Keywords: Iris, microstrip patch antenna, bandwidth, MOM-GEC, HFSS, measurement, dual-resonance, antenna array, Floquet theory. v Iris, antenne patch microruban, bande passante, MOM-GEC, HFSS, mesure, résonancedouble, réseau périodique d'antennes, théorie de Floquet. vi

prove my research skills, refine my methodology, and deepen my understanding of the scientific and technological issues related to my field of study. I am

List of Tables

 Table 3.
Introduction 0.1 Thesis objective

The X-band is a frequency range from 8 to 12 GHz. It is most often used for radar applications as it has a short wavelength which allows for higher resolution imaging for target identification and discrimination. It is used for civil, military and government radar applications including weather monitoring, air traffic control, maritime vessel traffic control, defense tracking and vehicle speed detection for law enforcement.

In addition to radar, the X-band is also used for satellite and terrestrial communications. The frequency range from 8.175 to 8.215 GHz is usually used by meteorological satellites for monitoring weather conditions. The ITU has also allocated 10 to 10.5 GHz for amateur radio operations and 10.45 to 10.5 GHz for amateur satellite operations.

The X-band has extremely low rates of atmospheric attenuation in comparison to frequencies above 10 GHz, which makes this band suitable for the harshest weather conditions, providing exceptionally high link availability and weather monitoring radar solutions.

In fact, communication systems in the X-band require the design of antennas characterized by higher gains and wide bandwidths. The microstrip patch antenna presents a good candidate for its low weight, its small size, the ease of its manufacture and its integration, etc. However, one of the major disadvantages is its low gain and its low bandwidth which is of the order of a few percent (2% to 4%).

Multiple techniques have been used to widen the bandwidth of microstrip antennas, namely: the thick substrate, the partial substrate removal, the superstrate cover [1][2], the parasitic patches coupled to the main patch or stacked patches [3], the patch with metallic rings, the defected ground structure, the reflecting layer, the electromagnetic band gap as material [4], the merging of several antenna resonance modes [5][6], etc.

The microstrip patch antenna's type of alimentation has an impact on its bandwidth as well. The iris feed technique is used to increase the patch bandwidth in order to get over the issue of narrow bandwidth [7][8]. The iris-fed microstrip patch antenna is equivalent to an antenna having two patches, one of which radiates toward the open end of the rectangular waveguide and the other toward open air. Therefore, this structure combines appealing characteristics of a microstrip antenna, like low profile, light weight, compact size, and simple integration in electronic circuits, with those of a waveguide, like high power handling capacity and low losses (resistive) [9]. In this method, the waveguide's mouth is directly fitted with the microstrip antenna. As a result, the most energy is radiated.

From the above literature survey, this work proposes to model a microstrip patch antenna fed by a waveguide using an end-wall iris through ground plane since it has not been widely used in X-band. The iris is sized so that it is resonant and subsequently the antenna operates under the fusion of two modes relating to the iris and the patch [10], resulting in a wideband radiation characteristic with dual resonance.

Regarding to the state-of-art works for the iris-fed patch antenna's electromagnetic modelling, it turns out that the current trend is to use simulation tools such as HFSS and CST which are essentially based on methods that require a 3D mesh [11]. Other works have been based on numerical models as in [7] and [8] which used the modal expansion model and the method of moments respectively. These integral methods are best suited to perform an electromagnetic modelling of planar microwave structures. However, when the complexity of the structures increases, the resolution becomes complicated and this requires more memory space and computation time.

Several research works [12][13] have focused on the use of hybrid methods to improve the conventional methods. In fact, their major interest lies in the ability to solve complex structures whereas neither a single numerical method nor commercial software can model them correctly.

In this context, this work is oriented towards modelling the proposed antenna based on MOM-GEC numerical technique. This latter handles complex electromagnetic structure as no mesh is required, thus, a gain is achieved in terms of computational time and memory storage.

The antenna is modelled by the hybrid MOM-GEC method [14][15][16]. Equivalent circuits (GEC) are introduced in the development of the formulation of integral methods in an attempt to transpose field problems into equivalent circuit problems that are generally easier to deal with, thus, a reduction in both memory requirements and computation time is achieved.

The problem thus formulated; its numerical resolution is done by applying the method of Galerkin known by the simplicity of its formulation. This procedure is in fact general and it can be applied to study several electromagnetic problems, namely: the uniaxial discontinuity in waveguides, the multiaxial diffraction discontinuities in free space, the antenna radiation, the circuits comprising localized sources, etc.

In fact, the proposed hybrid method is extended to become a multi-scale method in order to study the electromagnetic aspect of an iris-fed microstrip patch antenna array while taking into account the coupling between the array's elements. Therefore, a solution lies in the use of modal analysis, based on Floquet theorem [17][18], in an infinite and periodic array while limiting to the effective calculation of the physical quantity, field or current, over a single period by introducing the appropriate boundary conditions.

The techniques for solving each electromagnetic problem essentially differ from each other in the development of the operators involved. Admittedly, the theoretical basis and the approach to be followed remain unchanged. This is due to the fact that the methods used are in all cases integral methods.

Thesis Organization

This manuscript is divided as follows:

Chapter 1 provides an introduction and an overview of microstrip patch antenna namely its characteristics, its advantages, its disadvantages as well as the techniques of its feeding.

The main aim of Chapter 2 is to present a survey of the bandwidth enhancement techniques of microstrip patch antenna. Chapter 3 presents a formulation which makes it possible to perform a rigorous electromagnetic calculation while minimizing the execution time and the memory resources required. One solution consists in hybridizing various methods together in order to benefit from the performance of each in their chosen field. For this reason, the method of moments (MOM) hybridized with the Generalized Equivalent Circuits Method (GEC) is proposed.

In chapter 4, a microstrip patch antenna fed by a waveguide using an end-wall iris through ground plane has been modelled based on the MOM-GEC method. Prototype antenna is designed and implemented as proof of concept. To confirm the accuracy and the efficiency of the proposed approach, a comparison with the High Frequency Structure Simulator (HFSS) and with measurement is held. Performances comparison with the X-band antennas existing in literature using different bandwidth enlargement techniques is realized in terms of compacity, -10 dB fractional bandwidth and complexity fabrication.

Chapter 5 tends to overcome the classic full-wave approaches' limitations for the analysis of a unidimensional periodic iris-fed patch antenna array by using the MOM-GEC method combined Floquet approach.

1 Chapter 1 ties of the substrate. The first practical antenna was developed by Munson [4] and Howell [5].

An Overview of Microstrip Patch Antenna

Since then, extensive research and development of microstrip antennas and antenna array have led to diversified application within the broad field of microwave antennas.

The Microstrip antenna has proved to be an excellent radiator for many applications because of its several advantages [6] as compared to conventional microwave antennas. This results its many applications over the broad frequency range from around 100 MHz to 100 GHz.

This chapter provides an introduction and overview of the microstrip patch antenna along with the features, advantages, limitations, and different types of feed techniques. Indeed, the feeding of the microstrip patch antenna using a waveguide using an end-wall iris through ground plane will be further presented in order to mention the advantages it has over conventional antenna feeding techniques.

Geometries of the basic microstrip antenna

The idea of microstrip patch antennas arose from utilizing printed circuit technology not only for the circuit components and transmission lines but also for the radiating elements of an electronic system. The basic structure of the microstrip patch antenna is shown in Figure 1.1. It consists of area of metallization supported above a ground plane by a thin dielectric substrate and fed against the ground at an appropriate location. The patch shape can in principle be arbitrary; in practice, the rectangle, the circle, the equi-triangle and the annular-ring are common shapes. Four feeding methods are shown in Figure 1.2 namely: the coaxial probe feed, the microstrip line feed, the aperture-coupled feed and the proximity feed. Electromagnetic energy is first guided or coupled to the region under the patch, which acts like a resonant cavity with open circuits on the sides. Some of the energy leaks out of the cavity and radiates into space, resulting in an antenna.

Microstrip patch antenna: Advantages and disadvantages

This section presents the advantages and disadvantages of using the microstrip patch as well as the different areas of its application.

Advantages

The advantages of microstrip patch antennas include their lightweight design, occupying minimal volume. Their low-profile planar configuration makes them suitable for various applications.

These antennas can easily conform to both planar and non-planar surfaces, offering flexibility.

Additionally, their ease of mass production contributes to a low fabrication cost. Implementing them on devices is straightforward, and they seamlessly integrate with other MICs on the same substrate.

These antennas enable both linear polarization and circular polarization through a simple feed mechanism. They can also be made compact, catering to the needs of personal mobile communication. Furthermore, they allow for dual-and triple-frequency operations, expanding their versatility. Acting as efficient radiators, they exhibit a low scattering cross-section. Mechanically robust when mounted on rigid surfaces, they are resistant to shock and vibration. These antennas are highly compatible as embedded components in handheld wireless devices. Moreover, the feed line can be easily fabricated simultaneously on the substrate, streamlining the manufacturing process.

A large number of commercial needs are met by the use of these antennas [7]. The various application includes the ubiquitous Global Positioning System (GPS), ZigBee, Bluetooth, WiMax, Wireless Fidelity (WiFi) and wireless communication systems. Navigational applications, such as asset tracking of vehicles as well as marine uses, have created a large demand for antennas. It finds extensive use in radio frequency identification (RFID) and radar system [8][9] in the area of manufacturing, transportation and medical [10]. Because of the advantages listed above, the microstrip patch antenna has also become the favorite of antenna designers for commercial mobile and wireless communication systems. Figure 1.6

shows a patch antenna in a cellular phone.

Disadvantages

There are several disadvantages associated with microstrip patch antennas:

(1) In its basic form, the microstrip antenna has a narrow impedance band-width, typically less than 5%. However, various bandwidth-widening techniques have been developed. Up to 50%

bandwidth have been achieved. It is generally true that wider bandwidth is achieved with the sacrifice of increased antenna physical volume.

(2) Due to the small separation between the radiating patch and its ground plane, the microstrip antenna can handle relatively low RF power. The average power considered safe is a few tens of watts.

(3) While a single patch element generally incurs very little loss because it is only about one-half wave long, microstrip arrays generally have larger ohmic loss than arrays of other types of antennas of equivalent aperture size. This ohmic loss mostly occurs in the dielectric substrate and the metal conductor of the microstrip line feed network and power dividing circuit.

Material consideration

The metallic patch is normally made of thin copper foil. The substrate material provides mechanical support for the radiating patch elements. It also maintains the required spacing between the patch and its ground plane. The substrate thickness for the basic geometry is in the range of 0.01 to 0.05 free-space wavelength.

The dielectric constant ranges from 1 to 10 and can be separated into three categories:

(1) Those having a relative dielectric constant (relative permittivity) in the range of 1.0 to 2.0.

This type of material can be air, polystyrene foam, or dielectric honeycomb.

(2) Those having a relative dielectric constant in the range of 2.0 to 4.0. This type of material consists mostly of Fiber-glass reinforced Teflon.

(3) Those with a relative dielectric constant between 4.0 and 10.0. This type of material can be ceramic, quartz, or alumina.

The most commonly used material is Teflon-based with a relative permittivity between 2 and 3.

This material is also called PTFE (Polytetrafluoroethylene). It has a structure very similar to fiberglass material used for digital circuit boards, but has a much lower loss tangent. Cost, power loss, and performance are trade-off considerations in choosing the substrate material, as illustrated by the following examples. For example, a single patch or an array of a few elements may be fabricated on a low-cost fiberglass material at the L-band frequency, while a 20-element array at 30 GHz may have to use higher-cost, but lower loss, Teflon-based material (loss tangent less than 0.005). For a large number of array elements at lower microwave frequencies (below 15 GHz), a dielectric honeycomb or foam panel may be used as a substrate to minimize loss, antenna mass, and material cost while having increased bandwidth performance. There are materials with relative dielectric constant higher than 10. The patch size is smaller for higher dielectric constant.

However, higher dielectric constant also reduces bandwidth and radiation efficiency.

Feed methods for single microstrip antenna

Many techniques can be utilized to feed the microstrip antennas and they may be classified into contacting feed techniques and non-contacting feed techniques. These techniques are briefly illustrated as follows.

Microstrip line feed

In this type of feed technique, a conducting strip is connected directly [10] to the edge of the Microstrip patch. The conducting strip is smaller in width as compared to the size of the patch.

This method is the easiest to fabricate as this feeding arrangement and radiating patch can be printed on same dielectric substrate. With the microstrip-line feed approach, an array of patch elements and their microstrip power division lines can all be designed and chemically etched on the same substrate with relatively low fabrication cost per element. However, the leakage radiation of the transmission lines may be large enough to raise the sidelobe or cross-polarized levels of the array radiation.

At the edge of a patch, the impedance is generally much higher than 50 ohms. To avoid impedance mismatch, sections of quarter-wavelength transformers can be used to transform a large input impedance to a 50-ohm line as illustrated in Figure 1.9. The feed arrangement to the patch may also have an inset cut in the patch as illustrated in Figure 1.10. The purpose of the inset cut in the patch is to match the impedance of the feed line to the patch without the need for any additional matching element. This is achieved by properly controlling the inset position. This is an easy feeding scheme, because it provides ease of fabrication and simplicity in modelling as well as impedance matching.

Coaxial probe feed

The coaxial feed or probe feed [11] is the most common techniques used to feed printed patch antennas (Figure 1.11). This feed can be given at any desired location within the patch to achieve impedance matching. The inner conductor of the coaxial connector extends through the dielectric and is soldered to the radiating patch, while the outer conductor is connected to the ground plane.

The coaxial feed or probe feed method gives low radiation loss. The main advantage of this feed is that it can be placed at any desired location inside the patch to match with its input impedance.

This feed method is easy to fabricate and has low spurious radiation. However, its major disadvantage is that it gives less bandwidth and is not so easy to use since a hole has to be introduced into the dielectric substrate. Also, the hole has to be drilled in the substrate and that the connector protrudes outside the bottom ground plane. It is not completely planar and this feeding arrangement makes the configuration asymmetrical. For thicker dielectric substrates, the increased coaxial feed or probe feed length gives the input impedance more inductive, resulting impedance matching problems. For thick substrates, which are generally employed to achieve broad band both the above methods of direct feeding the microstrip antenna have problems. In the case of a coaxial feed, increased probe length makes the input impedance more inductive, leading to the matching problem. For the microstrip feed, an increase in the substrate thickness increases its width, which in turn increases the undesired feed radiation. In large thickness dielectric substrate, the microstrip line feed and the coaxial feed suffer from problem with probe reactance and surface wave excitation. The indirect feed, as discussed below solves these problems.

Proximity-Coupled Microstrip-Line Feed

A configuration of this non-contacting microstrip feed used two-layer substrate with the microstrip line on the lower layer and the patch antenna on the upper layer. This feeding method [12] contains two dielectric layers i.e., one is a radiating patch layer and on lower layer feed line is fabricated with a ground plane on back side. The two substrates are separated by a common ground plane. The microstrip feed line on the lower substrate is electromagnetically coupled to the patch through a slot aperture in the common ground plane. The slot can be of any shape or size and these parameters can be used to improve the bandwidth. The radiation from the open end of the feed line does not interfere with the radiation pattern of the patch because of the shielding effect of the ground plane. This feature also improves the polarization purity [13].

The coupling aperture is usually centered under the patch, leading to lower cross polarization due to symmetry of the configuration. Generally, a high dielectric material is used for the bottom substrate and a thick, low dielectric constant material is used for the top substrate to optimize radiation from the patch. The notable feature of this feed configuration is wider bandwidth, which is primarily because of an inclusive increase in the thickness of the microstrip patch antenna. This technique has the provision of choosing separate substrate for the patch and the feed line in order to achieve an optimize performances. The major drawback of this method is difficulty in fabrication due to multiple layers which need proper alignment. Also, in this method the thickness of the antenna increases.

The open-ended microstrip line can also be used to feed a patch antenna through proximity coupling, as shown in Figure 1.12. For example, the open end of a 100-ohm line can be placed underneath the patch at its 100-ohm location. The open-ended microstrip line can also be placed in parallel and very close to the edge of a patch, to achieve excitation through fringe-field coupling, as shown in Figure 1.13. Both these methods avoid any soldering connection, which in some cases, could achieve better mechanical reliability.

Aperture-Coupled Feed

The aperture-coupled technique was first introduced by D.M. Pozar in 1985 [14]. It consisted of two substrates separated by a common conductive ground plane in between. The patch is situated on top of the substrate and the feedline is situated on the bottom substrate. The conductive ground plane between the feed and the patch antenna forces the electromagnetic radiation to be "directed" to the patch through the slot without any physically connection as illustrated Figure 1.14. In fact, this technique is used to avoid soldering connection as well as to avoid leakage radiation of the line to interfere with the patch radiation. In addition, by using a thick substrate, this feed method allows the patch to achieve a wider bandwidth compared to the coax probe feed.

Iris feed

The use of a waveguide as a feeder to excite the microstrip patch antenna through an end-wall iris was proposed by [15] and since then, this technique has been widely used for communication systems [15][16][17]. The iris-fed microstrip patch antenna illustrated in Fig. 1.15 is composed by a three-layer device which consists on a dielectric layer sandwiched between a radiating patch and an exciter waveguide. The exciter waveguide and the dielectric layer are separated by a common conductive ground plane in between. In fact, the presence of the conductive ground plane leads the electromagnetic radiation to be oriented from the exciter truncated waveguide to the patch through the coupling slot. I fact, the use of a waveguide as a feeder to excite the microstrip patch

(c)
antenna through an end-wall iris has main advantages over the transmission line or the coaxial cable namely: the low less losses, the high design accuracy, the mechanical strength and the wide bandwidth.

Conclusion

This chapter is a concise an overview of microstrip patch antenna namely its characteristics, its advantages, its disadvantages as well as the techniques of its feeding. In fact, Microstrip patch is excited by a variety of methods classified into two categories namely: the contact-feed (microstrip line and coaxial feeds) and the non-contact feed (aperture coupling and proximity coupling feeds).

In fact, MPA are used in numerous applications, ranging from biomedical diagnosis to wireless communications due to its many alluring features such as light in weight, low volume, low cost, low profile, small dimensions and ease of fabrication and conformity etc. However, narrow bandwidth and low gain are its major drawbacks. It is in this context that the next chapter will be interested in detailing the bandwidth improving techniques of the MPA antenna.

Survey of Various Bandwidth Enhancement Techniques for

Microstrip Patch Antenna

Introduction

Microstrip patch antennas, in their conventional form, are narrow-band structures. Their impedance bandwidth is typically 1-2%. This can be attributed to two factors: the resonant style of antenna (which makes the antenna radiate efficiently only over a narrow band of frequencies)

and the thin thickness of antenna, typically less than 0.05𝜆 0 .

This feature of conventional microstrip patch antennas makes them unsuitable in many applications where quite wide bandwidth is required. So, many researches have been done during the past few decades to overcome this limitation and several procedures have been proposed.

In this chapter, we will describe the most useful procedures proposed in the literature to increase the impedance bandwidth of microstrip patch antennas. In Section 2, the intrinsic techniques which can be used to increase the bandwidth of a single layer direct fed microstrip patch antenna are described. Section 3 investigates different feeding techniques used for bandwidth enhancement of microstrip patch antennas. In the proposed techniques, the antenna structure is multilayered and the patch is fed by a non-contact feed network. In Section 4, bandwidth enhancement by use of horizontally parasitic elements at the antenna aperture will be described. Sections 5 and 6 explore the stacked patches, a multilayered solution using vertically parasitic elements that can result in very wide bandwidths. In Sections 2-6, the focus is on linear polarized antennas. Finally, in Section 7, other techniques proposed in the literature including logperiodic array of patches, E-shaped patch, L-shaped feeding and microstrip monopole slotted antenna are introduced and investigated.

In this chapter, we tried to discuss about the principles of operation of presented antennas to give the reader insight into how these antennas work. Some examples are also given. The advantages and limitations of each method will be described and a comparison between them will be provided by giving the typical order of bandwidth that can be achieved using each method. By this way, in this chapter, it is tried to provide a designer's prospective of different techniques used for bandwidth enhancement of microstrip patch antennas.

Intrinsic techniques

Two intrinsic procedures can be applied to improve the bandwidth of a single layer direct fed microstrip patch antenna. One is increasing the substrate thickness and the other is decreasing the dielectric constant of antenna substrate (relative permittivity of near to one). This can also be observed in In fact, by increasing the thickness of the radiating patch substrate, the aperture size through which the fields are radiating to the space can be enhanced leading to easier impedance matching of antenna aperture which results in antenna bandwidth enhancement. On the other hand, by decreasing the substrate permittivity (making it closer to the permittivity of outer space), the reflection coefficient at the antenna aperture is reduced making it easier to impedance match which provides wider bandwidth for the antenna. It should be noted that as the thickness of the substrate is increased and the dielectric constant is reduced, the patch size becomes smaller which provides lower gain. In addition, the radiated power by feed network increases leading to more spurious radiation and higher cross-polarization level. Furthermore, the surface waves are more excited reducing the antenna radiation efficiency.

Taking the abovementioned limiting factors into consideration, besides the fact that there are limited values for thicknesses and dielectric constants provided by standard commercial substrates, the bandwidth enhancement by the mentioned intrinsic techniques cannot exceed 10%

[2] which is still not adequate for many applications like L-band radar which needs 19% bandwidth (1.4-1.7 GHz) or C-band satellite TV that requires 12.5% bandwidth (3.7-4.2 GHz). So, other techniques as mentioned at the following must be applied to enhance the bandwidth more.

The antenna performance can be improved by suppressing surface waves and reducing dielectric loss through partial substrate removal surrounding the antenna. This technique is considered similar to open air cavity since the removed substrate can be a large portion.

In [3], partial substrate removal surrounding a rectangular patch was compared to a conventional patch antenna, it turned out that the gain is enhanced up to 2.7 dB. The designed antenna is fed with aperture coupling technique and its geometry is shown in Figure 2.2.

Feeding techniques

The microstrip patch antenna's type of alimentation has an impact on its bandwidth as well. In all proposed feeding techniques presented in this section, the radiating patch is fed by a noncontact feed. In fact, by introducing a coupling mechanism between the feed and patch, a resonance is created at the vicinity of patch resonance which can result in antenna bandwidth enhancement if the feed and patch are well coupled.

Proximity coupled feed

Topology of proximity coupled feeding is depicted previously in Section 1.5. relative to the patch, a proper coupling between them can be obtained leading to expansion of antenna bandwidth. By this way, the bandwidth at the order of 8% can be achieved. In order to increase the bandwidth more, we can increase the thickness of radiating layer. However, this makes the coupling of power from the feed to the patch more difficult. To solve this problem, thicker substrate for the feed layer can be used. In this case, more power is coupled to the patch via open circuit stub, but at the same time, the spurious radiation caused by feed becomes more severe which can degrade the antenna efficiency. Another solution for this issue is applying a simple matching structure, for example, a quarter-wave transformer, to the feed line. By this way, i.e., using thicker laminate for the radiating layer besides applying matching circuit to the feed, the bandwidth can be doubled from 8 to 14% [1].

Aperture coupled feed

Topology of aperture coupled feeding is depicted previously in Section 1.5.4. In this structure, two resonances are provided, one by the patch and the other one by slot. If these two radiators are properly coupled, then the two corresponding resonances become close to each other leading to antenna bandwidth enhancement. The proper coupling between slot and patch can be obtained by tuning the dimensions of slot. By this way, the bandwidth at the order of 20-30% can be obtained. In [1], a sample is designed for mobile communication base station application with operation frequency of 1.9 GHz, bandwidth of 27% and 9 dBi gain.

In order to increase the bandwidth more, thick dielectric can be used for radiating layer. In this case, the slot size should be increased to make sure that the power is still coupled to the patch properly. In [4], 40% bandwidth has been reported using this simple technique.

In this technique, the shape and the size of the coupling slot can considerably affect the coupled power and consequently the antenna bandwidth and efficiency. In general, as the coupling power gets stronger by changing the slot shape, thicker substrate for the radiating layer can be used leading to wider impedance bandwidth. On the other hand, as the slot size decreases, the backward radiation which is dominantly caused by the slot decreases enhancing the antenna efficiency. In fact, by choosing the proposer shape for the slot, we try to provide the maximum coupling with the smallest size of the slot. By this way, we can more improve the bandwidth by increasing the radiating layer thickness or reducing its dielectric constant while the maximum efficiency is guaranteed.

One of the most used shapes for the coupling slot is a thin rectangle (

Iris feed

Topology of iris feeding is depicted previously in Section 1.5.5. The iris-fed microstrip patch antenna is equivalent to an antenna having two patches, one of which radiates toward the open end of the rectangular waveguide and the other toward open air. Therefore, this structure combines appealing characteristics of a microstrip antenna, like low profile, light weight, compact size, and simple integration in electronic circuits, with those of a waveguide, like high power handling capacity and low losses (resistive) [6]. In this method, the waveguide's mouth is directly fitted with the microstrip antenna. As a result, the most energy is radiated. In previous work [7],

the iris feed technique is used to increase the patch bandwidth in order to get over the issue of narrow bandwidth.

Modification of patch shape

The conventional microstrip antenna has a narrow bandwidth; one of the popular techniques to improve bandwidth is to modify the shape of the radiating patch. The slots are implanted on the patch to improve impedance matching, especially at higher frequencies. The current distribution on the patch is changed due to the cut slits, hence the input impedance and the current path length change [8]. Besides, by adding new slots in the patch, new resonance modes are created and couple to achieve wider bandwidth. The slots cut in the patch are of different shapes, namely the slots are the U-slots, H-slots, T-slots, E-slots, etc.

Many research works resort to using the slotted patch. For example, in [9], a double U-slot microstrip patch antenna with a wide operating bandwidth has been developed as illustrated in Figure 2.4a. The bandwidth is increased by 50% compared to the traditional U-slot antenna. In [10], the design of slotted patch illustrated in Figure 2.4b permits to achieve a total impedance bandwidth (<-10 dB) of 2.23 GHz (1.22-3.45 GHz) with a very decent gain of above 5 dBi (measured) and above 85 % efficiency over the functional range.

Parasitic elements on the single layer 2.5.1 Coupling via edges

In coplanar technique, additional patches are incorporated on a single plane above the substrate.

The excitation is given to a patch which is called the main patch or the active patch. The parasitic patch is excited if it is placed near to the active patch because of the direct coupling between them.

Wider bandwidth can be accomplished if the resonance of the active patch is close to the resonance of the parasitic patch. In fact, parasitic patches can be single patch or two symmetrical patches along the main patch edges.

By introducing two symmetrical parasitic patches horizontally coupled to the driven rectangular patch, the parasitic patches are considered coupled to driven patch via the radiating edges of driven patch. By introducing two symmetrical parasitic patches vertically coupled to the driven rectangular patch, the parasitic patches are considered coupled to driven patch via the non-radiating edges of driven patch. By mixing the two previous procedures, i.e., placing four patches around the driven patch coupled to it via both radiating and non-radiating edges, further bandwidth enhancement and more gain can be achieved.

In [11], a dual-slot coupled multiple dense dielectric (DD) patch antenna is proposed to achieve wide bandwidth (see Figure 2.5). The multiple DD patches operating at the quasi-TM11 mode and at the quasi-TM21 mode which is excited by the dual-slot. In addition, the dual-slot also functions as an effective radiator as with DD patches. Thus, five modes including two modes of DD patches and three modes of the dual slots can be obtained to support wide bandwidth. Compared with the reported DD patch antennas, the proposed design exhibits wide bandwidth and relatively high gain with a finite ground size. . Wide bandwidth of 81.9 % and relatively high gain of 9.27 dBi are obtained compared with the reported DD patch antennas. [12] proposes a broadband octagonal patch antenna with parasitic patches (see Figure 2.6). The proposed patch antenna is constructed with four parasitic patches around a central radiating octagonal element. This arrangement has improved the antenna bandwidth and gain when compared with that of conventional antennas.

Annular ring loaded shorted patch

In this structure, the driven element is the patch which is shorted to the ground plane and the parasitic element is the annular ring surrounding it as illustrated in Figure 2.7. The dimensions of both the shorted patch and the ring should be well-determined so that strong coupling between them is formed.

In [13], a circularly polarized antenna operating at 2.4-2.48-GHz industrial, scientific, and medical (ISM) band for biomedical applications has been proposed. The structure is developed from a simple pin-loaded patch antenna, whose circular polarization (CP) is mainly attributed to the load-

Stacked patches

In microstrip patch antennas, the term 'stacking' is used for the case where the driven patch is vertically coupled to another patch. By this solution, we do not have the problem of antenna large planar size (which was the issue for the solutions presented in the previous section in which the coupling between parasitic patches and the driven patch were established horizontally) solving the problem of grating lobes when the antenna is used in an array. However, in this technique, since the antenna structure is dual layered, the fabrication process is quite harder than the single-layered antennas presented in previous section. Using this technique, the bandwidth at the order of 20% can be obtained.

In multilayer structure, a patch is used above another one with superseding dielectric substrate in between. This makes two or more patches to share the same aperture area. In this configuration, lower dielectric constant material is usually added above the radiating patch to minimize the effective permittivity of the multilayer antenna and thus to increase the gain of the antenna.

There are two configurations of electromagnetically coupled multilayer patch antenna which are depicted in Figure. In [14], a multilayer structure is utilized to provide a wide bandwidth of 114% with two resonance frequencies within the band 3 GHz up to 11 GHz. The design presents additional C-slot in the ground plane and the bandwidth is enhanced more by etching E-slot on the parasitic patch as illustrated in Figure 2.9. At the center frequency 7 GHz, a gain of 12 dB is obtained.

Electromagnetic band gap: EBG

Metamaterials are artificial materials made using periodic metallic structures and dielectric whose electromagnetic properties (defined by electric permittivity (ε) and magnetic permeability (μ)) can be modified to a level beyond the available natural material. Materials with simultaneously negative permeability and negative permittivity are not presented in nature but these characteristics can be achieved using metamaterial. If patch antenna is fabricated on double-negative metamaterial in place of any dielectric substrate, the resonance of the patch will be shifted to higher frequencies with increased bandwidth. As negative μ and ε increase, resonant frequency increases.

There are numerous classifications for metamaterial structures in the literature and electromagnetic band gaps (EBG) are categorized as one of these classifications. EBG structures are artificial periodic dielectric materials that produce stop band and pass band properties. The periodic nature of EBG structure suppresses the propagation of surface waves within a certain frequency, called a band gap, which provides additional control of the electromagnetic waves compared to conventional filtering and guiding structures. Surface wave suppression provides a significant improvement of the maximum gain, bandwidth and efficiency when EBG is utilized in the microstrip antenna as the generation of surface waves degrades the radiation pattern and the antenna efficiency. Structures of EBG are realized by periodic configuration of dielectric substrate materials or metallic conductor as an artificial magnetic ground plane. In other words, EBG can be achieved by drilling a periodic pattern of holes in the substrate material or by etching a periodic configuration in the ground plane. The properties of EBG rely on the size, shape, symmetry and the used material in their configuration. In general, EBG can be classified into three types according to their geometric design which are one-dimensional transmission line, two-dimensional planar surfaces, and three-dimensional volumetric structure [15].

In [16], an electromagnetic band gap (EBG) structure, including a layer of a metallic ring on a layer of Rogers 4003C substrate, is used as illustrated in Figure 2.10. For a better design, a patch antenna with and without the EBG substrate has been simulated. The results show that the bandwidth can be improved up to 1.6 GHz in X-band by adding the EBG substrate.

Defected Ground Structure (DGS)

The geometry and dimensions of the ground plane have a significant effect on the excitation mode and the operating bandwidth of the microstrip antenna. Defected ground structures may be accomplished by cutting a defect of any configuration on the ground plane. This defect disturbs the current distribution in the ground plane and helps control input impedance and excitation of radio waves in the substrate. The ground defect may be varied accordingly from simple configuration to more complex configuration to achieve the optimum performance. However, the absence of the conducting ground may lead to the increase of the back lobe radiation to enhance the directivity.

Therefore, a reflecting layer behind the antenna is used to reflect back the back lobe radiation [17]. As an exemple, a miniaturised microstrip patch design is proposed in [18] for wireless communications at 10 GHz. Antenna miniaturisation is achieved here by loading a highly capacitive modified Minkowski fractal (type-2) defected ground structure (MFDGS-II) exactly beneath the center of the radiating patch (see Figure 2.12). This enables a patch size reduction of as high as 68% and an overall volumetric reduction of 84% along with an improvement in bandwidth and efficiency of the antenna.

Reflecting Layer

The utilization of reflector planes or partial reflective surfaces [START_REF] Kim | Backfire suppressed low profile aperture coupled stacked patch antenna backed by a high impedance surface (HIS) reflector for UHF RFID reader applications[END_REF][START_REF] Kissi | Reflector-Backed Antenna for UWB Medical Applications with On-Body Investigations[END_REF] is commonly used for maximizing the bandwidth of microstrip antennas. In this technique, half wavelength in terms of resonant frequency is a reasonable resonant cavity between the reflective surface and the metallic ground plane. However the cavity gap can be minimized to quarter wavelength utilizing an artificial magnetic conductor, but the thickness cannot be completely mitigated [START_REF] Ge | The use of simple thin partially reflective surfaces with positive reflection phase gradients to design wideband, low-profile EBG resonator antennas[END_REF].

To get better performance, metasurface reflector (MSR) structure is used to reflect the unwanted back radiation by artificially engineering the metallic reflector surface.

Log-periodic array of patches

In [START_REF] Hall | Multi-octave bandwidth log-periodic microstrip antenna array[END_REF][START_REF] Pues | Wideband quasi log-periodic microstrip antennas[END_REF][START_REF] Kakkar | Stagger tuned microstrip log-periodic antenna[END_REF], the idea of log-periodic antennas for providing very wide bandwidths has been applied to microstrip patches. In this technique, the microstrip patches are arranged in a log-periodic formation. The patches are series fed by a microstrip line either directly or indirectly.

In The radiation pattern of antennas shown in Figure 2.14 is broadside, unlike the conventional log-periodic dipole array which radiates at end-fire direction. Although by using log-periodic configuration, very wide bandwidths can be achieved, but the main beam direction scans vs. frequency makes it impossible to provide constant beam direction through the whole bandwidth.

Superstrate layer

Using superstrate as a cover above the antenna is a simple way to increase antenna performance, including broad side gain, bandwidth and directivity characteristic of antenna. The superstrate can have several forms, namely: metallic superstrate, metasurface superstrate, parasitic patches above foam, etc. In [START_REF] Samantaray | A Gain-Enhanced Slotted Patch Antenna Using Metasurface as Superstrate Configuration[END_REF], a superstrate configuration based metasurface (MS)

antenna has been designed that incorporates a fractal-shaped slotted patch having periodic arrangement of square patches along with a shorting via at its center and a couple of rectangular slots in the ground plane (see Figure 2.15). Another way to enhance the antenna's performances is to use a metal-ring superstrate as illustrated in Figure 2.16. The impedance bandwidth has been enhanced from 6% to 26% in [START_REF] Chandrashekar | A Wideband Design of Microstrip Patch Antenna Loaded with Metal Ring Superstrate[END_REF] and the co-polarized gain has been significantly raised at resonances in [START_REF] Gajera | Dual-Band Design of Microstrip Patch Antenna with Copper Ring Superstrate for X-Band Applications[END_REF].

Conclusion

In this chapter, a variety of procedures proposed in the literature to increase the impedance bandwidth of linear-polarized microstrip patch antennas are presented and discussed.

In Section 2, bandwidth enhancement by proper choosing of substrate is described and declared that up to 10% bandwidth can be obtained using this method taking this fact into consideration that other characteristics of antenna, say efficiency and backward radiation, may be affected.

In Section 3, it is shown that by applying indirect feeding mechanism and coupling the electromagnetic fields to the patch, bandwidth can be enhanced. Three methods including proximity coupled, aperture-coupled and iris coupled patches were introduced by which up to 10%, 30% and 27 % bandwidth can be achieved, respectively. For proximity coupled patches, if external matching structure is used, the achievable bandwidth can be doubled.

In Section 4, the modification of the patch shape has been investigated and several state-of-art works have been introduced. In Section 5, bandwidth enhancement making use of horizontally parasitic elements at the antenna aperture is described and stated that namely 20% bandwidth can be achieved using this single-layer technique at the expense antenna size enlargement which makes it unsuitable for use in array configurations. Section 6 explores the stacked patches, a multilayered solution using vertically parasitic elements that can result in very wide bandwidths (namely 50-70%). By this method, the antenna planar size is kept small making it interesting for use in arrays; however, due to its multilayered topology, it has problems with cost and fabrication complexity compared to single-layer solutions.

In Sections 8, 9, 10 and 11, other techniques proposed in the literature including the electromagnetic bad gap material, the log-periodic array of patches, the defected ground/patch technique and the superstrate layer are introduced and investigated.

Introduction

Throughout the 1980's to the mid 1990's, the patch antenna designer often had to perform full wave analysis for specific designs and developed his own computer code based on a numerical method to obtain results for the characteristics of a design, which can be validated by comparison with experiment. In fact, electromagnetic analysis methods can be grouped into two large families, these are differential methods and integral methods. The differential methods consist in discretizing the Helmholtz equation in space and lead to the solution of linear equations. However, they require a very large memory space and computation times. Integral methods make it possible to obtain the solutions of a propagation or discontinuity problem from the determination of the current or field distributions on a particular surface. Among the integral methods we distinguish, the Finite Element Method [1-2], the Finite Difference Method [3] and the Moments Method [4][5].

These methods are better from a computational time point of view since they use little memory space. However, the problem can only be solved for structures in a homogeneous medium.

Beginning in the mid 1990's, electromagnetic simulation software capable of solving general planar microstrip structures became commercially available at moderate costs. Thus, being proficient in using at least a few of these softwares becomes a must for antenna designers.

Table 3.1 shows a number of commercial electromagnetic simulation softwares. One class uses the method of moments in the numerical analysis, while another uses the finite difference time domain method. One software, HFSS, is based on the finite element method. Another software, PCAAD, is based on the cavity model. The segmentation method used in Micropatch is a variation of the cavity model. This chapter presents a formulation which makes it possible to perform a rigorous electromagnetic calculation while minimizing the execution time and the memory resources required. One solution consists in hybridizing various methods together in order to benefit from the performance of each in their chosen field. For this reason, the method of moments (MOM) hybridized with the Generalized Equivalent Circuits Method (GEC) is proposed.

The method of moments (MOM) is best suited to perform an electromagnetic study of planar microwave structures. Indeed, it makes it possible to reduce the dimension of the problem since it writes the initial boundary conditions in the form of an integral equation defined on the surface of the obstacle. The Generalized equivalent circuits (GEC) are introduced in the development of the formulation of the Moment method in an attempt to transpose an investigation problem requiring a 3D mesh to a 2D investigation problem that is generally easier to deal with. Thereby, the MOM-GEC hybrid method provides a gain in terms of computation time and a reduction in memory space.

Moment method

The integral representations were obtained by Stratton and Chu [6] from Maxwell's equations and Green's vector theorem. These representations express the fields scattered from the fields or surface currents. MoM has long been known in other disciplines of physics. As early as 1915, Galerkin, a mechanical engineer of Russian origin, proposed a numerical procedure for solving equations where the unknown is a function. Later, mathematicians demonstrated that the Galerkin approach can be extended to other classes of problems generically called the method of moments.

MoM was introduced for solving problems with antennas and electromagnetic scattering through metal walls in the 1960s by Harrington [7][8].

When the MOM method is applied, the current density on the antenna is considered to be the variable from which all the antenna parameters are derived. The current density is discretized into a set of appropriate elements also called a base function, where the amplitudes are unknown to be determined. The boundary conditions for electric and magnetic fields are enhanced on the surface of the antenna using the function test. Indeed, from the procedure known as the test procedure results a linear system of integral equations. The latter can be expressed as a matrix and the interaction between basic functions and test functions is taken into account. If the base functions and the test functions are chosen identical, the method is called the Galerkin method [9].

The method of moments consists in solving any inhomogeneous linear equation of the form:

L ̂f = g (3.1)
𝐿 ̂: linear operator on a Hilbert space Η f: unknown function of the problem.

g: known function (excitation)

However, the unknown f is expressed in a basis of N known functions called basis functions, noted {f n }, n= 1..N:

f ≈ ∑ a n f n N n=1 (3.2)
{𝑎 𝑛 }, n= 1,..,N: are constants of unknown weights.

Since the operator L is linear, equation (3.1) becomes:

∑ a n L(f n ) + Ԑ = g (3.3) N n=1
In order to minimize the residual error Ԑ, equation (2.3) is projected onto a second basis h n , n = 1,..., N; called the test function basis:

∑ a n 〈h n , L(f n )〉 + 〈h n , Ԑ〉 = 〈h n , g〉 (3.4)

N n=1

To minimize the error, make it orthogonal to h n : 〈h n , Ԑ〉 = 0. Therefore, equation (2.4) will take the following matrix form:

[A][X] = [G] (3.5)
A is the MoM matrix which is expressed by:

A = ( 〈h 1 , L(f 1 )〉 〈h 1 , L(f 2 )〉 . . . 〈h 1 , L(f N )〉 〈h 2 , L(f 1 )〉 〈h 2 , L(f 2 )〉 . . . 〈h 2 , L(f N )〉 . . . . . . . . . 〈h N , L(f 1 )〉 〈h N , L(f N )〉 … 〈h N , L(f N )〉) (3.6)
The unknown weighting constants:

X = [ a 1 a 2 . . a N ] (3.7)
The known function:

G = [

Generalized Equivalent Circuit theory

To simplify the resolution of Maxwell's equations, the method of equivalent circuits was proposed by Baudrand [10][11][12] in order to represent the integral equations by equivalent circuits.

This representation is used to express the boundary conditions of the unknown electromagnetic field using a single electrical circuit. The latter can be considered as a true electrical image of the structure to be studied since it faithfully describes the discontinuity as well as its environment:

-At the discontinuity level, the electromagnetic state is described by generalized test functions which are modeled by virtual sources that do not store energy. The introduction of these virtual sources makes it possible to represent all the transition relations imposed on the electromagnetic field when crossing a given surface.

-The discontinuity environment is expressed by an impedance operator (or admittance operator) which represents the boundary conditions on either side of the discontinuity surface.

-The wave that excites the discontinuity surface is represented symbolically by a source (localized or modal) of field or current called real since it delivers power.

The mathematical formulation of the method has been well defined in multiple research thesis such as [13][14][15]. Electromagnetic modeling by generalized equivalent circuits (GEC) extends Kirchhoff's laws generally used with the concept (V, I) to Maxwell's formalism (E, J). In order to apply Kirchhoff's laws analogously to the case of low frequencies, we must substitute the magnetic field by the current density J defined by

J H n
= where n is the normal to the surface of discontinuity. It should be mentioned that these equivalent circuits are generally associated with perfect interfaces where the electric field and the current density are defined on two complementary domains. To properly describe the different components of the equivalent circuit, it is necessary to give a symbolic representation (Figure 3.1) of the discontinuity and its environment.

MOM-GEC formalism

The flowchart of MOM-GEC process is detailed in this section.

Generalized Equivalent Circuit representation

In this section, we present the different components of the equivalent circuit proposed by MOM-GEC method.

Virtual sources

There are two types of virtual sources, the virtual sources in field 𝐸 𝑒 and the virtual sources in current 𝐽 𝑒 (see Figure 3.3.). The virtual field source must be adjustable so as to cancel the dual magnitude (the transverse current of this source) while the virtual current source must be adjustable so as to cancel the dual magnitude (the equivalent voltage applied to the terminals of this source).

Real sources

The excitation of the discontinuity surface is symbolically represented by a localized or modal field or current source known as real since it delivers power.

Localized source

The localized sources are defined on a surface (S) of small dimension and are characterized by a quasi-static voltage or current (given the dimensions). The most used mathematical source is known under the name "delta gap voltage" to designate a voltage source of amplitude V m applied in a gap of very small dimensions in the circuit plane. The localized source must have a non-zero dimension to be able to apply an electric voltage. In addition, this dimension must be negligible compared to the wavelength to ensure the uniqueness of the definition of the source and the nondependence of the charge. There are two types of localized sources illustrated in Figure 3.5: Localized current source and Localized voltage source. The localized electric field E 0 defined on a surface S according to the shaping function f 0 and an amplitude V 0 is written: E 0 = V 0 f 0 .

Modal source

The extended sources are modal sources of the electric or magnetic field type relating to the returned guide. These sources are used to study structures in homogeneous or inhomogeneous waveguides. According to the nature of guide, a modal base is defined. The latter can be subdivided into two groups, lower order modes (called active) and higher order modes (called passive). The discontinuity surface can be excited by one or more active modes. Higher order modes are localized and therefore serve to describe abrupt variations in the vicinity of discontinuities. These passive modes are superimposed closed by their mode impedances. There are two types of modal excitation sources mentioned in Figure 3.6. These sources deliver power to the guide, which is why they are said to be real (as opposed to virtual sources).

a. Electric field source

The transverse electric field E 0 i corresponding to the fundamental mode of the guide located on side i of the discontinuity surface (see Figure 3.6a). The excitation is written:

E 0 i = V 0 i f 0 (3.9)
The internal impedance 𝑍 𝑖 represents the contribution of the evanescent modes of the i-side guide in the distribution of the electromagnetic field at the discontinuity interface.

b. Current source

The current density of the fundamental mode of the guide is expressed:

J 0 i = I 0 i f 0 (3.10)
The internal admittance Y i of the source represents the contribution of the evanescent modes of the i-side guide in the distribution of the electromagnetic field at the discontinuity interface (see

Admittance or impedance operator

The environment of the discontinuity is expressed by an impedance operator (or admittance operator) which represents the boundary conditions on either side of the discontinuity surface.

The admittance operator describes the boundary conditions imposed on the electromagnetic field in the domain at the surface S. It is a dipole when a single surface is supported on the domain and it is of multipole type when S is the union of several sub surfaces (see Figure 3.7).

Single surface

The admittance operator connects the current density J ⃗ on the surface S and the tangential elec- tric field E on this surface by this relation:

J ⃗ = Y. ̂E ⃗ ⃗⃗ (3.11)
In some problems, we prefer to use the admittance operator rather than the impedance operator.

The reverse of the latter is called admittance. In addition, we can write:

E ⃗ ⃗⃗ = Z. ̂J ⃗ (3.12)
From equation (3.11), we can calculate J ⃗ on a surface S from the admittance operator and the tan- gential electric field given on this surface. Also, from (3.12), we can calculate the tangential electric field on S from the impedance operator and the knowledge of current distribution. As a result, a generalized Ohm's law appears in the concept of fields where the admittance operator is associated with the half-space formed by the domain ∑ and the surface oriented by the normal n ⃗⃗ pointed to ∑. The existence of the admittance operator (or impedance) depends on the existence of the tangential electric field and the current density. Thus, the uniqueness of the admittance operator depends on the uniqueness of the solution (E ⃗ ⃗⃗ , J ⃗ ) over S. The closure or limitation on ∑ of the surface S is important for the admittance operator Y ̂ to be defined. Otherwise, the current density depends on the values that the electric field would take at other points in the global domain.

Several sub-surfaces

The operators linking the transverse electromagnetic fields for homogeneous areas are directly deduced from the reduced admittances or from the quadrupoles of line length. For a space delimited by two surfaces, the admittance operator "Q" is represented by a matrix of operators linking the tangential electric and magnetic fields as illustrated in Figure 3.8.

Application of Kirchhoff and Ohm's laws

The integral equations derive from the equations that express the boundary conditions on the discontinuity surface. Consequently, only the tangential components of the electromagnetic field are to be considered. Let 𝑆 1 and 𝑆 2 be two surfaces infinitely close to the discontinuity surface (S), located on either side of the latter. The continuity conditions of the tangential components of the electromagnetic field on the surface (S) result in:

-An equality between the tangential field on the surface (S) and the tangential fields on both sides of (S):

12

E E E == (3.13)
This electromagnetic state is similar to that in an electrical circuit where the voltage is the same on both sides of the connections joining the left and right closures of the circuit.

-A passing relation of type:

J ⃗ ∧ n 1 ⃗⃗⃗⃗⃗ = H 1t ⃗⃗⃗⃗⃗⃗⃗ -H 2t ⃗⃗⃗⃗⃗⃗⃗ (3.

14)

Such that 𝐻 1𝑡 ⃗⃗⃗⃗⃗⃗⃗ and 𝐻 2𝑡 ⃗⃗⃗⃗⃗⃗⃗ are the tangential fields on either side of the surface, 𝑛 1 ⃗⃗⃗⃗⃗ is the normal to the surface directed to region (1).

The relation (3.14) of magnetic field discontinuity becomes:

J ⃗ = H 1t ⃗⃗⃗⃗⃗⃗⃗ ∧ n 1 ⃗⃗⃗⃗⃗ -H 2t ⃗⃗⃗⃗⃗⃗⃗ ∧ n 1 ⃗⃗⃗⃗⃗ (3.15a) J ⃗ = H 1t ⃗⃗⃗⃗⃗⃗⃗ ∧ n 1 ⃗⃗⃗⃗⃗ + H 2t ⃗⃗⃗⃗⃗⃗⃗ ∧ n 2 ⃗⃗⃗⃗⃗ (3.15b)
Where 𝑛 2 ⃗⃗⃗⃗⃗ is the normal to S directed towards region 2 such that 𝑛 2 ⃗⃗⃗⃗⃗ = -𝑛 1 ⃗⃗⃗⃗⃗.

By introducing the notion of equivalent current by J i ⃗ ⃗⃗ = H i ⃗⃗⃗⃗⃗ ∧ n i ⃗⃗⃗⃗ with i = 1, 2, the transition relation (3.16) is rewritten:

J ⃗ = -( J 1 ⃗⃗⃗ + J 2 ⃗⃗⃗ ) (3.16)
From equation (3.13), we deduce that any surface (S) is represented at the level of the electric circuit by two connections as illustrated in Figure 3.9.

Application of Galerkin Method

Once the translation of the integral equations in the form of an equivalent circuit is established, we apply the generalized Kirchhoff laws (in current and in voltage) and Ohm's law (notion of impedance / admittance operator) to the circuit got. Thus, one deduces a formal relation between the sources (virtual and real) and their dual quantities in the following form: (3.17)

Mathematical formulation

The determination of the formal relationship between the sources and their dual quantities in the case of the equivalent circuit is done by applying Kirchhoff's laws and Ohm's law to the circuit.

In general, this equation does not admit of known analytical solutions. The expressions of the equations, one must use one of the existing numerical methods. We will opt here for the method of Galerkin known by the simplicity of its formulation and the generalization of its application.

Electromagnetic problems boil down to solving the equation (3.1) knowing that g is a term representing excitement, f is the unknown of the problem (either an electric field or a magnetic field), L is an integrodifferential operator which is generally linear and is written as a function of the impedance operator Z ̂ or admittance Y ̂. The weighting coefficients are calculated using the equation (3.18) by:

[X] = [G][A] -1 (3.18)
In the case of a homogeneous waveguide, the electric field (|E〉) is decomposed on a basis chosen in an adequately noted way (f mn ). This base must be complete and check the boundary conditions.

The field is written in the following form: The electric field can be deduced from the current J by the following formula:

|E〉 = ∑
E(x, y) = ∑ [ ∑ (a p ⟨g p , f m,n TE ⟩ z m,n TE f m,n TE (x, y)) 𝐍 𝐛𝐚𝐬𝐢𝐬_𝐓𝐄 m,n + ∑ (a p ⟨g p , f m,n TE ⟩ z m,n TE f m,n TM (x, y)) 𝐍 𝐛𝐚𝐬𝐢𝐬_𝐓𝐌 m,n ] 𝐍 𝐭𝐞𝐬𝐭 p (3.24)

Modal impedance

We define the modal impedance for a single layer substrate and a multilayer substrate:

Single Layer Substrate

We define the impedances of the modes according to their nature TE and TM according to the layer and the dielectric permittivity 𝜀 𝑟 .

Z α=TE,TM = { z mn TE = jωμ 0 γ mn z mn TM = γ mn jωε 0 ε r (3.25) We define the admittances of the modes according to their nature TE and TM according to the layer and the dielectric permittivity ε r .

Multilayer Substrate

Indeed, since there are no metallic ribbons between the layers forming the substrate, these N layers can be seen as dipoles Q 2 , … . Q N-1 of lengths respectively h 2 , … h N-1 as shown in Figure 3.11. To calculate the admittance operator of region 2, we must precede in a recurring fashion. In fact, we start with:

1. Calculation of the admittance of the modes of the layer brought back to the level of the discontinuity interface between the layer N and N-1 knowing that this last layer is terminated by a short-circuit (the ground plane).

2.

Based on line theory, we calculate the admittance of the modes of the layer brought to the level of the discontinuity interface between the i-1-layer i-2 by:

Y i-1,𝛼=𝑇𝐸,𝑇𝑀 = y i-1,𝛼=𝑇𝐸,𝑇𝑀 . [ y i-1,𝛼=𝑇𝐸,𝑇𝑀 + Y i,𝛼=𝑇𝐸,𝑇𝑀 tanh(γ i-1 l i-1 )
Y i,𝛼=𝑇𝐸,𝑇𝑀 +y i-1,𝛼=𝑇𝐸,𝑇𝑀 tanh(γ i-1 l i-1 ) ] (3.30)

We repeat this recursive process until we bring all the layers of the substrate (region) back to the discontinuity interface S on which the metallic tape is printed, the last step of the iteration is the calculation of Y 2 from Y 3 by the relation:

4.

Based on line theory, we calculate the admittance of the modes of the layer brought to the level of the discontinuity interface between the i-1-layer i-2 by:

Y i-1,α=TE,TM = y i-1,α=TE,TM . [ y i-1,α=TE,TM + Y i,α=TE,TM coth(γ i-1 l i-1 ) Y i,α=TE,TM +y i-1,α=TE,TM coth(γ i-1 l i-1 ) ] (3.32)
We repeat this recursive process until we bring all the layers of the substrate (region) back to the discontinuity interface S on which the metallic tape is printed, the last step of the iteration is the calculation of Y 2 from Y 3 by the relation: (3.33)

Y 2,α=TE,TM = y 2,

Conclusion

This chapter presents the concept of MOM-GEC method, the aim of which is to simplify the resolution of electromagnetic problems by transposing the field problem into a circuit problem that is easier to handle. The problem thus formulated, its numerical resolution is done by applying the method of Galerkin known by the simplicity of its formulation and the generalization of its application. Throughout this thesis, this formulation will be used to model and solve all the posed electromagnetic problems.

Wideband Iris-Fed Microstrip Patch Antenna Under Operation of Dual-Resonance

Introduction

Multiple techniques have been used to widen the bandwidth of antennas in the X-band, namely:

the thick substrate, the partial substrate removal, the superstrate cover [1][2], the parasitic patches coupled to the main patch or stacked patches [3], the patch with metallic rings, the defected ground structure, the reflecting layer, the electromagnetic band gap as material [4], the merging of several antenna resonance modes [5][6], etc.

The microstrip patch antenna's type of alimentation has an impact on its bandwidth as well. The iris feed technique is used to increase the patch bandwidth in order to get over the issue of narrow bandwidth [7][8]. The iris-fed microstrip patch antenna is equivalent to an antenna having two patches, one of which radiates toward the open end of the rectangular waveguide and the other toward open air. Therefore, this structure combines appealing characteristics of a microstrip antenna, like low profile, light weight, compact size, and simple integration in electronic circuits, with those of a waveguide, like high power handling capacity and low losses (resistive) [9]. In this method, the waveguide's mouth is directly fitted with the microstrip antenna. As a result, the most energy is radiated.

From the above literature survey, this work proposes to model a microstrip patch antenna fed by a waveguide using an end-wall iris through ground plane since it has not been widely used in X-band. The iris is sized so that it is resonant and subsequently the antenna operates under the fusion of two modes relating to the iris and the patch [10], resulting in a wideband radiation characteristic with dual resonance.

The modular method which is the hybridization of the method of Moments (MOM) with the method of Generalized Equivalent Circuits (GEC) is proposed. This one is made by using a flexible formalism which reduces the resolution of the electromagnetic problem in a simple manipulation of the equivalent circuits. Therefore, a conversion of a 3D investigation problem into a 2D problem is realized.

This chapter is divided as follows. Section 2 introduces the proposed antenna. The mathematical formulation based on the hybrid approach will be detailed in section 3. Section 4 presents the validation of the antenna by full-wave simulations. To confirm the accuracy and the efficiency of the proposed numerical approach, a comparison with HFSS and with measurement is held. Finally, Section 5 concludes the chapter.

Antenna Design:

This section includes the description of the model as well as the antenna design flowchart and its manufactured model. The flowchart's steps are performed subsequently to design the proposed antenna. It should be noted that the computer code of the numerical method is translated with MATLAB and executed on an Intel Core i7 processor computer with a processing capacity of 55 GHz and 16

Design Flowchart

GB RAM. The results obtained by the HFSS simulator are also obtained by this computer.

Theoretical model

The antenna illustrated in Figure 4.2 is composed by a three-layer device which consists on a dielectric layer sandwiched between a radiating patch and a ground plane.

The centered thick iris, as shown in Figure 4. 3(a), is located between two virtual waveguides at (z=0), each of which is characterized by two electric side walls and two electric horizontal walls (which is identified by EEEE for four times Electrical as defined in Annex A). These two waveguides are: the entry virtual waveguide (Guide 1) and the exit virtual waveguide (Guide 2) which do not have the same height.

The microstrip patch is placed in the cross section of a rectangular waveguide at (z=𝐷 2 ) open to infinity. The boundary conditions are defined by perfect electrical walls on all sides as shown in

01 01 0 E V f = (4.2)
The auxiliary current source 𝐽 02 and the auxiliary field source 𝐸 02 are expressed respectively according the amplitude 𝐼 02 and 𝑉 02 and according the fundamental mode 𝑓 0 ′ of the modal base function 𝑓 𝑚𝑛 ′ :

02 02 0 ' J I f = (4.3) ' 02 02 0 E V f = (4.4)
Discontinuities 𝐷 1 and 𝐷 2 are identical so they have the same test function (𝑔 1 𝑝 ) 𝑝=1,..,𝑁 𝑡𝑒𝑠𝑡 which is similar to the fundamental mode in the iris. The virtual electric fields 𝐸 𝑒 1 and 𝐸 𝑒 2 are equals and expressed in terms of test function (𝑔 1 𝑝 ) 𝑝=1,..,𝑁 𝑡𝑒𝑠𝑡 and weighting coefficients 𝑥 𝑝 : The thick rectangular iris between two guides of different heights can be subdivided into three guides as illustrated in Figure 4.3.

-Guide 1

The admittance of the modal source representing the evanescent modes at the level of discontinuity 𝐷 1 of the exciter waveguide which is associated with the modal base function

(𝑓 𝑚𝑛 ) (𝑚,𝑛)={1,.
.,𝑁 𝑏 2 } is expressed by:

1 1 10 , TE TM ev mn ev mn outTE Y f y f      = =    (4.6) { 𝑦 𝑒𝑣 1 𝑇𝐸 = 𝛾 𝑚𝑛 1 𝑗𝜔𝜇 0 𝑦 𝑒𝑣 1 𝑇𝑀 = 𝑗𝜔𝜀 𝑟 𝜀 0 𝛾 𝑚𝑛 1 𝛾 𝑚𝑛 1 = √ ( 𝑚𝜋 𝑎 ) 2 + ( 𝑛𝜋 𝑏 ) 2 -𝑘 0 2 (4.7)
-Guide 2

The admittance of evanescent modes respectively related to the modal source at the level of discontinuity 𝐷 2 of the second waveguide which is associated with the modal base function (𝑓 ′ 𝑚𝑛 ) (𝑚,𝑛)={1,..,𝑁 𝑏 2 } is expressed by:

2 2 , '' ev mn ev mn TE TM Y f y f      = =    (4.8) { 𝑦 𝑒𝑣 2 𝑇𝐸 = 𝛾 𝑚𝑛 2 𝑗𝜔𝜇 0 𝑦 𝑒𝑣 2 𝑇𝑀 = 𝑗𝜔𝜀 𝑟 𝜀 0 𝛾 𝑚𝑛 2 𝛾 𝑚𝑛 2 = √ ( 𝑚𝜋 𝑎′ ) 2 + ( 𝑛𝜋 𝑏′ ) 2 -𝑘 0 2 (4.9)
-Guide 3

The line length between the two discontinuities 𝐷 1 and 𝐷 2 is represented by:

11 12 12 11 [] Y Y Q Y Y         =     (4.10)
The input admittance 𝑌 11 ̂ and the inverse transfer admittance 𝑌 12 ̂ of quadrupole associated with the modal base function (𝑓 ′ 𝑚𝑛 ) (𝑚,𝑛)={1,..,𝑁 𝑏 2 } are respectively expressed by:

11 11 , '' mn mn TE TM Y f y f      = =    (4.11) 12 12 ,
'' 2/ On side (2), the virtual current distribution 𝐽 𝑠 which is the dual of virtual electric field E, is expressed in terms of test function (𝑔 2 𝑝 ) 𝑝=1,..,𝑁𝑡𝑒𝑠𝑡 and weighting coefficients 𝑥 𝑝 :

mn mn TE TM Y f y f      = =    (4.12)
2 test N s p p p J x g =  (4.14)
The admittance of the dielectric layer is expressed by:

4 1 , ' coth( ) ' th mn th mn mn TE TM Y f y t f       = =    (4.15)
The admittance from the vacuum brought to the surface of the dielectric 𝐷 3 is expressed by: , '' Therefore, the virtual current sources 𝐽 1 and 𝐽 2 are expressed as:

s mn s mn TE TM Y f y f      = =    (4.
1 1 1 01 11 12 2
()

ev ev ev J J Y Y E Y E    = -+ + + (4.20) 22 2 02 11 12 1
()

ev ev ev J J Y Y E Y E    = -+ + + (4.21)
The formulation of equations system in matrix system can be written as follows:

ev e e ev J E J E Y Y Y E J J E Y Y Y                 =  -+          -+  (4.22)
2/ On side (2), as the dielectric is thin, it is equivalent to a layer surface impedance. Then, by applying Ohms and Kirchhoff's laws, the auxiliary electric field 𝐸 02 and the virtual electric source E are given by:

2 1 02 02 () t t s J E Y J J -  =+ (4.23) 2 1 02 () t s E Y J J -  =+ (4.24)
The total admittance of the circuit 𝑌 𝑡 2 ̂is the sum of 𝑌 𝑡ℎ ̂, 𝑌 𝑠 ̂ and 𝑌 𝑒𝑣 2 ̂.

The formulation of equations system in matrix system can be written as follows:

Application of Galerkin method

The resolution of electromagnetic equations is provided by (V, I) physical scales instead of using (E, J) physical scales. Therefore, Figure 4.4 is presented as illustrated in Figure 4.5. 1/ Side (1): The application of Galerkin method to solve the electromagnetic problem consists on:

• The projection of equation (4.18) on the excitation 𝑓 0 :

1 0 01 0 ,, e f E f E   =   (4.26) 0 01 0 1 ,, b test N N mn p p m p f V f f x g   =    (4.27)
The (f mn ) m=2..N 𝑏 form an orthonormal basis, i.e. 〈f mn , f 𝑚𝑛 ′〉 = δ 𝑚 𝑚 ′ then (4.27) is written:

01 0 1 , test p N p p V x f g =    (4.28)
• The projection of equation (4.19) on the excitation 𝑓 0

′ : 2 ' ' 0 02 0 ,, e f E f E   =   (4.29) 0 02 0 1 ', ' ', b test N N mn p p m p f V f f x g   =    (4.30)
The (f mn ′) m=2..N 𝑏 form an orthonormal basis, i.e. 〈f mn ′, f 𝑚𝑛 ′ ′〉 = δ′ 𝑚 𝑚 ′ then (4.30) is written:

02 0 1 ', test p N p p V x f g =    (4.31)
• The projection of equation (4.20) on the test functions g 1 𝑞 for q = 1, …, 𝑁 𝑡𝑒𝑠𝑡 with the cancellation of virtual current source 𝐽 1 :

1 1 01 0 1 11 1 1 12 1 , ,( ) , 0 test test q q p q p NN ev p p pp g I f g Y Y x g g Y x g    -  +  +  +   =  (4.32)
• The projection of equation (4.21) on the test functions g q for q = 1, …, 𝑁 𝑡𝑒𝑠𝑡 with the cancellation of virtual current source 𝐽 2 :

2 1 02 0 1 12 1 1 11 1 , ' , ,( ) 0 test
test q q p q p NN p ev p pp g I f g Y x g g Y Y x g    -  +   +  +  =  (4.33)
The matrix system (4.34a) is obtained from the equations (4.28), (4.31), (4.32) and (4.33):

1 1 01 01 01 01 01 02 1 0 1 11 1 1 12 1 1 0 1 12 1 1 11 1 0 0 , 0 0 0 0 ', 0 , 0 ,( ) , 0 0 , ' , ,( ) p p
p q p q p p q p q p p ev p ev fg I V fg I V x g f g Y Y g g Y g x g f g Y g g Y Y g                   =    -   +            -     +   (4.34a)
The matrix system (4.34a) can be written in the form (4.34b) (Annex C):

12 11 21 22 1 1 1 01 01 1 1 02 02 2 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] t t Z Z t t Z Z A D A B C A V I V I A C B B D B - - - -           =               (4.34b)
Noting that:

  X is the vector ponderation expressed by   test N p p Xx =  ,   A is the first excitation vector ex- pressed by   10 , p A g f =   ,   B is the second excitation vector expressed by   10 ,' p B g f =   ,   C is the first impedance matrix expressed by   1 12 1 , pq C g Y g  =   ,   1 D is the second impedance matrix expressed by   1 1 1 11 1
, ( )

p ev q D g Y Y g  =  +  and   2 D
is the third impedance matrix expressed by   , ( )

p ev q D g Y Y g  =  +  .
The expression of the input impedance 𝑍 𝑖𝑛 is deduced from (4.34b):

21 12 11 22 12
.

in ZZ ZZ ZZ =- - (4.35)
The magnitude in (dB) of the reflection coefficient is deduced from (4.35) as expressed in (4.36):

0 11 0 in in ZZ S ZZ - = + (4.36a)
The angle in (degree) of the reflection coefficient is deduced from (4.36a) as expressed in Where 𝑍 0 is the impedance of fundamental mode and  is the phase.

2/ Side (2):

The application of Galerkin method to solve the electromagnetic problem consists on:

• The projection of the equation (4.23) on 𝑓 0 ′ provides:

22 02 0 0 02 0 2 ', ' ', test p N t p t p V f Y f I x f Y g  =   +    (3.37)
• The projection of the equation (4.24) on 𝑓 0 ′ with the cancellation the virtual electric source E provides:

22 2 0 02 2 2 , ' , 0 test q q p N t p t p g Y f I x g Y g    +   =  (3.38)
We deduct the writing of the following matrix system (3.39) from (3.37) and (3.38):

2 2 2 1 2 3 1 2 1 2 1 1 2 2 1 2 1 2 1 2 2 2 2 2 1 0 0 0 2 0 2 0 2 02 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 ', ' ', ', ... ', , ' , , ... , 0 0 . , , ... , . . . .
f Y f f Y g f Y g f Y g V g Y f g Y g g Y g g Y g g Y g g Y g g Y g g                                      =        2 2 1 2 2 2 02 1 2 0 2 2 2 2 2 2 . . , ' , , ...
, N N N test test test test N test p t t t t I x x x Y f g Y g g Y g g Y g                                   (3.39a)
The matrix system (4.39a) can be written in the form (4.39b):

            02 02 01 11 0 t V I AA X AC      =            (4.39b) Noting that:   2 1 0 0 0 ,' t A f Y f  - =   ,   1 A is the excitation vector expressed by   2 ' 1 2 0 , pt A g Y f  = 
 , and   1 C is the impedance matrix which define the coupling between p and q dielectric elements and expressed by   2

N t p mn mn q mn C y g f f g - =     . 1 1 2 2 , ' ', b
From (4.39b), the expression 02 Z of is deduced:

11 02 0 1 [ ] [ ] [] [] t AA ZA C =- (4.40)
The amplitude 𝑉 02 is expressed as: ()

eq Z Z Z Z Z Z Z Z Z +- = + - + (4.42)
The amplitude 𝐼 02 is expressed as:

Computed Results

This section presents the steps of MOM-GEC flowchart which follow the step of extraction of the unknown quantities namely: the study of the convergence, the verification of the boundary conditions as well as the extraction of the characteristics of the antenna.

Convergence study

The convergence study is essential in a first step for the validation of the numerical method MOM-GEC. It is carried out through the determination of the number of functions test and the number of modes m and n (here supposed to be equal).

The number of modes 𝑁 𝑏 and the test function number 𝑁 𝑡𝑒𝑠𝑡 are fixed in such a way that the boundary conditions of the electromagnetic state of the antenna is verified. Once these parameters are found, they will be used in the rest of the work to extract the characteristics of the antenna.

The exact solution is provided only at the convergent state of the structure's parameter that we can consider it as steady state. The convergence of the input impedance and the antenna current is established for 𝑁 𝑡𝑒𝑠𝑡 =11 and 𝑁 𝑏 =300 as mentioned respectively in The simulated reflection coefficient of the proposed antenna has two resonance frequencies as illustrated in Figure 4.9(a); one appears at 𝑓 𝑟 1 =8.7 GHz and the other appears at 𝑓 𝑟 2 = 9.6 GHz, so the bandwidth is centered around 𝑓 0 = 9.15 GHz.

Boundary conditions verification

Antenna's characteristics

The frequency study on the phase of the reflection coefficient shows there are two modes whose resonance frequencies are 𝑓 𝑟 1 and 𝑓 𝑟 2 where the phase of the reflection coefficient is zero (see Figure 4.9(b). Note that each resonance is determined where the phase of the reflection coefficient is zero.

In order to identify each mode, a study was carried out on the iris alone and then on the total structure. As presented in Figure 4.10, the iris alone presents a first resonance at 9.6 GHz. By adding the dielectric and the patch above the iris, a second resonance frequency at 8.7 GHz appears at a frequency lower than 𝑓 0 . Due to the merging of the modes relating to the iris and the patch, a wideband radiation characteristic with dual-resonance is obtained. In order to show the calculation efficiency of the proposed method, execution time and storage memory required by MOM-GEC and HFSS simulator have been reported in Table 1. It has been found that MOM-GEC model is more efficient in terms of memory requirements and approximately four times faster than HFSS simulator software.

Antenna fabrication and comparison with various X-band antennas

The prototype of the fabricated iris-fed patch antenna as shown in A slight difference between the simulated and measured results is marked especially between 7GHz and 8GHz. This may be due to the manufacturing tolerance and the effect of the connector which was not taken into account in the design with HFSS and when modelling with the numerical method. Integrating the connector into MOM-GEC modelling and in the HFSS design could be a solution to further minimize this difference.

Comparison with literature

Performances characteristics comparison between the proposed antenna and the state-of-art works operating in the X-band is listed in Table 2.

The X band ranges from 8 GHz to 12 GHz, so an antenna system with a bandwidth covering the entire X band possesses a fractional bandwidth of 40%.

Without the addition of superstrate cover, the antenna bandwidth is equal to 5.8% and 11.1%

respectively in [14] and [16]. Hence, the interest of adding superstrate cover as a bandwidth expansion technique. Indeed, the superstrate cover can be either dielectric, metallic, metamaterial, frequency selective surface. In [3], a stacked microstrip patch antenna has been presented. The addition of parasitic patch above foam compared to the main patch increased its bandwidth from 7.54% to 16.2%.

The proposed antenna has smaller volume than [2][3][14][15][16]. Moreover, it has the highest impedance bandwidth and the simplest design fabrication. Due to its high bandwidth, its simple fabrication and its compactness, this antenna presents a promising candidate for X-band satellite communication, radar application, microwave power transmission and wireless communication.

The state-of-the-art works mentioned in Table 2 have used simulation tools such as HFSS and CST which are essentially based on methods requiring a 3D mesh. This work is characterized by studying the antenna structure based on an analysis method which makes it possible to convert a 3D investigation problem into a 2D investigation problem and subsequently to minimize the execution time and the memory resources required.

Conclusion

In this chapter, the modelling of dual-resonant antenna with closely spaced resonance frequencies for bandwidth enhancement has been investigated for X-band applications. Equivalent circuits (GEC) are hybridized with the method of Moments in an attempt to transpose field problems into equivalent circuit problems that are generally easier to deal with, thus, a reduction in both memory requirements and computation time is achieved. The problem thus formulated; its numerical resolution is done by applying the method of Galerkin known by the simplicity of its formulation.

This procedure is in fact general and it can be applied to study several electromagnetic problems, namely: the uniaxial discontinuity in waveguides, the multiaxial diffraction discontinuities in free space, the antenna radiation, the circuits comprising localized sources, etc. Moreover, the proposed hybrid method can be extended to become a multi-scale method. The techniques for solving each electromagnetic problem essentially differ from each other in the development of the operators involved. Admittedly, the theoretical basis and the approach to be followed remain unchanged. This is due to the fact that the methods used are in all cases integral methods.

Analysis of Periodic Iris-Fed Microstrip Patch antenna

Introduction

Modeling the propagation of electromagnetic fields from a periodic antenna array requires rigorous modal analysis methods. Moreover, in periodic antenna arrays, the interaction between the radiating elements leads to the existence of intercellular couplings which change the radio electric characteristics of the antenna. The design of antenna arrays therefore requires prior knowledge of the mutual coupling between the different radiating elements. There are several approaches to modeling periodic antenna arrays namely: "Full-wave" approaches and modular approaches.

Classic full-wave approaches such as the Method of Moments and the Method of Finite Differences are precise, efficient and reliable methods. However, they are not suitable for large array. It is therefore necessary to focus on more developed full-wave approaches such as the domain decomposition method (DDM) [1] and the multilevel fast multipole method (MLFMM) [2]. These methods are dedicated to analyzing large periodic structures. Admittedly, they require expertise in the use of simulation software by conducting an advanced convergence study on the cases studied.

The modular approaches consist in individually characterizing one or a few cells of the antenna array to deduce afterward the total radiation of the array. Several modular approaches have been proposed for modeling the elementary cell, namely: the isolated cell approach, the surrounded cell approach and the Scale Changing Technique (SCT) method [3]. The isolated cell approach neglects the mutual couplings between the lattice elements so it does not seem to be the interesting approach for the study of the behavior of large periodic lattices. The surrounded cell approach takes into account the real couplings between network elements. The disadvantage of this method is that the number of adjacent cells to consider depends on the nature of the coupling and if it is a high number, it results in a very long calculation time. The (SCT) presents an interesting alternative but the implementation is complex. Most of these methods are not easy to apply on antenna arrays with a large number of radiating elements. Moreover, the infinity with the periodic character of the array makes the problem more complicated.

This work tends to overcome the classic full-wave approaches' limitations by using a rigorous full-wave numerical method for the calculation of the mutual coupling between elements in a unidimensional periodic iris-fed patch antenna array. The MOM-GEC method, previously applied for modelling a single iris-fed microstrip patch antenna, is extended to become a multi-scale method in order to study the electromagnetic aspect of an iris-fed microstrip patch antenna array while taking into account the coupling between the array's elements. Therefore, a solution lies in the use of modal analysis, based on Floquet theorem, in an infinite and periodic array while limiting to the effective calculation of the physical quantity, field or current, over a single period by introducing the appropriate boundary conditions.

This chapter is divided as follows. In section 2, a survey on Floquet theory is introduced. In section 3, the methodology problem statement is detailed. In section 4, the mathematical formulation based the hybridized MOM-GEC combined to the unit-cell spectral decomposition approach is applied to a periodic five-element iris-fed antenna array. The numerical resolution is released along with the results in section 5.

Survey on Floquet theory for antenna arrays

This section includes the Floquet approach definition and the properties of periodic structures.

Floquet approach

Floquet's approach, also called infinite periodic approach, says that all the cells of the periodic network must be identical and reproduce to infinity according to the directions of periodicity of the antenna [4][5][6][7]. The excitations which are applied to them must also be of identical intensities and can undergo a constant phase shift between two adjacent cells according to one or the other of the two periodic directions. As a result, the spatial electromagnetic calculation of a periodic structure of infinite dimensions is reduced to a spectral calculation carried out on a single reference cell by applying periodicity conditions to it. These boundary conditions make it possible to obtain the behavior of the latter when it is surrounded by a uniform and infinite environment.

Properties of periodic structures:

This section presents the Network definition parameters as well as the radiation directions.

Network definition parameters

] = [ 1 0 0 1 ] (5.2a) [D] t . [B] = [ 1 0 0 1 ] (5.2b)
Which is equivalent to:

[D t ] -1 = [B] ( 5

Radiation directions

The weightings carried out by the excitations applied to the cells are restricted to phase gradient, they are written:

a m = a 0 e -j(rφ x + sφ y ) (5.4)

Noting that 𝜑 𝑥 𝑎𝑛𝑑 𝜑 𝑦 are the phase shifts applied between two adjacent cells respectively along the axes (ox) and (oy).

The phase shift couples {φ x , φ y } applied to the cell excitations are linked to the direction of propagation ϕ in the azimuthal plane by the relationship:

( k 0 sinθcosϕ k 0 sinθsinϕ ) = -[B] ( φ x φ y ) (5.13)
Noting that:

[B] = ( 1 d x 1 d y ) (5.14)
We obtain:

K ⃗ ⃗⃗ T = - φ x d x u x ⃗⃗⃗⃗⃗ + (-

Unit-cell spectral decomposition approach

The physical reality of the applications used cannot include an infinity of patterns. In the literature, various techniques have adopted approaches to study a network with a finite number of patterns, namely the approaches of: Roederer, Ishimaru, Skrivervik and Mosig, Parfitt, etc.

It is proposed in this chapter to apply the method of spectral modal decomposition to model a finite periodic antenna array with odd number of elements [8][9] as illustrated in Figure . 5.3. To determine the spatial electric field on any element of coordinates rd x from the origin in an infinite antenna array with a phase shift of α between consecutive elements, it suffices to apply the following relations (5.16) [10][11][12]:

, We note that E t,α(0) (y) is the spectral electric field of the reference central cell. The response of the method still provides us J t,α(0) (y) with the spectral current on the unit central cell.

All the elements are included in one main period of the guide with y L which delimits the overall structure. Thus, the discretization of spectral modes can be obtained by relying on a correspondence relationship (5.17) between the phase domain and the space of the patterns' positions in a finite lattice to ensure the discretization of the mode α [13-14]:

2 y p L k   →   →  (5.17) 2 k y k L   = with 1 22 yy NN k -   - (5.18)
The total electric field in the spatial domain on any element are calculated by equations (5.19):

Problem formulation

MOM-GEC hybridized with unit-cell spectral decomposition approach

The MOM-GEC method, previously applied for modelling a single iris-fed microstrip patch antenna in Chapter 4, is extended to become a multi-scale method in order to study the electromagnetic aspect of an iris-fed microstrip patch antenna array while taking into account the coupling between the array's elements. Therefore, a solution lies in the use of modal analysis, based on Floquet theorem, in an infinite and periodic array while limiting to the effective calculation of the physical quantity, field or current, over a single period by introducing the appropriate boundary conditions.

As the physical reality of the applications used cannot include an infinity of patterns, it is proposed to apply the method of spectral modal decomposition to model a finite periodic antenna array with odd number of elements The flowchart of MOM-GEC hybridized with unit-cell spectral decomposition approach is illustrated in Figure 5.5.

GEC of the unit central cell

The unit cell is composed by an iris excited by the fundamental 𝑇𝐸 10 mode and by a microstrip patch antenna excited by the TEM mode. Each virtual waveguide is defined by electrical periodic walls (EPEP) and is associated with a modal base function 𝑓 𝑚𝑛,𝛼 where m and n are integers related to the mode (see Figure 5

    =  (5.27)
In general, the admittance is written in the form:

, ( 1, 2,11,12, , ) , , ( 1, 2,11,12, , ) ,
Y f y f          = = = =    (5.28)
Noting that: 𝑌 𝑒𝑣1,𝛼 ̂ is the admittance of the modal source representing the evanescent modes at the level of discontinuity 𝐷 1 of the exciter waveguide which is associated with the modal base function (𝑓 𝑚𝑛,𝛼 ) (𝑚,𝑛)={1,..,𝑁 𝑏 2 } .

𝑌 𝑒𝑣2,𝛼 ̂ is the admittance of the modal source representing the evanescent modes at the level of discontinuity 𝐷 2 of the exciter waveguide which is associated with the modal base function

(𝑓 ′ 𝑚𝑛,𝛼 ) (𝑚,𝑛)={1,..,𝑁 𝑏 2 } .
𝑌 𝑡ℎ,𝛼 ̂ is unit cell's admittance of the dielectric layer.

𝑌 𝑠,𝛼 ̂ is unit cell's admittance from the vacuum brought to the surface of the dielectric D 3 .

Mathematical formulation

The same calculation, performed throughout Chapter 4, is applied to extract the features of the unit referenced cell in Chapter 5. The only difference consists in the expressions of the modal basis of the waveguide of type EPEP (Electric Periodic Electric Periodic) whose expressions are defined in Appendix D. To any parameter corresponding to the unit reference cell is added the index 𝛼 𝑘 with k=0. For example, the reflexion coefficient of unit reference cell is expressed by: , 0, , 0,

k k k k k in in ZZ S ZZ    - = + (5.29)
Then, the principle of the unit-cell spectral decomposition approach, previously defined in Section 5.3, is applied to extract the electric field and the current distribution in the spatial domain on any element of the array, as well as mutual impedance and the scattering parameter between different elements.

Results and discussion

This section presents several sub-sections, namely: a validation of the MOM-GEC method applied to the central unit cell, a validation of the hybrid approach applied to a network of five elements, a calculation of the mutual coupling between the different elements of the antenna array and the the CPU time and the memory storage required.

Validation of MOM-GEC formulation

The current distribution and the electric field of the unit reference cell have been illustrated

respectively in Figure 5.7 and Figure 5.8; indeed, they check well the boundary conditions and this is regarded as validation of the numerical method MOM-GEC applied on the reference cell.

Validation of MOM-GEC-AS formulation with unit-cell spectral decomposition approach

The current distribution and the electric field of the antenna array have been illustrated respectively in Figure 5.9 and Figure 5.10; indeed, they check well the boundary conditions and this is regarded as validation of the numerical method MOM-GEC.

Mutual coupling calculation

Table 5.1 presents the mutual coupling parameter calculation between array's elements based on the MOM-GEC formulation hybridized with unit-cell spectral decomposition approach. Table 5.2 presents the mutual coupling parameter calculation between array's elements based HFSS simulator. The results are extracted at the resonant frequency fr=8.7GHz. In fact, they have a slight difference which does not contradict the good agreement. From the predicted mutual coupling values shown in Table 1 and Table 2, it can be noticed that the antennas are strongly decoupled which confirms that the surface waves are not excited. It's not then needed to use techniques for minimizing the coupling effect. In fact, the results validate the numerical calculation since the antennas are spaced by a distance of 𝜆 2 .

CPU time and memory storage

This section presents an evaluation of the formulation by spectral decomposition compared to the The mathematical complexity of the electromagnetic problem depends on the number of unknowns. Therefore, the CPU time and the memory storage required by a computer to solve a problem is expressed as a function of unknown numbers of the problem.

The Ansys HFSS calculator is essentially based on the finite element method, thus requiring a 3D mesh. When calculating the mutual coupling parameters, the HFSS simulator requires a matrix inversion (P × P), knowing that P is the number of test functions. In fact, the storage memory and the number of operations of the conventional HFSS simulator depend on P as [15]:

Storage ≈ O(𝑃 2 )
(5.30)

Operations number ≈ O(𝑃 3 ) (5.31) Indeed, the total time required by the spatial formulation 𝑇 𝐻𝐹𝑆𝑆 to calculate the impedance matrix is expressed as follows [16]:

𝑇 𝐻𝐹𝑆𝑆 = (N × P) 2 𝑇 𝑚 +𝑇 𝑠 (5.32)

The total time required using the spectral formulation 𝑇 𝑠𝑝𝑒𝑐𝑡 is expressed as follows:

𝑇 𝑠𝑝𝑒𝑐𝑡 = 𝑁. 𝑃 2 . 𝑇 𝑚 +𝑇 𝑠 (5.33)

Noting that: 𝑇 𝑠 is the necessary time required by a computer for calculating the matrix filling.

𝑁 2 : Elements of matrix resulted of N iterations.

𝑇 𝑚 is the evaluation time of the multiplication operation required by a computer.

When the number of elements is limited, the difference between the two techniques in terms of computation time and memory space is not significant. Beyond seven elements, the difference between the two techniques becomes remarkable.

By application of the HFSS simulator, the calculation time as well as the required memory resources increase exponentially according to the number of network elements. This is explained by the fact that the number of unknowns and the number of operations of the problem is amplified when the number of network elements increases.

By applying the hybridized MOM-GEC formulation with spectral decomposition approach, the 3D investigation problem is converted to a 2D investigation problem. As a result, the number of operations is reduced and, moreover, the calculation time and the memory resources are reduced considerably compared to those obtained by applying a spatial formulation.

Conclusion

This chapter presents a Floquet modal analysis for the analysis of the electromagnetic one-dimensional array of finite size antennas. According to Floquet theory, every cell in the network is similar and can replicate itself indefinitely depending on the antenna's periodicity requirements. Although the cells' phase shifts differ linearly from one cell to the next, their amplitudes are equally stimulated. Therefore, the spatial electromagnetic calculation of the infinite antenna array reduces to a spectral electromagnetic calculation applied to a single unit cell.

A simple transformation based on the superposition theorem (in other cases cited Fourier Transformation) is required to go from the study of an infinite array of antennas to the study of a finite array of antennas. Several advantages are presented for modeling periodic circuits based on Floquet theory, for example:

• Reduction of electromagnetic calculation: Instead of studying the overall structure, it is possible to reduce the electromagnetic calculation to a single cell, which makes it possible to reduce the wave phenomena associated with the global structure on a unit cell (central cell).

• The reduction of computation time and memory spacing for a large structure of largely finite size.

This study remains valid with great success for studying various periodic or quasi-periodic configurations. After studying the one-dimensional case, it is quite easy to study a case with several directions such as the two-dimensional case and in particular the configuration of a two-dimensional antenna array of finite size.

Conclusion and Perspectives

The microruban patch antenna is the most promising solution for many applications, ranging from biomedical diagnosis to wireless communications because of being compact, low cost, conformal and suitable to integrate with Radio Frequency (RF) devices. In spite of these advantages, the main problems of microstrip antennas are narrow bandwidth and low gain which restrict them from many wireless communication systems. In this context, this thesis presents the electromagnetic modeling of iris-fed microstrip patch antenna with bandwidth enhancement under operation of dual-resonance for X-band applications.

Indeed, this work has several important contributions:

First, it presents an antenna prototype with bandwidth enhancement under operation of dualresonance. Prototype antenna is designed and implemented as proof of concept. Measurement demonstrates the single band with dual resonances type (8.7GHz and 9.6GHz) and the radiation bandwidth of (8GHz-10.6GHz) which are in decent agreement with both HFSS and MOM-GEC approach.). In order to show the impact of the iris feed technique on microstrip patch antenna's bandwidth, a comparison with some state-of-the-art works proposing various bandwidth enhancement techniques in the X-band is made. The proposed prototype, with a size of 1.76𝜆 0 ×1.65𝜆 0 ×1.29𝜆 0 , is more compact than several designs. It has a fractional bandwidth of 27.9%, thus, it presents the prototype with the widest bandwidth with the simplest design fabrication.

Second, it presents an accurate and reliable electromagnetic model of the proposed antenna based on the hybrid MOM-GEC. Equivalent circuits (GEC) are introduced in the development of the formulation of integral methods in an attempt to transpose field problems into equivalent circuit problems that are generally easier to deal with, thus, a reduction in both memory requirements and computation time is achieved. The problem thus formulated; its numerical resolution is done by applying the method of Galerkin known by the simplicity of its formulation.

This procedure is in fact general and it can be applied to study several electromagnetic problems, namely: the uniaxial discontinuity in waveguides, the multiaxial diffraction discontinuities in free space, the antenna radiation, the circuits comprising localized sources, etc.

Third, the proposed hybrid method is extended to become a multi-scale method. The hybridization of the MOM-GEC method and Floquet theory aims to analyze a periodic iris-fed microstrip patch antenna array. As a result, the spatial electromagnetic calculation of a periodic structure of infinite dimensions is reduced to a spectral calculation carried out on a single reference cell by applying periodicity conditions to it. Therefore, rigorous electromagnetic calculation while minimizing the execution time and the required memory resources is carried out.

Based on the conclusion drawn above, future work can be done in the following areas:

First, it should be noted that this work has focused on the design of a broadband antenna with a simple technique but it was not intended. Thus, the proposed solution consists in adding a superstrate below the antenna in order to be able to improve the gain as well as the bandwidth. Second, for X-band applications, the antenna size must be miniature. Therefore, future research should focus on further reducing the antenna size by discovering new methods.

Figure 1 . 3 .

 13 Figure 1.3. The picture of shuttle imaging radar antenna consisting of an array of microstrip patch antennas, (a) in the laboratory and (b) during flight (Jet Propulsion Laboratory, NASA). ......................................

Figure 1 . 4 .

 14 Figure 1.4. A 896-element microstrip antenna array (Ball Aerospace Systems Division, Boulder, Colorado). .....................................................................................................................................................................................................

Figure 1 . 5 .

 15 Figure 1.5. An array of microstrip antennas is made conformal to a wing shape surface (Air Force Research Lab./Antenna Technology Branch, Hanscom AFB, USA). ...................................................................................

Figure 1 . 6 .

 16 Figure 1.6. A patch antenna in a cellular phone. ....................................................................................................................

Figure 1 . 7 .

 17 Figure 1.7. A base station antenna array consisting of patches. .....................................................................................

Figure 1 . 8 .

 18 Figure 1.8. Patch antenna arrays mounted on the walls of a building. ........................................................................

Figure 1 . 9 .

 19 Figure 1.9. The geometry of a direct microstrip feed microstrip patch antenna (a) top view and (b) side view. .............................................................................................................................................................................................................

Figure 1 . 10 .

 110 Figure 1.10. The geometry of a recessed microstrip line feed microstrip patch antenna (a) top view and (b) side view. ............................................................................................................................................................................................

Figure 1 . 11

 111 Figure 1.11 The geometry of coaxial probe feed microstrip patch antenna (a) top view and (b) side view. ........................................................................................................................................................................................................................

Figure 1 . 12 .

 112 Figure 1.12. The geometry of a proximity coupled microstrip feed microstrip patch antenna (a) top view and (b) side view. ...................................................................................................................................................................................

Figure 1 . 13 .

 113 Figure 1.13. The geometry of a patch antenna fed by an adjacent microstrip line (a) top view and (b) side view. .............................................................................................................................................................................................................

Figure 1 . 14 .

 114 Figure 1.14. The geometry of an aperture coupled feed microstrip patch antenna (a) top view, (b) side view (c) pictorial view .........................................................................................................................................................................

Figure 1 . 15 .

 115 Figure 1.15. Iris-fed microstrip antenna (a) top view (b) bottom view. ....................................................................

Figure 2 . 1 .

 21 Figure 2.1 . Bandwidth trends of a direct fed single-layer microstrip patch relative to substrate permittivity and thickness. ................................................................................................................................................................

Figure 2 . 2 .

 22 Figure 2.2. Partial substrate removal surrounding the patch. .........................................................................................

Figure 2 . 3 .

 23 Figure 2.3. Different shapes used for the coupling slot: (a). thin rectangular, (b). longer rectangular, (c). wider rectangular (d). 'H'-shaped, (e). Bowtie-shaped and (f). hourglass-shaped. .................................................

Figure 2 . 4 .

 24 Figure 2.4. (a) C-shaped slotted antenna [9] (b) Circular patch antenna with horizontal slots. ......................

Figure 2 . 5 .

 25 Figure 2.5 . Dual-slot coupled multiple dense dielectric (DD) patch antenna [11]. ...............................................

Figure 2 . 6 .

 26 Figure 2.6. Multi-resonator patch antenna geometry [12]. ...............................................................................................

Figure 2 . 7 .Figure 2 . 8 .

 2728 Figure 2.7. Circular patch antenna with metallic rings . .....................................................................................................

Figure 2 . 9 .

 29 Figure 2.9. Stacked patch antenna having excitation patch with inset and parasitic patch. ..............................

Figure 2 . 10 .

 210 Figure 2.10. Patch antenna with and without the EBG substrate...................................................................................

Figure 2 . 11 .

 211 Figure 2.11. (a) Defected ground plane (b) Different shapes of defects. ...................................................................

Figure 2 . 12 .

 212 Figure 2.12. Geometry of double hexagonal DGS. ..................................................................................................................

Figure 2 . 13 .

 213 Figure 2.13. Fabricated antenna with partially reflective metasurface structure. .................................................

Figure 2 . 14 .

 214 Figure 2.14. Microstrip patches arranged in log-periodic array formation. (a) Indirect feeding, (b) direct feeding and (c) VSWR plot of a sample antenna designed in [26]. ..................................................................................

Figure 2 . 15 .

 215 Figure 2.15. (a) Fractal-shaped slotted patch with a metasurface superstrate (b) Fabricated prototype [26]. ..............................................................................................................................................................................................................

Figure 2 . 16 .

 216 Figure 2.16. Copper ring superstrate above the rectangular patch. .............................................................................

Figure 3 . 1 .

 31 Figure 3.1. Symbolic representation of the discontinuity surface (S) formed by two subdomains (𝑺𝟎𝟐: metallic domain) and (𝑺𝟎𝟐: dielectric domain). ......................................................................................................................

Figure 3 . 2 .

 32 Figure 3.2. Flowchart of MOM-GEC process. ............................................................................................................................

Figure 3 . 3 .

 33 Figure 3.3. Symbolic representation of virtual sources: (a) virtual field source, (b) virtual current source. ........................................................................................................................................................................................................................

Figure 3 . 4 .

 34 Figure 3.4. Symbolic representation of virtual sources in their domain of definition and their complementary domain. .....................................................................................................................................................................

Figure 3 . 5 .

 35 Figure 3.5. Symbolic representation of a localized voltage source. ...............................................................................

Figure 3 . 6 .

 36 Figure 3.6. Symbolic representation of the modal source of field in a guide. ...........................................................

Figure 3 . 7 .

 37 Figure 3.7. Equivalence between (a) the half-space (S+C) and (b) a dipole circuit................................................

Figure 3 . 8 .

 38 Figure 3.8. Equivalence between (a) the space (S+𝐶1+𝐶2) and (b) a quadrupole circuit. ..................................

Figure 3 . 9 .

 39 Figure 3.9. (a) Continuity relation of the tangential electric field, (b) Symbolic representation at the level of the electric circuit. ............................................................................................................................................................................

Figure 3 . 10 .

 310 Figure 3.10. (a) Relation of discontinuity of the tangential magnetic field, (b) Symbolic representation by the law of knots for the current. ......................................................................................................................................................

Figure 3 . 11 .

 311 Figure 3.11. Equivalent electrical circuit of a microstrip structure printed on a multilayer substrate. .......

Figure 4 . 1 .

 41 Figure 4.1. Antenna design flowchart .........................................................................................................................................

Figure 4 . 2 .

 42 Figure 4.2. (a) Iris-fed patch antenna (b) bottom view (c) top view (d) Side view: a=22.86, b=10.16, h=41.5, a′′=56.5, b′′=53, a′ = b′ =32, Wp =0.93, Lp=8, Ws =8.28, Ls =10.62, Wr=3.675, Lr=1.31; lf=6.12;Wf=0.94; t1(dielectric thickness)= 1.6, t (ground and patch thickness)=0.025, (all dimensions in millimeter),Ԑr(FR4) =4.4.. ................................................................................................................................................................

Figure 4 . 3 .

 43 Figure 4.3. (a) Microstrip patch shielded in rectangular waveguide defined by perfect electrical walls (EEEE) on all faces (b) Iris located between two virtual waveguides at (z=0), each of which is characterized by perfect electrical walls (EEEE) on all faces. ...........................................................................................

Figure 4 . 4 .

 44 Figure 4.4. GEC of the antenna using the MOM-GEC technique. ......................................................................................

Figure 4 . 5 .

 45 Figure 4.5. GEC of the antenna using the MOM-GEC technique using (V, I) physical scales. ..............................

Figure 4 . 6 .

 46 Figure 4.6. Input impedance a function of different number of 𝑁𝑏 and 𝑁𝑡𝑒𝑠𝑡. ......................................................

Figure 4 . 7 .Figure 4 . 8 .

 4748 Figure 4.7. Current distribution as a function of different number of 𝑁𝑏 and 𝑁𝑡𝑒𝑠𝑡. ..........................................

Figure 4 . 9 .

 49 Figure 4.9. (a) Magnitude in dB and (b) the angle in degree of reflection coefficient of the antenna compared to HFSS. ................................................................................................................................................................................

Figure 4 . 10 .

 410 Figure 4.10. Simulated reflection coefficients of the iris alone and the iris-fed antenna as a function of frequency. ..................................................................................................................................................................................................

Figure 4 . 11 .

 411 Figure 4.11. Photograph of the fabricated iris-fed microstrip patch antenna (b) Photograph of measurement set-up: Reflection coefficient measurement. ...............................................................................................

Figure 4 . 12 .

 412 Figure 4.12. (a) Magnitude in dB and (b) the angle in degree of simulated and measured reflection coefficient of the antenna. ..................................................................................................................................................................

Figure 5 . 1 .

 51 Figure 5.1. Positions of the nodes of the periodic array marked by basis vectors d1, d2 and vectors of the reciprocal basis b1, b2. ........................................................................................................................................................................

Figure 5 . 2 .

 52 Figure 5.2. Directions of radiation{θ,ϕ} in (oxy). ..................................................................................................................

Figure 5 . 3 .

 53 Figure 5.3. (a) Infinite unidimensional periodic iris-fed microstrip patch antenna array presentation with Floquet theory (b) unit cell: a=22.86, b=10.16, h=41.5, a′ =32, b′ =31, Wp=0.93, Lp=8, Ws=8.28, Ls=10.62, t1(substrate thickness) = 1.5, t (patch thickness =ground thickness) =0.025, (all dimensions in millimeter),Ԑr(FR4) =4. .....................................................................................................................................................................

Figure 5 . 4 .

 54 Figure 5.4. 1-D antenna array along (Oy) composed by five identical cells, 𝑑𝑦 periodically reproduced along (Oy) direction and positioned according k=-2, -1,0,1,2. .......................................................................................

Figure 5 . 5 .

 55 Figure 5.5. Flowchart of MOM-GEC hybridized with unit-cell spectral decomposition approach. .................

Figure 5 . 6 .

 56 Figure 5.6. (a) Unit cell's iris located between two virtual waveguides at (z=0), each of which is characterized by perfect electrical walls (EEEE) on all faces (b) Unit cell's patch shielded in rectangular waveguide defined by perfect electrical walls (EEEE) on all faces (c) GEC of a unit cell. .....................................

Figure 5 . 7 .

 57 Figure 5.7. Normalized current distribution of the unit reference cell obtained at convergence for 𝑁𝑡𝑒𝑠𝑡=11 and 𝑁𝑏=300. .......................................................................................................................................................................

Figure 5 . 8 .

 58 Figure 5.8. Normalized electric field of the unit reference cell obtained at convergence for 𝑵𝒕𝒆𝒔𝒕=11 and 𝑵𝒃=300. .....................................................................................................................................................................................................

Figure 5 . 9 .

 59 Figure 5.9. Normalized current distribution of 1-D antenna array composed by five identical cells, 𝑑𝑦 periodically reproduced along (Oy) direction and positioned according k=-2, -1,0,1,2 for 𝐿𝑦=135mm....

Figure 5 . 10 .

 510 Figure 5.10. Normalized electric field of 1-D antenna array composed by five identical cells, 𝑑𝑦 periodically reproduced along (Oy) direction and positioned according k=-2, -1,0,1,2 for 𝐿𝑦=135mm....

Figure 5 . 11 .

 511 Figure 5.11. (a) CPU time required (b) Memory storage by using MOM-GEC code and HFSS simulator in function of antenna's elements for mutual coupling calculation. ....................................................................................

1 .Table 4 . 1 .Table 4 . 2 .Table 5 . 1 .

 1414251 Some commercial electromagnetic simulation softwares. ........................................................................... Comparison of CPU time between the MOM-GEC method and the HFSS simulator (Intel Core i7 3-55GHz, 16 GB RAM). ......................................................................................................................................................................... Performances comparison of the proposed antenna with the state-of-art works operating in the X-band. ........................................................................................................................................................................................................ Mutual coupling parameter calculation between array's elements based on the MOM-GEC formulation hybridized with unit-cell spectral decomposition approach ...................................................................

1. 1 Introduction

 1 An astonishing growth in the area of broadband communication has been opened access in the wide extent research and application of microstrip antenna. The adaptability of microstrip antennas has added a new aspect to it. Microstrip patch antenna (known under the abbreviation of MPA) is one of the latest technologies in antennas and electromagnetic applications. It is widely used now days in the wireless communication system due to its simplicity and compatibility with printed circuit technology. The concept of microstrip antenna was first proposed by Deschamps [1] in 1953. Gutton and Baissinot presented a patent in on the microstrip in 1955. Early microstrip lines and radiators were specialized devices developed in laboratories. No commercially available printed circuit boards with controlled dielectric constants were developed during this period. So, this antenna didn't become practical till 1970s when it was developed further by Robert E. Munson [2-3]. Development during this decade was accelerated by other researchers by the availability of lowloss tangent substrate materials. Others factors for the development include improved photo-lithographic techniques, better theoretical modelling and attractive thermal and mechanical proper-

Figure 1 . 1 .

 11 Figure 1.1. The basic structure of microstrip antenna.

Figure 1 . 2 .

 12 Figure 1.2. Conventional common feeding methods of microstrip antenna.

Figure 1 .

 1 Figure 1.3 shows the picture of a Shuttle imaging radar antenna consisting of an array of microstrip patch antennas, in the laboratory (a) and during flight (b).Figure 1.4 shows a 896 ele-

Figure 1 .

 1 4 shows a 896 element microstrip antenna array for remote sensing applications. In Figure1.5, an array of microstrip patch antennas is made conformal to a wing shape surface.

Figure 1 . 3 .

 13 Figure 1.3. The picture of shuttle imaging radar antenna consisting of an array of microstrip patch antennas, (a) in the laboratory and (b) during flight (Jet Propulsion Laboratory, NASA).

Figure 1 . 4 .

 14 Figure 1.4. A 896-element microstrip antenna array (Ball Aerospace Systems Division, Boulder, Colorado).

Figure 1 . 5 .

 15 Figure 1.5. An array of microstrip antennas is made conformal to a wing shape surface (Air Force Research Lab./Antenna Technology Branch, Hanscom AFB, USA).

Figure 1 . 6 .

 16 Figure 1.6. A patch antenna in a cellular phone.

Figure 1 .

 1 Figure 1.7 shows a base station antenna array consisting of patches. Figure 1.8 shows patch antenna arrays mounted on the walls of a building for satellite television reception.

Figure 1 . 7 .

 17 Figure 1.7. A base station antenna array consisting of patches.

Figure 1 . 8 .

 18 Figure 1.8. Patch antenna arrays mounted on the walls of a building.

Figure 1 . 9 .

 19 Figure 1.9. The geometry of a direct microstrip feed microstrip patch antenna (a) top view and (b) side view.

Figure 1 . 10 .

 110 Figure 1.10. The geometry of a recessed microstrip line feed microstrip patch antenna (a) top view and (b) side view.

Figure 1 . 11

 111 Figure 1.11 The geometry of coaxial probe feed microstrip patch antenna (a) top view and (b) side view.

Figure 1 . 12 .

 112 Figure 1.12. The geometry of a proximity coupled microstrip feed microstrip patch antenna (a) top view and (b) side view.

Figure 1 . 13 .

 113 Figure 1.13. The geometry of a patch antenna fed by an adjacent microstrip line (a) top view and (b) side view.

Figure 1 . 14 .

 114 Figure 1.14. The geometry of an aperture coupled feed microstrip patch antenna (a) top view, (b) side view (c) pictorial view

Figure 1 . 15 .

 115 Figure 1.15. Iris-fed microstrip antenna (a) top view (b) bottom view.

Figure 2 . 1 ,

 21 in which the bandwidth trends of a direct fed single layer microstrip patch relative to substrate permittivity and thickness are shown.

Figure 2 . 1 .

 21 Figure 2.1 . Bandwidth trends of a direct fed single-layer microstrip patch relative to substrate permittivity and thickness [1].

Figure 2 . 2 .

 22 Figure 2.2. Partial substrate removal surrounding the patch [3].

3 .

 3 The fields are coupled to the patch via the open end of stub. By tuning the location of open end of feed line

Figure 2 .

 2 3(a)) by which strong coupling can be given with a simple design. However, stronger coupling can be achieved by making it longer or wider (as shown in Figure2.3(b and c)). The 'H'-shaped and bowtieshaped slots (Figure2.3(d and e)) can provide stronger coupling than the rectangular slot since the fields magnitude in rectangular slot has sinusoidal variation; however, in 'H'-shaped and bowtie-shaped slots, it is quite uniform. By combining the 'H'-shaped and bowtie-shaped slots, i.e., making an hourglass-shaped slot shown in Figure2.3(f), even more uniform field distribution across the slot and consequently stronger coupling can be achieved[2, 4].

Figure 2 . 3 .

 23 Figure 2.3. Different shapes used for the coupling slot: (a). thin rectangular, (b). longer rectangular, (c). wider rectangular (d). 'H'-shaped, (e). Bowtie-shaped and (f). hourglass-shaped [2, 5].

Figure 2 . 4 .

 24 Figure 2.4. (a) C-shaped slotted antenna [9] (b) Circular patch antenna with horizontal slots [10].

Figure 2 . 5 .

 25 Figure 2.5 . Dual-slot coupled multiple dense dielectric (DD) patch antenna [11].

Figure 2 . 6 .

 26 Figure 2.6. Multi-resonator patch antenna geometry [12].

  ing of shorting pins and L-shaped open-end slot. Based on this, pin-loaded annular ring (PLAR) with dual-mode operation is adapted to introduce another resonance. In addition, two closely coupled rectangular patches are added to bring these two resonant frequencies close to each other in muscle. Furthermore, by introducing arc-shaped slots, CP property and improved impedance matching are achieved with size miniaturization. This proposed structure shows a simulated impedance bandwidth of 8% and a wide axial-ratio (AR) bandwidth of 19.1%.

Figure 2 . 7 .

 27 Figure 2.7. Circular patch antenna with metallic rings [13].

2 . 8 .

 28 The lower patch is fed with coaxial cable while the upper patch is excited by electromagnetic coupling. Patches are fabricated on different substrate materials and air gap is introduced between the two materials.

Figure 2 . 8 .

 28 Figure 2.8. Electromagnetically coupled multilayer structure (a) normal geometry (b) inverted geometry.

Figure 2 . 9 .

 29 Figure 2.9. Stacked patch antenna having excitation patch with inset and parasitic patch.

Figure 2 . 10 .

 210 Figure 2.10. Patch antenna with and without the EBG substrate [16].

Figure 2 .

 2 Figure 2.11 depicts a view of a periodic defected ground plane.In fact, defected ground structure and EBG have similar microwave circuit properties to provide a band stop or band pass like filter. A large area is required to construct the periodic pattern EBG implementation. In addition, defining the unit element of EBG is difficult whereas defected ground structures can be one or few etched shapes which do not need for a large implementation area and it can easily define the unit element.

Figure 2 . 11 .

 211 Figure 2.11. (a) Defected ground plane (b) Different shapes of defects.

Figure 2 . 12 .

 212 Figure 2.12. Geometry of double hexagonal DGS [18].

[ 22 ]

 22 presents a partially reflective metasurface (PRMS) structure for dual-polarized dipole antenna with a high gain and wide broadband. Stable radiation patterns are realized by using quadrilateral bottom reflector. The phase bandwidth of PRMS structure reflection presented in this paper matches well with the antenna operating bandwidth. The gain of the dual-polarized antenna is improved to 10.2dBi at 1.8GHz by adjusting the size and height of PRMS structure. Meanwhile, the bandwidth is expanded (61.2%).

Figure 2 . 13 .

 213 Figure 2.13. Fabricated antenna with partially reflective metasurface structure [22].

Figure 2 .

 2 figuration.

Figure 2 . 14 .

 214 Figure 2.14. Microstrip patches arranged in log-periodic array formation. (a) Indirect feeding, (b) direct feeding and (c) VSWR plot of a sample antenna designed in [26].

Figure 2 . 15 .

 215 Figure 2.15. (a) Fractal-shaped slotted patch with a metasurface superstrate (b) Fabricated prototype [26].

Figure 2 . 16 .

 216 Figure 2.16. Copper ring superstrate above the rectangular patch [28].

Figure 3 . 1 .

 31 Figure 3.1. Symbolic representation of the discontinuity surface (S) formed by two subdomains (𝑺 𝟎𝟐 : metallic domain) and (𝑺 𝟎𝟐 : dielectric domain).

Figure 3 . 2 .

 32 Figure 3.2. Flowchart of MOM-GEC process.

Figure 3 . 3 .

 33 Figure 3.3. Symbolic representation of virtual sources: (a) virtual field source, (b) virtual current source.

Figure 3 . 4 .

 34 Figure 3.4. Symbolic representation of virtual sources in their domain of definition and their complementary domain.

Figure 3 . 5 .

 35 Figure 3.5. Symbolic representation of a localized voltage source.

Figure 3 .

 3 Figure 3.6b).

Figure 3 . 6 .

 36 Figure 3.6. Symbolic representation of the modal source of field in a guide.

Figure 3 . 7 .

 37 Figure 3.7. Equivalence between (a) the half-space (S+C) and (b) a dipole circuit.

Figure 3 . 8 .

 38 Figure 3.8. Equivalence between (a) the space (S+𝐶 1 +𝐶 2 ) and (b) a quadrupole circuit.

Figure 3 . 9 .

 39 Figure 3.9. (a) Continuity relation of the tangential electric field, (b) Symbolic representation at the level of the electric circuit.

Figure 3 . 10 .

 310 Figure 3.10. (a) Relation of discontinuity of the tangential magnetic field, (b) Symbolic representation by the law of knots for the current.

Based on ( 3 . 4 )

 34 The current density J represented on the basic functions is expressed as follows:J(x, y) = ∑ [ ∑ (a p ⟨g p , f m,n TE ⟩f m,n TE (x, y))

26 )= 3 .

 263 Each dielectric has a propagation constant expressed in the spectral domain as a function of the geometric quantities and of the different characteristic elements of the middle by the following relation: 10 8 m𝑠 -1 : celerity of light m, n: natural numbers respectively denoting the indices relating to the modes. a: the dimension of the waveguide along the x axis b: the dimension of the waveguide along the y axis 𝜇 0 : Magnetic permittivity of vacuum Ԑ 0  Electrical permittivity of vacuum 𝜀 𝑟𝑖 :  Relative permittivity of the area  Angular pulsation equal to 2𝛑fWhen the layer of the structure is terminated by a short-circuit (presence of a metal wall), the admittance of the modes reduced to the level of the discontinuity interface containing the metal strip is:Y α=TE,TM = y mn ,α=TE,TM coth (γ m,n h) (3.28) ℎ : the layer height And if the layer of the structure is terminated by an open circuit (absence of metal wall), the admittance of the modes brought back to the level of the discontinuity interface containing the metal strip is: Y ,α=TE,TM = y mn ,α=TE,TM tanh (γ m,n h i ) (3.29)

Figure 3 . 11 .

 311 Figure 3.11. Equivalent electrical circuit of a microstrip structure printed on a multilayer substrate.

Y 2 , 3 .

 23 α=TE,TM = y 2,α=TE,TM . [ y 2,α=TE,TM + Y 3,α=TE,TM tanh(γ 2 l 2 ) Y 3,α=TE,TM +y 2,α=TE,TM tanh(γ 2 l 2 ) ] Calculation of the admittance of the modes of the layer brought back to the level of the discontinuity interface between the layer N and N-1 knowing that this last layer is terminated by an open-circuit.

Figure 4 .

 4 Figure 4.1 illustrates the design process, adapted from the concept due to John Huang and DavidPozar[11].

Figure 4 . 1 .

 41 Figure 4.1. Antenna design flowchart

Figure 4 .

 4 Figure 4.3(b).

Figure 4 . 3 .

 43 Figure 4.3. (a) Microstrip patch shielded in rectangular waveguide defined by perfect electrical walls (EEEE) on all faces (b) Iris located between two virtual waveguides at (z=0), each of which is characterized by perfect electrical walls (EEEE) on all faces.

Figure 4 . 4 .

 44 Figure 4.4. GEC of the antenna using the MOM-GEC technique.

1 /

 1 On side(1), the current source 𝐽 01 and the field source 𝐸 01 are expressed respectively according the amplitude 𝐼 01 and 𝑉 01 and according the fundamental mode 𝑓 0 of the modal base function 𝑓 𝑚𝑛 :

Figure 4 . 5 .

 45 Figure 4.5. GEC of the antenna using the MOM-GEC technique using (V, I) physical scales.

Figure 4 . 6 and

 46 Figure 4.7.

Figure 4 . 6 .

 46 Figure 4.6. Input impedance a function of different number of 𝑁 𝑏 and 𝑁 𝑡𝑒𝑠𝑡 .

Figure 4 . 7 .

 47 Figure 4.7. Current distribution as a function of different number of 𝑁 𝑏 and 𝑁 𝑡𝑒𝑠𝑡 .

Figure 4 .

 4 Figure 4.8 illustrates the current and electric field distributions of the antenna which are consistent with electromagnetic field theory. In Figure 4. 8(a), the current density distribution is shown at the metallic patch. In Figure 4.8(b), the normalized electric field peak is at the edge of the patch in the x direction and the minimum values are at the dielectric.

Figure 4 . 8 .

 48 Figure 4.8. Normalized current distribution of the antenna (b) Normalized electric field of antenna obtained at convergence.

Figure 4 .

 4 Figure 4.9 illustrates both the magnitude in dB and the angle in degree of reflection coefficient of the antenna. To confirm the accuracy and the efficiency of the proposed numerical approach, a comparison with HFSS is held. It shows a slight difference between results due to different numerical techniques used.

Figure 4 . 9 .

 49 Figure 4.9. (a) Magnitude in dB and (b) the angle in degree of reflection coefficient of the antenna compared to HFSS.

Fig. 8 .

 8 Fig. 8. (a). Normalized current distribution of the antenna (b) Normalized electric field of antenna obtained at convergence.

Figure 4 . 10 .

 410 Figure 4.10. Simulated reflection coefficients of the iris alone and the iris-fed antenna as a function of frequency.

Figure 4 .

 4 11(a). The antenna is excited by a rectangular waveguide of the Oritel TGN R100/WR90 type with 22.86 mm × 10.16 mm × 41.5 mm dimension, also extended wall dimension is 56.5 mm ×53mm.The results are measured by an Agilent HP 8510C vector network analyzer (VNA) as shown in

Figure 4 .

 4 Figure 4.11(b).

Figure 4 . 11 .

 411 Figure 4.11. Photograph of the fabricated iris-fed microstrip patch antenna (b) Photograph of measurement setup: Reflection coefficient measurement.

Figure 4 .

 4 Figure 4.12 illustrates both the magnitude in dB and the angle in degree of simulated and measured reflection coefficient of the antenna. Measurement demonstrates the single band with dual resonances type (8.7GHz and 9.6GHz) and the radiation bandwidth of (8GHz-10.6GHz).

Figure 4 . 12 .

 412 Figure 4.12. (a) Magnitude in dB and (b) the angle in degree of simulated and measured reflection coefficient of the antenna.

Figure 5 .

 5 Figure 5.1 illustrates the positions of the nodes of the periodic array marked by basis vectors {d 1 ⃗⃗⃗⃗⃗ , d 2 ⃗⃗⃗⃗⃗ } and vectors of the reciprocal basis {b 1 ⃗⃗⃗⃗⃗ , b 2 ⃗⃗⃗⃗⃗ }.

  .2c) The constituent terms of the matrix [B] define the steps of the network. The position of a node (m) of the direct network is identified by a position vector: P ⃗ ⃗⃗ (m) P ⃗ ⃗⃗ (m) = rd 1 ⃗⃗⃗⃗⃗ + sd 2 ⃗⃗⃗⃗⃗ (5.3a) P ⃗ ⃗⃗ (m) = r d x u x ⃗⃗⃗⃗⃗ + s d y u y ⃗⃗⃗⃗⃗ (5.3b) Noting that 𝑑 𝑥 and 𝑑 𝑦 are the inter-cellular periods respectively along the (ox) and (oy) directions.

Figure 5 . 1 .

 51 Figure 5.1. Positions of the nodes of the periodic array marked by basis vectors {d 1 ⃗⃗⃗⃗⃗ , d 2 ⃗⃗⃗⃗⃗ } and vectors of the reciprocal basis {b 1 ⃗⃗⃗⃗⃗ , b 2 ⃗⃗⃗⃗⃗ }.

  T = k 0 sinθ cosϕ u x ⃗⃗⃗⃗⃗⃗ + k 0 sinθ sinϕ u y ⃗⃗⃗⃗⃗⃗ (5.15b)In this case, for a couple of phase shifts {φ 1 , φ 2 }, the elevation angle θ depends on the frequency.

Figure 5 . 2 .

 52 Figure 5.2. Directions of radiation{θ,ϕ} in (oxy).

Figure 5 . 3 .

 53 Figure 5.3. (a) Infinite unidimensional periodic iris-fed microstrip patch antenna array presentation with Floquet theory (b) unit cell: a=22.86, b=10.16, h=41.5, a ′ =32, b ′ =31, W p =0.93, L p =8, W s =8.28, L s =10.62, t 1 (substrate thickness) = 1.5, t (patch thickness =ground thickness) =0.025, (all dimensions in millimeter),Ԑr (FR4) =4.

Figure 5 . 4 .

 54 Figure 5.4. 1-D antenna array along (Oy) composed by five identical cells, 𝑑 𝑦 periodically reproduced along (Oy) direction and positioned according k={-2, -1,0,1,2}.

S

  By applying the principles of FFT, the calculation of the mutual impedance and the scattering parameter between two elements of coordinates id y and jd y in spatial domain are respectively defined in (5.20) and (5.21):  are respectively the diagonal operator of input impedance and the scattering parameter related to spectral mode.

Figure 5 . 5 .

 55 Figure 5.5. Flowchart of MOM-GEC hybridized with unit-cell spectral decomposition approach.

  .6). Begin Data unit cell Extraction of the GEC of the reference central cell Application of Kirchhoff and d'Ohm Laws Galerkin-MOM: extraction of unknown quantities Determination of basic and test numbers Convergence? Results of unit cell in spectral domain fin no yes Unit cell spectral decomposition approach Results in spatial domain

Figure 5 . 6 .

 56 Figure 5.6. (a) Unit cell's iris located between two virtual waveguides at (z=0), each of which is characterized by perfect electrical walls (EEEE) on all faces (b) Unit cell's patch shielded in rectangular waveguide defined by perfect electrical walls (EEEE) on all faces (c) GEC of a unit cell.

Figure 5 . 7 .

 57 Figure 5.7. Normalized current distribution of the unit reference cell obtained at convergence for 𝑁 𝑡𝑒𝑠𝑡 =11 and 𝑁 𝑏 =300.

Figure 5 . 8 .

 58 Figure 5.8. Normalized electric field of the unit reference cell obtained at convergence for 𝑵 𝒕𝒆𝒔𝒕 =11 and 𝑵 𝒃 =300.

Figure 5 . 9 .

 59 Figure 5.9. Normalized current distribution of 1-D antenna array composed by five identical cells, 𝑑 𝑦 periodically reproduced along (Oy) direction and positioned according k={-2, -1,0,1,2} for 𝐿 𝑦 =135mm.

Figure 5 . 10 .

 510 Figure 5.10. Normalized electric field of 1-D antenna array composed by five identical cells, 𝑑 𝑦 periodically reproduced along (Oy) direction and positioned according k={-2, -1,0,1,2} for 𝐿 𝑦 =135mm.

  spatial formulation by the HFSS simulator in terms of execution time and memory space. The computation time and the memory storage required for calculation the mutual coupling parameters in function of antenna array's elements are illustrated respectively in Figure 5.11(a) and Figure 5.11(b).

Figure 5 . 11 .

 511 Figure 5.11. (a) CPU time required (b) Memory storage by using MOM-GEC code and HFSS simulator in function of antenna's elements for mutual coupling calculation.

Table 5 . 2 .

 52 Mutual coupling parameter calculation between array's elements based HFSS simulator ..........

Table 3 . 1 .

 31 Some commercial electromagnetic simulation softwares.On the numerical level, the classical methods (FEM, FDTD, IE, MoM, TLM, etc.) are based on a uniform or adaptive spatial discretization of the domain of the entire structure.

	Software name	Theoretical model	Company
	PCAAD	Cavity model	Antenn design associates
	Ensemble	Moment method	Ansoft
	IE3D	Moment method	Zland/Mentor Graphic
	Momentum	Moment method	Agilent
	EM	Moment method	Sonnet
	PiCasso	Moment method	EMAG
	FEKO	Moment method	EMSS
	Micropatch	Segmentation	Microstrip design, Inc
	Microwave studio	FDTD	CST
	Fidelity	FDTD	Zland/Mentor Graphic
	HFSS	Finite element	Ansoft

  If the base (f mn ) is orthonormal 〈f mn , f m ′ n ′ 〉 = δ mn m ′ n ′ , the impedance operator is defined by: Z ̂= ∑ |f mn 〉z mn 〈f mn |If the base (f mn ) isn't orthonormal 〈f mn , f m ′ n ′ 〉 ⧧ δ mn m ′ n ′ , the impedance operator is defined by:

		z mn	i mn |f mn	(3.19)
	mn
	𝑖 𝑚𝑛 : The current weightings that correspond to the modes of the guide
	𝑧 𝑚𝑛 : Modal impedances
	𝑓 𝑚𝑛 : Decomposition basis
	Thus, we can deduce the modal decomposition of the current density |J〉 from:
	|J〉 = ∑ i mn |f mn	(3.20)
	mn
	Z ̂= ∑ mn	|f mn 〉z mn 〈f mn | |f mn 〉〈f mn |	(3.22)
	𝛿 𝑚𝑛 𝑚

mn

(3.21) ′ 𝑛 ′ is the the Kronecker Delta and 𝑧 𝑚𝑛 is the impedance mode.

  α=TE,TM . [ y 2,α=TE,TM + Y 3,α=TE,TM coth(γ 2 l 2 ) Y 3,α=TE,TM +y 2,α=TE,TM coth(γ 2 l 2 ) ]

Table 4 . 1 .

 41 Comparison of Execution time between the MOM-GEC method and the HFSS simulator (Intel Core i7 with a processing capacity of 2.7 GHz and 8 GB RAM).

		Execution time (in seconds)	Memory requirements (GB)
	HFSS	Proposed method	HFSS	Proposed method
	1276	340	3.04	0.18

Table 4 . 2 .

 42 Performances comparison of the proposed antenna with the state-of-art works operating in the X-band.

	Ant	Bandwidth enhance-ment technique	Dimension (W×L×T in 𝞴 𝟎 𝟑 )	Band type	-10 dB fractional bandwidth (GHz; %)	Design Complexity	Modelling technique
	[14]	Metallic superstrate	2.54×3.6×0.92	Single*	7.7-9.7; 23%	(+)	HFSS
	[15]	Dielectric superstrate	2.4×2.4×1.2	Single	8.75-11.25; 25%	(+)	CST
	[3]	Parasitic patch above foam	2.33×3.03×0.79	Single*	7.85-9.25; 16.2%	(+)	CST
	[16]	Dielectric superstrate	16.6×13×0.62	Single*	9.4-10.8; 13.86%	(+)	HFSS
	[1]	Compact Partially Re-flective Surface	2.2×2.2×0.54	Single	8.8-11.4; 25%	(++)	HFSS
	[2]	AMC ground plane	3.33×3.33×0.35	Single	7.7-9.5; 20%	(++)	NA
		and a FSS superstrate
	[4]	Metamaterial substrate	1.25 × 0.9 ×0.04	Single*	9.2-11; 17.8%	(++)	NA
	[17]	Cavity backed struc-	0.46×2.1×0.025	Single	9.4-10.5; 11%	(+)	HFSS
		ture
	[10]	Air cavity	1.25×1.25×0.37	Single	7.7-10; 25%	(-)	NA
	This	Resonant iris as feed-	1.76×1.65×1.29	Single*	8-10.6; 27.9%	(-)	MOM-GEC
	work	ing technique					+ HFSS
	Ant: Antenna; W×L×T: Width×Length×Thickness; SIW: Substrate integrated waveguide; Single*: Single with dual-
	resonance; AMC: Artificial Magnetic Conductor; FSS: Frequency Selective Surface; (++): Very high; (+) High; (-) Low;
	NA: not mentioned; HFSS: High Frequency Simulation Software; CST: Computer Simulation Technology.

  𝐸 01,𝛼 is the real field source 𝐸 01 expressed according the amplitude 𝑉 01,𝛼 and the fundamental mode 𝑓 0,𝛼 of the modal base function 𝑓 𝑚𝑛,𝛼 . It is expressed as: 𝐸 02,𝛼 is the auxiliary electric field source expressed according the amplitude 𝑉 02,𝛼 and the fundamental mode 𝑓 0,𝛼 ′ of the modal base function 𝑓 ′ 𝑚𝑛,𝛼 :The virtual electric fields 𝐸 𝑒 1,𝛼 and 𝐸 𝑒 2,𝛼 on side (1) are equals and expressed in terms of test function (𝑔 1 𝛼𝑝 ) 𝑝=1,..,𝑁 𝑡𝑒𝑠𝑡 and weighting coefficients 𝑥 𝑝,𝛼 : The virtual current density source on side (1), dual of the unit cell virtual electric field 𝐸 𝛼 , is expressed in terms of test function (𝑔 2 𝛼𝑝 ) 𝑝=1,..,𝑁 𝑡𝑒𝑠𝑡 and weighting coefficients 𝑥 𝑝,𝛼 :

		02, E 	=		02, V f 	0, 	'	(5.25)
							N
									test
		1 , e E 	,  =  p	x	p	, 	1, g 	p	(5.26)
						N
								test
		J							x	g
		s	,						p	,	2,	p
							p	,
			01, J I f 01, 0,    =	(5.22)
		01, E 	=	01, V f   0,	(5.23)
	𝐽 02,𝛼 is the auxiliary current source expressed according the amplitude 𝐼 02,𝛼 and the funda-
	mental mode 𝑓 0,𝛼	′ of the modal base function 𝑓 ′	𝑚𝑛,𝛼 . It is expressed as:
			J	02, 	=	02, 0, ' I f  	(5.24)

  ,

	imp	imp	ev	ev	th	s	mn	imp	imp	ev	ev	th	s	mn
						TE	TM

Table 5 . 1 .

 51 Mutual coupling parameter calculation between array's elements based on the MOM-GEC formulation hybridized with unit-cell spectral decomposition approach

	Position k	-2	-1	0	1	2
	-2	-27.5	-22.37	-26.88	-26.88	-22.37
	-1	-22.37	-27.5	-22.37	-26.88	-26.88
	0	-26.88	-22.37	-27.5	-22.37	-26.88
	1	-26.88	-26.88	-22.37	-27.5	-22.37
	2	-22.37	-26.88	-26.88	-22.37	-27.5

Table 5 . 2 .

 52 Mutual coupling parameter calculation between array's elements based HFSS simulator

	Position k	-2	-1	0	1	2
	-2	-26.3	-21.93	-25.9	-25.9	-21.93
	-1	-21.93	-26.3	-21.93	-25.9	-25.9
	0	-25.9	-21.93	-26.3	-21.93	-25.9
	1	-25.9	-25.9	-21.93	-26.3	-21.93
	2	-21.93	-25.9	-25.9	-21.93	-26.3

  The test functions g p is expressed by: The test functions g p is expressed by:

							0
	𝑓 𝑚𝑛 𝑇𝐸 10 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =		-√ 𝑎𝑏 2	sin [	𝜋 𝑎	𝑥]
		{				0
																						g (p,q)x (x, y) = 0
	g p,q ⃗⃗⃗⃗⃗⃗⃗⃗⃗ =	g (p,q)y (x, y) = { sin [	pπ d	(x -( 2 a	-	d 2 ))] for x ϵ [	a 2	-	d 2	,	a 2	+	d 2	] , y ϵ [	b 2	-	D 2	,	b 2	+	D 2	]
	{																				0	if not
	• Scalar Product:
	𝑓 𝑚𝑛 𝑇𝐸 ⃗⃗⃗⃗⃗⃗⃗ = 𝑓 𝑛 𝑇𝑀 ⃗⃗⃗⃗⃗⃗⃗⃗ = ⟨g p , f m ⟩ = (-√ { 𝑓 𝑚𝑛𝑥 𝑇𝐸 = 𝑏√ 𝑚 2 𝑛 𝑎 2 + 𝑓 𝑚𝑛𝑦 𝑇𝐸 -𝑚 𝑎√ 𝑚 2 𝑎 2 + 𝑛 2 𝑏 2 √ 𝑛 2 𝑏 2 2𝜏 𝑚𝑛 √ 2𝜏 𝑚𝑛 𝑎𝑏 𝑎𝑏 sin [ cos [ 𝑚𝜋 𝑎 𝑥] cos [ 𝑚𝜋 𝑎 𝑥] sin [ 𝑛𝜋 𝑏 𝑦] 𝑛𝜋 𝑏 0 { 𝑓 𝑚𝑛𝑥 𝑇𝑀 = 𝑚 𝑎√ 𝑚 2 𝑎 2 + 𝑛 2 𝑏 2 √ 2𝜏 𝑚𝑛 𝑎𝑏 cos [ 𝑚𝜋 𝑎 𝑥] sin [ 2 ab ) ∫ ∫ g p × sin [ b 2 + D 2 a 2 d + 2 mπ 𝑦] x] dx dy pressed by: 𝑓 𝑚𝑛 𝑇𝐸 𝑓 𝑚𝑛𝑥 𝑇𝐸 = a b 2 -D 2 a 2 d -2 ⃗⃗⃗⃗⃗⃗⃗⃗ = { 𝑓 𝑚𝑛𝑦 𝑇𝐸 𝑎√ 𝑓 𝑛 𝑇𝑀 ⃗⃗⃗⃗⃗⃗⃗⃗ ⟨g p , f m ⟩ = (-√ 2 ab ) ∫ sin [ pπ d (x -( a 2 -d 2 ))] a 2 + d 2 a 2 -d 2 sin [ mπ x] dx a ⏟ 𝑛𝜋 𝑏 𝑦] 𝑓 𝑚𝑛𝑥 𝑇𝑀 = -𝑛 𝑏√ 𝑚 2 𝑎 2 + 𝑛 2 𝑏 2 √ 2𝜏 𝑚𝑛 𝑎𝑏 sin [ 𝑚𝜋 𝑎 𝑥] cos [ 𝑏 𝑦] 0 { 𝑓 𝑚𝑛𝑥 𝑇𝑀 = 𝑏√ 𝑚 2 𝑏√ -𝑚 𝑚 2 𝑛 𝑎 2 + 𝑎 2 + 𝑛 2 𝑏 2 √ 𝑛 2 𝑏 2 2𝜏 𝑚𝑛 √ 2𝜏 𝑚𝑛 𝑎𝑏 𝑎𝑏 sin [ cos [ 𝑎 2 , 𝑎 2 𝑚𝜋 ] 𝑎𝑛𝑑 𝑦𝜖 [-𝑎 (𝑥 + 𝑎 2 )] sin [ 𝑏 2 𝑛𝜋 , 𝑏 2 𝑏 (𝑦 + ] is ex-𝑏 2 )] 𝑚𝜋 𝑎 (𝑥 + 𝑎 2 )] cos [ 𝑛𝜋 𝑏 (𝑦 + 𝑏 2 )] 0 b 2 D + 2 × ∫ dy b 2 -D 2 ⏟ 0 -𝑛 𝑚 2 𝑏 2 𝑎 2 + 𝑛 2 √ 2𝜏 𝑚𝑛 𝑎𝑏 sin [ 𝑚𝜋 𝑎 (𝑥 + 𝑎 2 )] cos [ 𝑛𝜋 𝑏 (𝑦 + 2 )] 𝑏 = 𝑛𝜋 𝑓 𝑚𝑛𝑥 𝑇𝑀 = 𝑚 𝑎√ 𝑚 2 𝑏 2 𝑎 2 + 𝑛 2 √ 2τ mn ab cos [ mπ a (x + a 2 )] sin [ 𝑛𝜋 𝑏 (𝑦 + 𝑏 2 )] A 𝐷
	τ mn = {	2 if m and n ⧧ 0 1 if m or n = 0													τ mn = {	2 if m and n ⧧ 0 1 if m or n = 0
	𝑓 𝑚𝑛 𝑇𝐸 𝑚0 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =	{ -√ 𝑎𝑏 2	0 sin [ 𝑚𝜋 𝑎 0	𝑥]											𝑓 𝑚𝑛 𝑇𝐸 𝑚0 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =	-√ 𝑎𝑏 2 {	0 sin [ 𝑚𝜋 𝑎 0	(𝑥 +	𝑎 )] 2
	𝑓 𝑚𝑛 𝑇𝐸 0𝑛 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = { √	2 𝑎𝑏	sin [ 0	𝑛𝜋 𝑏	𝑥]													𝑓 𝑚𝑛 𝑇𝐸 0𝑛 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = { √	2 𝑎𝑏	sin [ 0 𝑛𝜋 𝑏	(𝑥 +	𝑎 2	)]
	g (p,q)y (x, y) = sin [ for x ϵ [ pπ d a 2 -(x -( 2 a d 2 , a 2 + d -2 ] , y ϵ [ d 2 ))] b 2	-	D 2	,	b 2	+	2 D	]		g (p,q)y (x, y) = sin [ pπ d	(x +	d 2 ))] for x ϵ [-	d 2	,	d 2	] , y ϵ [-	D 2	, +	D 2	]

B. Center of the waveguide

Figure A2. Reference position. The modal functions of metallic waveguide EEEE for a propagation in (z) 𝑓𝑜𝑟 𝑥𝜖 [-

The structure is fed by an iris which is formed by an obstacle pierced with a rectangular coupling slot and terminating a rectangular waveguide. The presence of the conductive ground plane leads the electromagnetic radiation to be oriented from the exciter waveguide to the patch through the coupling slot. The feeder waveguide is excited by the fundamental mode TE 10 and the microstrip patch antenna is excited by the quasi-TEM mode.

MOM-GEC formulation

The mathematical formulation's steps of the MOM-GEC method are detailed in this section.

Generalized equivalent circuit

The antenna is composed by the iris and a microstrip patch interconnected through the auxiliary sources (AS) [12][13] as illustrated in Figure 4.3. Noting that the external waveguide is not taken into account when modelling the structure since the excitation by the 𝑇𝐸 10 mode is done through the internal waveguide.

Thus defined, the electromagnetic problem also becomes periodic up to a phase term. For each pair of phase shifts {φ x , φ y }, the distribution of the electromagnetic field in the network is written as an identical complex amplitude for all cells, multiplied by the excitation phase of each cell.

E(x + rd x , y + sd y ) = E(x, y) e -j(rφ x + sφ y ) (5.5)

The phase gradient applied to the second member of the relation can be associated with a horizontal wave vector K ⃗ ⃗⃗ T defined in the plane (Oxy) of the network such as:

(5.6b)

𝜙 is the azimuth angle in the plane (Oxy) of the array.

• Let's search for K ⃗ ⃗⃗ T :

We express K ⃗ ⃗⃗ T in the reciprocal basis {b 1 ⃗⃗⃗⃗⃗ , b 2 ⃗⃗⃗⃗⃗ } :

Then,

By identification :

Thus :

The equivalences between the expressions of the vectorK ⃗ ⃗⃗ T give:

Network radiation is possible if the component k T of the wave vector, which is the consequence of the phase shifts φ x and φ y applied by the excitations, is less than k 0 . In this case, the zenith angle defining the direction of radiation is deduced by: k 0 sinθ = k T (5.12)

Appendix A: Modal Functions Definition

Appendix C: Quadrupole Calculation

We can simplify the following system to obtain the matrix operator of the discontinuity:

The equations (C.1) and (C.2) give:

The equation (C.3) gives:

|A B| ( I 0 I 01

From (C.5), we deduce the vector of ponderation X:

We replace the expression of X previously found in (C.6) in the equation (C.4):

We obtain from (C.7), the matrix operator of the discontinuity:

Appendix D: List of publications