Electromagnetic and Experimental Modeling of Waveguide-Based Spatial Power Combining Systems

(1)

Abstract

HICKS, CHRIS WAYNE. Experimental and Electromagnetic Modeling of Waveguide-Based Spatial Power Combining Systems. (Under the direction of Michael B. Steer.)

Recent technological advancements and demands for high power sources at microwave and millimeter-wave frequencies have initiated extensive theoretical and experimental research in the area of quasi-optical and spatial power combining. The work described here was motivated by the necessity to develop a modeling en-vironment for the electromagnetic analysis of planar quasi-optical and spatial power combining systems, in order to understand physical fundamentals and provide a basis for the design process. Two types of planar quasi-optical (QO) and spatial power combining systems are investigated.

Propagation in a QO parallel plate system is investigated with the aim of establishing the mode structure and characteristics of the modes. Theoretical elec-tromagnetic properties of a Gauss-Hermite beammode expansion was developed, and verified experimentally, for the prediction of the resonant frequencies of the structure and beammodes dispersion behavior. The system was designed, fabricated, tested, and showed good agreement between the experimental and theoretical results. In addition, a QO parallel-plate stripline-slot amplifier system was designed, tested and compared to a QO open HDSBW amplifier system with Vivaldi-type antennas. Experimental results verify that a QO parallel-plate stripline-slot amplifier proposed in this dissertation can be modeled using Gauss-Hermite beammodes.

(2)

(3)

Experimental and Electromagnetic

Modeling of Waveguide-Based Spatial

Power Combining Systems

by

CHRIS WAYNE HICKS

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

ELECTRICAL ENGINEERING

Raleigh

2002

APPROVED BY:

Co-Chair of Advisory Committee

(4)

Dedication

I dedicate this work to my wife, Shirley D. Hicks, my daughter, Christen A. Hicks, and my mother, Annie B. Hicks. I also dedicate this work to my late father, Andrew Hicks Jr. who provided me with the vision to dream dreams.

I also dedicate this work to God who gave me the mental and physical strength to pursue and obtain my Ph.D. degree.

(5)

Biographical Summary

Chris Wayne Hicks was born on September 8, 1962 at Kindley Air Force Base, Bermuda while his father was enlisted in the United States Navy. Chris attended high school at South Florence High School in Florence, South Carolina. He received his B.S. degree in Electrical Engineering at the University of South Carolina, Columbia, South Carolina in May 1985 and a M.S. degree in Electrical Engineering from North Carolina Agriculture and Technical State University in Greensboro, North Carolina in May 1994. Since June 1985, he has been employed at the Naval Air Systems Command (NAVAIR) where he currently works for the RF Sensors Division in Patuxent River, Maryland. In 1995, he enrolled in North Carolina State University to pursue his Ph.D. degree in electrical engineering where he worked as a Research Assistant for the Electronics Research Laboratory in the Electrical and Computer Engineering Department. Chris received two one-year NAVAIR training fellowships to pursue his graduate studies. He is a member of the Institute of Electrical and Electronic Engineers (IEEE) Microwave, Theory Technique Society and a member of the Antenna and Propagation society. He is also a member of the Etta Kappa Nu honor society.

(6)

Acknowledgements

I would like to thank my academic advisor Dr. Michael B. Steer for his support and guidance during my graduate studies. I also thank Dr. James Mink for his wisdom, patience and for serving on my advisory committee. I thank Dr. Alexander Yakovlev for serving on my advisory committee and teaching me the fine art of electromagnetics. I also thank Dr. Gianluca Lazzi, Dr. Robert J. Nemanich, Dr. James F. Kauffman, and Dr. James Harvey for serving on my advisory committee.

I would like to thank Dr. Huang-Shen Hwang and Dr. Todd Nuteson for their many discussions on two-dimensional and three-dimensional quasi-optical power combining research. I would like to thank Dr. Ahmed Khalil and Dr. Mostafa Abdulla for their discussions on electormagnetics. I would like to thank Mrs. Jaee Patwardhan-Naik for her moral support and encouragement. I would like to thank Dr. Mete Ozkar for assisting me with measurements and Dr. Carlos Christofferson for his computer expertise.

Finally, I would like to thank my wife, Shirley and my daughter, Chris-ten, for their loving support.

(7)

List of Tables

3.1 Selected resonance frequencies of the parallel-plate resonator system. 40

(13)

List of Figures

1.1 RF components sub-area military essential electronics. . . 2

1.2 A 3-D grid quasi-optical power combining system. . . 3

1.3 A 2-D dielectric slab quasi-optical power combining system. . . 4

1.4 A 2-D cascade quasi-optical system. . . 6

1.5 Aperture-coupled stripline-to-waveguide transition. . . 7

2.1 Free space quasi-optical grid amplifier. . . 15

2.2 Concave and convex lens dielectric slab power combining systems. . . 17

2.3 X-Band waveguide spatial power combining system. . . 18

3.1 Passive 2-D quasi-optical power combining system with concave lenses; (a) open structure and (b) closed structure. . . 26

3.2 The 2-D HDSBW system with convex/concave lenses and 4×1 MES-FET amplifier array. . . 27

3.3 Input and output coupling antennas on RT/Duriod substrate: (a) MESFET Vivaldi amplifier; and (b) MMIC stripline-slot amplifier. . . 28

(14)

3.4 Electric-field wave model for 2-D power combining system. . . 29

3.5 Test configuration for the confocal parallel-plate resonator system. . . 36

3.6 Reflection coefficient at the input to a confocal 2-D parallel-plate

resonator system. . . 37

3.7 Electric-field mode profile at 6.898 GHz for the parallel-plate confocal

system. . . 38

3.8 A plot of the magnitude of S₁₁ and S₂₁ for selected resonance

fre-quencies of the parallel-plate resonator system. . . 39

3.9 The concave-lens system configuration for a unit-cell Vivaldi-based

amplifier. . . 41

3.10 The concave-lens system configuration for a unit-cell amplifier. . . 42

3.11 Amplifier gain for a unit-cell MMIC amplifier; (a)Vivaldi cascade

MMICs and (b) single stripline-slot MMIC. . . 43

4.1 Geometry of a closed-boundary waveguiding structure containing

aper-tures and conducting strips in the presence of an impressed electric

current source. . . 47

4.2 An aperture-coupled electric-magnetic layered waveguide transition. . 53

4.3 Geometry of a rectangular waveguide. . . 54

4.4 An aperture-coupled stripline-to-waveguide transition. . . 64

(15)

5.1 Full view of waveguide transitions: (a) single slot-strip-slot; (b)

dou-ble slot-strip-slot; (c) doudou-ble slot-strip-slot one two shifted slots; (d)

triple slot-strip-slot; and (e) single slot-strip-slot with two strips. . . . 83

5.2 Top view of waveguide transitions: (a) single slot-strip-slot; (b)

dou-ble slot-strip-slot; (c) doudou-ble slot-strip-slot one two shifted slots; (d)

triple slot-strip-slot; and (e) single slot-strip-slot with two strips. . . . 84

5.3 Geometry of a centered transverse slot between two rectangular

waveg-uides: a) full view; b) top view. . . 86

5.4 Magnitude of S₁₁ versus varying centered transverse slot lengths

be-tween two rectangular waveguides: MoM (solid line), and published

experimental and simulation results (Fig. 2b curves (1) and (3)

in [72]) (dashed line). . . 87

5.5 MoM (solid line) and HFSS (dashed line) comparison of the scattering

parameters at the center of a transverse slot between two rectangular

waveguides: (a) magnitude of S₁₁ and S₂₁; (b) magnitude and phase

of S₁₁. . . 88

5.6 Geometry of an single slot-strip-slot waveguide transition.. . . 90

5.7 MoM (solid line) and HFSS (dashed line) comparison for the

scat-tering parameters for the single slot-strip-slot waveguide transition.

Magnitude and phase: (a) S₁₁; and (b) S₄₁. . . 91

scat-tering parameters for the double slot-strip-slot waveguide transition.

(a) Magnitude and (b) phase. . . 93

(16)

scat-tering parameters for the double slot-strip-slot waveguide transition

with two shifted slots . Magnitude and phase: (a) S₁₁; and (b) S₄₁. . 94

scat-tering parameters for the triple slot-strip-slot waveguide transition.

Magnitude and phase: (a) S₁₁; and (b) S₄₁. . . 95

scat-tering parameters for the single slot-strip-slot waveguide transition.

Magnitude and phase (a) S₁₁, and (b) S₄₁. . . 97

5.12 Full and top view: (a) series 1 × 2 slot-strip-slot coupler array; and (b) series 1 × 2 slot-strip-slot coupler array with breaks in the strip. . 98 5.13 Input scattering parameter S₁₁ for the series 1 × 2 slot-strip-slot

coupler array; (a) magnitude and (b) phase. MoM (solid line), HFSS

0.001 dB (dashed line) and HFSS 0.005 (dotted line). . . 99

5.14 MoM results for the magnitude of S₁₁ for the series 1 × 2 slot-strip-slot coupler array; (a) ε₂ = 1.0 (solid line) and ε₂ = 2.2 (dotted

line). . . 100

5.15 Reverse coupling for the series 1 × 2 slot-strip-slot coupler array,

ε₂ = 1.0, MoM (solid line) and HFSS (dotted line). . . 101

5.16 Input scattering parameter S₁₁ for the series 1 × 2 slot-strip-slot coupler array with two strips; (a) magnitude and (b) phase. MoM

(solid line), HFSS 0.001 dB (dashed line). . . 102

5.17 MoM simulation for the reverse coupling for the series 1×2 slot-strip-slot coupler array, ε₂ = 1.0. . . 103

(17)

5.18 Full and top view of a 2 × 2 slot-strip-slot array . . . 105

5.19 MoM simulation of a 2 × 2 slot-strip-slot waveguide-based array; ε₂ = 1.0 (solid line) and ε₂ = 2.2 (dashed line); S₁₁ (a) magnitude, (b) phase. . . 106

5.20 Reverse coupling for the series 1×2 slot-strip-slot coupler array,ε₂ = 1.0, MoM (solid line) and HFSS (dotted line). . . 107

A.1 The geometry for: (a) the dzdz0 integration for self-coupling for a single slot; (b) the dzdz0 integration for the coupling between two slots; and (c) the dx0 integration for all slots. . . 126

A.2 Testing and basis functions for the longitudinal strip: (a) completely overlapping case; (b) partially overlapping case; and (c) non-overlapping case. . . 130

A.3 Longitudinal strip overlapping case (a) z0 < zi, and (b) z0 > zi. . . 131

A.4 Longitudinal strip testing and basis functions do not overlap . . . 134

A.5 Longitudinal strip test and basis functions partially overlap. . . 136

A.6 Crossed strip and slot testing and basis functions: (a) non-overlapping case; and (b) overlapping case. . . 140

(18)

Chapter 1 Introduction

1.1 Motivation and Objective of This Study

Military and civilian applications require significant power at microwave and millimeter-wave frequencies [1]. Medium-to-high power levels are needed for applications such as communications, active missile seekers, radar, and millimeter-wave imaging. To meet this need, klystrons, traveling-wave tubes, and gridded tubes are generally utilized as shown in Fig 1.1. However, tubes are bulky, costly, require high operat-ing voltages, and have a short lifetime. As an alternative, solid-state devices offer several advantages such as lightweight, smaller size, wider bandwidths, and lower operating voltages. Lower costs also result because systems can be constructed us-ing planar fabrication techniques. However, as the frequency increases, the output power of solid-state devices decreases due to their smaller physical size. Therefore, to achieve sizable power levels that compete with the power levels generated by vacuum tubes, many solid-state devices must be combined in an array configuration

(19)

CHAPTER 1. INTRODUCTION 2

utilizing spatial power combining techniques [2]. The analysis, modeling and design procedure for spatial power combiners is not well developed. While many spatial power combining topologies have been investigated, it is not clear which topology is optimum. One form of spatial power combining is quasi-optical (QO) power

com-Potential IVHS Potential IVHS EHF Satcom Smart Weapons Commercial Telecommunications Commercial Comm. Base Stations Seekers

EW Phased Array MCMs (Broad Instantaneous Bandwidth)

DBS Missile Radar Array Acquisition Target Satcom Illuminator Space Object ID Non Coop. Target Rec.

ISAR

Target Discrimination

Air Defense, Surveillance & Fire Control Radar

SHF Satcom Smart Weapons EW Jammers Satcom Illumin. Target Recognition Armor Protection EW Jammers LMDS Surveillance & Multi-Functional Radar 100 50 30 10 5 3 1 0.5 0.1 1.0 106 105 104 103 102 101

Military

Commercial

Frequency (GHz)

Average Power (W)

MICROWAVE

POWER TUBES

SOLID STATE DEVICES

Source: 1998 OSD S&T TARA

RF Components Sub-Area

Military Essential Electronics

Figure 1.1: RF components sub-area military essential electronics.

(20)

combining is preferred because the field structure is described in terms of Gaussian beam modes [3]. Quasi-optical power combining couples the power from an array of solid-state amplifier or oscillator devices utilizing wavebeam principles. Microwave lenses are utilized to provide periodic re-focusing of the wavebeams to combine power in a single paraxial mode over many wavelengths. QO systems are designed to have cross-section dimensions of 2 to 10 or more wavelengths. Consequently, component tolerances are greatly relaxed along the transverse and longitudinal directions. As a result, significant area is made available for numerous solid-state devices and control components to be included within the structure to achieve the desired output power. Recent work pioneered at North Carolina State University has demonstrated a

vi-Polarizer

Input Output

Polarizer Array

Lens Amplifier Lens

Input Horn _Far-Field

Y X

Z

Figure 1.2: A 3-D grid quasi-optical power combining system.

(21)

there is no electric field perpendicular to the ground planes, the magnetic field at the ground planes is also zero. Thus the currents in the ground planes are to a first approximation are also zero. Consequently, this system has the potential for very low loss at millimeter-wave frequencies. Open systems have demonstrated the ability to combine power from a source array. However, using this configuration, excessive scattering losses were identified. Major losses include radiation losses from the dielectric top and sidewalls, beam confinement to within the dielectric slab, and scattering losses of the Vivaldi antenna. With the antenna located at the top inter-face, the high field distribution causes perturbations and variations which make it difficult to predict and control the phase distribution of the array. It was determined that this was due to the scattering of the field which was made more significant as the field was not strongly guided. Better performance was obtained with the array in the bottom ground plane. Still fields extend into the region above the slab in this open system. In an effort to improve the performance of the 2-D open system, an

SIGNAL GENERATOR DETECTOR

LENSES AMPLIFIERS

RECEIVING HORN ANTENNA TRANSMITTING

HORN ANTENNA

d1 d2 d3

Figure 1.3: A 2-D dielectric slab quasi-optical power combining system.

(22)

(23)

b

c a

QUASI-OPTICAL 2D POWER COMBINING SYSTEM

AMPLIFIER PLANE

AMPLIFIER ARRAYS INPUT COUPLING

NETWORK

OUTPUT COUPLING NETWORK

GROUND PLANE

INPUT PLANE

MICROWAVE ABSORBER PHOTONIC MATERIAL Pin

Pout OUTPUT PLANE

AMPLIFIER PLANE

Figure 1.4: A 2-D cascade quasi-optical system.

(24)

Port 3

Port 4

Output

Port 1

Input

Dielectric Between Waveguides

Port 2

Upper Waveguide

MMIC

Bottom Waveguide

Input Slots Output Slots

εr Stripline

(25)

results are obtained and compared with a commercial microwave simulator (Agilent HFSS and Ansoft HFSS) for a few simplified representative structures, including various configurations of planar arrays of slotted waveguide transitions coupled to a stripline.

1.2 Dissertation Overview

Chapter 2 presents the literature review for experimental 2-D and 3-D quasi-optical and waveguide-based spatial power combining systems, and discusses the numerical and analytical techniques utilized to model quasi-optical and waveguide-based spa-tial power combining systems.

Chapter 3 presents the Gaussian beammode theory for the 2-D quasi-optical parallel-plate waveguide and resonator. Next, the beammode theory is verified by designing and fabricating a 2-D quasi-optical parallel-plate cavity and stripline-to-slot quasi-optical amplifier. Lastly, the experimental results are pre-sented and discussed.

In Chapter 4, the electromagnetic model for an aperture-coupled stripline-to-waveguide transition is investigated. A general formulation for the scattered elec-tric and magnetic fields inside an arbitrary closed structure is developed and the dyadic Green’s functions for a rectangular waveguide are derived. Next, an integral equation formulation for electric and magnetic surface current density discretized via the MoM is developed for the stripline-to-waveguide transition. Lastly, the MoM matrix system equations utilized to calculate the unknown current coefficients and scattering parameters are discussed.

(26)

transitions are simulated and compared to a commercial simulator.

Chapter 6 summarizes the work presented in this dissertation and dis-cusses conclusions and future work.

1.3 Original Contributions

The original contributions for this research are:

• The development of Gaussian beammode theory that predicts the resonant frequencies for a 2-D QO parallel-plate cavity as described in Section 3.3.

• Successful experiment and design of a 2-D quasi-optical parallel-plate cavity that agrees with theory as described in Section 3.4.

• Experiment and design of a waveguide 2-D QO dielectric power combining

structure as described in Section 3.4.

• The development of an MoM electromagnetic simulator for a stripline-to-waveguide transition for a stripline-to-waveguibased power combining system as de-scribed in Chapter 4.

• Provide insight in the electromagnetic coupling behavior of a stripline-to-waveguide transition for a stripline-to-waveguibased power combining system as de-scribed in Chapter 6.

1.4 Publications

(27)

• C. W. Hicks, H. Hwang, M. B. Steer, J. W. Mink, J. Harvey, “Spatial power combining for two dimensional structures,” IEEE Trans. Microwave Theory Tech., Vol. 46, pp. 784–791, June 1998.

• A. B. Yakovlev, A. I. Khalil, C. W. Hicks, A. Mortazawi, M. B. Steer, “The generalized scattering matrix of closely spaced strip and slot layers in waveg-uide,” IEEE Trans. Microwave Theory Tech., Vol. 48, pp. 126–137, Jan. 2000.

• H. S. Hwang, C. W. Hicks, M. B. Steer, J. W. Mink, and J. Harvey, “A quasi-optical dielectric slab power combiner with a large amplifier array,”IEEE AP-S International Symp. and USNC/URSI National Radio Science Meeting Dig., pp. 482–485, June 1998.

• M. B. Steer, T. W. Nuteson, C. W. Hicks, J. Harvey, and J. W. Mink, “Strate-gies for handling complicated device-field interactions in microwave systems,”

Proc. PIERS Symp., July 1996.

• J. Harvey, M. B. Steer, H. Hwang, T. W. Nuteson, C. W. Hicks, and J. W. Mink, “Distributed power combining and signal processing in a 2D quasi-optical system,”Proc. WRI International Symp. on Directions for the Next Generation of MMIC Devices and Systems, Edited by N.K. Das and H.L. Bertoni, Plenum Press: New York, NY, pp. 75–82, September 1997.

(28)

• A. B. Yakovlev, A. I. Khalil, C. W. Hicks, and M. B. Steer, “Electromagnetic modeling of a waveguide-based strip-to-slot transition module for application to spatial power combining systems,” IEEE AP-S International Symp. and USNC/URSI National Radio Science Meeting Dig., pp. 286–289, July 1999.

(29)

Chapter 2 Literature Review

2.1 Introduction

In this chapter, a literature review of quasi-optical and waveguide-based spatial power combining systems is presented. This literature review consists of two main sections. The first section reviews the history of quasi-optical power combining systems and a discussion of experimental free space and waveguide-based spatial power combining systems is presented. The second section reviews the numerical modeling techniques that are utilized to model quasi-optical and waveguide-based spatial power combining systems.

(30)

CHAPTER 2. LITERATURE REVIEW 13

2.2 Background

Spatial power combining has emerged as a promising technique for combining power in free space at millimeter and sub-millimeter wave frequencies [3]- [5]. The output power of individual solid-state devices in a planar array is combined to produce moderate-to-high power levels. It is desirable to utilize a single solid-state amplifier. However, as frequency increases, the output power levels become low due to the 1/f2 fall-off of available power [6]. By utilizing power combining techniques, light-weight, reliable, and low cost amplifiers and oscillators can be potentially designed to meet the demand of military and civilian applications. The lack of available power at millimeter wave frequencies has delayed the development of novel circuits and systems. If optical techniques such as diffraction and optical elements such as lenses are incorporated into the system, the term quasi-optical power combining is preferred. Quasi-optical power combining indicates that the power from an array of solid-state devices are combined utilizing wavebeam principles. Optical lenses are utilized to provide periodic refocusing of the beam and to combine power in a single paraxial mode. The large transverse and longitudinal dimensions of quasi-optical structures provide significant area for the active MMIC devices and control components to be included within the structure.

(31)

quasi-optical architectures. The military demands high-performance MMICs for ship, ground, and airborne radars, missile seekers, and satellite communications links at microwave and millimeter wave frequencies. Other popular schemes uti-lized for power combining are chip-level and circuit-level power combining [7]. More specifically, microstrip (or similar transmission line structures) are used to realize combining either on a chip or in a package. These methods are successful at low and RF frequencies. However, at millimeter waves these choices become undesirable because dielectric, radiation, and conductor losses increase rapidly and degrade sys-tem performance. At the present, chip and circuit level power combining schemes are not capable of achieving reliable and efficient moderate-to-high power levels at millimeter wave frequencies. In this literature review, the most successful classes of systems, 3-D, 2-D quasi-optical and waveguide power combining systems that appear to be the most promising are reviewed [8].

2.2.1 Free-Space Quasi-Optical Amplifiers

(32)

gain of 10 dB at 10 GHz. The most recent and successful grid amplifier was de-signed by the California Institute of Technology and Rockwell Science Center. A 1-cm square 16 × 16 monolithic Gallium Arsenide (GaAs) pHEMT grid amplifier on a single chip was realized and produced 5 watts at 38 GHz [14].

ACTIVE GRID SURFACE OUTPUT POLARIZER

INPUT POLARIZER TUNING SLAB

E

INPUT BEAM

OUTPUT BEAM

Figure 2.1: Free space quasi-optical grid amplifier.

2.2.2 Quasi-Optical 2-D Dielectric Power Combining

Two-dimensional (2-D) quasi-optical technology offers an alternative approach that utilizes a dielectric substrate to combine power from a planar resonator or amplifier array (Fig.2.2). The novel planar waveguide structure was first proposed by Mink and Schwering [15]. The structure was designed to propagate Gauss-Hermite beam modes so that the structure was termed a Hybrid Dielelectric Slab Beam Waveg-uide (HDSBW). The advantages of two-dimensional power combining are that it is more amenable to photolithographic fabrication techniques, more compatible with MMIC technology, with reduced size and weight, and with improved heat-handling capability.

(33)

demonstrate the 2-D QO power combining system. The first experimental system design was an open cavity dielectric slab resonator based on the transverse electric (TE) mode of propagation [16], [17]. The resonator incorporated a curved reflector that was placed at the beam waist of the quasi-optical modes propagating in a grounded dielectric slab. An oscillator array consisting of four MESFET oscillators were injection locked to combine the power from the oscillating elements. For the first time, a resonance frequency was detected at 7.4 GHz with a 3 dB linewidth of less than 3 kHz [18]. Next, NCSU experimentally designed and demonstrated a viable 2-D quasi-optical dielectric power combining amplifier system. The system consisted of concave and convex lenses, MESFET or MMIC devices, and Vivaldi-type antennas operating in an open waveguide configuration as shown in Fig. 1.3. The amplifier array was placed underneath the slab with the array placed between the transmitting and receiving optical lenses. The system was designed to amplify the quasi-optical TE propagating modes in the dielectric slab. An experimental 4×1 amplifier array generated 11 dB and 4.5 dB of amplifier gain and system gain, respectively, at 7.1 GHz, and the single MMIC Vivaldi-type antenna produced 24 dB of amplifier gain at 8.4 GHz [19]. Incorporating a 5×4 MMIC 2-D quasi-optical amplifier array, the system produced 30 dB of amplifier gain, 14 dB of system gain, and 14.7 dBm of output power at 8.828 GHz [20].

(34)

convex lenses

dielectric slab

ground plane

dielectric slab

ground plane

concave lenses

FET ampilifiers

guided input waves _{guided output waves}

guided output waves guided input waves

Figure 2.2: Concave and convex lens dielectric slab power combining systems.

2.2.3 Waveguide Spatial Power Combining

(35)

obtained at 8.1 GHz. The waveguide spatial power combiner is successful because it is designed to propagate only the dominant TE₁₀mode. Eliminating the problem of higher order modes is regarded as a major reason for the success of this system. Higher power levels could be achieved with this system by choosing different MMIC amplifier technologies, such as Silicon Carbide (SiC) or Gallium Nitride (GaN) based transistors and MMICs as the amplifying devices.

WR42 waveguide opening

Waveguide Opening designed to accommodate the six cards antenna array

Incident Wave

Horn Antenna

Figure 2.3: X-Band waveguide spatial power combining system.

Lockheed Martin-Sanders demonstrated a V-Band spatial power com-bining system [25]. Based on a tray concept, the system utilized 17 solid-state tray assemblies with each tray populated with 16 MMICs and radiating dipoles. The 16

(36)

2.3 Numerical Modeling of Spatial Power

Com-biners

2.3.1 Background

The fundamental understanding of quasi-optical power combining systems has pri-marily been investigated experimentally. Several experimental free space, dielectric quasi-optical power combiners, and waveguide spatial power combiners have been successful at demonstrating the fundamental concepts of generating usable output power levels using spatial and quasi-optical techniques [27] - [35]. Although great strides have been made, to date, quasi-optical/spatial power combining systems have not yet out-performed conventional power combiners. In order to capture the full potential of quasi-optical/spatial systems to generate high power levels, numerical modeling and computer aid engineering tools are needed to fully understand these systems and to provide the basic tools for design [26]. The development of com-puter models helps to reduce the cost and time associated with experimental work, and assist with designing efficient quasi-optical/spatial power combining systems. Modeling a quasi-optical/spatial power combining system is complex and challeng-ing [36], [37]. There are several major system components that must be modeled such as the input and output sources, which are typically waveguide horns with op-tical lenses inside, the input and output antennas with associated transmission lines and control components, and the active integrated amplifier circuitry. In addition, the propagation of the Gaussian wave-beam in free space [38] or in a dielectric slab must be accurately modeled.

(37)

require-CHAPTER 2. LITERATURE REVIEW 20

ment for intuitive schemes for developing customized electromagnetic (EM) analysis and schemes for reusing electromagnetic models. The alternative is to use a general purpose package using volumetric girding so that arbitrarily complex structures can be modeled. These packages include those using the finite element method (FEM) in the frequency domain and the finite-difference time domain (FDTD) method in the time domain, to obtain the field through a structure at the nodes of the grid. However, these volumetric packages are unable to model the electrically large EM structures as encountered with quasi-optical and waveguide-based spatial power combiners.

2.3.2 Modeling Free-Space Power Combining Systems

The grid structure has been modeled using the unit-cell approach and by utilizing full-wave electromagnetic modeling techniques. The unit-cell approach models a single element of the array subject to special boundary conditions along the unit-cell edges [39]. The unit-cell approach assumes that all of the elements of an infinite array are identical. Utilizing this concept, a full-wave electromagnetic model based on the method of moments was utilized to relate the electric fields to the surface current density on the electric strips using Galerkin’s method in the spectral domain [40]. Using this simulator, the driving point impedance of several unit-cells such as dipoles, cross dipoles, and bow-ties were determined. In this analysis, the grid array was composed of a dielectric with or without metal on both sides of the dielectric surfaces.

(38)

to model the entire quasi-optical grid amplifier [41]- [44]. A special dyadic Green’s function was derived to model input and output horns, lens, polarizers, dielectric layers, and the quasi-optical amplifier grid [45]. The electric and magnetic fields derived from Gauss-Hermite wave beams were utilized to develop the dyadic Green’s function that is composed of two parts; one part that computes the paraxial fields (quasi-optical modes) and another part that computes the non-paraxial fields. The Green’s function was used to develop a mixed spectral and spatial domain method of moment technique. The MoM simulator successfully simulated a 5 × 5 and a 10× 10 grid array with horns, lenses and polarizers to determine the near and far fields [46].

2.3.3 Modeling Waveguide-Based Spatial Power Combining

Systems

Waveguide-based structures are playing an increasingly important role in spatial power combining systems. Numerical techniques and classical electromagnetic tech-niques can be used to model waveguide-based power combining systems. The electro-magnetic boundary value problems for adjacent waveguides separated by a coupling aperture, have been studied for years. Stevenson was one of the first pioneers to successfully develop Green’s functions and integral equations for slotted waveguides problems [47]. Stevenson developed analytical expressions to solve for the narrow and thin longitudinal (shunt) and transverse (series) resonant slots in an infinite conducting ground plane. Oliner expanded his work to include the rotated, shifted and the finite thickness of the resonant slots and developed fairly accurate models based on the variational technique [48].

(39)

value problem for a rectangular waveguide coupling into another waveguide or free space using numerical techniques [49]. They used the method of moments by expand-ing the basis functions with pulse functions and testexpand-ing with Dirac-delta functions. Later, Lyon and Sangester expanded the method of moment analysis by including the finite thickness of the waveguide walls which results in a thick slot [50]. Si-nusodial functions were used as the basis and testing functions because they more accurately represent the electric field in the slot.

(40)

rectangular waveguide with H-walls for the narrow walls, and a rectangular cavity in order to calculate the higher-order modes within the structure.

2.3.4 Waveguide Dyadic Green’s Function

(41)

2.4 Conclusion

(42)

Chapter 3 Two-Dimensional Parallel-Plate

Resonator

3.1 Introduction

Two-dimensional (2-D) dielectric power combining technology offers significant ad-vantages. The 2-D hybrid dielectric-slab beam waveguide (HDSBW) is amenable to photolithographic definition and fabrication, and is compatible with MMIC tech-nology [15]. The novel 2-D HDSBW has reduced size, weight, and improved heat removal capability which results in lower costs. Two-dimensional quasi-optical sys-tems previously fabricated are open planar structures which consist of a ground plane and a dielectric slab (with geometry shown in Fig. 3.1(a)) [17], [20]. Open systems have demonstrated the ability to combine power from an amplifier array. In this section, we document the lessons learned with the open structure, particularly

(43)

CHAPTER 3. TWO-DIMENSIONAL PARALLEL-PLATE RESONATOR 26

through understanding of radiation mechanisms associated with excessive scattering losses. The closed 2-D slab beam waveguide configuration shown in Fig. 3.1(b) is an effort to minimize radiative losses. Beam-mode theory and experimental charac-terization of this configuration are discussed here.

(b) (a)

> εslab lens

ε

z w

s

x

y d

phase transformersεlens

dielectric slab _slab

bottom ground plane

top ground plane

bottom ground plane

ε

Figure 3.1: Passive 2-D quasi-optical power combining system with concave lenses;

(a) open structure and (b) closed structure.

3.2 Principles of Operation

(44)

ˆ

x-direction is that of a parallel-plate waveguide dominant mode. In both systems the field distribution in the ˆy-direction corresponds to a wave beam-mode (Gauss-Hermite), which is guided by the lenses through periodic reconstitution of the cross-sectional phase distribution. The guided modes are either TE or TM-polarized modes with respect to the direction of propagation. The 2-D HDSBW principle is

00 00 00 00 00 00 11 11 11 11 11 11 000 000 000 000 000 000 111 111 111 111 111 111 or

E E" E" E""

Convex/Concave Phase Transformer Port Input d1 d2 Output Port Pout Amplifier Array d3 Pin

Figure 3.2: The 2-D HDSBW system with convex/concave lenses and 4× 1

MES-FET amplifier array.

(45)

through signal as shown in Fig. 3.4. Consequently, a growing traveling wave-beam mode is established within the guiding structure resulting in an increased output power.

MESFET

MMIC

ε_r

r

ε

(a)

(b)

Figure 3.3: Input and output coupling antennas on RT/Duriod substrate: (a)

MES-FET Vivaldi amplifier; and (b) MMIC stripline-slot amplifier.

3.3 Beam-Mode Theory In a Closed-Boundary Slab

Beam Waveguide

(46)

Ein,3

Eout,1

Eout,2

Eout,3

Eout,4

Ein _Eout

Ein,4

φ t Ein,2

Eth Ein,1

Ground Plane

Ein

THROUGH WAVES AMPLIFIED

WAVES _{Eout =} φt +_Ea φ_a

+ φ

Σ

n=1 4

Ea,n _a,n =

Ea φa

Eth

Figure 3.4: Electric-field wave model for 2-D power combining system.

axial fields in the waveguide. Once the axial components are found, the transverse fields are derived from Maxwell’s equations. By applying orthogonality conditions, the fields are normalized and the normalized power in the parallel-plate waveguide is computed. The general solution of the transverse and longitudinal fields for the guiding structure is obtained from Helmholtz equations in a source field region [62]:

∇2_H¯_(¯_r_{) +}_k2_H¯_(¯_r_{) = 0}_, _(3.1)

∇2_E¯_(¯_r_{) +}_k2_E¯_(¯_r_{) = 0}

where k = ω/c√²r with c being the velocity of light in free space. The waves propagating in the waveguide are classified as Transverse Electric (TE) and Trans-verse Magnetic (TM) waves with the corresponding transTrans-verse and longitudinal field components. The eigenmodes of electric and magnetic fields can be expressed in the following form,

¯

(47)

¯

E_mn± = (¯emn±ezmnaˆz)e∓jβmnz

where ¯emn and ¯hmn are transverse vector functions, whileezmn and hzmn are longi-tudinal scalar functions. The time dependence in the form of ejωt _{is assumed and} suppressed. The termβmn is the phase constant and m and n are the mode indices for the x and y mode variations, respectively. The TE-mode solution is obtained when hzmn = 0 while the TM-mode solution is obtained when ezmn = 0. All the transverse fields can be expressed in terms of the longitudinal components. Substi-tuting (3.2) into (3.1) results in the representation of transverse vector functions of TE modes,

¯

hmn = −

jβmn

k_c2 ∇thzmn, (3.3)

¯

emn = Zhˆaz×h¯mn,

and TM modes,

¯

emn = −

jβmn

k_c2 ∇tezmn, (3.4)

¯

hmn = Yeˆaz×e¯mn

where Ye = jk0Y0/βmn, is the scalar wave admittance of TM modes and Zh =

jk₀Z₀/βmn is the scalar wave impedance of TE modes; Z0 = (1/Y0) and Y0 are the intrinsic impedance and admittance of free space. The boundary conditions for the parallel-plate quasi-optical structure (Fig. 3.1(b)) are

TM−modes : ezmn = 0 at x= 0 andx=d, (3.5)

TE−modes : ∂hzmn/∂y= 0 at y=−∞and y= +∞. (3.6)

Assuming the axial components for the parallel-plate guiding structure take the following form, where a functional dependence on (x)-variable can be separated from (y, z)-dependence,

TM−modes : ezmn =AmnXm(x)Qn(y, z), (3.7)

(48)

where Amn and Bmn are the electric and magnetic field normalization factors, re-spectively, yet to be determined. Utilizing the boundary conditions (3.5), (3.6), a functionXm(x) is expressed in terms of eigenfunctions of one-dimensional Laplacian operator having forms of sin(mπx/d) and cos(mπx/d), while the function Qn(z, y) describes the slow variation in the ˆy-direction and it is defined as [2],

Qn(y, z) =

1

q

Y n!√π

(1 +ν_mn2 )14 ·Hen

Ã√ 2y yzmn ! · exp  ₋ Ã y yzmn

!₂



± expj



νmn

Ã

y

yzmn

!₂

−µn+1 2

¶

tan−1(νmn)

 

(3.9)

where

νmn=

z

βmn

Y2, yzmn =Y2(1 +νmn2 ),

and

Y2 =

 

q

(2−D/F)F D

βmn   1 2 .

In (3.9) D is the distance between the reflecting surfaces and F is the focal length of the lenses. The function Qn is composed of Hermite polynomials which form a complete set of orthonormal eigenfunctions of the Fourier transform operator.

By substituting the axial equations into the scalar wave equations, it is found that the phase constant βmn is defined as

βmn =

q

k_x2+k_y2−k2 (3.10)

wherek_x2 = mπ_a and

k2_y =

Ã

Q00_mn(y, z)

Qmn(y, z)

!

(βmn+ 1) +

Ã

Q0_mn(y, z)

Qmn(y, z)

!

(49)

In (3.11),

Q00_mn(y, z) = ∂

2_Q

mn(y, z)

∂y2 , Q

0

mn(y, z) =

∂Qmn(y, z)

∂y .

By substituting the axial fields (3.7) into (3.4), the electric-field compo-nents of the TM modes are obtained as follows,

ezmn = Amnsin

µ_mπx

d

¶

Qn(y, z), (3.12)

exmn = −Amn

Ã

jβ

kc

! µ_mπ

d ¶ cos µ_mπx d ¶

Qn(y, z),

eymn = −Amn

Ã jβ kc ! sin µ mπx d ¶

∂Qn(y, z)

∂y ,

and the magnetic-field transverse components are

hxmn = Yeeymn, (3.13)

hymn = −Yeexmn.

The magnetic-field components of the TE modes are similarly obtained in the fol-lowing form,

hzmn = Bmncos

µ_mπx

d

¶

Qn(y, z), (3.14)

hxmn = −Bmn

Ã

jβ

kc

! µ_mπ

d ¶ sin µ_mπx d ¶

Qn(y, z),

hymn = −Bmn

Ã jβ kc ! cos µ_mπx d ¶_∂Q

n(y, z)

∂y ,

and the electric-field transverse components are

exmn = Zhhymn, (3.15)

eymn = −Zhhxmn

(50)

3.3.1 Orthogonality of Fields

After the fields for the parallel-plate quasi-optical structure have been determined, the fields can be normalized by satisfying the orthogonality relationship. The field orthogonality of TM and TE modes is defined by

Z d

0

Z _∞

−∞¯emn·¯e ∗

m0n0dxdy = δmm0δnn0, (3.16)

Z _d

0

Z _∞

−∞

¯

hmn·¯h∗m0n0dxdy = δmm0δnn0.

The normalization factor needed to satisfy the TE and TM orthogonality relations (3.16) is obtained by evaluating the following integrals

Z _d 0 cos µ_mπx d ¶ cos Ã

m0πx

d ! dx=                    d

2, m=m0 6= 0

d, m =m0 = 0

0, m6=m0

(3.17) Z _d 0 sin µ_mπx d ¶ sin Ã

m0πx

d ! dx=                    d

2, m=m0 6= 0

0, m=m0 = 0

0, m6=m0

and solving

Z _∞

−∞ Ã

∂Qn(y, z)

∂y

! Ã

∂Qn0(y, z)

∂y

!

dy (3.18)

Now substituting the electric and magnetic fields (3.12) through (3.15) and orthog-onal relations (3.16) into (3.14) the result becomes

Z _d

0

Z _∞

−∞¯emn·¯e ∗

m0n0dxdy =

Z _d

0

Z _∞

−∞

¯

hmn·¯h∗m0n0dxdy (3.19)

= Dmnδmm0δnn0

(51)

and the TM and TE orthogonality relations will be satisfied. From here it will be assumed that all the transverse fields are normalized. The electric and magnetic fields of the mn th mode propagating in the +ˆz-direction are represented as [62]

¯

E_mn+ = (¯emn+ ¯ezmn)e−jβmnz, (3.20) ¯

H_mn+ = ³¯hmn+ ¯hzmn

´

e−jβmnz_, _(3.21)

and those for the mnth mode propagating in the −zˆ-direction are

¯

E_mn− = (¯emn−e¯zmn)ejβmnz, (3.22) ¯

H_mn− = ³−h¯mn+ ¯hzmn

´

ejβmnz_, _(3.23)

where ¯E_mn+ , ¯H_mn+ and ¯E_mn− , ¯H_mn− represent electric and magnetic fields of forward and backward traveling waves, respectively. The total electric and magnetic fields propagating in the +ˆz-direction are then expressed in the form of TE and TM eigenmode expansion,

¯

E+ = X

m

X

n

amnE¯mn+ , (3.24)

¯

H+ = X

m

X

n

amnH¯mn+ , (3.25)

and in the−zˆ-direction the total fields are

¯

E− = X

m

X

n

bmnE¯mn− , (3.26)

¯

H− = X

m

X

n

bmnH¯mn− , (3.27)

whereamn and bmn are expansion (amplitude) coefficients which can be determined from the Lorentz reciprocity theorem solving the excitation problem.

3.3.2 Power Normalization

(52)

between modes. The general expression for the power carried inside a parallel-plate quasi-optical structure in the +ˆz-direction is defined by

Pmn =

1 2Re Z _d 0 Z _∞ −∞ ¯

E+×H¯∗+·ˆazdxdy. (3.28)

Since the fields are orthogonal, the normalized power in the QO structure can be found from the TM and TE orthogonal relationships. The normalized power prop-agating in the +ˆz-direction is found to be

pmn =amna∗m0n0δmm0δnn0 (3.29)

wherepmn = 2PmnYe for the TE modes and pmn= 2Pmn/Zh for the TM modes.

3.3.3 Verification

The theory of the closed system was verified by testing a confocal parallel-plate cavity system. The geometry for the parallel-plate resonator is shown in Fig. 3.5. The width and length are denoted bya = 30.48 cm and b = 30 cm, and the radius of curvature is denoted by r = 60.96 cm. The upper and lower ground planes are separated by a dielectric (Rexolite material with the dielectric permittivity ²r = 2.57) of thickness t = 1.27 cm. A L-shaped coaxial probe normal to the ground plane was used to excite the cavity. The resonance frequencies for the parallel-plate cavity are calculated from

βmz−

µ

n+1

2

¶

tan−1νmn=qπ. (3.30)

Solving the above equation for the resonance frequencies of the parallel-plate cavity structure gives the following result,

fmn =

c

2π√²r

·     

qπ+ (n+ 1₂) tanhz/

q

(2−D/F)F Di z

2

+

µ_mπ

d

¶₂_

 

1 2

(53)

where m,n, and q are the mode indices for the ˆx, ˆy, and ˆz directions respectively.

Reflector Curved L-shaped

Antenna

t

ε

z x Planer

Reflector

Ground

Planes r

b

y

z a

Figure 3.5: Test configuration for the confocal parallel-plate resonator system.

Measurements were taken using a Hewlett Packard 8510C Vector Network Analyzer to measure S₁₁ of the resonator. A L-shaped coaxial probe normal to the ground plane was utilized to excite predominantly the TM modes inside the cavity. The TM_m,n,q modes were selected because m=0 signifies the dominant mode inside a parallel-plate cavity. Fig3.6 shows a dispersion behavior of the reflection coefficient

(54)

as shown in Fig. 3.6.

.

|S11|

Frequency (GHz)

5.0 5.5 6.0 6.5 7.0 7.5 8.0 0

0.2 0.4 0.6 0.8 1

s

Figure 3.6: Reflection coefficient at the input to a confocal 2-D parallel-plate

res-onator system.

3.3.4 Mode Profile

(55)

indicating that the theory and measurements are in excellent agreement. The

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 20 40 60 80 100 120 140

Distance (mm)

Theory Run #1 Run #2

|S11| (Reflection Coefficient)

Figure 3.7: Electric-field mode profile at 6.898 GHz for the parallel-plate confocal

system.

(56)

0 0.2 0.4 0.6 0.8 1 1.2

4.000 4.188 4.375 4.563 4.750 4.938 5.125 5.313 5.500 5.688 5.875 6.063 6.250 6.438 6.625 6.813 7.000

Frequency

S11 Magnitude (dB)

0 0.005 0.01 0.015 0.02 0.025

S21 Magnitude (dB)

(1) q=20, n=3, 6.408GHz (2) q=20, n=1, 6.254 GHz (3) q=21, n=1, 6.558GHz (4) q=21, n=3, 6.719GHz (5) q=22, n=0, 6.869GHz

(1)

(2) (3) (4)

(5)

|S11|

|S21|

Figure 3.8: A plot of the magnitude ofS₁₁andS₂₁for selected resonance frequencies

(57)

Table 3.1: Selected resonance frequencies of the parallel-plate resonator system.

Measured Calculated Error n q

Frequency Frequency (MHz)

(GHz) (GHz)

6.869 6.868 0 1 22

6.719 6.715 4.0 3 21

6.558 6.561 3.0 1 21

6.408 6.408 0 3 20

6.254 6.254 0 1 20

3.4 Experimental Results

3.4.1 Open System Configuration

(58)

6.5 cm × 1.5 cm. Two E-plane horns were designed and fabricated to efficiently transmit and receive the required wavebeam. Two experiments were performed for the open system. The first experiment utilized a 4× 1 MESFET Vivaldi amplifier array as shown in Fig. 3.2, and the second one employed a single MMIC Vivaldi amplifier located under the dielectric slab (see Fig.3.9). A measure of the relative energy coupled to the amplifier array was obtained by switching the amplifier bias levels off and on while measuring the output power,Pout. The system performance for the active Vivaldi-based amplifier array was determined by the system gain and amplifier gain. This provided an indication of the incident signal that passes through the dielectric as an undisturbed traveling wave. Fig. 3.10 shows the total

Pin Pout

20 cm 20 cm

5.5 cm

Eamp Eth+ Eamp

Ein

CONCAVE LENS

Eth=~Ein

9.7cm METAL WALL

Figure 3.9: The concave-lens system configuration for a unit-cell Vivaldi-based

am-plifier.

(59)

of 4.5 dB occurred at Pin = -15 dBm while the measured amplifier gain on-to-off was 11 dB.

-50 -40 -30 -20 -10 0 10

-40 -30 -20 -10 0 10

Pout (dBm)

Pin (dBm)

CONCAVE-LENS SYSTEM AMP OFF

AMP ON

Figure 3.10: The concave-lens system configuration for a unit-cell amplifier.

3.4.2 Vivaldi Unit Cell

(60)

metal wall were placed 5.5 cm and 9.7 cm, respectively, from the input horn. A concave lens was placed in the middle of a 40 cm dielectric slab. The slit in the metal wall allowed for only input power of the amplifier to go through the system so that the amplifier gain could be measured. The amplifier gain was determined by switching the bias voltage on and off, while measuring the power difference detected by the receiving horn. The amplifier gain indicated that more than 20 dB of gain was produced from 7 GHz to 10.5 GHz with a maximum gain of 24 dB at 8.4 GHz. The gain from 9.5 GHz to 10.5 GHz is shown in Fig.3.11.

5 10 15 20 25

9.5 9.75 10 10.25 10.5

Amplifier Gain (dB)

Frequency (GHz)

Vivaldi amplifier Slot amplifier #1

Figure 3.11: Amplifier gain for a unit-cell MMIC amplifier; (a)Vivaldi cascade

MMICs and (b) single stripline-slot MMIC.

3.4.3 Unit Cell of a Slot Antenna

(61)

in the bottom ground plane under a 1/8 inch Rexolite dielectric slab. In a similar manner, a metal wall was placed 8 mm from the middle of the input and output slot antenna where the MMIC was located. Two H-plane horns were designed to transmit and receive power and to vertically polarize the electric field, in order to achieve the maximum coupling to the slots. The slots were located λ/4 apart and the slot width and length wereλ/10 and λ/2, respectively.

Fig. 3.11 compares the unit cell Vivaldi-based and slot amplifier gain. The nominal gain of the MMIC at 10 GHz is 10 dB. Different gains were achieved because the Vivaldi was used in a cascade configuration while the slot utilized only one MMIC. Another difference is that the Vivaldi-based amplifier was tested over a wider frequency range than the slot. The Vivaldi-based amplifier reached 20 dB of gain whereas the gain of the slot antenna with one MMIC reached 10 dB.

3.5 Conclusions

(62)

Chapter 4 Waveguide-Based Slot-Strip-Slot

Transitions

4.1 Introduction

In this chapter, an electromagnetic modeling environment is developed for an aperture-coupled stripline-to-waveguide transition. This transition is the fundamental build-ing block for two-dimensional spatial power combinbuild-ing amplifier arrays and in turn for planar quasi-optical power combining systems. A full-wave electromagnetic model is developed for a structure that couples a waveguide to a stripline through a set of slots and from the stripline through another set of slots into a second waveg-uide. The system modeling is based on an integral equation formulation for the induced electric and magnetic surface current densities resulting in a coupled set of integral equations discretized utilizing the method of moments (MoM). The

(63)

CHAPTER 4. WAVEGUIDE-BASED SLOT-STRIP-SLOT TRANSITIONS 46

tered electric and magnetic fields are expressed in terms of dyadic Green’s functions and the electric and magnetic surface currents. The surface currents are discretized by overlapping piecewise sinusodial subdomain basis functions in order to accu-rately model narrow longitudinal strips and transverse slots. In this formulation, a MoM matrix includes all possible self and mutual coupling effects between the slots and strips. The transition is excited with the TE₁₀ dominant waveguide mode and the scattering parameters are calculated from the forward and backward coupling coefficients in the waveguide regions.

4.2 General Electromagnetic Formulation and Dyadic

Green’s Functions

A general electromagnetic formulation for a closed-boundary waveguiding structure containing arbitrarily shaped apertures and conducting strips (see Fig. 4.1) is de-veloped in this section. The formulation is based on the integral representation of incident and scattered electric and magnetic fields in terms of dyadic Green’s functions. Dyadic Green’s functions represent the electric and magnetic fields at an observation point inside a volume due to an arbitrarily oriented point source. Fig.4.1 shows an arbitrary volumeV enclosed by the surface ˜S=S∪Sm, where S represents an electric-type boundary surface and Sm represents the surface of aper-tures (magnetic-type surface). The volumeV encloses an impressed electric volume current source ¯Jimp⊂Vimp and an induced electric current source ¯Jindon the surface of conducting stripsSe (electric-type surface). The total electric ¯E(¯r) and magnetic

¯

H(¯r) fields inside the closed regionV are obtained as a superposition of the incident electric ¯Ei(¯r), and, magnetic, ¯Hi(¯r), fields due to ¯Jimp and the scattered electric,

¯

(64)

CHAPTER 4. WAVEGUIDE-BASED SLOT-STRIP-SLOT TRANSITIONS 47

nˆ

,,

s

H

E ,

m

S

imp

V

imp

J

ind

J

e

S

i i

H

E ,

V

Figure 4.1: Geometry of a closed-boundary waveguiding structure containing

aper-tures and conducting strips in the presence of an impressed electric current source.

The total fields due to ¯J = ¯Jimp+ ¯Jind inside the homogeneous volumeV satisfy the following vector wave equations [61]:

∇ × ∇ ×_E¯_(¯_r₎₋_k2_E¯_(¯_r_{) =} ₋_jωµ_J¯_(¯_r₎_, _(4.1)

∇ × ∇ ×_H¯_(¯_r₎₋_k2_H¯_(¯_r_{) =} _{∇ ×}_J¯_(¯_r₎_, _(4.2)

whereωis the radial frequency,k=ω√²µis the wavenumber in the media of volume

V,²=²r²0 and µ=µrµ0 are the primitivity and permeability of the media with ²r andµr are being the relative primitivity and permeability of the material media,²0,

µ₀ are the primitivity and permeability of free space. The time dependence in the form ofe+jωt _{is assumed and suppressed.}

The corresponding dyadic wave equations are derived by letting G=e= ¯

Ewˆ, G=m= −jωµH¯wˆ, and ¯Jwˆ = jωµ

=

I δ(¯r −r¯0) where ˆw = ˆx, ˆy, ˆz, and ¯r =

(x, y, z) and ¯r0 = (x0, y0, z0) are the positions of the observation and the source

point, respectively. Substituting these representations into (4.1) and (4.2) results in

∇ × ∇×G=e (¯r,r¯0)−k2 =Ge (¯r,r¯0) =

=

I δ(¯r−r¯0), (4.3)

∇ × ∇×G=m (¯r,r¯0)−k2 =Gm (¯r,r¯0) = ∇×

=

Electromagnetic and Experimental Modeling of Waveguide-Based Spatial Power Combining Systems

Abstract

Experimental and Electromagnetic

Modeling of Waveguide-Based Spatial

Power Combining Systems

Dedication

Biographical Summary

Acknowledgements

Contents

List of Tables

List of Figures

Chapter 1

Introduction

1.1

Motivation and Objective of This Study

Military

Commercial

MICROWAVE

POWER TUBES

SOLID STATE DEVICES

RF Components Sub-Area

Military Essential Electronics

Port 3

Port 4

Output

Port 1

Input

Dielectric Between Waveguides

Port 2

Upper Waveguide

Bottom Waveguide

1.2

Dissertation Overview

1.3

Original Contributions

1.4

Publications

Chapter 2

Literature Review

2.1

Introduction

2.2

Background

2.2.1

Free-Space Quasi-Optical Amplifiers

2.2.2

Quasi-Optical 2-D Dielectric Power Combining

2.2.3

Waveguide Spatial Power Combining

2.3

Numerical Modeling of Spatial Power

Com-biners

2.3.1

Background

2.3.2

Modeling Free-Space Power Combining Systems

2.3.3

Modeling Waveguide-Based Spatial Power Combining

Systems

2.3.4

Waveguide Dyadic Green’s Function

2.4

Conclusion

Chapter 3

Two-Dimensional Parallel-Plate

Resonator

3.1

Introduction

3.2

Principles of Operation

3.3

Beam-Mode Theory In a Closed-Boundary Slab

Beam Waveguide

3.3.1

Orthogonality of Fields

3.3.2

Power Normalization

3.3.3

Verification

3.3.4