The conventional geometry of a plate microstrip resonator is made up of a single metallic patch, which is printed on a monolayer dielectric substrate. Its arrangement is simple and easy to make, but it is limited in its functional abilities. Many searches have been realized to improve the bandwidth and the gain of the microstrip resonators. Among the various configurations proposed in the open literature, the stacked geometry seems to be very promising. By appropriate design, it is able to provide the operation in dual frequency mode, wide bandwidth enough and high gain. The theoretical investigations of structures composed of two stacked antireflection coatings, enhanced metallic coatings are available in the literature, however, for the stacked configurations involving three metallic coatings or more, not to exact or approximate analysis was conducted due to the complexity of the structure.
1. Introduction
The microstrip antenna, originally consisting of a substrate layer with a radiating element, which is printed on a side and a ground plane on the other, is currently in evolution in the structures to dielectric multilayer substrates
[1

7]
. The use of multiple dielectric layers offers additional degrees of freedom to the designer who can control and optimize the performance of the antenna such as the bandwidth, the aperture of the radiation beam, the gain and others. Also, the tendencies towards plate antenna arrays indicate that microstrip antennas in the future will be largely based on multilayer technology. Given this increasing technological significance of multilayer substrates in the of plate antennas domain, to analyze these antennas the models developed should be able to hold the multilayer structures.
In the first subsection, we study the resonance characteristics of a conventional circular microstrip resonator (single patch on a substrate monolayer). In the second subsection, we complete the analysis in
[7

9]
. And on the behavior of the dualfrequency microstrip resonator formed by a stack of two circular patches, in examining the behavior of a dualfrequency microstrip resonator formed by a stack of two circular discs. In the third subsection, we give for the first time, numerical results concerning the behavior of a dualfrequency microscript resonator formed by a stack of three circular disks. Finally, the conclusions drawn from this study are summarized in Section 5.
2. Theory
The problem to solve is shown in (
Fig. 1.
) where we have three circular stacked microstrip patches of radii:
a_{1}
,
a_{2}
and
a_{3}
, fabricated on a multilayer substrate (Although we show that this theory is valid for an arbitrary number of patches). The circular discs and the ground plane are assumed to be perfect conductors of a negligible thickness and the layers are of infinite extension. The multilayer medium consists of
N
dielectric layers having uniaxial anisotropy with the normal optical axis to the patch. Thickness of each layer:
d_{j}
=
z_{j}
–
z_{j1}
(
j
=1,2,…, N) is characterized by the vacuum permeability
μ
_{0}
and the permittivity tensor
which is of the form:
Geometry of three circular discs stacked in a multilayer dielectric medium containing isotropic and / or anisotropic material
ε
_{0}
is the freespace permittivity and diag stands for the diagonal matrix with the diagonal elements appearing between the brackets. Eq. (1) can be specialized to the isotropic substrate by allowing
ε_{x j}
=
ε_{z j}
=
ε_{r j}
. The circular patch is embedded in the stratification at the interface plane
z
=
z_{p}
.
All fields and currents are in harmonic regime with the e
^{iωt}
dependence.
Consider
the areal density current on the circular disk of radius
a_{i}
. As well,
the value of the transverse electric field at the plane of the patch of radius
a_{i}
. In reason of the symmetric revolution of the multilayer medium around the z axis in
Fig. 1
, when the Helmholtz equations for the longitudinal components of the
E_{z}
and
H_{z}
fields are solved in cylindrical coordinates in each layer of this medium, it turns out that the
E_{z}
and
H_{z}
dependence on the
φ
coordinate is of
e^{ikφ}
type (where k is an integer). As a result,
J^{i}
(
ρ
,
φ
) and
E^{i}
(
ρ
,
φ
,
z_{pi}
) can be written as:
Following mathematical reasoning similar to
[10

11]
. We obtain a relationship, in the spectral data, between
J
^{1}
(
ρ
,
φ
) ,
J
^{2}
(
ρ
,
φ
) ,
J
^{3}
(
ρ
,
φ
) and
E
^{1}
(
ρ
,
φ
,
z_{p}
_{1}
) ,
E
^{2}
(
ρ
,
φ
,
z_{p}
_{2}
) ,
E
^{3}
(
ρ
,
φ
,
z_{p}
_{3}
) , respectively, as:
and
are, respectively, the Hankel transform vectors of
and
; the
nm
part of the spectral dyadic Green function is given by
with
In Eq. (8)
is the matrix representation of the
j^{th}
layer in the representation (TM, TE), is given by
In Eq. (10),
and
are, respectively, the propagation constants of the TE and TM waves in the
j^{th}
layer. They are defined by the following equations:
For any number of layers in the stacked configuration, the new explicit expression, shown in Eq. (7), allows easy calculation of dyadic Green functions via simple matrix multiplication. It is also important to note that the expression (7) is valid for stacked structures with more than three patches (four patches and more). Now, that we have the necessary dyadic Green functions, the RitzGalerkin procedure may be applied to Eqs. (4), (5) and (6) leading to a homogeneous system. This homogeneous system has nontrivial solutions when:
where
is the impedance matrix of the homogeneous system. Equation (13) is an equation to own
ω
, from which the characteristics of the stacked structure shown in
Fig.1
can be obtained. In fact, if we denote:
ω_{nm}
=
2π
(
f_{r}^{nm}
+
if_{i}^{nm}
) (
n
= 0,±1, ±2,...;
m
= 1,2,3,...) the set of complex roots of equation (13). In this case, the amounts
f_{r}^{nm}
refer to the resonant frequencies of the resonant modes of the plate circular microstrip; the amounts
BW_{nm}
= 2
f_{i}^{nm}
/
f_{r}^{nm}
refer to the bandwidth and quantity
Q_{nm}
=
f_{r}^{nm}
/ 2
f_{i}^{nm}
designate quality factors
[12]
. Since
f_{r}^{nm}
=
f_{r}^{n, m}
and
f_{i}^{nm}
=
f_{i}^{n, m}
, without loss of generality, in what follows, we always assume that
n
≥ 0 . Note that this result is not valid in the case of ferritebased materials; they are a part of the multilayer substrate. This behavior is attributed to the nonreciprocal ferrite and is predicted by the cavity model of a circular plate which is printed on a ferrite substrate
[13
,
14]
.
3. Approximation of Current Densities on the Circular Discs
Current densities on the three circular disks are approximated using basis functions formed by the complete orthogonal set of the TM and TE modes of a cylindrical cavity of radii
a_{i}
(
a_{1}
for the first disc,
a_{2}
the second disc and the third disc
a_{3}
) with lateral magnetic and electrical walls at the top and the base. These modes of currents, which, are nonzero only on the three circular disks are given by:
n
= 0, +1, +2, ...
Ψ
_{np1}
(
ρ
) (
p
_{1}
= 1,2,...,
P
_{1}
) ,
Ψ
_{np2}
(
ρ
) (
p
_{2}
= 1,2,...,
P
_{2}
) and
Ψ
_{np3}
(
ρ
) (
p
_{3}
= 1,2,...,
P
_{3}
) correspond to the resonant cavity TM modes and
Φ
_{nq1}
(
ρ
) (
q
_{1}
= 1,2,...,
Q
_{1}
) ,
Φ
_{nq2}
(
ρ
) (
q
_{2}
= 1,2,...,
Q
_{2}
) and
Φ
_{nq3}
(
ρ
) (
q
_{1}
= 1,2,...,
Q
_{3}
) correspond to the cavity resonant TE modes.
The constants:
β
_{n p1}
,
β
_{n p2}
,
β
_{n p3}
,
α
_{n q1}
,
α
_{n q2}
, and
α
_{n q3}
correspond to
p
_{1}
th,
p
_{2}
th,
p
_{3}
th,
q
_{1}
th,
q
_{2}
th et
q
_{3}
th zeros of
,
,
,
J_{n}
(
α
_{n q1}
) = 0 ,
J_{n}
(
α
_{n q2}
) = 0 and
J_{n}
(
α
_{n q3}
) = 0 , respectively.
4. Numerical Results and Discussion
Although the Fullwave analysis presented in this part is able to give numerical results for the different modes. In what follows, we just present numerical results concerning the mode (TM11). We divide this section into three subsections. In the first subsection, we study the resonance characteristics of the conventional circular microstrip resonator (single patch on a monolayer substrate). In the second subsection: we complete the analysis in
[7

9]
and on the behavior of the dualfrequency microstrip resonator formed by a stack of two circular patches, by examining the behavior of a dualfrequency microstrip resonator formed by a stack of two circular discs. In the third subsection, we present numerical results concerning the behavior of a dualfrequency microstrip resonator formed by a stack of three circular disks.
 4.1 Conventional circular microstrip resonator
In the case of a conventional circular disk (see
Fig. 2
), convergent results for the resonant frequency, the bandwidth and the quality factor are obtained with (
P
= 5,
Q
= 4) . To validate the proposed method for the case of a circular disk printed on a single monolayer substrate, we compare in
Tables 1
and
2
our numerical results with the results of Fittage curves
[15]
. The radius of the disc is
a
= 0.5 cm. In the
Table 1
the substrate’s material is the Duroid, while in
Table 2
, it is the Plexiglas. From the two tables, it is clear that our calculations of the resonance frequency and of the quality factor coincide with those of Chew
[15]
. The numerical results indicate that the increase of the thickness of the substrate results in a decrease of the operating frequency and the quality factor of the resonator. This means that the increase in thickness can increases the bandwidth initially very close.
Geometry of a circular microstrip printed on a monolayer dielectric substrate.
Comparison of the resonance frequencies and quality factors calculated with the data from the literature, for a circular line resonator formed on a substrate made of Duroid;a=0.5cm,εr= 2.32.
Comparison of the resonance frequencies and quality factors calculated with the data from the literature, for a circular line resonator formed on a substrate made of Duroid; a =0.5cm, ε_{r} = 2.32.
Comparison of the resonance frequencies and quality factors calculated with the data from the literature, for a circular line resonator formed on a substrate made of Plexiglas;a=0.5cm,εr= 2.6 .
Comparison of the resonance frequencies and quality factors calculated with the data from the literature, for a circular line resonator formed on a substrate made of Plexiglas; a =0.5cm, ε_{r} = 2.6 .
 4.2 Resonator formed by a stack of two circular discs
A more complex configuration, consisting of two circular discs in a stacked configuration (see
Fig. 3
), offers new performance that are not usually obtained by the singledisk configuration single dielectric. These performances include high gain, widebandwidth, dual frequency characteristics.
Geometry of a stack composed of two circular discs made of a bilayer dielectric substrate.
In order to validate the proposed theory, in the case of two stacked circular discs, numerical results were obtained for the parameters used in the Long and Walton experiment
[16]
. The two dielectric substrates in
Fig. 3
are the same (
ε
_{r1}
=
ε
_{r2}
= 2.47 and
d_{1}
=
d_{2}
= 750
μm
) and the radius of the lower disc is of
a_{1}
=18.9mm, while that of the higher one is to be variable. In
Table 3
, we report both (lower
and greater
) resonances. It is observed from these comparisons that the agreement between theory and experiment is excellent.
Lower and upper resonance mode TM11 of a strip line resonator formed by a stack of two circular discs microstrip
Lower and upper resonance mode TM11 of a strip line resonator formed by a stack of two circular discs microstrip
Finally, it is important to note that from the different executions of the developed program, we conclude that the two resonators constituting the stacked structure, which determine the behavior of the dual frequency resonator depends on the relative sizes of the circular discs. In the case, where the radius of the upper disk is larger than that of the lower disc, less resonance is associated with the resonator formed by the upper disc and the ground plane, and the upper resonance is connected to the lower disk. The lower resonance is very close to the resonance frequency of the upper disk isolated. In the case, where the radius of the upper disk is smaller than that of the lower disc, the lower resonance is associated with the resonator formed by the lower disc and the ground plane and the upper resonance is associated with the resonator formed by the two discs circular. Now, the lower resonance is very close to the resonance frequency of the lower disk isolated (i.e. the lower disc substrate in the configuration substratesubstrate). These behaviors are in agreement with those found theoretically for the case of a structure stacked microstrip formed of two circular patches
[7

9]
.
 4.3 Resonator formed by a stack of three circular discs
In this subsection, we apply the formulation of the Section II to study the resonance characteristics of the stacked geometry shown in
Fig. 4
.
Geometry of the stacked microstrip configuration used in the experiment of Revankar and Kumar [17]
In order to have the characteristic of an adjustable resonance, two adjustable air gaps are inserted into the configuration. The first air gap is located just on the first circular disc, while the second air gap is located on the second circular disc. The first disk (of radius
a_{1}
) is printed on a substrate (of thickness
d_{1}
and relative permittivity
ε
_{r1}
).
The second disc (of radius
a_{2}
) is printed on a substrate (of thickness
d_{3}
and relative permittivity
ε
_{r3}
). The third disc (of radius
a_{3}
) is printed on a substrate (of thickness
d_{5}
and relative permittivity
ε
_{r5}
).
Numerical results are obtained for the parameters used in the Revankar and Kumar experiment
[17]
, for
a_{1}
=
a_{2}
=
a_{3}
= 1.65cm,
ε
_{r1}
= 2.33 ,
ε
_{r3}
= 2.45 ,
ε
_{r5}
= 2.2 ,
d_{1}
= 1.58mm,
d_{2}
= 4mm,
d_{3}
= 0.762mm and
d_{4}
= 0.508mm.
The antenna is therefore characterized by the variation of the air separation
d_{4}
. With respect to Microstrip resonators having single disk, the phenomenon of spurious resonances is much more noticeable in microstrip resonators with stacking configurations. The calculated resonance frequencies of the TM
_{11}
mode are shown in
Table 4
and are compared with measured values of Revankar and Kumar
[17]
. The presence of parasitic disks introduces two resonances: lower and higher resonances. Note that the measured resonance frequencies
and
given in
Table 4
are obtained from the curves “Return loss”. It is observed that the difference between our calculated resonance frequencies and the measured data is estimated by 6.06% at most. Therefore, a good agreement between theory and experiment is achieved.
Lower and upper resonance mode TM11 of a strip line resonator formed by a stack of two circular discs microstrip
Lower and upper resonance mode TM11 of a strip line resonator formed by a stack of two circular discs microstrip
Finally, it is important to note that the presence of three circular discs in a stacked configuration of
Fig. 4
indicates the presence of three resonant cavities. This led us to automatically think that the stacked structure in
Fig. 4
has three resonances. Although the numerical results in
Table 4
indicate the presence of only two resonances, we think that it is important to examine, more carefully, the resonance of a microstrip structure formed by a stack of three metal patches.
5. Conclusion
In this paper, we have presented a numerical model of a microstrip resonator formed by a stack of arbitrary number of patches. It combines both precision and computing speed and allows the calculation of: resonant frequencies, bandwidths and quality factors. The circular discs are produced in a medium containing isotropic and / or anisotropic multilayer nonmagnetic materials. The contributions in this paper can be summarized as follows:

• The new explicit formulas have been developed for the calculation of spectral dyadic Green's functions of microstrip resonator formed by a stack of an arbitrary number of the circular discs. These circular discs are embedded in a multilayered dielectric medium containing isotropic and / or anisotropic materials.

• By transforming the system of Cartesian axes towards the (TM, TE) representation, the new formulas of dyadic Green spectral functions are valid for both circular geometries for rectangular geometries.

• The origin of the behavior of a dualfrequency microstrip resonator formed by a stack of two circular microstrip disks has been explained in detail. Indeed, the resonant cavities responsible for lower and upper resonances were clearly marked.

• Numerical results related to the behavior of a dualfrequency microstrip resonator formed by a stack of three circular discs were given and argued.
In the triple Frequency operation mode, where the antenna operates efficiently in three distinct frequencies, is sometimes of an extreme need in some civil or military applications. For the research of structures to this type of operation, we suppose that the patterns formed by a stack of three metal patches can response to this need. It is very interesting to examine carefully the resonance configurations with three stacked metallic patches.
BIO
F. Chebbara was borne in 1978 in Batna, Algeria. Has Doctoral degree in Electronics from University of Batna, has been a teacher at the University of Ouargla since 2010. His research interstes are: Power electronics, antennas, microbande and transmission.
T. Fortaki was borne in Constantine, Algeria. Has Doctoral degree in Electronics from University of Constantine, has been a teacher at the University of Batna since 2001. He is now Professor. His research interstes are: Study of microwave devices, Study applications of microwave devices, Realization of microwave devices, Power electronics, antennas, microbande and transmission.
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