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I Edited by

I

WE1 CHEN KAI FONG LEE

A WILEY-INTERSCIENCE PUBLICATION JOHN WlLEY & SONS, INC.

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This text is printed on acid-free paper. Copyright 0 1997 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further

information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012.

Library of Congress Cataloging-in-Publication Data

Advances in microstrip and printed antennas / edited by Kai Fong Lee and Wei Chen.

p. an. - - (Wiley series in microwave and optical engineering) "A Wiley-Interscience publication."

Includes bibliographical references (p. ). ISBN 0-471-04421-0 (alk. paper)

1. Microstrip antennas. 2. Printed circuits. I. Lee, Kai Fong. 11. Chen, Wei, 1959- . 111. Series.

TK7871.6.A394 1997

621.381'331 -- dc20 96-39032

Printed in the United States of America 1 0 9 8 7 6 5 4 3

Contents

Contributors

Preface

1 Probe-Fed Microstrip Antennas K. F. Lee, W Chen, and R. Q. Lee

1.1 Introduction

1.2 Full-Wave Analysis of Multilayer Multipatch Microstrip Antennas

1.2.1 Introductory Remarks 1.2.2 Conventions and Definitions 1.2.3 Basic Formulations 1.2.4 Green's Functions

1.3 Spectral Domain Full-Wave Analysis of Probe-Fed Rectangular Microstrip Antennas

1.3.1 Formulation 1.3.2 Basis Functions

1.3.3 Multiple Feeds and Shorting Pins 1.3.4 Attachment Modes

1.4 Representative Numerical and Experimental Results 1.4.1 Single Patch

1.4.2 Single Patch in Multidielectric Media 1.4.3 Coplanar Parasitic Subarray 1.4.4 Two-Layer Stacked Patches 1.5 Rectangular Patch with a U-Shaped Slot 1.6 Concluding Remarks

References

xiii

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2.4.2 Coplanar Microstrip Subarrays 2.4.3 Offset Dual-Patch Microstrip Antennas 2.4.4 Two-Layer Microstrip Antennas with Stacked

Parasitic Patches

2.5 Infinite Arrays of Aperture-Coupled Multilayer Microstrip Antennas

2.5.1 Skewed Periodic Structure and Floquet Modes 2.5.2 Infinite Array of Microstrip Antennas with Air Gaps 2.5.3 Infinite Array of Dual-Patch Microstrip Antennas 2.6 Conclusions

Appendix: Fourier Transforms of Expansion and Test Functions Acknowledgments

References

3 Microstrip Arrays: Analysis, Design, and Applications John Huang and David M . Pozar

3.1 Introduction

3.2 Analysis Techniques for Microstrip Arrays

3.2.1 Review of Microstrip Antenna Analysis Techniques 3.2.2 Full-Wave Moment Method Analysis

3.2.3 Calculation of Mutual Coupling 3.2.4 Infinite Array Analysis

3.2.5 The Active Element Pattern 3.2.6 Waveguide Simulators 3.3 Design Methodology

3.3.1 Array Configuration Design 3.3.2 Patch Element Design

3.3.3 Power Division Transmission Line Design 3.3.4 Microstrip Reflectarray Design

3.4 Applications 3.4.1 Military Applications 3.4.2 Space Applications 3.4.3 Commercial Applications 4.4.1 Orthogonal Patches 4.4.2 Multipoint Feeds 4.4.3 Single-Point Feeds . 4.5 Dual Polarized Patches

4.5.1 Triangular Patch with Right- and Left-Hand Circular Polarization

4.6 Microstrip Spirals

4.6.1 Operation of the Spiral Antenna 4.7 Special Substrates and Active Antennas 4.8 Dual and Circularly Polarized Arrays

4.8.1 Patch Arrays

4.8.2 Microstrip Line Arrays 4.8.3 Sequentially Rotated Arrays 4.9 Conclusions

References

5 Computer-Aided Design of Rectangular Microstrip Antennas David R. Jackson, Stuart A. Long, Jeffery T . Williams, and Vickie B. Davis

5.1 Introduction

5.2 CAD Model for Rectangular Patch Antenna 5.3 CAD Formulas for Resonance Frequency 5.4 CAD Formulas for the Q Factors

5.4.1 Dielectric and Conductor Q Factors

5.4.2 Relation Between Surface-Wave and Space-Wave Q Factors

5.4.3 Space-Wave Quality Factor 5.5 CAD Formula for Bandwidth

5.5.1 CAD Formula 5.5.2 Results

5.6 CAD Formula for Radiation Efficiency

169 170 177 183 184 184 185 186 188 188 188 190 217 217 223 223 224 231 234 234 235 237 242 243 243 246

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CONTENTS ix

5.6.1 CAD Formula 5.6.2 Results

5.7 CAD Formula for Input Resistance 5.8 CAD Formula for Probe Reactance 5.9 Results for Input Impedance 5.10 Radiation Patterns

5.10.1 Infinite Substrate 5.10.2 Truncated Substrate 5.1 1 CAD Formula for Directivity 5.12 Conclusions

Appendix A: Derivation of the p Factor

Appendix B: Radiation Formulas for HED and H M D References

6 Multifunction Printed Antennas J. R. James and G. Andrasic 6.1 Introduction

6.2 Printed Antenna Design Freedom

6.3 Multifunction Antenna Design Opportunities and Recent Advances

6.3.1 Choice of Substrate Materials and Their Design Potential

6.3.2 Innovative Use of Superstrates 6.3.3 Printed Conductor Topology 6.3.4 Quest for Feeder Simplicity 6.3.5 Conformality

6.3.6 Integration of Antennas and Circuits 6.4 Possible Future Developments

6.4.1 Impact of New Materials 6.4.2 The Application Drivers 6.5 Conclusions

References

7 Superconducting Microstrip Antennas

Jeffery T. Williams, Jarrett D. Morrow, David R. Jackson, and Stuart A. Long

7.1 Introduction

7.2 Basics of Superconductivity

7.2.1 General Properties of Superconductors 7.2.2 High-Temperature Superconductors

7.2.3 Characteristics of High-Temperature Superconductors

7.3 HTS Microstrip Transmission Lines and Antennas 7.3.1 Superconducting Transmission Lines and

Feed Networks

7.3.2 Superconducting Microstrip Patch Antennas 7.4 Design Considerations

7.5 Experimental Results 7.6 Summary

Appendix References

8 Active Microstrip Antennas Julio A. Navarro and Kai Chang

Introduction

The Early History of Integrated Antennas Diode-Integrated Active Microwave Antennas Transistor-Integrated Active Microstrip Antennas Diode Arrays for Spatial Power Combining Transistor Arrays for Spatial Power Combining System Applications

Conclusions and Future Trends Acknowledgments

References

9 Tapered Slot Antenna

Richard Q. Lee and Rainee N . Simons Introduction

Basic Geometries Design Considerations Fundamentals Analytical Methods

9.5.1 Analysis of Uniform Slotline by the Spectral Domain Approach 9.5.2 Far-Field Computation Feeding Techniques Characteristics of TSA 9.7.1 Radiation Characteristics 9.7.2 Impedance Characteristics 9.7.3 Bandwidth Characteristics 9.7.4 Field Distributions Tapered Slot Antenna Arrays Active Tapered Slot Antenna Array

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10.4.1 Spatial Discretization 10.4.2 Source Excitation 10.4.3 Phased Array Excitation 10.4.4 Extrapolation Techniques 10.4.5 Impedance

10.4.6 Absorbing ~oundaries' 10.4.7 Radiation Pattern 10.4.8 Distributed Computing 10.4.9 Dielectric Loss Tangent 10.5 Single Patch Modeling

10.5.1 Impedance of a Patch Antenna Mounted on a Moderately Thick Substrate

10.5.2 Impedance of a Patch Antenna Mounted on a Thick Substrate

10.5.3 Effect of a Finite Ground Plane on Impedance and Radiation Pattern

10.6 Analysis of a Two-Layer Stacked Patch Antenna 10.7 Design of a Compact Broadband Antenna 10.8 Conclusions

References

1 1 Analysis of Dielectric Resonator Antennas K . M . Luk, K .

W

h u n g , and S. M . Shum 11.1 Introduction

11.2 Analysis of Aperture-Coupled Hemispherical DR Antenna 11.2.1 Problem Formulation

11.2.2 Moment Method Solution

11.2.3 Derivation of DR Antenna Green's Function GZy

11.2.4 Evaluation of Yi,,

11.2.5 Single-Cavity-Mode Approximation

11.2.6 Single-Cavity-Mode Radiation Field of the DR Antenna

References Index

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Contributors

C. Andrasic j. S. Dahele

School of Engineering and Applied School of Engineering and Applied

Science Science

Royal Military College of Science Royal Military College of Science Cranfield University Shrivenham

Shrivenham, Wilts SN6 8LA Wilts SN6 8LA

England England

T. M. Au Vickie B. Davis

Center of Wireless Communications D~~~~~~~~~ of ~ l ~and ~ ~ ~ i ~ ~ l National University of Singapore Computer Engineering

Singapore University of Houston

Houston, TX 77204

Kai Chang U.S.A.

Department of Electrical Engineering Texas A&M University

College Station, TX 77843 U.S.A. Supriyo Dey Electromagnetic Communication - - - - Laboratory

Siva Chebolu University of Illinois, Urbana-

Celwave Champaign

Division of Radio Frequency Urbana, IL 61801

Systems, Inc. U.S.A.

Phoenix, AZ 85034

U.S.A. P. S. Hall

School of Electronic and Electrical

W. Chen Engineering

Cooper Energy Services University of Birmingham Mount Verson, O H 43050 Edgbaston, Birmingham B15 2TT

U.S.A. England

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U.S.A.

J. R. James

School of Engineering and Applied Science

Royal Military College of Science Cranfield University

Shrivenham, Wilts SN6 8LA England

K. F. Lee

Department of Electrical Engineering University of Missouri-Columbia Columbia, MO 6521 1

U.S.A.

Richard Q. Lee

NASA Lewis Research Center 21000 Brookpark Road Cleveland, O H 44135 U.S.A.

K. W. Leung

Department of Electronic Engineering City University of Hong Kong Kowloon

Hong Kong

Stuart A. Long

Department of Electrical and Computer Engineering University of Houston Houston, TX 77204 U S A .

Jarrett D. Morrow

Department of Electrical and Computer Engineering University of Houston Houston, TX 77204 U.S.A.

Julio A. Navarro

Boeing Defense and Space Group Seattle, WA 98124

U.S.A.

David M. Pozar

Department of Electrical and I Computer Engineering

University of Massachusetts, Amherst

Amherst, MA 01003 I

U.S.A.

S. M. Shum

Department of Electronic Engineering City University of Hong Kong

Kowloon I

Hong Kong

Rainee N. Simons

NASA Lewis Research Center 21000 Brookpark Road Cleveland, OH 441 35 U.S.A.

John Svigelj

Texas Instruments, Inc. 2501 University Drive MS 8019 McKinney, TX 75070

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Preface

Since the late 1970s, the international antenna community has devoted much effort to the theoretical and experimental research on microstrip and printed antennas, which offer the advantages of low profile, compatibility with integrated circuit technology, and conformability to a shaped surface. The results of this research have contributed to the success of these antennas not only in military applications such as aircraft, missiles, and rockets but also in commercial areas such as mobile satellite communications, the direct broadcast satellite (DBS) system, global positioning system (GPS), remote sensing, and hyperthermia. While many of the results of the late 1970s and 1980s were summarized in the Handbook of Microstrip Antennas, edited by J. R. James and P. S. Hall in 1989, the research on microstrip and printed antennas has continued unabated in the 1990s. In addition to advances in conventional topics, there have been new research areas. The purpose of this book is to update and to present new information on microstrip and printed antennas since the two-volume hand- book was published. The contributors are all active researchers and well known in the field.

Chapters 1-4 deal with recent advances in conventional topics. These include accounts on recent results on probe-fed microstrip antennas and aperture- coupled microstrip antennas; analysis, design, and applications of microstrip arrays including the recently developed configuration known as microstrip reflectarray; and dual and circularly polarized planar antennas. Most of the topics in Chapters 5-11 are relatively new. They were not covered in the 1989 Handbook. These include the development of computer-aided design (CAD) formulas for the rectangular patch; the concept, development, and future possi- bilities of multifunction printed antennas; microstrip antennas made of high- temperature superconducting materials; active microstrip antennas; and tapered slot minted antennas. C h a ~ t e r 10 discusses the finite-difference time-domain method of analysis which is becoming popular due to its ability to handle complex configurations and to generate the characteristics of the patch over a broad band of frequencies with a single simulation. The book ends with a chapter

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Antennas

K. F. LEE, W. C H E N , and R. Q. LEE

1.1 INTRODUCTION

One of the common methods of feeding a microstrip antenna is by means of a coaxial probe. The basic configuration is shown in Figure 1.1, where a single metallic patch is printed on a grounded substrate. A number of designs have evolved from the basic configuration. Figure 1.2 shows a design in which a fed patch is surrounded by closely spaced parasitic patches, which can have the effect of improving the impedance bandwidth and the gain of the antenna. Such a configuration is referred to as a coplanar parasitic subarray. Figure 1.3 shows cases where the metallic patch is embedded in a multilayered dielectric media. In Figure 1.3a, a superstrate or dielectric cover is used to protect the patch against . environmental hazards. If a naturally occurring dielectric layer such as ice is formed on top of the cover, the three-layer configuration of Figure 1.3b results. Figure 1 . 3 ~ shows a one-superstrate two-substrate geometry, as, for example, when an air gap is introduced between the substrate and the ground plane t o alter the resonant frequency of the antenna. Figure 1.4 shows the two-layer stacked geometry consisting of one fed patch and a parasitic patch on another layer. These stacked patches are popular for providing wide bandwidth characteristics. Another wideband microstrip antenna is the rectangular patch with a U-shaped slot (Figure 1.5). In recent years, the various linearly polarized probe-fed micro- strip antennas depicted above have been extensively studied. It is the purpose of

Advances in Microstrip and Printed Antennas, Edited by Kai Fong Lee and Wei Chen ISBNO-471-04421-0 0 1997 John Wiley & Sons, Inc.

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l---

,

Conducting patch

I 1 1

Ground plane Coax feed

(b)

FIGURE 1.1 Basic configuration of the probe-fed microstrip antenna. (a) Top view, (b) side view.

this chapter to give a coherent account of recent work in this area. The materials to be presented are based mainly on the authors'research. Related work by others will be referenced but not described in detail.

We shall be concerned with rectangular patches only. However, the methods of analysis can be extended to other geometrical shapes; and many qualitative features are not dependent on whether the patches are rectangular or circular, which are the two most commonly used shapes in practice.

In Section 1.2, a general full-wave analysis of multilayer multipatch microstrip antennas is presented. The application of the analysis to probe-fed rectangular microstrip antennas is described in Section 1.3. Representative numerical and experimental results for configurations 1.1-1.4 are given in Section 1.4. Experi- mental results of the U-slot patch are described in Section 1.5. The chapter ends with some concluding remarks.

(a)

Parasitic Fed Parasitic

Coax feed

(b)

FIGURE 1.2 Geometry of coplanar parasitic subarray. (a) Top view, (b) side view.

Superstrate Patch Superstrate 1

Substrate Suwrstrate 2

T r

Coax feed (a) Coax feed (b)

i r

Superstrate

.T

Substrate Coax feed (c)

FIGURE 1.3 Microstrip antenna in multidielectric media. (a) Patch with superstrate, (b) patch with two superstrates, (c) patch with one superstrate and two substrates.

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FIGURE 1.4 Geometry of two-layer stacked patches.

Coax feed

I

,

wtcn

FIGURE 1.5 Rectangular patch with a U-shaped slot. (a) Top view of the patch, (b) side view.

Air or foam

1

1

1.2 FULL-WAVE ANALYSIS OF MULTILAYER MULTIPATCH MICROSTRIP ANTENNAS

\

Ground plane

1.2.1 introductory Remarks

Consider first the basic form of the microstrip antenna shown in Figure 1.1. Although the antenna appears simple and is easy to fabricate, obtaining electro- magnetic fields which satisfy all the boundary conditions is a complicated task. For this reason, simplified approaches such as the transmission line model and

simple is that the substrate thickness is assumed to be much smaller than wavelength so that the electric field has only a vertical (z) component which does not vary with z. From this it follows that:

1. The fields in the cavity are T M (transverse magnetic).

2. The cavity is bounded by magnetic walls (H

,,,,,,,,,,

= 0) on the sides. 3. Surface wave excitation is negligible.

4. The current in the coaxial probe is independent of z.

The coaxial probe is modeled by a current ribbon of a certain width, which is a free parameter chosen to fit the experimental data.

The cavity model has the advantages of being simple and providing physical insight. Design information for rectangular, circular, annular, and triangular patches can be obtained with relative ease [3].

There are a number of limitations to the cavity model even if the thin substrate condition is satisfied. The magnetic wall boundary condition leads to resonant frequencies which do not agree well with experimental observations, and an ad hoc correction factor has to be introduced to account for the effect of fringing fields. The width of the current ribbon used to model the coaxial probe is another ad hoc parameter. The model cannot handle designs involving parasitic elements, either on the same layer or on another layer. It cannot analyze microstrip antennas with superstrates. When the thickness of the substrate exceeds about 2 % of the free space wavelength, the cavity model results begin to become inaccurate, due to the breakdown of (1)-(4). For these reasons, more accurate analyses of the microstrip antenna based on solving Maxwell's equations subject to the boundary conditions are clearly of interest. These are known as full-wave models and they are the topics of many papers that appeared in the recent literature. In this section we present our version of full-wave analysis which can be applied to a variety of multilayer multipatch microstrip antennas.

1.2.2 Conventions and Definitions

(1) ej"' convention: Throughout this chapter, all fields are assumed to have

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6 PROBE-FED MICROSTRIP ANTENNAS

Using this definition, we now introduce the following notations for the reaction: (2) Fourier transform: The Fourier transform pair is defined as

+ ....

( J , , M,) = -

J

B(;j;).G2dv (1.11)

where

1.2.3 Basic Formulations Usually it is more convenient to first find the localized Fourier transform

and then convert it to the standard form. We define the localized Fourier transform as

In this section, we shall set up the basic formulation for general layered structures based on the electric field integral equations (EFIE).

1.2.3.1 Simple Region. We define a simple region as a region consisting of layered materials bounded by free space or metallic conducting surface(s) (Figure 1.6). It also contains radiating elements and feeding elements which are assumed to be approximately represented by a set of current expansions; that is, the current on the radiating elements and feeding elements is expanded into a set of basis functions:

where

Eo

is a local reference point. y(xs) is related toy1(<) by

-

+

(3) Reaction: The reaction of source 1 (J,, MI) to source 2

(T2,

G,) is defined

where are the basis functions and Cj are the corresponding coefficients. A simple region may connect to the outside through aperture coupling. For the aperture coupling case, the electrical field in the aperture(s) is expanded into basis functions,

-

where (El, HI) are the @Ids produced by source 1, is the electric current density of source i, and Mi is the magnetic current density of source i.

In this chapter, we will assume all media to be homogeneous and isotropic. For these media, the Reciprocity Theorem holds, which states that

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.Conducting plane

FIGURE 1.6 Examples of simple regions. Conducting patches embedded in multilayer dielectric media bounded by (a) two conducting planes, (b) one conducting plane. In (c), there are no conducting boundaries other than the patches.

where

Ga

=

Ea

x i and

li?,

=

Ek

x 6. 6 in the inward normal. Note that this expansion represents the magnetic current over all the aperture(s).

If we assume all.the metal sheets to be perfect conducts, the electric integral equation is established by forcing the total tangential electric field to vanish on all the metallic surfaces inside the region:

This is basically an integral equation. We will convert it to a set of linear equations using the Method of Moments. Using the current expansions, we have

This is one of the most important equations in our formulation. We call it fundamental equation one.

If the metal sheets are not perfect conducts, we have to apply the impedance boundary condition

where Z , is the surface impedance @/square meter). Equations (1.19)-(1.21) become

Equation (1.26) is the generalization of fundamental equation one to nonperfect conductors.

To define the problem completely, we also need the condition of continuity of tangential magnetic fields across the aperture. The total tangential magnetic field in the inner side of the aperture(s) is

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Testing it using

Gm,

we get

We call this equation fundamental equation two. Note that to apply the fundamen- tal equation two, we also need to know the total tangential magnetic field on the other side of the aperture which usually is done by applying the same equation to the region on the other side of the aperture.

The two fundamental equations can also be written in matrix form. Define:

and

The fundamental equation one becomes

and the fundamental equation two is

Combining the two equations, we have

1.2.3.2 Complex Region. A complex region is a region consisting of two or more simple regions coupled through apertures (Figure 1.7). Let us label the simple regions sequentially from top down starting with region 0. Region k is on the top of region k

+

1 with a common boundary which we label as interface k. The electric field in the apertures on interface k can be expanded as

Conducting

planes

FIGURE 1.7 Examples of complex regions.

On the side of region k, ii = 2, and the magnetic current over the apertures is

where

GL1

=

E,

x h and

Gj:

=

zk

x 2.2 is the inward normal.

On the side of region k

+

1, h = - i, and the magnetic current over the apertures is

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k,,, = k , sin 0 cos cp

k,,, = k , sin 0 sin cp

k,,, = ko ICOS

el

For region k

+

1,

Using Eqs. (1.81) and (1.83) of the next section, the above equation can be written as

Equating (1.41) with (1.43) we obtain

1.2.4 Green's Functions

1.2.4.1 Spatial Domain Green's Function. In the spatial domain, the electric and magnetic fields and their sources are related through the dyadic Green's functions according to

which can be called fundamental equation three.

1.2.3.3 Excitations. As part of the feed modeling, we usually can specify an electric current or a magnetic current as a known incident source. We will still include this incident current in our total electric or magnetic current expansion [Eq. (1.15) or (1.17)], but its corresponding coefficient is a known constant. So, in our fundamental equations, some terms can be evaluated to constants, and those terms can be moved to the right-hand side to form an excitation vector and all other terms with unknown coefficients will be moved to the left-Land side. The equations can then be solved for the unknown coefficients.

-

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Layer 0 lnterface 0

Let FE layer n and 7 ' ~ layer m. The spatial electric and magnetic Green's functions are given by [4]

Layer 1 lnterface 1 and lnterface k-1 Layer k lnterface k Layer k+l

FIGURE 1.8 A planarly layered medium.

where 7 is the field point and 7 is the source point,

7

and

7,

are the electric and the magneticcurrent densities, respectively (the symbol J, is tsed in this section rather than M to highlight the symmetry of the equations), G, and G , are the electric dyadic Green? function and the magnetic dyadic Green's function, respectively, E , and H e are the electricjeld ancmagnetic field, respectively, produced by the electric current, and E m and H , are the electric field and magnetic field, respectively, produced by the magnetic current.

Consider the planarly layered geometry shown in Figure 1.8. We assume that there are N

+

2 layers (including free space and/or boundary conductors) labeled layer 0 to layer N

+

1 and the interface between layer k and layer k

+

1 is labeled interfaces k.

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Because A:, is given in Eq. (1.66), starting from A:,, all the A:, can be computed. In the equations above, R", and R", are the so-called generalized reflection coefficients [4] at the upper and lower boundaries of layer 1, respectively. They are also defined by the following recursive relations:

1.2.4.2 Spectral Domain Green's Functions. Transforming Eqs. (1.51) to (1.54), we obtain the following equations relating the sources and fields in the spectral domain:

where Rf:*, and RTEl are the Fresnel reflection coefficients defined by

Sm

=

-

j w

l,.

E,G; 3 , dz'

- -

For surface current independent of z, the above equations can be simplified:

If one layer, say layer N

+

1, is a conductor, then E , should be replaced by ~ / j w . So,

The spectral domain electric and magnetic Green's functions can be obtained by applying the Fourier transform to Eqs. (1.55) and (1.56). After some

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18 PROBE-FED MICROSTRIP ANTENNAS

manipulation, their components are obtained as follows:

1 -k: F T M

- j w p m G ~ =

---

6(z - z') 'kmzw&n m e r n

2. Find all the generalized reflection coefficients iteratively for all the layers using Eqs. (1.70)-(1.73).

3. Find the necessary B: and BE using Eqs. (1.64) and (1.65). 4. Find the coefficients A:, iteratively using Eqs. (1.66) and (1.69). 5. Find the necessary scalar waves F",sing - Eq. (1.63) or (1.67). 6. Find the Green's functions of interest.

The above procedure can be used to find the dyadic Green's functions analyti- cally. It can also be used to find the dyadic Green's functions numerically. When it is applied numerically, it can handle an arbitrary number of layers, so it is well suited to be used in a general-purpose program.

1.3 SPECTRAL D O M A I N FULL-WAVE ANALYSIS O F PROBE-FED RECTANGULAR MICROSTRIP ANTENNAS

In this section, we shall narrow our focus to the specific case of rectangular microstrip antennas with a coaxial feed.

1.3.1 Formulation

6,

can be obtained by applying the following replacements to Eqs. (1.89)-(1.98) above

C e

+

C m

(1.99)

&-'P (1.100)

In summary, to obtain the dyadic Green's functions, follow these steps: 1. Find all the Fresnel reflection coefficients for all the interfaces using

Eqs. (1.74)-(1.80).

The configuration of the antennais shown in Figure 1.9. We first consider the case when only one probe feed is used. The formulation will be extended to allow for multiple feeds later. Since the region above the ground plane is a simple region, the fundamental equation one and two-that is, Eqs. (1.22) and (1.30)-can be applied here.

The total current on the patch and probe is expanded into

where

Zatd,,

is the basis function on the patch, f,, is the basis function on the probe, and is the attachment mode, a special basis function used to ensure the continuity of current a t the patch-probe junction. This will be discussed in detail in Section 1.3.4. The field inside the coaxial cable can be expressed as

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Layer N

Ground plane

Coax feed

FIGURE 1.9 Geometry of probe-fed rectangular microstrip antenna in planarly layered medium.

where

go

and

Go

are the fields of the TEM mode,

gl

and

GI

(1 > 0) are the fields of the higher modes, and z, is the z-coordinate -of the ground plane which we take as a reference point. The fields at the aperture ( z = 2,) are

The magnetic current over the aperture is given by

Equation (1.30) can be written as

Assuming

Eo

and

go

to be the known incident fields, we can solve the above equationsfor the unknown coefficients. The input impedance Z, of the antenna is

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derived from the reflection coefficient To by the equation

where Zo is the characteristic impedance of the coaxial cable.

If the dimension of the coaxial cable is much smaller than the dimension of the patch, use of higher-order modes is generally not necessary. If only the TEM mode is used, Eqs. (1.1 14)-(1.118) become

And the fundamental equation two can be written as

From Eq. (1.127), the input impedance can be expressed as

which is the same as the reaction formula used in Chen et al. [ 5 ] .

FIGURE 1.10 Overlapping piecewise rooftop basis function for currents in (a) x-

direction and (b) y-direction. 1.3.2 Basis Functions

The basis functions on the patches are taken to be overlapping piecewise rooftop functions as shown in Figure 1.10.

The basis functions on the probe are taken to be a set of overlapping triangular functions as shown in Figure 1.1 1.

To allow for azimuthal variation of the current on the probe, the basis functions on the probe are written as

-

f

b.s

),= ;, sin k q

{nk,j

ices

xr

}

- ra)

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FIGURE 1.1 1 Overlapping triangular basis function.

where J,(k, r,) is the kth-order Bessei function of the first kind, and cos a = kJk,

sin a = kdk,

1.3.3 Multiple Feeds and Shorting Pins

1.3.3.1 Multiple Feeds. In this section we will extend our formulation t o handle multiple feeds. It is assumed that only the TEM mode is required to model the fields inside the coaxial cables. Let the total number of probe feeds be P.

The fields inside the cables can be written as

where v; represents the intensity of the incident waves and v; represents the intensity of the reflected waves. The magnetic current over the apertures is given by

=

C

(up+

+

vvp)

Gp

(1.135)

p = 1

where ~p,t,h2 is the basis function on the patch, f,,,, is the basis function on the

probe, and fat,,,, is the attachment mode, a special basis function used to ensure

the continuity of current at the patch-probe junction. Now, Eq. (1.22) can be written as

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26 PROBE-FED MICROSTRIP ANTENNAS

Equation (1.30) can be written as

Solving Eqs. (1.138), (1.140), (1.143), and (1.145) for v; in terms of v; yields

where [s] is the s-matrix (scattering matrix).

One way to obtain the s-matrix numerically is to let

then solve for v, for each set of v i , which will be the corresponding column of the s-matrix.

1.3.3.2 Shorting Pins. Shorting pins are sometimes used to modify the behavior of patch antennas. For example, it can be used to suppress some modes of the patch current. Shorting pins can be modeled much the same way as probes. Modeling shorting pins is actually simpler, because they d o not couple t o the outside directly. The same basis function used on a probe can be used on a pin. For a shorting pin connecting the ground plane and a patch, attachment modes need to be used on the pin-patch junction. For a shorting pin connecting two patches, attachment modes need to be used at both pin-patch junctions.

1.3.4 Attachment Modes

The "attachment modes" are special types of basis function used to model the current in the vicinity of the probe-patch junction and to ensure the continuity of the current at the probe-patch junction. Part of a n attachment mode will exist on the probe surface (see Figure 1.12) and part of it will be on the patch surface.

Let us first find the current existing on the top of a magnetic wall cavity [6] excited by a uniform filament of current at (xi, y;). This current will be used in our attachment mode definition later.

FIGURE

function.

1.1 2 Attachment mode on the probe: half-triangular

The dimension of the patch that the probe attaches to is a x b. The fields inside the cavity are found to be

m n ( x + t ) cos m n ( i p + f ) cos nn(Y+t) cos nn(y;+t) ' COS a b b (1.149) a and where 1 p=O and p = m or n.

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COS

..(xi

+;)

,n ..(Y +

f)

cos

..(.

+

f)}

a b b (1.152)

By expanding cos ax into a Fourier series, we have I

m

6 , cos mx l~ cos ax

C

-(-I)"=

m=O m -a a sin ax ' - R l x < a (1.153)

Taking derivative a t both sides of the above equation, we have

-5

Em sin mx

sin ax

( - I ) " ' = - 7 ~ - - x 5 x

<

x (1.154)

,,,=,,

m 2 - a 2 sln ax '

Using these two equations, the sum with respect to m in Eq. (1.151) can be carried out to yields where sin Z ( x

+

x i )

+

sin Z [a

-

( x i - x ) ] , x < x ; sin Za sinZ(x+x;)-sinZ[a-(x-xi)], x,xi sin Za

cos Z ( x

+

xi)

+

cos Z(a

-

Ix

-

xkl)

5,

=

Z sin Za

k,a Z cos Zx'

-

jk, ejkxxb

+

j sin- ---I - COS-

2 sin Za/2

jkx,b k, cos Zxb k,a jk, sin Zx'

sin

-

+

+ n b 2

[

+

z

sin ~ a / 2 2

z

cos ~ a / ; cos

Y ] }

However, the above series is very slowly convergent. To avoid computation problem due to slow convergence, we separate the_series into two components. One is a fast convergent regular series denoted by

f

~ g u ' a r . The other is a slowly

convergent series which contains the singular behavior of

yp

at ( x i , yb) and denoted by

7$"9""'.

We write

where

S,, = Z cos Zxbjsin Za/2 (1.163)

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If xb < a/2, S,, and C , , decay exponentially. When xb approaches a/2, these terms decay more slowly. However, unless the feed is at a corner of the patch, one can always avoid this problem by choosing the coordinates wisely.

The last step in Eq. (1.165) is obtained using partial fraction expansion. From Eq. (1.153), we have

Hence, A(u) can be summed up in closed form:

-

j cos(uyb) . k b sin(uyb) k b

-

-- s m + + j u ub COS -2 sin

-

ub 2 2 COS

-

2 Note that Hence

The slowly convergent series (1.165) has thus been summed in closed form. Note that the first term in Eq. (1.172) is the current induced on two infinite parallel conducting plates by a uniform filamentary current.

Now we can define our attachment mode. Let (xp, yp) to be the center of the probe. The probe parts of the attachment modes are chosen to be

sin k q

whose Fourier transform is

where f OF (2) is a half-triangular function shown in Figure 1.1 1.

The patch parts of the attachment modes have to be chosen so that the total currents are continuous at the junction; that is, the currents that flow out of the junction are equal to the currents that flow into the junction from the probe. Using the patch current of a magnetic cavity which is given by Eq. (1.161) above, we define our patch part attachment modes t o be

where

x ~ = x p + r a c o s ( p

-

z p i s the patch current of the cavity and (m,, n,) is theresonant mode of the cavity. We subtract the resonant mode from the current

T p

in our attachment mode definition because the resonant mode is large and is all over the patch but the unknown patch current can be modeled by other basis functions on the patch, and the attachment mode is only intended to be used to ensure the continuity of the total current so we d o not want the resonant mode to be included in the attachment mode. Since the resonant mode is continuous all over the patch, subtracting it from the patch current will not affect the current continuity.

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From Eqs. (1.180) and (1.181), the following equations can be derived [they are merely linear combinations o f Eqs. (1.180) and (1.181)]:

C ~ C , ~ U X ~ cos vyb] = jk J , ( k s r,) e"' cos vy, cos ky

(1.184)

S k [ e j u x k o s vyb] = jk+ ' 3 , ( k s r,) e P X p sin vy, sin ky (1.185) Ck[ejuxb sin vyb] = jk Jk ( k , r,) ejuxp sin vy, cos ky (1.186)

S, [ejux"in vyb] =

-

jk + J k ( k , r J ejUxp sin vy, sin ky (1.187)

Sk[sin(uxl,

+

9,) cos(vy;

+

€J2)l

= Jk(k,ra) sin ky cos

(1.191)

7;;

= - ~ ~ + + ' k ~ ~ J ~ ( k , r , ) s i n k c c ( P k ~ + j k y )

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Z

cos(ZXp

+

F)

s;,

= sin Za/2

z

sin (Zx,

+

F)

S" Zn - -

-

sin Za/2

z

cos(Zxp

+

F)

C" - ' 2" -I cos Za/2

sin (k, b/2)

+

j sin(kayp) cos(kyb/2)} (1.200) cos (k, b/2)

A"(ka) = -

L

{

Jin(ka 'P) sin (k, b/2) - j cOs(knyp) C O S ( ~ , b/2) k, sm(k, b/2) cos (k, b/2)

REPRESENTATIVE NUMERICAL A N D EXPERIMENTAL RESULTS 35

(1.205)

cos a = kdk,

sin cc = ky/k,

For the simplified case in which azimuthal variation is not taken into account, the patch part attachment mode is given by

f""

=

xe

= - j k i 2 JO(kpro)(fkx

+

jk,)

- k i 2 J,(& k0rJ ~ ' ( k ~ ( i k , k ,

+

jk;)e-jkyYp

Z cos Zx, sin kxa/2 Z sin Zx, cos kxa/2 '

(

sin za/2 + j cos z a / 2

1.4 REPRESENTATIVE NUMERICAL A N D EXPERIMENTAL RESULTS Based on the formulation of Sections 1.2. and 1.3, a computer code has been written for probe-fed multilayer multipatch rectangular microstrip antennas. Special attention was paid to branch cuts and poles of the Green's function, avoidance of redundant reaction computations, and the spectral integral.

In this section, representative numerical results are preiented for the single patch, single patch in multidielectric media, coplanar subarrays, and two-layer stacked patches. Experimental results are included to validate the numerical results.

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Conducting patch

FIGURE 1.13 Probe-fed rectangular patch on a grounded substrate.

1.4.1 Single Patch [I 51

1.4.1.1 Introductory Remarks. Figure 1.13 shows a rectangular patch with dimensions a x b printed on a grounded substrate of thickness d and relative permittivity 8,. The patch is fed by a coaxial probe a t (x,, y,) with inner radius r, and outer radius r,. The radiation patterns of the antenna obtained from full-wave analysis are basically the same as those obtained from the cavity model [3]. Here, we are interested in the input impedance of the antenna.

The input impedance of this antenna has been studied by many authors using full wave analysis [7-151. In reference [9], the driving source is taken to be the probe current, which is assumed to be a constant filament. N o effort was made to ensure current continuity at the probe-patch junction, nor was the singular nature of the current at the junction modeled by the basis functions. In later papers by Aberle and Pozar [lo, 111, the driving source is taken to be a delta-gap voltage a t the base of the probe. Current continuity at the probe-patch junction is ensured by the use of an attachment mode. However, the numerical results obtained by Aberle and Pozar are still based on the assumption that the probe current is uniform across the thickness of the substrate.

In this section, numerical results based on the theory presented in Sections 1.2 and 1.3 are presented. This theory differs from that of Aberle and Pozar [lo, 111 in

Sommerfeld integrals. In particular, the studies of Hall and Mosig [8] and of Zheng and Michalski [I21 are valid for electrically thick substrates.

1.4.1.2 Comparison with ExperimentalResuIts. We first check the validity of our analysis and computer code by comparing our numerical results with the measured data given in Figure 8 of Hall and Mosig [8]. The feed is located at

x, = 0 and y, = 0.5b - F (see Figure 1.13). The comparison is shown in Figurel.14. It is seen that the agreement is excellent, giving confidence to our theory. The agreement appeared to be comparable to the corresponding comparison between measured data and the theoretical results of Hall and Mosig [8] shown in their Figure 8. The difference between their approach and ours is that they worked in the spatial domain while we worked in the spectral domain.

Next, we compare theoretical results with the measurements given in Schaubert et al. [14]. One difficulty encounteredis that the value of the substrate loss tangent, tan 6, was not given in reference [14]. Using tan 6 = 0.0006 and 0.003, we obtained the results shown in Table 1.1 for their cases l b and 2b, the antenna parameters ofwhich are given in Table 1.2. As far as resonant frequencies are concerned, the calculated values are insensitive to tan6 and agree with experimental values to within 2.5%. As far as the resonant resistance is con- cerned, it is seen that as t a n s increases from 0.0006 to 0.003, the resonant resistance decreases by 17.9% and 12.7% for cases l b and 2b, respectively. This shows that when comparison with experiment is made, it is important to use the true value of tan 6, which, however, was not given in reference

1141.

Table 1.1 also shows the moment method results of Schaubert et al. 1141, which were obtained assuming the probe current to be a constant and no mention was made of the value of tan 6 used in their computations. Our results are based o n a model which uses multiple expansion functions t o take into account both the axial and azimuthal variations of the probe surface current. It is found, however, that the results obtained with and without azimuthal variations are almost identical, provided that the diameter of the probe is much less than the patch dimensions.

1.4.1.3 Changes of Impedance as Substrate Thickness Increases. In this section we results-illustrating the change to the input impedance as the substrate thickness increases, keeping the feed location and the other parameters

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38 PROBE-FED MICROSTRIP ANTENNAS

FIGURE 1.14 Measured [8] and calculated input impedance data for an a = 39.22-mm

by b = 46.30-mm rectangular patch with a coaxial feed at F = 15.43 rnm using the present formulation. d = 7.976 mm, E, = 2.484, tans = 0.0006, r, = 0.635 mm, r, = 2.1 mm. x x x

Calculated data, A f = 0.04 GHz. mmm Measured data [a], A f = 0.04 GH..

TABLE 1.1 Comparison of Measured Data from Schaubert et al. [I41 and Calculated Results Our Calculated Results Measured R,(R) Calculated Results t a n 6 = t a n s = C141 Case f,(GHz) R,(R) f,(GHz) 0.0006 0.003 f, R m

TABLE 1.2 Parameters for Cases 1 b and 2b of Schaubert et al. [I 41

REPRESENTATIVE NUMERICAL A N D EXPERIMENTAL RESULTS -Advanced model 70.0- R -X -

- - - -

Advanced model Af = 0.02 GHz 227.00 228.00 229.00\*\230.00 231.00~.'232.00 233.00 234.00 235.00 -lO.O1 x107 Frequency (Hz) (a) 70.07 R -X----. Advanced model f: 2.14-2.26 GH Af = 0.03 GHz 50.0- ? 30.0- E 214.00 216.00 218.00 220.00 222.00 224.00 226.00 Frequency (Hz) (b)

FIGURE 1.15 Input impedance for a fixed feed position computed using the simple model and the advanced model presented in rectangular coordinates and in Smith Charts. a=3.0cm, b=2.0cm, &,=10.2, tan6=0.001, F=0.75cm,r,=0.22mm, R,= 1.40mm. (a) d = 0.127 cm, (b) d = 0.254cm, (c) d = 0.381 cm, (d) d = 0.508 cm, (e) d = 0.635 cm.

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207.00 209.00 211.00 213.00 215.00 217.00 219.00 22i.00 Frequency (Hz) x107 (C) 1.86 GHz t

-

t.Advanced mode Simple mod f: 1.86-2.22 GHz 80.0

'

\ 70.0 0 0 - _ / - 0.- 0 50.0 ___.--. K 40.0 30.0 20.0

-

. \

. .

... . + - . . . _ _ . I I I I 186.00 190.00 194.00 198.00 202.00 206.00 210.00 214.00 218.00 222.00 Frequency (Hz) .lo7 (d) 0.0 I I 182.00 186.00 190.00 194.00 198.00 202.00 Frequency (Hz) xlo7 (el FIGURE 1.15 (continued)

fixed. The calculations are for a patch with a = 3.0 cm, b = 2.0 cm, fabricated on a substrate with E, = 10.2 and tan 6 = 0.001. The input impedance is computed for five thicknesses, beginning with d = 0.127cm and increasing at multiples of 0.127cm. For the TM,, mode, these thicknesses correspond to approximately the range 0.011,-0.0411,. The probe inner and outer radii are r, = 0.43 m m and R, = 1.40 mm, respectively. The feed position is pegged at F = 0.75 cm, which yields an input resistance of about 50R for d = 0.127 cm. The computations are performed using both the model assuming a uniform probe current (simple model) and the model which uses multiple expansion functions for the probe current (advanced model). The results are shown in Figures 1.15a-e, in which the impedances are presented both in rectangular coordinates and in Smith Charts. The following features are noted:

The results for the simple model are reasonably accurate for case a only. Significant deviations from the advanced model begin to appear in case b. For case e the simple model is totally inaccurate.

At the frequency where the input resistance is maximum, there is significant inductive reactance even for case a. The reactance curve increases as the substrate thickness increases, and it does not become negative for cases c, d, and e. For these cases, a resonant frequency cannot be defined in terms of

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42 PROBE-FED MICROSTRIP ANTENNAS REPRESENTATIVE NUMERICAL AND EXPERIMENTAL RESULTS

FIGURE 1.16 Impedance bandwidth (VSWR= 2) as a function of substrate thickness when the feed is located at the optimum position. a = 3.0cm, b = 2.0cm, e, = 10.2, tan S = 0.001, r, = 0.22mm, R, = 1.4mm.

zero input reactances and it is more meaningful to define it as the frequency at which the input resistance is maximum.

3. The computed impedance bandwidth (VSWR= 2) are 0.56%, 1.13%, 1.33%, and 0.59% for cases a, b, c, and d, respectively. For case e(d = 0.635 cm), the VSWR = 2 circle does not intersect the impedance loci. This shows that, if the feed position is fixed, the impedance bandwidth does not increase monotonically with substrate thickness.

1.4.1.4 Optimization of Feed Position for Maximum Bandwidth. The results presented in Section 1.4.1.3 are for a fixed feed position. If the feed position is allowed to be optimized for maximum bandwidth, we obtain, using the advanced model, the result of Figure 1.16, which shows that the bandwidth does increase as d increases. The optimal feed positions are F = 0.74,0.71,0.67,0.63, and 0.55 cm for the five values of d. The impedance loci for d = 0.635 cm ( E 0.041 1,) when F is

at 0.55cm is shown in Figure 1.17, which is considerably changed from that of Figure 1.15e. Similar changes also occur for the other four thicknesses.

1 A.2 Single Patch in Multidielectric Media [la, 221

1.4.2.1 Introduction. Microstrip antennas with more than one dielectric layers are of theoretical and practical interest. Several configurations of such

FIGURE 1.1 7 Smith Chart display of the input impedance obtained using the advanced model for d = 0.635cm when the feed is placed at the optimum location (F = 0.55 cm). a=3.0cm, b=2.0cm,&,= 10.2, tan6=0.001,rp=0.22mm,Rp= 1.4mm.

antennas are shown in Figure 1.18. In Figure 1.18a, a superstrate or dielectric cover is used to protect the patch against environmental hazards. If a naturally occurring dielectric layer such as ice is formed on top of the superstrate cover, the three-layer configuration of Figure 1.18 b results. Figure 1.18 b can also represent the situation when an unwanted air space exists between the substrate and the superstrate(&,, = 1). If an air gap is deliberately introduced between the substrate and the ground plane to alter the resonant frequency of the antenna, we have the one-superstrate two-substrate geometry shown in Figure 1.18~. In this section the results of our study of these configurations using the formulation of Sections 1.2 and 1.3 are presented.

1.4.2.2 Rectangular Patch with Dielectric Cover (Superstrate). A number of authors have studied the effect of a protective dielectric cover on a probe-fed rectangular patch antenna [16-201.

Figure 1.19 shows calculated input impedance of a rectangular patch with ~ , ~ = ~ , ~ = 2 . 6 4 , tanb2=tan63=0.003, d3=0.159cm, a=7.62cm, b = 11.43cm, r,, = 0.0635 cm, x, = 1.52cm, and y, = 0.385 cm for different values of the dielec- tric cover thickness d,. The cased, = 0 corresponds to the uncovered antenna, for

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FIGURE 1.18 Microstrip antennas with multidielectric layers. (a) Rectangular patch with a superstrate. (b) Rectangular patch with two superstrates. (c) Rectangular patch with a superstrate and two substrates.

dl = 0.795 cm 0.159 cm dl=Ocm

\

\ -

(NO cover)

V

1.13 1.15 1.17 1.19 1.21

Frequency (GHz)

FIGURE 1.1 9 Input impedance Z = R

+

j X versus frequency for three values of cover thickness d,. = E , ~ =2.64, tansl = tan~5~=0.0()3, d2=0.159cm, a=7.62cm, b=

11.43cm, r, = 0.0635 cm, x, = 1.52 cm, y, = 0.385 cm. Crosses indicate the experimental results for the case of no cover [3].

FIGURE 1.20 Input impedance Z = R

+

j X versus frequency for three values of cover relative permittivity. dl =d2 =0.159cm, &,,=2.64, tanC52=0.003,a=7.62cm, b= 11.43 cm,

r, =0.0635cm, x,= 1.52cm, y, =0.385cm.

which measured results are available in Richards et al. [2]. The measured data are indicated as crosses in Figure 1.19. It is seen that the dielectric cover decreases the resonant frequency. However, neither the resonant resistance nor the imped- ance bandwidth changes appreciably as dl varies from 0 to 0.795 cm.

Figure 1.20 shows the calculated input impedance for different dielectric cover relative permittivity E,,. As E,, increases from 1 (no cover) to 2.64 to 13.2, the resonant frequency de&eases,8ccompanied by a slight increase in the resonant resistance. There is again no significant change in the impedance bandwidth.

In Chen et al. [20], computer-aided design formulas based on curve-fitting a database of moment method results are presented for the resonant frequencies of the TM,, and TM,, modes of a rectangular patch with a superstrate. The formulas, in polynomial form, can be used for a wide range of substrate and superstrate thicknesses and permittivities.

1.4.2.3 Rectangular Patch with Two Superstrates. In this section we present numerical results for a rectangular patch with two superstrates. The first super-

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46 PROBE-FED MICROSTRIP ANTENNAS REPRESENTATIVE NUMERICAL A N D EXPERIMENTAL RESULTS

TABLE 1.3 Rectangular Patch with Two Superstrates: Figure 1 .I 6b with a = 3.375 cm, b=2.25cm, xp=O, yp=0.35cm, +,=3.2 (ice), &,=2.32, d2=0.76mm, &,,=4.0,

d, = 1.55 mm dl (mm)

f

o I @Hz) BW(%) HPBW 0.0 3.150 1.4 141" x 78" 1 .O 3.100 1.4 132" x 78" 2.0 3.070 1.4 125" x 78" Bare patch (dl = d2 = 0) 3.205 1.4 148" x 78"

strate is a dielectric cover for environmental hazard protection. The second superstrate represents ice formed on top of the cover. The geometry is that shown in Figure 1.18 b with the following parameters: a = 3.375 cm, b = 2.25 cm, x, = 0, yP=0.35cm,~,, =3.2(ice),e,,=2.32,d,=0.76mm,~,,=4.0,d,= 1.55mm.The results for resonant frequency, bandwidth (VSWR = 2), and half-power beam- widths are shown in Table 1.3 for three values of dl and for the bare patch (dl = d, = 0). The impedance loci and the normalized patterns for the four cases are shown in Figures 1.21 and 1.22, respectively.

The following features are noted:

1. The dielectric cover lowers the resonant frequency of the antenna by about 1.7%.

2. A second superstrate lowers the resonant frequency further: by 1.6% for dl = 1.0 mm and 2.5% for dl = 2.0 mm. Aside from a shift in frequencies, the shapes of the impedance loci remain essentially the same.

3. The superstrates d o not cause any noticeable change in impedance band- width,

4. The HPBW in the x-z plane decreases from 148" for the bare patch to 125' for the case with d, = 0.76mm and dl = 2.0mm. The HPBW in the y-z

plane remains at 78'.

1.4.2.4 Rectangular Patch with Superstrate and an Unwanted Air Space. In this section we present results illustrating the effect of an unwanted air space whichexists between the substrate and the superstrate plane. The geometry is still given by Figure 1.18 b, and the antenna parameters are a = 3.375 cm, b = 2.25 cm, x,=O, yp=0.35cm, &,,=2.32, dl=0.76mm, &,,=4.0, d3=1.55mm, &,,=I. The resonant frequencies and bandwidths for two values of thickness of the air space d, are shown in Table 1.4. It is seen that there is a slight increase of the resonant frequency due to the presence of the air space. However, the increase is only about 0.3 % as d, varies f;om 0 to 0.2 mm. The bandwidth remains at 1.4%.

FIGURE 1.21 (a) Impedance loci for rectangular patch with two superstrates: a=3.375cm, b=2.25cm, xp=O, yp=0.35cm, ~ , ~ = 2 . 3 2 , E,, =4.0, d2 =0.76mm,

d, = 1.55mm, E,, = 3.2, dl = 0. (b) Same as (a) except dl = 1.0mm. (c) Same as (a) except d, = 2.0 mm. (d) Bare patch (dl = d, = 0).

1.4.2.5 Rectangular Patch with a Superstrate and Two Substrates. In this section we present results for the case of a rectangular patch with one superstrate and two substrates. Of particular interest is when an air gap is deliberately introduced between the substrate and the ground plane. As in the case of no superstrate [21], it is expected that the resonant frequency of the antenna can be altered by changing the width of the air gap.

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d, = 0.8 mm, d , = 2.54mm. The measured and computed resonant frequencies

-

Bare patch (dl = d2 = 0 )

-

- -

-

- - -

-

dl= 0

....

..*.

.

...

. .

. .

.

dl = 1.0 rnm

.-.--.-.

dl = 2.0 mrn

FIGURE 1.22 (a) Normalized pattern in the x-z plane for rectangular patch with two superstrates. The antenna parameters are the same as in Figure 1.19. (b) Normalized pattern in y-z plane for rectangular patch with two superstrates. The antenna parameters are the same as in Figure 1.19.

(b)

FIGURE 1.22 (continued)

TABLE 1.4 Rectangular Patch with a Superstrate and an Unwanted Air Space: Figure 1.18b with a = 3.375~1~1, b= 2.25 crn, xp= 0, yp = 0.35 crn, E, = 2.32, d, =0.76rnrn,~,=l (air),&,=4.0,d3=1.55rnm

4

(mm) f o ~ (GH4 BW (%) 0 3.150 1.4 0.1 3.157 1.4 0.2 3.161 1.4

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50 PROBE-FED MICROSTRIP ANTENNAS

TABLE 1.5 Rectangular Patch with a Superstrate and Two Substrates: Figure 1 .l8c with a = 3.325cm, b=2.25cml xp=Ol yp=0.5cm, &,,=2.32, d2= 0.76mm,&,,=4.0,d3=1.55mm

TABLE 1.6 Rectangular Patch with Two Superstrates: Figure 1.18b with a = 2.0cm, b= 1.446 cm, xp = 0, yp=2.35mm, E,, =2.53, &,, =2.17, d, =0.8rnm,

E.? = 6.0, d- = 2.54 mm - d l (mm) Theory Measurement 0.0 3.725 3.660 1.0 3.674 3.615 2.0 3.641 3.585

for three values of dl are shown in Table 1.6. The close agreement gives

confidence to our theory. The discrepancy is most likely due to the tolerances associated with the values of the relative permittivities.

1.4.3 Coplanar Parasitic Subarray

1.4.3.1 Introductory Remarks. One way of increasing the bandwidth of the microstrip patch antenna is to introduce closely spaced parasitic patches on the- same layer as the fed patch. This has been a subject of considerable interest, both experimentally and theoretically. Wood [23] showed that the impedance band- width of a rectangular patch antenna can be enhanced by an adjacent patch. This was supported by the analysis of Mosig and Gardiol[24]. Aanandan et al. [25] studied a broadband design in which a rectangular patch was broken up into several gap-coupled rectangular strips. Another class of configurations consists of one fed rectangular patch and two or more parasitic patches placed symmetri- cally with respect to the fed element, forming a coplanar microstrip parasitic subarray. Figure 1.23 shows several configurations that have been studied: the radiating-edge (Figure 1.23 a) and nonradiating-edge (Figure 1.23 b) gap-coupled three-element subarrays and the five-patch cross (Figure 1.23~). Gupta [26] showed that the impedance bandwidth can be enhanced by using parasitic patches smaller than the fed patch. Lee et al. [27], MacKinchen et al. [28], and

Parasitic

REPRESENTATIVE NUMERICAL A N D EXPERIMENTAL RESULTS

FIGURE 1.23 Coplanar parasitic subarray configurations. (a) Radiating-edge-coupled three-element subarray, (b) nonradiating-edge-coupled three-element subarray, (c) five- patch cross.

Staker et al. [29] used parasitic patches of the same size as the fed patch and obtained improved gain characteristics. Planar arrays of five-patch crosses were also built and tested [30].

In this section the basic characteristics of the coplanar microstrip parasitic subarrays shown in Figure 1.23 are presented based on the spectral domain full-wave analysis of Sections 1.2 and 1.3 [31].

1.4.3.2 NumericalResults. The results for the input impedance of a radiating- edge-coupled three-element subarray for five gap widths are shown in Figure 1.24. The dimensions are given in the inset of Figure 1.24. As expected, the gap width is a sensitive parameter controlling the nature of the coupling. The results for case 5 (widest spacing) are basically those of the single patch without parasitics. As the gap width decreases, the curve for R broadens and becomes doubled humped for cases 1 and 2. Case 2 appears to be the one with the best bandwidth characteristics.

The results for a nonradiating-edge gap-coupled three-element subarray are shown in Figure 1.25. Here case 1 appears to yield the best impedance character- istics. The results for a five-patch cross are shown in Figure 1.26. For this configuration, case 2 again yields the best impedance characteristics.

In the above example, the impedance bandwidth obtainable is about 5-6%. By optimizing the various parameters, Gupta et al. [26] have reported that 10-20% bandwidths are obtainable. However, pattern degradation over the frequency band has been observed.

1.4.3.3 Comparison with Experiment. In Figure 1.27 we compare the theoreti- cal input impedance results obtained from our analysis for a three-element linear array given in reference [26] and compared with his experimental data given in

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Frequency (GHz)

-100

l

I I I I I I I 1

3.1 3.2 3.3 3.4 3.5

Frequency (GHz)

FIGURE 1.24 Input impedance Z = R +jX of a radiating-edge-coupled three-element subarray in which the parasitic patches are smaller than the fed patch, for five gap spacings. Substrate relative permittivity = 2.64, loss tangent = 0.001. Substrate thickness =

0.159cm.

-.-.-,

d = 0.0825 cm; ----, d = 0.165 cm; ---, d = 0.33 cm; --,

d=0.66m;- , d = 1.32cm.

Figure 9.44 of Gupta [26]. It is seen that, aside from frequency shift of about 3%,

the agreement between the computed and measured R and X is very reasonable. The shift in frequency may be due to the uncertainties in the values of substrate relative permittivity and substrate thickness.

Figure 1.28 shows the H-plane radiation patterns computed a t three fre- quencies for one of the three-element linear subarrays in reference 1291. T h e pattern a t 4.73 GHz compares well with the measured pattern given in Figure 2

of reference [29]. 3.1 3.2 3.3 3.4 3.5 Frequency (GHz) 0

l

I I I I I I I I 3.1 3.2 3.3 3.4 3.5 Frequency (GHzI

FIGURE 1.25 Input impedance Z = R

+

j X of a nonradiating-edge-coupled three- element subarray in which the parasitic patches are smaller than the fed patch. Substrate loss tangent, permittivity, thickness, and the gap spacings are the same as in Figure 1.24.

-.-.-

, d = 0.0825cm; ----, d 0.165 cm;

.

---, d 0.33 cm;

.

. . . a , d = 0.66cm;

-, d = 1.32 cm.

1.4.4 Two-Layer Stacked Patches [38]

1.4.4.1 introductory Remarks. The stacked electromagnetically coupled patch (EMCP) antenna shown in Figure 1.29 has been shown experimentally to provide a n impedance bandwidth which is significantly larger than that of a single patch [32]. A number of theoretical papers o n the probe-fed E M C P

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54 PROBE-FED MICROSTRIP ANTENNAS Frequency (GHz) 2.605 3.9 crn 2.605 cm I /0.45 ern 2.605 crn 2.605 crn 0 -50

1

I I I I I I I I 3.1 3.2 3.3 3.4 3.5 Frequency (GHz)

FIGURE 1.26 Input impedance Z = R

+

jX of a five-patch cross in which the parasitic patches are smaller than the fed patch. Substrate permittivity, loss tangent, thickness, and the gap spacings are the same as in Figure 1.24.

-.-.-,

d=0.0825cm;

---,

d = O.165cm; ---, d = O.33cm;

...,

d = O.66cm;

-

, d = 1.32cm.

antenna have also appeared in recent years [33-371. The theories in references [33-371 all assumed a uniform probe current. Also, except for reference [35], no attachment mode was used to ensure current continuity at the probe-patch junction. In this section we present some theoretical results obtained for the

stacked EMCP using the formulation of Sections 1.2 and 1.3 [38].

REPRESENTATIVE NUMERICAL AND EXPERIMENTAL RESULTS 2.605 crn 0.165 cm 2.605 cm 1.95 cm 0.45 crn

-

Present theory 0

l

I I I I I I I I 3.1 3.2 3.3 3.4 3.5 Frequency (GHz)

\

-

Present theory

---

Measurement [261 -50

1

I I I I I I I I 3.1 3.2 3.3 3.4 3.5 Frequency (GHz)

FIGURE 1.27 Comparison of the theoretical input impedance results obtained from our theory for a three-element linear array given in Gupta [26] with the experimental data given in Figure 9.44 of Gupta [26]. Substrate relative permittivity = 2.55; loss tangent = 0.001, substrate thickness = 0.159 cm.

To establish contact with measurement, we first compare the predictions of our model for the antenna depicted in Figure 8 a of Barlately et al. [33] with their experimental data. This is shown in Figure 1.30. The close agreement gives confidence to our theory. (A corresponding comparison between the theoretical predictions of Barlately et al. and experimental data was given in Figure 10 of Barlately et al. [33].)

References

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