A Numerically Efficient Full Wave Analysis of Circular Resonators Microbandes Stacked Involving Multimetallisations

Abstract

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 anti-reflection 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 sub-section, 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 dual-frequency microstrip resonator formed by a stack of two circular patches, in examining the behavior of a dual-frequency microstrip resonator formed by a stack of two circular discs. In the third sub-section, we give for the first time, numerical results concerning the behavior of a dual-frequency 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: a1, a2 and a3, 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: dj = zj – zj-1 (j =1,2,…, N) is characterized by the vacuum permeability μ0 and the permittivity tensor which is of the form:

Fig. 1.Geometry of three circular discs stacked in a multilayer dielectric medium containing isotropic and / or anisotropic material

ε0 is the free-space 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 εx j = εz j = εr j . The circular patch is embedded in the stratification at the interface plane z = zp.

All fields and currents are in harmonic regime with the eiωt dependence.

Consider the areal density current on the circular disk of radius ai . As well, the value of the transverse electric field at the plane of the patch of radius ai . 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 Ez and Hz fields are solved in cylindrical coordinates in each layer of this medium, it turns out that the Ez and Hz dependence on the φ coordinate is of eikφ type (where k is an integer). As a result, Ji(ρ,φ) and Ei(ρ,φ,zpi) can be written as:

Following mathematical reasoning similar to [10-11]. We obtain a relationship, in the spectral data, between J1 (ρ,φ) , J2 (ρ,φ) , J3 (ρ,φ) and E1 (ρ,φ,zp1) , E2 (ρ,φ,zp2) , E3 (ρ,φ,zp3) , 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 jth 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 jth 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 Ritz-Galerkin 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: ωnm = 2π(frnm + ifinm) (n= 0,±1, ±2,...; m = 1,2,3,...) the set of complex roots of equation (13). In this case, the amounts frnm refer to the resonant frequencies of the resonant modes of the plate circular microstrip; the amounts BWnm = 2finm / frnm refer to the bandwidth and quantity Qnm = frnm / 2finm designate quality factors [12]. Since frnm = fr-n, m and finm = fi-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 ferrite-based materials; they are a part of the multilayer substrate. This behavior is attributed to the non-reciprocal 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 ai (a1 for the first disc, a2 the second disc and the third disc a3) 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, ...

Ψnp1 (ρ) (p1 = 1,2,...,P1) , Ψnp2 (ρ) (p2 = 1,2,...,P2) and Ψnp3 (ρ) (p3 = 1,2,...,P3) correspond to the resonant cavity TM modes and Φnq1 (ρ) (q1 = 1,2,...,Q1) , Φnq2 (ρ) (q2 = 1,2,...,Q2) and Φnq3 (ρ) (q1 = 1,2,...,Q3) correspond to the cavity resonant TE modes.

The constants: βn p1 , βn p2 , βn p3 , αn q1 , αn q2 , and αn q3 correspond to p1th, p2th, p3th, q1th, q2th et q3th zeros of , , , Jn (αn q1) = 0 , Jn (αn q2) = 0 and Jn (αn q3) = 0 , respectively.

 

4. Numerical Results and Discussion

Although the Full-wave 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 sub-section, we study the resonance characteristics of the conventional circular microstrip resonator (single patch on a monolayer substrate). In the second sub-section: we complete the analysis in [7-9] and on the behavior of the dual-frequency microstrip resonator formed by a stack of two circular patches, by examining the behavior of a dual-frequency microstrip resonator formed by a stack of two circular discs. In the third subsection, we present numerical results concerning the behavior of a dual-frequency 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.

Fig. 2.Geometry of a circular microstrip printed on a monolayer dielectric substrate.

Table 1.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.

Table 2.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 .

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, wide-bandwidth, dual frequency characteristics.

Fig. 3.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 (εr1 = εr2 = 2.47 and d1 = d2 = 750 μm) and the radius of the lower disc is of a1 =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.

Table 3.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 substrate-substrate). 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 sub-section, we apply the formulation of the Section II to study the resonance characteristics of the stacked geometry shown in Fig. 4.

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 a1) is printed on a substrate (of thickness d1 and relative permittivity εr1 ).

The second disc (of radius a2) is printed on a substrate (of thickness d3 and relative permittivity εr3 ). The third disc (of radius a3) is printed on a substrate (of thickness d5 and relative permittivity εr5 ).

Numerical results are obtained for the parameters used in the Revankar and Kumar experiment [17], for a1 =a2 = a3 = 1.65cm, εr1 = 2.33 , εr3 = 2.45 , εr5 = 2.2 , d1 = 1.58mm, d2 = 4mm, d3 = 0.762mm and d4 = 0.508mm.

The antenna is therefore characterized by the variation of the air separation d4. 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 TM11 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.

Table 4.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 dual-frequency 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 dual-frequency 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.