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'1 ggudvlivu i . ,flle.\ltfl1.5lv\(§\ufiltsufiil.h . \flfikzfilzfln‘w .t-vi‘uiifivilil (331).) .3;- a}... s a . . : . . t. 1.. : 3.36.1...x1x. 321).... 2.32....1‘...‘.qutiiillituonnxifl$o§iq?%§ owl—Una... . . . .. J... 3.3V . v. r z ..x. .V. .. \‘ix .\(\‘ 2 Iii. vfiwiu. 411.th ..,.. . u . . ... .1. . {7..- ....>.\.-... It “3 “a“. ' Ly: LIBRARY Michigan State University This is to certify that the dissertation entitled ELECTROMAGNETIC INTERACTIONS IN INTEGRATED ELECTRONIC CIRCUITS presented by Michael John Cloud has been accepted towards fulfillment of the requirements for Ph.D. degree in Electrical Engineering SW D9 «a beam M 519/;th F [Limd Major profegcik Date . ' / MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES .——. \— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. ‘FINES will be charged if book is returned after the date stamped below. ELECTROMAGNETIC INTERACTIONS IN INTEGRATED ELECTRONIC CIRCUITS By Michael John Cloud A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1987 Copyright by MICHAEL JOHN CLOUD 1987 ABSTRACT ELECTROMAGNETIC INTERACTIONS IN INTEGRATED ELECTRONIC CIRCUITS By Michael John Cloud Electromagnetic waves in millimeter-wave integrated circuits are studied. Dielectric film and cover regions overlay a conducting half- space in the configuration modeled, forming a nonuniform background for integrated devices. Emphasis is placed upon the microstrip tran- smission line in this environment. This research exploits an integral-Operator description of the system. Constructed through Hertz potentials, an electric field inte- gral equation (IE) quantifies microstrip surface currents. Complex plane analysis in the axial Fourier transform domain leads to the rigorous identification of discrete and continuous eigenvalue Spectra. Discrete modes are associated with simple pole singularities, while the continuous spectrum arises from branch-cut integrals. These modal spectra are subsequently linked to natural and forced solutions of the IE. Discrete wave modes are associated with the homogeneous IE, which is solved by the moment method for electrically-thin microstrip. The current density function is expanded in both subsectional and entire- domain basis functions. Results clearly validate: (1) the dominant axial current approximation for narrow strips, and (2) the edge Michael John Cloud singularity for axial currents. DiSpersion characteristics are given for several modes; those for the fundamental mode compare favorably with results already in the literature. Solution of the forced IE, at points along the complex-plane branch cut, yields spectral components of strip radiation modes. The forcing function is taken to be the electric field impressed by a vertical monopole current that resides in the film layer. Preliminary moment-method results are given. The transform-domain IE is also a basis for the study of coupling phenomena in axially-uniform multi-strip systems. System pr0pagation modes are characterized by coupled currents sharing simple-pole singularities. Perturbation approximations apply for loose coupling, leading to an overlap-integral description of the coupling phenomenon. Finally, a novel approach to the numerical evaluation of Sommerfeld integrals, using the Fast Fourier Transform, is advanced. ACKNOWLEDGMENTS The author is grateful to major professor Dennis P. Nyquist and co-thesis advisor Byron C. Drachman for their guidance throughout this research. In addition, he wishes to thank guidance committee members K.M. Chen and E.J. Rothwell for their encouragement and support. Funding for the project was provided by the Office of Naval Research, grant # N0014-86-K-0609. LIST OF FIGURES ................................................ viii I. INTRODUCTION ................................................ 1 II. BASIC EM FORMULATION FOR INTEGRATED ELECTRONICS ............ 4 2.1 Introduction and Geometry ............................ 4 2.2 Symmetric-slab Interpretation ........................ 6 2.3 Homogeneous Dielectric Slab Guide .................... 6 2.4 Hertz Potential Representation for EM Fields ......... 7 2.5 Boundary Conditions for Hertz Potential .............. 8 2.5.1 Conditions at Film/Cover Interface ............ 9 2.5.2 Conditions at Film/Conductor Interface ........ 11 2.6 Integral Representation for Hertz Potential .......... 13 2.7 Hertz Potential Green's Dyad ......................... 17 2.8 Reflected Green's Dyad for Equal Permittivities ...... 19 2.9 Green's Dyad Source-point Singularity ................ 20 2.10 Conclusion ........................................... 23 III. PROPER MODE SPECTRUM FOR MICRDSTRIP LINE .................. 24 3.1 Introduction ......................................... 24 3.2 General Integral Equation for Electric Field ......... 24 3.3 General EFIE in the Fourier Transform Domain ......... 26 3.4 Identification of PrOpagation-mode Spectrum .......... 27 3.5 Forced and Unforced EFIE Solutions .................. 30 3.6 Conclusion ........................................... 31 IV. DISCRETE MICRDSTRIP MODES .................................. 32 4.1 Introduction ......................................... 32 4.2 Limiting Case of Thin Microstrip ..................... 32 4.3 Coupled IE's for Current Components .................. 34 TABLE OF CONTENTS vi 4.4 Dominant Axial Current Approximation ................. 4.4.1 Solution by Pulse Galerkin's Method ........... 4.4.2 Physical Implications of Surface-wave Poles ... 4.4.3 Computational Methods ......................... 4.5 Solution of Coupled Integral Equations ............... 4.5.1 Expansion in Entire-domain Basis Functions .... 4.5.2 Galerkin's Method Testing ..................... 4.5.3 Simultaneous Equations for Unknown Expansion Coefficients ........................ 4.5.4 Computational Methods ......................... 4.6 Results .............................................. 4.7 Equal Permittivities and the TEM Mode ................ 4.8 Conclusion ........................................... V. THE CONTINUOUS SPECTRUM ..................................... 5.1 Introduction .......................................... 5.2 Microstrip Excitation ................................. 5.3 Green's Function for Vertical Current in Film ......... 5.4 Dominant Axial Current Approximation .................. 5.5 Wavenumber Parameters Along Branch Cut ................ 5.6 Moment Method Solution of Spectral EFIE ............... 5.7 Results ............................................... 5.8 Conclusion ............................................ VI. COUPLING BETWEEN ADJACENT MICROSTRIP LINES ................. 6.1 Introduction .......................................... 6.2 Geometry of Multi-Strip System ........................ 6.3 Transform-domain EFIE for Pr0pagation Modes ........... 6.4 Testing Operation ..................................... 6.5 Approximations for Loose Coupling ..................... 6.6 Illustration For Two Strips ........................... 6.7 Overlap Integral For Coupling Coefficient ............. 6.8 Overlap Integral For Thin, Narrow Strips .............. 6.9 Results and Conclusion ................................ VII. CONCLUSION AND RECOMMENDATION ............................. APPENDIX A ...................................................... APPENDIX B ...................................................... BIBLIOGRAPHY .................................................... vii 59 6O 82 91 LIST OF FIGURES Figure Page 1. Basic form of high-speed integrated circuit .............. 3 2. Geometry and medium parameters ........................... 5 3. Symmetric slab dielectric waveguide ...................... 5 4. Principal wave of Hertz potential ........................ 15 5. Transmitted and reflected potential waves ................ 15 6. Impressed and scattered fields ........................... 25 7. Branch cuts in complex uZ plane .......................... 29 8. Integration contour closure ............................... 29 9. Thin microstrip line ..................................... 33 10. Fundamental mode dispersion characteristics .............. 51 11. Fundamental mode dispersion characteristics .............. 51 12. Fundamental mode dispersion characteristics .............. 52 13. Fundamental mode axial currents .......................... 52 14. Fundamental mode axial currents .......................... 53 15. Fundamental mode transverse currents ..................... 53 16. Example of axial and transverse currents ................. 54 17. Comparison of axial and transverse current amplitudes .... 54 18. Propagation eigenvalue vs. nf ............................ 55 viii 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. Current amplitude at strip center vs. nf ................. 55 First odd mode dispersion characteristics ................ 56 First odd mode axial currents ............................ 56 Second even mode dispersion characteristics .............. 57 Second even mode axial currents .......................... 57 Example of second odd mode axial current ................. 58 Microstrip excitation .................................... 61 Locations of U2 relative to branch cut ................... 65 Branch points in complex ux plane ........................ 65 Solutions of the forced EFIE ............................. 69 Parallel microstrips ..................................... 71 Overlap integral vs. strip separation .................... 79 Numerical integration rules .............................. 83 Complex uZ plane phasors ................................. 90 CHAPTER I INTRODUCTION The subject of high-frequency, planar integrated circuits is becoming increasingly important. Rapid evolution in this technology suggests a need for new, more rigorous methods of analysis. This dissertation addresses the media configuration indicated in Figure 1. A dielectric film layer resides between two other regions: perfectly conducting ground plane, and dielectric cover. Conducting devices (e.g., microstrip lines [1], patch antennas [2]) are integ- rated over the film/cover interface. High-speed integrated electronic circuits operating at microwave and millimeter wave frequencies use microstrip connections. The strip is an attractive transmission line for applications requiring high packaging densities [3]. Commonly, the analysis of circuit performance proceeds through a combination of low-frequency electric circuit and quasi-TEM transmi- ssion line theories. These are only approximate at high frequencies and circuit densities. In modern configurations electromagnetic phe- nomena play key roles in terms of both losses and coupling between adjacent integrated devices. Except for an integral equation analysis by Wu [4] of the princi- pal propagation mode, the isolated microstrip line is studied primari- ly via the quasi-TEM electrostatic approximation. Existing quasi-TEM formulations are typified by the variational method of [5]. Microstrip open-end and gap discontinuities are examined by full- wave analysis in [6]. The finite microstrip segment finds many appli- cations as a radiating element; this problem is treated, for example, in [7,8,9,10]. This research exploits a conceptually exact integral-operator description of the integrated circuit, with primary emphasis on the microstrip transmission line. The seven chapters of this dissertation treat various parts of the microstrip problem. The goal of Chapter 2 is to develop and emphasize certain parts of the relevant EM theory. Chapter 3 formulates an electric field integral equation, which is the basis for all subsequent work; later in Chapter 3, Fourier transform and complex plane analyses quantify the proper microstrip eigenmode spectrum. Discrete modal surface currents are detailed in Chapter 4. Chapter 5 treats the continuous spectrum. Finally, a simple coupled- mode theory for parallel microstrips is advanced in Chapter 6. Some words about notation here might be helpful. All integrals, unless limits are explicitly shown, are over the entire real line. As is standard in most electrical engineering works, the symbol j denotes the elementary imaginary number. Vector quantities are embellished with arrows, and are also slightly boldfaced for emphasis. Dyadic quantities (also slightly boldfaced) appear with a double overbar. Finally, a simple-harmonic time dependence is assumed, but supressed, throughout the dissertation. cover dielectric integrated devices / 1’ 4’ / film dielectric /// / /'/ conducting half-space Figure 1. Basic form of high-speed integrated circuit. CHAPTER II BASIC EM FORMULATION FOR INTEGRATED ELECTRONICS 2.1 Introduction and Geometry The electromagnetic description of integrated electronics em- braces the theory of EM waves in layered dielectric media. An integ— ral operator method is required here, because boundary conditions-of differential-operator methods are inseparable for many device geomet- ries [11]. Integral-operator formulations are also well-adapted to techniques of numerical solution. Hertz potentials, expressed as Fourier integrals, are employed in the present analysis; the challenge faced in this chapter is to derive practical boundary conditions for their components. Consider the setting displayed in Figure 2. A perfectly conduc- ting half-space is overlayed with a dielectric film having refractive index nf and thickness t; the film is, in turn, covered by a dielec- tric having refractive index nc. All materials are assumed nonmagne— tic and infinite in extent. For a suitable geometry, choose a rectangular coordinate system with z-axis parallel to the microstrip waveguiding axis; x and y axes are tangent and normal, respectively, to the cover/film interface. Define unit vector E as locally tangent to all conducting device surfaces. See Figure 2. / / / / [ x f nanf z z/ x’ /’ // /’ ./ perfect conductor Figure 2. Geometry and medium parameters. 2t guiding region nf cover surround nc Figure 3. Symmetric slab dielectric waveguide. 2.2 Symmetric-slab Interpretation The cover/film/conductor complex constitutes the background envi- ronment of an integrated device such as the microstrip line. It is appropriate to think of this layered-media background structure's behavior as a symmetric-slab dielectric waveguide. Classical image theory implies a guiding region dielectric having thickness 2t, of course, because all phenomena have mirror images through the plane at y=-t. Some interpretations in subsequent chapters are based upon this idea; therefore, a brief summary and discussion of the symmetric-slab guide is given in the next section. 2.3 Homogeneous Dielectric Slab Guide Assuming axial dependence of electromagnetic fields as exp(-sz), where B is a phase constant, a transverse-longitudinal decomposition of Maxwell's equations is achieved in standard guided-wave theory [12]. Longitudinal (Q-directed) field components satisfy resulting two-dimensional Helmholtz equations, and serve as generating functions for transverse field components. This formulation, applied to the symmetric slab guide of Figure 3, gives the well-known slab field distributions. Quasi-standard symmetric-slab terminology exists. The guide is composed of a guiding region embedded in an unbounded dielectric cover surround (see Figure 3). Transverse-electric (TE) and transverse- magnetic (TM) modes are identified for which the z-component of elec- tric and magnetic fields, respectively, are zero. Surface waves are discrete, guided-wave modes propagating axially and decaying exponen- tially in the y-direction; these modes exhibit no transverse - ‘1‘»5'" (y directed) power flow. Distinguished by oscillatory, standing-wave solutions in the guiding region, surface waves arise physically from total internal reflection when nf>nc. It is conventional to further classify them as even or odd in their transverse field components. Radiation modes are spectral-component fields exhibiting oscillations in both guiding region and surround; such modes do allow transverse power flow into the surround. Phase constant B for a surface-wave mode is restricted to the existence interval (kc,k), where k and kc are the guiding region and surround wavenumbers. For radiation-modes, 82 is confined to be less than kg. These results follow directly from the mode type defini- tions. The fundamental surface-wave mode is the TMO, an even mode with no low-frequency cutoff regardless of guiding region thickness. Its characteristic equation is tan(Kt) = nZY/nEK (2.1) where n and nc are guiding region and surround refractive indices, and K and Y are the guiding region and surround eigenvalues defined by 2: 2 2_ 2 2= 2_ 2 2 K n ko B and Y B "cko° 2.4 Hertz Potential Representation for EM Fields Electric and magnetic fields are obtained through the electric- type Hertz [13] potential 2 (notation following [14]) as E = k2? + vv-E (2.2a) -> + H = jweVX Z (2.2b) The Hertz potential satisfies an inhomogeneous vector Helmholtz equa- + tion with source term -J/jme; this term represents time-harmonic equivalent polarization sources. the electric and magnetic H = I ll field intensities are 2 -> x - k Zx+(a/ax)v-z kZZ ( ) E = + O y y a/ay V 2 -> z = k Zz+(a/az)v~z 3w6[(3/8y)Zz-(3/32)Zy] jwe[(a/az)zX-(a/ax)zz] jwe[(3/3X)Zy-(3/3y)Zx] Expanded in rectangular components, (2.3a) (2.3b) (2.3c) (2.4a) (2.4b) (2.4c) These expressions are used in section 2.5 to obtain boundary condi- tions for components of Hertz potential. 2.5 Boundary Conditions for Hertz Potential Consider the EM field produced by time-harmonic current sources embedded in the cover medium. are linked by boundary conditions applicable at y=0. Fields in the film and cover regions conditions for Hertz potential are consequences of familiar conditions on tangential components of electric and magnetic intensities: at the film/cover interface Ecx=Efx’ Ecz=Efz’ ch=fo and HCZ=Hfz; at the film/conductor interface Efx=0 and EfZ=O. Development of Hertz poten- tial boundary conditions proceeds one interface at a time, beginning with the film/cover. 2.5.1 Conditions at Film/Cover Interface Consider the y=0 interface. Enforcing continuity of tangential electric and magnetic fields in terms of equations (2.3) and (2.4), with definition of the refractive index ratio Nfc=("f/nc)2’ results in kEZcx-kgzfx = (a/axW-(If-ZZ) (2.5a) kZZ -k22 = (a/aziv-(‘z’ i ) (2 5b) c cz f fz f c ‘ (a/ay)[Zcz-Nfczle = (a/az)[ch-Nfczfy] (2.5c) (a/ax)[ch-Nfczfy] = (a/ayltzcx-Nfczfx] (2.5d) These conditions on potential are impractical, because many components appear in each. Following the meritable plan of Sommerfeld [15], consider the independent effects of each current-source component, noting that not all potential components are required to represent the EM field due to each component of current. Then construct, by super- position, boundary conditions for a general current source having all three components. Case (a): 3 = 9Jy Conjecture that the total fields are determined by 2392y in each region. In this instance the general boundary conditions (2.5) reduce significantly. Equality of tangential derivatives (i.e., those with reSpect to x and z), for all x and 2 at y=O, leads to the following conclusions: ch = Nchfy (2.6a) (a/ay)[Z ] = 0 (2.6b) cy‘zfy Case (b): 3 = QJX A naive conjecture that i=§2x is adequate in this case leads to a mathematical contradiction; therefore, the boundary conditions cannot be satisfied. Include the normal component of i as well, ie., let E¥QZX+9Zy, giving (a/ay)[zcx-ufczfx1 = o (2.7a) (S/By)[ch-ny] = (a/aXILfo‘Zcx] (2.713) Case (c): 3 = 902 This case is similar to (b); let 2e922+92y and arrive at (a/ay)[ZCZ-Nfc2fz] = O (2.8a) (8/3y)[ch-ny] = (3/32)LZfZ-Zcz] (2.8b) 10 , + _ A A A Case (d). J - xetny+sz General boundary conditions at y=O are obtained by combining the outcomes of cases (a), (b), and (c). The final results, which show coupling between normal and tangential potential components, are: Zcx = Nchfx (2.9a) ch = Nfczfy (2.9b) ZCZ = Nchfz (2.9c) (a/ay)[Zcx-Nfc2fx] = 0 (2.9d) (a/ay)[ZCz-Nfc2fz] = O (2.9e) (a/ay)[ZCy-Z ] = (l-Nfc)[(a/ax)zfx+(a/az)zfz) (2.9f) fy 2.5.2 Conditions at Film/Conductor Interface Next, consider the interface at y=-t. Vanishing tangential com- ponents of electric intensity imply 2 + - kfzfx + (a/axW-Zf - 0 (2.10a) 2 + - Again, each component of current density receives separate treatment. 11 + A Case (a): J = ny .) Let Za92y represent the EM fields. Setting Efx=0 and Efz=0 at the conducting interface yields 2 (a /a xay)zfy 2 (a /azay)Z1_.‘y II C II C (2.11a) (2.11b) From vanishing of both tangential derivatives in (2.11) everywhere on the interface, Case (b): 3 = xe +A A Let =x2xty2y. get + A Case (c): J = sz +A A Let Z=zZzty2y. fo = 0 (i.e., rotation of coordinates), to arrive at Zfz = 0 12 (2.12) Implement the same conditions as in case (a) to (2.13a) (2.13b) Use the results of (b), with radial slab symmetry (2.14a) (a/ay)2fy = o (2.14b) * A A A Case (d): J = xetny+sz General boundary conditions for y=-t are secured by linearity; specific contributions to azfy/ay are summed, as are contributions (all zero) to fo and Zfz° The results, = O (2.15a) fo Zfz = O (2.15b) (a/ay)ny = o (2.15c) show a lack of coupling between normal and tangential potential compo- nents. 2.6 Integral Representation for Hertz Potential Integral representations for components of potential are created via the two-dimensional Fourier transform or, alternately, by conti- nuous superposition of plane waves with suitable spectral amplitudes. Care is taken not to transform on y, however, since substitutions into boundary conditions at y=0 and y=-t are imperative. Let F{ } and F'1{ } denote two-dimensional Fourier and inverse Fourier transform operations. The spatial/spatial-frequency domain variable pairs are (x,ux) and (z,uz), with the transforms defined as FM} = Q =ff q(x,y,z)exp[-j(uxx+uzz)]dxdz (2.16a) 13 2 F'1{Q} = q = (1/2n)d/J{ Q(ux,y,uz)exp[j(uxX+UzZ)]deduz (2.16b) It should be stressed that the ensuing development is valid only for sources in the cover medium. In a later chapter on the continuous spectrum, expressions for waves excited by sources in the film layer are needed; however, that development need not be as general as the present one, and is of more limited and specific interest. Study of sources in the film layer is therefore deferred to Chapter 5. Primary potential waves satisfy the inhomogeneous Helmholtz equa- tion 2; 22 _ * v cp+kc cp - -J/AC (2.17) where Ac=jwec. Primary waves are those excited in a hypothetical unbounded region having permittivity of the cover (see Figure 4). The solution of (2.17) proceeds by standard Green's function techniques, giving the principal Green's function: exp[jE-(;-;')]exp[-pc|y-y'|]d2L Gp(r,r') = (2.18) 2Aopc where Ao=(2n)2. Here p§(L)=L2-kg is a wavenumber parameter, E=§ux+9uz is a vector 2-d spatial frequency, and d2L=duxduZ. It is noted that (2.18) is just an integral representation for elementary, outward- traveling, spherical waves. By weighted superposition, the primary 14 Figure 4. Principal wave of Hertz potential. overall reflected wave primary wave overall transmitted wave ////////.// Figure 5. Transmitted and reflected potential waves. 15 potential wave is chm = (I/Ac)jvJ(r )Gp(r,r')dv' (2.19) Scattered (reflected or transmitted) potential waves are, by definition, source-free. A generic form can be derived to represent the scattered Hertz potentials. Their components (designated by sub- script i=x,y,z) satisfy 2 s 2 s - v Zia + k Zia - O (2.20) where a=c,f for cover and film regions. Substitute, using (2.16), for 2i: in terms of F'1{F{Zi:}}. Pass the x and z derivatives of the Laplacian through integrals and carry them out on the integrand; these operations are easy due to the integrand's exponential form. Then the Fourier transform theorem gives an ordinary differential (in the y- variable) equation for F{Zi:}, which has solutions 5 T s T Filiafldl} = NialLleprpay) (2.21) l where 2- 2 + - 2 2 ‘ pa-pa(L)-L -ka IS the transverse wavenumber parameter. The prototype solution for scattered waves of Hertz potential is: 5+ 5+ .++ 2 Zfi1(r) = (1/Ao) wia(L)exp(jL'r)exp[fpay]d L (2.22) Finally, requiring that traveling waves suffer decay, total 16 solutions for Hertz potential are formed. In the cover, [ J1. ]] eprjE- (F1? )JexpE-pc |y-y' I JdZLdV Zci = 'T"— AC v 2A0pc + (l/Ao)/f Nem-(E)exp(jE-;)exp[-pcy]d2L (2.23) where "cri(t) is the overall reflected—wave amplitude spectrum in the cover. In the film, 2.. = (l/Ao) l]exp(j-L';){Wfti(I)exp[pfy]+wfri(I)exp[-pfy]}d2L (2.24) where wfti(I) and wfri(I) are unknown amplitude spectra for waves transmitted and reflected in the film. Waves affiliated with these spectral coefficients are suggested by Figure 5. 2.7 Hertz Potential Green's Dyad Nine boundary conditions derived in section 2.5 provide for evaluation of the overall spectral amplitudes “W". Their application and subsequent algebraic reduction lead to concise statement [11] of results in Green's dyadic form. It is.convenient to express Hertz potentials in terms of a Green's dyadic G(F,F') composed of principal and reflected parts: ' .. ‘3’}:- _l_ ‘_- ...-3.001;.“ ..;L:_.;$:‘.-.' {C = my] §(F,F')-3(?')dv' (2.25) V E(F,F') = a (F?) +6 (E?) (2.26) Principal Green's dyad GP is simply the scalar principal Green's function multiplied by the unit dyad I=29r99+92. Vector directions of primary potentials are therefore determined solely by orientations of source currents. The more intricate reflected Green's dyad Gr embo- dies cross-coupling effects produced by the cover/film interface: -++ , A A A.A A A Gr(r,r ) = xGrtx+QGrtzry[(a/ax)Grcx+Grny+(3/82)Grc9] (2.27) Subscripts t, n, and c stand for tangential, normal, and coupling, respectively; these signify how vector components of current produce other components of Hertzian potential. The reflected Green's dyad components are .+ + +| ' 2 G R eXp[jL°(r-r )]expE-p (y+y )]d L rt t c Grn =f] Rn (2.28) G C 2A p rc o c with reflection and coupling coefficients [11] as follows. Rt = At/Zh (2.29a) Rn = An/Ze (2.29b) 18 C = 2pC(Nfc'1)/Zezh where At = pC-pfcoth(pft) An = Nfcpc-pftanh(pft) Zh = pc+pfcoth(pft) Ze = Nfcpc+pftanh(pft) and 2 _ 2 2_ 2 pC uX'HJZ kC p: = (2.29c) (2.30a) (2.30b) (2.30c) (2.30d) (2.31a) (2.31b) Note that Zh(L)=O and Ze(L)=O are, respectively, eigenvalue equa- tions for TE and TM surface-wave modes of the layered surround; solu- tions L=Lb of these characteristic equations are named background poles. 2.8 Reflected Green's Dyad for Equal Permittivities It is instructive to inspect the reflected Green's dyad in the situation where nf=nc. The refractive index ratio Nfc equals unity, 19 resulting in C=0 and Grc=o' The reflected Green's dyad reduces to Gr=§GrtQ+§Grn9+QGrt2 with pf=pc' With some algebraic simplification, it is shown to be proportional to the dyad -99t99-22. Moreover, a phase factor expE-cht] appears explicitly, giving the expected propa- gation delay as a function of the distance from source to conducting plane. In this special case, therefore, the reflected Green's dyad predicts classical equivalent image currents. The image is placed at a distance t below the conducting plane, with its normal (y) component in the same direction as the primary source, and tangential (x and 2) components reversed. 2.9 Green's Dyad Source-point Singularity A thorough investigation of the Green's function source-point singularity is worthwhile, as it brings out an interesting physical interpretation involving charges and currents. For completeness, salient points of the development for a surface-current source distri- bution are given in this section. Let currents, described by density R(;), be located on a surface S. S has outer boundary contour Co. Assume that R is a continuous function everywhere on S except at the boundary CD of patch Sp. Let G be the outward-directed unit normal vector from S, in the plane of S. First, consider the electric field associated with the principal Green's function only. Despite the double-integral formulation of equation (2.18), it is also known that Gp can be represented simply by Gp=exp(-jkCR)/4wR, + + O I I o I where R=|r-r'| IS the distance from source paint to observation p01nt; 20 this reveals the source-point singularity strength experienced when F=F'. Due to the singularity, integrals involving Gp must be done as improper integrals, taking care not to allow |F-;'|=0 to occur. To this end it is necessary to further exclude from S a limitingly small circle, of radius d, centered on the location F; call the excluded region Sd with boundary Cd, noting that G is directed into Sd. From equations (2.2a), (2.25) and (2.26), the principal electric field component is -) 2 +->+-> +->->-> E = (1/A ){k G (r,r')K(r')dS' + vv. G (r,r')K(r')dS'} P C C P P S S (2.32) where, in the limit as d approaches zero, S is composed of the two portions S={S-Sd-Sp}+Sp. It is immediately clear that, in (2.32), the first term is related to vector potential produced by current. The second term relates to scalar potential produced by electric charge; this is evident after passing the divergence operator through the spatial integral, with subsequent identification of the second term as the gradient of a scalar potential. Manipulation of the scalar poten- tial integrand yields v.[ep(F,?'))‘<’(?')] = ep(F,F') v-EG') +vep(?}')-E(;') = — v'ep(F,?')-E(F') = - v'.[cp(?,F')E(F')] + Gp(;,;') v'-E(F) (2.33) 21 The divergence theorem in primed coordinates now applies to the first term to give contour integrals of functions h'-R (clearly equivalent line charges) around the three contours Co, GP, and Cd. The second term, by the surface continuity equation, yields superpositions over surface charges on S. It is found, moreover, that as d approaches zero the contribution from the (artificially introduced) line charge at Cd vanishes. Treatment of the reflected component of electric field is initia- lly not as easy. This is due to the more complicated dyadic form of the reflected Green's function. From equation (2.27) with E=QKX+2KZ, the reflected scalar potential integrand becomes v-[Gr-R] = [lea/3x)Grt+Kz(a/az)Grt] + [Kx'a/BX)'3/ay)Grc+Kz'3/az)'3/3y)Grc] (2.34) The right-hand side can be written VtG °R, where ér=Grt+(3/ay'6rc and Vt is the transverse gradient operation involving only partial deriva- tives on x and 2. Referring to equation (2.33), this is of exactly the form required to carry through steps analogous to those for Gp. Besides offering a convenient physical picture of equivalent sources that produce electric fields, the source-point singularity development outlined here represents an alternative formulation for later problems in terms of unknown charges and currents (instead of just unknown currents). Removing derivatives from the Green's fun- ction and placing them onto currents is a highly recommended method of 22 avoiding increasing the Green's function singularity; more is said of this, in connection with integration by parts, in Chapter 4. 2.10 Conclusion Electromagnetic fields, maintained by currents over layered con- ductor/film media, are given by integro-differential operations on the Hertz potential Green's dyad developed in this chapter. The Green's dyad formulation accounts rigorously for layered media effects. The EM fields are, in addition, subject to further boundary conditions imposed by any conducting devices resting over the film. In the following chapter, such boundary conditions are carried out for the infinite microstrip line, resulting in an integral equation for un- known surface currents on the strip. 23 CHAPTER III PROPER MODE SPECTRUM FOR MICROSTRIP LINE 3.1 Introduction Propagation modes of axially-uniform microstrip lines are quanti- fied based upon a general electric field integral equation (IE). IE construction proceeds through the Hertz potential Green's dyad, equa- tion (2.26), developed in Chapter 2. A subsequent complex plane analysis reveals naturally the proper spectrum for microstrip modes, including the discrete and continuous spectra. 3.2 General Integral Equation for Electric Field Suppose microstrip excitation is provided through an impressed electric field E' maintained by primary current source 3e. The impre- ssed field induces a system of surface currents, described by density N, on conducting strip lines. Induced currents maintain a scattered field Es. See Figure 6. At any point E, the total field is given by superposition: E=Ei+Es. The boundary condition at a conductor, EOE=O, implies E°ES(;)=-€°E'(;) at the microstrip surface; this is the basis for constructing an electric field integral equation (EFIE) ++ for unknown surface currents K(r). Assuming time-harmonic excitation, ta: + We] §(?.F')-E(F')ds' = -AC’t-E'(F) (3.1) S 24 /// / / /.// Figure 6. Impressed and scattered fields. 25 Particularily well-suited to moment method solutions, this EFIE is the mathematical foundation for all tapics addressed in this thesis. It is Fourier-transformed in the next section. 3.3 General EFIE in the Fourier Transform Domain For axially-invariant microstrip systems, A 2 t . ‘ -++ ++ A+i+ t-{kc + vv-} dl dz'GLp,p';z-z')-K(p',z') = -Act~E (p,z) C (3.2) + where p is the 2-d transverse (x-y plane) position vector, and C denotes periphery contours of microstrip conductors. Take a Fourier transform on z, noting the axial (z) integral is purely convolutional, to arrive at a two dimensional EFIE: in: _+ 56.}. c 3(;,Z')-k’(3';uz)di' = -Ac’€-3'(Z.uz) (3.3) where U2 is the transform variable for z. Transformed quantities in (3.3) are stated in lower case letters; furthermore, it is understood that the differentiation theorem has mapped partial derivatives with respect to z into multiplications by juz (new symbol 6). Scalar components of the principal and reflected Green's dyadics transform to Jf’expLjux(x-x')]expf-PCIY‘Y'Ildux g _ 41rpc (3.4) 26 (3.5) 9“ I (St) expEjux(x-x')]expE-pc(y+y' )]duX grn n grc C 41rpc These expressions are useful for applications in Chapters 4, 5, and 6. 3.4 Identification of Propagation-mode Spectrum The propagation-mode Spectrum is identified from inverse trans- forms of solutions to the 2-d EFIE for R, equation (3.3). Inverse Fourier transforming, ++ _1+ ++ . K(r) = F {k} = (1/2W) k(p,uz)exp(juzz)duZ (3.6) Consider wavenumber parameters pc and pf, found by taking square roots of p2 and p%; they are given generically by = 2 2_ 2 1/2 = 2 2 1/2 P (Ux+uZ k ) (ux+Y ) (3.7) where v = (ug-kzll/z = [(uz-k)(uz+k)]1/2 (3.8) Care is taken to choose correct branch cuts for these potentially multivalued functions 0f 02 so decaying wave solutions are generated, i.e., SO Reipc}>0 and Re{pf}>O. As pc and pf exist within an integra- tion spanning all "X, ux=0 will occur; this imposes Re{ih}>0 and R6{Yf}>0. The square roots Tc and vf have branch point singularities at uz=tkc and uz=tkf. However, all functions of pf in the integral 27 representation for the Green's dyad are even functions; see equation (2.30). Consequently branch points at fkf are not implicated; only those at fkc are relevant. Considering yc in detail, 2 Yc = (uzr' 2 2 u -k 2+k ?)+j2(u u zi cr c1 zr zi'k k ' (3'9) cr ci where subscripts r and i denote real and imaginary parts. It is understood that kci=Im{kc}O implies largfvc}|<900, or larngE}|<180°, and there- fore the boundary line for the proper Riemann sheet is the negative real axis defined by Im{vg}=0 and Re{YE} ax1s (where uz1 kc1 zr cr sentation of the suitably cut uz-plane, specialized to the low-loss limit. The singularities of the transform-domain surface current are: (l) branch points at fkc through dependence on the Green's dyad, and (2) surface-wave poles at fuzm; m=1,2,...,p,..., M. With appropriate hyperbolic branch cuts along Cb’ the real-axis Fourier inversion integral is deformed into the complex uZ plane and subjected to Cauchy's Integral Theorem [16]. Let semicircles of infinite radius in the upper and lower half-plane provide alternatives for closure of the 28 . r_;. 33‘: ‘5'. ‘m‘amt * ‘ ‘ zi Figure 7. Branch cuts in complex uZ plane. Figure 8. Integration contour closure. 29 integration contour. See Figure 8. No contribution from the chosen infinite semicircle should be made. From J/. I exp(juzz)duz =./' k exp(-uziz)exp(juzrz)duz (3.11) C C the correct selection of closing contour is to choose: (1) upper half plane closure if z>z', and (2) lower half plane closure if z __ z-(g-k) - '9p+9rt)kz (4.4b) Coupled IE's for components of surface current, each holding for x in (-w,w), are then: 2 w . w= + . kc ‘1'-" (9p+9rt)kxdx + (a/ax) v- -w g-kdx = O (4.5a) 2 w . - w... + ' kC .l'-w 'gp+9rt)kzdx + JUZ v-J{ -w gokdx = O (4.5b) where, again, evaluation at y=y'=0 is understood. In working with equations (4.5a) and (4.5b), the immediate 34 interchange of derivative and integral operations is tempting. How- ever, such exchanges are often associated with problems of integral convergence; this occurs particularly in the second term of (4.5b). Interchanges are therefore done with caution, and are avoided in the case of (4.5b). 4.4 Dominant Axial Current Approximation Expectedly [3], |kx|<<|kz|. In this approximation both IE's cannot be satisfied, so discard the equation arising from the ex boundary condition. The remaining IE, equation (4.5b), with = A. A 9.x = [yauzgrculgpmrtnkz (4.6) is enforced at yay'=0 for all x in (-w,w). Passing the divergence operator through the integral, V°9°k = Juztagrclay + (gp+grt)]kz (4.7) The EFIE is " 2 2 Jf-w ['Yc(9p+9rt)‘"z(3grc/3Y)]kzdx = 0 (4.8) where-y§=u§-k§. Further simplification proceeds via symmetry. Write the LHS integral as a sum of two integrations: one spanning (-w,0), the other (O,w). Then make a change of variable u=-x' in the first integral. By physical symmetry kz(x) must be either even or odd about 35 the microstrip centerline, prompting decomposition of kZ(x) into even and odd modes: kz(x) for even modes kz(-x) = Skz(x) = (4.9) -kz(x) for odd modes where S=1 for even modes and S=-1 for odd modes. The EFIE is now enforced over the right half of the strip, (O,w): ['[A 2 J { 'Y 9 +9 )-u 39 /8y 5 O c p rt 2 rc (x' replaced by -x') + [-Y§(gp+grt)-u§ang/3y]}kz(x')dx' = 0 (4.10) with y=y'=0 understood subsequent to differentiation. The EFIE is now written compactly: w J{' [T(x,x')+ST(x,-x')]kz(x')dx' = 0 (4.11) 0 where _a I 2 2 - Yc'ngQrt)+uz(39rc/3y'|(y=y.=o) (1/4wd jfexpEjux(x-x')]{v§[(1+Rt)/pc]-u§C}dux (4.12) 36 Let - 2 2 _ 2 2 U — [Yc(1+Rt)/pc]-uzC - 2[vc-uzpc(Nfc-1)/Ze]/Zh (4.13) and dr0p multiplicative constants to get w Jfg'Jf U{exp[jux(x-x')]+Sexp[jux(x+x')}duxkz(x')dx' = 0 (4.14) 4.4.1 Solution by Pulse Galerkin's Method In preparation for moment method [17] solution, rewrite the EFIE slightly: w Jli ‘jLUexp(juxx){exp(-juxx')+Sexp(juxx')}duxkz(x')dx' = O (4.15) 0 Subdivide (0,w) into N partitions, each of width 2h, and let the center point of the n'th partition be x The n'th partition is n' (xn-h, xn+h). A set of N nonoverlapping pulse functions, defined by 1 . . . if (xn-h)IYiI’ arg{p} is essentially zero. For qurI|0|'/2 pc = +j[0-u§]1/2 if |ux|<|Q|1/2, uz=uzl (5.10) . 2 1/2 . 1/2 - -J[Q-Ux] 1f |ux|<|Q| , uz-u22 - 2 2 where Q-kc-uz. 5.6 Moment Method Solution of Spectral EFIE EFIE numerical solution takes place pointwise along the uz-plane branch cut. Explicitly, the equation to be solved is: w . [ [UexpEjux(x-x')]kzduxdx' = -211jkc’z\oe'(x,0,uz) (5.11) -w where U = [-Y2+u2p (N -1)/z ]/z (5 12) c z c fc e h ' In general it is hard to solve Fredholm equations of the first kind, such as (5.11). Since the first-kind equation doesn't lead to itera- tive solutions, moment method is exploited here to gain preliminary results. Galerkin's method with pulse functions for expansion and testing (see section 4.4.1) gives 66 2 . _ §KZPIUT exp[.]ux(xm-xn)]duX - juzexp(-juzzm) /” cheXp[j0x(xm-xm)] dux (5.13) 2Nfc prchpf+pccoth(pft)] where T=[sin(uxh)]/ux. 5.7 Results Figure 28 shows solutions of (5.13) for various values of parame- ter "2' The equation is solved for the strip dimensions and physical parameters as indicated in the figure; for simplicity the excitation monopole is located at (xm=0,zm=0), the geometrical strip center. These functions are individual contributions to the current of the micro- strip continuous spectrum. Note the behavior of the spectral components as uZ moves out on the branch cut: the currents increase, pass what appears to be reso- nant level, and finally begin to oscillate while decreasing in ampli- tude. Indeed, examining the uZ dependence of each term in equation (5.13) does lead one to expect the amplitude decrease. The resonant phenomenon is probably associated with the combined effects of incident and reflected electric fields. At values of U2 well past the "resonance", the impressed field is primarily responsible for driving the currents; therefore, it is expected that for large uz the currents will oscillate in a pattern resembling e'. In addition to the U2 values given in Figure 28, which all fall on the imaginary-axis part of the branch cut, u values along the z real-axis section of the cut were investigated. The continuous 67 spectrum functions associated with the latter uz are found to resemble curves 1 and 2 of Figure 28. These additional curves are omitted from the graph to avoid confusion. Finally, one senses a disadvantage of moment method solution to this problem: very fine partitioning would be needed to continue this solution process ad infinitum. Perhaps, for future work, iterative methods of solution (for example [22]) are worth investigating. 5.8 Conclusion Microstrip modes belonging to the continuous spectrum are quanti- fied in this chapter as forced solutions of the transform-domain EFIE. The problem is solved for fields impressed by vertical currents under the strip; furthermore, transverse currents are neglected for simpli- city. Numerical results, gained by the moment method, are presented. It is important to note that the functions shown do not represent the final solution to the continuous spectrum problem; rather, the solution process is just beginning. Once a feasibly quick method of solution for equation (5.8) is developed, it remains to substitute the spectral components into (3.12). 68 io o T LAMBDA-0.05. w LAMBDA-o.18 f=1.5._Nc-1.0. N =10 'o o 1 Uz=0+j2 2 Uz-O+j4 3 Uz=0+j6 4 Uz=0+17 5 Ust+19 6 Uz=0+110 .0 00 o .0 o o NORMALIZED CURRENT AMPLITUDE O O C 8 8 8 OIIIIIJLALLIIAAIIIALIIIJLII114411 N (k. a) 1F I.” [mlTIITI—IIITTUYVT'] .0 0.2 0.4 0.6 0.8 1.0 NORMALIZED x COORDINATE (X/W) Figure 28. Solutions of the forced EFIE. 69 CHAPTER VI COUPLING BETWEEN ADJACENT MICROSTRIP LINES 6.1 Introduction Coupling phenomena are of great importance in the design of high- speed integrated circuits. These effects are undesirable in some circuits (e.g. crosstalk in communication hardware); in others the design itself depends on their existence (e.g. microwave couplers). Microstrip coupling problems are treated by various approximate methods in the literature. For example, [23] uses a "directly-coupled parallel-plate ideal waveguide model" with quasi-static parameters. The so-called LSE model of [24] is another quasi-TEM approach. The transform-domain EFIE provides a basis for studying COUpling phenomena in axially-uniform multi-strip systems. The problem is formulated in detail here, and preliminary numerical results are obtained for a simple case. 6.2 Geometry of Multi-Strip System Consider N parallel microstrip lines as shown in Figure 29. Let Cn denote the n'th microstrip periphery contour, and In the current density on the n'th strip. Unit vector Em is tangent everywhere to the m'th strip surface. As in previous chapters, the strips are integrated over a dielectric film layer of thickness t. 70 Figure 29. Parallel microstrips. 71 6.3 Transform-domain EFIE for Propagation Modes Formulation of this problem parallels earlier chapters. However, the metallic cross-section over which the EFIE is enforced must be collectively the set of strip cross-sections. Moreover, the electric Green's dyad 3e proves valuable in formulation and early theoretical stages of the present problem. Based on the tangential electric field boundary condition, formulate the EFIE as the system N . A = .+ ?I-.-+-. +' I = .. A 0+1 9 THE c_ge(p,p 1 k.n(p.. .0210) Actm e (p.02) (6.1) n. which holds for all 3 on cm; m=1,2,3,.‘..,N. Again, note Ac=jmec. System propagation modes are characterized by coupled currents sharing uz-plane simple polesl Furthermore, it is important to note that propagation-mode coupling becomes significant only when the iso- lated microstrip mode prOpagation constants are nearly equal. By arguments analogous to those for the isolated microstrip, when U2 is the propagation-mode eigenvalue associated with the simple-pole uz-plane singularity, the natural eigenmode currents satisfy a system of homogeneous transform-domain integral equations: A N ‘ = + ->. + +4 . - tm-Z ge(p-,p )-'kn(p )dl - 0 (6.2) n-1 Cn holding for all E’on Cm; m=1,2,3,...,N. Note that the electric Green's dyad depends implicitly upon "2' 72 6.4 Testing Operation In equation (6.2), in place of the simple pre-dot Operation with A tm, apply the testing operator -> + f C dl (mow) III for m=1,...,N. Embedded in this operator is EmO’ the prOpagation-mode current of the m'th strip (when isolated from other strips). Since EEO is tangent everywhere to strip surfaces, this testing operation is a suitable replacement for the original Operation; its motivation becomes apparent in the following section. After testing, + + N __, ++ + + f .11 km0(p)° z: ge(p,p')'kn(p’)dl' = 0 (6.3) Cm n=1 Cn for m=1,...,N. 6.5 Approximations for Loose Couplipg_ In weak, nearly-degenerate coupling, propagation constant 02 remains close to the (approximately equal) values uzno of the isolated strips. For the electric Green's dyad, take the leading two terms in a.Taylor's series expansion about uznO' 9e = 9e u + (uz'uzn0"d9e/duz' u = ge0+(uz"'zn0)9e0 (6‘4) zn0 znO Substitute into the EFIE system, (6.3), and exploit the reciprocal 73 property [25] of the electric Green's dyad. Retain only leading, non- vanishing terms: C"I Cm - gjcn d1 k n(10' ) [m 390(1) .16) km0(p)dl = o (6.5) where the defining EFIE for isolated eigenmode current Rm0.has-been exploited, and the summation now excludes n=m. As a perturbation approximation, assume kn =a nknO where an is a constant. That is, the functional form of this approximation resem- bles an isolated strip current: only the amplitude is allowed to differ. Again invoking Green's dyad reciprocity gives amc mn""z um0)+ Zcmna n = 0; m=1,...,N (6.6) where " _, 1+ +.=I.+TI ,+ (4;!) (07) cm - [ch [Cdl km0(.p)«-ge.o(.p,p- )4kmo . . a m m - + + = ++. + +1 ' cmn - d1 km0(p)o . 9eO(P’P‘)°knO'p )dl (6.7b) Cm . Cn and the summation over n excludes n=m. Equation (6.6) is an algebraic system for the unknown uz. Coefficients (6.7) dictate the location of 74 u relative to the nearly identical values of u z sz' 6.6 Illustration For Two Strips Letting N=2 and solving algebraic system (6.6) for U2 gives UZ = 01:02 (6.8) where Ql=(uzlo+U220)/2 is the average of the isolated strip propaga- tion constants and 02 = [Q§+Q§Jl/2 (6.9a) Q3 = (”210'u220)/2 (5'9”) Q4 = (C12C21I/(011622I (6.9CI Thus two shifted, coupled-mode eigenvalues result when N=2; they are symmetrically placed about the average of the isolated strip values. 6.7 Overlap Integral For Coupling Coefficient Equation (6.70) is converted to a more useful form by recogni- zing = ++ + +' + + “/Ic 9e0(p,p')-kho(p )01' = Acen0(p) (6.8) 11 where gnO is the electric field maintained everywhere by surface currents on the n'th strip. Substituting into (6.70), 75 + + + + c“m = Ac [c km0(.p)oen~0(.p)dl (6.9) m . which is explicitly an overlap integral, providing a physical picture in which currents on the m'th strip link with fields of the n'th strip. Equation (6.9) is also computationally better than (6.7b), because enO is easily written in terms of the Hertz potential Green's dyad: 3n0(;) = (l/ACHI‘CZ: + 65-} fC§(;.;")°Eno(-;')dl' , (6.10) n The overlap integral formulation is physically significant be— cause, referring to equations (6.6) and (6.7), the amount of oVerlap bears directly upon the strength of coupling between any two micro- strip lines. The remainder of this chapter is therefore devoted to overlap integrals for the fundamental modes of thin and narrow strips. 6.8 Overlap Integral For Thin, Narrow Strips Let x=pi be the center location of the i'th strip; the strip extends over (pi'wi’pi+wi)° Invoke the dominant axial current appro- ximation by neglecting transverse currents on narrow strips. Then for thin microstrip lines, (6.9) becomes ( pm+wm1 Cmn = AC 'l' km0(x)[2-3ho(x)]dx (6.11a) (Pm-wm) 76 where (pn+wn) 2.3n01x) = (172.0%) kn0(x') -Uexp[ux(x-x')]duxdx' (p -w ) " " (6.11b) and u = [-Y2+u2p (N -1)/z 1/2 (6 11c) c z c fc e h ° Note that equation (6.11c) has uz=uzn0' To approximate fundamental mode coupling, let the current on the i'th strip be represented by the leading term of equation (4.31a). That is, take 2 1 2 1.0(x) = ai/{l-[(x-pi)/wi] } / (6.12) for x in (pi'wi’pi+"i" and substitute into (6.11a) and (6.110). Some preliminary numerical results, obtained under the approxima- tions of this section, are given next. 6.9 Results and Conclusion Figure 30 depicts an example of overlap integral behavior vs. separation between two parallel microstrips. The overlap integral is (done numerically by Simpson's rule. Physical parameters chosen for 77 the example are given in the text of the figure. A best-fit exponen- tial curve appears along with the data points. Coupling decreases rapidly with increasing distance in this case. These preliminary results are given for illustration; obviously many other cases await treatment (e.g., effect of changing strip and film dimensions, effect of altering medium parameters, effect of including more terms in the current expansion, and coupling between higher-order modes). 78 Exponential Best Fit: y = 2.93 exp(-27.6x) Nf=1.5. Nc=1.0 T/LAMBDA=0.05 W1 /LAMBDA=0.05 w2/LAMBDA=0.0S 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 SEPARATION BETWEEN STRIP CENTERS (WAVELENGTHS) Figure 30. Overlap integral vs. strip separation. 79 CHAPTER VII CONCLUSION AND RECOMMENDATION Electromagnetic waves and their interactions in integrated cir- cuits form a challenging area for investigation. The EFIE approach departs from other, less cogent methods of approximate analysis; it is appealing due to its simplicity and conceptual exactness. It predicts the existence of both discrete and continuous microstrip spectra, and permits their numerical calculation. In addition, it facilitates the study of important electromagnetic coupling phenomena. Nevertheless, because of the high computational demands of even simple problems, it is doubtful that the EFIE will soon supplant faster methods for design work. Chapter 2 of this dissertation outlines in somewhat general form the electromagnetic theory relevant to high-speed integrated cichits. This formulation applies not only to infinite strips, but also to various patch antennas and integrated devices composed of finite strips. The successful application of this theory to the circular patch, with initial emphasis on the low-order resonant modes, is currently underway . The rigorous identification of discrete and continuous microstrip (eigenvalue spectra via complex plane analysis, as presented in Chapter .3, is believed to be new. Some observations made in that chapter are (probably fundamental to knowledge of the microstrip as an open- 80 boundary waveguiding structure. That chapter also provides a crucial link between the theory and its application, viz., that of associating the homogeneous IE with discrete modes and the forced IE with the continuous spectrum. Moment methods, attempted with various expansion functions, per- form well in solving the homogeneous microstrip IE of Chapter 4. Fundamental mode eigenvalues match those published by other authors, thereby providing solid validation for the EFIE approach. Higher- order mode solutions are easily generated once their determinantal zeros are located; however, these zeros are found only by somewhat fortuitous searching. Certainly, with continued effort, more higher- order modes will be located; research should be encouraged to catalog these modes completely. The continuous spectrum is a topic on which, to date, only sparse information is available. Chapter 5 brings forward some preliminary results for consideration. Forthcoming research, perhaps using combi- nations of different numerical methods, will perforce shed much light on this new area. Coupling between parallel microstrips is addressed in Chapter 6. Of course, much more work remains in this area. Other topics of interest (besides those already mentioned in section 6.9) include COUpling between nonparallel strips, and coupling from strips to adjacent printed antennas. 81 APPENDICES APPENDIX A APPENDIX A ACCURATE EVALUATION OF SOMMERFELD INTEGRALS USING THE FAST FOURIER TRANSFORM An approach to real-axis numerical evaluation of Sommerfeld-type [26] integrals is advanced in this appendix. The method combines the Fast Fourier Transform (FFT) algorithm with Simpson's rule. Refe- rences such as [27,28,29] discuss the FFT for periodic functions only. A principal advantage is that a single FFT call gives many sample values of an integral with respect to an integrand parameter. Such samples could be immediate goals, as when the parameter is a spatial coordinate variable. In other cases, the Sommerfeld integral is a Green's function occurring in a spatial integral: here, too, sample values are desired. To estimate an integral such as b f ...... a partition [a,b] into N subintervals [ti’ti+1] with ti=a+hi, i=0,1,2,...,N-1. The step size h is (b-a)/N. Figure 31 outlines integration rules [30], provided f(t) is sufficiently smooth. Rule names, corresponding to numbers in the figure, are: (1) rectangular, (2) trapezoidal, (3) corrected trapezoidal, and (4) Simpson's. (Note: N is even for Simpson's rule.) Observe that rectangular and corrected 82 Integration Rule Error N-I - 1. h 2 {(61) h(b-a)f'(n)/2 1:0 N-I 2. h [MEI-«s 2 f(t1)] -h2(b-a)f"(n)/12 2 i=1 11-1 2 - 3. h [Mi—£0314 121 mg] + 87915.) - rm] h“(b-a)1'(”)(n)/720 3:130!) + «(61) + 22(12) + . . . + {(16)} _h4(b.a)f(1V)(n),lso .5 Figure 31. Numerical integration rules. 83 trapezoidal rules are identical for integrands periodic on [a,b], since f(a)=f(b) and f'(a)=f'(b). Therefore, a rectangular rule ap- proach to integrating periodic functions really gives corrected trape- zoidal rule accuracy. The FFT algorithm computes the discrete Fourier transform (DFT) 11-1 Sk+1= "2:30 An+1exp(j21rkn/N); k=0,1,2,...,N-1 of the sequence A1,...,AN; this is viewed as numerical integration of T I(x) = [0 g(t)exp(jxt)dt as follows. A change of variables transforms the integration limits to span [0,210. Partition this new interval into N equal segments. By rectangular rule, 11-1 I(xk) = h 2: g(nh)exp(j21rnk/N). n=O where h=T/N and xk=20k/T. That is I(xk)=hSk+1, where An+1=g(nh). Observe again that for g(t) periodic on [0,T] the above scheme is corrected trapezoidal rule; however, if g(t) is aperiodic the method is only rectangular rule. In this instance the simple step of letting A1=g(0)+g(hN), A2=4g(h), A3=29(2h), A4=4g(3h), ..., conforms to Simpson's rule within a factor of h/3. 84 To show the improvement, consider the following example. Let g(t)=exp(-t2/2), which is proportional to its own transform. Truncation at T=20 is adequate with N=210. Direct FFT with unweighted integrand sample values (i.e., rectangular rule) produces an answer correct to one significant figure. The Simpson's rule scheme gives 13 digit accuracy according to [31]. The error introduced by truncating an improper integral having infinite upper limit is bounded by proper choice of T, the stopping point of numerical integration. Assuming g(t) decays monotonically for large t, the truncation error is an alternating series due to the oscillatory integrand. Then it is possible to choose a minimum T to limit truncation error, since replacement of an alternating series by its n'th partial sum causes an error less than the (n+1)'th contribu- tion to the sum. To this end, take T at a zero-crossing of the oscillating integrand. The error of a truncated integral, or resolution error, is asses- sed via error terms from Figure 31. For the specific integrand studied, these are proportional to powers of (Tx/N). Therefore, as higher order methods are used, N is chosen carefully in relation to x and T (the Nyquist sampling rate is one guide). Another way is to use a Rhomberg-type integration, doubling the partition number (each step reusing 01d computations) until convergence is met. A single call to FFT gives values for the next iteration in Rhomberg's method simulta- neously for all sample points {xk}. As a relevant example suppose the Sommerfeld-type integral 85 103,9) = fexnIa‘xt);;p(-plyl)dt 2=t2+92 where p , is needed at 2M evenly-spaced values xk in [0,2W]. This example is prompted by application of the point-matching Galerkin's moment method to a microstrip problem: viz., evaluation of the 2-d principal Green's function. Let T=(20M)L, where L is an integer large enough for good approximation. Taking the real part of the integral changes the lower limit to zero. By its definition, xk=2nk/T=k/LM; k=O,1,2,...,N-1. Choose N for desired resolution, and recover every L'th evaluation by the FFT. Results from this technique are identical to those obtained by other methods (e.g. adaptive Simpson's rule); moreover, the FFT/Simpson's rule method is much faster since a single FFT call returns a needed sequence of integrals for {xk}. Some algorithms already using the FFT might benefit from this slight modification. It also holds promise for evaluation of Sommerfeld-type integrals. 86 APPENDIX B APPENDIX B SURFACE-WAVE CONTRIBUTIONS T0 REFLECTED HERTZ POTENTIAL Integral representations of reflected Green's dyad components, in the axial transform domain, take the form [[Wi(L)/41rpc]exp[jux(x-x )]exp[-pc(y+y )]dux where Ni(L) is a generic weighting factor. Note Re{pc}>0 is needed for waves decaying in the cover (y>0). Since "i and pc are even functions of Ux, the integration interval is halved by symmetry; an integration exclusively over Re{ux}>0 implicates only right-half ux plane poles. ‘Rational function W1(L) carries TE and/or TM surface-wave poles; i.e., “f(L)=Ai(L)/Zi(L) where Zi(L)=O are characteristic equations for even TM or odd TE modes of the symmetric-slab background. Let L=Lb be surface-wave eigenvalues, i.e., solutions of Zi(L)=O. Then L2=u2+u2 locates corresponding ux-plane poles, where u is the propagation z eigenvalue leading to nontrivial solutions of the homogeneous EFIE. It is desired to have ux as an analytic function of Lb and "2' Since 2= 2 2=_ 2_ 2 = . 1/2 ux L -U2 ((12 L ), ux tJ[(uz-L)(UZ+L)] . As only RHP poles are implicated, a criterion for choice of proper branch cut and algebraic sign is Re{uxb}=0. This means |arg(ux)|<90°. or larg(u§)|<180°, 2 defining a uX plane region for ux bounded by the negative real axis. 87 It is necessary to approach the lossless case from a low-loss limit standpoint; wavenumbers (and related quantities such as Lb) are then complex: br Lbi uzr+ uzi'+2".'L brLbiu zr uzi' The criteria to meet are 2 _ - Im{uxb} I 2'LbrLbiu zr Uzi) I 0 2 _ 2 2 2 2 Re{uxb} ' (Lbr'Lbi uzr+uzi' < 0 The first is the hyperbolic branch cut uzi=LbrLbi/uzr' The second indicates prOper hyperbola branches as those asymptotic to the real axis. Define (uz-Lb)=r+exp(ja+) and (uz+Lb)=r'exp(ja') as uz-plane phasors directed from be, respectively, to u See Figure 32. Then 2' ux=fj(rIr')exp[j(a++a')/2]. Since Lbi is small in the parameter regime of interest, take luzil small with U2" near Lbr' So a" is practically 0°, as is seen graphically, while -360°0. Analytical pole contributions to an otherwise numerical integration are of the form = lim]: [f /Z-X(L)]du d->0 where f=duxbr+d and e=uxbr'd' By Taylor series expansion, Zi(L) near 88 the pole is approximately 21(L'=Zi'Lb'+'ux -u x1bIZ. (uxb), where the first term vanishes by definition. Assuming the rest of the integrand is continuous at "xb’ f D = [f (u be/Z' (u xb)] 111110 [e du xx/(u "'be But f f f du xx/(u '“xb' = I do xx/[(u 'uxbr' 'juxbi] e e ln{exp[j[fn - 2tan'1(uxbi/d)]]} ijn - 2tan'1(uxbi/d)] where qubiI is understood to be small. Proper choice of algebraic sign in the last expression depends on the sign of uxbi’ and precludes crossing the branch cut Of the ln function (negative real axis). The argument of the bracketed quantity must fall between -1800 and 180°. For uxbi0’ choose the minus sign. 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