēŽå)
##### (13)
L╬Ė(m,n)=A(m,n)┬Ę(Mx(m,n)ŌĆēcosŌĆē╬ĖŌĆēcosŌĆēŽå+My(m,n)ŌĆēcosŌĆē╬ĖŌĆēsinŌĆēŽå-Mz(m,n)ŌĆēsinŌĆē╬Ė)
##### (14)
LŽå(m,n)=A(m,n)┬Ę(-Mx(m,n)ŌĆēsinŌĆēŽå+My(m,n)ŌĆēcosŌĆēŽå)
where A(m, n) is the surface area of each mesh as
##### (15)
A(m,n)=2ŽĆN┬ĘŌł½tmintmaxR2ŽĆ(t-sin(2t)2)(RiŽĆ┬Ę1-cos(2t)-2t-sin(2t)/2+(-L22sin(t)))dt
##### (16)
tmax=arccos(1-2(L2-zm)L2)
##### (17)
tmin=arccos(1-2(L2-zm-1)L2)
##### (18)
zm=m┬ĘL2M
where, L2 and R2 are the length and the radius of the Von Karman radome, respectively. The radiated power can be obtained from the Poynting vector (PŌāŚ) as
##### (19)
PŌåÆ=12(Re(EŌåÆ├ŚHŌåÆ*))
where EŌāŚ and HŌāŚ are the radiated electromagnetic fields at the observation points.

### III. Numerical Results

To check the validity of our analysis, we consider the radome with a dielectric constant (╔ør) of 1, which means (that the array has no radome). Fig. 5 illustrates the radiation pattern of the Von Karman radome-enclosed waveguide slot array antenna designed for the Ka-band. We compare our calculated result with the radiation pattern of the slot array antenna. As the mesh number increases to 400 ├Ś 400, the results of the ray tracing technique and the case of the slot antenna without a radome show a good agreement. Note that the mesh size should be less than ╬╗/4 (Table 1).
Fig. 6 shows the radiation pattern of the Von Karman radome-enclosed waveguide slot array antenna for the different gimbal tilt angles (╬▒x). The design parameters of the radome are presented in Table 2, and Table 3 shows the transmission loss of the radome. Two possibilities can account for the gain reduction. The transmission loss and the sidelobe level decrease as the tilt angle increases because of the enhancement of reflection loss at smaller incident angles. Note that the image robes are observed at nearly ╬Ė = ŌłÆ40┬░ in Fig. 6 because of the internal reflection within the radome. Fig. 7 illustrates the surface currents of the radome for the different gimbal tilt angles (╬▒x). Large current distributions can be seen near the each tilt angle and the angle of the image robe.

### IV. Conclusion

We have analyzed the electromagnetic radiation from a slot array antenna enclosed by a Von Karman radome using the ray tracing technique and HuygensŌĆÖs principle. The radiation patterns of the radome-enclosed waveguide slot array antenna were calculated to illustrate the electromagnetic behaviors of the radomes. Our method is useful to estimate the electromagnetic characteristics, such as the radiation patterns and the BSE, of the radome-enclosed waveguide slot array antenna.

### ACKNOWLEDGEMENTS

This work was supported by Hanwha Systems.
##### Fig.┬Ā1
Waveguide slot array antenna enclosed by a Von Karman radome. ##### Fig.┬Ā2
Analysis procedure of the radome-enclosed waveguide slot array antenna. ##### Fig.┬Ā3
Electric field of the rectangular slots. ##### Fig.┬Ā4
Mesh and ray generation. ##### Fig.┬Ā5
Radiation pattern of the radome-enclosed waveguide slot array antenna (╔ør = 1). ##### Fig.┬Ā6
Radiation pattern of the radome-enclosed waveguide slot array antenna. (a) ╬▒x = 10┬░, (b) ╬▒x = 20┬░, and (c) ╬▒x = 30┬░. ##### Fig.┬Ā7
Surface current of the radome. (a) ╬▒x = 10┬░ (side view), (b) ╬▒x = 10┬░ (top view), (c) ╬▒x = 20┬░ (side view), (d) ╬▒x = 20┬░ (top view), (e) ╬▒x = 30┬░ (side view), (f) ╬▒x = 30┬░ (top view). ##### Table┬Ā1
Number of mesh and mesh size (outer surface)
Number of mesh (M ├Ś N) Mesh size (z ├Ś Žå)
80 ├Ś 80 0.87╬╗ ├Ś 0.061╬╗ ŌłÆ 1.3╬╗
100 ├Ś 100 0.70╬╗ ├Ś 0.041╬╗ ŌłÆ 1.0╬╗
200 ├Ś 200 0.35╬╗ ├Ś 0.012╬╗ ŌłÆ 0.50╬╗
400 ├Ś 400 0.17╬╗ ├Ś 0.0037╬╗ ŌłÆ 0.25╬╗
##### Table┬Ā2
Design parameter of the Von Karman radome in the case of Fig. 5
Parameter Value
Inner length (L1) 66.9╬╗
Outer length (L2) 68.0╬╗
╔ør 3.41
##### Table┬Ā3
Transmission loss and the sidelobe level of the Von Karman radome-enclosed waveguide slot array antenna
Gimbal tilt angle (╬▒x)

10┬░ 20┬░ 30┬░
Transmission loss (dB) ŌłÆ3.10 ŌłÆ3.35 ŌłÆ1.64
Sidelobe level (dB) ŌłÆ9.29 ŌłÆ16.55 ŌłÆ20.44

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### Biography

Jihyung Kim obtained his B.S. and M.S./Ph.D. degrees in electrical engineering from Ajou University, Suwon, Korea, in 2009 and 2016, respectively. Since 2016, he has been a researcher at Hanwha Systems. His research interests include the analysis of aperture array antennas and radomes. ### Biography

Hokeun Shin received his B.S. degree in electrical and computer engineering from Ajou University, Suwon, Korea, in 2015. He is currently undertaking his M.S. and Ph.D. course at the Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea. His research interests include the analysis of radomes and radar cross section. ### Biography

Sung Chan Song obtained his B.S. and M.S. degrees in avionics engineering from Korea Aerospace University, Goyang, Korea, in 2001 and 2003, respectively. From 2002 to 2015, he worked at Samsung Thales. Since 2015, he has been a researcher at Hanwha systems. His research interests include antennas, electromagnetic wave numerical analysis, and radar systems. ### Biography

Yong Bae Park received his B.S. (summa cum laude), M.S., and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1998, 2000, and 2003, respectively. From 2003 to 2006, he was with the Korea Telecom Laboratory, Seoul, Korea. In 2006, he joined the School of Electrical and Computer Engineering, Ajou University, Suwon, Korea, where he is currently a professor. His research interests include electromagnetic field analysis and electromagnetic interference and compatibility. Editorial Office
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