Download Resonant Frequencies in the Open Microstrip Structures

Resonant Frequencies in the Open Microstrip
Structures Placed on Curved Surfaces
Rafal Lech and Adam Kusiek
Gdansk University of Technology
Faculty of Electronics, Telecommunications and Informatics
80-233 Gdansk, Poland
Abstract—The paper presents the research on open microstrip
structures placed on curved surfaces such as cylindrical, elliptical
or spherical. The numerical analysis of investigated structures is
based on expansion of electric and magnetic fields into suitable
function series. Utilizing the continuity conditions the boundary
problem is formulated which is solved with the use of method of
moments. The investigated structures find application in antenna
technique for the construction of conformal antennas.
Index Terms—Microstrip structure, conformal antennas, resonant structure, method of moments.
I. I NTRODUCTION
Conformal structure, i.e. the structures which shape is
matched to the surface of the object on which they are embedded or integrated, are ideally suited due to the aerodynamic
reasons for the use on the wings of the aircraft, missiles, highspeed trains or boats. The most popular application of these
structures are conformal antenna. The ability to place these
antennas on curved surfaces means that they can also find
applications in communication systems located in the historic
buildings, towers or columns without disturbing their aesthetics and local architecture. In addition to the high integrity
of these structures, the undisputed advantages and benefits
that can be gained by using these antennas (relative to the
planar antennas) are e.g. larger angular coverage, possibility
to control the beam without distortion of characteristics, wider
operation band and lower RCS.
However, there are various reasons for which the conformal
antennas are not widespread. For example, the realization
technology of conformal antenna is much more difficult than
the planar antennas and is not yet fully mature. In addition, the
market lacks universal design tools of this type of antennas.
Due to the size of the designed antenna and its analysis method, the antennas are commonly divided into the
electrically small and electrically large antennas. Electrically
large antennas are those whose surface dimensions are many
times larger than the wavelength, and the surfaces of the
electrically small antennas are comparable to the wavelength.
For both these categories, the time domain and frequency
domain methods are utilized. The most commonly used and
most effective approach is to use hybrid methods. Electrically
small antenna are often analyzed using the mode matching
method [1], method of moments [2]–[5], the finite element
method often combined with other approaches [6], or the
finite-difference time-domain method [7]. Electrically large
978-1-5090-2214-4/16/$31.00 ©2016 IEEE
Fig. 1: Investigated structures: cylindrical, elliptical and spherical.
antennas are mainly analyzed using high frequency approaches
that are based on the asymptotic techniques allowing for
finding the approximate solution [8].
In this article the problem of determining the resonant
frequency of the microstrip structures, located on cylindrical,
spherical and elliptical surfaces is considered. The curved
surfaces are made of metal and coated with a dielectric
material. For the case of elliptical surfaces both the structure
made of confocal ellipses (for which the thickness of the
dielectric layer is irregular) and the structure made of nonconfocal ellipses (where the dielectric layer is of constant
thickness) are considered. In order to solve the problem the
fullwave approach based on method of moments is utilized.
Depending on the geometry of the investigated structure the
expansion of electric and magnetic fields into suitable function
series is used.
II. D ESCRIPTION
OF THE STRUCTURE AND ANALYSIS
METHOD
Schematic views of the structures under consideration are
shown in Fig. 1. In each case, the structure can be divided
into two regions: region 1 as a dielectric layer with a relative
permittivity εr covering a metallic object (cylinder, ellipse or
sphere) and region 2 located outside of the structure (open
space). Microstrip patch is located on the outer surface of the
dielectric - at the interface between region 1 and 2.
For a cylindrical structure the z components of electric and
magnetic fields take the form:
Fz(κ) =
Z∞
1
2π
dkz ejkz z
m=−M
−∞
M
X
ejmφ ×
(1)
F
Hm
(kκρ ρ)AF
mκ + Jm (kκρ ρ)Bmκ
(1)
F
where F = {E, H}, AF
mκ i Bmκ are unknown coefficients,
(1)
Jm (·) and Hm (·) are Bessel
p and Hankel functions, respectively, of order m, kκρ = ω 2 µ0 ε0 εrκ − kz2 , for κ = {1, 2}
denoting the region.
For an elliptical structure the z components of electric and
magnetic fields take the form:
Fz(κ)
Z∞
1
=
2π
X
ν
L
X
(ν)
(ν),F
Mcl (u, qκ,ξ )Al,κ,ξ cel (v, qκ,ξ )+
l=0
!
(ν)
(ν),F
Msl (u, qκ,ξ )Bl,κ,ξ sel (v, qκ,ξ )
l=1
(ν),F
(ν),F
(2)
(ν)
where Al,κ,ξ and Bl,κ,ξ are unknown coefficients, Mcl (·)
(ν)
and Msl (·) are even and odd Mathieu functions, respectively,
of order l and kind ν = {1, 4}. cel (·) and sel (·) are even
and odd angular Mathieu functions, respectively, of order l.
√
Parameter qκ,ξ = (kκ2 − kz2 )d2ξ /4, where kκ = ω µ0 ε0 εrκ for
region κ = {1, 2} and interface ξ = {1, 2}. It should be noted
that for nonconfocal ellipses each ellipse has a different focal
length dξ (ξ = 1 for the ellipse describing the metal surfaces
of the core, and ξ = 2 for the ellipse describing the interface
between the dielectric layer and the region outside), therefore
the parameter q in region 1 would have different values in the
equations describing the boundary conditions on these ellipses
(interfaces). In the analysis for this case the addition theorems
for Mathieu functions [9] need to be utilized.
For a spherical structure the tangential components of
electric and magnetic fields take the form:
(κ)
Ft
=
2
M
X
X
M X
zn(i) (kκ r)Mtmn (θ, φ)AE,κ
imn +
i=1 m=−M n=m
′
zn(i) (kκ r)Ntmn (θ, φ)AH,κ
imn
between the regions. J is an unknown current distribution on
the surface of microstrip patch and ia denotes a normal versor
to the patch surface. In order to solve the above problem the
patch current should be expanded in a series of current basic
functions:
P
X
J=
ap Jp
(5)
p=1
dkz ejkz z ×
−∞
L
X
where Pnm is an associate Legandre polynomial.
In every case of the above-described structures in order to
determine the resonance frequency the boundary conditions
must be satisfied for the tangential field components at both
interfaces:
(1)
Et = 0 on the surface of metallic core and
(
(2)
(1)
Et − Et = 0
(4)
(2)
(1)
ia × (Ht − Ht ) = J
(3)
H,κ
i
where AE,κ
imn and Aimn are unknown coefficients, zn (·) is
t
spherical Hankel function of kind i and order n, Mmn (θ, φ)
and Ntmn (θ, φ) are vector functions defined as follows:
∂Pnm (cos θ)
jm m
Mtmn =
Pn (cos θ)iθ −
iφ ejmφ
sin θ
∂θ
m
∂P
jm
n (cos θ)
Pnm (cos θ)iφ −
iθ ejmφ
Ntmn =
sin θ
∂θ
where Jp are the appropriate basis functions and ap are unknown coefficients of the expansion. In the proposed approach,
the current basis functions are defined using the cavity model
[4], [5]. In the case of complex shape of the patch, in order to
determine the field distribution in the cavity the FDFD method
is used.
Substituting (5) to (4) and using the orthogonality properties
of the respective functions (in different coordinate systems),
we can determine the relation between the unknown coefficients in the external region and unknown coefficients of
the current. Here, we can utilize the condition that the inner
(2)
product of the tangential electric field Et and current on the
surface of the patch J is equal to zero:
ZZ
(2)
Et J∗ ds = 0
(6)
Sp
where ∗ denotes complex conjugate and Sp is the surface of
the patch.
Then, using the method of moments, and assuming the
current basis functions as testing functions (Galerkin procedure), we obtain a homogeneous system of equations. A nontrivial solutions are the resonant frequencies of a structure
under investigation and current coefficients that allow for the
determination of the current distribution on the microstrip
patch.
The procedure described above can be developed for the
analysis of multilayer structures having any number of layers
constituting the substrate (located under the patch), and any
number of layers constituting the superstrate (located above
the patch) as presented in [4].
III. N UMERICAL
RESULTS
This section presents the examples of analyzed conformal
structures with calculated resonant frequency and radiation
characteristics.
A. Cylindrical structures
The first example is a cylindrical structure with a metal
core of radius r = 5 cm with a single dielectric substrate
with relative permittivity εr1 = 2.32 and thickness h = 0.795
mm and with a single superstrate made of the same dielectric
material with thickness t. Between the substrate and the
superstrate the rectangular patch of dimensions L = 3 cm
and W = 4 cm is placed. Fig. 2 shows the geometry of the
structure and the calculated resonance frequencies of the TM01
mode as a function of thickness t of the supestrate. As one
can see from the results the presence of a dielectric supestrate
lowers the resonant frequency of the structure. The results are
consistent with those presented in [2].
and the effect of the thickness of the whole substrate begins
to dominate, which results with the decrease of resonance
frequency. The results are consistent with the results presented
in [3].
B. Elliptical structures
An example of the elliptic structure is a microstrip rectangular patch of dimensions L = 3 cm and W = 4
cm located on a single dielectric substrate with a relative
permittivity εr1 = 2.32 and thickness h = 0.795 mm covering
the elliptical metal core of dimensions amax = 5 cm and
amin ∈ (0.3, 0.999)amax. This example assumes two types
of elliptical structures. The first structure is made of confocal
ellipses. In this case, the two ellipses (one representing the
outline of the metal core and the other is the interface between
the substrate and air layer) have the same ellipticity which
simplifies the numerical analysis. However, the thickness of
the dielectric layer is irregular, causing a problem in the
manufacturing process. The second structure has a dielectric
layer with a constant thickness, thus it is made with the use
of non-confocal ellipses. Fig. 4 shows the geometries of the
structures and the calculated complex resonant frequencies as
a function of the ellipticity of the structure.
Fig. 2: The geometry of the structure with single substrate
and single superstrate and resonant frequencies in function of
substrate thickness. Solid line - this method; Circles - results
form [2].
Fig. 4: Resonant frequencies of TM10 mode in function of
structure ellipticity and the geometry of the elliptical structures
(confocal and non-confocal) with single substrate. Solid and
dashed lines - this method; Circles - HFSS.
Fig. 3: The geometry of the structure with double substrate
and resonant frequencies in function of air layer thickness.
Solid line - this method; Circles - results form [3].
Another example is a rectangular patch with dimensions of
L = 8 cm and W = 16.8 cm placed on a double substrate
with a relative permittivity εr1 = 1 and εr2 = 2.32 (substrate
with air gap) covering the metal core of radius r = 20 cm.
Fig. 3 shows the geometry of the structure and the calculated
resonance frequencies for the TM01 mode as a function of the
thickness of the air gap at h = 0.795 mm. The obtained results
shows that the increase of the air gap increases the resonant
frequency of the structure which is caused by a decrease in
the effective permittivity of the area under the patch. For
thicker air gap the effective permeability changes slightly
In the literature, often it is assumed that for small thicknesses of the dielectric layer one can approximate constant
thickness of the substrate by confocal ellipses. The resonant
frequency results shows that such an assumption, even for thin
layers of the substrate, is incorrect, since with the increasing
ellipticity the resonant frequencies vary considerably.
C. Spherical structures
The first example of a spherical structure is a circular
patch placed on a single dielectric substrate with a relative
permittivity εr2 = 2.32, covering a metallic sphere of radius
r = 50 mm. Characteristics of the resonance frequency as
a function of thickness of the substrate are shown in Fig. 5.
As can be seen with increasing thickness of the substrate the
resonant frequency decreases.
The last example is a rectangular patch with four cut slits
placed on a dielectric substrate with a relative permittivity
Fig. 5: The geometry of the spherical structure and resonant
frequencies of TM11 mode in function of substrate thickness.
Solid line - this method; Circles - HFSS.
εr1 = 3.5 and thickness h = 2 mm, covering a metallic sphere
of radius r = 50 mm. The structure geometry and the obtained
resonant frequencies for the first two modes (TM10 and TM01 )
as a function of the length of the slots ∆θs are shown in Fig. 6.
From the presented results it can be seen that the increase
of the length of the slots causes a decrease in the resonant
frequency of the structure.
Fig. 6: The geometry of the spherical structure and resonant
frequencies of TM10 and TM01 modes in function of slot
lengths. Solid line - this method; Circles - HFSS.
IV. C ONCLUSION
A method for determining the resonant frequency of microstrip open structures placed on curved surfaces was described. Several examples of multilayer cylindrical structures,
confocal and non-confocal elliptical structures and spherical
structures were presented. The obtained results are consistent
with those found in the literature and the results of simulation
using a fullwave electromagnetic simulator HFSS.
ACKNOWLEDGMENT
This work was supported from sources of National Science
Center under grant decision no. DEC-2011/01/D/ST7/06639.
R EFERENCES
[1] P. Persson and R. G. Rojas, ”Efficient technique for mutual coupling
calculations between apertures on a PEC circular cylinder covered with a
dielectric layer,” Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2001,
vol. 3, pp. 252255
[2] S.-Y. Ke and K.-L. Wong, ”Input impedance of a probe-fed superstrateloaded cylindrical-rectangular microstrip antenna,” Microw. Opt. Technol.
Lett., vol. 7, pp. 232236, 1994
[3] K.-L. Wong et al., ”Analysis of a cylindrical-rectangular microstrip
structure with an airgap,” IEEE Trans. Microw. Theory Tech., vol. 42,
no. 6, pp. 10321037, Jun. 1994
[4] R. Lech et al., ”An Analysis of Probe-Fed Rectangular Patch Antennas
With Multilayer and Multipatch Configurations on Cylindrical Surfaces”
IEEE Transaction On Antennas and Propagation, vol. 62, no. 6, pp. 29352945, June 2014
[5] R. Lech et al., ”An Analysis of Elliptical-Rectangular Multipatch Structure on Dielectric-Coated Confocal and Nonconfocal Elliptic Cylinders”
IEEE Transaction On Antennas and Propagation, vol. 63, no. 1, pp. 97105, Jan 2015
[6] T. Rylander and A. Bondeson, ”Application of sta-ble FEM-FDTD hybrid
to scattering problems,” IEEE Trans. Antennas Propag., vol. 50, no. 2,
pp. 141144, Feb. 2002
[7] H. Mang and X. Xiaowen, ”Full-wave analysis and wideband design of
probe-fed multilayered cylindrical-rectangular microstrip antennas,” IEEE
Trans. Antennas Propag., vol. 52, no. 7, pp. 17491757, Jul. 2004
[8] P. Persson and L. Josefsson, ”Calculating the mutual coupling between
apertures on a convex circular cylinder using a hybrid UTD-MoM
method,” IEEE Trans. Antennas Propag., vol. 49, no. 4, pp. 672677, Apr.
2001
[9] A. K. Hamid and M. I. Hussein, ”Electromagnetic scattering by two
parallel conducting elliptic cylinders: Iterative solution,” in Proc. IEEE
Int. Symp. Electromagn. Compat. (EMC), May 16, 2003, vol. 1, pp. 2124