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Micro Strip Patch Antenna

Disclaimer: This work has been submitted by a student. This is not an example of the work written by our professional academic writers. You can view samples of our professional work here.

Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.

Published: Fri, 02 Mar 2018

Chapter 1

Introduction

The project which we have chosen to do as our final year project for the under graduate program involves the characterization of micro strip patch antenna.

In this project we have carried out simulations of different types of antennas, which include dipole, monopole and patch. The purpose of designing all of these is to gain knowledge and experience in the designing of antennas for different purposes by using commercially available CEM. The frequency band, which we have chosen as our relevant band, is the GSM-900 band, which is of wide use in the cellular network. The purpose of choosing this band is to gain valuable knowledge of this frequency band.

Antennas are a fundamental part of every system in which wireless or free space is the medium of communication. Basically, an antenna is a transducer and is designed to transmit or receive electromagnetic waves. It is a transducer as it converts radio frequency electrical currents into electromagnetic waves. Common applications of antennas include radio, television broadcasting, point-to-point radio communication, wireless networks and radar. A detailed study of antennas is discussed in chapter two and chapter three of this report.

The CEM software’s that we have used for the designing include XFDTD® provided by Remcom Inc. and CST Microwave Studio®, which is a full wave, 3-Dimensional, Electromagnetic simulation software and CST Microwave Studio®. XFDTD® utilizes a numerical electromagnetic code for antenna design, that is, the finite difference time domain technique (FDTD). Finite-difference time-domain (FDTD) is a popular computational electrodynamics modeling technique.

The first antenna structure modeled is the dipole. A dipole antenna consists of two conductors on the same axis with a source at the center. It is also modeled in XFDTD® by following the procedure provided by the software and mentioned in the Appendix. The results are verified by comparing with analytical papers of (lambda/2) dipole. After completing this, the next goal is to model the micro strip (patch) antenna which is one of the main focuses of this project. It comprises of a metallic patch bonded to a dielectric substrate with a metal layer bonded to the opposite side of the substrate forming a ground plane. This metal layer is very thin. Hence, it can be fabricated very easily using printed circuit techniques. Therefore, they are inexpensive to manufacture and are easily integrate able with microwave integrated circuits.

The software modeling is carried out in XFDTD® and on CST Microwave Studio®. The verification of the results with the experimental results obtained leads to the final phase and the conclusion of the project.

1.1 Purpose

The purpose of this project is to gain knowledge and experience about computational electromagnetic, as it applies to antenna design. It was also our sole purpose to gain experience in fabrication and experimental characterization of micro strip patch antennas. To achieve these objectives we used two commercially available CEM software’s, XFDTD® and CST Microwave Studio®, to design a micro strip patch antenna for 900 MHz. We also gained experimental experience by characterizing the return loss of this patch antenna using the vector network analyzer.

1.2 Project Scope

1.2.1 Description

We will study some basic types of antennas; extending basic knowledge of antenna to complex antenna designs such as micro strip patch antennas and also modeled them on antenna design and simulation software. This report has been divided into a number of chapters each discussing a different stage of the project. They are briefly described below:

Chapter 2 describes the fundamentals of antennas and thoroughly discusses the theory of fundamental parameters and quantities of antenna. In this chapter the basic concept of an antenna is discussed and its working is explained. Some critical performance parameters of antennas are also discussed. Finally, some common types of antennas are also discussed for understanding purposes.

Chapter 3 discusses the important characteristics of antennas as radiators of electromagnetic energy. These characteristics are normally considered in the far field as the antenna pattern or radiation pattern of an antenna is the three-dimensional plot of its radiation at far field. It also discusses the types of antenna patterns in detail. Some important mathematical equations are also solved in this chapter for the better understanding of how an antenna works.

Chapter 4 discusses in detail the modeling of the half wave dipole and micro strip patch antenna using XFDTD®. It describes the modeling of the antenna, the feeding, and the resultant plots obtained. Furthermore it concludes with comparison of the results obtained with the simulations already available in the software.

Chapter 5 discusses the theory, calculations involved and the fabrication of the micro strip (patch) antenna in detail. The calculations for the dimensions of the rectangular patch in detail are in this chapter. Also, this chapter describes the results obtained through simulation of the model on the software CST Microwave Studio®.

Chapter 6 discusses conclusions drawn from the whole project.

Chapter 2

Antenna Fundamentals

In this chapter, the basic concept of an antenna is discussed and its working is explained.

Next, some critical performance parameters of antennas are discussed. Finally, some common

types of antennas are introduced. The treatment for these is taken from the reference [4], [6] and [9].

2.1 Introduction

Antenna is a metallic structure designed for radiating and receiving electromagnetic energy. An antenna acts as a transitional structure between the guiding devices (e.g. waveguide, transmission line) and the free space. The official IEEE definition of an antenna as given by Stutzman and Thiele [9] is as follows: 

“That part of a transmitting or receiving system that is designed to radiate or receive electromagnetic waves”.

2.2 How an Antenna radiates?

In order to understand how an antenna radiates, we have to first know how radiation occurs. A conducting wire radiates because of time-varying current or an acceleration or deceleration of charge. If there is no motion of charges in a wire, no radiation will occur, since no flow of current occurs. Radiation will not occur even if charges are moving with uniform or constant velocity along a straight wire. Also, charges moving with uniform velocity along a curved or bent wire will produce radiation. If charge is oscillating with time, then radiation will occur even along a straight wire as explained by Balanis [4].

The radiation pattern from an antenna can be further understood by considering a voltage source connected to a two-conductor transmission line. When a sinusoidal voltage source is applied across the transmission line, an electric field is generated which is sinusoidal in nature. The bunching of the electric lines of force can indicate the magnitude of this electric field. The free electrons on the conductors are forcefully displaced by the electric lines of force and the motion of these charges causes the flow of current, which leads to the creation of a magnetic field.

Due to time varying electric and magnetic fields, electromagnetic waves are created which travel between the conductors. When these waves approach open space, connecting the open ends of the electric lines forms free space waves. As the sinusoidal source continuously creates electric disturbance, electromagnetic waves are generated continuously and these travel through the transmission line, the antenna and are radiated into the free space.

2.3 Near and Far Field Regions

The field patterns of an antenna, change with distance and are associated with two types of energy radiating and reactive energy. Hence, the space surrounding an antenna can be divided into three regions.

Figure 2.1: Field regions around an antenna

The three regions that are depicted in above figure are described as:

2.3.1 Reactive Near-Field Region:

In this region the reactive field dominates. The reactive energy oscillates towards and away from the antenna, thus appearing as reactance. In this region, energy is stored and no energy is dissipated. The outermost boundary for this region is at a distance

λ (2.1)

where R1is the distance from antenna surface, D is the largest dimension of the antenna and λ is the wavelength.

2.3.2 Radiating Near-Field Region:

This region also called Fresnel region lies between the reactive near-field region and the far field region. In this region, the angular field distribution is a function of the distance from the antenna. reactive fields are smaller in this field as compared to the reactive near-field region and the radiation fields dominate. The outermost boundary for this region is at a distance

(2.2)

where R2is the distance from the antenna surface.

2.3.3 Far-Field Region:

The region beyond is the far field region also called Fraunhofer region. The angular field distribution is not dependent on the distance from the antenna in this region. In this region, the reactive fields are absent and only the radiation fields exist and the power density varies as the inverse square of the radial distance in this region.

2.4 The Hertzian Dipole

A hertzian dipole or infinitesimal dipole, which is a piece of straight wire whose length L and diameter are both very small, compared to one wavelength. A uniform current I is assumed to flow along its length. Although such a current element does not exist in real life, it serves as a building block from which the field of a practical antenna can be calculated (Sadiku [6]).

Consider the hertzian dipole shown in figure. We assume that it is located at the origin of a coordinate system and that it carries a uniform current. i.e. I=I˳ cosωt. The retarded magnetic vector potential at the field point, due to dipole is given by

(2.3)

Where [I] is the retarded current given by

(2.4)

Where β=ω/u=2π/λ, and u=1/ the current is said to be retarded at point under consideration because there is a propagation time delay r/u or phase delay.

By substitution we may also write A in phasor form as

t(2.5)

Transforming this vector in Cartesian to spherical coordinates yields

Where

But

(2.6)

We find the E field using

(2.7)

(2.8)

Where,

A close observation of the field equations reveals that we have terms varying as The 1/ term is called the electrostatic field since it corresponds to the field of an electric dipole. This term dominates over other terms in a region very close to the hertzian dipole. The is called the inductive field, and it is predictable from the from the Biot Savart law. The term is important only at near field, that is, at distances close to the current element. The 1/r term is called the far field or radiation field because it is the only term that remains at the far zone, that is, at a point very far from the current element.

Here, we are mainly concerned with the far field or radiation zone (βr˃˃1), where the terms in can be neglected in favor of the 1/r term. Thus at far field,

(2.9)

The radiation terms of and are in time phase and orthogonal just as the fields of a uniform plane wave. The near and far zone fields are determined respectively to be the in equalities We define the boundary between the near and far zones by the value of r given by . where d is the largest dimension of the antenna.

The time average power density is obtained as

)

(2.10)

Substitution yields time average radiated power as

But

And hence above equation becomes

If free space is the medium of propagation, η=120 and

(2.11)

This power is equivalent to the power dissipated in a fictitious resistance by current

That is,

(2.12)

Where is the root mean square value of I. From above equations we obtain

Or

(2.13)

The resistance is a characteristic property of the hertzian dipole antenna and is called its radiation resistance. We observe that it requires antennas with large radiation resistances to deliver large amounts of power to space. The above equation for is for a hertzian dipole in free space.

2.5 Half Wave Dipole Antenna

The Half Wave dipole is named after the fact that its length is half of the wavelength i.e. . It is excited through a thin wire fed at the midpoint by a voltage source connected to the antenna via a transmission line. The radiated electromagnetic field due to a dipole can be obtained if we consider it as a chain of hertzian dipoles (Sadiku [6]).

λ/2 I z

x y

I

Figure 2.3: Half Wave Dipole

The magnetic Vector potential P due to length dl of the dipole carrying a phasor current is

(2.14)

We have assumed a sinusoidal current distribution because the current must vanish at the ends of the dipole. Also note that the actual current distribution on an antenna is not precisely known. It can be determined by using Maxwell’s equations subject to the boundary conditions on the antenna by a mathematically complex procedure. The sinusoidal current assumption approximates the distribution obtained by solving the boundary value problem and is commonly used.

O

Y

X

Z

Figure 2.4. Magnetic field at point o

If r >> â„“, then

Hence we can substitute in the denominator of the first equation where the magnitude of the distance is needed. In the numerator for the phase term, the difference between β and β is significant, so we will replace by . We maintain the cosine term in the exponent while neglecting it in the denominator because the exponent involves the phase constant while the denominator does not. So,

(2.15)

Using the following integrating equation,

Applying this equation gives on (2.15)

Since and the above equation becomes,

Using identity = 2cos x, we obtain

(2.16)

We use in conjunction with the fact that to obtain electric and magnetic fields at far zone as

(2.17)

The radiation term of and are in time phase and orthogonal.

We can obtain the time-average power density as

(2.18)

The time average radiated power can be determined as

In the previous equations has been substituted assuming free space as the medium of propagation. The last equation can be written as

Changing the variables, and using partial fractions reduces the above equation to

Replacing with in the first integrand with in the second results in

(2.19)

Solving the previous equation of yields value of . The radiation resistance for the half wave dipole antenna is readily obtained from the following equation and comes out to be.

(2.20)

Chapter 3

Antenna Characteristics

In the previous chapter we have discussed the basics of antennas and the elementary types of antennas. Now we will discuss the important characteristics of antennas as radiators of electromagnetic energy. These characteristics are normally considered in the far field and are as follows. And have been treated from the references [4], [6] and [9].

3.1 Antenna Patterns

The Antenna Pattern or Radiation Pattern of an antenna is the three-dimensional plot of its radiation at far field. There are two types of Radiation Patterns of antennas. The Field and the Power Pattern.

3.1.1 Field Pattern

When the amplitude of the E-field is plotted, it is called the Field Pattern or the Voltage Pattern. A three dimensional plot of an antenna pattern is avoided by plotting separately the normalized versus for a constant which is called an E-Plane pattern or vertical pattern and the normalized versus for called the H-plane pattern or horizontal pattern. The normalization of is with respect to the maximum value of the so that the maximum value of the normalized is unity as explained by Sadiku [6].

For Example, for the hertzian Dipole, the normalized comes out to be,

(3.1)

Which is independent of From this equation we can obtain the E-plane pattern as the polar pattern of by varying from 0 to 180 degrees. This plot will be symmetric about the z-axis. For the H-plane pattern we set so that , which is a circle of radius 1.

3.1.2 Power Pattern

When the square of the amplitude of E is plotted, it is called the power pattern. A plot of the time-average power, for a fixed distance r is the power pattern of the antenna. It is obtained by plotting separately versus for constant and versus for constant.

The normalized power pattern for the hertzian dipole is obtained from the equation.

(3.2)

3.2 Radiation Intensity

The Radiation intensity of an antenna is defined as

(3.3)

Using the above equation, the total average power radiated can be expressed as

(3.4)

(3.5)

Where dΩ= is the differential solid angle in steradian (sr). The radiation intensity is measured in watts per steradian (W/sr).

The average value of is the total radiated power divided by ; that is,

(3.6)

3.3 Directive Gain

The directive gain of an antenna is a measure of the concentration of the radiated power in a particular direction

It can also be regarded as the ability of the antenna to direct radiated power in a given direction. It is usually obtained as the ratio of radiation intensity in a given direction to the average radiation intensity, that is

(3.7)

may also be expressed in terms of directive gain as

(3.8)

The directive gain depends on antenna pattern. For the hertzian dipole as well as for the half wave dipole is maximum at and minimum at . Hence they radiate power in a direction broadside to their length. For an isotropic antenna, . However, such an antenna is not in reality but an ideality.

The directivity D of an antenna is the ratio of the maximum radiation intensity to the average radiation intensity. D is also the maximum directive gain

So,

(3.9)

Or,

(3.10)

For an isotropic antenna, D=1, which is the smallest value that D can have. For the hertzian dipole, as derived in equation (3.7)

For half wave dipole,

Where, η=120 and

(3.11)

3.4 Bandwidth (Impedance Bandwidth)

By definition Bandwidth of an antenna is the difference between the highest and the lowest operational frequency of the antenna.

Mathematically,

(3.12)

If this ratio is 10 to 1, then the antenna I classified as a broadband antenna.

Another definition for Bandwidth is:

Where,

.

3.5 Gain

We define that G is the actual gain in power over an ideal isotropic radiator when both are fed with same power. The reference for gain is the input power, not the radiated power. This efficiency is defined as the ratio of the radiated power () to the input power ().

The input power is transformed into radiated power and surface wave power while a small portion is dissipated due to conductor and dielectric losses of the materials used. The power gain of the antenna as

(3.13)

The ratio of the power gain in any specified direction to the directive gain in that direction is referred to as the radiation efficiency of the antenna i.e.

(3.14)

Antenna gain can also be specified using the total efficiency instead of the radiation efficiency only. This total efficiency is a combination of the radiation efficiency and efficiency linked to the impedance matching of the antenna. Hence, from equation 3.14

(3.14(a))

3.6 Polarization

The definition for polarization can be quoted from Balanis [4] as:

“Polarization of a radiated wave can be expressed as “that property of an electromagnetic wave describing the time-varying direction and relative magnitude of the electric field vector; specifically, the figure traced as a function of time by the extremity of the vector at a fixed location in space, and in the sense in which it is traced, as observed along the direction of propagation.” Polarization then is the curve traced by the end point of the arrow representing the instantaneous electric field. The field must be observed along the direction of propagation.”

3.7 Return Loss

The Return Loss (RL) is the parameter which indicates the amount of power that is lost to or consumed by the load and is not reflected back as waves are reflected which leads to the formation of standing waves. This occurs when the transmitter and antenna impedance do not match. Hence, the RL is a parameter to indicate how well the matching between the transmitter and antenna has taken place.

The RL is given as:

(3.15)

For perfect matching between the antenna and transmitter, RL = ∞ and Γ = 0 which

means no power is being reflected back, whereas a Γ = 1 has a RL = 0 dB, which implies that

all incident power is reflected. For practical applications a RL of -9.54 dB is acceptable.

Chapter 4

Modeling of Half-Wave Dipole & Micro Strip Patch Antenna Using XFDTD®

4.1 Introduction

For the purpose of modeling and simulation of antennas we have used modeling software’s, which are widely used in industries. These software’s are specially used for the purpose of electromagnetic (EM) modeling, which refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment.

The first such software brought into use is XFDTD®. It is a three-dimensional full wave electromagnetic solver based on the finite difference time domain method. It is fully three-dimensional.  Complex CAD® objects can be imported into XFDTD® and combining and editing can be done within XFDTD® using the internal graphical editor. It is a powerful software which offers a lot of options to its users.

This software has been initially used for modeling of basic antennas to get familiarity with interface and working of the software. Dipole is one of such basic antennas with a simple structure; as the name suggests dipole antenna consists of two wires on the same axis with a source applied at the center point.

In this chapter, we begin with the analysis of a half-wave dipole antenna by derivation of field equations and the MATLAB® plot. After the analysis the modeling is done using XFDTD®. Finally, all the results are matched by plotting the data in MATLAB®.

4.2 Derivation of Vector Magnetic Potential

We begin with the derivation done in chapter 2 for of the radiated fields for a half-wave dipole antenna in equation 3.11 which gives us the following expression for

(4.11)

4.2.1 MATLAB® Plots of Half Wave Dipole Antenna

The expression can be plotted in MATLAB® using the following code

clear all;

theta = [0:360]*pi/180;

F = cos((pi/2)*cos(theta))./(0.0000001 + sin(theta));

Pn = F./max(F);

Pn=abs(Pn);

title (‘POLAR PLOT OF HALF WAVE DIPOLE ‘)

polar(0,1); hold on;

polar (theta,Pn,’r’);

The MATLAB® generated plot of normalized electric field for half-wave dipole for above code is as follows

Figure 4.1: MATLAB® plot for Normalized Electric Field

4.3 Modeling of Half Wave Dipole Using XFDTD®

4.3.1 Introduction

XFDTD® is a full wave, 3D, Electromagnetic Analysis Software. XFDTD® used solid, dimension based modeling to create geometries. To create geometry, library objects and editing functions may be used. Modeling of half-wave dipole antenna was carried out in XFDTD® to test the software’s capability of generating far field radiation pattern. And also to get in depth knowledge of XFDTD® before using it for the modeling of patch antennas, which is the foremost objective of this project.

4.3.2 Validity of Model

As in the previous section the electromagnetic theory of half-wave dipole was studied and its mathematical equations for normalized radiated field was derived and plotted. This plot will be our reference plot while doing the modeling of half-wave dipole.

4.3.3 Modeling of Half Wave Dipole

As we know the length of a half-wave dipole antenna should be half the wavelength of the operating carrier wave frequency. Thus the dipole modeled in XFDTD® has the following specifications:

  • Length of 30cm
  • Frequency used 1 GHz
  • Thin wire was used to create the dipole
  • Source was attached in the middle

Figure below shows the geometry of dipole being modeled in XFDTD®.

Figure 4.2: XFDTD® geometry of Half-Wave Dipole

4.3.4 Results

The far fields of dipole antenna were calculated by XFDTD® and plots were obtained for far field versus both Phi and Theta, as shown in Figure 4.3 & Figure 4.4. The results matched with the theoretically established results.

Figure 4.3: Far Field vs. Theta Figure 4.4: Far Field vs. Phi

4.3.5 Plotting XFDTD® Results in MATLAB®

The data for far fields from XFDTD® was exported and matched with the theoretical results in MATLAB® for the purpose of confirming the results. Help was taken from the XFDTD® reference manual to learn how to export far field data.

The XFDTD® file was copied and the extension was changed to ‘.dat’ and name was changed to ‘XFTDT.dat’ Next this file was read by MATLAB® using the MATLAB® code provided

[angle1, a1, c1, d1, e1] = textread(‘XFDTD.dat’,’%f %f %f %f %f’, 361);

angle1=angle1*pi/180;

q=find(c1<-9);

c1(q)=-9;

c1=c1+9;

m=max(c1);

c1=c1./m;

polar(angle1,c1,’g’)

The MATLAB® result is shown n figure below.

Figure 4.8: XFDTD® radiation pattern in MATLAB®

The experimentally produced curve qualitatively matches with our theoretical calculations. The shape of the curve is similar to the theoretical description, whereas the scale is different. For the purpose of confirming this result, the data of this curve is also exported into MATLAB® to be compared with previously simulated results.

4.4 Modeling of Micro Strip Patch Antenna Using XFDTD®

4.4.1 Introduction

After gaining confidence on the design of dipole antenna by comparing its results with the simulations and the results obtained from MATLAB®, we use the same computational software for the modeling of micro strip patch antenna.

4.4.2 Validity of Model

For the modeling of micro strip patch antenna, a paper of IEEE “Application of Three-Dimensional Finite-Difference Time Domain Method of the Analysis of Planar Micro strip Circuits” is reproduced. This paper is used as a reference so that the results could be compared in order to check the validity. The result of our exercise confirms the results of the IEEE paper; this takes us to design a micro strip antenna of our desired parameters. This training will help us gain the expertise over the computational software, which can be used for the modeling of multiple different antennas.

4.4.3 Modeling of Micro Strip Patch Antenna

The antenna is designed for the frequency range from 0 GHz (dc) to 20 GHz. The dimensions used for the antenna centers it at 7.8 GHz. Although its results at the higher frequencies are also examined for the accuracy, the parameters for the antenna are given below:

  • Duroid substrate is used with =2.2
  • Thickness is 1/32 inch=0.794mm
  • Length = 12.45mm
  • Width = 16mm
  • Transmission line feed is used and is placed at 2.09mm away from the left corner.

With these specifications the center frequency comes out to be 7.8 GHz and this can be verified from the link www.emtalk.com/mpaclac.php

Figure 4.5 shows the geometry of micro strip patch modeled in XFDTD®.

Figure 4.5 Geometry of the micro strip patch antenna

4.4.4 Results

The S11 plot of micro strip patch antenna was calculated by XFDTD®, as shown in Figure 4.6 & Figure 4.7 is the plot of the IEEE paper. This gives us the comparison between the two.

Figure 4.6 obtained from the XFDTD®

Figure 4.7: Results of S11 parameters from published IEEE Papers

Chapter 5

Micro Strip Antennas

5.1 Introduction

These days there are many commercial applications, such as mobile radio and wireless communication, where size, weight, cost, performance, ease of installation, and aerodynamic profiles are constraints and low profile antennas may be required. To meet these requirements micro strip antennas can be used. These are low profile antennas and are conformable to planar and non-planar surfaces. These are simple and inexpensive to manufacture using modern printed circuit technology. They are also mechanically robust and can be mounted on rigid surfaces. In addition, micro strip antennas are very versatile in terms of resonant frequency, polarization, pattern and impedance as explained by Balanis [4].

5.1.1 Basic Characteristics

Micro strip antennas consist of a very thin metallic strip or patch placed a small fraction of a wavelength above a ground plane. The micro strip patch is designed so its pattern maximum is normal to the patch hence making it a broadside radiator. This is accomplished by properly choosing the mode or field configuration of excitation beneath the patch. End-fire radiation can also be accomplished by judicious mode selection. For a rectangular patch, the length L of the element is usually . The conducting micro strip or patch and the ground plane are separated by the substrate (Balanis [4]).

There are numerous substrates that can be used for the design of micro strip antennas and their dielectric constants are usually in the range of . The substrate that we are using in our designs has a value of 4.6.

Often micro strip antennas are also referred to as patch antennas. The radiating elements and the feed lines are usually photo etched on the dielectric substrate. The radiating patch may be square, rectangular, thin strip, circular, elliptical, triangular or any other configuration.

Arrays of micro strip elements with single or multiple feeds are used to achieve greater directivities.

5.1.2 Feeding Methods

There are numerous methods that can be used to feed micro strip antennas. The four most common and popular are the micro strip line, coaxial probe, aperture coupling and proximity coupling. In our designs we have selected coaxial probe as our method of feeding the Micro strip antenna. Following is a brief explanation of coaxial feeding as explained by Balanis [4].

Coaxial-line feeds, where the inner conductor of the coax is attached to the radiation patch while the outer conductor is connected to the ground plane are widely used. The coaxial probe feed is also easy to fabricate and match, and it has low spurious radiation. However is has narrow bandwidth and it is more difficult to model.

5.2 Rectangular Patch

The rectangular patch is one of the most widely used configurations of Micro strip antennas. It is very easy to analyze using either the transmission line model or the cavity model, which have higher accuracy for thin substrates as explained by Balanis [4]. In our desig


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