The Pennsylvania State University The Graduate School College of Engineering THEORY OF ELECTROMAGNETIC WAVES IN ANISOTROPIC,

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1 The Pennsylvania State University The Graduate School College of Engineering THEORY OF ELECTROMAGNETIC WAVES IN ANISOTROPIC, MAGNETO-DIELECTRIC, ANTENNA SUBSTRATES A Dissertation in Electrical Engineering by Gregory Allen Talalai 2018 Gregory Allen Talalai Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2018

2 The dissertation of Gregory Allen Talalai was reviewed and approved by the following: James K. Breakall Professor of Electrical Engineering Dissertation Advisor, Chair of Committee Victor Pasko Professor of Electrical Engineering Julio V. Urbina Associate Professor of Electrical Engineering Michael T. Lanagan Professor of Engineering Science and Mechanics Kultegin Aydin Professor of Electrical Engineering Head of the Department of Electrical Engineering Signatures are on file in the Graduate School. ii

3 Abstract The unique electromagnetic properties of magneto-dielectric composite materials, including ferromagnetic particle composites, ferromagnetic film laminates, and crystal-oriented ferrites have yet to be extensively leveraged in the design of modern microwave devices and antennas. By surveying the relevant literature, we first compile the relevant information on the dynamic permeability of magneto-dielectric materials, focusing on the various formulations of Snoek s laws, their achievable permeability spectra, and the sources of anisotropy in a passive material medium. In subsequent chapters, an electromagnetic theory is developed for the investigation of planar antennas printed on anisotropic, magneto-dielectric substrates. For the case of a substrate whose in-plane permittivity and permeability are isotropic, expressions are derived for the surface wave modes. Similar to an isotropic substrate, the anisotropic magneto-dielectric substrate admits of a finite series of surface wave modes, which exist only provided the frequency of operation is above their respective cutoff frequencies. In most cases, the primary TM mode has no cutoff frequency. However, contrary to an isotropic substrate, the primary TM surface wave mode is suppressed completely in the anisotropic substrate provided the in-plane permeability, and out-of-plane permittivity are equal to 1. The more general theory for a fully anisotropic substrate can be developed indirectly through an investigation of wave propagation and radiation along principal axes and planes of the material. TE and TM decompositions of the electromagnetic field may be given for waves propagating within principal planes, but are impossible for arbitrary directions of propagation. An eigenvector approach shows how to resolve the electromagnetic field solutions into types, but these classifications are not, in general, TE/TM. For planar antennas the anisotropic magnetic response of a magneto-dielectric substrate enables circularly polarized radiation, even for single-fed linear antennas. Furthermore, the cutoff frequencies for the surface wave modes become directionally dependent. Dyadic Green s functions are derived for a grounded, infinite, anisotropic magneto-dielectric substrate using spectral domain techniques. Utilizing the dyadic iii

4 Green s functions, the surface wave excitations and radiation patterns of short electric dipole antennas are calculated. The principal plane patterns for a dipole over a substrate possessing in-plane anisotropy are also derived. A method of moments computer program is developed for the numerical investigation of the input impedance and efficiency of microstrip dipoles printed over an anisotropic magneto-dielectric substrate. Example results are given to illustrate the effect of the anisotropic properties of the magneto-dielectric substrate on the dipole s resonant length, impedance, and efficiency. Notably, the results indicate that a higher efficiency is obtained if the permittivity of the substrate in the direction normal to the air-substrate interface is small, an effect attributable to the suppression of the primary TM surface wave mode. From these results, we also conclude that the permeability in the direction normal to the air-substrate interface is unimportant for thin substrates. Consequently, no penalty is paid for utilizing anisotropic magnetic materials such as the crystal-oriented ferrites, which only possess an in-plane permeability. The superior frequency-permeability Snoek product of the oriented magnetic materials compared to traditional isotropic materials, combined with the unimportance of the out-of-plane permeability, lead to the conclusion that anisotropic magneto-dielectric materials are potentially better-suited for antenna applications than their isotropic counterparts. iv

5 Table of Contents List of Figures Acknowledgments viii xi Chapter 1 Overview and Introduction Overview of Prior Studies on the Application of Magnetic Materials to Antenna Design Contributions to Knowledge The Layout of this Dissertation Chapter 2 Permeability of Anisotropic Magnetic Materials Ferromagnetism from Magnetic Domain Rotation Equation of Motion for Internal Magnetization Easy-Plane Crystalline Anisotropy and Demagnetization Fields Crystal-Oriented Ferrites Ferromagnetic Laminates Ferromagnetic Particle Composites Chapter 3 Electromagnetic Wave Propagation in Anisotropic Magnetodielectric Media Maxwell s Equations in Anisotropic Media Plane Wave Solutions TEM Wave Propagation Along Principal Axes TE and TM Wave Propagation Within Principal Planes TE and TM Wave Propagation in Arbitrary Directions For Magneto-dielectric Medium With In-Plane Isotropy v

6 3.2.4 Hybrid Mode Wave Propagation for Arbitrary Directions of Propagation Surface Wave Modes of the Grounded Substrate Surface Waves in Substrates with In-Plane Isotropy TE and TM Surface Wave Propagation Within Principal Planes Chapter 4 Electromagnetic Radiation in the Presence of an Anisotropic Magneto-dielectric Substrate The Plane Wave Spectrum, and Formulation of the Green s Function Problem Dyadic Green s Functions for Substrate with In-Plane Isotropy Asymptotic Evaluation of the Radiation Field by Stationary Phase Excitation of Surface Waves Radiation over a Generally Anisotropic Substrate Necessary Conditions for the Excitation of Circularly Polarized Radiation Fields Principal Plane Radiation Chapter 5 Method of Moments Analysis of Microstrip Dipoles Spectral Formulation of the Electric-Field Integral Equation Basis Functions and Source Model Numerical Integration Input Impedance Analysis Anisotropic Effects Chapter 6 Conlusions and Future Work 128 Bibliography 132 Appendix A The Spectral Dyadic Green s Function in the Complex Plane 138 Appendix B Method of Moments MATLAB Code 141 vi

7 Appendix C Comparisons to FEKO for an Isotropic Substrate 149 vii

8 List of Figures 2.1 A magnetic specimen is divided up into domains with disordered orientations in the absence of an external magnetic field A uniformly magnetized sphere A uniformly magnetized infinite slab A boundary value problem for the change in permeability due to a small embedded magnetic domain Curve fits to measured permeability of Nickel ferrites [36]. Solid and dashed lines of matching colors represent the real and imaginary parts, respectively, of the permeability for a particular Nickel ferrite sample of density d. For the imaginary permeability, Im(µ g ) is plotted so that only positive values are needed for the vertical axis Curve fits to measured permeability of Barium ferrites with easy plane anisotropy. Solid and dashed lines of a particular color give the real and imaginary parts of the permeability, respectively, for a Barium ferrite with X percent Zinc doping Fitted curves for the in-plane permeability of crystal-oriented easy plane ferrites. The solid and dashed lines of a particular color correspond to the real and imaginary permeabilities of a particular ferrite. The sources for the data are indicated in the legend The directional permeabilities of a crystal-oriented easy plane ferrite [30]. Black and red curves are the in-plane and out-of-plane directional permeability, respectively. Solid and dashed lines are the real and imaginary parts, respectively A biphasic composite material A curve fit to the measured permeability spectrum along the in-plane hard axis of a ferromagnetic laminate of single domain films [42] The spheroid viewed along ẑ (left panel) and along ŷ (right panel) Comparison of spherical particle permeability and effective permeability at 45% particle concentration viii

9 3.1 The geometry of a TM plane wave in anisotropic magneto-dielectric media. In the figure, ψ < π /2, hence the medium depicted must have ε z > ε t. If instead, ψ > π /2, then ε z < ε t Propagation coefficients of complex waves versus the free space wavenumber Anisotropic magneto-dielectric Substrate Geometry Electric field lines for the T M 0 surface wave mode in an anisotropic magneto-dielectric substrate. The field lines pictured are obtained by the choice of parameters: ε t = 5 ; ε z = 6 ; µ t = 10 ; k 0 = 2π ; h = Magnetic field lines for the T E 0 surface wave mode in an anisotropic magneto-dielectric substrate. The field lines pictured are obtained for: ε x = 10 ; µ z = 6 ; µ y = 5 ; k 0 = 4π ; h = Source-Excited magneto-dielectric Antenna Substrate Polar plots of the E-plane (blue traces) and H-plane (red traces) electric field patterns of a short dipole over thin substrates. The radial magnitude axis is in logarithmic units, and normalized to 0 λ db. All plots are computed with a thin substrate with h =.05 µtεt. Panels (a) and (b) show that µ z has no effect. Panels (c) and (d) show the influence of ε z in a non-magnetic thin substrate. Panels (e) and (f) show that ε z has no influence on patterns when the substrate is strongly magnetic Polar plots of the E-plane (blue traces) and H-plane (red traces) electric field patterns of a short dipole over thick substrates. Plots λ are computed setting h =.5 µtεt Original contour C 1 from to +, and contour C 2 deformed around branch cut for k z1. During the contour deformation, the surface wave pole at k tp is crossed Contours for integration of radiated and total power. In the figure, k m is defined as max ( ) k 0 µt ε z, k 0 εt µ z Efficiency of short dipoles versus substrate thickness and electromagnetic properties. Cusps on plots are indicative of crossing a cutoff point for a higher order surface wave mode Overlapping triangular subdomain basis functions Integration contour for calculation of Z mn Input impedance versus length for a microstrip dipole. For this calculation, W = 0.01λ 0, ε = 2.45I, µ = I, and h = 0.2λ 0. Note that W and h are rescaled at each value for the length ix

10 5.4 Resonant length of microstrip dipoles versus the permittivity of an isotropic substrate. For this calculation, W = 0.01λ 0, ε = ε r I, µ = I, and h = 0.2λ Resonant length of microstrip dipoles versus the permeability of an isotropic substrate. For this calculation, W = 0.001λ 0, ε = I, and h = 0.2λ Input impedance versus length for a microstrip dipole over a magnetodielectric substrate with ε r = 3 and µ r = 50 and W = h = L/ VSWR versus length for a microstrip dipole over a magneto-dielectric substrate with ε r = 3 and µ r = 50 and W = h = L/ Efficiency of the microstrip dipole lying over a substrate with µ r = 50 and ε r = Comparison of input impedance of microstrip dipoles with and without dielectric anisotropy. We set W = 0.01λ 0 and h = 0.1λ Comparison of efficiency of microstrip dipoles with and without dielectric anisotropy. We set W = 0.01λ 0 and h = 0.1λ Comparison of directive gain with and without dielectric anisotropy. 127 A.1 Branch cuts for k z1 (given by dashed red lines) C.1 Input impedance, comparison of our model to FEKO C.2 Directive gain, comparison of our model to FEKO C.3 Segmentation of the microstrip dipole in FEKO x

11 Acknowledgments This research was made possible by the Walker graduate assistantship program at the Penn State Applied Research Laboratory, and the Brown Graduate and Society PS Electrical Engineering Fellowships. My sincerest thanks are owed to my advisor Dr. James Breakall for his willingness to oversee the completion of my research. His advisory approach involved a minimum of distractions, and allowed the work to be completed in a timely fashion. I would also like to thank Dr. Steven Weiss, for encouraging me to pursue a doctoral degree, and for introducing me to many of the mathematical techniques that make this research possible. I offer additional thanks to my committee members Dr. Julio Urbina, Dr. Victor Pasko, and Dr. Michael Lanagan for constructive suggestions for improving this dissertation. I am grateful to my parents for their consistently wise advice, and for encouraging me to pursue a career in engineering. Lastly, I would like to acknowledge the loving support of my wife Theresa. Her confidence in me has never waivered. Most amazingly, during the last six months, she took upon the task of caring for our newborn son through the night, allowing me to dedicate the time fully to writing. xi

12 Dedication For my wife Theresa and son Levi. xii

13 Chapter 1 Overview and Introduction The incorporation of exotic materials into antenna designs remains a frontier for interesting theoretical research. The most general linear material is the bianisotropic medium, which requires 36 different constitutive parameters to specify completely. Under the conditions of reciprocity, 16 of the 36 parameters are independent [1]. If electromagnetic chiral materials are excluded, and a coordinate system is chosen to be aligned with the material s 3 principal axes, then only 6 parameters remain [2, pp ]. The specification of these parameters defines the anisotropic magnetodielectric material. In an anisotropic magneto-dielectric we presume the material may be described by permittivity and permeability tensors of the form µ = ˆxˆxµ x + ŷŷµ y + ẑẑµ z (1.1) ε = ˆxˆxε x + ŷŷε y + ẑẑε z (1.2) the main assumption being that both the magnetic and dielectric principal axes are aligned with the same set of Cartesian coordinate axes ˆx, ŷ, and ẑ. Empirical data seems to support this assumption. For instance, see [3]. In Chapter 2, we discuss the sources of anisotropy in the permeability tensor for ferromagnetic materials. It is now well-established that antennas built with high-permittivity dielectrics exhibit poor impedance bandwidth [4]. This is due, in part, to the inherent size reductions, which can be associated through Chu s theory [5] with a higher radiation Q factor. We might expect the introduction of a high permeability (instead of permittivity) would offer no advantage. After all, we often still expect a size reduction. However, in many cases, a large permeability implies fundamentally different behavior than a large permittivity. For example, consider that the sign on 1

14 the reflection coefficient for normally incident plane waves on a magnetic material is positive, not negative as is the case for a dielectric material. Furthermore, the asymptotic behavior of a magnetic material as µ r is that of a perfect magnetic conductor (PMC). From image theory, an antenna positioned over a PMC ground plane can be placed arbitrarily close to the ground plane, thus minimizing the physical space required by the antenna. Moreover, the resonant length of a dipole over a PMC ground plane is still approximately λ 0 /2. Thus, in actuality, the introduction of permeability need not necessarily result in a size reduction along a physical dimension that would impact bandwidth. As we will show in Chapter 4, a magnetic ground plane with large values of µ r will possess excellent wave-trapping capability. In fact, its wave trapping capability far exceeds a non-magnetic substrate with an equivalent value of permittivity. Thus, a planar antenna positioned over a magnetic substrate will usually deliver a large percentage of its power into trapped surface waves in the substrate, which leads to low efficiency. It would seem that materials with comparable values of permittivity and permeability are the best bet. The behavior of an antenna positioned over a substrate with arbitrary values of permittivity and permeability, though, is very difficult to predict from intuition alone. Precise models need to be formulated, and numerical analysis needs to be conducted. We may add additional degrees of freedom by allowing the material s permittivity and permeability to vary over different directions. As will be demonstrated in Chapter 2 of this dissertation, a large class of materials is obtainable that possesses anisotropic magnetic properties. One of the principal conclusions of this research is that the anisotropic magnetic materials are potentially better suited for planar antenna designs than their isotropic counterparts. We can enumerate at least 2 advantages of using anisotropic magnetic materials. The first and most important relates to Snoek s law. We review in Chapter 2, why anisotropic magnetic materials inherently possess a larger Snoek product, and consequently why they can possess a greater value of permeability at higher frequencies, than their isotropic counterparts. For thin magnetic substrates, we also show in Chapter 5 that the permeability normal to the air-substrate interface is not important. Thus, if we sacrifice permeability in the normal direction in exchange for a larger Snoek product for the in-plane permeability, we are getting all the intended benefit without any downside. Hence, if for no other reason, anisotropic 2

15 magnetic materials are preferable. Another intriguing possibility involves the application of anisotropy in the permittivity of a material. It is shown, for instance, in Chapter 3 that the primary TM surface wave mode cannot propagate if the in-plane permeability and normal permittivity are equal to 1. Subsequently, in Chapter 5, we show that even for smaller values of the normally directed permittivity, that the primary TM surface wave is suppressed to a degree, leading to improved efficiency of planar antennas as radiators. Our principal contribution is in the development of precise electromagnetic models that include a general form of magneto-dielectric anisotropy. Since the search for effective antenna designs now invariably involves some amount of optimization, formulating precise numerical models incorporating new degrees of freedom has become an essential component of theoretical progress in the field of antennas. 1.1 Overview of Prior Studies on the Application of Magnetic Materials to Antenna Design The most popular and prevalent antenna that makes use of magnetic materials has to be the ferrite rod or loopstick antenna. The design calls for many turns of wire coiled around a cylindrical ferrite core. The resultant antenna possesses a radiation resistance that far surpasses a simple small dipole of a comparable size, and is easily more efficient [6]. However, the antenna is still much less efficient than an actual resonant dipole of the appropriate half-wavelength size. Moreover, the nonlinear nature of the magnetic material is such that it can only present a permeability to fields of a relatively small magnitude. The nonlinear nature thus makes the antenna unsuitable for use in a broadcast station requiring large transmission power. In a low-frequency channel where low-profile, low cost mobile receivers are desired, an electrically small ferrite rod antenna that is as efficient as possible, though still not very efficient, is acceptable, especially since the difference can be made up by increasing transmission power and using a physically large antenna at the central transmitting station. Indeed, the ferrite rod antenna is in common use for mobile AM radio receivers. There are other applications, in higher frequency bands, where similar require- 3

16 ments would suggest we look again to magnetic materials for solutions that admit of similar tradeoffs. Moreover, if the application does not call for an extremely large amount of transmit power, then a magnetic material antenna solution can be considered also for transmission. In the remainder of this section, we consider some instances of antennas designed with magnetic materials that have appeared in the literature. An important distinction that must be made, is whether the magnetic material is operated with or without an applied variable DC magnetic field bias. With a magnetic field bias applied to a ferrite or other ferromagnetic material, its properties change with the amplitude and direction of the applied bias. Furthermore, wave propagation becomes nonreciprocal, and the permeability matrix must contain nonzero off-diagonal entries, making analysis difficult. Several publications exist treating the biased ferrite substrate for patch and microstrip antenna geometries [7] [8] [9] [10]. It has been demonstrated in these investigations, often experimentally, that the biasing of the ferrite enables some control over the various characteristics of antennas, including the beam direction, resonance frequency, and principle polarization. As a consequence of non-reciprocity, an antenna with different transmit and receive radiation patterns is possible [10]. In this dissertation, we consider only unbiased magnetic materials. For an unbiased, isotropic magneto-dielectric material, existing research can be roughly sorted into 2 categories. In the first, the magneto-dielectric material replaces the conventional dielectric material as the substrate for a planar antenna, usually a square or rectangular patch antenna, implemented in microstrip. A rectangular patch antenna printed over a ferrite substrate was investigated in [11], while a planar inverted-f antenna (PIFA) using magneto-dielectric material has been analyzed in [12]. In both cases, their results indicate that larger bandwidth may be obtained when permeability is larger than permittivity. Hansen and Burke gave a zero-th order analysis of the patch over a magneto-dielectric substrate [4], concluding that the bandwidth of the patch is controlled by the permittivity and permeability of the substrate according to BW = 96 µ r ε r ( h λ 0 ) 2 ( µr ε r ) (1.3) 4

17 A main result of this formula, as discussed in the paper, is that for a small ε r µ r, the bandwidth of a patch printed over a ε r = µ r substrate is roughly the same as the bandwidth of the patch situated over the corresponding dielectric substrate where the permittivity is still ε r but the permeability is µ r = 1. Yet, the resonant length is reduced by a factor of µ r. Thus an additional size reduction factor could theoretically be achieved without paying an additional bandwidth penalty. The second primary utilization of a magneto-dielectric material is as a coating on an antenna s conductive backplane. In this case, the antenna is not printed directly over a substrate lying over the ground plane. Instead, there is a certain amount of free space between the antenna element and the ground plane. Normally, this distance is required to be large to avoid poor behavior, as the image currents through the backplane tend to cancel the radiation. With the magneto-dielectric coating, it is reasoned, the reflection of the impinging radiation from the antenna element will be associated with a reflection coefficient of a positive sign. Thus, the returning radiation need not, in theory, pose any substantive problems in terms of the input impedance, even as the antenna element is subsequently brought very close to the ground plane. For instance, Breakall designed a broadband planar dipole antenna positioned over a magneto-dielectric-coated ground plane, that exhibited a decreased spatial profile [13]. A similar design for a planar archimedean spiral over a magneto-dielectric-coated ground plane has been considered by other researchers [14]. For the archimedean spiral, the magnetic material was chosen to satisfy µ r = ε r, and to be lossy. Thus, their intention was apparently to use the coating as an absorber, and to take the 3 db hit to directive gain in the forward direction, in exchange for an antenna with a low spatial profile exhibiting large bandwidth. Comparatively less has been written on the use of anisotropic magneto-dielectric materials in antennas, though, several antenna designs have been reported. An interesting antenna developed by Metamaterials, Inc., and reported by Mitchell and Weiss, utilizes several designed anisotropic magneto-dielectric materials with varying values of permeability and permittivity along different directions [3]. The large permeability of the materials along certain axes is leveraged, by orienting several material samples under the various arms of a planar crossed dipole according to the local direction of the magnetic field generated by the currents flowing on the antenna. For each material sample, a significant permeability is exhibited along 5

18 one axis only. Thus this axis is aligned with the anticipated direction along which the magnetic field should have the largest magnitude. The two arms of the crossed dipole are loaded with disparate magneto-dielectric materials, designed to achieve a relative phasing between the radiated fields from the arms, which enables circularly polarized radiation to be obtained. A separate antenna concept has been reported by Mitchell [3]. In his work, a resonant cavity is loaded at a chosen location with an anisotropic magneto-dielectric material. The cavity has a slot cut into it, turning the cavity into an antenna whose properties are perturbed by the presence of the magneto-dielectric material loading. An anisotropic transverse resonance theory is proposed to explain the antenna s functioning, and is supported by calculations using the Finite Difference Time Domain (FDTD) method. Our analysis for the anisotropic magneto-dielectric substrate is modeled, in part, after the analogous treatments that have been given for anisotropic dielectric substrates. For instance, a numerical analysis of a patch printed on an anisotropic dielectric (µ r = 1) substrate has been conducted by Pozar [15]. His analysis shows that the permittivity along the in-plane and out-of-plane directions exert various influences on the resonant length of the patch. A general matrix-eigenvector theoretical method is given by Krowne for the derivation of Green s functions for geometries involving anisotropic layered materials [16]. 1 Based on Krowne s work, more detailed numerical analyses have been given for anisotropic dielectric substrates with arbitrary alignment of its principal axes by Pettis and Graham [17] [18]. These studies demonstrated further, that the resonant lengths, radiation patterns, and input impedance of microstrip dipoles and patches are influenced by the anisotropy. In our opinion, the most serious gaps in understanding the properties of planar antennas printed over anisotropic magneto-dielectric substrates arise from a lack of known results regarding the surface wave modes. For instance, little is currently published regarding the cutoff frequencies, or mode patterns, for surface waves propagating in an anisotropic substrate. This information is critical to understanding the dependence of the surface waves on the substrate anisotropy. The surface wave excitation of a substrate is a principle cause of low efficiency in planar antennas, and moreover, is responsible for degradational effects associated with 1 The theoretical formulation given in Chapter 3 of this disseration makes similar use of eigenvectors as suggested by Krowne. 6

19 surface-wave-mediated mutual coupling between elements functioning within an antenna array. A primary goal of this dissertation is to integrate a treatment of the surface waves (in the anisotropic substrate), into a more complete analysis of planar printed antennas. Our analysis will enable some conclusions to be drawn, regarding what our preferences should be for the properties of magneto-dielectric materials intended for use in planar antenna design. 1.2 Contributions to Knowledge The novel contributions of this dissertation are contained primarily in the theoretical explorations of surface wave propagation in the anisotropic magnetodielectric substrate, and its role in controlling the efficiency and gain of printed microstrip dipole antennas. Several publications have resulted from research related to the composition of this disseration. An Army Research Laboratory technical report [19], and two theoretical papers have been published concerning the solution of electromagnetic boundary value problems [20] [21]. A paper concerning the method of moments analysis of microstrip dipoles printed on anisotropic magnetodielectric substrates, as discussed in Chapter 5, is in preparation [22]. While the topic is not treated here, from the author s research on magneto-dielectric media, attention was drawn to work on refraction through periodic media comprising densely spaced magneto-dielectric spheres, resulting in a paper on the theoretical explanation for negative refraction [23]. In this dissertation, the following results are developed: Snoek s laws are derived for uniaxial ferrites, easy-plane ferrites, ferromagnetic laminates, and ferromagnetic particle composites. A new derivation is given for the permeability tensor of a magnetic domain for a ferrite with easy-plane crystalline anisotropy. A concise eigenvector formulation is developed for the plane-wave solutions of Maxwell s equations in an anisotropic magneto-dielectric medium. Analytical expressions are derived for the trapped surface wave modes in a magneto-dielectric substrate with in-plane isotropy. Inequalities bounding 7

20 the propagation coefficients, formulas for the cutoff frequencies, and field line plots for the modal profiles are provided. From the bounding inequalities for the surface wave propagation, it is proven that the primary TM surface wave can be suppressed in a substrate for a certain choice of directional permittivity and permeability. Dyadic Green s functions are found for the calculation of the electromagnetic fields due to impressed surface current distributions lying over an anisotropic magneto-dielectric substrate. The far-field radiation formulas are utilized to prove that circularly polarized radiation may be obtained from a linearly-polarized current distribution, such as that supported by a single-feed microstrip dipole, by exploiting in-plane anisotropy in a magneto-dielectric substrate. A method of moments algorithm is given, along with the computer program in an appendix, for the calculation of the current distribution on a centerfed microstrip dipole printed over an anisotropic magneto-dielectric antenna substrate. The algorithm calculates input impedance, efficiency, and directive gain, and includes provisions checking that the solutions satisfy Poynting s theorem. From a method of moments analysis of microstrip dipoles, it is shown that highly permeable substrates effect tradeoffs in an antenna design between efficiency and bandwidth. An argument is developed, and supported by numerical results, that magnetic anisotropy may be disregarded in the case of an in-plane isotropic material. Consequently, anisotropic materials may be utilized where the out-of-plane permeability is equal to 1, without affecting performance. Numerical results are given, which show that the efficiency of microstrip dipoles may be improved by controlling the dielectric anisotropy of the substrate. This effect is attributed to the suppression of the transverse magnetic (TM) surface wave. 8

21 1.3 The Layout of this Dissertation In Chapter 2, we investigate known information regarding passive anisotropic properties of magnetic materials, including crystal-oriented ferrites, ferromagnetic laminates, and particle composites. The various Snoek s laws are derived, and from the models for the permeability spectra, extracted best-fit curves of measured permeability spectra are shown. In Chapter 3, the analytic expressions for surface wave modes are developed starting from Maxwell s equations. An analysis of the surface wave modes cutoff frequencies, and plots of their modal profiles are given. In Chapter 4, dyadic Green s functions are introduced, and from them, the radiation patterns of short dipoles are calculated. Starting from the dyadic Green s function, the surface wave excitations by a short dipole are also analyzed in the complex plane, and the efficiency, defined as the proportion of power delivered to the radiation field versus the total power (which includes surface wave power), is calculated for various values of permittivity and permeability along the in-plane and out-of-plane directions. In Chapter 5, a method of moments algorithm is detailed for the solution of the currents on a microstrip dipole positioned over the anisotropic magneto-dielectric substrate. A method is given for the separate calculation of radiated and surface wave power, so that efficiencies can be compared for various values of the substrate properties. In Appendix A, we give an explanation of the complex square roots appearing in the dyadic Green s function. A code listing for the method of moments MATLAB program is given in Appendix B. Finally, validation of the MATLAB program against commercially available software, for the restricted case of an isotropic substrate, is given in Appendix C. In this dissertation, e jωt time variation is assumed. Vectors are written in bold font. Unit vectors are indicated with a hat, as in â. Dyadic/matrix quantities are denoted using a double over-bar, as in A. Square brackets are reserved mainly for matrices, such as A B. SI units are used throughout the dissertation. C D 9

22 Chapter 2 Permeability of Anisotropic Magnetic Materials In Chapters 3 and 4 we study electromagnetic propagation and radiation of waves in anisotropic magneto-dielectric substrates. In this chapter, we focus specifically on the permeability tensor. The goal of this chapter is to establish that a large class of materials is obtainable, that for a proper model of electromagnetic wave propagation to be given, requires we assign a permeability tensor to the material of the general form µ = ˆxˆxµ x + ŷŷµ y + ẑẑµ z (2.1) While it is possible to produce artificially magnetic substrates utilizing effective medium techniques [24], the resulting magnetic properties are weak at low frequency, and highly dispersive at frequencies where the magnetic properties are strong. We focus instead on naturally ferromagnetic materials. In order to understand the possibilities and limitations inherent in working with ferromagnetic materials, we give a treatment in this chapter of the dynamic permeability tensor of macroscopically anisotropic magnetic materials. We describe the theory for the gyromagnetic response of magnetic domains to external magnetic fields in Section 2.1. We consider the effects of crystalline anisotropy fields, in both uniaxial and easy-plane crystals. We also incorporate demagnetization fields arising from the finite boundaries of the domain. In Section 2.2, we then discuss how the anisotropy fields lead to macroscopically anisotropic permeability in crystal-oriented ferrites. In Sections 2.3 and 2.4, we discuss how uniaxial crystalline anisotropy, and demagnetization fields lead to anisotropic 10

23 permeability in ferromagnetic-laminate and particle composites. By its very nature, this review must borrow heavily from existing literature on magnetic materials. Our intention is to give a reasonably self-contained treatment, at a theoretical level appropriate for the curious antenna engineer, who would like a basic grasp of the achievable properties of magnetic materials, and especially to know their fundamental limitations. We do not consider the incremental permeability of materials that are in a remanent magnetized state. Additionally, we assume that materials are not subjected to an active DC magnetic field bias. Moreover, although eddy currents play an important role in placing additional limits on the obtainable performance of certain magnetic materials, particularly the laminate and particle composites, we omit its consideration here, for simplicity. We also use the word ferromagnetic in a wider sense than is now usual, and take it to include both metallic alloys of Ni, Co, and Fe, and ferrimagnetic materials, such as ferrites. 2.1 Ferromagnetism from Magnetic Domain Rotation Ferromagnetic materials are a rather small class constituting only several elements in pure form and some of their compounds. In fact, at room temperature, and in pure chemical form, only Ni, Fe, and Co are ferromagnetic [25]. At lower temperatures, other elements become ferromagnetic, including several elements from the lanthanide series of the periodic table [25]. Metal alloys and certain oxides of these and other elements are also ferromagnetic. The various oxides of Ni, Fe, and Co are collectively referred to as ferrites. The addition of non-metallic elements into the crystal structure of ferrites endows them with low electrical conductivity, and consequently, ferrites have found more widespread use in electromagnetic devices operating at microwave frequencies. All materials respond to externally applied magnetic fields, but in most situations common to our everyday experience, these responses are small and inconsequential. In a paramagnetic material, electron spins experience a torque force that tends to reorient their magnetic dipole moments toward external fields. However, each electron responds more or less independently to the magnetic field. The force on a single isolated electron is small, and the tendency of this force to reorient the electron spin magnetic moment is easily nullified by thermal vibrations. 11

24 A ferromagnetic material also derives its magnetic response from the spin magnetic moment of the electron. However, the ferromagnetic material is uniquely characterized by the existence of a strong coupling force of quantum mechanical origins. This force tends to spontaneously align magnetic spin moments of neighboring electrons. Coupled electrons behave like coherent magnetic dipoles with larger magnetic moments. The torque force exerted upon the larger magnetic moments can overcome the quashing effect of thermal vibrations. Ferromagnetism is explained in terms of the magnetic domain model. In this model, the coupling forces among neighboring electrons leads to spontaneous groupings of electrons within small regions, called domains. Within each domain, the magnetic spin moments of the electrons are aligned, and thus the domain is fully magnetized. Each domain s magnetization is relatively stable and constant over time, due to the presence of effective magnetic fields internal to the domain. These internal effective fields result from a variety of internal forces, and dictate the direction of magnetization in each domain. In a macroscopically demagnetized ferromagnetic material, the individual domains are magnetized along varying directions, as shown in Figure 2.1. Upon adding contributions from many domains, magnetized along various axes, there results no net observable magnetization. Figure 2.1. A magnetic specimen is divided up into domains with disordered orientations in the absence of an external magnetic field. At oscillation frequencies above about 100 MHz, the primary mechanism leading to significant initial permeabilities is the rotation of domain magnetization under the action of an external magnetic field. The magnetization in each domain experiences a torque force that can cause a precessional movement, or wobble, about its fixed axis. This occurs in a way that adds up to a net observable magnetization in a macroscopic material specimen. 12

25 2.1.1 Equation of Motion for Internal Magnetization A basic understanding of ferromagnetism starts with a consideration of the magnetic response of a domain to an external oscillating magnetic field. Since each domain is composed of individual magnetic dipoles arising from the spin moments of electrons, we begin by considering the force felt by the electron in an applied magnetic field B. If we treat the electron classically like a distributed sphere of charge spinning on its axis, we would obtain for its spin magnetic dipole moment, m = el 2m e (2.2) where e is the charge of an electron, and m e is its mass. L is the vector angular momentum of the electron s spin, which in a classical description would be given by L = ẑiω 0 (2.3) where I is the electron s rotational inertia and ω 0 its angular velocity. Quantum mechanics, however, shows that the true spin magnetic moment of an electron deviates from this classical prediction by a constant of proportionality termed the Landé factor g [25], m = ge 2m e L (2.4) The ratio of the spin magnetic moment to the angular momentum is termed the gyromagnetic ratio γ. γ = ge 2m e (s T) 1 (2.5) If the electron is exposed to a uniform static magnetic field B, a torque force is produced given by T = m B (2.6) This torque force acts on the angular momentum of the electron according to the classical equation of motion for the rotation of a rigid body [26] T = dl dt (2.7) 13

26 Combining (2.4)-(2.7), we obtain dm dt = γm B (2.8) Magnetic domains consist of many coupled electrons contributing aligned magnetic dipole moments m k. The number of atoms in each domain varies, but is often on the order of or atoms [2]. It is therefore permissible to treat each magnetic domain as a continuous distribution of magnetic dipole moment density. Thus, we define a magnetization density vector [2] M = lim v 0 n v m k k=1 ( v) (2.9) and let (2.8) pass in this limit to the vector point relation dm dt = γm B (2.10) By definition, the magnetic field vectors B and H existing within the magnetic domain relate to this magnetization according to B = µ 0 (H + M) (2.11) where µ 0 = 4π 10 7 (H/m) is the permeability of free space. We find, upon substituting (2.11) into (2.10), and noting that M M = 0, dm dt = γµ 0 M H (2.12) which is an equation of motion for the magnetization within an isolated domain. In order to utilize (2.12) to investigate dynamic magnetization responses, it is necessary to make assumptions regarding the static components of magnetization and magnetic field initially existing within the magnetic domain before any external oscillating fields are applied. In a magnetic domain, we will assume that there exists a nonzero static component of magnetization M S which is fully saturated and initially aligned along some axis. There are two effective internal magnetic fields we will discuss. In this section, we will incorporate the anisotropy field of a uniaxial 14

27 crystal, which is a simple configuration for which calculations are analogous to a macroscopically saturated ferromagnet. In the following section, we will incorporate demagnetization fields, and anisotropy fields for other types of crystals. An effective static magnetic field H A, termed the anisotropy field, has been shown to effectively model the consequences of internal crystalline forces [27]. The existence of this effective field is not surprising, since in a crystal, atoms will be packed closer together along certain crystalline axes than others, and the coupling force between electron spins is sensitive to the distance between electrons. In a uniaxial crystal, the crystalline forces can be modeled to first order by an anisotropy field that always points along a particular preferential crystal axis, independent of the static magnetization [28]. Thus for a uniaxial crystal, it is reasonable on the basis of (2.12), to assume under equilibrium conditions that M S and H A are initially parallel to each other, for otherwise a torque force would tend to reorient M S toward H A, contrary to the assumption of equilibrium. Let us suppose, in addition to M S and H A, a time varying RF magnetic field H RF is externally applied to the domain. We expect this time varying magnetic field to induce a time varying magnetization response M RF within the domain. Thus, in (2.12) we set M = M S + M RF (2.13) H = H A + H RF (2.14) obtaining dm S dt + dm RF dt = γµ 0 (M S H A +M S H RF +M RF H A +M RF H RF ) (2.15) Inspecting the four terms on the right hand side of (2.15), we observe that M S and H A are parallel, and thus M S H A = 0. Second, the nonlinear term M RF H RF, for applied fields of small strength, can be assumed to have negligible magnitude in comparison to the remaining terms and may be ignored. On the left hand side of (2.15), by definition, dm S dt = 0. Therefore, (2.15) reduces to dm RF dt = γµ 0 (M S H RF + M RF H A ) (2.16) As (2.16) is now a linear equation, frequency domain analysis is permissible. Letting 15

28 the field vectors H RF and M RF have an e jωt time variation, (2.16) becomes jωm RF = γµ 0 (M S H RF + M RF H A ) (2.17) If we set M S = ˆnM S and H A = ˆnH A, then (2.17) can be solved to obtain M RF = χ m H RF (2.18) where χ m is the magnetic susceptibility tensor and χ m = Utilizing the definition of the permeability tensor ω 0ω m (I ˆnˆn) jωω m ˆn I (2.19) ω0 2 ω2 ω0 2 ω2 ω m = µ 0 γm S (2.20) ω 0 = µ 0 γh A (2.21) B RF = µ 0 µ H RF (2.22) We obtain the permeability tensor for a magnetic domain in the form 1 µ = I + χ m (2.23) µ = µ d (I ˆnˆn) jκ dˆn I + ˆnˆn (2.24) where µ d = 1 + ω 0ω m ω 2 0 ω 2 (2.25) κ d = ωω m ω 2 0 ω 2 (2.26) Typically, this result is derived for a macroscopic magnetic material specimen that has been saturated in a DC bias field. Under these conditions, the specimen consists of only a single magnetic domain. In our case, we consider only demagnetized 1 In cartesian coordinates, I = ˆxˆx + ŷŷ + ẑẑ. It is essentially the identity matrix / tensor, and acts like the number 1 in dyadic / tensor algebra. The quantity ˆn I is readily understood as an operator upon an arbitrary vector. Consider that (ˆn I) A = ˆn (I A) = ˆn A. 16

29 materials. Hence, an isolated domain is assumed to remain microscopic in size. We note though, that a domain can be microscopic in size, yet still contain billions of electrons. We are therefore justified in assigning local values of permeability to various domains in the form given by (2.24) with ˆn = ˆn i, where ˆn i is the direction of magnetization in the ith domain. The theoretical derivation of the bulk permeability tensor for a macroscopic sample consisting of many domains will in general depend on the statistical distribution of orientations of the individual domains. We can make two important observations regarding (2.24). First, we observe that the permeability tensor can be written in a coordinate-free manner. This emphasizes the useful property that the tensor takes the same form in either cartesian or cylindrical coordinates. In either coordinate system, we take ˆn to be the ẑ axis. Second, we note that the permeability tensor cannot be diagonalized by any geometric rotation of coordinate axes. That is, there exists no particular set of stationary axes in which the tensor becomes diagonal. This property follows from the fact that the tensor is not symmetric. However, if a circulating (time-varying) coordinate system is used, such as û 1 = 1 2 (ˆx + jŷ), û 2 = 1 2 (ˆx jŷ), û 3 = ẑ, then (2.24) becomes B = (û 1 û 1 (µ d + κ d ) + û 2 û 2 (µ d κ d ) + û 3 û 3 ) H (2.27) and the relationship between the B and H field vectors is expressed in terms of a diagonal tensor. Note that the above trick works equally in the cylindrical coordinate system if one chooses û 1 = 1 2 (ˆρ + j ˆφ), û 2 = 1 2 (ˆρ j ˆφ), and û 3 = ẑ. 2 This strategy of diagonalizing the permeability tensor for a magnetic domain usually results in less work in derivations. In the formulation of the equation of motion, we assumed that the precessional movement of the magnetization about its anisotropy field axis was undamped. In order to fit curves to realistic experimental data, it is necessary to incorporate loss into the model. Equation (2.12) can be modified by adding a phenomenological 2 This follows, for instance, because ˆρ + j ˆφ = (ˆx + jŷ)e jφ. Letting â = 1 2 (ˆx + jŷ) and ˆb = 1 2 (ˆρ + j ˆφ), we see that â ˆb = 1, and hence the vectors are parallel in complex vector space. 17

30 term to account for damping forces [26], yielding dm dt = γµ 0 M H + α ˆM dm dt (2.28) where α is a dimensionless damping coefficient. Note that the added term α ˆM dm dt simplifies to jωα ˆM S M RF. This term can be combined with the factor M RF H A in the following fashion. γµ 0 (M RF H A ) + (jωα ˆM S M RF ) = (ω 0 + jωα)ˆn M RF (2.29) where ˆn is the shared direction of the static magnetization and internal anisotropy field. Thus, from (2.29), once the permeability has been derived in the lossless form, loss can be added into the model by replacing ω 0 with ω 0 + jωα Easy-Plane Crystalline Anisotropy and Demagnetization Fields In the previous section, we derived the magnetic permeability tensor for an isolated magnetic domain with a fixed DC magnetization along a direction that was parallel to its internal effective magnetic anisotropy field. We considered a domain with uniaxial anisotropy [27], in the sense that the anisotropy field H A points along a single axis. This axis, call it ˆn, is referred to as the easy axis of the magnetic domain. Other axes are called the hard axes. Note from (2.24) that the permeability along the easy axis is µ n = 1. The crystalline anisotropy field can take other forms in different kinds of crystals. For example, there are a class of ferrites that possess an easy plane for magnetization [29] [30]. The axis perpendicular to this plane is called the crystal axis, and it is the hard axis for the specimen. The appropriate anisotropy field H A, that accounts for the relevant forces in this crystal are more complicated than for the uniaxial crystal. It turns out that the anisotropy field will depend on the orientation of the dynamic, RF component of the magnetization with time [31]. In [31], an interesting indirect derivation of the effective anisotropy field is given, that was found to be in essential agreement with measurements. Alternatively, direct formulas are provided, without derivation, by Schlomann for the complete permeability tensor [29]. We will proceed by postulating an anisotropy field that 18