# Simulation, design and fabrication of microwave ferrite components for monostatic radar applications

код для вставкиСкачатьSIMULATION, DESIGN AND FABRICATION OF MICROWAVE FERRITE COMPONENTS FOR MONOSTATIC RADAR APPLICATIONS A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy with a Major in Electrical Engineering in the College of Graduate Studies University of Idaho by Ryan Seamus Adams July 2007 Major Professor: Jeffrey L. Young, Ph.D. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3281286 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and im proper alignm ent can adversely affect reproduction. In the unlikely event that the author did not send a complete m anuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3281286 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii AUTHORIZATION TO SUBMIT DISSERTATION This dissertation of Ryan Seamus Adams, submitted for the degree of Doctor of Philosophy with a major in Electrical Engineering and titled ’’SIMULATION, DESIGN AND FABRICA TION OF MICROWAVE FERRITE COMPONENTS FOR MONOSTATIC RADAR APPLI CATIONS,” has been reviewed in final form. Permission, as indicated by the signatures and dates given below, is now granted to submit final copies to the College of Graduate Studies for approval. Major Professor *^ 1 — Cf~J Dat e ffrev LxYoun Committee Members Date_ Richard B. Wells Date David N. Mcllroy ^ David H. Atkinson Date Department Administrator Date IJ-O ? Brian KrJohnson Discipline’ss uiscipnne College Dean A /j1 v A f f Jffi -------- Date----- 7/// j0*{ Final Approval and Acceptance by the College of Graduate Studies Date Margrit von Braun Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract A theory encompassing ferrite materials and their application to microwave circuits is presented herein. Several aspects of these materials and the devices that incorporate them are discussed. In particular, the behavior of infinitely extended microwave ferrites in the presence of a radio frequency (rf) signal is reviewed. The conditions for Faraday rotation and birefringence are outlined and the electromagnetic wave response is quantified. Further, magnetostatic modes and spin waves are discussed which require that the rf electric intensity be identically zero. A time-domain simulation routine is presented that accounts for electromagnetic wave prop agation within finitely extended ferrite materials. This routine is fully second-order accurate. The routine is validated by comparing the network parameter data of a stripline phase shifter against the data obtained from a popular frequency domain formulation. Excellent correlation is observed, thus establishing that the routine is consistent with other simulation approaches. A review of the Bode-Fano criterion is presented wherein the fundamental matchability of an arbitrary load impedance function is addressed. This criterion is represented by integral equations that provide a relationship between the bandwidth and tolerance of match of an arbi trary impedance function. This theory is applied to the matchability of three port circulators by appealing to the concept of a required load impedance for perfect isolation. When applied to circulators in this manner, the realized bandwidth can exceed the fundamental limit in certain cases, but it is shown that the Bode-Fano criterion still provides a good “rule of thumb” regard ing the matchability of a given circulator device. Three circulators are presented in validation of the Bode-Fano criterion, each of which utilize ferrite materials that exhibit high crystalline anisotropy, and hence they are “self-biased” devices. Simulation data is provided for the fre quency response of these circulators which demonstrate approximately 5% bandwidth centered around 23 GHz. A unique circulator topology which consists of a rectangular ferrite region and orthogonal ports is presented. This topology arises from the theory of the perfect isolation impedance that allows for any arbitrary shape of the ferrite region and port locations. The Bode-Fano criterion is applied to this device and the achieved response correlates quite well the the fundamental lim its. In addition to this device, two circulator/antenna systems are presented which demonstrate that systems which incorporate circulators can be designed as a whole, rather than by design ing individual components to some interface specification. Simulation and measured data are presented for each of these circulator devices and systems. One additional antenna/circulator system is presented which accomplishes system design without the use of interconnecting matching networks; system design is accomplished by vary ing the circulator geometrical and material parameters. Once again, the Bode-Fano criterion is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. applied to this system with good correlation of the achieved response to the fundamental limit. Simulation data is presented as validation of this design approach. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V Vita The author received from the University of Idaho the BSEE (cum laude) and BS Applied Mathematics degrees (cum laude) in 1999, the MSEE degree in 2005 and the doctoral degree in Electrical Engineering in 2007. He was formerly a graduate research assistant with the de partment of Electrical and Computer Engineering at the University of Idaho. He is currently serving the Department of Electrical and Computer Engineering at the University of North Carolina-Charlotte as an Assistant Professor. His research interests include electromagnetic wave propagation in complex media and high frequency circuits. His research awards include third place finalist in the 2007 IEEE Antennas and Propagation International Symposium, Hon olulu, Hawaii student paper contest. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to gratefully acknowledge the direction and encouragement of my advisor, Dr. Jeffrey L. Young. He provided the right mix of direction and freedom to pursue interesting ideas that made my graduate school experience very rewarding. I and my family owe him a debt of gratitude. I would also like to acknowledge the contributions of Mr. Benton O ’Neil for his coax to microstrip transition designs and some of the circulator pucks that were used in this document. Also, his wealth of knowledge in making accurate microwave measurements made this project much easier. I also have appreciated our brainstorming sessions and technical discussions thank you! The MRCI support staff have provided tremendous help to me over the last three years. Without the assistance of Ray Anderson, Beth Cree and Karen Cassil, this dissertation may never have happened. Finally, I would like to acknowledge the Office of Naval Research for their financial assis tance in the form of research grant number N00014-06-1-0416. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication This thesis is lovingly dedicated to my wife, Tracie. I could never have accomplished so much in so short a time without her support and encouragement. I would be nothing without her. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viii Contents 1 2 3 4 5 A b s tr a c t.................................................................................................................................... iii V i t a .......................................................................................................................................... v A cknow ledgem ents................................................................................................................ vi D edication................................................................................................................................. vii Introduction 1 1.1 B ack g ro u n d .................................................................................................................. 1 1.2 Dissertation O u tlin e ..................................................................................................... 2 Wave Processes in Microwave Ferrites 5 2.1 High-Level Ferrite D escription.................................................................................. 5 2.2 Generalized Propagation in Infinitely Extended F e rrite s ....................................... 8 Time Domain Simulation of Wave Propagation in Ferrite Materials 22 3.1 Derivation of Time-Stepping E q u a tio n s ................................................................. 23 3.2 Numerical R esu lts......................................................................................................... 27 Bode-Fano Criterion 32 4.1 Physical R ealizability.................................................................................................. 33 4.2 Limitations on Tolerance and B andw idth................................................................. 41 4.3 The Design of Simple Matching N etw o rk s.............................................................. 48 4.4 Matching Network S y n th e sis ..................................................................................... 55 Fundamental Matchability of Three Self-Biased Circulators 60 5.1 Circuit M o d e lin g ......................................................................................................... 61 5.2 Fundamental Limits on M atch ab ility........................................................................ 67 5.3 Realized Circulator R esponse..................................................................................... 69 5.4 Concluding R e m a r k s .................................................................................................. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 7 A Microstrip, Rectangular Ferrite, Circulator 6.1 Circulator D e s ig n .......................................................................................................... 76 6.2 Measured R e s u lts .......................................................................................................... 82 Integration of a Microstrip Circulator with a Planar Yagi Antenna of Several Di rectors 8 9 A B 75 87 7.1 System Integration ...................................................................................................... 93 7.2 R esu lts............................................................................................................................. 95 Circulator System Design Without Matching Networks 104 8.1 Antenna A n a ly s is .......................................................................................................... 8.2 Green’s Function for System D e s ig n ............................................................................. 108 8.3 Simulation Results ...................................................................................................... Conclusion 104 110 115 9.1 C o n trib u tio n s ................................................................................................................ 115 9.2 Future W o rk ................................................................................................................... 118 Finite-Difference Time-Domain Analysis of Debye Materials 120 A .l F o rm u latio n ................................................................................................................... 121 A.2 R esu lts............................................................................................................................. 124 A.3 C o n clu sio n s................................................................................................................... 126 Losses in Extraordinary Mode Propagation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 X List of Figures 2.1 Resonance frequency in the infinite medium as a function of wave number. The regions indicated are (I) electromagnetic propagation, (II) magnetostatic modes and (III) spin-wave modes. The curves for the extraordinary waves are com puted from Eqns. (2.68) and (2.86). In this plot H a = AnMs = 2000 Oersteds and t f = 12....................................................................................................................... 18 3.1 Yee grid cell for this new scheme................................................................................ 27 3.2 Stripline phase shifter validation topology. The darker shaded region represents the ferrite.......................................................................................................................... 28 3.3 A comparison of S 2\ with HFSS for effective internal field equal to zero 29 3.4 Return loss of the device as a function of applied field when / = 13 GHz. . . . 30 3.5 Phase shift of the device as a function of applied field when / = 13 GHz. . . . 31 4.1 A purely reactive matching network to be determined............................................. 33 4.2 A network equivalent to Z l .......................................................................................... 33 4.3 Equivalent network to that of Figure 4.1..................................................................... 34 4.4 Flow diagram of the overall network N of Figure 4.3.............................................. 34 4.5 Example of four zeros of T i......................................................................................... 36 4.6 Example of four zeros of Tim that are derived from the function shown in Figure 4.5 4.7 An example of a branch cut in the complex plane caused by a singularity of ln ( /( 2 ) ) ............................................................................................................................ 4.8 39 A depiction of the contour C and evaluation point sa for the Cauchy integral theorems........................................................................................................................... 4.9 36 42 The closed contour used to evaluate the Cauchy integral. Z a is a branch point of the function ln ( l/T im) and creates a branch cut that extends to infinity. . . . 43 4.10 An n-elem ent high-pass ladder network.................................................................... 45 4.11 An n-elem ent low-pass ladder network..................................................................... 46 4.12 The optimum frequency response................................................................................ 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xi 4.13 The matching problem under consideration................................................................ 49 4.14 Typical frequency behavior of Ti with a Chebyshev approximation...................... 50 4.15 A plot of the first 5 Chebyshev polynomials............................................................... 51 4.16 Typical frequency behavior of t using a Chebyshev response.................................. 51 4.17 Location of the poles of T i(z )T i(—2 ) for networks of 3 and 4 elements.............. 54 4.18 Location of the zeros of r i ( z ) r 1(—z) for networks of 3 and 4 elements.............. 54 4.19 Network to be synthesized utilizing Fano’s design equations.................................. 56 4.20 Reflection coefficient versus frequency for the low reflection coefficient design. 58 4.21 Reflection coefficient versus frequency for the wide bandwidth design................. 59 5.1 The geometry of a self-biased ferrite puck.................................................................. 61 5.2 A comparison of the impedance data of the parallel RLC model with simulated data associated with the 2000 G material.................................................................... 5.3 A comparison of the impedance data of the parallel RLC model with simulated data associated with the 2250 G material.................................................................... 5.4 62 A comparison of the impedance data of the parallel RLC model with simulated data associated with the 2500 G material.................................................................... 5.5 62 63 The lumped element equivalent circuit used to approximate the impedance func tions of Figures 5.2, 5.3 and 5.4.................................................................................... 64 5.6 The admittance data of the impedance function shown in Figure 5.2...................... 64 5.7 Admittance data of the impedance function shown in Figure 5.3............................. 65 5.8 Admittance data of the impedance function shown in Figure 5.4............................. 65 5.9 The topology of the microstrip matching network. Through tracesshare a com mon centering line; the stubs are open-circuits 5.10 Self-biased circulator utilizing ferrite material with 4 ttM s = 2000 G .................... 70 71 5.11 Scattering parameters and the Fano bandwidth limit of the 2000 G circulator shown in Figure 5.10...................................................................................................... 71 5.12 Self-biased circulator utilizing ferrite material with AirMs = 2250 G.................... 72 5.13 Scattering parameters and the Fano bandwidth limit of the 2250 G circulator shown in Figure 5.12...................................................................................................... 73 5.14 Self-biased circulator utilizing ferrite material with 4-k M s = 2500 G.................... 73 5.15 Scattering parameters and the Fano bandwidth limit of the 2500 G circulator shown in Figure 5.14...................................................................................................... 74 6.1 A rectangular ferrite region for use in a microstrip c ir c u la to r .............................. 76 6.2 Required load impedance for port 1 to assure perfect c ir c u la tio n ....................... 78 6.3 Required load impedance for ports 2 and 3 to assure perfect circulation............. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xii 6.4 Parallel RLC model approximation centered at 12.5 GHz of the impedance func tion of Figure 6.2............................................................................................................. 79 6.5 Parallel RLC model approximation centered at 9.5 GHz of the impedance func tion of Figure 6.2............................................................................................................. 79 6.6 Parallel RLC model approximation centered at 12.5 GHz of the impedance func tion of Figure 6.3............................................................................................................. 80 6.7 Parallel RLC model approximation centered at 9.5 GHz of the impedance func tion of Figure 6.3............................................................................................................. 80 6.8 The topology of each matching network. Through traces share a common cen tering line; the stubs are open-circuits........................................................................ 81 6.9 A photograph of the fabricated circulator in its test fixture....................................... 82 6.10 Simulated return loss for the circulator with square topology.................................. 83 6.11 Measured return loss for the circulator with square topology.................................. 84 6.12 Simulated insertion loss for the circulator with square topology............................. 84 6.13 Measured insertion loss for the circulator with square topology............................. 85 6.14 Simulated isolation for the circulator with square topology..................................... 85 6.15 Measured isolation for the circulator with square topology...................................... 86 7.1 Single director planar Yagi-Uda antenna..................................................................... 88 7.2 E-plane pattern comparison for the baseline, double director and triple director antennas............................................................................................................................ 89 7.3 H -plane pattern comparison for the baseline, double director and triple director antennas............................................................................................................................ 89 7.4 Input impedance of the baseline antenna...................................................................... 90 7.5 Double director planar Yagi-Uda antenna.................................................................... 91 7.6 Normalized simulation antenna pattern for the two director antenna at15 GHz. 91 7.7 Normalized simulation antenna pattern for the three director antenna at 15 GHz. 92 7.8 Traditional microstrip matching network topology using unbalanced stubs. . . . 94 7.9 Balanced microstrip impedance equalizer.................................................................. 94 7.10 Circulator/antenna system with two director elements............................................. 96 7.11 Circulator/antenna system with three director elements........................................... 97 7.12 Measured hysteresis curve of T T 1-3000..................................................................... 98 7.13 Measured peak-to-peak linewidth of TT 1-3000 at 10 GHz.................................... 99 7.14 Measured (solid lines) and simulated (dashed lines) scattering parameters of the two-director element system with a 2575 Oe applied bias....................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 xiii 7.15 Measured (solid lines) and simulated (dashed lines) scattering parameters of the three-director element system with a 2575 Oe applied bias.........................................100 7.16 Measured scattering parameters of the two-director element system with a 2291 Oe applied bias................................................................................................................ 100 7.17 Measured scattering parameters of the three-director element system with a 2280 Oe applied bias...................................................................................................... 101 7.18 A comparison of measured return loss for two fabricated circuits with two di rector elements................................................................................................................ 101 7.19 A comparison of measured insertion loss for two fabricated circuits with two director elements............................................................................................................. 102 7.20 A comparison of measured return loss for two fabricated circuits with three director elements............................................................................................................. 102 7.21 A comparison of measured insertion loss and isolation for two fabricated cir cuits with three director elements..................................................................................... 103 8.1 Topology of antenna on alumina.................................................................................... 105 8.2 Simulated input impedance of the antenna of Figure 8.1 with a substrate thick ness of 0.5 mm................................................................................................................ 106 8.3 Simulated E - and H -plane antenna patterns of the antenna of Figure 8.1 with a substrate thickness of 0.5 mm ........................................................................................ 106 8.4 Simulated input impedance of the antenna of Figure 8.1 with a substrate thick ness of 0.75 mm...................................................................................................................107 8.5 Simulated E - and 77-plane antenna patterns of the antenna of Figure 8.1 with a substrate thickness of 0.75mm....................................................................................... 108 8.6 Simulated input impedance of the antenna of Figure 8.1 with a substrate thick ness of 1.0 mm................................................................................................................ 109 8.7 Simulated E —and 77-plane antenna patterns of the antenna of Figure 8.1 with a substrate thickness of 1.0mm..................................................................................... 109 8.8 Variables associated with system design without matching networks..........................I l l 8.9 Puck design to be used with the 0.75mm thickness antenna..........................................112 8.10 Impedance function to assure perfect isolation for port 1 and its RLC circuit model equivalent impedance function..............................................................................112 8.11 Impedance function to assure perfect isolation for ports 2 and 3 and its RLC circuit model equivalent impedance function................................................................. 113 8.12 Overall system consisting of the ferrite puck of Figure 8.9 and the antenna of Figure 8.1 with a 0.75mm thick dielectric.......................................................................113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xiv 8.13 Simulation scattering parameters of the system of Figure 8.12...................................114 A .l Graphical representation of the simulated topology...................................................... 125 A.2 Reflection coefficient data associated with the individual relaxations of metha nol. Data obtained from the integration scheme and from HFSS are presented. Case n, for n = 1, 2,3, corresponds to a single pole response associated with r n and es„; see Table 1......................................................................................................... A.3 126 Reflection coefficient data associated with the individual and collective relax ations of methanol. Data obtained from the integration scheme and from HFSS are presented. Case 4 is a single pole response associated with ri and esi . Case 5 is a two pole response associated with t i , ea\, 72 and es2. Case 6 is a three-pole response associated with t \, esi, r 2, es2, r 3 and es3. See Table 1 for parameter values.................................................................................................................................... 127 A.4 Reflection coefficient data associated with the two-pole response of water. Data obtained from the integration scheme and from HFSS are presented. See Table 1 for parameter values........................................................................................................ 128 B .l Losses associated with the extraordinary mode propagating through a ferrite for various values of effective internal field when f = 20 GHz, er = 12.0 and A H = 3000 Oe when measured at 55 GHz.................................................................................131 B.2 A comparison of the approximation of Eqn. (B.13) with the exact solution of Eqn. (B.3) for the following parameters: A H = 3000 Oe, Fmeas = 55 GHz, H int = 0. Oe, AnMs = 2500 Gauss, er = 12..............................................................................132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 Introduction Ferrite devices such as circulators and phase shifters play a crucial role in monostatic RADAR systems. Many papers have been presented to describe the behavior of these devices at mi crowave frequencies in the past four decades, each with its own strengths and weaknesses. Unfortunately, no concise theory has yet been presented that describe all of the nuances of these devices, particularly for circulators. 1.1 Background In 1964, Bosma [1] presented a trans-impedance Green’s function that describes the strip-line junction circulator. With this formulation, the cylindrical shaped ferrite is assumed to have perfect electric boundaries on the top and bottom and perfect magnetic boundaries on the radial edge. The cylinder is also assumed to be very thin, which implies that the fields experience no variation in the 2 -direction and the problem is two dimensional. As with all Green’s function problems, the source port consists of an infinite number of delta functions that are integrated to determine the overall response. This Green’s function was then modified by various researchers. For example Krowne and Neidert [2], Newman and Krowne [3] and Young and Johnson [4] extended the original formulation to account for the inhomogeneous demagnetizing field in the ferrite [5]; Young and Sterbentz [6] improved convergence of the function by removal of the weak singularity in the formulation. The most recent modification to the Green’s function was made by O ’Neil [7] to extend the function to account for microstrip fringing. Although the Green’s function is a computationally robust design tool, errors still exist in the solution. One method that can be used to overcome the errors inherent in the Green’s function ap proach is to employ a full-wave solver to compute the network parameters of the ferrite device. Several attempts have been made to develop a reliable simulation scheme. The frequency do main approach has been implemented successfully, and is readily available from commercial Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 sources (i.e. Ansoft’s HFSS). Most of the existing time-domain schemes are extensions of the approach first presented by Kane Yee in 1966 [8], which is now the standard approach for time-domain simulation. When implementing ferrites into the scheme, two approaches have been considered - recursive convolution (RC) and direct integration (DI). The RC methods use the time-domain Polder model for the susceptibility matrix in conjunction with the constitutive convolution integral as the governing equations. Schuster and Luebbers [9] and others achieved first or second order schemes that admit some artificial dissipation. The DI schemes implement the Gilbert equation directly, but some use backward differences [10] or interpolation [11] and hence the solution includes artificial dissipation. Others [12] have chosen to combine the Gilbert equation with Ampere’s law and the magnetic constitutive relation into a single equation. This formulation appears to be second-order accurate, but the bias field direction is required to point in one of the cartesian coordinate directions. To affect a circulator circuit design, three approaches have been presented. The first ap proach was introduced by Fay and Comstock [13] in which the operation of the junction cir culator was described in terms of contra-rotating waves within the cylindrical ferrite material. These waves set up a static field structure within the ferrite cavity; circuit design is affected by rotating the static field structure to isolate one of the ports. The second approach was introduced by Wu and Rosenbaum [14] in which a rule of thumb is given that relates magnetic saturation to ferrite radius and frequency of operation when the effective internal field is set to zero. Although this rule of thumb does hold in general, it does not assure that the resultant design is optimal in terms of bandwidth or tolerance of match. The final approach was presented by Young et. al. [15] in which the Green’s function is coupled with a search algorithm to determine the optimal geometrical and material properties. An accurately simulated ferrite coupled with appropriate material and geometrical parame ters can be included in a system design which integrates the circulator with other system com ponents. Optimal system design utilizing networks with three ports has not been addressed in the open literature. However, Bode [16] presented a theory that describes the optimal design of two port networks that provide impedance matching between a simple load impedance and a generator. Fano [17] generalized the theory of Bode to accomplish impedance matching for a general load impedance that consists of lumped elements. This theory was then validated by many including Tanner [18], Lopez [19] - [22] and Hansen [23]. 1.2 Dissertation Outline This dissertation addresses the optimal design of ferrite systems that are used in monostatic RADAR applications. A novel method is presented that simulates electromagnetic wave prop Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 agation within ferrite materials in the time-domain. Also, the theory of Bode and Fano is reviewed, and the optimal matching criterion are applied to three port circulators. The appli cability of this criterion to three port circulators is assessed. Following this discussion, three chapters are presented wherein circulator systems are designed and evaluated. To set the stage for the discussion of ferrites throughout the dissertation, Chapter 2 is in cluded to provide background on the nature of electromagnetic waves, magnetostatic modes and spin waves in ferrites. Infinitely extended ferrite materials can support several modes of electromagnetic wave propagation. In particular, Faraday rotation and birefringence are sup ported. Faraday rotation occurs when the electromagnetic wave propagates in the direction of the magnetic bias, and is characterized by a rotation of the polarization of the wave. Birefrin gence occurs when the electromagnetic wave propagates transverse to the direction of magnetic bias and consists of two distinct modes: the ordinary mode and the extraordinary mode. The ordinary mode occurs when the rf magnetic intensity points in the direction of magnetic bias and is characterized by a wavenumber that is unaffected by the magnetization of the material. The extraordinary mode occurs when the rf magnetic intensity points transverse to the direction of magnetic bias and exhibits a wavenumber that is different from that of the ordinary mode. Also discussed in Chapter 2 are two other types of waves that can be supported within ferrite materials, both of which occur when the rf electric intensity is negligible. For moder ate wavenumbers, the rf magnetic intensity interacts directly with the magnetic dipoles of the ferrite material according to the Walker equation [24], Modes of this type are called magnetostatic modes and each represents an energy storage mechanism. When the wavenumber is very large, so that the wavelength is on the order of the interatomic spacing, the interaction of the rf magnetic intensity with the magnetic dipoles results in a propagating wave called a spin wave. Simulation of the effects of electromagnetic wave propagation in ferrites in the time do main is the topic of Chapter 3. In this chapter, the Yee scheme [8] serves as the starting point. Faraday’s law and Ampere’s law (assuming D = eE) are discretized in time using central dif ferences, and the Gilbert equation [25] is invoked to include the effects of the ferrite in the solution. To complete the scheme, the constitutive relation of B = //0(H + M ) is invoked. Spa tial discretization is accomplished by colocating the individual elements of the magnetization vector with the elements of magnetic intensity and magnetic flux density. The scheme is vali dated by comparison of network parameters of a simulated phase shifter with the same device simulated with a popular frequency domain finite element solver. Good correlation between the data obtained from simulation and benchmark data is observed. The work of Fano [17] is reviewed in Chapter 4 to place the remainder of the dissertation in context. This work defines the fundamental limits of matchability of a two port network under the assumption that the load impedance and matching network consist of lumped, passive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 and reciprocal elements. With these assumptions, it is shown that a relationship exists between tolerance of match and bandwidth leading to the conclusion that a better match implies narrower band performance, or conversely, a broader bandwidth implies a worse match. The design and implementation of self-biased circulators has been a goal of many re searchers [26] [27]. Three self-biased circulator designs are presented in Chapter 5 and the theory of Fano is applied to these devices. When the fundamental limits are computed for these devices, it is discovered that broader bandwidths are attainable in practice, although the fun damental limits still provide a good rule of thumb for the matchability of a given circulator device. Chapter 6 presents a circulator that deviates from the traditional shape of Bosma [1] by using a rectangular shaped ferrite region with ports that diverge at right angles from each other. For analysis purposes, each port of this circulator is designed to terminate with a standard 50fl termination. The network parameters of this device indicate that two of the isolation specifi cations can be designed to have a very broad band performance for a 20 dB specification. The third isolation specification is necessarily narrow as shown by the fundamental limit calculation. Chapter 7 presents two antenna/circulator systems that each employ a different antenna ele ment. The substrate Yagi antenna of Kaneda et. al. [28] is invoked and modified by incorporat ing a different dielectric constant and additional director elements. The single director antenna of Kaneda exhibits a broad beamwidth in the //-p la n e of 170°. The double director antenna exhibits an H -plane beamwidth of 140° and the triple director antenna exhibits a beamwidth of 120° in the H - plane. Further, the maximum gain of the antenna increases with each additional director element, with an overall improvement of approximately 1.5 dB. The ferrite material chosen for the system design was characterized with a ferromagnetic resonance testing system and vibrating sample magnetometer to validate the commercially published values. The scatter ing parameters of the systems were measured with the ferrite material fully saturated; the results demonstrated that design goals were met. Each system was then magnetically tuned to optimize performance, with optimal results occuring when the ferrite is not fully saturated. Each of the system designs discussed thus far incorporate a circulator and other compo nents with matching networks providing impedance equalization. Chapter 8 presents an ap proach wherein an antenna element is integrated with a circulator without the need of inter connecting matching networks. Impedance equalization is accomplished by simply varying the ferrite geometrical and material parameters. A modest 20 dB bandwidth of 6% is achieved in simulation. Verification of this design through fabrication and measurement has not yet been accomplished. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Chapter 2 Wave Processes in Microwave Ferrites The interaction of microwave signals with ferrite materials is somewhat unique, and typically not discussed outside of the ferrite community. However, these materials are integral to the contributions of this dissertation. Consequently, this chapter is included to introduce the con cepts and notation of ferrite materials, as well as to provide a framework for the rest of the dissertation. 2.1 High-Level Ferrite Description A ferrite is a material that contains a net macroscopic magnetic moment in a particular direction, which is primarily due to electron spin and secondarily due to electron orbital motion. Both spin and orbital motion can be modeled in terms of equivalent currents that give rise to a net magnetic field. If the number of electrons spinning in one direction are greater than the number spinning in the opposite direction within the material, a net magnetic field will be observed on a macroscopic scale. This net magnetic moment of the ferrite is quantized by the magnetization vector M . 2.1.1 Biasing Considerations To align the magnetic dipoles in a particular direction, either an externally applied dc magnetic intensity is required, or the material must have high crystalline anisotropy. These effects, as well as the demagnetizing field [5], combine to form the effective internal field H a [25]. For ellipsoidal ferrites, H 0 = H ext + Hk — N • 47tM s , (2.1) where N is the demagnetizing dyad which is a function of sample shape, Hk is the effective in ternal field due to crystalline anisotropy and H ext is the externally applied dc magnetic intensity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 Other effective internal fields can be introduced [25] to provide a more complete description, but the terms included in Eqn. (2.1) are sufficient for most applications. Typically, the ferrite is used in microwave applications in a saturated state, which suggests that all moments are aligned in a single direction. For this reason we choose to use the saturation magnetization 47tM s in Eqn. (2.1). 2.1.2 Units In this dissertation two systems of units will be employed. When addressing biasing, or dc, quantities, the Gaussian system is employed. In this system, the magnetic intensity, H , is ex pressed in units of Oersteds (Oe) and magnetic saturation, 47tM s, and magnetic flux density, B , are both expressed in units of Gauss. When referring to rf quantities, the MKS system is employed. In this system, the magnetic intensity, H , and magnetic saturation, M s, are both expressed in units of A/m and magnetic flux density, B , is expressed in units of Tesla (T). 2.1.3 Material Parameters The following material parameters are commonly used when referring to ferrites: • Linewidth (AH): a measure of resonance losses, with units of Oersteds (Oe). • Gyromagnetic Ratio ( 7): a constant roughly equal to 2.8 MHz/Oe. • Relative Permittivity (e /): a measure of the electric polarizability. • Dielectric Loss Tangent (tan <5): a measure of the dielectric losses. From these material parameters, we can define the Larmour precession frequency, u>0 = 2tt^ H 0, and the magnetization frequency, u m — 2tt^(4:TtM s). 2.1.4 Polder Model When electromagnetic waves propagate though a ferrite, such waves are solutions to M axwell’s equations, which in source-free media are given by V x E = dB at (2.2) and <9D V x H = — . (2.3) at In Eqns. (2.2)and (2.3), E is the rf electric intensity, H is the rf magnetic intensity, D is the rf electric flux density and B is the rf magnetic flux density.We then canapply the constitutive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 relations as usual. The relationship between E and D is given by D = eE, where e is the permittivity of the material. The relationship between B and H is given by B = fiQ(H + M ) , (2.4) where //0 is the permeability of free space. Hence, the two curl equations are dependent on the three unknown vectors E, H and M . In order to solve the equations for ferrite media, a third vector equation is required, namely the Gilbert equation [25] as given by dM = —/z07 M x H dt a dM | M| M x ^ r (2.5) In this equation, a is the phenomenological loss term and is derived from the linewidth, A H according to a = ---------, (2.6) ^^meas where ujmeas is the frequency at which the linewidth is measured. Eqn. (2.5) represents a phenomenological model that can be used in conjunction with Maxwell’s curl equations to describe electromagnetic wave behavior within a ferrite material. If all of the dipoles within the material are aligned in the same direction, say the z-direction, the Gilbert equation of Eqn. (2.5) can be used to define a magnetic susceptibility dyad, [x], as [29] M = [x] H = X.xx Xxy 0 Xyx Xyy 0 0 0 0 (2.7) and for an ejuJt time dependence when a = 0, the elements of the susceptibility dyad are given by Xxx = Xyy ( 2 . 8) u j2 — u j2 and Xxy JUJLOm UJ2 — u 2 Xyx (2.9) Eqn. (2.7) can be combined with Eqn. (2.4) to become Ai B =JL H = ~JK 0 0 ( 2 . 10) A^ 0 0 A*o where // is called the Polder dyad [30], The elements of the Polder dyad are given by A* /^ ° ( 1 3 " Xxx) A4o ( 1 " E X y y ) At ° ( 1 " b U!2 ~ CO2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.11) 8 and K Jk-oXxy — J fJ’o X y x — /L (2 .12) when a is again assumed to equal zero. When a is nonzero, we replace tu0 in Eqns. (2.8) (2.12) with uj0 + jacu. 2.1.5 Other Supported Waves When wavenumbers are very large, the wavelength becomes very small. As the wavelength ap proaches the interatomic spacing, a phase shift can occur between neighboring magnetic dipoles. This phase shift can induce propagating modes that consist exclusively of the rf magnetic in tensity and the exchange coupling between neighboring dipoles; the electric field is negligible in this case. These lattice supported waves (i.e. non-electromagnetic waves) are called “spin waves” and convey energy. When we account for spin waves, we must modify the Larmour precession frequency, u>0, to include the exchange coupling effects. For plane wave propagation of the form e~jk'rejujt, we define the modified precession frequency, u>r, as [24] u)r — u)Q T uje x Q? k^ (2.13) where u>ex = 2-k^ H ex, a is the interatomic spacing and k is the wavenumber. Note that H ex is the effective internal dc field associated with the exchange energy. If the wavelength is small, but still large relative to the interatomic spacing, static modes can be supported. In this case, the rf magnetic intensity interacts directly with the dipoles, and the electric intensity is negligible as before. This phenomena is known as “magnetostatic modes” and represents energy storage only; energy propagation does not occur. 2.2 Generalized Propagation in Infinitely Extended Ferrites Before addressing the topic of circulators in finite ferrite samples, it behooves us to first consider wave propagation in unbounded samples in order to ascertain the various wave mechanisms that can be supported. Such mechanisms include Faraday rotation, birefringence, magnetostatic waves and spin waves. We will outline the governing equations for each mode as well as the conditions whereby each can be analyzed. A brief outline of the ensuing material can also be found in Lax and Button [24], Section 7-5. We begin with Maxwell’s curl equations, which are given by (2.14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and V x H = ju e E , (2.15) where we have assumed an e]u3t dependence. If we take the curl of Eqn. (2.15), we obtain V x V x H = j u e V x E = u 2ep, ■H . (2.16) Since V x V x H = V (V • H ) - V 2H , V (V • H ) - V 2H = o;2e^ • H . (2.17) H = H 0e - j k r , (2.18) Let where k is the wavevector that is to be determined in this analysis. Then V (V • H ) = - k ( k • H ) (2.19) V 2H = - k 2H , (2.20) k = n k, (2.21) and where k 2 = k • k. If where n is the unit vector in the direction of wave propagation, then from Eqn. (2.19) V (V • H ) = —fc2n ( n • H ). (2.22) By inserting Eqns. (2.20) and (2.22) into (2.17) and eliminating the common propagation factor e _jk r, we obtain —k 2n ( n • H 0) + A:2H 0 = u)2e]I • H 0 (2.23) k 2 [H0 —n (n • H 0)] = u;2e/ZH0. (2.24) Ho — Hox^-x ~"F f^oy&-y “i” (2.25) or Let z&z and n = n x a x+ n ySLy + n z az = sin 0 cos <fra.x + sin 9 sin (f>&y + cos 6az , (2.26) where 9 and 4>define the direction of wave propagation relative to the standard spherical coor dinate system. For simplicity, but without loss of generality, the plane of propagation is chosen Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 such that 0 = 0 (i.e. x z - plane). For a z-bias, recall that the Polder tensor is given by Eqn. (2.10). The scalar equivalent of Eqn. (2.24) is then k 2[Hox — sin #(sin 9Hox + cos 9Hoz)\ = ui2e[pLHox + jn H ,° y J (2.27) k2H oy = u 2e[—j K H ox + fiHt (2.28) k 2[Hoz — cos 9{sm9Hox + cos 9Hoz)\ = uj2efioH 0Z. (2.29) In matrix form we have —jLU2eK —k 2 sin 9 cos 9 Hox juj2tK k 2 — u 2^e 0 H 0y —k 2 s'm9cos9 0 k 2 sin2 9 —w2e/x0 0 = 1 1 ** k 2 cos2 9 — u)2/j,e 0 (2.30) 0 A non-trivial solution is obtained when the determinant of the above matrix is set to zero, i.e. when, (.k2 cos2 9 — u)2fj,e)(k2 —a)2fj,e)(k2 sin2 9 — uj2efi0) +ju)2€K[jKui2e(k2 sin2 9 — u;2e/x0)] — fc2sin0cos0[fc2sin0cos0(A:2 —w 2/ie ) ] = 0. (2.31) This equation is equivalent to —io2fj,ek4 sin2 9 — k 4ui2jjie sin2 9 cos2 9 + u>ApL2e2k 2 sin2 9 —k 4uj2efi0 cos2 9 + to4e2/j,fj,0k 2 + k 2uj4t 2ii^i0 cos2 9 — Lu6e3fi2ii0 — u>4e2n2k 2 sin2 9 +L) €' k + k 4 ui. 2 lie sin 2 a9 cos„2 9 = 0. . j , ________■ ______ (2.32) Collecting like terms of k4, k 2 and k°, we can couch the previous equation as a quadratic in k 2: ak4 + bk2 + c = 0, (2.33) where - 2 / vi — co2iiQt cos 2/i0 a = —uj2lie sin —u 2^ e s m 2 9 — uj2fi0e( 1 —sin2 9) —uj2e[(fj, — fi0) sin2 9 + fia] = -k2 — - 1 ) sin2 9 + 1 flo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.34) 11 and k 2 = u!2/j,0e. Note, kQ is not the free space wavenumber; it is the wavenumber associated with a non-magnetized dielectric of permittivity e. Also, from Eqn. (2.32) b = uj4/j,2€2 sin2 9 + cj4/r/r0e2 cos2 9 — u 4e2K2 sin2 9 + aj4€2/j,/j,0 = u>4e2[fi2 sin2 9 + ^ 0(2 —sin2 9) — k 2 sin2 9) 2 — sin 9 H u 4 e2 fia Hi (2 —sin 9 ) ----- sin 9 Ho Hi -l— sin2 9 + — (2 —sin2 9) — sin2 9 Ho Hi H, = kt y = kt H Ho IH 2 o sin2 9 + 2 — k4 j^O Hi (2.35) and c = —u}6c3h 2Ho + ui6eSK2Ho K , ,6^3//3 u) e Ho El . hI I h K H Hi Hi (2.36) The solution to Eqn. (2.33) is then ;2 K — —b ± \/b2 — 4ac 2a ------------------------- (2.37) . This equation represents the dispersion relation of a plane wave propagating in an arbitrary direction in a z-directed saturated ferrite under the non-restrictive assumption that <j) = 0. From this relationship one can ascertain the u — k dependency. Fortunately, this equation can be simplified somewhat by computing b2 from Eqn. (2.35): b2 = k» H H K21 Hi Ho H iJ sin4 9 + 4k sA sin2 9 - 4k » A sin2 9 - 4k * ^ sin2 9 + 4 ^ -2k lo'. Hi H Hi Hi (2.38) and —4ac from Eqns. (2.34) and (2.36): —4 ac = 4k, = 4k, H Ho sin2 9 - sin2 9 + 1 ■ 2a K H ——sin 9 Hi K El Hi hI ^ s i n 2 9 + ^ - ^ Sin2 9 + ^ s i n 2 9 - ^ Hi Hi Hi Hi (2.39) H i\ From Eqns. (2.38) and (2.39), it follows that b2 — 4 ac = k\8 o = k:o8 \ y _ H_ .h I y .h I K2 ' 2 K K ■ 2 o sin4 9 + 4 kl — - — sm 9 h I. .Hi Hi 2 _ K2 ' flo h I. sin4 9 + 4kl 1— cos2 9. Hi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.40) 12 Thus, using Eqns. (2.34), (2.35), (2.37) and (2.40), we obtain 1/2 (2.41) i k2 ( U? _ JL _ ^o cos2 9 \ MS Mo Mo/ sin4 9 + 4 M 4Mo - ^Mo- 4Mo sin2 9 + 2-^ Mo ± kl - l ) sin2 9 + 1 i 2 This equation gives the dispersion relation as a function of the ferrite parameters // and k and direction of propagation 9 as a function of lu Let us now find the field components H and E. From the second row of Eqn. (2.30), (2.42) jKU!2eHox + (k 2 —a)2/j,e)Hoy = 0, in which case _ ^ oi — k 2 - u 2 fj,e . o k 2f i a ° V — w 2/ i / i 0 e o * * o y (2.43) Since k Q= _ ' k o2fi - k 2fl0' x ~~ j k 2K = 3k-o ' v 1 - (kZ/k2){ii/n0) K,(k2/ k 2) H o y (2.44) At this stage we note that we are free to set a value for one of the three components of H 0. Following Lax and Button, we choose to let H oy = k 2J k 2, (2.45) in which case H 0x — j 1 - (k2 J k 2) ( ^ h i 0) k/ h 0 (2.46) Now from the third row of Eqn. (2.30), —k2 sin 9 cos 9Hox + (k2 sin2 9 — ui2en0)H 0Z = 0, (2.47) then . fc2sin 0 co s# . H ot — I - ~ ! o “ 7*77 1 Ht 02 1 k 2 sin2 9 - k 2 1 °x sin 9 cos 9 H ox sin 9 —k 2j k 2 • (2.48) Inserting Eqn. (2.46) into (2.48), we obtain Hoz= j '1 - (k0/ k ) 2(ii/fioy k/ h0 sin 9 cos 9 sin2 9 — k 2j k 2_ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.49) 13 Hence, our solution for H , using Eqns. (2.45), (2.46) and (2.49), along with (2.18) and (2.25), is l - ( f c o / f c ) 2 (/n/^o) 3 k/ ho H = s-jk-r k 2J k 2 3 l - ( k 0/ k ) 2 {n/Ho) k / ho (2.50) sin 6 cos 9 s in 2 O—k ' v / k 2 To find the corresponding expressions for E, let E have a similar form as H : E = E 0e~j k r . (2.51) Then from Eqns. (2.15), (2.18) and (2.51), —j k x H 0 = joje'Eo. (2.52) And thus, from Eqn. (2.21) and (2.25) (2.53) u;eE0 = fcH0 x n. For 4>= 0, we observe from Eqns. (2.25) and (2.26) that H 0 x n = [Hoxax + H oyaiy + H ozaz] x [sin 9ax + cos 6az] = —Hox cos 9ay — H oy sin 6az + H oy cos dax + H oz sin day . (2.54) Matching like components, we note that k E ox = — cos 0 Hoy. cue (2.55) But Hoy = k 2/ k 2 and k ue k ujy/fiot k0 toe k fjit^ kD (2.56) so that r-i k 0 Ik-o Q E qx — ^ \ l cos 0. (2.57) Next consider the y-component of Eqns. (2.53) and (2.54): k E ov = — \Hoz sin 6 — H ox cos 9\ toe = (2.58) \ j ^ - [ H oz s in 9 - H ox cos9\. K0 V € Since H 0 is given by Eqn. (2.50), Eqn. (2.58) is identical to 1 - (k0/ k ) 2(n/no) k /H o k nn^. k0 \ e J 1 - (k0/ k ) 2(fi/fio) k / ho sin2 9 cos 9 sin2 9 — k l / k 2 —cos I (K / k f cos 9 sin 9 — k 2/ k 2_ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.59) 14 Finally, for the z-com ponent of Eqns. (2.53) and (2.54), we observe that E oz = = — k H oy sin 9 uie k0 [ K . 0 ~ r \ — sin e /. k V e (2.60) Assembling the various components into one vector, i.e. using Eqns. (2.51), (2.57), (2.59) and (2.60), we conclude that cos 9 E = kQ IJl, cos 6 sin 2 6 —k 2/ k 2 K/flo -jkr (2.61) — sin# To avoid dealing with singularities when 9 = 90°, however, we choose to renormalize E and H by (sin2 9 — k l / k 2)1!2. From Eqns. (2.50) and (2.61), l - ( k 0 / k ) 2 (iJ./fio) (sin2 6 — k2/ k 2)ll 2 k//i 0 H = (2.62) (ko/k )2(sin B - k 2J k 2)1/2 l-(f c p sin 6 cos 6 (sin 2 e- fc2/f c2 )V 2 / k ) 2 ( f i /n o ) k/)1o and cos #(sin2 9 — k 2/ k 2)1^2 E = tl [K k y e J l - ( f c o / f c ) 2 Ql/Vo) k/ ho cos 6______ (sin2 6 o~ik r — k o / k 2 ) 1/ 2 (2.63) —sin#(sin2 9 — k l / k 2)V2 This completes the formal solution to the problem of an arbitrarily directed plane wave in a saturated ferrite biased in the z-direction. We now want to consider Eqn. (2.41) for the two special cases: 9 = 0° (longitudinal to bias, Faraday Rotation) and 9 = 90° (transverse to bias, Birefringence). Case 1: Faraday Rotation When 9 = 0°, Eqn. (2.41) reduces to k2 n 1 , . k TJ = — ± — = — (M ± «)• kQ /i0 f^o j-^o (2.64) Now recall that li = Ho (2.65) 1+ and UJ UJ m K = llo I - j 2 .■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2 .66 ) 15 By inserting Eqns. (2.65) and (2.66) into (2.64) we obtain UJrUJm UJUJm UJ2 — UJ2 (2.67) or k2 L 1+ uir ± Uj) (2 .68) u thus k 2 = k J2 l + — - ----\ u 0 + uiexa2k 2 uj (2.69) (2.70) Solutions of uj in terms of k involve solving a cubic in uj (i.e., to3 terms). This is difficult to do analytically and is best done by considering a linearly polarized wave as a superposition of RHCP and LHCP waves; for example see [29]. A numerical solution is fairly straightforward and is presented in an ensuing section. Case 2: Birefringence When 9 = 90°, there are two possible values for k in Eqn. (2.41), depending on the choice of sign. If we choose the minus sign, (2.71) (2.72) Since k 2 = k%, then sin2 6 — k2/ k 2 = sin2 9 — 1 = —cos2 9 = 0, (2.73) since 9 = 90°. For this case Eqns. (2.62) and (2.63) reduce to 0 H = e —j k x 0 (fio - (2.74) fJ,)/K and (2.75) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 This is clearly the ordinary TEMX wave; it propagates through the ferrite as if the ferrite were isotropic with an effective permeability of / i Q. This is because the rf H field is parallel to the biasing H field, and as such, the rf field exerts no net torque on the dipoles. If we choose the plus sign, then from Eqn. (2.41), k2 ^ / h I - M/Vo - «2/Mo + /VAio _ /i2 - K2 ^ Me n/Ho fJ'O HHo (2.76) And from Eqns. (2.62) and (2.63) i —OWMW mq) (1 k/Mo j H o / H e ) 112 - —j k x H = {.H o/H e)(l H o/H e) ^ (2.77) 0 and 0 f~lo E = s/ H D- j k x 0 ^ (2.78) - ( 1 - Ho/He)1/2 This is clearly the extraordinary TEX wave; it propagates with a component of H that points in the direction of propagation. Now consider Eqn. (2.76) more closely by inserting (2.65) and (2.66) into (2.76): H_ k2 Kj i ' 1 Ho H* UJ2 — UJ2 + UJr UJm UJ2 — UJ2 '( u j2 - u j2 + ( u j2 - 2 , ,2 (uJ2 - UJ2 + Ulrujm)2 — U32U),m ( c j2 - U 2 )(UJ? ~ UJ2 + UJr LUm ) UJr UJm ) 2 UJ2 + UJ2 UJ2 m UJr UJm ) 2 N D' (2.79) Note: N = (ui2 - u 2)2 + 2u rujm(u2r - u?) + uj2ru m 2 - u 2uj2m = {uj2 — UJ2)[uJ2 — UJ2 + 2UJrUJm + = {u2 r - u j 2)[{ujr + u m)2 - u 2} (2.80) and thus, k2 k2 (u r + ujm)2 — uj2 UJ2 - UJ2 + UJrUJm ' (2.81) That is, for 9 = 90°, 1 k 2 = U 2 (ujr + UJm)2 - UJ2 ii0t UJ2 + LUr UJn UJ‘ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.82) 17 This equation is a bi-quadratic (i.e., u j4 terms) and solutions for uj in terms of k can be found. That is, Eqn. (2.82) is equivalent to uj'4 — bu2 + c = 0 (2.83) where k 2 b = (u r + ujm)2 H-----A*0e (2.84) and k 2 ( u >2 + u ru m ) (2.85) Hot Hence, ur,2 (b ± V b2 - 4c)/2. ( 2 .86 ) This equation is the dispersion relation for u) as a function of k for the extraordinary mode. The fact that both ordinary and extraordinary modes can be supported for the 0 = 90° case gives rise to the notion of birefringence of modes. 2.2.1 Supported Wave Types Eqn. (2.41) provides the generalized dispersion relationship for waves in a ferrite under the as sumption of Landau-Lifschitz-Gilbert dynamics. From this equation we were able to ascertain various forms for specialized cases, such as Faraday rotation and birefringence. However, there are other cases that are also of interest. For example, there are cases when the electric intensity is equal to zero within the material; the magnetic intensity can interact with the magnetic moments to provide an energy storage mechanism similar to an LC circuit in low frequency applications. The modes of energy storage are called magnetostatic modes. Similarly, if the wavenumber is large enough, the intaraction of the rf magnetic intensity with the magnetic dipoles can sup port energy propagation, and hence a wave structure is supported. These waves are called spin waves. In this section we present the necessary conditions associated with the three fundamental modes supported by a saturated ferrite: electromagnetic modes, magnetostatic modes and spin wave modes. A typical dispersion plot of u >versus k is shown in Figure 2.1 for the various cases explored herein. There are several interesting features to this plot. First consider the case when k ~ k a. In this case, both H and E are important and the wave process is electromagnetic. Second consider the case when k / k a 1. From Eqn. (2.50), we see that jHo/K II ~ 0 j ( n o/ k ) cot# Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.87) 18 18.0 15.0 | 12.0 o C 05 1l-H IX o0 fi cd 90 §<Z) 6.0 01 o— o Faraday Rotation a ------ a Faraday Rotation a o Birefringence Q □ Birefringence 3.0 0.0 W avenum ber, k (1/m) Figure 2.1: Resonance frequency in the infinite medium as a function of wave number. The regions indicated are (I) electromagnetic propagation, (II) magnetostatic modes and (III) spinwave modes. The curves for the extraordinary waves are computed from Eqns. (2.68) and (2.86). In this plot H a = 47xMs = 2000 Oersteds and = 12. Then V x H = —j k x (Hxax + H za 2) ( 2 .88 ) and for 4>= 0, V x H « -jk sin 9ax + cos 9az] x [j— ax + j — cot 6az K K ■i r ./^o ^ ./^o /) -j kr = —j k J — cos u — j — cos u aye L = K n 0 . (2.89) This is the condition for magnetostatic modes, in which case, from M axwell’s Equations, E « 0. Thus, for “large” wavenumbers, the electric field is negligible and the phenomena is a function of H and M only. In the magnetostatic region k? /k l » 1, which suggests that the denominator Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 of Eqn. (2.41) is essentially zero. That is, — - 1 ) sin2 0 + 1 ^ 0 . (2.90) Inserting Eqn. (2.65) in the above equation, we obtain UJrOJ-m. . 2 sin 0 + 1 « 0 u>2 — UJ2 (2.91) or u? = uj2 + uoruim sin2 0 = u r(u)r + uim sin2 0) w , . , , . „21„2 , . . - 2 , = (u!0 + LUexa, 2 ,k„ 2 )(uj0 +uiexa k +a>m sin 6). Clearly, when u a (2.92) ojexa2k 2, uj is independent of k. Hence, duj vg = group velocity = — sa 0 die (2.93) Thus, in the magnetostatic region we have a zero slope, as suggested in Figure 2.1. Also, since the phase velocity vp is given by vp = u>/k, we see that vp « 0 well into the magnetostatic band [31]. Finally, from Eqn. (2.13), we observe that exchange effects will begin to be noticed when i2k 2 > O.lwo (jJoxQ> (2.94) This inequality defines the spin wave region of Figure 2.1. For iron, typical parameter values are: a « 5 x 10_8cm, f ex « 14 THz (i.e., H ex « 5 x 106 Oe), uiex = 27Tf ex, f 0 ~ 5.6 x 109 (i.e., H 0 w 2000 Oe), in which case k ~ t/ ^ Va ~ 1-3 x 105 r/cm. (2.95) Jex Using the previous discussion as a guide, we are in a position to classify the various regions of the dispersion plot, and write the governing equations for each wave type, assuming z directed bias, small signal conditions and full saturation. Electromagnetic Waves: k ~ ka jujg0H = —V x E —jujgjs/i. jueE =V x H (2.96) (2.97) juiMx = —ui0My + ujmHy (2.98) ju>My =i0oM x - u mHx (2.99) Mz = 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.100) 20 Magnetostatic Modes: k / k D 1 and k <C lo0/(a?ujcx) V x H =0 JU)AflX ( 2 .1 0 1 ) UJ0A/[y juiMy iOmHy lu0M x u mHx Mz =0 (2 . 102) (2.103) (2.104) Spin Waves: k > u 0/{ a2u ex) V x H =0 jU>M X = (2.105) - U J 0M y + UJm H y + UJeXCl2 V 2 M y juiMy = u 0M x - uJmH x - ujexa2V 2M x M z =0 (2.106) (2.107) (2.108) We close this section by noting that a scalar formulation for the magnetostatic modes can be obtained by recognizing that V x H « 0 and by letting (2.109) H = VV>. In the present context, r4> is the Walker potential [24]. Next recall that B = /jq(H + M ) and V • B = 0, in which case V • H + V • M = 0. (2.110) Since V • H = V • V'tp = V 2i>, then from Eqn. (2.110) d M x dM „ dx dy vV + = 0. ( 2 . 111 ) Now we have previously indicated that Eqns. (2.102) and (2.103) can be solved to yield " M x \xx My Xxy Xxx Xxy Xxy H x Xxx Hy Xxy dijj/dx Xxx _ dip/dy _ ( 2 . 112) where \ xx and Xxy are given by Eqns. (2.8) and (2.9). By means of a spatial derivative of each term in Eqn. (2.112), d M x dx 8M y dy d 2j) d 2j> Xxy ' dx2 dxdy d2ip d2j> Xxx Xxy d xd y dy2 ’ Xx ~ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.113) (2.114) 21 Substituting Eqn. (2.113) and (2.114) into (2.111), we obtain (2.115) or ( d2rip (1 + x J ( 8^ d2ip\ d2ip (2.116) + a ^ ) + f c 5‘ = a This is known as Walker’s Equation. It is understood that ip is a phasor with an ejuJt dependence. Consider wave solutions of the form ip — ipDe~jk'r (2.117) where ip0 is a phasor with ejujt dependence. Then from Eqns. (2.116) or (2.115) ~ ( kl + kl + kl)^o - X xxikl + 1 _ kl + kl k l)lpo = 0. (2.118) Thus, kl + kl +kV Xxx (2.119) Using geometrical considerations, we note that kl + kl1 = sin22 f), kl + kl + kl ( 2 . 120 ) Since U J o ^ rn n o ? X.XX ( 2 . 121 ) — UJZ it follows from the previous three equations that UJ 2 — UJ 2 Of \ o • 2 ------------— sin 0 ( 2 . 122) or UJ 2 2 , -2/1 = U J 0 + U J 0^ T n S m V, (2.123) which agrees with Eqn. (2.92) when exchange effects are neglected. When 9 = 90°, uj2 = u>2-\-u0ujm which defines the upper range of the magnetostatic manifold. When 6 - 0°, Ul = LOa, which defines the lower range of the magnetostatic manifold; see Figure 2.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 Chapter 3 Time Domain Simulation of Wave Propagation in Ferrite Materials The previous chapter dealt with waves in unbounded ferrite media, which allowed for a closedform analytical solution. This development provides insight into the behavior of waves in fer rites, but is insufficient to describe the wave behavior within bounded media. When boundaries are present, the solution is often difficult to obtain analytically and a numerical algorithm is necessary. A time-dom ain scheme can be derived based on Maxwell’s equations of Eqns. (2.2) and (2.3) coupled with the Gilbert equation of Eqn. (2.5). Two popular approaches to a time-domain simulation scheme incorporating ferrites have already been considered to date - recursive con volution (RC) and direct integration (DI). The RC methods use the time-domain representation of the elements of the Polder model of Eqn. (2.10) in conjunction with the constitutive convo lution integral as the governing equation for the ferrite. Depending on whether the convolution integral is discretized using piecewise constant or piecewise linear interpolation, Schuster and Luebbers [9] and others were able to achieve stable schemes that yielded first or second or der solutions. Both solutions, however, admit a certain amount of artificial dissipation, even when the ferrite is lossless. Other researchers, such as [10] - [12], adopted various direct in tegration methods, whereby the Gilbert equation is used as the starting point. In these cases, one-sided differences, central differences and/or central averages are invoked to discretize the Gilbert equation. As with the RC methods, the DI methods that employed one-sided differences also yielded data that are corrupted by artificial dissipation. The scheme considered in this chapter continues the development of the direct integration methodologies by following a similar line of thought advocated by [12]. However, instead of rewriting the torque equation in terms of the magnetic flux density B and the magnetic intensity H , as was done by [12], we opt to write the torque equation in terms of the magnetic flux density Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 B and the magnetization M . By doing so, we leave the torque equation in its natural state variable form, which allows us then to discretize it using central differences and averages. The resulting discretized equation is couched in a compact form using dyadic and vector operations and is valid for any direction of ferrite field bias. To validate the proposed integration methodology, a three-dimensional, stripline phase shifter is considered in which the field bias points in the direction of propagation. This de vice is particularly attractive for our purposes, since various biasing fields need to be considered to see the phase shift effect. Using a solution produced by a frequency-domain, finite-element solver as a standard, we show near perfect agreement between that standard and the data pro duced by the proposed scheme. Moreover, from a device performance point of view, a full 180° phase shift can be observed by adjusting the effective internal field from 0 Oe to 2800 Oe; the return loss remains below 20 dB and the transmission loss remains below 0.5 dB over this same range. 3.1 Derivation of Time-Stepping Equations Within the ferrite material, the magnetization M and magnetic intensity H are related to each other by the Gilbert equation of motion, including the phenomenological loss term [25], From Eqn. (2.5), we recall that ¥ = -M w M « TT H - j «s j ™ M x¥ . (3.1) The vectors M and H consist of both dc and ac components. The dc component of H is the total effective internal dc magnetic intensity; the dc component of M is the internal dc magnetization. For purposes of this work, we are only interested in electromagnetic waves. Spin wave effects are particularly excluded due to the absence of exchange coupling effects in Eqn. (3.1). The ac magnetic flux density is related to the magnetic intensity and magnetization as [29] H ac = — B ac - M ac, fJ'O (3.2) so the total magnetic intensity within the ferrite is given by H = H dc H B ac —M ac. (3.3) In this work we define a vector Larmor precession frequency uJa and vector magnetization fre quency u>m such that UJo — Ho'yUrfc — UJ0x<\x + CtJ0y3.y -f- UJqz3.z (3.4) and UJm fJ'o'y'^A-dc UJrnx3.x -|- LOrrly^Jy -|- UJrrlzH.z . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.5) 24 Eqn. (3.1) can then be expressed as <9M at dM.ar N _ 1_ ,, , a , , _ . - —i//o (Hdc —\( M ac * +1 ■M oi ° 7i (Mdc \* ■ “■cic +1 M ac) x v **ac H ■ ^Ba ac c — M-a- ac) a cj —— i-*/ri ‘■ ■ ‘■ac) a c j 'xv at ii o |M | /io7 M dc x H(/c - 7 M ac x ujo 3“ ^ a <9Mac M ac x B ac + t_ ■. — |a;m| Ot _ x u jm If we assume that |M ac| < |M dc|, |H ac| < |H rfc|, B ac x ]\dac x (Jo a ,, 5 M ac - TT-prMac x —— . |M | at r \, at (3.6) || H dc and |M dc| = M s, then Eqn. (3.6) becomes <9Mac o. Ot 1 = — fia B ac _ x u m - M ac _ Xujm - M ac x a _ uj0 + |a;m| <9Mac Ot x _ u im . (3.7) In so far as the equation of motion given by Eqn. (3.1) is valid, Eqn. (3.7) represents the relationship between M and B within a fully saturated ferrite under small-signal conditions. This equation, as cast in terms of dc biasing fields and ac signal fields will be the basis for the subsequent discretization procedure. For the purposes herein, we will regard the Gilbert form of the Landau-Lifschitz model to be phenomenologically true in the form given by Eqn. (3.7) [24], Along with Eqns. (3.2) and (3.7), the equations governing the electromagnetic problem are: ^ dt = - V x H ac e (3.8) = - V x E ac, (3.9) and Y where E ac is the ac electric intensity and B ac is the ac magnetic flux density.These equations, along with Eqn. (3.7) can be temporally discretized using both central differences and averages as follows: K t1'2= K :1'2+ b :acc +i V x H "c, 7 = b ac ^ - ^ 1 v x e :ac: i/2,’ M " +1 = M " + — ac - 5t Ho /M a n ) ( M " ^ 1 + M "c + B qA V 2 J -l- M " \ n ac + ac ) x u j 0 + —— (M ”c+1 - M "c) x cJro, " +1 V h (3.10) / (3.12) |^ m | : 7 = T B^+1 - (3.13) f-lo Here 5t is the numerical time-step and B "c, for example, is the value of B ac at time ndt . It should be noted that M ac, H ac and B ac are all temporally synchronized in the aforementioned scheme. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 Equations (3.10) - (3.13) are valid for any numerical scheme that employs both temporal central differences and averages. To obtain an implementation algorithm associated with Eqn. (3.12), we must first solve for M " ^ 1. To do this, we define the following dyads couched in matrix form: 0 = % 0 Umx Wmv -Umx 0 1 Mu —Mly - M lz 1 Mu —M u 1 l m 2z ~ M 2y - M lz 1 m 2x II ^m z 1 £1 Umz (3.14) (3.15) and 2 = m M 2y M 2x (3.16) 1 The elements of M \ and M 2 are ir fit i , M i n — ” \^mn £ aLiJmn___ _ i ij^ *^>2/j ^ OJm. \ n) (3.17) and A/ T n— ^ i I \ ,-v y^ran “r ^on/ a ijJ m n \LOr> , n = x,y,z. (3.18) Hence, M 1 is given by 1 + M lx M \ xM i y — M \ z M \ ZM \ X -T M \ y M \ xM i y + M \ z 1 + M^y M XxM i z — M \ y M i yM \ z + M \ x where A = 1 + M \ + M?ly +' l z'. M iy M iz — M i x 1 + M \z With these definitions, Eqn. (3.12) can be expressed as 2 ______ M ”c+1 = M j (3.19) A ____ i / R n~^ -I- R n • M 2 • M " + — M 1 ■WT 1 ac ac (3.20) /i o where, for example, M i • M ” represents a dyadic-vector dot product operation. The cartesian Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 components of Eqn. (3.20) are then M ”+1 = ^ [ ( i + M l ) - M 2z (MixMiy - M u ) + M 2y (M u M 1x + M ly)\ M x" + ^ [M2z ( l + M l ) + (M lxM iy — M u ) — M 2x (M \ ZM \ X + M i y)] M y [—M 2y ( l + M l ) + M 2x ( Mi xM \y — M u ) + (M u M u + M ly)] M " +■ ^ H T- [—u mz ( M lxM ly — M \ z) + u>my ( M lzM lx + M ly)] £?£+1/2 ft 0 ^ + —V K , (1 + M 2X) - o,mx ( M u M u + M ly)] B ”+1/2 fto L \ 3-----T- [—1 ftol^ (l + M l ) + wmx ( M ixM i y — M u)] B z +1^2, (3.21) M y +l = — [(M ixM iy + M l2) — M 2z ( l + M 2y) + M 2y (M i yM u — M ix)\ M " + ^ [M2z (M i xM \ y + M u ) + ( l + M 2y) — M 2x (M iyM u — M ix)] M™ + — [—M 2y (M \ xM ly + M \ z) + M 2x ( l + M l ) + (M i y M i z — M Jx)] M z 3 T" [ _ u m2 ( l + M 2y ) + LUm y ( M i y M u ~ M i x )] £ ? £ + 1 / 2 fto L \ £ H fto T- [wmz (MlxMly + Z A M u ) ~ tOmx ( M i y M u — M u )] By +1/ 2 £ -r [—uimy (M ixMiy + M u ) + u mx ( l + M 2y)] B z +1^2 H (3.22) fto ^ and M ; +1 = — [(M ixM lz — M ly) — M 2z (M iyM i z + M u ) + M 2y ( l + M 2Z)] M " + ^ [-^2z ( M ixM u — M iy) + (M iyM i z + M u ) ~ M 2x ( l + M 2Z)] M y + ^ [ ~ M 2y (M ixM u — Miy) + M 2x ( M i y M u + M ix) + ( l + M 2l)] M z t" [—u mz (M iy M iz + M u ) + u my ( l + Mil)] 5 " +1/ 2 H fto ^ T" [^rnz ( M ixM i z + Mly) — U!mx ( l + M l ) ] B y +1/2 H fto lA £ H — [—Lomy ( M u M u — Miy) + u)mx ( M i y M u + M u ) ] B z +1^2. fto lS (3.23) To complete our treatment, we choose to employ the Yee cell of Figure 3.1 in conjunction with spatial central differences and averages. With this cell arrangement, Eqns. (3.10) - (3.11) can be spatially discretized as in the traditional Yee scheme [8]. Since the components of the magnetization vector have been placed coincident with the magnetic flux density and magnetic intensity, Eqn. (3.13) requires no spatial averaging. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Bz,Mz Hx,Bx,Mx Figure 3.1: Yee grid cell for this new scheme. With the grid chosen according to Figure 3.1, Eqns. (3.21) - (3.23) clearly show that four point spatial averaging is necessary in order to compute the update equations for magnetization. For example, we see in Eqn. (3.21) that M " +1 depends on the previous values of M ” , M ", B y +1^2 and _B"+1/2, each of which is not located at the same point in the grid as M " +1. Hence, each of these components require four-point spatial averaging in order to compute the correct value at the location of M x within the grid. 3.2 Numerical Results To validate the previously developed scheme, consider the stripline phase shifter topology shown in Figure 3.2. This problem is an appealing validation geometry for the algorithm pre sented herein, as it provides a mixture of different materials (substrate, PEC and ferrite), allows a comparison of several different values of effective internal field and requires an applied bias in a direction different from bias normal to the plane of the circuit. The geometrical parameters chosen for this validation are W f = 2.9mm, L f = 24mm, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 W = 1.1mm and h = 2mm. The ferrite material chosen is Trans-Tech TTVG-1000 which has a dielectric constant of t f = 14, magnetic saturation 4irMs = 1000 G and linewidth A H — 10 Oe, as measured at 9.4 GHz. The surrounding material is Trans-Tech D-16 Mg-Ti which has a dielectric constant of ef/ = 16. From these parameters, oom = 1.76 x 1010 rad/s and a = 0.237 x 10-4 . The width of the trace is set to achieve a 250 characteristic impedance. Wf—► IT Figure 3.2: Stripline phase shifter validation topology. The darker shaded region represents the ferrite. The stripline’s TEM waves are excited and detected per the method described in [32]. This approach allows for a much smaller domain size than would otherwise be required as the source and terminal planes of the simulation can be placed arbitrarily near the ferrite/dielectric interface without loss of accuracy. The pulse chosen to characterize the circuit is gaussian of width a, where a = 6.065 x 1010 1/s; the pulse is of the form e~*2//“2. To determine appropriate cell size, we must first consider the wavelength within the ferrite material. Per Chapter 2, A = 2ir/k and k = uj^//iee, where /j,,, = fij — k 2/ n / \ /if is given by Eqn. (2.12) and n is given by Eqn. (2.12). At an operating frequency of 13 GHz, A = 6.3mm for an effective internal field of zero, and A = 8.1mm for an effective internal field of 3000 Oe. To achieve superb accuracy, the cell size is chosen to be less than A/40 which for the shortest wavelength is 0.158mm. For convenience in accomodating the correct dimensions described previously, the following cell dimensions were chosen: 5X — 0.15mm, 6y = 0.1mm and 6Z = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 0.1mm. The overall domain size was chosen to be 100 cells in the y-direction, 180 cells in the .E-direction and 20 cells in the z-direction. The timestep was chosen utilizing a CFL stability criteria of 0.5 so that St ~ 0.1 ps. To validate the algorithm presented herein, a comparison of the real and imaginary transmit ted scattering parameters of the phase shifter was made with Ansoft’s High Frequency Structure Simulator (HFSS), as shown in Figure 3.3. In this plot, we see very good correlation between the two data sets, thus indicating that for the fully saturated and low power case, the algorithm is consistent with frequency-domain formulations. • — • Re{S21} fdtd ■-----■ lm{S21} fdtd —♦R e{S21} hfss — * lm(S21) hfss 0.5 c \j C O 0.0 -0.5 - 1.0 10.0 12.0 13.0 Frequency (GHz) 14.0 15.0 Figure 3.3: A comparison of S2i with HFSS for effective internal field equal to zero. The simulated return loss of this device at 13 GHz is shown in Figure 3.4. Using a 20 dB specification requirement, we note that this device is usable in a 25fl system for an effective internal field less than 2900 Oe. The simulated phase shift at 13 GHz is shown in Figure 3.5 and shows that for a reasonable effective internal field strength, this device can realize greater than 180° phase shift. To achieve variation in the effective internal field as shown in Figs 3.4 and 3.5, variation of the externally applied bias is necessary. To determine what the appropriate external field should be, we assume a homogeneous ferrite, in which case, Hint = H appue(i — (4 n M sN yy), where N yy is the demagnetization factor. Due to the geometrical shape of the ferrite material, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 m - 2 0 .0 (/) 0) © E (0 aj Q. I a) -4 0 .0 Return Loss Insertion Loss -6 0 .0 0.0 1000.0 2000.0 3000.0 Effective Internal Field (Oe) Figure 3.4: Return loss of the device as a function of applied field when / = 13 GHz. the demagnetizing factor, as given by Joseph and Schlomann [5], is approximately zero, hence the effective internal field is equal to the externally applied bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 200.0 100.0 0.0 - 100.0 - 200.0 0.0 1000.0 2000.0 3000.0 Effective Internal Field (Oe) Figure 3.5: Phase shift of the device as a function of applied field when / = 13 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Chapter 4 Bode-Fano Criterion The simulation scheme outlined in the previous chapter represents one method that may be used to compute network scattering or impedance parameters of a device containing ferrite materials. This scheme can be incorporated into the typical system design approach which requires ferrite devices to be designed to some standard interface specification for incorporation with adjoining components. The simulation scheme may also be used with alternative design approaches such as those of the ensuing chapters wherein system design is accomplished by considering the system as a whole. To successfully follow these approaches, knowledge of optimal limits is necessary to ensure design specifications are attainable and provide insight as to which of various designs are likely to attain the broadest bandwidth, for example. To set the groundwork for the system designs that will be discussed later, an overview of the seminal work of Bode [16] and Fano [17] is presented. In this chapter, criteria will be reviewed which describe the optimal transfer of power from a generator to a load. The overall system to be analyzed is shown in Figure 4.1 and consists of a source with internal impedance of R g, a load impedance, Z tl, and some unknown matching network, N " . The conditions for optimal power transfer can be couched exclusively in terms of the reactive network associated with the load impedance. This is done by considering the conditions of physical realizability for the reflection coefficient, which are exclusively depen dent on the load impedance. The conditions of physical realizability lead directly to a statement pertaining to the limitation of power transfer (tolerance) or bandwidth. A tradeoff is presented: either a tolerance is chosen and the bandwidth is limited, or the bandwidth is chosen and the tol erance is limited. A simple procedure is also presented that can result in a synthesized matching network, N" , which approximates the optimum condition. For this analysis, we make the following assumptions: • The source for the two port network is assumed to have a purely real input impedance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 N” Reactive Two-Port Network (to be determined) Figure 4.1: A purely reactive matching network to be determined. Reactive Two-Port Network (defined by ZL) Figure 4.2: A network equivalent to Z l • The load impedance must consist of a finite number of linear passive lumped elements. • The matching network must consist of linear passive lumped elements (i.e. we exclude distributed elements such as transmission lines, cavity resonators, etc.). • The matching network is assumed to be lossless and reciprocal. 4.1 Physical Realizability Darlington showed [33] that any arbitrary impedance can be expressed as a lossless two port reactive network, N ' , terminated with a purely resistive load, as shown in Figure 4.2. The overall problem to be solved can then be viewed as a pair of reactive two port networks, N ' and N " , connected in series with one port terminated in a pure resistance and the other connected to a source to create the overall two port network, N , as shown in Figure 4.3. For convenience, we will consider the reflection coefficient as seen by the load resistance rather than Tm. We note that since the network N ' is derived from the load impedance, the reflection coefficient F x cannot be physically measured. The scattering matrices, S, S' and S" associated with the networks N , N ' and N " respec- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 out N’ Reactive Two-Port Network (to be determined) Reactive Two-Port Network (defined by ZL) Figure 4.3: Equivalent network to that of Figure 4.1. source r2 v Network S >^ri load Figure 4.4: Flow diagram of the overall network TV of Figure 4.3. tively, are defined as S = S' = Ti t t r2 r; a if r' (4.1) (4.2) and S" = r; t” 1r "2 (4.3) where Ti, T7, and T" are the reflection coefficients closest to the load for network N , TV' and TV", respectively. Similarly, r2, V2 and are the reflection coefficients closest to the source and t, if and t" are the transmission coefficients for network N, N ' and TV", respectively. For example, for the overall network TV, the reflection and transmission coefficients are defined in terms of a flow diagram per Figure 4.4. (Note: Although the terms Ti and T 2 seem reversed in Figure 4.4, Fano viewed T j as the “input” from the perspective of the load. We maintain that perspective in this treatment.) For the purposes of this analysis, the reflection coefficient of the overall network, Tj, is given by Ti = T( ffo ut Z m d + R Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.4) 35 where Z out is the equivalent impedance of the overall network. In general, I \ can also be written as a rationalized polynomial in the s-domain. In factorized form let p _ rL = i (s - zi) (a nj=i (s - P j ) 1 _ rrJ ( ----------V’ ( ' where s = a + jut is the complex Laplace variable and z, and pj are the zeros and poles of I \ ; I and J are the total number of zeros and poles of Ti, respectively. To highlight the fact that the zeros of T i can be located in either the left or right half s-plane, an equivalent representation for Ti is 11 ( s - r 1) ( s - r 2 ) . . . ( s - Z i ) ( s - / 2 ) - - - ^ -------------j---------- T7-------1----------------(s —Pi) (s - p 2) • • • (4.6) where k is the ith zero of F ! that occurs in the left half plane and r, is the r th zero of Ti that occurs in the right half plane. For reasons that will become apparent in the ensuing development, we choose to also define a modified reflection coefficient, r lm, that is equal to r : with one exception: each of the zeros of Ti that occur in the right half plane are moved to a symmetric location in the left half plane; the line of symmetry is the imaginary axis. That is, let (s + r*) (s + r * ) . . . ( s - Zi) (s - l2) ■.. 1m ( s - p 1) (4.7) ( s - p 2) . . . By construction, this new function, T Vm has the same magnitude on the imaginary axis. For example, let T i have four zeros as shown in Figure 4.5. Then T lm has four zeros that are located as shown in Figure 4.6. Next consider the transmission coefficient, t, of of the overall network N which is given by t't" 1= i- r ^ r " ' (4‘8) It is clear from this equation that every zero of t' is also a zero of t, with the same multiplicity, as long as the denominator is not equal to zero. Even when the denominator is zero, every zero of t' is a zero of t, with the same multiplicity, as a result of pole-zero cancellation.To prove this latter statement,we first note that when t' = 0 then |r'2| = 1, sinceN ' is lossless by definition. Then for the denominator in Eqn. (4.8) to be zero, we must also have |T"| = 1. But, N " is also lossless, in which case t" = 0. From these observations it is clear the zero of the numerator due to t" = 0 is annihilated by the corresponding zero of the denominator when if = 0. As for Ti, it is easy to show that (t'f ri = r i + r " r ^ f " ’ (4-9) where we see that at every zero of t' ri = r;. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4 .10) 36 Figure 4.5: Example of four zeros of T JO Figure 4.6: Example of four zeros of rlmthat are derived from the function shown in Figure 4.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 Let us consider the it')2 term in Eqn. (4.9) a little further by letting t, = ( s - z [ n s - z ' y (4.11) (s - P{)u . .. where the Z t are the zeros and the Pi are the poles of the transmission coefficient t'\ n , m , u , . . . are the multiplicities of the associated zeros and poles. From Eqn. (4.11) it follows that 2 ( s - Z [ ) 2" ( s - Z > ) 2m... [ } Hence, when s is evaluated at a zero of d fcr ! dsk (4.12) (s - P[)2u . .. it follows from Eqns. (4.9) and (4.12) that d kF[ dsk a=Z'i for k = 1, 2 , . . . , K, (4.13) s= Z ’ where K = 2n — 1 if i = 1, K = 2m — 1 if i = 2, etc. We see from Eqns. (4.10) and (4.13) that when the system is evaluated at a zero of t' , the reflection coefficient T ! of the overall network and its first K derivatives do not depend on the applied matching network. For convenience later on, we prefer to work with the function l n ^ / T i ) rather than with T i directly. We can show through arguments similar to those above that d k ln(l/rx) dsk d k in (i/r;) dsk s= Z'i for k = 1,2 , , jK.»., j (4.14) s= Z' when s is evaluated at a zero of (t') 2. Next, recall that any function f ( s ) that is analytic at s = s0 can be expressed as a Taylor series about the point s0 according to f { s ) = f ( s a) + (s - + ...+ So) df(s) ds + (s - So)2 2! d 2f ( s ) ds2 (s - s 0)" <9"/(s) n\ ds n (4.15) The region of convergence for the previous expansion is the circle centered at s D of radius r, where r is the distance between s 0 and the closest non-analytical point of ln ( l/F i) . The Taylor series expansion of ln( 1/ F i ) about Z[, is given by — >lo + A i(s — Zl) + A 2( s — Zl)2 + j43(s — Z-)3 + . . . , In (4.16) where 1 dk for k = 0 ,1 ,2 ,___ At = - - < 1 n k\ d s k m i From this last equation and Eqn. (4.14) we can also say y. k 1 dk n k\dsk i n Vrk ) } , for k = 0,1, 2 , . . . , K. s= Z' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.17) (4.18) 38 Given that A k of Eqn. (4.18) is explicitly determined by V i, we see that these are the first K Taylor coefficients of I \ . Hence, it follows from these latter three equations that (4.19) Here H.O.T. denotes Higher Order Terms and is used to illustrate those terms in the Taylor series of l n ( l / r i ) that are different from the Taylor series of l n ( l / r j ) . Since the Taylor coeffi cients are evaluated exactly at a zero of t', Eqn. (4.9) implies that Eqns. (4.16) and (4.19) hold everywhere within the region of convergence for the Taylor series. The significance of Eqn. (4.19) is important: the Taylor coefficients A k are derived from our direct knowledge of T, associated with the reactive portion of the load N ' . These coefficients in turn can be used to partially construct the response of Ti for the overall network, regardless of the matching network N " that is used. Thus, we argue at this point, as Fano did, that the response of N is directly correlated to the response of N ' and conditions for maximum bandwidth or maximum power transfer can hence be deduced. How this is done is explored in subsequent subsections. Before we do so, however, it is illustrative to point out that the function In{f {z)) exhibits a somewhat unique response when the argument is complex. Specifically, when z is evaluated at a branch point or branch cut of ln (/(z )), the function ln (/(z )) is non-analytic. The route that the branch cut takes is arbitrary, but the Taylor series is only convergent in those regions that do not contain a branch cut. For example, consider Figure 4.7 which shows the branch point and a possible branch cut of the function ln (/(z )). O f the two regions shown in the figure, the Taylor series will only converge in the region Ri , and will not converge in the region R 2 since it is bisected by the branch cut. Singularities of the function l n ( l / | r x|) only occur at the poles of r 1; which for passive networks are restricted to the left half plane. In order to ensure that the Taylor series converges, we choose to restrict the ensuing analysis to the right half plane, and choose all of the branch cuts to travel to infinity entirely within the left half of the complex plane. 4.1.1 Zero of t ' at the origin When a zero of t' of multiplicity n occurs at the origin, all of the even derivatives of ln( 1 ), and hence of l n ( l / r i ) , are equal to zero, up to and including the 2(n — l ) th derivative. Hence, the Taylor series expansion of ln ( l/T i) about the point s = 0 + jO is given by In = A° + A°lS + A°s3 + . . . + A°2n_1s2n~1 + H.O.T. (4.20) The zeroth coefficient, A°0, is not important to the ensuing analysis. To determine the remain ing coefficients A \ , A ^ , . . . in this expansion, we begin with the first coefficient, A\ , which is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Singularity in ln(iU)} Branch cut induced by the singularity at Z0 Figure 4.7: An example of a branch cut in the complex plane caused by a singularity of ln ( / ( z ) ) . computed using A° = i ain(i/rx) ds 1! = Ti s=0 (4.21) ds 8= 0 where T , is given by Eqn. (4.5). The derivative on the right hand side of Eqn. (4.21) can be written as 9( 1/ Ti ) = d f (s - p i ) (s - p 2) ( s - p 3) . . . ds d s \ ( s - zi) (s - z2) (s - z3) . . . (4.22) or dj l / Tr) = [(s - P 2 ) (s - P 3 ) ■■ •] [(s - Zi) 1 { s - z 2) 1 ( s - z 3) X..-] ds + [(s - P i) (s - (s - Z2) ~ l (s - z3)- 1 . . .] + - [(s - P i) (s - p3) • • •] [(s - Z i)-2 (s - Z2) ~ l (s - Z3 ) " 1 • • •] - P i ) (5 - E(s - p 3) • • •] [(s - Z \Y l p3) • • •] [(s - Z i) - 1 (s - z 2) - 2 ( s - Z3)” 1 • • •] - (4.23) Hence, + Ti 1 + 1 ... - ds s — pi s — p2 s —z1 When s is evaluated at the origin, Eqn. (4.24) becomes ^(1/ro r1 ds s - z2 + j_+ j_+ s=0 Pi zl P2 (4.24) (4.25) 22 and from Eqn. (4.21) A°= (4.26) ,«=i 3 =1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Here we recall that I and J are the total number of zeros and poles respectively of IV This procedure can be extended to compute each of the coefficients of Eqn. (4.20) up to A \ n_x as ^ + i = 2F T z*~(2k+1)~ ^ 1 7(2fc+1)) f o r k = o ’ i ’ 2’ - > n - 1- (4 -27> Hence, when a zero of t’ occurs at the origin the coefficients of the Taylor series representation of h ^ l/T x ) are expressed exclusively in terms of the individual poles and zeros of IY Finally, let R be the number of zeros of T 1 that lie in the right half plane and L be the number of zeros of T 1 that lie in the left half plane so that R + L = I. Eqn. (4.27) can then be written as ^ = u k ~ l ( ' t i."(2‘ +1> + E \t= l ^ ~ X > Y +1)) for * = 0 . 1. 2 , j =1 i=l 1. J (4.28) In future development, we will also have need of a Taylor series representation of Tim, which is given by Eqn. (4.7). To that end, we write ■ Replacing 1° ■r, „3 1 . /1° 1 „2n— J ~ Aom + A \ ms + A l ms + . . . + ^4(2n_i)ms 71 In + H.O.T. (4.29) with —r* in Eqn. (4.28), we obtain Y Y f e ^ (2fc+1) + E ( - r *)_(2fc+1) - E J4 ( 2 f c + l ) m i= 1 \t= l ^ for k = 0 , 1 , 2 , . . . , n - 1, H j=1 / (4.30) which is equivalent to A° - (2k+l)m " 2k Y I \ E / —(2fc+l) _ Y - ^ ^ _ ( 2fc+l) _ ^ f E c (“ + , ) - X > t ‘“ + i ) - E Vi=l i =1 \ j =i t= 1 - E ^"(2fc+l) j >7 j =1 r 7 (2fc+1)- E ^ 7 (2fc+1)) forfc = 0 , l , 2 , . . . , n - l . j=1 *=1 (4.31) / We recognize at this stage that several terms in Eqn. (4.31) are identical to the terms in Eqn. (4.28). For passive and lumped networks, the zeros are always either purely real or are conjugately matched [34]. Hence the Taylor coefficients of Tim can be expressed in terms of the Taylor coefficients of Ti to yield A \2k+l) m = 2 E R r 7 (2fc+1) for k = 0 , 1 , 2 , . . . , n - 1. i=1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.32) 41 4.1.2 Zero of t ' at Infinity We can also consider a zero of t' of multiplicity n at infinity, where the Taylor expansion of ln(l/ri) about the zero is given by In ( E ) = A0°° + + A3°°s-3 + . . . + J4 ^ _ 3s " (2n_3) + A?n^ s - ( 2n-V + . . . Once again, the even coefficients are equal to zero up to and including (4.33) The odd coeffi cients are given by ATk+1 = 2 f c T l \»=i ~ T , p T +1) forfc = 0 ) l , 2 , . . . , n - l . j =i / (4.34) As before, the z, and pj are the zeros and poles of T t respectively. Following a similar develop ment to that of the zeros at the origin, we can write 2 4(2*+!)m = R E r “ +1 for fc = 0,1, 2 , . . . , n - 1. (4.35) i= l where r, are the zeros of Ti that lie in the right half plane. The coefficients, A ^ k+1^m, are the Taylor coefficients associated with the modified reflection coefficient, Tlm 4.2 Limitations on Tolerance and Bandwidth 4.2.1 Cauchy’s Integral Relations Before we continue with the analysis of optimal matching, an understanding of two mathemati cal relations known as Cauchy’s integral relations [35] is needed. Theorem 1 (Cauchy’s Integral Formula) Let f ( s ) be analytic in a simply con nected domain D, and let C be a simple closed positively oriented contour that lies in D , as shown in Figure 4.8. I f s a is a point that lies interior to C, then lM = - k l ' T * Z i" ( 4 ' 3 6 ) A positively oriented contour is one that is evaluated in the counter-clockwise sense. Eqn. (4.36) states that the function f ( s ) can be evaluated at any point, s a, that exists within the domain exclusively from the knowledge of the function on the boundary. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.8: A depiction of the contour C and evaluation point s0 for the Cauchy integral theo rems. Theorem 2 (Cauchy’s Integral Formulae for Derivatives) Let f ( s ) be analytic in the simply connected domain D, and let C be a simple closed positively oriented contour that lies in D. If s a is a point that lies interior to C, then (4.37) fo r n = 0 , 1 , 2 , . . . From the second Cauchy theorem, we note that we can also compute the nth derivative of f ( s ) at any point, s 0, that exists within the domain directly from knowledge of the function on the boundary. 4.2.2 Conditions of Realizability To see how these integrals may be used in the current analysis, consider two cases: first a zero of multiplicity n at the origin, followed by a zero of multiplicity n at infinity. Zero of t' at the origin Equation (4.37) can be applied to the function ln ( l/T im) when s = 0 as In ( f t ) ^ = j2n d 2fc+1 In s 2(fc+i) (2fc + l)! d s 2k+1 I (4.38) s=0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 s-plane j(B Figure 4.9: The closed contour used to evaluate the Cauchy integral. Z 0 is a branch point of the function ln ( l / F i m) and creates a branch cut that extends to infinity. where the integration is chosen to follow the closed right handed contour consisting of the imaginary axis from +00 to —00 and the semicircle at infinity surrounding the right half plane, as shown in Figure 4.9. In the vicinity of s — 0, the function ln ( l/T lm) can be expanded in a Taylor series according to Eqn. (4.29) over a radius of convergence extending from s = 0 to the nearest singularity of ln ( l/T lm). Eqn. (4.38) can be written as / Q2k+1 s2(fc+i) ds = r2 k + Y ) j <9 s 2 fc+ l [ ^ 0 m + ^ lm S + ^3mS + ••• + A 0(2k+l)ms2k+l + A {2k+3)ms2k+3 + H.O.T.] s=Q . (4.39) A term -by-term differentiation of the right hand side of Eqn. (4.39) and an evaluation at s — 0 yields _ i_ ln (l/r lm)ds. = A ^ +1)m (4.40) As depicted in Figure 4.9 the integration contour consists of two parts: an integration along the imaginary axis and an integration along the arc of infinite radius. Since the integrand along this arc is zero, _1_ f s - 2(-k+1)l n ( l / T lm)ds = j 2 tt J c j f 2 i r J +QO (jw )_2(fc+1) ln ( l /T im)d(jaj). (4.41) The function l n ( l / r lTO) can be expanded as ln( i ) =ln(irb)+j9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 4 ' 4 2 ) 44 where 6 is the phase of l / T i m. The phase is an odd function in u>and will integrate to zero, so Eqn. (4.41) becomes f ~ OO 1 — / 1 r+ o o 0'a»)-2't+1»ln (l/r lra)d(iw) = J +QO I n tl/lr ,!)^ ^ ). JZ 7T 7 - 0 0 (4.43) Note that ln( 1 / |Tim |) = l n ( l / | r 11) by definition and is an even function of lu, so the previous equation becomes r+ oo 1 - - 7T ] / J 1 ( j r 2(fc+1)^ " 2(fc+1) l n ( l / | r a |) r f ( — oo ju ) = ^ r+ o o ( - l ) fc- / u T 2 (fc+1>10( 1/ 1^ 1) ^ . JO (4.44) We now combine Eqn. (4.32), (4.40) and (4.44) to yield rr*+oo +oo / u-^^hnil/lr+ dw Jo = | ( - l ) fc ^A °fc+1 - - j _ y ] r -(2fc+1)^ for k = 0 , 1 , 2 , . . . , n - 1 (4.45) This equation is equivalent to the first equation listed in Fano’s Table I, where this formulation deviates from that of Eqn. (4.44) by a factor of two. We have chosen to adhere to the published equation of Fano for the ensuing analysis. Zero of t' at infinity An equation similar to Eqn. (4.45) can be derived when the impedance function function con tains a zero of transmission at infinity through similar mathematical steps to the previous case. The resultant equation is Jo w 2fcl n ( l / | r i | ) d o ; = | ( - l ) fc ^ 4 ~ + i - ^ ^ X J r 2fc+1j for k = 0 , 1, 2,..., n - 1, (4.46 which is equivalent to the second equation listed in Fano’s Table I. 4.2.3 Network Examples Eqns. (4.45) and (4.46) represent the foundation of the present analysis. These equations define the conditions of physical realizability in terms of the magnitude of the reflection coefficient. To obtain the best match, we must determine the function, |T i|, which satisfies the set of equa tions formed by the appropriate integral relations. A couple of examples should serve to help understand how this works. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 n-l n-2 Figure 4.10: An n-elem ent high-pass ladder network. Our first example is the special case of a network N ' which has all of its zeros of transmis sion at the origin (i.e., a high pass network). One example of this type of network is the high pass ladder shown in Figure 4.10. The integral relations that apply when the zero has multiplicity 3 are R \ u 2ln(l/|r,|)<ia> = ^ I y i ; - 2 ^ r , 1 I , u fJo 4ln(l/|ri|)dw = ~ Ia 3° - | j S (4.47) (4.48) and R r+ o o ln(l/|ri|)dw = 1 IJo -5 \ A °5 - I J 2 r ~i (4.49) t=l These three coupled equations represent the conditions of physical realizability. The function |Til and the zeros r, must be chosen such that each equation above is simultaneously satisfied. As an additional example, consider the special case of a network, N ' , with all of the zeros of transmission at infinity (i.e., a low pass circuit). One example of this case is the ladder network shown in Figure 4.11. Again when the zero has multiplicity 3, the integral relations are given by / R \ U r - 2 ^ r f Jo f Jo , (4.50) i= 1 7r / 2 R w2ln(l/|r1|)do; = - - U r - 3 (4.51) 1 and /Jo (4.52) UJ i=i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 N’ f yT 5 T x ■ o m n L n -2 Li L, C: ■ ^ T inLrn^ C n -1 C n-3 r Figure 4.11: An n-elem ent low-pass ladder network. Again, these equations represent a coupled set that must be simultaneously satisfied. One equation for each reactive element in N ' must be satisfied simultaneously. Unfortu nately, satisfying the simultaneous equations for a large number of elements is easier said than done. In fact, according to Fano, “When the network N ' consists of three or more elements, the problem of determin ing the optimum tolerance of match becomes much more difficult, and no general solution has been obtained.” Comments Some pertinent information can be ascertained at this point, particularly by examining Eqn. (4.50). The left hand side of this equation represents the area under the curve ln ( l / |F i |) versus frequency. When no zeros of I \ occur in the right half plane, defines the area represented by the integral. The most efficient way to use this area, from a bandwidth point of view, occurs when l n ( l / |r i |) is kept constant over the desired frequency band and is set to zero everywhere else. This situation is shown in Figure 4.12. If u>c is the desired bandwidth, the best possible tolerance is given by 1 ln = |Fi 2U}r 1 (4.53) or ln 1 |Fi (4.54) In other words, the product of bandwidth and h ^ l / l F i l ) ^ ^ is constant, which suggests that as one increases, the other must decrease proportionally. Thus, a low specified value of |F i| necessarily decreases the bandwidth; a large specified value of u c necessarily increases |F i|. 4.2.4 Summary The following are the key equations of the current chapter: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 i inCi/ro [ln(l/T,)] max C0f CO Figure 4.12: The optimum frequency response. • When all of the zeros of t' are at the origin, the coefficients of the Taylor series expansion of T i are given by *2fc+l 1 a 2fc+i l n (jJr ) (2 fc + l)! d s2k+l 2k Zi (2fc+l) -(2fc+l) -E*. = j i s=0 (4.55) The (2 k 4-1 )th derivative of ln( 1 /T i) evaluated at a zero of t' is independent of the applied matching network, for all 2k + 1 that are less than twice the multiplicity of the zero of t'. Eqn. (4.55) shows the conditions of physical realizability of Ti, since it relates the poles and zeros of Ti to a function that is only dependent on the load impedance. For this case, the limitations on tolerance and bandwidth are given by -(2fc+l) 7T I, (r. du = —(—1) 12 k + 1 2k + t E - (4.56) • When all of the zeros of t' occur at oo, the coefficients of the Taylor expansion of Tj are given by <91n 1 4°o _ 2fc+1 (2k + 1 )! (A) = l/s=0 S T r \ i(=E1 ^ ‘ - Ij= > r )/ . 1 (« 7 ) For this case, Eqn. (4.57) gives a relationship between the poles and zeros of Tx and a function that is exclusively dependent on the load impedance. The limitations on tolerance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 and bandwidth are given by (4.58) • When the load has only a single reactive element that induces a zero at oo, the best possi ble tolerance is given by (4.59) m ax This simple equation illustrates the relationship between tolerance and bandwidth. If either bandwidth or tolerance is specified, then the theoretically optimal value of the other is given by this equation. 4.3 The Design of Simple Matching Networks The theory presented thus far can be illustrated with the example of a load impedance that consists of a series inductor resistor combination. We wish to design matching networks with a low pass response (i.e. networks whose passband extends from dc to some cut-off frequency, u>c). Hence, for our purposes, the ideal form of the return loss is shown in Figure 4.12. 4.3.1 Hyperbolic Identities At this point, we wish to present several important mathematical identities that may be unfamil iar to the reader. First, we wish to present an alternative definition for cos-1 (z) and sinh-1 (z), which are (4.60) and (4.61) Other hyperbolic identities that will prove useful are (4.62) cos(jx) = cosh(x) cosh2(x) = sinh2(x) + 1 (4.63) smh2(z) = (1/2) cosh(2x) — 1/2 (4.64) cosh(x) + cosh(y) = 2cosh((a: + y ) / 2) cosh((x —y ) j 2) cosh(2A) —cosh(2.B) = 2sinh(A + B ) sinh(A — B) cosh(A) = 0 => A = m: odd integer sinh(A) = 0 =*> A = jrm r, m: integer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.65) (4.66) (4.67) (4.68) 49 Reactive Two-Port Network (to be determined) R Figure 4.13: The matching problem under consideration. 4.3.2 Low Pass Networks The problem to be addressed is shown in Figure 4.13, where we will assume that all impedance values are normalized to a common resistance of lfl. In other words, the load resistor and the generator resistance are each assumed to equal 117, and the remaining impedances are scaled accordingly. For the problem shown in Figure 4.13, we recognize that the load impedance is given by Z l = R + s L = 1 + s L x where L x is the normalized inductance (L x = L /R ) . The load reflection coefficient is given by Eqn. (4.4), which gives us r L = (Z L - 1) / ( Z L + 1) = S h / i 2 + S L X). (4.69) The coefficient A f' is hence 2 + sL\ sL\ In (4.70) 1/ 5 = 0 and the reflection coefficient, as given by Eqn. (4.53), is limited to ln 1 = IrTi n ljc L \ irR u cL x (4.71) which is analogous to the celebrated result presented by Bode [16] (Note that when the load is a parallel RC combination, A ^ 1 = 2/R C ). We want the matching network to approach this theoretical limit, using a finite number of tuning elements. To do so, we choose a Chebyshev polynomial, C u(uj/ ujc), as a form of the solution. This will induce a response that oscillates between two specified values in the pass band, and approaches unity in the stop band, as shown in Figure 4.14. 4.3.3 Filter design with Chebyshev Polynomials The Chebyshev polynomial of the first kind of order n is defined by [36] Cn(x) = cos (n co s_1(a;)) for |x| < 1 = cosh (n co sh _1(a:)) for |x| > 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.72) 50 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Figure 4.14: Typical frequency behavior of Ti with a Chebyshev approximation. Figure 4.15 shows a plot of the first 5 Chebyshev polynomials, which shows that these functions are bounded by ± 1 when \x\ < 1, and approaches ± oo for \x\ > 1. In order to use this function in filter design, we first set x = ju / u i c, which locates the cutoff frequency, ujc. Next we wish to set the passband ripple to an acceptable value. This is done by recognizing that 0 < |Cn(a;)|2 < 1 for |x| < 1. (4.73) We can multiply this equation by a variable, e2 to get 0 < e2 iC ^a:)!2 < e2, (4.74) where e represents the passband ripple. We wish to apply this function to the transmission coefficient, which should be near unity. Consequently, we modify Eqn. (4.74) to allow us to set the nominal passband value by adding the parameter 1 + K 2 to each side to obtain 1 + K 2 < 1 + K 2 + e2 |C„(a:)|2 < 1 + K 2 + e2. (4.75) The final step in this type of filter is to define the transmission coefficient of the network N according to o\t\2 = ------------1 + X 2 + e2 \Cn{x)\2 (4.76) The overall response and the effects of e and K on the filter response are shown in Figure 4.16. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 5.0 4.0 3.0 2.0 c u 0.0 - 1.0 - 2.0 «C1 ■e C2 -3 .0 -4 .0 -5 .0 -1 .5 - 1.0 0.0 -0 .5 0.5 1.0 1.5 Argument o f the Chebyshev Polynomial Figure 4.15: A plot of the first 5 Chebyshev polynomials. (1+K2+ e2)1/2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 4.16: Typical frequency behavior of t using a Chebyshev response. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.6 52 4.3.4 Optimal Matching with Chebyshev Polynomials We can apply Eqn. (4.76) to the present analysis by recognizing that |F i|2 = 1 \t\2. The magnitude of the reflection coefficient is then given by 1 1 + K 2 + cos2 (n co s-1 (jz)) + cos2 ( n cos 1(jz)) 1+J f 2 + cos2 (n co s_1(jz )) ’ (4.77) where j z = j o j / u c. The argument of the cos2 terms can be expanded from Eqns. (4.60) and (4.61), to become n c o s_1(jz) = n(ir/2 — j sinh_1(z)). Hence cos2 jn s in h 1(-z)) = ^7177 cos — j cos (jn s in h X(2)) — sin ( — j sin (jn s in h x(z)) V 2 J (4.78) We now consider the cases of even and odd values of n. Even values of n When the order of the Chebyshev polynomial is an even number, Eqn. (4.78) is given by n7r (— \ j7isin h _x(z)J = cos2 (jn s in h _1(z)) = cosh2 (n sin h _1( 2;)) . (4.79) For convenience, we choose to define two parameters, a and b, that satisfy the following rela tionships sinh 2{nb) = iC2/e 2 (4.80) sinh2(na) = (1 + K 2)/<?. (4.81) and Equations (4.77) and (4.79) - (4.81) can be combined so that the reflection coefficient is ex pressed as follows: 2 sinh2(nb) + cosh2 (n sinh-1 (z)) (4 82) sinh2(na) + cosh2 (n sin h - 1 (z)) In light of Eqns. (4.63) - (4.65), this equation is equivalent to 2_ cosh [n (fr + sinh~1(z))] cosh [77 (fe - sinh~x(z))] cosh [n (a + sinh_x(z))] cosh [n (a — sinh_1(z))] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ ^ 53 Odd Values of n When the order of the Chebyshev polynomial is an odd number, Eqn. (4.78) is given by / Yb'jjcos2 \ j n s in h - 1 ( 2:)J = sin2 (jn s in h - 1 (z)) = —sinh2 (n sin h J(z)) . (4.84) Eqns. (4.77), (4.80) - (4.81) and (4.84) can be combined so that the reflection coefficient is given by 2 sinh2(n6) — sinh2 (n sin h -1 (,z)) sinh2 (na) — sinh2 (n sin h -1 (z)) ’ 1 ^ ^ which is equivalent to | F i |2 = sinh [n (b + sinh- 1 (z))] sinh [n (b - sinh- 1 (z))] g sinh [n (a + sinh-1 ( 2:))] sinh [n (a — sinh- 1 (z))] Eqns. (4.83) and (4.86) can be written in compact form as iTil2 = cosh(re(b— sinh 1(^)))cosh(n(6-|-sinh *(2))) cosh(n(a— sinh-1(2)))cosh(n(a+sinh~1(2:))) z = j x sinh(re(b— sinh-1(2:)))sinh(n(6+sinh~1(z))) sinh(re(a— sinh-1(2)))sinh^n(a+sinh"1(^))) z = j x n: even (4.87) n: odd To determine the poles of Eqn. (4.87), we use Eqns. (4.67) and (4.68) to give z = , sinh ( ± a ± j ^ ) Psinh ( ± a ± j n: even, m: odd g n: odd, m: integer In a similar way, the zeros are given by f sinh (± 6 ± J t t 1) n: even, m: odd z0 = < ) 2n{ [ sinh (±b ± j ^ ) n: odd, m: integer (4.89) The poles lie on an ellipse centered at the origin with axes equal to 2 cosh a and 2 sinh a as shown in Figure 4.17. The zeros also lie on an ellipse centered at the origin, but with axes equal to 2 cos b and 2 sinh b as shown in Figure 4.18. Since the system consists of passive elements, the poles of T i must lie in the left half of the complex plane. Consequently, we choose the poles of Eqn. (4.88) that have a negative real part as the poles of IY We also recognize that all of the poles of Tlm must lie in the left half plane. Hence, we choose those zerosof Eqn. (4.89) for which the real part is negative. The coefficient, A™, can be computed from Eqn.(4.34) which, for this choice of zeros and poles, is given by A T = Y J zi ~ Y J P j = 1 =1 J=l sinh(a) —sinh(6) \ /0 n)/ sin(7r/2 v ' ' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.90 54 a J® 2cosh(a) 2sinh(a) 3 Element Network 4 Element Network Figure4.17: Location of the poles of F i(z)F i(—z) for networks o f3 and 4 elements. A j® 2cosh(b) 2sinh(b) 3 Elem ent Network 4 Elem ent Network Figure 4.18: Location of the zeros of T i(;z )ri(—z) for networks of 3 and 4 elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 To determine the maximum value of the reflection coefficient, (Ti j2, we differentiate Eqn. (4.87) with respect to z and set that result equal to zero. This yields, (4.91) cosh(na) We wish to minimize this value, when Af" is given by Eqn. (4.90). The result is tan h (n a) cosh(a) tanh(nfr) cosh( 6) ’ (4.92) the parameters a and b are determined by solving Eqns. (4.90) and (4.92) simultaneously. 4.4 Matching Network Synthesis In the previous section, a methodology was presented for designing the transfer function (or reflection coefficient) for matching networks that approach the fundamental limit with a Cheby shev type response. In this section, we complete the network design by discussing the synthesis problem. We choose to design a three element matching network, so that the network, S, has a total of four reactive elements. We can derive the Taylor series coefficients for this network from Eqns. (4.34) and (4.88) (4.89) as ^ ^ c (\ 5inh(a)~ , h(i,))J sin( 7r /,2f n) AOO _ ^ “ _ (2 ) too a^ .3 ( sinh(3a) - sinh(36) ^ /n'i- 4 = (2 )-v 1, 3 sin( 37r / 2 n) + < « 3> sinh(a) - sinh( 6)^ sm (n /2 n) ) ’ f sinh(5d) —sinh(56) i sinh(3a) —sinh(36) y+ 5 sin( 57r / 2 n) sin( 37r / 2 n) sinh(a) —sinh( 6) \ + 2 L; ■ sin( 7r / 2 n) J (4'94) 5 (4.95) and A™ = - ( 2 ) “ V / sinh(7a) ~ sinh(7^) + sinh(5a) - sinh(56) 7 sin( 77r / 2 n) sin( 57r / 2 n) sinh(3a) - sinh(3fr) sinh(a) - sinh(fr) \ sin( 37r / 2 n) sin( 7r / 2 n) J At this stage it is convenient to define the following intermediate variables A °° a 3 = (2)2— ^ V' (A ff 4 °° 1 3’ 1 a 5 = (2)4— ^ - V' (A ff 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.97) (4.98) 56 Figure 4.19: Network to be synthesized utilizing Fano’s design equations. and A? a 7 = (2)e {A™)1 1 7' (4.99) The elements of the network are then given by [17] 2 Lx (4.100) A f’ C2 = .h i L3 = (4.101) «3’ a 3Li 1 + CK3 (4.102) 25. a3 and Lx ( l + a 3 C4 = (4.103) a3 1+ a3 05 I 03 (s)‘ 21 a3 where the lumped reactive elements are arranged as shown in Figure 4.19. We now recognize that two possible approaches can be taken for network design. We can choose to optimize either tolerance or bandwidth, but not both simultaneously. In what follows we will present two network designs, one optimizing on reflection coefficient and the other optimizing on bandwidth. For both cases, we assume that the load impedance consists of a 100/iH inductor in series with a lf l resistor. 4.4.1 Low Reflection Coefficient In this section we choose to design as broad a bandwidth as possible while ensuring a reflec tion coefficient metric of |T! | = 0.01, which corresponds to -20 dB. To do so, we solve Eqn. (4.71) for ujc, where for this example, u c = 6821.9 rad/s. This bandwidth represents the theo retical maximum bandwidth attainable for this load impedance and this tolerance. The actual bandwidth will not exceed this optimal value. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 To determine the reactive element values of the network in Figure 4.19, we consider Eqns. (4.90) and (4.92). These equations are combined to find that a - 1.3317 and b = 0.00226016. We now compute the actual maximum reflection coefficient from Eqn. (4.91) as well as the values of L ,, C2, L 3 and C4 from the equations above. When we do so, the following values are computed: JTi I = 0.0097 « 0.01, L: = 100//H, C2 = 207.716/xF, u = 211.675/xH, C4 = 97.824/rF. The theory presented previously made the assumption that both the source resistance and load resistance had a value of 1(2. However, in order to assure proper match, an ideal transformer is required to convert the source resistance from 1(2 to a value consistent with the chosen reflection coefficient. Rather than including an ideal transformer in the solution, we will compute the required resistance at the source to assure the desired tolerance. When there are four reactive elements in the network, the reflection coefficient reaches is maximum value at zero frequency. So, we can compute the required source resistance by evaluating the reflection coefficient when oj — 0, which implies that the source resistance is given by Rg = (4.104) From which we compute R g = 1.02(2. The reflection coefficient of this network is shown in Figure 4.20, where we note that the realized bandwidth is equal to 4357 rad/s. 4.4.2 Broad Bandwidth In this section we choose a bandwidth of u c = 10,000 rad/s and compute the tolerance. To determine the optimum reflection coefficient for this bandwidth, we solve Eqn. (4.71) for |T i|; we obtain |T| = 0.0432, which corresponds to -13.6 dB. We recognize that the actual reflection coefficient will not be lower than this value. We now use Eqns. (4.90) and (4.92) to determine that a = 0.7268 and b = 0.0271285. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 0.0 aOJ 3 < D O U a 5 o<u 53 u - 10.0 - 20.0 -30.0 Oh -40.0 ■©Actual -h Limit -50 .0 0.0 2000.0 4000.0 6000.0 8000.0 Frequency (rad/s) Figure 4.20: Reflection coefficient versus frequency for the low reflection coefficient design. actual reflection coefficient and matching network values are computed to be |rx| = 0.10957, Li - 100/iH, C2 = 128.754/xF, L3 = 162.804/iH, CA = 74.94//F R g = 1.24612Q. The reflection coefficient over frequency for this network is shown in Figure 4.21, where we note the synthesized reflection coefficient of -9.6 dB across the band. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 0.0 -5 .0 o u -1 0 .0 c o oa> Cu o£ -15.0 ■OActual -a Limit - 20.0 0.0 10000.0 Frequency (rad/s) 5000.0 15000.0 Figure 4.21: Reflection coefficient versus frequency for the wide bandwidth design. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Chapter 5 Fundamental Matchability of Three Self-Biased Circulators The previous chapter discussed the Fano limits of two port matching networks as first presented by Bode [16] and Fano [17]. These limits were derived assuming lumped matching networks and load impedance functions. In this chapter we investigate the applicability of these Fano limits to load impedances that are not lumped, and distributed matching networks consisting of transmission line sections in the context of wideband, high isolation ferrite circulator design. One of the principle goals of several research groups is to create ferrite devices that are “self-biased”, meaning they do not require an external magnet to align the magnetic dipoles within the material. The typical material that is used to create such devices is barium ferrite (BaFe), which typically has an anisotropy field of about 16,300 Oe. Single crystal BaFe has a saturation magnetization of 4,700 Gauss, and a linewidth of about 25 Oe when measured at 55 GHz [37]. Researchers at the University of Idaho have successfully created polycrystalline BaFe that has an effective internal field of around 17,000 Oe, 4 ttM s & 2500 Gauss, and a remanent field of « 2250 Gauss. The linewidth has not been measured, but a value of 3,000 Oe when mea sured at 55 GHz is assumed. Using these values, and a relative permittivity of 12, we designed three circulator pucks, the values of 4ttM s were 2,000 G, 2,250 G and 2,500 G. Each has the same geometrical features as follows: radius of 3mm, coupling angle of 0.03 rad and dielectric thickness of 0.5mm. Figure 5.1 shows the geometry of this device, which was provided by [7]. In the ensuing analysis, the matchability of these devices will be evaluated. To do so we invoke the perfect isolation conditions as presented by Young et. al. [15]. With this theory, it is assumed that the three-port network parameters are known from measurement, simulation or other means; the conditions for perfect isolation can be deduced directly from the network parameters. For the source at port one, port two is perfectly isolated if the load reflection Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 Figure 5.1: The geometry of a self-biased ferrite puck. coefficient, T3, and impedance, Z 3, are given by r,= s 21 S 2 1 S 33 — S :ilS 2:i (5.1) and Z 2 3 Z 31 — Z)2 11-^Z-3 3 (5.2) ^21 where 5V, and Z tJ are the network scattering and impedance parameters. The perfect isolation condition when the source is at port 2 or port 3 is similar. The impedances associated with the perfect isolation conditions of the three devices considered in the present analysis are shown in Figures 5.2 - 5.4. If the material is lossless, these impedances provide a very good figure of merit for the impedance seen looking into the ferrite puck. However, most materials under consideration present some losses; hence, we would expect an analysis based on this metric to be approximate. 5.1 Circuit Modeling We wish to evaluate the matchability of these circulator topologies in terms of the theory of Chapter 4. In order to determine the optimal matching conditions, we first need to develop an equivalent second-order model to approximate each impedance function in Figures 5.2, 5.3 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 75.0 50.0 25.0 O 0.0 0 — 0 Real - Actual a------ a Imag - Actual o o Real - Model A A Imag —Model -2 5 .0 -5 0 .0 22.0 23.0 24.0 25.0 Frequency (GHz) Figure 5.2: A comparison of the impedance data of the parallel RLC model with simulated data associated with the 2000 G material. 75.0 50.0 25.0 0.0 ■s R eal ■a Im ag R eal ■ A Im ag - -2 5 .0 -5 0 .0 21.5 22.0 22.5 A ctual A ctual M odel M odel 23.0 23.5 24.0 24.5 25.0 Frequency (G H z) Figure 5.3: A comparison of the impedance data of the parallel RLC model with simulated data associated with the 2250 G material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 75.i 50 j 25J •o < au. ■© R eal ■a Im ag Real - A Im ag - -2 5 J A ctual A ctual M odel M odel -50.i 21.5 22.0 22.5 23.0 23.5 24.0 24.5 Frequency (G H z) Figure 5.4: A comparison of the impedance data of the parallel RLC model with simulated data associated with the 2500 G material. 5.4. The circuit model that we will use to approximate the load impedance is the parallel RLC circuit shown in Figure 5.5. The values of the lumped parameters must be determined to create an acceptable approximation to the actual impedance functions. The input admittance to the model is given by + j y jjC — G in + j B i , — (5.3) We can manipulate this equation in the following ways dBin duj = c^ + 1 (5.4) uj2 L and B.m UJ =c - 1 (5.5) uj 2 L The effective capacitance and inductance can then be solved according to dBin c = \ dui + Bin (5.6) UJ and d B in B„m (5.7) to* d u UJ Thus, we can determine the effective capacitance and inductance at a particular frequency by L = knowing the measured or computed susceptance and its derivative as a function of frequency. The admittance functions associated with the impedance functions of Figures 5.2 - 5.4 are shown in Figures 5.6 - 5.8. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 R in Figure 5.5: The lumped element equivalent circuit used to approximate the impedance functions of Figures 5.2, 5.3 and 5.4. 0.0 ■o Conductance -a Susceptance - 0.1 5.0 7.0 9.0 11.0 13.0 15.0 Frequency (GHz) Figure 5.6: The admittance data of the impedance function shown in Figure 5.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 0.20 0.10 < L> O 0.00 0.10 ■€> Conductance ■a Susceptance - 0.20 5.0 7.0 9.0 11.0 13.0 15.0 Frequency (GHz) Figure 5.7: Admittance data of the impedance function shown in Figure 5.3. 0.20 0.10 < D O § t! 0.00 1 < - 0.10 ■0 Conductance a Susceptance - 0.20 5.0 7.0 9.0 11.0 Frequency (GHz) 13.0 15.0 Figure 5.8: Admittance data of the impedance function shown in Figure 5.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 5.1.1 2000 G Material The input admittance for the 2000 G material is shown in Figure 5.6. The derivative of the susceptance at a center frequency of 23.25 GHz is equal to dB in/duj = 4.29 x 10-12. Also, B in /u = —3.86 x 10“ 14 and G m = 0.01515. Hence the lumped element values of the model are computed to be C = 2.123 x 10~12F L = 2.168 x F T 11 H R = 66.5ft A comparison of the input impedance of this second order circuit model versus the simulated result is shown in Figure 5.2. Note that the model provides reasonably good correlation with simulated data over the frequency span from 22.5 GHz to 23.5 GHz. 5.1.2 2250 G Material The input admittance for 2250 G material is shown in Figure 5.7. The derivative of the sus ceptance at the center frequency of 23 GHz is equal to d B in/duj = 4.23 x 10-12. Also, B in /u = —5.4 x 10-14 and Gin = 0.01487415. Hence the lumped element values are com puted to be C = 2.0863 x 10“ 12 F L = 2.2373 x 10"11 H R = 67.231 ft A comparison of the input impedance of this second order circuit model versus the simulated result is shown in Figure 5.3. Note that the model provides reasonably good correlation with simulated data over the frequency span from 22.25 GHz to 23.25 GHz. 5.1.3 2500 G Material The input admittance for the 2500 G material is shown in Figure 5.8. The derivative of the susceptance at the center frequency of 22.75 GHz is equal to d B in/dui = 4.5776 x 10~12. Also, B in/u) = —8.46 x 10-14 and Gin = 0.016175. Hence the lumped element values are computed to be C = 2.2465 x 10~12 F L = 2.0995 x 10~n H R = 62.181ft Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 A comparison of the input impedance of this second order circuit model versus the simulated result is shown in Figure 5.4. Note that the model provides reasonably good correlation with simulated data over the frequency span from 22 GHz to 23.25 GHz. 5.2 Fundamental Limits on Matchability The circuit model of Figure 5.5 has one zero at the origin and one at oo, and hence it has a bandpass characteristic. From the theory of chapter 4, the following two equations must be simultaneously solved in order to determine the optimum response: r -(A ) 7r 2 1 (5.8) d u o — —A?° — 7Ts r and r -U t2 lne^ jf i)j ) ^ (5.9) = \ A °X ~ nSr 1 The coefficients A \ and A™ are given by d in A\ = (A) (5.10) ds s= 0 and d in A? = GO (5.11) d± t=o respectively. Recall that sr = ar + juor is a zero of T1 that resides in the right half of the complex plane. This zero must originate from the matching network and hence it can be chosen arbitrarily. For simplicity, we choose to let the right half plane zero occur on the real axis, so sr = o r. The reflection coefficient associated with the load is then given by r; = s2L C + 1 \ + s2L C + 2 s V (5-12) which is equivalent to n \s z +__________ LC (5.13) + — + LC To determine the Taylor coefficient A \, we differentiate Eqn. (5.12) with respect to s and set the result equal to zero. This yields d in (*) (s2L C + 1)(2L + 2sL C ) - (1 + 2sL + s2L C ){2sL C ) ds (1 + s2L C + 2sL )(s2L C + 1) s= 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.14) s= 0 68 which implies that A \ = 2L. Similarly, to determine the Taylor coefficient A f , we differentiate Eqn. (5.13) with respect to 1 /s and evaluate it at 1 /s = 0. This yields. d In (L C + 1 /s 2)(2L + 2 /s) - (1 /s 2 + 2L / s + L C) ( 2 / s ) (1 /s 2 + L C + 2L /s ) { L C + 1 /s 2) ds (5.15) l/s=0 1/ 5 = 0 which implies that A^° — 2 /C . The equations to be solved are then _ L ^ = | _ Wr (5.16) and r - - " ( n) 7T . du) = ttL |T i|y . (5.17) oy Since the load impedance has a zero of transmission at the origin as well as one at 00 , we choose a center frequency of operation given by toc. We also assume that the frequency span is symmetric about this center frequency, or l o h and lo l = 2cuc — l o l , where l o h is the high frequency limit is the low frequency limit of the bandwidth. To be consistent with the development of chapter 4, let In ,UJi < LO < U!h In 1 |r i 0 (5.18) , elsewhere where [ l n ( l /|r i|) ] ma:r is the maximum allowable value of the function ln ( l/|T i|) . Since the integrand in Eqn. (5.16) is a constant function between u>l and u>h , and zero elsewhere, Eqn. (5.16) becomes r, /' 1 y \ "1 = 7r — —(7r (u)H —U)L) In I c m ax (5.19) or ( 1 2 (loc — lol) In A' liril). 7r = '1 C m ax ~ a\ (5.20) Similarly, Eqn. (5.17) becomes (\ U“1 l “ LO “ h )) In = |Ti (5.21) 7T or to l ) vL O l( 2 lO c In — U>L) = 7T (5.22) lr i We solve Eqn. (5.19) for ar as 1 Cr = 7; G (Uc - 7T iOL )^ In |r i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.23) 69 Eqns. (5.22) and (5.23) can be combined to create a relationship between bandwidth and toler ance. If we choose a desired value for | r 11 and wish to compute the optimum bandwidth, the relationship becomes m ax — 7T —Q L iJ m ax 2 itL ujc + 2nu>c C m ax (5.24) m ax This equation is a cubic polynomial in the unknown variable, u l , which defines the optimum bandwidth for the specified tolerance. If we choose a desired bandwidth and wish to solve for optimum tolerance, the relationship is given by f 8u cu L _ In \ iril + + 7T 7T 7r i In - 4 - ------ —p + c c iril ,2 f 7TLeo2 27TL/USlUJq C C A o jl o j2 L — Q L u j c l ) l2 + 2 L ijj\ } n + 2nL0cu)L — tclol S = 0. This equation is a quadratic polynomial in the variable [ln(l/|ri|)]moI, which (5.25) specifies the optimum tolerance for a given bandwidth. We now apply this theory to the example impedances shown in Figures 5.2, 5.3 and 5.4. For each impedance function, we choose the center frequency to be between 23 and 23.25 GHz. We also choose the tolerance to be |I \ | = 0.01 (i.e. 20 dB), and wish to solve for the widest possible bandwidth for each impedance function. 5.3 Realized Circulator Response Matching networks were designed according to Figure 5.9 following the approach of [38], op timizing return loss, isolation and insertion loss of each design. Also, the Fano limitation on bandwidth for each design was computed according the the theory presented previously. Com parison data of the actual response and the theoretical limit are presented. It should be noted at this point that both the perfect isolation metric of [15], and the theoretical limitation, are figures of merit only. These tools are used to provide insight into the relative merits of different designs, but they do not provide a rigorous definition in every case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 T 6-S ■ ' ZT2 L7 Source LNTPLN W3 ■ Z-T4 •W4 Figure 5.9: The topology of the microstrip matching network. Through traces share a common centering line; the stubs are open-circuits. 5.3.1 2000 G Circulator For the 2000 G circulator, the following parameter values were computed: W3 = 0.48mm, W4 = 0.11mm, W5 = 0.33mm, W6 = 0.11mm, W7 = 0.18mm, L3 = 2.54mm, L4 = 0.56mm, L5 = 2.10mm, L6 = 4.57mm, L7 = 1.67mm. Sections one and two were omitted for this design. This circuit is shown in Figure 5.10, and the simulated response, together with the Fano bandwidth limit, is shown in Figure 5.11. From this latter figure, we see that the Fano limit on bandwidth is narrower than the achieved bandwidth. Even so, the limit did in fact provide a good estimate of the actual response. The reason for the discrepancy between the Fano limit and the actual re sponse is the losses associated with the ferrite material. The perfect isolation impedance of [15] was derived assuming a lossless ferrite material, but the material currently under consideration presents a significant amount of loss. These losses smooth out the actual response of the circu lator, and hence widen the attainable bandwidth. Additionally, the model itself assumes a strict second-order function, and does not capture the actual functional relationship over frequency. Finally, the Fano matchability theory was derived assuming the matching network consists of lumped elements, but in this case the matching network is distributed and hence the Fano limits do not strictly apply. 5.3.2 2250 G Circulator For the 2250 G circulator, the following parameter values were computed: W3 = 0.48mm, W4 = 0.1 1mm, W5 = 0.33mm, W6 = 0.11mm, W7 = 0.18mm, L3 = 2.54mm, L4 = 0.43mm, L5 = 1.92mm, L6 = 4.73mm, L7 = 1.62mm. Sections one and two were omitted for this design. This circuit is shown in Figure 5.12, and the simulated response, together with the Fano bandwidth limit, is shown in Figure 5.13. As with the previous circuit, the Fano limit provides a good estimate of the achievable response. Although, we also see a narrower bandwidth limitation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.10: Self-biased circulator utilizing ferrite material with 47tM s = 2000 G. 0.0 - 10.0 - 20.0 CQ 73 C , Il on S ll -0 S21 o -3 0 .0 -» S 3 1 A L im it -4 0 .0 22.0 22.5 23.0 23.5 24.0 Frequency (GH z) Figure 5.11: Scattering parameters and the Fano bandwidth limit of the 2000 G circulator shown in Figure 5.10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 Figure 5.12: Self-biased circulator utilizing ferrite material with AitM s = 2250 G. than we actually achieve as a consequence of the losses of the ferrite material. 5.3.3 2500 G Circulator For the 2500 G circulator, the following parameter values were computed: W3 = 0.48mm, W4 = 0.11mm, W5 = 0.33mm, W6 = 0.11mm, W7 = 0.18mm, L3 = 2.54mm, L4 = 0.43mm, L5 = 2 .17mm, L6 = 4.64mm, L7 = 1.839mm. Sections one and two were omitted for this design. This circuit is shown in Figure 5.14, and the simulated response, together with the Fano bandwidth limit, is shown in Figure 5.15. Again we see a good prediction of realizable performance, with a narrower bandwidth limitation than was actually achieved. 5.4 Concluding Remarks This chapter considered the fundamental matchability of three circulator systems. In each case, the Fano limits provided a good predictor o f performance. For example, the Fano limit of the 2250 G design exhibited the widest bandwidth of the three designs presented, and this same design achieved the broadest bandwidth in practice. Although the realized response exceeded the limit in each case, the theory of Chapter 4 provides a good rule of thumb for relative match ability of different devices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 o.o -10.1 BQ T3 -2 0 J t/3 ■e -30.1 -b S ll S21 -» S 3 1 ■ A L im it -40.1 22.0 22.5 23.0 23.5 24.0 Frequency (G H z) Figure 5.13: Scattering parameters and the Fano bandwidth limit of the 2250 G circulator shown in Figure 5.12. Figure 5.14: Self-biased circulator utilizing ferrite material with AttM s = 2500 G. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 0.0 - 10.0 - 20.0 -o S I 1 -a S21 -» S 3 1 -A L im it -3 0 .0 -4 0 .0 22.0 22.5 23.0 23.5 24.0 Frequency (G H z) Figure 5.15: Scattering parameters and the Fano bandwidth limit of the 2500 G circulator shown in Figure 5.14. The Fano limits derived in Chapter 4 were based on the assumption that the ideal response was flat in the passband and zero elsewhere. The achieved responses in Figures 5.11, 5.13 and 5.15 do not exhibit a flat response in the passband however. This “ripple” in the passband is a consequence of the limited degrees of freedom in the impedance matching network. As more degrees of freedom are added to the network, the response can become more flat in the passband. Further, it should be noted that any measurement below -20 dB represents very small numbers; minute fluctuations would appear to be very large. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Chapter 6 A Microstrip, Rectangular Ferrite, Circulator The traditional junction circulator is comprised of a ferrite - typically circular, hexagonal or triangular in shape - surrounded by a uniform dielectric. Microstrip traces are then patterned on the top with ports emanating at equal intervals in the azimuthal direction [14]; a circulator with these features was discussed in the previous chapter (see Figure 5.1). Even though this approach to circulator design is well understood, it presents a problem with regards to circuit layout. Namely, the ports diverge from the ferrite region 120° apart, which complicates integration with other system components, such as antennas or amplifiers. The perfect isolation condition presented by [15] allows the ferrite region to have any ar bitrary shape, and ports emanating at any convenient angle. One such unique geometry is pre sented in this chapter, where the ports of the circulator diverge from a rectangular ferrite region at right angles. This topology eases the burdens of the layout designer by allowing more dense integration of accompanying components. While the topology presented in this chapter solves the layout problem, it changes the over all functionality of the traditional circulator. Since the geometry is not symmetric, the properties of the device are not symmetrical i.e., isolation between ports 1 and 2 may not have the same bandwidth as isolation between ports 2 and 3, etc. This fact adds complexity to the design, and the theory of Chapter 5 is extended to apply to a device that exhibits unique isolation impedances for each port. Further, the fundamental limits of matchability are used to select a frequency span of operation that is away from resonance rather than near resonance as in the previous cases. With this approach, very promising results are obtained, such as a 15 dB isolation specification from 5 to 15 GHz between two of the ports. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Dielectric | Glued Notch Ferrite Port 3 Port 1 Microstrip ; Figure 6.1: A rectangular ferrite region for use in a microstrip circulator 6.1 Circulator Design In this section we consider a rectangular ferrite region embedded in a uniform dielectric, as shown in Figure 6.1. We will assume that the values of the ferrite’s geometry and magnetic properties are known a priori. For this device, Trans-Tech T T 1-2000 bulk magnesium ferrite is chosen. This ferrite has the following parameters: 4irMs = 2000 G, A H = 300 Oe, e/ = 12.4, tan 5f = 0.00025, and H c = 1.6 Oe. The applied field chosen for this device is 1710 Oe, which creates an internal field of approximately zero. The ferrite is 3.1 mm wide and 5.37 mm long and is embedded in a uniform dielectric made up of D4 Cordierite, which has a dielectric constant of 4.5, tan 6 = 0.0002 and thickness equal to 0.5 mm. The ferrite/dielectric combination is clad with a copper ground plane; copper traces are patterned on top so that each port is centered on its respective edge of the ferrite. The microstrip traces comprising each port have widths of 2.51 mm. The terminal planes for the simulation of the ferrite response are chosen to coincide with the ferrite/dielectric boundary of each port. The substrate/ferrite is fabricated from a continuous sheet of D4 material with the appropri ate thickness. A notch is made in the material of width equal to that of the ferrite, and the ferrite is glued in the notch. A piece of D4 is then glued next to the ferrite to fill the remainder of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 notch. With the geometry and material parameters established, the required load impedances for perfect circulation, Z \, Z 2 and Z3 can be determined per [38] using S-parameters. For example, <7 _ v ' ^ ( ‘S 'll + 1) - S z \S u Zl " Z ° S 32(S U - 1) - S 3lS 12’ /c_ 1N (6A) where Z 0 is the characteristic impedance of the port. For traditional circulators with symmet rically divergent ports, the required load impedance for perfect circulation is the same for all ports. Since the current topology does not present rotational symmetry, unique load impedances are required for each port. To find the .S-parameters of the ferrite puck, and hence the circu lation impedances, a full-wave, electromagnetic solver, such as [32], is used to simulate the circuit of Figure 6.1. The simulated impedance data Z \ is plotted in Figure 6.2. In this figure we see that reso nance occurs around 12 GHz and the resonant impedance is about 55 Ohms. The imaginary part ranges from about -30 Ohms to 20 Ohms in the range 5 < / < 20 GHz. When this impedance function is evaluated near resonance, or 12.5 GHz, the lumped model of Figure 5.5 has the fol lowing values: R =5612, L = 157 pH and C = 1.102 pF; a comparison of the actual and model impedance functions isshown in Figure 6.4. These values yield a limitation onbandwidth of 15.2% for a tolerance of -20 dB. This bandwidth is reasonable, however a broader bandwidth can be attained if we choose a center frequency of 9.5 GHz. At this center frequency, the lumped model of Figure 5.5 has the following values: R = 34.5812, L = 274.13 pH and C = 556.6 fF; a comparison of the actual and model impedance functions is shown in Figure 6.5. The funda mental limitation on bandwidth for these elements is 31.6%. Hence, we have chosen to design matching networks around a frequency of 9.5 GHz. The impedance data Z 2 is identical to that of Z 3, and is shown in Figure 6.3. When the impedance function is evaluated at 12.5 GHz, the lumped element model of Figure 5.5 requires the following values: R = 16.4712, L = 303.5 pH and C = 442.3 fF; a comparison of the actual and model impedance functions is shown in Figure 6.6. These values allow a broad limitation on bandwidth of 82.4%. When the center frequency is chosen to be 9.5 GHz, the lumped element model of Figure 5.5 has the following values: R = 17.612, L = 357.5 pH and C = 325.7 fF; a comparison of the actual and model impedance functions is shown in Figure 6.7. These values yield a limitation on bandwidth of 72.6%, which is narrower than the bandwidth with a center frequency of 12.5 GHz. However, the bandwidth at 9.5 GHz is significantly broader than the bandwidth associated with port 1 of this device, so we choose to operate at this frequency, rather than resonance. The fundamental limits on bandwidth for each port are summarized in Figures 6.2 and 6.3. With Z i impedance data known from Figure 6.2, a suitable matching network that uses Z \ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 60.0 Theoretical -20dB bandwidth when fc=12.5 GHz 40.0 Theoretical -20dB bandwidth when fc=9.5 GHz 20.0 - 20.0 - < D O T9U 3 & a - 0----- 0 Real Imaginary -4 0 .0 10.0 15.0 20.0 Frequency (GHz) Figure 6.2: Required load impedance for port 1 to assure perfect circulation 40.0 Theoretical ,-20dB fe»!2.5<SHz "O <0 o. - 20.0 - 20.0 - -4 0 .0 - -60.0 5.0 10.0 15.0 20.0 Frequency (GHz) Figure 6.3: Required load impedance for ports 2 and 3 to assure perfect circulation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11.0 11.5 12.0 12.5 13.0 Frequency (GHz) Figure 6.4: Parallel RLC model approximation centered at 12.5 GHz of the impedance function of Figure 6.2. 40.0 < D O § -a u I* 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 Frequency (GHz) Figure 6.5: Parallel RLC model approximation centered at 9.5 GHz of the impedance function of Figure 6.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 7.0 9.0 11.0 13.0 15.0 Frequency (GHz) Figure 6.6: Parallel RLC model approximation centered at 12.5 GHz of the impedance function of Figure 6.3. Frequency (GHz) Figure 6.7: Parallel RLC model approximation centered at 9.5 GHz of the impedance function of Figure 6.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 W6 L6 •W2 L7 Source LNTPLN W1 W5 T3 W7 T7 W3 fiv T5 ' ZT4 L4 W4 Figure 6.8: The topology of each matching network. Through traces share a common centering line; the stubs are open-circuits. as a load and transforms it into a 50 Ohm input impedance is designed. For our purposes, the microstrip topology of Figure 6.8 is invoked for the matching network. The search algorithm outlined in [38] is used to choose the appropriate lengths and widths to optimize the transducer power gain of the matching network. The resulting lengths and widths for this port are: W 3 = 1.2000 mm, W 4 = 0.2157 mm, W 5 = 0.3794 mm, W 6 = 0.2157 mm, W 7 = 2.5100 mm, L3 = 0.5037 mm, L4 = 6.9490 mm, L5 = 0.2630 mm, L6 = 4.5436 mm, and L7 = 1.2184 mm. Lines T1 and T2 were not initially used for the design since the additional degrees freedom require more computational resources to determine; the results were deemed adequate and hence lines T1 and T2 were not used in the final design. With Z -2 impedance data known from Figure 6.3, a suitable matching network that uses as a load and transforms it into a 50 Ohm input impedance can also be designed. As before, the microstrip topology of Figure 6.8 is invoked for the matching network. The search algorithm outlined in [38] is again used. The resulting lengths and widths for this port are: W 3 = 1.2000 mm, W 4 = 0.2157 mm, W 5 = 0.3794 mm, W 6 = 0.2157 mm, W 7 = 2.5100 mm, L3 = 0.5037 mm, L4 = 6.9490 mm, L5 = 0.2630 mm, L6 = 4.5436 mm, and L7 = 1.2184 mm; and again lines T1 and T2 are not deemed necessary for this design. The matching networks and the ferrite region are next conjoined to form the circulator and the resulting circulator is validated using simulation tools. In this work the validation gives results that meet with our expectations, and modifications to the design are deemed unnecessary. The device can now be fabricated; see Figure 6.9 for a photograph of the circulator in its test fixture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 Figure 6.9: A photograph of the fabricated circulator in its test fixture. 6.2 Measured Results Figure 6.10 shows simulated data and 6.11 shows measured data of the return loss of all three ports. The differences between simulation and measured data is attributed to inadequate coaxial to microstrip transitions on each port. Note the marked difference in each return loss measure ment; port 1 presents narrower band performance than do the other two ports. This difference is attributed to the high Q impedance data of Figure 6.2 verses the low Q impedance data of Figure 6.3. Figure 6.13 shows data associated with the three insertion loss measurements. This device exhibits moderately broad band performance for each of these three measurements, although the insertion loss between ports 2 and 3 is the broadest. Figure 6.14 shows simulated data and 6.15 shows measured data of the isolation of all three ports. These plots are perhaps the most interesting, as two of the three isolation measurements are quite broad band. Isolation between ports 2 and 1 is particularly broad, due in part to the low return loss in the out-band. Using a 15 dB specification, the bandwidth is 10 GHz over the range from 5 to 15 GHz. While two of the isolation measurements exhibit very broad band performance, the third measurement presents a very narrow band response. In addition, the isolation data exhibits very deep nulls of 30 dB or more in the passband. This suggests that the required load impedances are virtually the exact impedances for perfect isolation at specific frequencies. For all three ports in the 9 to 11 GHz band, the return loss is less than 15 dB, insertion loss is approximately 1 dB and isolation is less than 14 dB. Thus, for a 20% bandwidth, the square Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 7.0 9.0 11.0 13.0 15.0 Frequency (GHz) Figure 6.10: Simulated return loss for the circulator with square topology, ferrite circulator can be used in many traditional applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 -10.0 - CQ •o - 20.0 - o -J u Pi Ports 2 & 3 Theoretical Bandwidth at -20 dB -30.0 - -40.0 - -50.0 Port 1 Theoretical Bandwidth at -20 dB 9.0 11.0 Frequency (GHz) 15.0 Figure 6.11: Measured return loss for the circulator with square topology. - 10.0 - 20.0 CQ -d o eo O Ports 2 & 3. Theoretical Bandwidth. at -20 dB -30.0 -40.0 - -50.0 Port 1 Theoretical Bandwidth at -20 dB 9.0 11.0 Frequency (GHz) 15.0 Figure 6.12: Simulated insertion loss for the circulator with square topology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 7.0 9.0 11.0 13.0 15.0 Frequency (GHz) Figure 6.13: Measured insertion loss for the circulator with square topology. Ports 2 & 3 Theoretical Bandw idth at -20 dB / Port 1 Theoretical Bandw idth at -20 dB 9.0 11.0 Frequency (G H z) Figure 6.14: Simulated isolation for the circulator with square topology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 Ports 2 & 3 Theoretical Bandw idth at -20 dB Port 1 Theoretical Bandw idth at -20 dB -10.0 03 T3 - 20.0 - 3 0 .0 - -4 0 .0 9.0 11.0 15.0 Frequency (G H z) Figure 6.15: Measured isolation for the circulator with square topology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Chapter 7 Integration of a Microstrip Circulator with a Planar Yagi Antenna of Several Directors The integration of circulators with other components within a system, say an antenna, is typi cally considered separately from the rest of the system as in the previous chapter. The circulator is often designed to a standard interface specification (e.g. 500), as is the adjoining circuitry. Each component is then combined to form the system. Although this approach will yield ad equate results, other approaches can provide greater performance and flexibility. We present in this chapter a methodology that considers the design of a planar antenna with a microstrip circulator as a single, integrated component. The antenna topology considered is the planar Yagi-Uda array [28]. To validate the proposed methodology, two integrated antenna/circulator systems are presented. Simulation data are provided for three substrate antennas: baseline, two director and three director antennas. The baseline antenna is derived from the Yagi-Uda antenna presented by Kaneda et. al. [28]. This antenna exhibits a fairly narrow beamwidth in the .E-plane, but the //-p la n e beamwidth is very wide. The performance of this baseline antenna topology is enhanced through the use of additional director elements that provide a significant improvement in H —plane beamwidth. The directivity of the antenna is also improved as a consequence of the modifications presented in this chapter. When these systems are utilized for communications or RADAR applications, the improved gain compensates for the inherent insertion loss associated with the circulator. Baseline Antenna We use as a baseline a modified form of the antenna of Kaneda; modifications were made to account for our choice of substrate thickness (0.5mm) and substrate permittivity (4.5). This antenna topology is shown in Figure 7.1 and has the following geometrical parameters: S sub = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 Figure 7.1: Single director planar Yagi-Uda antenna. 1.2mm, Ldir = 3.3mm, Wdir = 0.6mm and S (ur = 2.4mm. All other geometrical parameters are identical to that of Kaneda, except for minor changes in the miters. This antenna was simulated to determine pattern and impedance information; Figures 7.2 and 7.3 show the E -plane and //-p la n e patterns respectively and Figure 7.4 shows the input impedance to the antenna. The patterns exhibit the following performance: E -plane beamwidth is 87° with sidelobe levels of 16.5 dB; antenna gain is 5.12 dB; //-p la n e beamwidth is greater than 180° with no sidelobes. The real part of the input impedance undulates between 25 and 50 Q in the 12 to 17 GHz band. The imaginary part undulates between 20 and -20 fI in the same band. Provided that the circulator presents to the antenna an impedance that is close to these values, a good match between the circulator and the antenna can be designed. This antenna exhibits adequate performance in the Ku-band; however, the //-p la n e beam width is quite broad. To allow tradeoffs in system design, we modify the topology to include additional director elements, which improves H -plane beamwidth as well as antenna gain, as described next. Double Director Antenna Consider the effect of an additional director element to the geometry of the baseline an tenna, as shown in Figure 7.5. In order to be consistent with the baseline antenna, we chose the following modified geometrical parameters: S sub = 1.2mm, L ^r = 3.3mm, Wdtr = 0.6mm, Sdir Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 10.0 5.0 0.0 -5 .0 c < - 10.0 ©------ © 1-Director □ •--■ e 2-Director o— -o 3-Director -1 5 .0 - 20.0 0.0 60.0 120.0 180.0 240.0 300.0 360.0 E-plane angle (deg) Figure 7.2: E-plane pattern comparison for the baseline, double director and triple director antennas. 7.0 n. \ 5.0 -o 3.0 c < - G © 1-Director B - - - B 2-D irector 1.0 * - — ■* 3-D irector -3 .0 0.0 60.0 120.0 180.0 H -Plane angle (deg) Figure 7.3: H -plane pattern comparison for the baseline, double director and triple director antennas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 1 0 0 .0 75.0 'w E 50.0 .c O 8 c 25.0 (0 T3 CD Q. E 0.0 -25.0 o □ -50.0 10.0 o Re{Zin} □ lm{Zin} 11.0 12.0 13.0 14.0 15.0 16.0 Frequency (GHz) 17.0 18.0 19.0 Figure 7.4: Input impedance of the baseline antenna. = 2.4mm and S<nr-dir = 2.4mm. The remaining parameters are identical to those of the base line antenna. The E -plane and //-p la n e antenna patterns were simulated with results shown in Figures 7.2 and 7.3. From these plots, we note the following: E -p lan e beamwidth is 93° with sidelobe levels of 13.9 dB; antenna gain is 5.86 dB; //-p la n e beamwidth is 140° with no sidelobes. The input impedance to this antenna was approximately the same as that of the baseline antenna, and hence the simulated impedance is not shown. The polar pattern of this antenna is shown in Figure 7.6, where we note a 20 dB difference between co-polarization and cross-polarization in the E-plane. Triple Director Antenna The final antenna under consideration uses a third director element to the geometry of Figure 7.5. Again, we maintain consistency with the previous antennas by choosing the following modified geometrical parameters: S sub = 1.2mm, L * r = 3.3mm, Wdir = 0.6mm, Sdtr = 2.4mm and Sdir—dir = 2.4mm. The E -plane and //-p la n e patterns of this antenna were also simulated and the data are shown in Figures 7.2 and 7.3. The following features are noted from these plots: E -plane beamwidth is 76° with sidelobe levels of 13.58 dB; antenna gain is 7.05 dB; //-p la n e beamwidth is 105° with no sidelobes. The input impedance of this antenna was also very similar to that of the baseline antenna, and is not shown. This topology yields best overall gain, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.5: Double director planar Yagi-Uda antenna. C ross-polarization \ -30 Copolarization Figure 7.6: Normalized simulation antenna pattern for the two director antenna at 15 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 Cross-polarization Copolarization -180 Figure 7.7: Normalized simulation antenna pattern for the three director antenna at 15 GHz. narrowest beamwidth in both principle planes, at the expense of slightly higher sidelobe levels. The polar pattern of this antenna is shown in Figure 7.7, where we note a 20 dB difference between co-polarization and cross-polarization in the .E-plane. Antenna Comparison Now that we have designed and simulated the responses of three substrate antennas, we have several tradeoffs to consider. These tradeoffs can best be seen from the comparison data of Figures 7.2 and 7.3. First, the baseline antenna is the most compact and has the lowest sidelobe levels; however, it has lower gain and broader H -plane beamwidth. The double and triple director antennas each exhibit improved antenna gain and much narrower beamwidth in the H -plane at the expense of higher sidelobe levels in the E -p lan e as well as a larger footprint. These tradeoffs can be exploited in overall system design. For example, the insertion loss (which is due to material loss in the ferrite) of a circulator is typically on the order of 0.5-1 dB. An improvement in antenna gain by greater than 1 dB can offset the negative effects of the circulator in the overall system. Circulator Design To design systems consisting of the antennas presented previously and a circulator, the fer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 rite circuit needs to be described in terms of some network parameters. We do this by appealing to the approach of Bosma [1], who provided the approximate impedance parameters of the fer rite circuit. With this approach, the ferrite region is assumed to be a homogeneous closed cavity with mixed PEC and PMC boundary conditions, and hence a trans-impedance G reen’s function can be defined as follows [6]: C U .Y’l ZM ^ f a) | Z/ V 2 7 r/'(7 / a) 2tr ^ Z f A W sa ) e ^ - ^ I'J n f a) + ( n « / / i / ) / n( 7 / o ) / ( 7 / o) I n {l f a ) e - ^ - ^ t ^ (7/a) - (nK/nf)In{r/fa)/('yfa) ' The Green’s function relates the vertical electric field everywhere in the ferrite to the azimuthal magnetic field on the walls of the ferrite cavity. The appropriate impedance parameters can be determined from these fields with the following equation Z ij= G ttu tfW . (7.2) Although this approach is well-known and extensively used to design circulators, certain errors still exist in the solution. To obtain a more accurate set of network parameters, the geom etry should be simulated via a full-wave electromagnetic solver. However, full-wave solvers take a very long time to arrive at the desired solution when compared with the time required for the green’s function. Hence, a two step process is required: first, the Green’s function is repeatedly invoked to find a solution that is approximately correct (say, 20%), and second a full-wave solver is used to validate that the ferrite circuit is adequate, and to generate more accurate network parameters for use in the system integration. Once the network parameters are accurately known, the ferrite region can be regarded as a lossless, nonreciprocal, three-port, linear network (LNTPLN) as presented in [15]. With this network theory representation, the power flow through the network can be evaluated and metrics such as return loss, isolation and insertion loss at each port can be determined. This is described next. 7.1 System Integration To create the integrated components, complex-to-complex impedance equalizers need to be designed to provide the interface between each component of the system to optimize return loss, isolation and insertion loss. This approach of designing the overall system potentially allows for better performance and flexibility than the common approach of isolating each component from the rest of the system and designing each port to match some standard characteristic impedance, such as 5 0 fl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 ■W2 L5 - Z-T2 Source LNTPLN W1 W5 T1 W3 L4 W4 Figure 7.8: Traditional microstrip matching network topology using unbalanced stubs. ■Z-T4 W4 L4 ■W2 L2 Source LNTPLN W1 W5 Q v . W3 L2 T5 ' Z-T4 ■W2 L4 W4 Figure 7.9: Balanced microstrip impedance equalizer. The traditional approach to designing microstrip impedance equalizers consists of a stan dard topology of alternating through traces and individual stubs, as shown in Figure 7.8. For most applications, this approach is sufficient. However, when driving an antenna like the ones described previously with an equalizer like that of Figure 7.8, we note that a field imbalance will occur in the transverse plane of the microstrip. When this signal is presented to the antenna, the balun will be less effective and hence the radiation and input impedance will be affected. Con sequently, we have chosen to design our impedance equalizers with electrically balanced stubs as shown in Figure 7.9. To design the equalizer networks, we follow the design procedure outlined in [38] with a few modifications. Rather than use the transducer power gain as a metric in conjunction with the circulator impedance, we choose to compute the power flow through the network. For example, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 we know that two of the equalizers are terminated with a 5017, purely real impedance, and the third equalizer is terminated with the impedance function shown in Figure 7.4. We choose a set of lengths and widths for each of the elements shown in Figure 7.9 and compute Ti, T2 and r 3, which are the reflection coefficients that load the circulator at ports 1, 2 and 3 respectively. Using these reflection coefficients, we can compute the insertion loss at port 1 according to [7] RLi = (l-ir^xi-ir^i2) (7.3) the insertion loss between ports 3 and 1 according to IL U = K z 0i ( i - | r 3|2) ( i - ITij2) l i - r ^ i 2 -'o3 VT (7.4) and the isolation between ports 2 and 1 according to l/~ I S 21 = 2 z0l (i-|r2|2)(i-|ri|2) Z 02 |1 —TiTinjI2 (7.5) where r in)1 is the input reflection coefficient at port 1 of the circulator when ports two and three are loaded with r 2 and r 3 respectively. Here Z ai, Z o2 and Zo3 are the port characteristic impedances. We can compute the return loss, insertion loss and isolation for each of the remain ing two ports in a similar manner. We then compare the computed return loss, insertion loss and isolation with the specificied acceptable values. If the computed values satisfy the specification over a broader bandwidth than any previous topology, we keep the design; if not we choose a new set of lengths and widths for the topology of Figure 7.9 and reiterate. 7.2 Results Two antenna/circulator systems were designed according to the procedure outlined above. Each design utilized a different antenna element as shown in Figures 7.10 and 7.11. For each sys tem, impedance equalizers were required for the two ports not interfacing with the antenna to convert from a standard 50 Q termination to an impedance that optimizes metrics of return loss, isolation and insertion loss. For the circulator/antenna port, the impedance equalizer is one of a com plex-to-complex impedance match, where the input impedance of each antenna is mod ified to optimize these same metrics. We stress that this approach is fundamentally different from designing the circulator and antenna as separate 50 Q devices. The two director system was fabricated on 0.5mm thick Trans-Tech D4 material, which has a dielectric constant of 4.5. The ferrite material selected was Trans-Tech TTI-3000, which has the following properties: AttM s — 3000 G, £/ = 12.9, ta n <5 = .0005 and radius = 1.75mm. The externally applied field required for these material properties and geometry to assure a zero Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 Figure 7.10: Circulator/antenna system with two director elements. internal field was set at 2575 Oe. External biasing for this material was provided by a laboratory electromagnet. The externally measured hysteresis loop for this material is shown in Figure 7.12, where the “as designed” bias point of 2575 Oe is shown at the edge of saturation. The peak-to-peak linewidth of the ferrite material was also measured with results shown in Figure 7.13. The peak to peak linewidth, at 10 GHz, is shown to be 140 Oe, which corresponds to a 3 dB linewidth of 238 Oe. The published linewidth from Trans-Tech is 228 Oe as measured at 9.4 GHz. The impedance equalizers connected to the two 50fl ports of the two director system were identical and have the following geometrical parameters: W \ = 0.94mm, W 2 = 0.216mm, L 2 = 1.142mm, W 3 = 1.097mm, L 3 = 1.067mm, W 4 = 0.216mm, L4 = 4.846mm, W 5 = 2.255mm and L5 = 4.219mm. The matching network connecting the circulator to the tw o-director antenna has the following geometrical parameters: W \ - 1.2mm, L \ = 1.201mm, W 2 = 0.216mm, L 2 = 1.214mm, W 3 = 2.196mm, L 3 = 2.372mm, W 4 = 0.216mm, L4 = 6.983mm, W 5 = 2.255mm and L5 = 4.719mm. The three director system was also fabricated on 0.5mm thick Trans-Tech D4 material, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.11: Circulator/antenna system with three director elements. and the ferrite material was Trans-Tech TTI-3000 with a radius of 1.75mm. Once again, the externally applied bias field was set at 2575 Oe. The matching networks connected to the two 50f2 ports were identical to those of the two-director system. The matching network connecting the circulator to the three-director antenna has the following geometrical parameters: W \ = 1.2mm, L i = 1.101mm, W 2 = 0.216mm, L2 = 1.427mm, W 3 = 1.561mm, L3 = 6.473mm, W 4 = 0.216mm, L 4 = 11.47mm, W 5 = 2.255mm and L5 = 4.719mm. The measured and simulated data for the final fabricated systems are shown in Figures 7.14 and 7.15. The measured return loss for each device is about 25 dB; the measured isolation varies from 20 dB to 35 dB across the band of interest. The insertion loss is about 0.8 dB and is quite flat across the band. These measured results correlate quite well to results of simulation. Moreover, we regard isolations and return losses in excess of 20 dB to be quite good and in excess of 25 dB to be superb. Although the plots shown in Figures 7.14 and 7.15 illustrate the results of the as designed systems, improved performance may be attained by varying the applied bias field. For example, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 4 0 0 0 .0 2000.0 C/5 c/5 3 O C O ed 0.0 v As Designed Optimized _N §> CO S3 - 2000.0 - 4 0 0 0 .0 - 1 5 0 0 0 .0 - 5 0 0 0 .0 5 0 0 0 .0 1 5 0 0 0 .0 E x te rn a lly A p p lie d F ie ld (O e) Figure 7.12: Measured hysteresis curve of T T 1-3000. Figure 7.16 shows the measured scattering parameters of the two-director antenna system with an applied bias of 2291 Oe rather than that of the previous measurement of 2575 Oe. These tuned parameters exhibit a very good isolation metric of 30 dB over a bandwidth of around 6%. Similarly, Figure 7.17 shows the measured scattering parameters of the three-director antenna system with an applied bias of 2280 Oe rather than that of the previous measurement of 2575 Oe. These parameters show a very good isolation metric of 30 dB over a relatively narrow bandwidth of 2.1%, and a superb isolation of 40 dB over a narrow bandwidth of 0.8%. Hence by varying the applied bias field, we can dynamically optimize these systems for a variety of applications. The optimized bias point is shown on the measured hysteresis curve of Figure 7.12, which shows that the material is in a partially saturated state. This partial saturation is significant in light of the commonly held belief that losses increase as the material moves out of saturation. The data shown in Figure 7.16 and 7.17 illustrate that this assumption does not always hold, at least for our application. At 15 GHz, for example, the insertion loss changes from 0.8 dB to 0.68 dB for the two director system and from 0.72 dB to 0.68 dB for the three director system. Hence losses have reduced, rather than increased, when the ferrite material is partially saturated. To illustrate the repeatability of these designs, two circuits were fabricated of each system design. The measurements of each individual circuit is illustrated in Figures 7.18 - 7.21. The level of correlation in these plots points to the repeatibility of the design. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 0.0008 ■AHpp 0 .0 0 0 6 i-i a> *o Oh 0 .0 0 0 4 <t> bH o 0.0002 y) «: 0.0000 - 0.0002 - 0 .0 0 0 4 - 0 .0 0 0 6 4 5 0 0 .0 5 0 0 0 .0 5 5 0 0 .0 6 0 0 0 .0 6 5 0 0 .0 Applied Field (Oe) Figure 7.13: Measured peak-to-peak linewidth of TT1-3000 at 10 GHz. 0.0 Insertion Loss - 10.0 ■Isolation 1 s03 - 20.0 'Return Loss - 3 0 .0 - 4 0 .0 1 4 .0 1 5 .0 1 6 .0 Frequency (GHz) Figure 7.14: Measured (solid lines) and simulated (dashed lines) scattering parameters of the two-director element system with a 2575 Oe applied bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 0.0 Insertion Loss 10.0 Isolation CQ T3 -3 0 .0 'Return Loss -4 0 .0 13.0 14.0 15.0 16.0 Frequency (GHz) Figure 7.15: Measured (solid lines) and simulated (dashed lines) scattering parameters of the three-director element system with a 2575 Oe applied bias. 0.0 - 10.0 - 20.0 9.7% B W 6% B W -3 0 .0 « S ll a S12 o S21 -A S22 -4 0 .0 14.0 14.5 15.0 15.5 16.0 Frequency (GHz) Figure 7.16: Measured scattering parameters of the two-director element system with a 2291 Oe applied bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 0.0 OQ T 3 - 15.8% BW, /'“S 20.0 <U 2.1% BW- 0.8% BW -40.0 o S ll ■a S12 S21 A S22 -60.0 13.0 15.0 Frequency (GHz) 14.0 16.0 17.0 Figure 7.17: Measured scattering parameters of the three-director element system with a 2280 Oe applied bias. 0.0 - 10.0 o------ ©S ll- c k tl -30.0 □------- □ S 22-cktl o------- o S l l —ckt2 A-------A S22—ckt2 -40.0 13.0 15.0 14.0 16.0 Frequency (GHz) Figure 7.18: A comparison of measured return loss for two fabricated circuits with two director elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 0.0 - 10.0 - 20.0 /—\ CQ c/a t-H <D S CL, -3 0 .0 -e S12-cktl -a S21-cktl o S12-ckt2 ■A S21-ckt2 -40 .0 13.0 14.0 15.0 16.0 Frequency (GHz) Figure 7.19: A comparison of measured insertion loss for two fabricated circuits with two director elements. 0.0 - 10.0 - 20.0 CQ 3 c/a c/a 3 B 3<D d ■o S l l- c k t l -3 0 .0 a S 22-cktl o S ll-c k t2 ■A S22-ckt2 -4 0 .0 13.0 14.0 15.0 16.0 Frequency (GHz) Figure 7.20: A comparison of measured return loss for two fabricated circuits with three director elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 0.0 - 10.0 - 20.0 ■e>S12-cktl -a S21-cktl ■o S12-ckt2 •a S21-ckt2 -3 0 .0 -4 0 .0 13.0 14.0 15.0 16.0 Frequency (GHz) Figure 7.21: A comparison of measured insertion loss and isolation for two fabricated circuits with three director elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 Chapter 8 Circulator System Design Without Matching Networks The systems presented thus far have focused on the design of microstrip matching networks with a given topology. While this approach accomplishes the task of system integration, it may not be the most efficient method in terms of real estate and losses. To provide a more efficient system design, we consider the system as a whole and use the power equations of Eqns. (7.3) - (7.5). In this case, however, we do not apply matching networks between the individual components in the system. Instead, we choose to modify the circulator parameters directly to affect a good match. The Green’s function model for the circulator shown in Eqn. (7.1) [4] contains the following variables: thickness (h ), radius (a), coupling angle 0/0, port location (6), magnetic saturation (47tMs), ferrite dielectric constant (e/), FMR linewidth ( A H ) and surrounding material (ej). By adjusting these variables appropriately, we can create a system that incorporates both a circulator and antenna, and any other component, without the need of interconnecting matching networks. 8.1 Antenna Analysis In designing a system with this method, we wish to use alumina as the dielectric since this material is easily found with very low surface roughness. This material has a dielectric constant of 9.9, rather than 4.5 which was used with the previous systems. Consequently, a new antenna design and analysis is required. Consider the topology of Figure 8.1 which is the geometry chosen for this system. This trace topology is very similar to that of Kaneda et. al. [28]; the only differences are in the feeding structure and miters. We will investigate the response of this antenna using three variations of substrate thickness: 0.5mm, 0.75mm and 1.0mm. The antenna used in this system will be operated in the vicinity of 10 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 Figure 8.1: Topology of antenna on alumina. For each of the antennas presented, the lumped element model of Figure 5.5 is employed to allow an approximate calculation of the Fano limit. This lumped element model is not used in the system design procedure itself. Dielectric Thickness of 0.5mm With a dielectric thickness of 0.5mm, the input impedance to the antenna is shown in Figure 8.2. The real part of this impedance function oscillates around 270 over the frequency range of 9 - 1 2 GHz; the imaginary part varies from +10D to —10Q over the same span. At the center frequency of 10 GHz, the lumped element model of this impedance function has the following values: R = 27.89f2, L = 339.2 pH and C = 832.3 fF; a comparison of the actual and model impedance functions is shown in Figure 8.2. From these model values, the fundamental limit on bandwidth of 49% can be obtained. The simulated antenna pattern at 10 GHz is shown in Figure 8.3, which shows a maximum gain of 4.8 dB, an E -plane beamwidth of 96° with -19 dB sidelobes and an E -p lan e beamwidth of 158° also with -19 dB sidelobes. Dielectric thickness of 0.75mm With a dielectric thickness of 0.75mm, the input impedance to the antenna is shown in Figure 8.4. The real part of this impedance function oscillates around 35D across a frequency range of 9.5 GHz - 12 GHz and varies a bit more gradually than the impedance function of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 Frequency (GHz) Figure 8.2: Simulated input impedance of the antenna of Figure 8.1 with a substrate thickness of 0.5 mm. 10.0 0.0 - 10.0 ■€>E - Plane -a H - Plane - 20.0 0.0 60.0 120.0 180.0 240.0 300.0 360.0 Angle (deg) Figure 8.3: Simulated E - and H -plane antenna patterns of the antenna of Figure 8.1 with a substrate thickness of 0.5 mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Frequency (GHz) Figure 8.4: Simulated input impedance of the antenna of Figure 8.1 with a substrate thickness of 0.75 mm. Figure 8.2. The imaginary part varies from +1017 to —10(2 over the same band, and also varies more gradually than the impedance function of Figure 8.2. At the center frequency of 10 GHz, the lumped element model of this impedance function has the following values: R = 35.7617, L = 508.1 pH and C = 560.9 fF; a comparison of the actual and model impedance functions is shown in Figure 8.4. From these model values, the fundamental limit on bandwidth of 57% can be obtained. The simulated antenna pattern at 10 GHz is shown in Figure 8.5, which shows a maximum gain of 4.65 dB, an E -plane beamwidth of 107° with -23 dB sidelobes and an //-p la n e beamwidth of 154° with -23 dB sidelobes. Dielectric Thickness of 0.1mm With a dielectric thickness of 1.0mm, the input impedance to the antenna is shown in Figure 8.6. The real part of this impedance function varies from 4217 to 27fI and is quite flat across the band from 9 GHz - 12 GHz; the imaginary part varies from +1017 to —14.517 over the same band. At the center frequency of 10 GHz, the lumped element model of this impedance function has the following values: R = 41.817, L = 520.2 pH and C = 554.2 fF; a comparison of the actual and model impedance functions is shown in Figure 8.6. From these model values, the fundamental limit on bandwidth of 50% can be obtained. The simulated antenna pattern at 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 10.0 0.0 'S < - 10.0 E - Plane ■o H - Plane ■o -20.0 0.0 ■ 1 60.0 ■ 1 120.0 ■ 1 180.0 ■ 1 240.0 ■ 1 300.0 ■------ 360.0 Angle (deg) Figure 8.5: Simulated E - and H -plane antenna patterns of the antenna of Figure 8.1 with a substrate thickness of 0.75mm. GHz is shown in Figure 8.7, which shows a maximum gain of 4.49 dB, an .E-plane beamwidth of 119° with -28 dB sidelobes and an E -p lan e beamwidth of 144° with -28 dB sidelobes. Summary As shown in Figures 8.2, 8.4 and 8.6, the fundamental limit on bandwidth of the 0.75mm thick dielectric is the broadest of the three considered in this chapter. Further, Figures 8.3, 8.5 and 8.7 indicate that the 0.75mm dielectric antenna exhibits the narrowest beamwidth in the E -plan e. For these reasons, we choose to integrate the 0.75mm antenna with a circulator to complete the system. 8.2 Green’s Function for System Design To design systems with no matching networks, we need to choose circulator parameters that will create a reasonable system response. To do so, we choose to find the system response as a function of these parameters using the Green’s function of Eqn. (7.1) [4]. However, a word of caution is appropriate at this point. The Green’s function provides a good design tool, as it is computationally robust, but the results are approximate. Consequently, using the Green’s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Frequency (GHz) Figure 8.6: Simulated input impedance of the antenna of Figure 8.1 with a substrate thickness of 1.0 mm. 10.0 0.0 'S < - -© E - Plane -h H - Plane 20.0 -3 0 .0 0.0 60.0 120.0 180.0 240.0 300.0 360.0 Angle (deg) Figure 8.7: Simulated E - and H -plane antenna patterns of the antenna of Figure 8.1 with a substrate thickness of 1.0mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 function exclusively for system design will only create a design that operates near the desired value, but may not meet specifications. Further modifications may be required with a full-wave solver prior to fabrication. The first step that must be accomplished is to fully characterize the components that will connect to the circulator, i.e. the antenna presented in the previous section. This characterization can come from a full-wave solver, measurement, or other source. We recognize that to accu rately characterize the devices, we must select the port characteristic impedance and substrate thickness. The characteristic impedance and thickness equates to a microstrip trace width. For the characterization to be valid, this trace width must be kept consistent throughout the rest of the design sequence. Since the trace widths emanating from the circulator are fixed by the loading devices, we have effectively limited our degrees of freedom. In particular, the trace width at a given port is given by w = 2 asin (0 ), (8.1) where a and 0 are defined previously. We can rearrange Eqn. (8.1) to provide the following relationship between radius and coupling angle ( 8 .2 ) That is, we can vary either radius or coupling angle, but not both. The following variables can be used to affect a system design: radius, a, port angle location, d2 and 03, interconnecting trace lengths, Li - L3, and saturation magnetization, 4 ttM s. The geometrical parameters are shown in the diagram of Figure 8.8. 8.3 Simulation Results A system was designed utilizing the theory presented above. The puck design is shown in Figure 8.9, and the geometrical parameters are: radius = 2.6 mm, thickness = 0.75mm, 0 i = 0.233 rad, 02 = 0.14 rad, 03= 0.14 rad, 02 = 2.15 rad and 03 = 2.15 rad. The ferrite material chosen for this system is Trans-Tech TT1-1000 which has the following material parameters: 47rM s = 1000 Gauss, A H = 120 Oe as measured at Frneas = 9.4 GHz, e = 11.6 and ta n S = 0.00025. This puck was simulated with a commercially available finite-element solver to yield the perfect isolation condition of Figure 8.10 for port one and 8.11 for ports two and three. The values of the lumped element model of the impedance function of Figure 8.10 are given by: R = 48.5612, L = 258 pH and C = 691 fF. These values provide a fundamental limit on bandwidth of 24%. The values of the lumped element model of the impedance function of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Port 2 Port 1 Port 3 Figure 8.8: Variables associated with system design without matching networks. Figure 8.11 are given by: R = 56.24 fi, L = 292 pH and C = 582 fF. These values provide a fundamental limit on bandwidth of 22%. Hence the impedance function associated with ports 2 and 3 is the most restrictive of bandwidth. The overall system design is shown in Figure 8.12; the simulated response is shown in Figure 8.13. In this design Li = 2.9mm; L 2 and L3 are arbitrary since ports 2 and 3 are loaded with their respective characteristic impedances. Figure 8.13 shows that the device realizes a -20 dB bandwidth of 6.4% as compared with the theoretical limit of 22%. This data demonstrates that system integration without matching networks is indeed possible, albeit at the expense of bandwidth. The limitation in achieved bandwidth is attributed to the limited degrees of freedom to accomplish the match. If additional degrees of freedom were available, a broader bandwidth would be attainable. Also, the RLC model impedance function does not correlate well with the actual impedance function of the antenna as shown in Figure 8.4. Hence, the fundamental limit shown in Figure 8.13 wider than is actually attainable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 Port 1 1.2 mm 2.15 rad 5 rad / Port 3 P o rt 2 \ 0.742 mm Figure 8.9: Puck design to be used with the 0.75mm thickness antenna. 8.0 9.0 10.0 11.0 12.0 Frequency (G H z) Figure 8.10: Impedance function to assure perfect isolation for port 1 and its RLC circuit model equivalent impedance function. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Frequency (GHz) Figure 8.11: Impedance function to assure perfect isolation for ports 2 and 3 and its RLC circuit model equivalent impedance function. Figure 8.12: Overall system consisting of the ferrite puck of Figure 8.9 and the antenna of Figure 8.1 with a 0.75mm thick dielectric. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 - 10.0 - 20.0 « S ll ■e S12 -©■ S21 ■A S22 -3 0 .0 -< Limit -4 0 .0 8.0 9.0 10.0 11.0 12.0 Frequency (GHz) Figure 8.13: Simulation scattering parameters of the system of Figure 8.12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 Chapter 9 Conclusion This dissertation has addressed microwave ferrites and their application to microwave devices that are commonly used in monostatic RADAR systems. Several contributions to the general body of knowledge have been presented in several sub-areas associated with microwave ferrites, including numerical modeling of electromagnetic waves in complex media, new antenna topolo gies, and the design of systems that incorporate ferrite circulators and antennas. Improvements in each of these areas help increase capabilities of monostatic RADAR systems, and in a similar way, communication systems. 9.1 Contributions 9.1.1 Numerical Modeling Numerical modeling of electromagnetic wave propagation within complex media, particularly that of ferrites, is a significant area of current research in the electromagnetic community. Ad vances in this area are continuously pursued for two different reasons: to understand how elec tromagnetic waves propagate in bounded ferrites and to improve the quality of the numerical solution. This dissertation presented two schemes that fall into the latter category, they are: • A finite-difference time-domain scheme that can be used to simulate the effects of elec tromagnetic wave propagation in ferrite materials. • A finite-difference time-domain scheme that can be used to simulate the effects of wave propagation in materials exhibiting Debye relaxation processes. An abbreviated analysis of the errors associated with this scheme was also presented. These schemes allow for the simulation of electromagnetic waves within two types of com plex media with second order accuracy, thus adding accurate functionality to the existing tim e- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 domain simulation schemes. 9.1.2 Substrate Antennas Substrate antennas are very appealing topologies since they can be integrated directly with mi crostrip elements within a system without the need to convert any signal to some other trans mission line type (such as coax). This dissertation introduces the following new substrate Yagi antenna topologies: • A new topology is presented that utilized two director elements and a dielectric constant of 4.5. • A new topology is presented that utilized three director elements and a dielectric constant of 4.5. • A new topology is presented that utilized a single director element, a dielectric constant of 9.9 and thickness of 0.5mm. • A new topology is presented that utilized a single director element, a dielectric constant of 9.9 and thickness of 0.75mm. • A new topology is presented that utilized a single director element, a dielectric constant of 9.9 and thickness of 1.0mm. Each of the substrate Yagi antennas presented herein are easily integrated with other system components using a microstrip transmission line topology. 9.1.3 A Novel Circulator Topology Circulators are three port devices that are typically assumed to have rotational symmetry. This assumption allows for a simplified analysis, but it is not necessary. The concept of a perfect isolation impedance, as presented in [15], allows for more flexibility in circulator design. For example, the ports and ferrite shape can be chosen to allow easier integration with other system components. One example of this concept was presented herein: • A novel three port circulator topology was presented wherein the ferrite region is rectan gular in shape and the ports diverge at 90° angles from each other. This circulator exhibits very good isolation performance for two of the ports, and narrow per formance for the third port. Thus, for certain applications this device is an appealing option. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 9.1.4 Self-Biased Circulators The design of self-biased circulators that utilize high crystalline anisotropy materials has been the goal of researchers for the last few decades. Researchers at the University of Idaho have recently developed a new material that exhibits the high crystalline anisotropy effects necessary for self-biased circulator design. This dissertation presents simulation data for three possible designs utilizing this material: • One circulator was designed utilizing a saturation magnetization of AnM s = 2500 Gauss. • One circulator was designed utilizing a saturation magnetization of AnM s = 2250 Gauss. • One circulator was designed utilizing a saturation magnetization of 4 n M s = 2000 Gauss. Each of these devices may possibly be fabricated utilizing the new materials. 9.1.5 Antenna/Circulator Systems Whenever circulators are designed in a system, they are typically designed separately to some standard interface specification (i.e., 50Q). This approach leads to adequate system integrations, however it is not optimal in terms of performance or real estate. In this dissertation, a novel approach was presented where the system was designed in total, thus reducing the real estate by removing redundant matching networks. The following antenna/circulator systems were presented in this dissertation: • Two antenna/circulator systems were presented; system design was accomplished utiliz ing the power equations of [7] to affect an overall system design. • An antenna/circulator system was presented wherein no matching networks are used; sys tem integration was accomplished exclusively by varying the circulator material and geo metrical parameters. These systems offer validation of the approach that the circulator need not be designed sepa rately from the rest of the system. 9.1.6 Losses in Ferrites Through experimental and simulation results, we have discovered that losses in ferrite devices tend to increase as saturation magnetization increases. It is commonly assumed that losses in ferrite devices increase when the material is partially saturated. However, experimental evidence indicates that the reverse is possible. The following findings were presented in this dissertation: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 • A relationship was described between losses in ferrite materials and magnetic saturation, where the losses are found to be directly proportional to saturation magnetization (see Appendix B). • Losses do not necessarily get worse when the ferrite material is partially saturated, they can actually improve. These two findings allow for a much more complete understanding of electromagnetic wave propagation in ferrite materials, and allows for greater flexibility in designing circulator devices; full saturation is no longer a strict requirement. 9.1.7 Bode-Fano Theory A theory describing the fundamental matchability of an arbitrary load impedance function was presented by Fano in 1950. This theory has typically been applied to the matchability of electri cally small antennas. The matchability of three port circulators has not been addressed, and until the theory of [15], it was believed that the theory of Fano cannot be applied to these devices. However, with the understanding of the load impedance for perfect isolation, the following contribution was made: • The Bode-Fano criterion was applied to three port circulators. It was discovered that the Bode-Fano criterion does not provide a strict limit on the matchability of circulators, but they do provide a good metric for the relative matchability of such devices. 9.2 Future Work In Chapter 7 we found that device losses do not necessarily increase when the ferrite is partially saturated, and device functionality can actually improve. This suggests that if the appropriate theory could be developed to predict the response of partially saturated ferrites, a whole new class of devices could be designed. Such devices would require lower applied bias fields, with correspondingly smaller external bias magnets. Further, the development of self-biased devices would be significantly simpler to accomplish since the remanent magnetization would no longer need to be equal to the saturation magnetization. One of the most significant problems facing designers of monostatic RADAR systems that utilize phased array antennas is the integration of multiple circulator devices within the same array. The traditional ferrite circulator consists of three ports that diverge from the ferrite at 120° angles. When multiple circulators are present on the same substrate, the port divergence Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 tends to drive the array performance. One step toward easing the burden of array designers is presented in Chapter 6, where a device is presented that utilizes orthogonal ports. This device would be much simpler to integrate with other circulators on the same substrate and hence the array performance would no longer be dependent on the proximity of neighboring circulators. The two topics recently discussed represent a couple of possible ideas for work that could be accomplished with the foundation of the present dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 Appendix A Finite-Difference Time-Domain Analysis of Debye Materials Chapter 3 presented a time-domain scheme that can be used to simulate the effects of electro magnetic wave propagation in ferrite materials. Much of this approach is directly applicable to materials that exhibit a Debye-type response. The material in this appendix was developed following an approach similar to the one outlined in Chapter 3. Over the past decade or so there has been significant interest in the simulation of electro magnetic waves propagating through dispersive media. In the case where the media is biological or water-based [39] [40], the dispersive process is one of multiple relaxations giving rise to a low-pass, frequency-domain response between the electric flux density D and electric intensity E. Couched in terms of the frequency dependent permittivity e, this low-pass response is given by = e er e Q, where e0 is the permittivity of free-space and M (A .l) Here rm is the mth relaxation constant, is the permittivity at infinite frequency and e sm is the permittivity at zero frequency for the mth relaxation process. In this chapter a medium that responds in accordance with the description of Eqn. (A. 1) is regarded as an M th-order Debye medium. It is not the purpose of this chapter to review the various time-domain numerical algorithms that have been postulated, tested and reported in the literature for electromagnetic waves in De bye media; one such review can be found in [41]. However, it suffices to say that at present, when the algorithms are time-domain in nature, the time integration methodologies fall under one of two titles - recursive convolution and direct integration. The former starts with the con volution representation between the electric flux density and the electric intensity; the kernel of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 21 the convolution integral is the inverse Fourier transform of Eqn. (A .l). The convolution integral is then discretized in conjunction with M axwell’s equations to yield a consistent scheme, which may be first- or second- order accurate, depending on whether the electric intensity is regarded as piecewise constant or piecewise linear within the time interval; as an example, see [42]. The latter, i.e., direct integration, discretizes Maxwell’s equations and the state equations that govern the relaxation processes; Eqn. (A .l) is not directly part of the scheme, but is a result derived from the state equations. These direct integration methods are typically second-order accurate due to the employment of central differences and averages. As demonstrated in [41], the direct integration schemes of Kashiwa et al. [43] and of Joseph et al. [44] have identical error characteristics; the direct integration scheme of Young [45], al though second-order, is inferior due to a small, but nonphysical term, in the effective permittiv ity. Nonphysical terms also are present in the recursive convolution methods, regardless of the order of accuracy [41], The key attribute of the method of Young is the ability to treat the multiple relaxation pro cess in the general formulation; whereas the method of Joseph requires a reformulation of the scheme any time a relaxation term is added or taken away from the model. The scheme of Kashiwa is also general, but being semi-implicit, the formulation is more cumbersome and not as easy to implement with existing finite-difference, time-domain (FDTD) standard codes. In this chapter, a new algorithm is presented that retains the simplicity of Young’s original scheme while maintaining the same superior accuracy of Kashiwa and Joseph. A .l Formulation The governing equations that describe the multiple relaxation process associated with Debye media have been fully described in the literature [46]. Couched in terms of the polarization vec tor P , the electric intensity E and the electric flux density D , the pertinent equations governing an M th-order relaxation process are p = — [(eSm - eoo)e0E - P m] (A.2) and (A.3) where the subscript m denotes the mth relaxation. Moreover, r m is the relaxation time, esm is the static value of the permittivity, e,*, is the permittivity at infinite frequency and eD is the free-space permittivity. For purposes of a numerical algorithm, Eqn. (A.2) can be couched in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 terms of the polarization vectors and D by means of Eqn. (A.3): M dPrr dt (A.4) 1= 1 where cum = l / r m and OCm. — C-sm €qo €r>n7~rr (A.5) In matrix form, Eqn. (A.4) is equivalent to d? = T > -A -3> : dt (A.6) here CP = [ P i , P 2,-- - ,P m ]4, £> = K D , a 2P>, ■■• , a MD]*and Oil + U>1 A = a1 Oi\ Oi2 Oi2 a M + OJ2 OCm £*2 (A.7) OiM + ' ' ' With the polarization equation so specified, we are now in a position to consider the tempo ral advancement of Maxwell’s equations, which, for this treatment, is given in generalized form for non-magnetic media: 3D = V x H dt and 31T „ (A.8) ^ (A.9) E. dt By definition H is the magnetic intensity and /i0 is the permeability of free— space. Invoking = -V X second-order central differences and averages, we can replace the continuous equations, i.e., Eqns. (A.3), (A.6), (A.8) and (A.9), with Dn+l/2 = Dn-l/2 + ^ '<rm+l/2 1tnn-l/2\ y x jjn (A. 10) /'T>n+l/2 1mn-1/2 (A .l 1) ■gn+1/2 _ \ / (j-jn+1/2 ^M ' pn+1/2\ j e° e°° \ and m=l m (A. 12) J Rn+1 = Rn_ _LV x En+l/2^ (A 13) o Here 5t is the temporal time-step; the superscript n denotes the time nSt . To finalize our treat ment, it is desirable to algebraically manipulate Eqn. (A .ll) such that y n+1/ 2 appears exclu sively on the left hand side and T "-1/2 on the right. Doing so, we obtain r y n + 1/2 = ( rj + S tA -1 r/ rn\ i / (£)n+l/2 <pn-l/2' ) • ? n~1/2 + ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A. 14) 123 where 3 is the identity matrix. Since the scheme is cast in terms of matrices, there exists signif icant flexibility in handling an arbitrary number of relaxations for any given Debye medium. The spatial discretization of the curl operators can be accomplished in a number of ways. For purposes herein, we opt to discretize them in accordance with the standard FDTD procedure in conjunction with the Yee grid [8]. To appreciate the accuracy of the previous numerical scheme, consider a plane wave prop agating through an unbounded, homogeneous Debye material. Assuming that all field com ponents under investigation vary like ejult~jk 'r, where k is the wave-vector and ui is the tim eharmonic angular frequency, we, per [41], can correlate continuous derivatives with numerical operators as follows: d_ dt jn (A .15) and V -» - j K = - j { K xa x + K yay + K zaz) (A. 16) where (A. 17) and (A. 18) here i denotes either x , y or z and k%denotes the component of k in the direction of i. Clearly, as the time step and grid cell size reduce to zero, the aforementioned numerical operators reduce to their continuous counterparts. In addition to the discretization of the derivatives, the numerical scheme also employs temporal averages. For this reason, we also need to invoke the temporal averaging operator A, which is given by (A. 19) For infinitesimally small time steps, A takes on a value of unity. With these numerical operators in place, Eqn. (A. 10) - (A. 13) are replaced with jQ D 0 = —j K x H 0, (A.20) (A.21) (A.22) and jQ,fj,0H 0 = j K x E 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.23) 124 In the previous equations D 0 denotes the constant amplitude vector of the phasor D , i.e., D — D 0eJa,t_k r ; similar definitions apply for the other wave vectors. We also note from Eqns. (A.20) and (A.23) that K • (K x H 0) = 0 and K • (K x E 0) = 0, thus suggesting that D 0 and H c are solenoidal, i.e., K • D 0 = 0 and K •H 0 = 0. Using these solenoidal relationships, we can obtain the following dispersion relationship by eliminating the various amplitude vectors that appear in Eqns. (A.20)-(A.23): K • K = f I2fj,0ere0 where M / e- = + £ (A.24) , . ," " o (A -25) Clearly, as St —> 0, we observe that er takes on the value of its continuous counterpart of Eqn. (A .l). Specifically, the use of temporal differencing and averaging has the manifested effect of shifting the relaxation time; it does not introduce any additional dissipation or non-physical effects as in the direct integration method of Young [45]. For reasonably small time steps, the effective relaxation time T'm can be deduced by retaining the first two terms of the Taylor series representation of r mQ/A. Doing so, we obtain 1 + 1 ( SJ0L 3 V 2 (A.26) The shift in relaxation time is seen to be frequency dependent and second-order, but the same for all relaxations. As for the remaining part of the dispersion analysis, it follows one-for-one the analysis of the traditional FDTD scheme [47]; further discussion is unnecessary. Equation (A.25) is identical with the semi-implicit method of Kashiwa et al. and the direct integration method of Joseph et al. when M = 1 [41]. The strength of the methodology presented herein lies in its systematic approach when multiple relaxations are present in the medium and the simplicity of incorporating the scheme into existing solvers. A.2 Results As a validation test case, consider a rectangular cavity filled with a homogeneous Debye ma terial - namely methanol or water. The fields within the cavity are excited by a stripline trans mission line, as shown in Figure A. 1. The geometrical parameters chosen for this validation are W = 2 mm, H = 2 mm, L = 1.5 mm, W /eed - 0.8 mm, iT/eed = 0.8 mm and L /eed = 0.3 mm. The trace width is 0.2 mm and the dielectric constant of the stripline structure is 8. The stripline’s TEM waves are excited and detected per the method described in [32]. This approach allows for the very short L /eed without loss of accuracy. The excitation pulse is chosen to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 Figure A .l: Graphical representation of the simulated topology. gaussian of width a, where a = 1.9 x 1011 1/s; the pulse is of the form e~t2/ “2. For simulations in which the material is either water or methanol, the grid parameters are 8X = 5y = 5Z = 5 = 0.05 mm. The timestep was chosen utilizing a CFL stability criteria of 0.5 so that St ~ 48 fs. The electrical parameters of water and methanol are listed in Table 1; see also [48] and [49]. compound esl n (ps) ts2 T2 (PS) water 87.57 17.67 6.69 0.9 methanol 32.5 51.5 5.91 7.09 ^s3 r 3 ps ^oo 3.92 4.9 1.12 2.79 Table A .l: Electrical parameters of water and methanol. Three sets of results were generated to validate the multi-term Debye model. These are shown in Figures A.2, A.3 and A.4. The quantity in consideration is the magnitude of the reflection coefficient, as measured at the terminal plane of the stripline and as a function of frequency. All cases considered were validated using an independent numerical solver, i.e., Ansoft’s High Frequency Structure Simulator (HFSS). And, as these three figures suggest, the degree of correlation between the data associated with the scheme discussed herein and that of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 HFSS is very good. Consider Figures A.2 and A.3 more closely. In Figure A.2, we are interested in seeing the effect of the individual relaxations on the reflection coefficient response; in Figure A.3, we are interested in seeing how multiple relaxations taken in sets of one, two and three impact the response. These plots demonstrate that all relaxations, whether individually or collectively, materially change the response. From this data the conclusion can be reached that the time integration scheme presented herein is valid for any number of relaxations. 0.0 - 2.0 CD "O -4 .0 - 6.0 - 8.0 c o o CD FDTD FDTD FDTD HFSS HFSS HFSS CD DC - 10.0 - 12.0 0.0 5.0 - Case Case Case Case Case Case 1 2 3 1 2 3 10.0 15.0 20.0 Frequency (GHz) Figure A.2: Reflection coefficient data associated with the individual relaxations of metha nol. Data obtained from the integration scheme and from HFSS are presented. Case n, for n = 1, 2,3, corresponds to a single pole response associated with r„ and esn; see Table 1. A.3 Conclusions A time integration scheme for M axwell’s equations in conjunction with Debye relaxation pro cesses has been presented, analyzed and validated. The presented scheme is valid for any num ber of relaxations and is adaptable to allow the addition or deletion of relaxations, as needed or wanted for the material model, without any reformulation. The error analysis manifests the second-order accuracy of the scheme and reveals that the truncation errors are identical to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 0.0 - m ■o 2.0 - 4 .0 C g o i*— <x> o - 6.0 c o o _g - 8.0 O 0) cc - 10.0 - 12.0 FD TD FD TD FD TD H FSS H FSS H FSS 0.0 - 5.0 Case Case Case Case Case Case 4 5 6 4 5 6 10.0 15.0 20.0 Frequency (GHz) Figure A.3: Reflection coefficient data associated with the individual and collective relaxations of methanol. Data obtained from the integration scheme and from HFSS are presented. Case 4 is a single pole response associated with ti and esi. Case 5 is a two pole response associated with Ti, e.,i, r 2 and es2. Case 6 is a three-pole response associated with n , es i, r 2, es2, r 3 and es3. See Table 1 for parameter values. second-order errors of [43] and [44], Finally, validation results provide the necessary confi dence in the time integration methodology for any number of relaxations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 0.0 - 10.0 CD ~o c Q) [o o O -2 0 .0 c o ■*—1 o Q) 0) H — IT -30.0 FD TD H FSS -40.0 0.0 5.0 10.0 15.0 20.0 Frequency (GHz) Figure A.4: Reflection coefficient data associated with the two-pole response of water. Data obtained from the integration scheme and from HFSS are presented. See Table 1 for parameter values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 Appendix B Losses in Extraordinary Mode Propagation As discussed in Chapter 7, the model for a circulator assumes a bias direction transverse to the plane of wave propagation, or the bias is z-directed. Furthermore, it is assumed that the rf electric intensity is also z-directed and the rf magnetic intensity is in the xy-plane. In other words, propagation occurs transverse to bias and the magnetic intensity is also transverse to bias, which is the criteria for birefringence, specifically extraordinary mode propagation. Hence, an analysis of losses in extraordinary mode propagation gives insight into losses within a circulator. The wavenumber for the extraordinary mode is given by Eqn. (2.76), or k = uJejT e = u J f i 0e0er . V A*° , (B .l) where /ie is the effective permeability of the extraordinary mode and is given by II2 fie = - K 2 ---------. (B.2) A wave propagating with the wavenumber of Eqn. (B .l) has an attenuation constant given by fatten ktft Iw S ^c^c\/ I V k-o J UJ-\J~jio€o€rItTl \ a I ( ✓. I V Mo J (B.3) For bias in the z-direction, the Polder dyad of Eqn. (2.10) is used. The elements of the dyad are given by ( 1 ^ = V Lu0u jm o j2 \ - to 2 ) = f uj 2 - u )2 + u w m \ ^ ' ( } and (B-5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 where exchange effects are neglected. The phenomenological loss term, a , is included by re placing uj0 with ojQ+ j a u everywhere in Eqns. (B.4) and (B.5), which become (B.6) and (B.7) respectively. The effective permeability, fie, can then be computed via Eqn. (B.2) when losses are included. To deduce the losses associated with a ferrite material, Eqn. (B.3) is invoked in conjunction with Eqns. (B.2), (B.6) and (B.7) with the following material properties: linewidth of 3000 Oe as measured at 55 GHz and relative dielectric constant of 12.0; the operating frequency is chosen to be 20 GHz. The attenuation constant, a Qtten, was computed for values of 4irM s from 0 to 5000 Gauss and effective internal field, H int, from 9000 to 15000 Oe with results shown in Figure B .l. Note that the values of effective internal field are high enough that the ferrite material operates above resonance. Consequently, the losses improve for increasing values of H int since the frequency of the electromagnetic wave is farther from ferromagnetic resonance. One interesting feature that can be seen from Figure B. 1 is that losses increase for increasing values of 4irM s. This result can be understood qualitatively by considering the cause of ferro magnetic losses. FMR losses are incurred when energy is transmitted to the magnetic dipoles within the material, thus increasing the energy state of the individual particles. Saturation mag netization (or 4irM s) is a measure of the number of magnetic dipoles per unit volume. Hence when more dipoles exist, more energy can be transmitted into the material per unit volume, and less will propagate through it. Analytical justification that losses increase with increasing values of 47tMs is obtained by assuming that a 2 -C 1, lu0 — 0 and defining u>mn = u>m/u>. This implies that Eqns. (B.6) and (B.7) become /i Ho (1 (B.8) J Q^mn) and (B.9) respectively. For this situation, the effective permeability is given by ra n ra n H° I 1 j&^mn 1 ^ ran jocujmn J jc ^ ^ m n ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.10) 131 18.0 •---------- 1---------- ■---------- 1---------- ■---------- 1---------- >---------- r 16.0 ■ 14.0 ■ - & 10.0 |a 8.0 < 6.0 a _o o ts o — o 9,000 Oe □--------□ 10,000 Oe ♦------- * 11,000 Oe A-------- A 12,000 Oe <------- <13,000 Oe V------- V 14,000 Oe ►——► 15,000 Oe 4.0 2.0 0.0 0.0 1000.0 2000.0 3000.0 Magnetic Saturation (Gauss) 4000.0 5000.0 Figure B .l: Losses associated with the extraordinary mode propagating through a ferrite for various values of effective internal field when f = 20 GHz, er = 12.0 and A H = 3000 Oe when measured at 55 GHz. The following simplifying assumptions are made: u>mn < 1 and a 2Afnn « 1, in which case the effective permeability becomes He = /io ( 1 - ^ ‘L n - ja ^ m n ~ 3 ^ m n ) Ho ( l ujmn) 1 jotuimn l + AL mn (B.l 1) Note that the Taylor series approximation of the function \ / l + x is y /\ + x ~ 1 + x / 2, so the function fwi j -\// i e / /x01 is approximately equal to [J^ e a u J rnn f, ~ p ~ ( 1 + WD This equation combined with Eqn. (B.3) gives the attenuation constant as ^, fatten ~ r r .—- /^o^o^r ^ ( ^ ^ f\ J y 1 ~ (1 + a;mn) ^mn (! - WL ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /p i on 132 Hence we see that the attenuation constant is directly proportional to the phenomenological loss term, a, and the magnetization frequency, u>m = /j.0j 47tM s; as saturation magnetization increases, wave attenuation increases. Validation of the approximations associated with Eqn. (B.13) is accomplished by comparison with Eqn. (B.3) and is shown in Figure B.2; good corre lation is observed for all values of ujmn > 1. 150.0 ■o Exact ■a Approximate 5 100.0 "a. 6 a ts s c _o ID 50.0 0.0 0.0 4.0 6.0 2.0 Normalized Frequency (CO/GOm) 8.0 Figure B.2: A comparison of the approximation of Eqn. (B.13) with the exact solution of Eqn. (B.3) for the following parameters: A H = 3000 Oe, Fmeas = 55 GHz, H int - 0. Oe, 4ttM s = 2500 Gauss, er = 12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 Bibliography [1] H. Bosma, “On stripline Y-circulation at UHF,” IEEE Trans. M icrowave Theory Tech., vol. MTT-12, n o .l, pp. 61-72, 1964. [2] K. M. Krowne and R. E. Neidert, “Theory and Numerical Calculations for Radially Inhomogeneous Circular Ferrite Circulators,” IEEE Trans. M icrowave Theory Tech., Vol 44, No. 3, pp. 419-431, March 1996. [3] H. S. Newman and C. M. 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