# Extended spatial domain and hybrid time-domain analyses for microwave circuits

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U n iv e r s it y o f C a l if o r n ia Los Angeles EX TEN D ED SPATIAL DOM AIN A N D H YBRID TIM E-DOM AIN ANALYSES FOR MICROWAVE CIRCUITS A dissertation subm itted in partial satisfaction of the requirem ents for the degree Doctor of Philosophy in Electrical Engineering by D o n g so o Koh 1997 t i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9811450 UMI Microform 9811450 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Copyright by Dongsoo Koh 1997 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The dissertation of Dongsoo Koh is approved. Tony F.-C . Chan Behzad Razavi Yahya R ahm at-Sam ii Tatsuo Itoh, C om m ittee Chair University of California, Los Angeles 1997 u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To m y family and friends iii i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able of Contents 1 I n t r o d u c t io n .................................................................................................. 1 2 E x te n d e d sp a tia l d o m a in a n a l y s i s ................................................... 6 2.1 Spatial Green’s functions on the conductor-backed dielectric slab . 2.2 The m ixed-potential integral equation (M PIE) and the m ethod of moments (MoM) fo rm u la tio n s.......................................................... 3 7 21 2.3 Incorporation of lum ped resistors into the spatial domain analysis 36 2.4 Analysis of W ilkinson power divider and c o m b in e r...................... 40 H y b rid tim e -d o m a in a n a l y s i s ............................................................. 49 3.1 Finite difference tim e-dom ain m e t h o d ............................................... 50 3.2 Hybrid analysis using the FDTD and F E T D ...................................... 72 3.2.1 FETD f o r m u la tio n ....................................................................... 74 3.2.2 Hybridizing the FDTD analysis and the FETD analysis . . 79 3.2.3 Applications of the hybrid a n a l y s i s ......................................... 83 4 C o n c lu s io n ..................................................................................................... 97 A E v a lu a tio n o f e le m e n ta l m a trix [Ke\ for th e stiffn ess m a tr ix . . 100 B E v a lu a tio n o f e le m e n ta l m a trix [F/] for th e load v e c t o r ............. 109 R e f e r e n c e s .....................................................................................................................118 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L is t of F ig u r e s 2.1 Conductor-backed dielectric slab.............................................................. 2.2 The am plitude of normalized Green’s functions, G and G q. . . . 20 2.3 Geom etry for the triangular-patch-pair basis function....................... 22 2.4 Normalized area coordinates for a local coordinate system .............. 28 2.5 Geometric param eters associated with the line integral along A jS 7 for calculating Ai and B{............................................................................ 32 2.6 Sample points for seven quadrature rule................................................ 33 2.7 Voltage gap source applied onto the sth patch.................................... 35 2.8 New type of triangular patch pair for the lumped resistor connection. 36 2.9 Example of resistor connection................................................................. 2.10 Schematic diagram for a microstrip power combiner / divider. . . 36 38 2.11 Surface current on the T-junction with and w ithout resistor. . . . 39 2.12 The analyzed m icrostrip Wilkinson power divider stru ctu re. 41 2.13 New type of triangular patch pair for the matched excitation. ... . . 42 2.14 M atched excitation and load for the m icrostrip................................... 43 2.15 M agnitude of I x along the microstrip.................................................... 43 2.16 Scattering param eters of the Wilkinson power divider...................... 46 2.17 Current distribution on the Wilkinson power divider (P o rt 1 exci ta tio n )............................................................................................................. 47 2.18 C urrent distribution on the Wilkinson power divider (P o rt 2 exci ta tio n )............................................................................................................. v I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 ! 3.1 The FDTD analysis..................................................................................... 51 3.2 The FDTD unit cell..................................................................................... 53 3.3 Bondwire interconnect structure (3-D view)............................... 56 3.4 Bondwire interconnect structure (top and side view)............... 56 3.5 Frequency responses of a bondwire interconnect structure............... 57 3.6 Bondwire interconnect 58 3.7 M atching stub design for a bondwire interconnect structure. 3.8 R eturn losses of bondwire interconnects with different bondwire structure with matching stubs............. ... lengths............................................................................................................. 3.9 59 60 R eturn losses of bondwire interconnects with different bondwire heights............................................................................................................. 61 3.10 Microstrip-to-waveguide transition structure (3-D view)................... 62 3.11 Microstrip-to-waveguide transition structure (Side, top, and back view)................................................................................................................ 63 3.12 FDTD domain decom position................................................................... 65 3.13 Interface between the region I and the region II in the FD TD domain decomposition m ethod ................................................................. 66 3.14 R eturn loss of the microstrip-to-waveguide transition structure with 8 mil neck.............................................................................................. 68 3.15 R eturn loss of the microstrip-to-waveguide transition structure with 9 mil neck.............................................................................................. 69 3.16 The housing structure for the microstrip circuitry (3-D view). . . 70 3.17 The housing structure for the microstrip circuitry (Top view). . . 70 3.18 The spectrum of the housing stru ctu re................................................... 71 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.19 The prism elem ent....................................................................... 75 3.20 The FD TD and FET D interface region................................. 80 3.21 Com m unication between FD TD and FETD field............... 81 3.22 The regular brick elem ent......................................................... 82 3.23 The via hole grounded m icrostrip........................................... 83 3.24 FET D meshes for the via holes............................................... 85 3.25 FD TD staircasing model for the 0.6 mm diam eter via hole 86 3.26 |S 2 i| of the via hole grounded m icrostrip.............................. 87 3.27 IS2 1 I of the via hole grounded m icrostrip.............................. 88 3.28 The FD TD modeling for the two via holes.............................. 90 3.29 |52i| of the grounded m icrostrip w ith two via holes.............. 91 3.30 IS2 1 I of the via hole grounded m icrostrip................................. 92 3.31 The waveguide with an iris of finite thickness......................... 94 3.32 Normalized inductive iris susceptance vs. iris w idth............. 95 3.33 ISnl of the waveguide w ith an inductive iris........................... B .l Boundaries of the FETD dom ain............................................... permission of the copyright owner. Further reproduction prohibited without permission. L is t of T ables 2.1 Surface wave poles and residues (freq. = 30 GHz)............................... 21 2.2 Ai and B{ for complex image term s (freq. = 30GHz).......................... 21 2.3 Param eters for three point quadrature m ethod..................................... 30 2.4 Param eters for seven point quadrature rule........................................... 34 viii i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A cknow ledgm ents The author would like to express his sincere gratitude to his advisor Professor Tatsuo Itoh for his guidance, encouragement, and patience during this research. A debt of gratitude is owed to Professors Yahya Rahmat-Samii, Behzad Razavi, and Tony F.-C. Chan for serving in the committee. The author would like to th an k Professor Ruey-Beei Wu at N ational Taiwan University, who visited UCLA in 1994-95, for his helpful discussions. T he author is very grateful to Dr. Hong-bae Lee for his fruitful discussions during the course of this research. The author would like to thank Dr. Jon Gulick a t Hughes A ircraft Company, for his valuable advice. In addition, the author would like to express his appreciation to all the fellow students and visiting scholars in Professor Ito h ’s group, for their assistance and helpful discussions. The author also thanks TRW for partially funding this research. A sincere appreciation is given by author to his parents for their constant support and encouragement. T he author also would like to extend a special gratitude to Ms. Esther C. Roe and Mr. Kimin Cha for their sincere help when he started his graduate study in America. Finally, the author would like to thank his friends, Mr. George Kondylis and Ms. A nastasia Karaglani for their friendship during his graduate study in UCLA. ‘ ix i I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V it a 1967 Born, Seoul, Korea 1985-1989 B.S., Electronic Engineering Sogang University, Seoul, Korea 1989-1991 M.S., Electronic Engineering Sogang University, Seoul, Korea 1991-1992 Army, Korea 1993-1993 Member of technical staff Korea Telecom, Seoul, Korea 1993-present Graduate Student Researcher Electrical Engineering Departm ent University of California, Los Angeles 1995-present Teaching Assistant Electrical Engineering Departm ent University of California, Los Angeles P u b l ic a t io n s Dongsoo Koh, Ruey-Beei Wu, and T atsuo Itoh, “Spatial domain analysis of the full-wave effect of a lumped resistor in m icrostrip power combiners and dividers,” Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Asia-Pacific Microwave Conference Proceedings, pp. 52-55. Taejon, Korea, Oct. 1995. Dongsoo Koh, Ruey-Beei Wu, and Tatsuo Itoh, “A hybrid spatial domain analysis of th e W ilkinson power divider,” Proceedings o f the 26th European Microwave Conference, pp. 760-762, Prague, Czech Republic, Sept. 1996. Dongsoo Koh, Hong-bae Lee, Bijan Houshmand, and Tatsuo Itoh, “A hybrid analysis using FDTD and FETD for locally arbitrarily shaped structures,” Pro ceedings o f the 13th annual review o f progress in applied computational electro magnetics, pp. 119-124, Monterey, CA, Mar. 1997 Dongsoo Koh, Hong-bae Lee, and Tatsuo Itoh, “A hybrid full-wave analysis of via hole grounds using finite difference and finite elem ent time domain m ethods,” IEE E M T T -S International Microwave Sym posium , pp. 89-92, Denver, CO, June 1997. Juno Kim , Dongsoo Koh, and Tatsuo Itoh, “A novel broadband flip-chip inter connection,” accepted to be published in the 6th Topical Meeting on Electrical Performance o f Electronic Packaging, San Jose, CA, Oct. 1997. Hong-bae Lee, Dongsoo Koh, Tatsuo Itoh, Frank J. Villegas, and H. A. Hung “O ptim um shape design of matching stubs based on full-wave analysis,” accepted to be published in Asia-Pacific Microwave Conference Proceedings, Hong Kong, China, Dec. 1997. xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i i Dongsoo Koh, Hong-bae Lee, and Tatsuo Itoh, “A hybrid full-wave analysis of via hole grounds using finite difference and finite element time domain m ethods,” accepted to be published in IEEE Trans. Microwave Theory and Tech., Dec. 1997. xii i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fI A bstra ct of the D is s e r t a t io n EXTENDED SPATIAL DOM AIN AND H YBRID TIME-DOMAIN ANALYSES FOR MICROWAVE CIRCUITS by D on gsoo K oh Doctor of Philosophy in Electrical Engineering University of California, Los Angeles, 1997 Professor Tatsuo Itoh, Chair The development of integration technology for microwave circuits is based on accurate and efficient full-wave analysis and design tools. A full-wave spatial dom ain analysis is developed to analyze arbitrarily shaped planar microwave cir cuits. By introducing a lumped element incorporation scheme, the spatial domain analysis is extended to analyze the microwave circuits having lumped elements such as Wilkinson power dividers. In addition, the proposed matched load and excitation scheme replaces the conventional voltage gap source excitation m ethod in the spatial dom ain analysis. Three-dimensional general structures can be ana lyzed more efficiently using the finite difference time-domain method. Bondwire interconnect and microstrip-to-waveguide transition structures are studied using the finite difference time-domain method. Finally, a hybrid time-domain analysis using the finite difference and finite element time-domain methods is proposed to analyze three-dimensional locally detailed and curved structures. The validity of this hybrid m ethod is demonstrated using waveguide iris problems and via hole grounded microstrips. xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 1 Introduction The development of integration technology for microwave circuits such as mi crowave integrated circuits (MICs) and monolithic microwave integrated circuits (MMICs) has brought the same advantages as those of the low frequency inte grated circuits, such as improvement of system reliability, small volume, light weight, and mass productivity. This integration technology is based on pre dictable and precise analysis and design tools, because optim ization techniques such as trim m ing and tuning cannot be employed after fabrication [1, 2]. Since complex integrated circuits cannot avoid surface wave excitations, radiations, and couplings a t high frequency operations, full-wave analysis and design tools are necessary. Full-wave analyses solve the wave equations considering tim e varying elec trom agnetic fields with given boundary conditions [3]. A num ber of different m ethods are available for solving the wave equations, and they can be classified in many ways, for example, frequency domain or time domain analyses, or inte gral equation based or differential equation based formulations. The finite difference method (FDM) solves the differential forms of the wave equations directly [4] and the finite element method (FEM) solves the field solu tions using the variational forms of the wave equations [5]. The integral equations can be derived from the application of the vector Green’s theorem on the wave equations and the m ethod of moments (MoM) has been widely employed to solve 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the integral equations in electromagnetic problems [6]. The full-wave analysis m ethods mentioned above are based on the frequency dom ain, from which the steady-state responses can be obtained. W ith the rapid development of fast and large memory digital com puters, th e direct solution m ethods of the differentialform wave equations in the time domain have been introduced. Time-domain analyses give the transient response information as well as the wide-band fre quency response inform ation which can be obtained using the Fourier transform of the tim e dom ain responses [7]. The finite difference tim e-dom ain (FDTD) technique is one of the m ost extensively used m ethods due to its numerical effi ciency and sim plicity in the form ulation and im plem entation [8, 9]. The integral equation m ethod can be one of the m ost general and rigorous electrom agnetic analysis m ethods [10]. Usually, electric or magnetic current dis tributions on the microwave circuits are unknown param eters in the integral equa tions and the m ethod of moments (MoM) technique is used to solve the integral equations. T he kernel of the integral equation, Green’s function, can be modified to simlify the integral equation by taking account of the boundary conditions. Most of the microwave circuits in the MIC / MMIC are modeled using the planar layered structures. The Green’s functions on the planar layered structures are derived in closed-forms in the spectral domain [11]. The spatial Green’s functions can be obtained by taking inverse Hankel transformations of the spectral domain G reen’s functions, which become th e well-known Sommerfeld integrals [12, 13]. The spatial dom ain analysis needs to calculate the singular integrals when the field points correspond to the source points, but this m ethod can be applied to more arbitrarily shaped structures than the spectral dom ain m ethod [14]. In addition, the num ber of unknown param eters are generally smaller than those of FDM and FEM , because the unknown parameters are defined only on the surface of the microwave circuits, while those of FDM and FEM should be defined in the 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. entire com putational volume including the microwave circuits. Therefore, the spatial dom ain analysis can be one of most efficient full-wave analysis m ethods for arb itrarily shaped planar structures. An extended spatial domain m ethod is proposed to analyze the arbitrarily shaped planar microwave circuits having lumped elements. T his m ethod makes it possible to incorporate the lum ped element effects into the spatial dom ain analysis. The current trend in microwave circuits is making use of lum ped ele m ents [1]. W ith the development of photolithography and thin film techniques, the size of the lumped elements has been reduced to be used in the microwave frequency band. The spatial domain analysis in this dissertation is based on the m ixed-potential integral equation (M PIE) and employs triangular patch pair ba sis functions which are suitable for arb itrarily shaped structures [15]. This spatial dom ain analysis is extended to include lum ped element effects by introducing new triangular patch pair basis functions. In addition, the proposed m atched load and excitation scheme replaces the conventional voltage gap source excitation m ethod in the spatial domain m ethod. The new method is applied to analyze the W ilkin son power combiner and divider. The interconnect structures of different substrates like bondwires and the transition structures between two different guided-wave structures are neces sary for the integration of microwave circuits [16]. These structures are not planar structures any more. The integral equation formulations for the threedimensional structures become very complicated because all the nine elements of the dyadic Green’s functions need to be calculated. Finding G reen’s func tions is not an easy task as well because the Green’s functions cannot generally be expressed in closed forms [17]. T h e finite difference tim e-dom ain (FD TD ) m ethod and the finite element m ethod (FEM) can be effectively applied to the three-dim ensional structures [7, 18]. B oth methods are based on the differen- 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tial forms of wave equations and do not require Green’s functions. The finite difference tim e-dom ain (FDTD) m ethod using Super absorbing M ur’s 1st order absorbing boundary condition (ABC) is developed in this dissertation [19]. In addition to time dom ain responses, wide-band frequency responses can be ob tained using the tim e dom ain d a ta through the Fourier transform in the finite difference tim e-dom ain (FDTD) analysis [8]. A domain decom position m ethod for the finite difference tim e-dom ain (FDTD) method is also developed to save memory in case the com putational dom ain does not have a regular box shape. T he finite difference tim e-dom ain (FDTD) method has a num ber of advan tages in analyzing three-dim ensional microwave structures [20]. M aking use of the uniform mesh in the finite difference time-domain (FDTD) algorithm does not require any special mesh generation scheme and storage for the mesh. However the finite difference time-domain (FD TD ) analysis has difficulty in dealing with curved structures and needs very fine mesh in the entire com putational dom ain for locally detailed structures. Several methods have been developed to over come these difficulties, including th e FDTD algorithm in curvilinear coordinates [21]-[23], the discrete surface integral (DSI) method [24], the locally conformed FD TD algorithm [25], a hybrid F V T D /F D T D algorithm [26] and so on. Recently, a hybrid m ethod was developed to model the locally curved structures by incor porating the finite element m ethod (FEM ) into the finite difference tim e-dom ain (FD TD ) m ethod [27]. This m ethod utilizes the advantage of the m ore flexible FEM while retaining all the advantages of the FDTD method. However, the FD TD and FEM mesh m atching m ethod in the interface region introduces diffi culty in the mesh generation of the FEM region. This dissertation proposes a new finite difference time-domain (FD TD ) and finite element tim e-dom ain (FETD ) hybrid m ethod by introducing an interpolation scheme for com m unicating the FDTD field and the FETD field. In this method, one can avoid th e effort of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fitting the FET D mesh to the FDTD cells in the interface. T his hybrid m ethod is applied to the via hole grounded microstrips and the waveguide w ith an iris of finite thickness. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PT E R 2 Extended spatial dom ain analysis The extended spatial dom ain analysis is proposed to analyze the arbitrarily shaped planar microwave circuits having lum ped elements. This m ethod in troduces an incorporation scheme of the lum ped element circuit equations into the sp atial dom ain analysis formulated by the m ixed-potential integral equation (M PIE). The m ixed-potential integral equation (M PIE) is obtained by apply ing th e boundary conditions on the electric field integral representations based on vector and scalar potentials [14, 28]. T he m ixed-potential integral equation (M PIE) has an advantage over the standard electric field integral equation (EFIE) in th a t the M PIE Green’s function has weak singularities due to the introduction of th e surface charge distribution which is related to the surface current by a two-dimensional continuity equation [29]. T he approach of the extended spatial dom ain analysis for microstrip mi crowave circuits can be divided into four steps: 1. o b tain the spatial Green’s functions on the conductor-backed dielectric slab, 2. derive the mixed-potential integral equation (M PIE) of the current distri bution on the conductor by enforcing the boundary condition such th a t the to ta l tangential electric field is zero on the conductor surface, 3. express the current with suitable basis functions and apply the m ethod of m om ents (MoM) to obtain a m atrix equation of the expansion coefficients, 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. » 4. incorporate the lumped element effect into the spatial domain analysis by m odifying the system of equations based on circuit characteristics. 2.1 S p a tia l G reen ’s fu n ction s on th e con d uctor-b ack ed di ele c tr ic slab • (x,y,z) conductor Figure 2.1: Conductor-backed dielectric slab. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Spatial Green’s functions on the conductor-backed dielectric slab as shown in Fig. 2.1 can be obtained by taking inverse Hankel transform ations of the spec tral dom ain G reen’s functions, which become the well-known Sommerfeld inte grals [12, 13]. It is time-consuming to find the spatial G reen’s functions on the conductor-backed dielectric slab by directly evaluating the associated Sommer feld integrals. Recently, an efficient method based on the Sommerfeld identity and the P rony’s m ethod has been proposed to obtain the closed-form Green’s functions [30]. In this dissertation, these closed-form G reen’s functions are used to improve the efficiency of the numerical com putation. Instead of using the P rony’s m ethod, more systematic m atrix pencil m ethod [31, 32] is employed to get asym ptotic exponential functions of the spectral G reen’s functions. The G reen’s function of an electric dipole on the conductor-backed dielectric slab can be derived using the electric Hertzian potential fl [12, 13]. Using the electric H ertzian potential fl and the Lorentz gauge, the magnetic and electric field intensities can be expressed as H = juje0€rV x II, ( 2 . 1) E = (V V • + k2) ri, ( 2 .2 ) and the electric H ertzian potential 11 becomes the solution of the inhomogeneous vector Helmholtz equation, (V 2 + fc2) n = - ( j W 0er ) - ‘ / . (2.3) In free space, th e electric Hertzian potential n has the sam e direction as the excitation current density J . The electric Hertzian potential II in the inhomoge- 8 ' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. neous planar layered structure has only the vertical component for th e vertical current excitation. However, the horizontal current excites the H ertzian potential having the vertical component as well as the horizontal component. Since planar microstrip structures are the structures of interest, this dissertation focuses on the horizontal current excitation case. For the x directed current, J = xld x '6 (x )6 (y )6 (z — z1), (2.4) the Hertzian potential will have x and z components [12]: II = Uxx + n Zz. (2.5) By exploiting the homogeniety of the planar structure along the x an d y direc tions, two-dimensional Fourier transform defined as OO n= OO J J n e x p [ - j( k xx + kyy)]dxdy, — OO — OO can be applied to the vector Helmholtz equation (2.3) and the result is where ! i 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.6) ri k zi = sjk f - k l - fcjj, Im { k zi) < 0 , i = 0,1, (2.9) and k f = u 2ii0e0eri, I 0 = {jujtQer l ) lIdx'. ( 2 . 10 ) The Fourier-transformed Helmholtz equations (2.7) and (2.8) have the general solution which satisfies the radiation condition, expressed as The unknown coefficients in (2.11) and (2.12) can be found by applying the boundary conditions of electromagnetic fields along the interface between air and the dielectric substrate as well as the ground plane of the substrate. The calculated results are shown as e-jkzo\z-z'\ _ noz — I q e-jk zo(z+z') kx (K - 1)(1 - e~^4k:lh)e~:’kz0<‘z+z^ 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.13) nl2 = /0 kx (K - 1)(1 - e- l2k--lh)e -jk'-oz' [e>*slS + e- jk'-l(:+2h)] j K (kzl + kzQ)(k:l + K k z0)(l + f if ^ e - - ' 2*=i/l)(l - jf c jf f ilje-J2* ^ * ): (2.16) where AT=^. ^Or (2.17) Now the spatial vector H ertzian potential can be obtained using an inverse Fourier transform , OO OO H = ^ / j ^ exp \j(h xx + kyy)\dkxdky, (2.18) — OO — OO which can be transform ed into an inverse Hankel transform using the following procedure. By the substitutions, kx = —kpcos£ (2.19) ky = —kpsinti and x = p COS(f) ( 2 . 20 ) y = p sincf) (2.18) becomes 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n = / / S e - ^ coa^ k pdkpdi 4^2 —7T 0 i 00- r -i *■ = — / ft 27r 7 0 = — L [ e ~ JpkpCOS^ ~ ^ d f 27r J kpdkp —7T 1 °° ~ 2tt J 11 o 1 00 — ^ fi H ^ \ k pp)kpdkp. ( 2 . 21 ) W ith the relationship between the vector potential .4 and the vector Hertzian potential ft, A = juje0erHo n , ( 2 . 22 ) the x component of the vector potential due to the x-directed electric dipole can be obtained using (2.13) and (2.21) as A xx = ou Afo f Id x ' Air j 2 k so J , - j k zo \ z - z ' \ _ ( k - i - k - n \ , e - j 2 k zlh \ k z i + k zp j ___________ ^ - j k . 0 ( z + z ' ) l i f k : l - k :0 \ p - j 2 k z l h 1 ^ \ k zi+kz0J e H g \ k pP)kpdkp. (2.23) Therefore, the spatial Green’s function, Gx£ due to the rr-directed electric dipole of u n it strength becomes [30] 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 00 G“ = / 1 jn2 kTz0 (2-24) — OO where f k ; l - k :p \ R te e ~ j 2 k z lh \ ------------• = 1 + ( f cf f e) (2.25) e-m " k T he scalar potential of a point charge of the x-directed electric dipole can be derived using [33] 1 dGq _ ju f d G f j u dx' k2 [ d x d G z* ' + - od zf - • (2.26) ' (2 2 7 > The Fourier transform of (2.26) gives d < = 7 e { G~f - tk M 2 2 °' Gxx and G z£ above the substrate can be calculated from (2.13) and (2.14) using (2.22), and the substitution of Gx£ and G zx into (2.27) gives [30] G = ——^— \ \ e - jk^ z- z^ + e0 j 2k zo *-<• J 4- R qe~jh^ z+z'A , J where R = _________________ 2A;0(1 - K )( 1_- _e~**klth)_________________ ’ (fei +*«o)(fc,i +K k, o)(l + - !tT R to e~J2’“'h'> 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.28) By applying the sim ilar procedure as (2.21) to (2.28), the spatial scalar Green’s function can be obtained as i ^ l H G" = 4 J j2 00 + (Rte + Ha’\ kpp)kpdkp. — OO (2.30) Finding the spatial Green’s functions, (2.24) and (2.30) by direct integrations can be tim e-consum ing because of the oscillatory behavior of the integrands. However, the spatial Green’s functions can be efficiently calculated by making use of the Sommerfeld identity [11, 12], ,-jk o r 1_ J2 <JU / d h j £ H $ \ k pp )e -* - (2.31) where k2 - (2.32) and r = yj p1 + z2, p = yjx2 + y 2. (2.33) T he Sommerfeld identity has the physical meaning th at the spherical wave can be the infinite sum m ation of the cylindrical waves propagating in the p direction m ultiplied by plane waves propagating in the z direction, over kp [11]. The procedure for obtaining the spatial Green’s functions is divided into three 14 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. steps [30]. First, quasi-dynamic image terms can be extracted by assuming zero frequency condition such as k0 = ss 0. (2.34) W ith the approximation (2.34), R te and R q in (2.25) and (2.29) become R teo = - e ~ ]2k-lh « _ e-J2h-oh, (2.35) _ P (1 - „ - * * ..* ) __ P( 1 - e-W"*") q0 1 - P e-i2k--'h ~ 1 - P e~i2kz°h ’ ^ ' where P = f 0r— l l r = 1— ! l . ^Or + e l r 1 + (2 .3 7 ) Cr W ith i?riEo and R q0, the quasi-dynamic image term s can be extracted from Gxx and G q using the Sommerfeld identity, which gives = GS + ^ 1 G " = G ,° + 4 ^ f f f (k ,p ) i k „ / —oo °° / (2.38) 1 ] 2 k f i {RTE + R q ~ R t e 0 " «*> )e ' 2‘ ,0<' +‘') f lo2\ k f p ) d k „ (2.39) where S I^ J’ 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 40) , - j k 0r0 G,}o = 47ren e—jkorn ' e ~jkorg + P — z - + ' £ P ’,~ '( P 2 - 1) n=L r0 (2.41) and r'Q = \jfp- + (z + z' + 2h)2 r'o = rn = \Jfp- -F (z + z' 4- 2nh)2. \JfP + (z + z')2 (2.42) These quasi-dynamic image terms dom inate in the near-field region. Next, the surface wave terms can be extracted by perform ing the integration using the residue theorem since the surface wave poles exist along the real axis of the complex kp plane for lossless spectral Green’s functions [34]. Gxx and G q can be rewritten as j G f = C% + G % „ + ^ ~ F A(k ,)e -ik^ » ^ H S - \ k eP)kedkp, (2.43) —oo G q = Gqo + Gq,sw + 1 °° 1 / — — Fq(kp) e 'jk^ ^ ’)HSl)(kpp)kpdkp, 47T€o — Joo jAkzO (2.44) using the surface wave terms, = £ ( - 2 T j) • £ p(TE) Gq,sw — . (2-45) R esqHQ \kppp)kpp, (—2ttj) p(TE,TXf) 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2-46) where R es a and Resq are expressed as e - j k z0( z + z ' ) Res a = jZ k zQ lim {kp - kpJRTE, (2.47) lim (kp —kpp)(RTE + Rq), (2.48) up“__L +kpp ft> ""pp kp— e - j k zQ( z + z f )' Resq = j 2 k z0 fCn“tKq which can be analytically calculated using the L’H opital’s rule. FA and Fq of the rem aining integrands of G“ and G q are FA(kp) = R t e — R teq — 5 3 p(TE) kp ^pp ^pp R e s Aj 2 k z0ejk^ z+z' \ (2.49) Fq(kp) = R t e + Rq — R teo — R qo — 53 p{T E ,T M) kp kpp kp -t- kpp R e s qj 2 k zOejk:o{z+z'). (2.50) The surface wave terms affect the far-field along the substrate. Lastly, the remaining integrals of Gxj£ and Gq can be calculated by asym ptotic expansions of FA and Fq using exponentials. In other words, if F a and Fq in (2.49) and (2.50) can be expanded as N F * iK ) = ' £ a i e - b‘k- \ (2.51) i=l N' F,{k„) = (2.52) i= l the Sommerfeld identity can be applied to the remaining integrals, which gives 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G XX _ (~\XX . /~1XX A ~ Gq AO + I /~1XX l* A * w + ^-r A , c i i GqO + GqSW-+- GqlCi, (2.53) (2.54) where /-in _ N p - j k o r; i= l '« V' n = yJfP + iz + z' — jb{)2, 1 * t e~ikari Gq,a = j47re0 — t-Y,=,l ai IT— ’ r, ri = 'Jp2 + (z + ~ ~ M ) 2v (2.55) (2-56) The complex amplitudes, a, and a', as well as the complex distances, r t and r[ can be obtained using numerical m ethods such as Prony’s m ethod [35] and the m atrix pencil method [31, 32]. By introducing a parametric equation for kzQ as k zo — Atq - j t + Q < t< T 0 (2.57) F .i and Fq in (2.49) and (2.50) can be rew ritten as N (2.58) i= l Fq(kp) = ' £ A l e * :=1 (2.59) with : Ai = aie_SiTo/(l+jTo), (2.60) B = 6y i+ £ o ) To (2.61) 18 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now, Ai and B{ can be obtained from Prony’s method or the m atrix pencil i m ethod by uniform samplings of F a and Fq a t t = n A t. The truncation point, To is set to be larger than in order to avoid the surface wave effect. The calculated complex image term s are related to leaky waves and dom inate in the interm ediate fields. The derived closed-form Green’s functions Gx£ and Gq in (2.53) and (2.54) were tested using a conductor-backed dielectric slab with the high dielectric con stant (er = 12.6) and the thick substrate (h = 1mm) [30]. Fig. 2.2 shows the calculation results of the closed-form vector and scalar Green’s functions. The closed-form G reen’s functions include the full-wave effects such as surface waves and leaky waves. Table 2.1 shows the surface wave poles and the corresponding residues for the 30 G H z excitation. The vector Green’s function, G1* has one T E mode surface wave, while the scalar Green’s function, Gq has one T E mode and one TM m ode surface waves. T he complex image terms take into account of the leaky wave effect. For the 30 G H z excitation, the leaky wave effect was considered using two complex image term s for both the vector and scalar G reen’s functions. Table 2.2 shows the .Aj’s and B,-’s for this case, calculated by the ma trix pencil m ethod. 19 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Freq. = 10 GHz (b) Freq. = 10 GHz o 3 a < * x x to O o> o — Closed-fqrjn 0 Q Numerical integration •2 0 (c) Freq. = 30 GHz -1 0 1 (d) Freq. = 30 GHz o 3 £ — a « o(A a <D a < < « x x « O’ (0 <2 O) o (5 o> o •2 0 (e) Freq. = 50 GHz ■2 0 (f) Freq. = 50 GHz 1 o |3 <a « x x CO g O) o 0 ■2 0 ■2 Iog(k0*rho) !og(kO*rho) Figure 2.2: The am plitude of normalized Green’s functions, G xj£ and G q. 20 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 r[ i Green’s function (30 GHz) surface wave mode pole (kpp) residue TE 1.666 k0 + 0 .3 6 9 + j0.531 x 10~14 TE 1.666 k0 -0.236 - j0.676 x 10“ 14 TM 2.641 k0 -0.351 + j0.436 x 10~14 Gq Table 2.1: Surface wave poles and residues (freq. = 30 GHz). G reen’s function (30 GHz) G f Ai Bi —1.9922 + j'1.7212 -0.1416 -jO .1 1 4 4 -1.2946 - j ‘0.4864 +0.3669 - j'0.7691 -3.6786 - jO.0567 -0.0086 + j'0.0052 -0.6099 - 70.6894 +0.0180 - j0.0736 Gq Table 2.2: .4, and Bi for complex image term s (freq. = 30 GHz). 2.2 T he m ixed -p oten tial integral eq u ation (M P IE ) and th e m eth o d o f m om ents (M oM ) form ulations Using the mixed potential formulation, E = —ju)A — V V, the mixed-potential integral equation (MPIE) can be derived by applying the boundary condition such th a t the tangential electric field on the m icrostrip conductor surface is zero, where A represents the magnetic vector potential and V represents the electric scalar potential. The mixed-potential integral equation (M PIE) is expressed as 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. h x E excitation = n x ju/G a * / - (G q * V • J ) (2.62) where C?a and G , are the vector and scalar Green’s functions, respectively, of the conductor-backed dielectric slab and * represents the convolution integral [36]. W hen applying the m ethod of moments (MoM), the triangular-patch-pair basis function r in T+ 2.4+ fn ( f) = 2A ^ Pn 0 (2.63) r in T~ otherwise is chosen, since it is well-suited for modeling arbitrary conductor shapes [15]. The basis function (2.63) is defined on the geometry shown in Fig. 2.3. ■-A4 Figure 2.3: Geom etry for the triangular-patch-pair basis function. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The unknown current distribution on the conductor surface can be expanded by (2.64) n=l where ln is the length of the common edge in the n th triangular patch pair, In is the total current flowing across the common edge, and N is th e to tal number of triangular patch pair criss-crossing the microstrip structures. T his introduction of the division of f n [f) by ln gives the unknown coefficient In w ith the unit in Ampere. This new scheme makes it easy to incorporate circuit equations of the lum ped elements into the spatial domain analysis. After taking the inner product of (2.62) w ith the same modified current basis functions as in (2.64), we arrive a t a system of equations, N 5 3 Z mnln = Kn^ms, ™, = 1, 2, . . . , N , (2.65) n=l where Z-mn = * fn if)) ~ V re?, * V • f n ( f) ) ) ) , (2.66) and Vm5ms represents the voltage gap source applied onto the s th patch. The evaluation of the m atrix elements, Zmn’s requires various num erical tech niques including singular integrals when the field points are close to th e source points. Using the basis function, (2.63), the first reaction integral term can be w ritten as 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (/»(»=), G W » ( d ) =U+4'*■fe 4 B*WU*"+£ 4 B^.)**)* +2A^It-P"'' (2I+4G -4(r1r"*^£fa'+2A^4 * ' (2.67) In order to calculate the second reaction integral term , the surface vector calculus identity can be first applied [15], which gives (L(f), v (a, . V • /> - ) ) ) = -(G , » V • V . /1(f)). (2.68) Next, using r in T+ V • / m(f) = < (2.69) r in T z 0 : otherwise (2.68) can be expressed as =-44 (44 -44 * +4 4 ( 4 4 G"(rT”)<is' _ 4 4 G’(r1f-)ds') d s (2.70) 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When calculating (2.67) and (2.70), the l / r 0 singularities within the G reen’s functions need to be extracted and calculated separately. In order to decide whether or not to extract the singularity, the unitless distance param eter crmn defined as m, n € 1, 2 , . . . , N , (2.71) can be used [37], where f£, and f£ represent the global position vectors from origin to the centroids of the m th and nth patches having areas .4m and .4n, respectively. Now, by defining the threshold a th and using the unit step function U(crth — crmn), the 1/ro singularities can be extracted from G reen’s functions, which gives the reduced Green’s functions : = r.40 + G A , s w + G A ,c i, ho U{aa1 where ^mn) (2.72) and r, where = Gq - 1 U (a th Omn) 47Te0 r0 Tq0 = Gq0 - 1 47Te0 ^mn) r0 (2.73) 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The unit step function U (ath — crmn) is nonzero only when amn is sm aller than (Tth- Using the reduced Green’s functions, (2.72) and (2.73), (2.67) becomes ( fm ( r ) iG A * fn (r)) -i 4 *•(*44 ™ ds'+w4 +Ui 4* -($44 +k 4 +i44 «•fe 4 + 4 4 *' G44 +£ 4 * * *’ (2.74) and (2.70) can be rewritten as ( / m W , V ( G ,* V - / ; ( r ) ) ) --4 4 (44 +4 4 ^ UJ* ^ ds')ds +4 4 (44 +4 414^44=^)* +44 ( 4 4 +4 4 i4£44m n)JJ' -ds' ds "-4-4 J/t’- (\ A4~ h/ n r,(f]f„)*' + i=/ A - hT ~ 1 47T£o a m n) U ( a th A r0 (2.75) 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since the integral terms having the reduced G reen’s functions are locally smooth, the double integrals of (2.74) can be reduced to the single int.pgra.ls as shown in ( fm { r ) ,G A * f n (r)) L-mln - c -f Po^mk P o^m k f -*f /" U (& th r - f 16tt.4+.4+ *Jt1m + m Jt + t*0 ( z c +1 — j ■kf ^ A^i ( ri* i f „ ) ^ d si' + f ^ ^ j , pPo^mk k m k [f tfr f[ • f ^ A ( f % \ f n)fa d s ' + + 4 ^ p" Jk*irt 'r")p’'d s + \16vA+A~ 6 i A ± A.k; k Pmp" Itk, ^ m k -c + (m Im k — c— f -JJ+ A . ' b f=r f s c —i ~ \ j„ i , ^ ( r . M /V * + U {(T th °m n) p - d■s /a ,s &mn) ---- ^---- p' dsds Po^mk p k m k [ f_p„ r— j fnf U ( ( T t h Jt 16ttA -.4+ .k Pm<'T+ Po^mk fL fL . ^r.4(rTlr A i r ^ l ^ „)K P ^ dM s' + + T ^ r p! I <r j4-4n k f a •■7r„ K 7'T167T/1-.4lb7T.4m.4n 7r~ ,k Pm' r" *in & m n ) -*+. , / , f t t * d. jr„ds'ds, r0 (2.76) where fjjf represent the global position vectors pointing the centroids of T * . In the sam e manner, (2.75) can be approximated as ( / l ( 0 , v ( c , . v - /> -} )) = - j1A+ t t Jt+ I r q, ( d lm ■ nkk f r . ) d s ' - ■■ u 47re0.4 + .4 + »-> / - c +1 -« \ jj i . lI imn k f f [ Jt+ Jt+ f ^ k r<{rMds+t ^ x k k ^■mk f 'At k \ , I ^mk f r’(r" K)d*+ toSTt* k / -xz— | — \ i / — ^f r,(rm Irjd, - Im ^ n f f k f r0 tmnn) ) , , / /, &< m UU\{G&t thh — - - ^ d s ’d s S -------- d s d s U {(7th ^mn) ^ & \& th , /, dS* ^m n) , / » dsds. (2.77) i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t ! W hen evaluating the nonsingular integrals in (2.76) and (2.77), it is convenient to em ploy the local coordinate system [15, 38, 39]. A 2 c A i §= 1 — A n A 3 A 2 11= —— An „ 5 = A3 A n 2 Figure 2.4: Normalized area coordinates for a local coordinate system. Fig. 2.4 shows the local coordinate system. Using the normalized area coordi nates, the source position vector r ' can be rew ritten as r' = £fi + r}f2 + <T3, (2.79) and th e surface integral over the nth triangular patch can be transform ed as 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [ JT n F(r')ds' = 2A n [ + -qf2 + (1 - £ - T])f3]d^drj. f JO J 0 (2.80) Therefore the nonsingular integrals in (2.76) and (2.77) can be calculated as j L P A(fa^\fa) fa d s ' = 2.4* [(f\ - r 3)/?r 4- (r2 - f 3)/„r + (r3 - r ^ ) /r] ***■n f ** * fi ^ A( fa ^ \rn) fa d s ' = -2 .4 * [(r4 - f 3) / ?r + (r2 - f 3)/„ r + (r3 ~ Kr] (2.81) (2.82) and L ^*n r ,(r S = |r - )ds' = 2 4 j / r (2.83) where / ' f a tr(f*J?)dt;dr, V = JO Jo V = / L (2.84) (2.85) Jo Jo / r = f l f lr> r ( f a l \r')d£dT). (2.86) Jo Jo In th e evaluation of (2.84), (2.85), and (2.86), a three point quadrature method is applied using the sample points (fa + ri)/2 , i = 1,2,3. Hence, (2.84), (2.85), and (2.86) can be calculated as V = E f r ( e . ’/) » ri 1=1 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.87) s a m p le p o in t (^,^7) ( ^ + n )/2 ( -3 ’ 6 ' '■ (f£ + r 2) / 2 (I 2) ' 6! 3' (f£ + T3) / 2 '‘6 ’ 6 ' Wi I 6 1 6 1 6 Table 2.3: Param eters for three point quadrature method. ( 2 . 88 ) i= l h = E m ,r iW ,. (2 .8 9 ) t= l Table 2.3 shows the sample points and corresponding (£, t j ) coordinates with weighting coefficients Wvs. The evaluation of the singular integrals in (2.76) can be performed by dividing the integrands into the removable singularity term s and the l / r 0 singularity terms as [40] U{<Jth ^mn) -4-j / j ------------------ pZds as ro ds = U (ath - crmn) f (? - f £ ) [ f ~ - r - ds + (P - f ^ ) [ ds' JT+ Jt + r - r'\ Ji It + \r —r'|_ = & (&th ^mn) ^ 1 ^"tiI ' S 2 “I" S 3 + ( r n j • r mi)iS4 {fnl 4“ ^m l) ' ^oj • (2.90) Similarly, 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f U -4- f U { a th Pm' k °~mn) n , , p* - a m n ) [5 , - f„4 • S 2 + S 3 + (f„ 4 • r m i ) S 4 - (f„ 4 + r m |) ■S s] = -U io th (2.91) f k Xf P m k — °m n ) - + j i j ^ ----- r 0---- «•** U {& th @Tnn) [S i ^*nl " *?2 "h ^ 3 “h (^nl ^'m4)‘^4 ( ^n l d* ^m4) ’ (2.92) f x— f & i& th k ^ ' k - O’mn) — j t j -----S-----^ U ( ( 7 th ^mn) ^3 “h (j~Ti4 ‘ ^*m4) *^4 ^*n4 ‘ *^2 (^n 4 ~F Cm-1) **5gJ . (2.93) Since the integrands of singular integral term s in (2.77) have only l / r 0 singular ities, the singular integrals can be calculated as [ [ Jt ± Jt£ ds'ds = S 4. (2.94) tq S i through S 5 are defined as N Si = Y l Wi(Psi ■Ai) :=1 (2.95) yv s3 S 2 = ^ 2 w iA i i=i N = Y ^ wi\T'si\2Bi i=L 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.96) (2.97) source p oint h A :S Figure 2.5: G eom etric param eters associated with the line integral along Aj S for calculating A., an d Bi. 32 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AT S4 = (2.98) Y l w i& i 1=1 N B5 — ^ ' WiBit s{, (2.99) 1=1 where 1 3 ( 2 . 100 ) 1 3 - //+ 4- P ^ ' ( 2 . 101 ) and the geometric param eters associated with Ai and Bi are defined in Fig. 2.5. In calculating Si through 5s, a seven point quadrature rule can be applied [41]. Fig. 2.6 and Table 2.4 show the sample points and the weighting coefficients. c3 c2 cl Figure 2.6: Sample points for seven quadrature rule. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sample point position Wi 1 rc 270 1200 2 rc + '/if+ l(rc3 ~ r c) 3 tc + 4 rc + ^ ( r c2 - r e) 5 rc - & = ± (r* ~ r c) 6 rc 7 rc ^ ^ ( r cl ^ ^ - r e) 1(rcl - r e) 155—\/l5 1200 l5 5 + \/ l5 1200 1(rc2 ~ r c) Table 2.4: Param eters for seven point quadrature rule. The RHS of (2.65). VmSms can be obtained from the moment m ethod proce dure using the voltage gap source applied onto the sth patch as shown in Fig. 2.7. When the voltage gap source, V* induces the electric field intensity, E l, the moment m ethod procedure gives — [ 2^4+ Jt + p i - E'ds - f p i ■E'ds 2A ~ Jt ~ ( 2 . 102 ) Since the induced electric field exists only in the gap, the surface integral on the patch reduces to the surface integral only on the gap. Hence, (2.102) becomes 34 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - o 1 2 I Vs A A A Figure 2.7: Voltage gap source applied onto the sth patch. A / 2 Zm T. , A / 2 Zm = ~7T f A /2t, A / 2 Zm f h + ^Td s + T T r 2Am o o | h +im v; ^ f y r l r n V , A+ 2 .4- 2 = K. [ [ h~~ds m oo (2.103) W hen the voltage gap source is excited on the sth patch, (2.103) can be concisely expressed as VmSms in the system of equation (2.65). 35 i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 In co rp o ra tio n o f lum ped resisto rs into th e sp a tia l do m ain an a lysis To include the lum ped resistor effect into the spatial dom ain full-wave analysis, a new type of triangular patch pair has been proposed as shown in Fig. 2.8. Since 3 3’ 2 2 ’ Figure 2.8: New type of triangular patch pair for th e lum ped resistor connection. Resistor Figure 2.9: Example of resistor connection. current flowing out of the edge of one patch m ust flow into the edge of the other 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t patch, it is reasonable to assum e a new type of triangular patch pair in which the common edges are connected through the resistor. Fig. 2.9 shows an example of connecting the new type of triangular patch pair in order to take account of the resistor effect. The new patch pair become the N + 1th patch pair when N patch pairs criss-cross the m icrostrip conductor portion. It is well known from circuit theory th at the induced voltage in the simple lum ped resistor is related to th e new patch current I^+i by the following equation, Vn +I = Zrlr ~ (R + j u L ) I tf+ 1- (2.104) This circuit equation (2.104) can be directly incorporated into the system of equations denoted by (2.65). As a result, the new system of equations, including the current distribution on the conductor and the lumped device current, becomes jV +1 5 3 Z'mnIn = VmSms, m = 1 ,2 ,-----N + 1, (2.105) n=l where the m atrix element Z'mn is the same as th a t defined in (2.66) except th a t the self element of the new patch now becomes Zn+i,n+i — Z n + i , n + i 4- Z r — %n + i ,n + i + R + j u L (2.106) because of the lumped resistor connected in series. The resistor effect was studied using a simplified T-junction m icrostrip power combiner / divider configuration as shown in Fig. 2.10. In order to investigate the 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C ) (d ) Figure 2.10: Schematic diagram for a microstrip power combiner / divider. (a) In-phase excitation without resistor. (b) In-phase excitation with resistor. (c) Out-of-phase excitation without resistor. (d) Out-of-phase excitation with resistor. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I (a) 4 2 (wiu)A 0 -2 -4 0 5 10 15 x(mm) (c) 4 1. 1 i.f 2 t/l t J (uuiu)A .........//I . * /\\ ? „ E 0 ............. . l i t ! ................................... 0 5 10 . . . » > f. \ 1,1 x(mm) -4 \ / 1 f . t t,t -2 15 t I r i .......................... 0 5 10 x(m m ) Figure 2.11: Surface current on the T-junction with and w ithout resistor. (a) In-phase excitation w ithout resistor. (b) In-phase excitation with resistor. (c) Out-of-phase excitation w ithout resistor. (d) Out-of-phase excitation with resistor (R = 10 f2, L = 0.1 nH). 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 resistor effect, both in-phase and out-of-phase excitations at the two sym m etrical arm s were sim ulated. In the case of in-phase excitation, the voltage difference in the resistor is zero because of the symm etry, and the resistor does not affect the field distribution. In contrast, the out-of-phase excitation would induce a voltage difference in the resistor. The resistor effect, thus, would be significant on the T-junction. T he T-junction was designed on a substrate with er = 2.33 and thickness of 20 mil. The voltage gap sources of 10 G H z and Vs — 1 V were applied as excitation sources as shown in Fig. 2.10. Fig. 2.11 depicts the surface current on the T-junction. In the case of in-phase excitation, a magnetic wall exists a t the y = 0 plane due to the even sym m etry. The resistor (R = 10 fi, L = 0.1 nH), even though present, is open-circuited on the PM C, and consequently, there is no difference between Fig. 2.11 (a) and (b). In the case of out-of-phase excitation, the y = 0 plane is equivalent to a perfect electric wall. If the resistor is in plase, a large current would flow through the resistor. As a result, the current distributions shown in Fig. 2.11 (c) and (d) behave differently. Fig. 2.11 (d) clearly shows th a t current flows through the resistor. 2.4 A n alysis o f W ilk in son p ow er divider and com b in er The proposed extended spatial dom ain analysis was applied to analyze an actual W ilkinson power divider. Power combiners and dividers are essential com ponents in microwave circuit integrations. Among various implementations, the threeport T-junction configuration is the m ost popular. Because all three ports of the lossless reciprocal junction cannot be m atched at the same time, im plem enting a lossy sim ultaneous im pedance-m atching junction with a lumped resistor becam e a well-known technique in planar power combiner and divider designs [42]. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A m icrostrip Wilkinson power divider is modeled as shown in Fig. V ss2 2.12. Z s2 w2 Vss i •( \ y \ Z r = R r+ jco L r V ssI " ( f \ j 0-wHi' V ss3 Z s3 Figure 2.12: The analyzed m icrostrip Wilkinson power divider structure. In order to consider the m atched excitation and load, another new type of the triangular patch pair is introduced in Fig. 2.13, in addition to the triangular patch pair for the resistor (Fig. 2.8). T he load can be regarded as a special case of Fig. 2.13 when the voltage source is shorted. The matched excitation and load 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 _ Vsi A/ Zi 2 Figure 2.13: New type of triangular patch pair for the matched excitation. scheme was tested using the microstrip line modeled in Fig. 2.14. Fig. 2.15 shows the calculation results. The m icrostrip line is designed to have the characteristic impedance, Z q = oOQ, (w = 1.464 m m ) on the 20 mil thickness substrate having the relative dielectric constant, er = 2.33. Both an open-ended and a matchedloaded (Rl = 50 Q) microstrip were simulated by applying the 10 G H z sinusoidal wave of 1 volt amplitude with the source impedance, R s = 50 fi. It verifies the new matched excitation and load scheme th a t the matched-loaded m icrostrip has about 1.15 standing wave ratio (SWR) while the open-ended m icrostrip has SW R close to infinity. The small variation in the current magnitude on the matchedloaded m icrostrip may be due to the imperfect 50 characteristic impedance of the microstrip. The moment method results were also compared with the circuit theory results. This comparison shows th a t the new matched excitation and load scheme can simulate the exact am ount of voltage excitation which is difficult with the voltage gap source excitation. The small deviations between the moment m ethod results and the circuit theory results may be caused by the 42 t i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w 'K\} Vs R1 Figure 2.14: Matched excitation and load for the microstrip. (w = 1.464 m m , I = 14.640 m m ) o p en -en d ed < Zl = 5 0 (ohm) E X — Method of Moments Circuit Theory x (mm) Figure 2.15: M agnitude of Ix along the microstrip. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. negligence of the fringing effect in the circuit theory. W ith the new m atched excitation and load scheme as well as the lum ped resistor connection scheme, the application of the method of moments to the W ilkinson power divider as shown in Fig. 2.12 gives the system of equations, W +5 £ Z'mJn = m = 1 , 2 , . . . , N + 5, K n, (2.107) n=L where N is the number of patch pairs on the conductor part and the additional five term s come from the four matched excitation patch pairs and the one lum ped resistor patch pair as shown in Fig. 2.12. Z ' ^ s include the contribution from the patch current, i.e., Zmn’s expressed as (2.66), and the induced voltage term s due to the current flowing through the resistor elements, if any. Let the m atched excitation patch pairs be the s i', s i ” , s2, and s3th patches, and the lum ped resistor patch pair be the r th patch in the system . The induced voltage is Vi — Z i l i , i = s i', s i ” , s2, s3, and r, (2.108) based on the circuit theory. Ztcan be directly incorporated into the corresponding diagonal term Za due to the equivalence between the physical interpretation of the impedance m atrix [Z] and Zt, and , consequently, Z'ii = Z ii + Z i. (2.109) The excitation voltage, Vssi’s are positioned on the RHS of (2.107). T he sim ulation procedure of the Wilkinson power divider of Fig. 2.12 is sim ilar to the actual measurement procedure [43]. In order to find the S u and S 2 1 , Port 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 is excited while ports 2 and 3 are term inated in m atched loads. Similarly, 5 12, S 22, and S 32 are characterized by excitation a t port a t ports 1 2 and matched term inations and 3. In order to investigate the effect of the lumped resistor on the Wilkinson power divider, the same structure w ithout resistor was also simulated. According to theory, the resistor contributes a simultaneous match of all three ports and provides a strong isolation between ports 2 and 3. The actual circuit was designed on a substrate of er = 2.33 and thickness t = 20 mil in X-band. A tiny 100 Q chip resistor, 20 mil by 20 m i l , was used as the lum ped resitor element. The calculated S-parameters of both cases were compared w ith measured results in Figs. 2.16 (a) and (b). The simulated and the experim ental results agree well in both cases, and one can see the improvement of S 22 and S 32 (isolation) in the power divider w ith resistor over the power divider without resistor. Fig. 2.17 and Fig. 2.18 show the current distribution on the Wilkinson power divider when the circuit is excited from port 1 and port 2, respectively. The resistor effect can be clearly observed in case port 2 is excited. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) w ithout resisto r I S 3 0---O 7 9 8 10 2 T , m easu i 11 12 13 freq u en cy (G H z) (b) with resisto r - ■ a - - - w - - - 0 ------ 0------ a------ IS12I IS22I _ ^)-"0 ^ 7 - IS32I 0 -""9 ” 9 8 10 11 12 13 frequ ency (G H z) Figure 2.16: Scattering parameters of th e Wilkinson power divider, (wl = vv2 = 1.44 mm, w3 = 0.9 m m , a = 4.896 mm, b = 0.36 m m , Z s\> = Z s\" = 100 £7, ^ 2 = %sz ~ 50 Q, Rr — 100 Q, L r = 0.1 n H ) 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 6 4 2 "E jr . 0 >. -2 -4 -6 ~8 0 5 10 15 x (mm) Figure 2.17: C urrent distribution on the Wilkinson power divider (P ort tion). 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 excita 8 6 4 2 eT E. 0 >* -2 -4 -6 ~8 0 5 10 15 x (m m ) Figure 2.18: Current distribution on the W ilkinson power divider (P ort 2 excita tion). 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3 Hybrid time-domain analysis The transient and wide-band responses of the three-dim ensional microwave struc tures can be effectively explored using the finite difference tim e-dom ain (FDTD) m ethod [20]. This dissertation develops the FDTD m ethod in order to analyze and design bondwire structures and microstrip-to-waveguide transition tructures by employing Super absorbing M ur’s 1 st order absorbing boundary condition (ABC) [19]. Especially, for effective analyses of microstrip-to-waveguide transi tion structures, a dom ain decomposition m ethod for the FD TD m ethod is de veloped, which reduces the memory requirements and the com putational time. In addition, a hybrid m ethod using the finite difference tim e-dom ain (FDTD) m ethod and the finite element time-domain (FETD ) m ethod is proposed for locally detailed and curved structures and applied to via-hole structures and waveguides with an iris of finite thickness. In the hybrid tim e-dom ain analysis, the FE T D m ethod is applied to the locally detailed and curved structures only and the FD T D m ethod is applied to the remaining regular structures. Therefore, this hybrid m ethod can be an optim um time-domain analysis for locally detailed and curved structures by taking advantage of both the F E T D flexibility and the FDTD efficiency. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 F in ite difference tim e-d om ain m eth od T he formulation of the finite difference time-domain (FDTD) m ethod starts from Maxwell’s two curl equations in a source free, isotropic, and lossless medium [8 , 9], dE 1_ f t- = 7 V x H dH . . (3*1} . - -ST a t = —n V x (3-2 E- After discretizing (3.1) and (3.2) using the central finite differencing both in tim e and space, electric and magnetic fields can be alternatively calculated based on the leapfrog scheme [44]. Fig. 3.1 shows the FDTD analysis. The electric field updating equations can be obtained from the finite difference approximations of the Maxwell’s equation based on A m pere’s law, (3.1): E 2 + '( i, j,k ) = E%(i,j,k) + C„ [H«+l/2(i,j + l,k) - H r l/2(i,3,k)] - Cz i [H^*'l2( i , j , k + l ) - H ; * ' l 2( i , j , k ) } , (3.3) £%*'(>,},k) = + Czi [ H r ' l2( i , } , k + 1) - - *)] + (3-4) + C* [flJ+'/2(>+ lj\k )~ H”+'l2(i, j, A)] - C ^ J £ +1* ( i J + l , * ) - i £ +I/ 2 (i,.;,* )] , 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.5) FDTD A n a ly sis F in ite d ifferen ce ap p roxim ation s o f M a x w e ll’s equations D iscretization o f the com p u tation al d om ain w ith th e F D T D unit c e ll T im e-step d ecisio n (stab ility con d ition) - Program m ing In itialization o f all field s (t= 0 ) G au ssian p u lse ex cita tio n U p d ate H fie ld (H n+I/2 ) U pdate E field ( E n+I ) A p p ly boundary condition^ n n-J-1 P ost p ro cessin g (F F T ,...) for freq u en cy resp o n ses Figure 3.1: The FDTD analysis. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! where r Cxi ~ At r Cyi ~ At At C zi ~ (3 ' 6) and the superscript n represents the tim e-step. Also, finite difference approxim a tions of the Maxwell’s equation based on Faraday’s law give the m agnetic field u p d atin g equations: H ^* '/2( i , j , k ) m *l'2{i,j,k) - D , { E ' ; a , j + i , k ) ~ E';(t.] . k)\ + A [ £ J ( i , J , * + l ) - £ ” ( i,J ,* ) ] , = k) - k + l) - £ ? ( i,;,f c ) ] + Ar K (> ' + l , i , * ) - £ ? ( ; , j.fc)], = H?-‘'2(ij,k) - D,[EZ(i + l , j , k ) - E 2 ( i , j , k ) } + Dy[E^{i,j + l , k ) - £ % ( i , j , k ) ] , (3.7) (3.8) (3.9) where Next, the structure modeling is obtained using box-shaped uniform meshes discretized by the FDTD unit cells. Fig. 3.2 shows the FDTD unit cell where the 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. electric field elements and the magnetic field elements are defined in the unit cell (i,j, k). The upper limit of the unit cell size is decided by the guided wavelength, Hx(i,j,k) node (ij,k ) / ^ l,j Hz(iJ,k) Ex(i,j,k) Hy(i,j,k) Figure 3.2: The FD TD unit cell. Ag of the m axim um frequency wave. Generally, Ax, Ay, and A z are chosen to be smaller th an Aff/10. Also, the decision of the tim e-step is im portant for the stable FD TD analysis. The maximum time-step is lim ited by the Courant stability condition [45]: 1 ( 1 1 1 \ -1/2 A t < ------ ( ——j "b — — 2 "b T ~2 ) > Umax \ A x Ay Az J (3.11) where vmax is the maximum velocity of wave in the com putational medium. The FD TD analysis is usually performed by exciting a Gaussian pulse at one port, expressed as 53 Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. I g (t) = exp [-(i - lo f / T 2] . (3.12) A Gaussian pulse (3.12) has a sm ooth transition in the time domain, which can reduce the numerical errors due to the finite difference approximations in the FD T D analysis. In addition, the spectrum of a Gaussian pulse has a Gaussian pulse form, which gives a salient advantages in studying frequency responses of microwave circuits. According to th e maximum frequency of interest, / max, T is determ ined as (3.13) Also, a m odulated Gaussian pulse can be used as an excitation, which results in shifting the center frequency from D C to / c: 9m(t) = g{t) ■sin(u>ct). (3.14) The field quantities are updated by the magnetic field updating equations ((3.7) - (3.10)) and the electric field updating equations ((3.3) - (3.6)) alter natively, and repeated until the responses at the observation points have zero steady states. For the efficient analysis, the coefficients, (3.6) and (3.10) can be calculated before the field updating routine is started. The coefficients, (3.6) can be precalculated including the dielectric material information (relative dielectric constants) an d stored in memory to reduce numerical operations. There are two boundary conditions to apply in the field updating routine. F irst, the tangential electric fields on the conductor should be set to zero. Sec- 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I I ondly, absorbing boundary condition (ABC) should be implemented for sim ulat ing the infinite real space with the finite com putational domain. This dissertation employs Super absorbing M ur’s 1 st order absorbing boundary condition (ABC), in which the error correction procedure is added to the M ur’s 1st order absorbing boundary condition (ABC) [19, 46]. Due to the point-wise formulation, the finite difference tim e-domain (FD TD ) m ethod can be easily applied to general three-dimensional structures. The bondwire structure which interconnects the microwave circuits on different substrates as shown in Fig. 3.3 and Fig. 3.4 can be studied using the FDTD m ethod ef ficiently. The different substrate height, finite substrates, and vertical portions of the bondwire structure make it difficult to analyze the bondwire interconnect structure using integral equation based analyses such as spectral domain and spatial domain analyses, which require Green’s function calculations. The bondwire structure connecting two 50 fi microstrip lines on two separate substrates with er = 2.33 was designed for the S-band operation. Fig. 3.5 shows b oth measured and simulated return losses of the bondwire interconnect struc ture. The FD T D m ethod gives a good prediction of the bondwire characteristics which resembles a series inductor. The FD TD m ethod was employed to design impedance m atching stubs for the bondwire structure connecting two 50 Q microstrip lines on two separate Alum ina substrates (er = 9.6). The bondwire structure is for Q-band operation with the center frequency, f c = 44 G Hz. Fig. 3.7 (a) shows the frequency responses of the bondwire structure without m atching stubs. In the required frequency band from 40 to 45 GHz, the calculated return loss is about 12 dB. Based on the re tu rn loss data, impedance matching stubs can be designed as shown in Fig. 3.6 using Smith ch art or commercial softwares such as LIBRA. Fig. 3.7 (b) shows th e improved return loss of the bondwire structure with the m atching stubs. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.3: Bondwire interconnect structure (3-D view). S id e V ie w «■ h b ----- erl T o p V ie w Figure 3.4: Bondwire interconnect structure (top and side view). 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S21 S11 -10 -1 5 -2 5 -3 0 -3 5 FDTD calculation .— -4 0 m easurem ent -4 5 frequency (GHz) Figure 3.5: Frequency responses of a bondwire interconnect structure. (erl = er 2 = 2.33, h i = h2 = 92 mil, hb = 40 mil, lb = 120 mil, lm = 80 mil, lg = 40 mil, w l = w2 = 92 mil, vvb = 80 mil) 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.6: Bondwire interconnect structure with matching stubs. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) frequency (GHz) (b) 0 •10 •20 ■30 -40 •50 -60 30 35 40 45 50 55 60 frequency (GHz) Figure 3.7: Matching stub design for a bondwire interconnect structure, (a) W ithout m atching stubs, (b) W ith matching stubs. (er i = er 2 = 9.6, h i = h2 = 5 mil, hb = 2 mil, lb = 8 mil, lm = 6 mil, lg = 3 mil, w l = w2 = 5 mil, wb = 3 mil, w pl = wp2 = 14 mil, lpl = lp2 = 5 mil) 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The sensitivity analyses for param eter changes in the bondwire structure were Bondwire Interconnection IS11I: lb: = 8 mil - - IS11I :l b = 1 2 mil -10 -1 5 -20 -2 5 freq(GHz) Figure 3.8: R eturn losses of bondwire interconnects with different bondwire lengths. also perform ed using the FDTD method. Fig. 3.8 shows the return losses of bondwire structures w ith two different bondwire lengths, lb's. As lb increases, the retu rn losses become worse due to the inductance increase. The larger dif ferences of the return losses in higher frequencies support this observation. The bondwire height effects were studied in Fig. 3.9. Changing of the bondwire height will m ostly affect the capacitance between the bondwire and the ground plane. 60 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As the bondwire height increases, the characteristic im pedance of the bondwire will be increased due to th e capacitance decrease. Because the capacitance is Bondwire Interconnection IS111: hb = 2 mil — IS111: hb = 3 mil - - IS11i: hb = 4 mil -10 -1 5 -20 -2 5 Figure 3.9: R eturn losses of bondwire interconnects w ith different bondwire heights. reversely propotional to th e distance between two conductors, larger difference in the return loss is observed between 2 mil and 3 mil high bondwire structures th an between 3 mil and 4 mil high bondwire structures. A nother structure of interest is the microstrip-to-waveguide transition struc ture as depicted in Fig. 3.10 and Fig. 3.11. Due to the complex three dimen- 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. microstrip dielectric waveguide Figure 3.10: Microstrip-to-waveguide transition stru ctu re (3-D view) 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Figure 3.11: Microstrip-to-waveguide transition structure (Side, top, and back view). 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sional geometry, it is quite challenging to find Green’s functions for this struc ture. Therefore, the differential equation based FD TD m ethod is more suitable for analyzing this structure th a n the integral equation based analyses. Since the microstrip section as well as the waveguide section are shielded, the com puta tional domain of the microstrip-to-waveguide transition structure can be divided into two regions by sharing an aperture as shown in Fig. 3.12. If the electromag netic fields in region I and region II are calculated separately, b u t communicate each other through the ap erture fields, the memory size and th e com putational time can be reduced dram atically compared with the regular F D T D m ethod of which com putational domain should be the dotted rectangular box. In other words, the memory saving as much as the portion excluding the dom ain I and II in the dotted rectangular box can be obtained since the regular FD TD m ethod requires box-shaped com putational domain due to the convenience of the array allocations. The domain decomposition algorithm can be explained using Fig. 3.13. Fig. 3.13 shows the interface between region I and region II, which corresponds to the aperture in Fig. 3.12. T he electric field and the m agnetic field components in region I and region II are assigned to the different arrays, b u t the interface electric field components are shared by correspoding two arrays for region I and region II. For example, the sam e value is assigned for E y ( i , j , k ) for region I and for E™ (i', j ' , h') for region II. When updating the electric field in region I, E y ( i , j , k — 1 ), E zl ( i , j — l , k ) , and E zl ( i , j , k ) can be updated by A m pere’s law using the magnetic field of region I only. Also, the electric field in region II, E + 1 ), E ^ i i ^ j ' — 1 , k' + 1 ), and E [ ! ( i' , j ', k' + 1 ) can be updated us ing the magnetic field of region II only. However, the interface electric field, E y ( i , j , k ) { E y { i ' , j \ k')) cannot be updated using the magnetic field in region I (II) only. In order to update E y ( i , j , k ) (E™(i1, j ' , k’) ) , H xl ! { i ' , j ' , k ' 4- 1 ) should 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.12: FDTD domain decom position Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i n X Figure 3.13: Interface between the region I and the region II in the FDTD domain decomposition method. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be employed with H * ( i,j ,k ) . Once E y ( i , j, k ) and E y ! {i',j', k’) are updated cor rectly, the magnetic fields in region I and region II can be updated by Faraday’s law. This m ethod can be extended to n domains an d used for the parallel com putation algorithm . The dom ain decomposition method for FD TD was employed to study the microstrip-to-waveguide transition structure. Fig. 3.14 shows the return loss of the microstrip-to-waveguide transition structure w ith 8 mil neck. The design center frequency is 44 GHz. The calculated results show good agreement with the HFSS (High Frequency Structure Simulator) results. The center frequency of the HFSS sim ulation with the patch length, pi = 44.5 mil falls into the middle of two FD TD results with the patch lengths, pi = 44 mil and pi = 45 mil. The return loss of the microstrip-to-waveguide transition stru ctu re with the 9 mil neck length was also depicted in Fig. 3.15. The FDTD results show good agreement with the HFSS result. As the patch length (pi) an d the neck length (fl) were increased, the center frequencies were shifted toward high frequencies. The FDTD m ethod can also be employed to stu d y the resonance characteris tics. Using the Discrete Fourier Transform (DFT) of the tim e response [47], the wide band frequency spectrum can be obtained. Fig. 3.16 and Fig. 3.17 show the housing structure for the microstrip circuitry. The housing structure is connected to the waveguides through the microstrip-to-waveguide transition structure. In order to find the resonance behaviors of the housing structure, both the source and the observation points were positioned inside th e housing structure. Fig. 3.18 shows the spectrum of the housing structure. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microstrip-to-Waveguide Transition -10 -1 5 3 -20 co - 2 5 -3 0 -3 5 FDTD (pi = 44 mil) -4 0 - - FDTD (pi = 45 mil) -- HFSS (pi =44.5 mil) -4 5 freq(GHz) Figure 3.14: R eturn loss of the microstrip-to-waveguide transition structure with 8 mil neck. (er=5.9, dt=7.4m il, hlh=94.4m il, h2h=47.2mil, hlw = 47m il, a=224m il, b=112mil, cw=11.3mil, fl= 8 mil, pl=44(45)mil, pw=47mil, pi=0mil) 68 v i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microstrip-to-Waveguide Transition -10 -15 £"-20 *o £0-25 -30 -35 FDTD (pi =44 mil) -40 - - FDTD (pi = 45 mil) - - HFSS (pi = 44.5 mil) -45 freq(GHz) Figure 3.15: R eturn loss of the microstrip-to-waveguide transition structure w ith 9 mil neck. (er=5.9, dt= 7.4m il, hlh=94.4m il, h2h=47.2m il, hlw = 47m il, a=224m il, b=112mil, cw=11.3mil, fl=9m il, pl=44(45)mil, pw=47mil, pi=0m il) 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 housing waveguides Figure 3.16: The housing structure for the microstrip circuitry (3-D view). ( u n i t : in) 0.03 0.045 0.090 0.150 0.195 0.245 0.280 0.045 0.240 0.580 0 145 0.790 0.885 0.825 0.985 0.205 Figure 3.17: The housing structure for th e microstrip circuitry (Top view). 70 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Spectrum of housing structure 5 4 3 2 1 0 U 50 60 70 80 110 90 100 frequency (GHz) 120 130 140 Figure 3.18: The spectrum of the housing structure. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 I 3.2 H ybrid an alysis using th e F D T D and F E T D The finite difference tim e-domain (FDTD) m ethod has a number of advantages in analyzing three-dimensional microwave structures due to its simplicity and numerical efficiency [8 , 9]. The conventional FDTD algorithm makes use of the uniform meshes and does not require any special mesh generation scheme and storage for the mesh. However, the use of box-shaped Cartesian coordinate uni form meshes in the conventional FDTD algorithm causes difficulties while dealing w ith curved structures and locally detailed structures. Typically, curved struc tures have been analyzed using the staircasing approximations [48], which requires finer meshes and dram atic increases in memory size as well as longer com puta tional tim e caused by using the smaller tim e-step size to satisfy the Courant sta bility condition [45]. The same problems can be encountered in employing veryfine meshes in the entire com putational dom ain for the locally detailed struc tures. Several m ethods have been developed to overcome these difficulties in the FDTD m ethod. The nonorthogonal FD TD algorithm has been introduced to solve uniform, uncurved, but oblique structures [2 1 ], and improved later to ana lyze three dimensional structures including curved shapes [22, 23]. Using covari an t and contravariant components of E and H fields to obtain a finite difference approxim ation of the integral forms of Maxwell’s equations gives an im portant advantage in th a t the nonorthogonal FD TD algorithm can have the same form as the conventional FDTD algorithm. However, this method still requires longer com putational time and larger memory size. The DSI (discrete surface integral) m ethod has been also proposed to analyze general structures [24]. This m ethod can be regarded as a generalization of the regular FDTD m ethod since it reduces to the conventional FDTD m ethod when structured orthogonal hexahedral grids 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are used. However, this m ethod needs large memory size due to the require m ent for the dual grid as well as the prim ary grid. A locally conformed FD TD algorithm has been studied for analyzing locally arbitrarily shaped structures ef ficiently [25]. This m ethod performs th e conventional FDTD leapfrog scheme and then corrects the field in order to take into account the m etal structures which do not conform to the FD TD mesh, by using the integral form of Maxwell’s equa tions. Different geometries require different correction procedures, which can be a disadvantage of this m ethod. T he finite element time-domain (FETD ) m ethod has also been developed to improve flexibility in modeling structures by retaining the advantage of tim e do main analysis [49]-[51]. Although this m ethod can have no geom etric lim itations in m odeling structures, it can be less efficient than the FD TD m ethod because it requires the system of equations to be solved for each tim e-step. Recently, a hybrid m ethod incorporating the F E T D m ethod into the FD T D m ethod was developed and applied to the electrom agnetic scattering problem of two dim en sional circular shaped dielectric cylinders [27]. By conforming only the circular structures using FETD while applying FD T D elsewhere, a trade-off between the FETD flexibility and the FDTD numerical efficiency can be obtained. However, the F D T D and FETD mesh m atching m ethod in the interface region can give difficulty in the mesh generation in the FETD region. This dissertation proposes a new FD T D and FETD hybrid m ethod by intro ducing an interpolation scheme for com m unicating between the FD T D field and the FE T D field in the interface region. In this method, the effort of fitting the FETD mesh to the FDTD cells in the interface can be avoided. The hybrid anal ysis proposed here employs the stan d ard FD TD method w ith Super absorbing M ur’s 1st order ABC (absorbing boundary condition) [19] and the FE T D m ethod using the 2nd order vector prism element [52, 53]. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This hybrid m ethod is applied to analyze single and multiple via hole grounds in m icrostrip as well as the waveguide with an iris of finite thickness. The via hole grounded microstrip stru ctu re is a three-dimensional problem having both cylindrical and rectangular boundaries, as described in detail later. Applying the FETD to the part of the FD T D region including via hole grounds and the FDTD elsewhere preserves the advantages of both FDTD and FE T D . In addition, changing the structure inside the FETD region does not require any change in the FDTD variables, which can be a great advantage in design sensitivity analysis because different structures can be analyzed by changing the FET D meshes only. In the same manner, when analyzing the waveguide with an iris of finite thick ness, FETD is applied to the iris region while FDTD is applied to th e remaining waveguide sections. This application shows th a t the hybrid analysis can handle locally detailed structures efficiently. 3 .2 .1 F E T D fo rm u lation S tarting from the source-free Maxwell’s two curl equations in a linear isotropic region, the vector wave equation can be obtained as &2E V x V x l + Me— — — 0 . (3.15) Applying th e weak form form ulation, or the Galerkin’s procedure to (3.15) gives r r f r J (V x E) • (V x W ) d v + J » e W • - ^ - d v = J W • (V x E x n)ds, * (3.16) where W is the weighting function defined using the 2nd order vector prism ele m ent [52, 53]. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he prism element is composed of 30 edge elements as shown in Fig. 3.19. 25 16 28 30 26 14 29 27 12 10 . 22 24 20 a 23 Figure 3.19: The prism element. W ith the shape function, the electric field inside the prism element can be inter polated as 3 E(r, t) = 3 [. Y l i Y l L iT(-bj 2 + c3£ ) f y ^ ( . t ) i +6{l-i) i=i t = i ^ U ^3 ^Lj--( biZ + Ci£} fyi£te(t)i+3+6(i—i)] i=i ^ 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 3 1=4 t= l — l)fyl£ye(t)i+6(l-l) + 3 fyl^ye{i)i+3+6(l—l) ] ? / i "b Xrf (3.1 ( ) x= l where £te(t)k = E ( r , t ) - t k, k = 1 , 2 ,...,18, £ye(t)k = E ( r , t ) - y k, k = 19,20, ...,30, (3.18) and fy 1 = L y l ( 2 L y l — 1) fy 2 = ^ L yi L y 2 fy 3 = L y 2{2Ly2 — 1) fy4 — Ly 1 /t/5 = Ly2 . (3.19) In (3.18), ik and yk are the unit vectors of the kth edge in the prism element (Fig. 3.19). Therefore £te{t)k and £ye(t)k represent the state variables of the fcth edge in FETD , which can be also interpreted as electric fields. In (3.19), Li s are the barycentric coordinates of a triangle, and Ly, ’s are the first order Lagrange interpolation polynomials between upper and lower triangular faces, which are expressed as [52, 53] a.i + U = + CiX ' 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 3 ' 2 0 ) (Li ZjXfc bi Xj Zfz'Xj Xfc (3.21) Ci = zk - Zj, and Lyi — V u-y 2 Ly2 = ly y - vi 2L ' (3.22) In (3.20), A represents the area of the triangular face, which can be calculated as - 4 1 1 1 Zi 22 23 Xi X2 ^3 (3.23) The variables, yu and yi in (3.22) are the y coordinates of the upper triangle and the lower triangle in the prism element and 2 ly corresponds to the difference between yu and yiIn the finite differencing of (3.16) in time, the unconditionally stable backward difference is used [51], which gives: AT' £ ( [ J f ] [ £ T = K l l B 1]’ + 2 K e=l ) [ - K H B '] " - 2) , where 77 / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.24) Kfj = | ( V x Wf) • (V x W f ) d v + Ff = J w ? ■(V x F‘ = w S . * i -* !* > ’ W f ■W ' d v x h)d s (3.25) N e is the num ber of the total edge elements and [Ee\n represents the state variable vector a t the tim e-step n. In (3.25), W f ’s represent shape functions for the 2nd order vector prism element, expressed as h Li — (bjZ H~ Cji'jfyi ^ i + 6 (/—1) li —L) "h Ci£)fyl for i = 1 ,2 ,3 , Wi+6 (i-i) I,^ii+3+6(Z—l) = and / = 1,2,3, and 1 = 4,5. Li(2Li — 1 ) f yi ^LiLjfyl for* = 1,2, 3, (3.26) The evaluation of the elemental m atrix [.K e] is shown in Appendix A. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 .2 .2 H y b rid izin g th e F D T D an alysis and th e F E T D an alysis The RHS of (3.24), the load vector of the FETD requires the knowledge of the electric field values for two previous time-steps, as well as the boundary values at the present time-step. Appendix B shows the evaluation of the elemental m atrix [Ff] for the load vector. The boundary values calculated from FDTD become the Dirichlet boundary conditions on the FETD boundary, and are used to solve the inner field of the FETD region. The FETD region is chosen to be a brick replacing the part of the FDTD region and includes the locally arbitrarily shaped structures. This choice of FETD region gives a great advantage when different arbitrarily shaped structures need to be analyzed because only the F E T D mesh change will be required without affecting the FDTD variables. Fig. 3.20 shows the FDTD and FETD interface region in the two dim en sional view. One cell size of FDTD region is to be overlapped in the FE T D region. In the FD TD time-marching procedure, E y n{ i , j , k —1 ), E z n(i,j, k), and E z n( i , j — 1 , k) are updated, but the FDTD boundary value, E y n(i:j , k ) can not be updated because k + 1 ) does not exist. The updated interface values are employed in the FETD through the surface integrals of (3.25) to cal culate the inner field in the FET D domain. The calculated inner field is used to update the FD TD boundary value, E y n( i , j , k ) . Once E y n( i , j , k ) is updated, the H x n+^(i, j, k) can be calculated from H field updating procedure of FD TD . This completes one time-marching procedure of the hybrid method. This hybrid m ethod makes use of an interpolation scheme in com m unicating between the FETD field and the FDTD field at the interface. Fig. 3.21 explains this communication method. After the FETD calculation, the FDTD interface field is interpolated using the FETD prism element shape function (3.17). O n the other hand, the FETD interface field can be interpolated using the regular brick 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FDTD r i FETD * Hx+L(ij-l,k) Ez (ij-l,k ) x G- Ez (ij,k) n + l/2 Hx (ij,k) Figure 3.20: The FDTD and FET D interface 80 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A . FETD — FDTD FDTD field is interpolated FDTD cell using the FETD prism element shape function. FETD mesh B. FDTD FETD FETD field is interpolated FDTD cell using the brick element shape function. FETD mesh Figure 3.21: Comm unication between FDTD and FETD field. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elem ent shape function with the FDTD interface field. Fig. 3.22 shows the brick N z2 N y3 N y4 N z4 N x3 N xl N zl Nx2 N yl 4 N x4 N y2 8 N z3 y Figure 3.22: The regular brick element. elem ent [18]. By substituting the FD TD interface field to the edge elements of the regular brick element, the electric field inside the cell can be interpolated as E(r, i) = i £ N itE ii + y ' £ 1=1 + z£ N i= l t= l where N ^ ’s are defined as /Ve ■ /vii (yu - y)(zu - z) l e le Lylz 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ' , (3.27) an d can be derived from (3.28) using the cyclical relationships of x, y, z. T he Z®, Z®, and Z® correspond to the A x, Ay, and A z of the FDTD cell. Now, the F E T D interface field can be easily calculated using (3.18). This scheme is m ore general th an the FD TD and FE T D mesh matching m ethod [27] because the m esh m atching effort is avoided. 3 .2 .3 A p p lic a tio n s o f th e hybrid an alysis FETDregion m r ; .r Wm. h 0 Figure 3.23: The via hole grounded microstrip. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w This hybrid m ethod was applied to characterize the cylindrical via hole grounds in m icrostrip. The via hole region is replaced by the FETD region as shown in Fig. 3.23. Since the microstrip and the ground plane coincide with the top and the bo tto m boundaries of the FETD region, the Dirichlet boundary conditions are applied to the top and the bottom as well as the via hole cylinder wall. This reduces the m atrix size in the FETD analysis and increases the com putational efficiency. T he param eters of the first analyzed via hole grounded microstrip structure are as follows: The via hole diam eter is 0.6 mm. The microstrip width is 2.3 mm. T he substrate thickness is 0.794 mm. Lastly, the substrate has a low dielectric constant (er = 2.32). Fig. 3.24 (a) shows the cross-sectional view of the F E T D mesh for the 0.6 mm diam eter via hole, which was employed in the hybrid m ethod. For the good quality triangular meshes, the Delaunay tessellation algorithm was used. The same structure was also simulated using the FD TD staircasing approximations. Fig. 3.25 depicts the cross section of the FDTD staircasing model of the via hole. In order to get the resolution in Fig. 3.25, the 2.3 m m wide microstrip was divided into 40 cells. In the hybrid method, only 6 cells were used for the microstrip in the FDTD region and 4 x 3 x 4 FDTD cells were replaced by the FETD region among the to tal 60 x 20 x 100 FDTD cells. Fig. 3.26 compares the |Soil’s of this via hole grounded microstrip calculated by this hybrid method, the mode m atching m ethod [54], and the FDTD staircasing approxim ations. A very good agreement was observed between the hybrid m ethod d a ta and the mode matching data. In the next step, the via hole grounds with 0.4mm diam eter as shown in Fig. 3.24 (b) were analyzed and the result is shown in Fig. 3.27. The hybrid m ethod gives a good prediction of |S 2 i| for various via hole diam eters. Practically, the ground effect of a large diam eter via hole can be obtained 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.5 0.5 -0.5 -0.5 (a) (b) 0.5 0.5 -0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 (c) (d) Figure 3.24: FETD meshes for the via holes. ((a) 2r = 0.6 mm, (b) 2r = 0.4 mm, (c) 2r = 0.3 mm, (d) 2r = 0.3 mm) 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - i.. 0.0575m m -y Figure 3.25: FDTD staircasing model for the 0.6 mm diam eter via hole. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Or -10 CM -1 5 -20 this method(FDTD + FETD) o — M ode Matching FDTD staircasing -2 5 freq(GHz) Figure 3.26: IS 2 1 I of the via hole grounded microstrip. (er = 2.32, w = 2.3 mm, h = 0.794 m m , 2r = 0.6 mm) 87 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Or x -5 x -10 m 13 CM CO -15 this method(2r = 0.6mm) -20 — this method(2r = 0.4mm) o Mode Matching(2r = 0.6mm) x Mode Matching(2r = 0.4mm) -2 5 8 10 freq(GHz) 12 14 16 18 Figure 3.27: |5 2 i| of the via hole grounded m icrostrip. (er = 2.32, w = 2.3 mm, h = 0.794 mm) 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 using multiple small diam eter via holes. To begin with, the two via hole problem was chosen as shown in Fig. 3.24 (c). Both via holes have the same diam eter ( 2 r = 0.3 mm). For reference, the two via hole grounds with the square via holes (a = b = 0.3 mm) were also analyzed using the FDTD method only. Fig. 3.28 shows the square via holes in the FD TD cells. The l u l l ’s of these via hole grounds are shown in Fig. 3.29. The result of the via hole grounded m icrostrip with two circular via holes is very close to th at of the via hole grounded m icrostrip with one circular via hole having larger diameter size ( 2 r = 0 .6 m m ). The via hole grounded m icrostrip w ith two square via holes shows lower IS2 1 I than one with circular ones (Fig. 3.29). This result is shown to be reasonable because the effective via hole area of the square via holes is larger than the circular via holes. In the next step, three via holes were analyzed for the grounded m icrostrip. Fig. 3.24 (d) shows the cross-sectional view of the three via hole grounds. IS^il of the three via hole grounds is a t least 3 dB less than IS2 1 I of the two via hole grounds over a wide frequency band (Fig. 3.30). At this moment, it is worthwhile to mention th at only the FE T D meshes were changed for analyzing four different via hole grounds w ithout affecting the FDTD variables. In Fig. 3.24, one can notice th at the outer boundaries of all the FETD meshes are fixed and correspond to 4 x 4 FDTD cells. For staircasing approximations, the cell size and the A t size need to be changed for different structures. Therefore, the hybrid m ethod is more suitable for investigating the design sensitivity of locally arbitrarily shaped structures. For exam ple, one can analyze circuit performance according to the variations of the design param eters, such as structure geometries. This hybrid m ethod makes use of the QMR (quasi minimal residue) iterative method [55] to solve the system of equations in FETD for each tim e-step. Solving the system of equations for every tim e-step can be inefficient. However, the FET D 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.1mm 0 .1mm ■ t r=- - b - I -=>1 Figure 3.28: The FD T D modeling for the two via holes. (er = 2.32, w = 2.3 m m , h = 0.794 mm, a = b = 0.3 mm) 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 0 -15 -20 — this method(two Was, 2r = 0.3mm) — this method(one w'a, 2r = 0.6mm) —- FDTD(two was, a=b=0.3mm) -25 freq(GHz) Figure 3.29: IS2 1 I of the grounded microstrip with two via holes. (er = 2.32, w = 2.3 mm, h = 0.794 mm) 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -10 -1 5 -20 — two via holes three via holes -2 5 freq(GHz) Figure 3.30: |f>2 i| of the via hole grounded microstrip. (er = 2.32, w = 2.3 mm, h = 0.794 mm) 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. region in this hybrid m ethod takes only small part of the entire dom ain and the overall com putational efficiency is not affected. For example, the num ber of edge elements for the 0.6 mm diam eter via hole ground is 448. The hybrid m ethod requires 23 Mbytes and takes 8 .1 seconds for one time-step running on a Sun SPARC station 20. In the mean time, the FDTD staircasing approxim ations (Fig. 3.25) using total 140 x 40 x 180 cells require 54 Mbytes and takes 7.2 seconds for one tim e-step running in the same machine. Since the A t size of the FD TD staircasing approxim ations is chosen to be 2.5 times less than the A t size of the hybrid m ethod (A t = 0.25 x 10- 1 2 Sec.), the com putational tim e of the FD TD staircasing is a t least two times longer than th at of the hybrid m ethod. The hybrid m ethod was also applied to the waveguide with an iris as shown in Fig. 3.31. The iris region is analyzed by FETD so that the iris thickness can be considered regardless of the FD T D Az size, while the remaining waveguide regions are characterized by FDTD. For simulation, the standard WR90 waveguide (0.9 inches x 0.4 inches) was chosen. The FETD volume corresponds to 14 x 32 x 4 com putational domain of FD TD . Fig. 3.32 depicts the normalized inductive iris susceptance versus the iris w idth when the iris thickness is zero. The normalized iris susceptance can be calculated from S u using Y CTo ~ 2S n e ^ L ~ l + Sn e> W L ' ( 3 ’2 9 ) where L is the distance from the iris to the reference plane for S u [56]. T he calculated results by the hybrid m ethod show very good agreement with the M arcuvitz’s curve [57], the m ethod of moments solutions [58], and the FD TD data. W hen the iris has a finite thickness, the flexible FETD m ethod can be effec tively employed, while the FD TD Az size is not affected. Fig. 3.33 shows the 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. | S n | of the waveguide w ith an iris of different thickness. The waveguide of an iris of 20 mil thickness was analyzed while the Az size of the FDTD domains is 28 mil. Figure 3.31: T h e waveguide with an iris of finite thickness. 94 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Marcuvitz curve Method of Moments FDTD FDTD+FETD 0.8 aoc SQ. © a 3 0.6 « (O 0.4 0.2 0.05 0.1 0.15 0.2 Iris Width (inches) 0.25 Figure 3.32: Normalized inductive iris susceptance vs. iris width. (Waveguide dimensions : a = 0.9” , b = 0.4” , Freq. = 9.375 GHz) 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.3 FDTD (t=Omil) - - FDTD + FETD (t=20mil) - - FDTD (t=28mil) -10 -1 5 6.5 7.5 8.5 freq(GHz) 9.5 10.5 Figure 3.33: |5 u | of the waveguide with an inductive iris. (Waveguide dimensions : a = 0.9” , b = 0.4” , w /a = 0.25) 96 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 Conclusion The im portance of full-wave analysis for microwave circuits has been increasing as microwave circuits employ integration technology such as microwave integrated circuits (MICs) and monolithic microwave integrated circuits (MMICs), which requires predictable and precise analysis and design tools. T he full-wave analyses solve the wave equations considering time varying electric and magnetic fields with given boundary conditions, and include all th e full-wave effects such as ra diations, couplings, and surface waves. The full-wave analyses can be divided into two categories: integral equation based formulations and differential equation based formulations. The integral equation based full-wave analyses are formulated using G reen’s functions and their unknown variables are defined only on the microwave structures themselves. Since G reen’s functions for free space or layered structures are well-known, the integral equation m ethods can be efficiently used for microwave circuits designed in free space or layered structures. However, for general structures having com plex boundary conditions, finding Green’s functions is not an easy task and the integral equation formulations become very complicated. On the other hand, the differential equation based formulations do not require G reen’s functions and can deal with general structures relatively easily due to their point-wise formu lations. However, the entire computational dom ain needs to be discretized by meshes defining unknown field values, which require large memory in general. 97 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Both integral equation and differential equation based full-wave analyses were studied in this dissertation. Among various integral equation analyses, the spa tial dom ain analysis was chosen to analyze arbitrarily shaped m icrostrip circuits. Two differential equation based full-wave analyses, finite difference tim e dom ain (FDTD) and finite element tim e domain (FETD) methods were developed for three dimensional general structures. Also, by combining the FDTD and the FETD methods, a hybrid m ethod was proposed to analyze locally detailed and curved structures. The spatial dom ain analysis has an advantage over the spectal domain analy sis in th a t it is more flexible in modeling arbitrarily shaped m icrostrip structures. The triangular patch pair basis functions were used for the spatial domain anal ysis. By introducing new triangular patch pair basis functions, the extended spatial dom ain analysis was proposed to incorporate lumped elements. In ad dition, the matched excitation and load scheme were also introduced, which is superior to the conventional voltage gap source excitation m ethod. The valid ity of the extended spatial dom ain analysis was shown by analyzing m icrostrip T-junction, open and loaded microstrips, and microstrip W ilkinson power divider. Three-dimensional microwave circuits such as the bondwire interconnect stru c tures and the microstrip-to-waveguide transition structures were studied using the finite difference tim e-dom ain (FDTD) m ethod employing Super absorbing M ur’s 1 st order absorbing boundary condition (ABC). The bondwire intercon nect structures have finite substrates with different heights, of which G reen’s functions become complex. In addition, the vertical com ponents of the bond wire gives difficulty in the integral equation formulation. waveguide transition structures have similar problems. T he m icrostrip-to- Therefore, differential equation based finite difference time-domain (FDTD) method is better for these structures. Besides, wideband responses and transient responses can be obtained 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. since the FD TD m ethod is based on the tim e dom ain analysis. Especially for microstrip-to-waveguide transition structures, a dom ain decomposition scheme was developed, which resulted in saving memory and com putational time. This domain decomposition algorithm can be employed for th e parallel computation for FDTD. The finite difference time-domain (FDTD) m ethod is numerically efficient, but it has difficulty in modeling arbitrarily shaped structures because of using box-shaped uniform meshes. In order to overcome these difficulties and model the locally arbitrarily shaped structures efficiently and accurately, a hybrid fullwave time-domain analysis was introduced by incorporating the finite element time-domain (FETD ) m ethod into the finite difference tim e-dom ain (FDTD) method. The finite element time-domain (FETD ) m ethod is the tim e domain formulation of the finite element m ethod which is based on the differential equa tion formulations, and has flexibility in modeling general structures. Therefore, the application of the FE T D method to the locally arb itrarily shaped structure and the application of the FDTD method to the rem aining regular structure will give advantages of both the FETD flexibility and the FD T D efficiency. This hy brid time-domain m ethod was verified by analyzing via hole grounded microstrip structures and the waveguide structures with an iris with finite thickness. Analyses of complex microwave integrated circuits require full-wave analyses in order to predict the circuit characteristics precisely, because the full-wave ef fects can be included in the analyses. Since full-wave analyses require rigorous formulations and lengthy com putational time in general, th e proper choice of the full-wave m ethod depending on the microwave circuit stru ctu re is very' im portant for efficient and successful analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P PE N D IX A Evaluation of elem ental matrix for the [ K e] stiffness matrix The elemental m atrix, [K e] is composed of 3 x 3 small m atrices as shown in (A .l), and has to tal 30 x 30 m atrix elements. [K'} = K n -f-A i i A £ A 10 +K 22 K n +A 21 K f, 12 k +A 22 K l3 + k 7 A l3 + A 23 +K& + A 23 K n + K 21 K 12 + K 22 K n + K 2l +K Kr> +I<t A 13 + A 23 4 -A 'T o K n + K 21 + A 22 Kyy + K 22 Kj-> K "F A 2 3 + k72 +K KTt K'fo K ir K Ts K l K is K K 12 13 K J0 &T> &To Kjn A 12 22 A l3 +K 23 A n A 12 +A 21 +A 22 KTs K fr KTs 13 23 K f9 K jo KTs k To Kn K 18 K n K is A ' 13 +A 23 a 19 A 10 ATig A 10 A '[2 - t - A >2 K 17 A ' 18 K n K is A 13 A ig A 10 K 19 K io K n + K 21 +A 22 K t + kT 2 K n + K 21 i <t A '12 2 + a >3 K n + K 21 K lt + K 22 Kn A ts K n K is k T> + K 22 K X3 + A >3 K \o A ho K K io K u +A 24 A '1 5 +A 25 KT + K js + + K14 + K '24 A 'is +Ao5 A m +A 24 + k7 KTt KTs K yr KTs k T KTo KTs KTo K \ +K% K 16 K26 K 16 +A 26 A ,4 +A 24 k a A 16 | +A 26 K h . Evaluation of f v(V x WT) ■(V x W ^ d v . 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A 15 +A 25 T +KJ5 Each elements in (A .l) can be calculated as following. 1 19 K is +A 25 K is +K26 I<u (t = 1 , 2 , 3, I = /(V x 1 , 2 , 3) {if = 1 i)) • (V x ^ ( 1 + fe ) ( w , 2 , 3, V = 1 , 2, 3) i) )c/i; = + 4^) / +X / (A.2) iT 12 (i = 1 , 2, 3, Z = 1, 2, 3) (i' = 1 , 2, 3, Z' = 1, 2, 3) / ( V x Wi+eft-!)) • (V x W ii+a+6(lt_l))dv = + / df a J ~d y d y + / f y ‘f y i ’d 'y (A-3) A" 17 (i = 1 , 2, 3, I = 1 / ( V X , 2, 3) (i# = 1 , 2 , 3, /' = 4, 5) W v + 6 ( i - l ) ) ’ ( V X W V + 6 ( / / _ l ))rZu = d fy t dy (A.4) • AT18 (z = 1, 2, 3, Z = 1, 2, 3) (z' = 1, 2, 3, Z' = 4, 5) /(V x tf i+ e p .! ) ) • (V x ' + 3 + 6 ( / '- l ) ) « Z u = - ^ [ ( i + &w)(bjbj> + CjCf ) + (1 + Sij’W j b i ' + CjCi»)] / - ^ - f y i ' d y (A.5) • K n [i = 1, 2, 3, Z = 1 , 2, 3) (*# = 1 , 2, 3, Z' = 1 , 2, 3) / ( V x Wi+3 + 6 (/_i)) • (V x l'V'i/+3+6(//_ 1))cZu = ^ - ( i + * * ,)( < * * + w / (A-6 ) 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • K w (Z = 1, 2, 3, I = 1, 2, 3) (Z7 = 1, 2, 3, V = 4, 5) / ( V X Wi+3+6(/-l)) - (V x Wi dfyl ■^SjAciCe +bibi>) / ^ f 1jVdy dy (A.7) • K l0 [i = / ( V X 1 , 2, 3, Z = 1 , 2, 3) (Z7 = t'V’i + 3 + 6 ( / _ i ) ) • ( V , 2 , 3 , Z7 = X V V t'+ 3 + 6 ( / ' _ l ) ) d u 5-jji) (bibi> + aa>) + ^ •[(1 + 1 (1 + , 5) 4 = + qcj/)] J -gjj-fyi'dy (A-8 ) • t f 14 (i = 1, 2, 3, Z = 4, 5) (Z7 = 1, 2, 3, Z7 = 4, 5) /" (V X PVrt + 6 ( / - l ) ) ■ ( V X W V + 6( ,/_ i) ) e fi; = 1 (4 ^ -1 ) •(CjCi< + 6j6,/) 4A J fyifyi'dy (A.9) AT1S (i = 1 , 2 , 3, Z = 4, 5) (Z7 = L 2, 3, Z7 = 4r 5) /(V X ^^ Wf-H6 (/—1 )) • (V W'ri/+3+6(//_1))cZu = X \djj’(CjCj>4“ 4" ^it'(CiCj' 4~ 6 j 6 j ') ] / fy ify i'd y (A-10) #16 (Z = 1, 2, 3, Z = 4, 5) (Z7 = 1, 2, 3, Z7 = 4, 5) / ( V x Wi-j-3 -i-6 (z—i)) • (V x Wii+z+e(i>-i))dv = 2^[(1 + d jj> )(c iC i> + bib{>) 4 - ( l 4- ^j't)(ci'Cj 4- 4- (1 4- bj Z)j/ ) 4- (1 ) (c,-Cj< 4- &,&■/') 4- ^it')(cjC_,' 4- bjby)} J fyifyi'dy ( A - 11) 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2. Evaluation of Jp- f v W t ■W ^dv. K 2l (i = 1 , 2 , 3, / = 1 , 2 , 3) (*' = 1 , 2 , 3. V = 1 , 2, 3) /iC f — -> if! "72 J ^»'+6(/-i) • W V+e^'-ijdi; = S t2 /ie lil? + SH ' ) ( bj bj ' + cJ cj ' ) J fyifyi'dy (A.12) • #22 (i = 1 , 2, 3, Z = 1, 2, 3) (i' = 1, 2, 3, I' = 1 , 2, 3) fie f f if! "72 j W»+6(l-l) • St2 1^1ft S ^ l + —1)cto = + c J Ci' ) J f y i f y i ' d y (A-13) • K 2Z (i = 1, 2, 3, Z = 1, 2, 3) (i' = 1, 2, 3, Z' = if! A , y 1 , 2, 3) ^ i + 3 + 6 ( i - l ) • W V + 3 + 6 (/'-i)d i; = St2 /ie ZjZj/ [(1 +6jj>)(bibii + aci>) J fyifyi’dy (A .14) • K 2A (i = 1, 2, 3, Z = 4, 5) (t' = fie f - 1 , 2. 3, Z' = 4, 5) -> 772 / ^ i + 6 ( i - l ) • W i'+ 6(Z '-l)C fr> = Ot */t; (A-15) • AT25 (* = 1 , 2, 3, Z = 4, 5) (*' = 1, 2, 3, I' = 4, 5) fJLE f ~S? J v — — i+ 6 (/ _ 1 ) ’ ^ +3+6(Z' —1) d v = /ie A ^ 2 45 (— 1 + <%' or j')) J fyifyi'dy (A .16) 103 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K 26 (z = 1, 2, 3, / = 4, 5) (i ' = 1, 2, 3, I' = 4. 5) fie J ? f Jv ^ + 3 + 6 (, _ l ) ’ delta ^^2 delta = = J fyi fyii dy 4 : if z= z' 2 : if z= j ' or z' = j (A-17) : if z 7^ z' and j 7 ^ / 1 Evaluation of 3 + 6 ( i '- l ) ^ f fyifyi'dy. - I = 1. /' = 1 / ~ 4£yi + l)dy — — /j, (A.18) - / = 1. /' = 2 J - I = l, /' = 4 ( 2 L y l L y2 — L y l L y 2 ) d y — - ^ z ly (A.19) 3 j LylLy2(4LylLy2 ~ 2Lyl ~ 2Ly2 + 1)^Z/ —~ ~ £ (A.20) - I = 1, /' = 4 1 /< 2 -^ y l — ^ y l ) d y = ~ly - / = l, /' = 5 J (2 ^ i^ y 2 —LyiLy2)dy —0 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A .21) - 1 = 2.1' = 2 / 1^L2ylL]2dy = ^ - 1 = 2 ,1 '= 3 J - 1 = - I = 2. V = 4 ( 2 L y i L y2 L y i L y2) d y — ^ l y J hy 2 , I' = I = 3 . / ' = 3 - I = 3 . / ' = 4 J = 3 , / ' = ^ liL y2 d y = / = 4 . I' = A L y i L y2d y ( “^ L y i L ^ ( A . 2 5 ) — - / j , ( A . 2 6 ) L y\L y2)dy — 0 ( A . 2 8 ) ( A . 2 9 ) ( A . 3 0 ) 5 / - . 2 4 ) 5 - / ( A 4 J - (A.23) ( 2 ^ 2 ^ 2) ^ 2/ — 4 J L yid y — 1 0 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j L M ly (A .31) 3 — / = 5, /' = 5 / L y2 2 d y = -ly (A .32) Evaluation of / = ^ ~ £ - d y . - I = 1. /' = 1 J - Z= 1. /' = C1 ~ 8 L j,i + 16Z -^)rf7/ = — (A .33) 2 J - i = l. /' = ^2 J2 ( ^ l ~ + ^ L y iL y 2 )d y = (A .3 4 ) 3 I ~ *~ ^ L y i L y 2 + 4 L y l)dy = — (A .3 5 ) 7y- (A .36) - I = 2,1' = 2 J ~ 2 L y iL y 2 + ^ "y 2) ^ = %ly - 1 = 2. I' = 3 J j2^ L y i L y 2 — L y i — 4 L 22 + L y 2 ) d y = - — 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A .3 7 ) - / = 3, /' = 3 J Evaluation of / ^ / (1 6 iy2 - SL y2 + 1)dy = — 4 2 y 6ly (A.38) f yUd y. - I = 1, /' = 4 (A.39) - / = 1, /' = 5 /■ I 1 J ^ ~ ( ^ y 2 — ^ L y i L y 2 ) d y = —- (A.40) - 1 = 2.1' = 4 (A.41) - I = 2.1' = o J J ~ ( ^ y i ^ 2 — L y 2 ) d y = —— r^ 2 2 (A.42) f 1 1 J ^~(4-£'2/i£y2 ~ L y i ) d y = - - (A.43) - I = 3, /' = 4 J 2ly {AL2y2 ~ Ly2)dy ~ I 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.44) In the evaluation of the above integrals, the following integral formula can be employed : J L (A.45) /W -O T ^ T I)!' where Ai and A2 are defined as Ai = 'thLf 21 and A2 = U~ UL. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( A -46 ) I A PPE N D IX B Evaluation of elem ental matrix [Fse] for the load vector The elemental matrix, [F/] is calculated from the surface integrals. It is also composed of 3 x 3 small matrices as shown in (B .l), and has total 30 x 30 m atrix elements. [Ff] = 1*3 1 F32 Fn F32 Fzi F32 F37 F38 F37 Fl8 Fla F36 Fza Fzb Fza Fzb f Fzh f 39 F 3 /1 Fn F32 Fn F32 Fzi F32 Fit Fl8 F 37 F38 Fl„ F3& F}a Fzb Fza Fzb Fi3 F zh Fig Fi/i Fn F32 F 31 Fzi Fzs F 37 F32 F 32 39 F 37 F38 Fla Fn Fza Fzb Fza Fzb F3g Fzh Fig F3A Fu F3i Fzi Fii Fzi Fzj Fzd Fze Fzd Fie Fzk Fzi Fzk Fzi F3 k Fzi Fzm Fzn Fzm F3„ Fa Fzj Fzi F*j Fzi Fzj Fzd Fze Fzd Fie Fn Fzi Fzk Fzi Fzk Fzi Fzm Fzn Fzm Fin Fig. B .l shows the boundaries of th e FDTD dom ain. The surface integrals become different from face to face. This Appendix will show the element calcu lations based on the Face I, III, and V. 109 t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * Evaluation of / Wi - (V x W ,-/ x n)ds. Face V Face ID Face I Face II Face VI Face IV Figure B .l: Boundaries of the FETD domain. 1. Face I : h = —z • F n (i = 1 , 2 , 3,1 = 1, 2, 3) (z' = L 2, 3, /' = 1, 2, 3) / Wi+6(/—i) • (V x x {-z))d s = -C j~- J Lifyifyi'ds (B.2) • FZ2 (i = 1, 2, 3 , I = 1, 2, 3) (i1 = 1, 2, 3, f = 1, 2, 3) ^ /■///* IVj+6(z_i)-(VxJ'F’t-/+3+6(j/_1) x ( —z))ds = —cj ~^2 J Lifyifyi'ds (B.3) • F 37 (i = 1, 2, 3, I = 1, 2, 3) (*' = 1, 2, 3, /' = 4, 5) J IFi+6(i_i) • (V x Wi*+6(z'_i) x ( —z))ds = 0 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.4) • F38 (i = 1 ,2 ,3 ,1 = 1, 2, 3) (z' = 1, 2, 3, /' = 4, 5) J ^t+6(i-i) (V x Wr,;/_(_3 +6(//_l) x ( —z))ds = 0 (B.5) • • F3a (z = 1 ,2 ,3 ,1 = 1, 2, 3) (i' = 1, 2, 3, V = 1, 2, 3) / • F3b / •"* _» I'l't f ^ + 3 +6(/_1) • (V x l'Vri/+6(i/_ 1) x (~z))ds = C j ~ J L j f yi f yi>ds (B.6) (i = 1 ,2 ,3 , I = 1, 2, 3) (i' = 1, 2, 3, V = 1, 2, 3) -* -* I I' f # I+3+6 ( i - i ) - ( V x ^ +3W _ 1)x ( - i ) ) r f s = a - £ J L jfyifyvd s (B.7) • F3g (z = 1 ,2 ,3 , I = 1, 2, 3) (i' = 1, 2, 3, I' = 4, 5) J W'i+3+6(i—l) • (V X l'Vj'+6(r-l) X {—z))ds • (B.8) PPj+3+6(/-l) * (V X HV-f3+6(J'-L) x {—z))ds = 0 (B.9) F3h(i = 1, 2, 3, / = 1, 2, 3) (ii' = 1, 2. 3, I' = 4. 5) J • = 0 F3i(* = 1, 2, 3, / = 4, 5) (z' = 1, 2, 3, I' = 1, 2, 3) J ^ i+6(;_ 1)- ( V x ^ w . l) x ( - i ) ) r f s = ^ j L t f L i - V U 'f y ^ d s (B-10) F3j (i = 1 ,2 , 3,1 = 4, 5) (i ' = 1, 2, 3, I' = 1, 2, 3) J W i + 6 (1- 1) • (V X PF,/+3 +6 (Z'-l) x ( —z))ds = J j f c j L _(2L__ 1)L jif^ d j j ^ ds (B n ) F3d (z = 1, 2, 3, I = 4, 5) (*' = 1, 2, 3, I' = 4, 5) J W i+ (Z- ) • (V X PPV+ (Z'- ) x (—z))ds = 6 1 6 1 ~ 2 K I Li{2Li ~ 1)(4Li' ~ W y 'fv 1"*8 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B -12) • F3e(i = 1, 2, 3, 1 = 4, 5) (i' = 1, 2. 3, I' = 4, 5) J Wi+ 6 (l-l) ■(V X WV+3+6(/'-l) — (-£))(fe = X J Li(2Li — 1){bi>Lj> + L i'bj')fyi f yi'ds (B.13) • F3k {i = 1, 2, 3,1 = 4, 5) (t# = 1, 2, 3, I' = 1, 2, 3) J H W « - i ) • (V x x (~z))ds = ^ J U L .U f y P - ^ d s (B.14) • F3l (i = 1 ,2 ,3 , I = 4, 5) (i' = 1, 2, 3, /' = 1. 2, 3) J ^i+3+6(i-l) • (V X Wi'+3+6(f'-t) X{ —z ) ) d s = _ 4 l ^ j L .L j L f f J h ' - ds (B. 15) • F3m {i = 1 , 2 , 3 , 1 = 4, 5) {%' = 1, 2, 3, V = 4r 5) J Wi+3+6(l-l) • (V X X J L iL ji4Le • (—z ) ) d s = 1) f ylf yPds (B.16) F3n(i = 1 ,2 ,3 ,1 = 4, 5) (i! = 1, 2, 3, V = 4, 5) J Wi+3+6(i-L) • (V X WV+3+6(/'-l) X (—z ) ) d s = —— J LiLj{bi>Lj’ + L i'bj')fyifyi'ds (B.17) 2. Face III : fi = —y • F31 (i = 1 ,2 , 3,1 = 1, 2, 3) {i! = 1, 2, 3, V = 1, 2, 3) -* - / IFl+6({_1)-(Vxl'FI/+6(/-_i) x(-y))ds = lit ( 1 +£«') (fyby 4- Cj Cj>)Sn t (B-18) 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ; • • F32 (i = 1, 2, 3, I = 1, 2, 3) (i' = 1, 2, 3, /' = 1, 2. 3) J Wi+ 6 ( /_ i) • ~8 • F 37 (z (V X W V + 3 + 6 ( /' - l) X ^+ (—y ) ) ds = + (B.19) = 1, 2, 3, I = 1, 2, 3) (»' = 1, 2, 3, /' = 4, 5) J ^i+ 6(/-i),( V x ^ i '+6(jf.1) x ( - y ) ) ds = (B.20) • F38 (i = 1, 2, 3, * = 1, 2, 3) (z' = L 2, 3, /' = 4, 5) J Wi+efz-L) • (V x Wi'+3+6(Z'-i) x ( —y))ds = g ^ [ ( l + M P A + cJct') + (1 + $ii')(bjbj> + CjCj')]6nSi' 4 (B.21) • F*. (z = 1, 2, 3, / = 1, 2, 3) (*' = L 2. 3, /' = 1, 2, 3) ‘ _ 8 A /” (^ X ^ i'+ 6(Z '-l) x {-y))ds = + CiCj>)S[iSi>i (B.22) • F3b (z = 1, 2, 3, Z = 1, 2, 3) (*' = 1, 2, 3, /' = 1, 2, 3) J Wz+ + (z-i) • (V x l^i'+ + (Z'-i) x (—y) ) ds = 3 6 3 6 g ^ ' (1 + 5jj')(bibi> + c,Cj')<5a^z'i (B.23) • F3g (i = 1, 2, 3, Z = 1, 2, 3) (z' = 1, 2, 3, Z' = 4, 5) J ^t+3+6(z-i) • (V x Wi'+6(z'-i) x ( —y) ) ds = k — Q^Sjii(bibi> + CiCi')SiiSi'4 113 / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.24) • F3h (i = 1, 2, 3, / = 1, 2, 3) (i' = 1, 2, 3, /' = 4, 5) / ^+3+6(/-i) • (v x W v+ z+ w -i) x (~ y ))d s = + City) + (1 -F Sji')(bibjt + CiCj')]5n di>4 ~ g ^ [(! + (B.25) • F3i- (i = 1, 2, 3, f = 4, 5) («' = 1, 2. 3, Z' = 1, 2, 3) / ^K+6(i-i) • (V x WV+ef*'-!) x (—y))ds = 0 (B.26) • F3j (i = 1, 2, 3, Z = 4, 5) (i' = 1, 2, 3, Z' = 1, 2, 3) J ^t+6((-i) • (V x M'V+3+6(i/_ l) x (—y))ds = 0 (B.27) • Fzd (i = 1 ,2, 3,1 = 4, 5) (i' = 1, 2. 3, V = 4, 5) /p W d • (V x VFi/+6(/<_L) x ( —y))ds = 0 (B.28) • Fze (i = 1, 2, 3, Z = 4, 5) (i' = 1, 2, 3, Z' = 4, 5) J Wi+6(/-i) • (V x $V +3 +6(/'-i) x (—y))ds = 0 (B.29) • Fzk (i = 1, 2, 3, I = A, 5) (*' = 1, 2, 3, Z' = 1, 2, 3) J Wi+3+6(j-i) • (V x WV+efJ'-i) x ( —y))ds = 0 • Fzi (i = 1, 2, 3, I = 4, 5) /l W - 1 ,' (B.30) (i' = 1, 2, 3, V = 1, 2, 3) x ^'+ 3 + 6 (^ -i) x (—y))ds = 0 (B.31) • F3m (i = 1 ,2 ,3 ,1 = 4, 5) (i' = 1, 2, 3, Z' = 4, 5) / * W i , ‘ x ^z'+6(i'-i) x ( —y))ds = 0 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.32) • F 3„ (z = 1, 2, 3, / = 4, 5) (i' = L 2, 3, V = 4, 5) J Wi+3+6(f-L) • (V X ^ +3+6(/'-l) X ( —y))ds = 0 (B.33) 3. Face V : h = x • F 31 (i = 1, 2, 3, / = 1, 2, 3) (z' = 1, 2, 3, I' = 1, 2, 3) t / • F32 ( z J I~l ~i c f W ^ + sji-d • (V X W i / + 6 (p _ i ) X i)d s (z j J L ifylf yl'ds (z ( z (B.36) = 1, 2, 3, I = 1, 2, 3) (*' = 1; 2, 3, /' = 4. 5) / ^i+6(z-i) • (V x ^i'+3+6(i'-i) x %)ds = 0 . F3a (B.35) = 1, 2, 3, / = 1, 2, 3) (*' = 1, 2, 3, /' = 4, 5) J ^ i+ 6{i-i) • (V x PFi»+6(//_1) x i)rfs = 0 • F38 (B.34) = 1, 2, 3, I = 1, 2, 3) (z' = 1, 2, 3, Z' = 1, 2, 3) • (V x Wi'+3+6{ir-i) x i) d s = - b • F37 J L ify ify i'd s = (B.37) = 1, 2, 3, I = 1, 2, 3) (*' = 1, 2, 3, /' = 1, 2, 3) J Wi+3+6(z—i) • (V x PFi/+6(z'-i) x x)d s = bi~^2 J L j f y[fyiids (B.38) • F36 (i = 1, 2; 3, / = 1, 2, 3) (z' = 1, 2, 3, /' = 1; 2, 3) / • F 3j? _* lli r ^Fi+3+6(z-i) • (V x W'i;+3+6^/_l) x x )d s = bi-^£ J Ljfy[fy[>ds (B.39) ( z = 1, 2, 3, I = 1, 2, 3) (z' = 1, 2, 3, Z' = 4, 5) J Wi+ + (Z-l) • (V 3 6 X WV+6 (Z'-l) x x )d s = 0 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.40) • F3h (Z = 1, 2, 3, I = 1, 2, 3) (i' = 1, 2, 3, Z' = 4, 5) J ^ i+ 3 + 6 (t-l) ’ (V IV? + 3 + 6 (i'-i) x i) d s = 0 X (B.41) • F3i (Z = 1, 2, 3, Z = 4, 5) (i' = 1, 2, 3, Z' = 1, 2, 3) J ^:+ 6(f-i)-(V xP F t'+6(Z'-i) x x ) d s = — J Li{2Li —l) L i'fyl- ^ - d s (B.42) • F3i (Z= L 2, 3, Z = 4, 5) (Z' = 1, 2. 3, Z' = 1, 2, 3) J W i+6 (/_ i) • (V X PVV+3+6(Z'-L) X £ ) d s = J Li(2Li - l)L r f J t e - d a ^ (B.43) • F3d (i = 1, 2, 3, Z = 4, 5) (Z' = 1, 2, 3, Z' = 4, 5) J W i+ ey-D • X ^V +6(t'-l) X *)<& = g - J L i( 2 L i - 1)(4L , - l ) f vlf yl'ds (B.44) F3e (Z= 1, 2, 3, Z = 4, 5) (Z' = 1, 2, 3, V = 4, 5) • J W i+6(l-l) AI ' ( V X P F j/+ 3 + 6 ( i '_ i ) X x)d s ~ = Li'cj')fyifyi'ds (B.45) . F3k (i = 1, 2, 3, Z = 4, 5) (*' = 1, 2, 3, V = 1, 2, 3) I ^ + 3 + 6 ( 1 —!) • (V x x )d s = X J L iL jL v fy ^ d s (B.46) • F3Z (Z = 1, 2, 3, Z = 4, 5) (Z' = 1, 2, 3, Z' = 1, 2, 3) y ^i+3+6(z-i) • (V x Wi'+3+6(t'-i) x x )d s = - ^ J L iL jL ffy i- ^ -ds (B.47) 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FZm (i = 1, 2, 3, I = 4, 5) (i' = 1, 2, 3, /' = 4, 5) J ^ i+ 3 + 6 (/-I) • (V X W i ' +6( l ' - 1) X f ) t / s = J L iL j(4 L i- ~ 1 )fy lfy l'd s (B.48) FZn (i = 1, 2, 3, I = 4, 5) (*' = 1, 2, 3, /' = 4, 5) J ^ i + 3+ 6 ( / - l ) • J 8 r A (V X H'V+3+6(r _ 1) X x)d s = L i ' cj ' ) f , j i f , j i ' d s 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.49) R eferences [1] K. C. G upta, R. Garg, and R. Chadha, Computer-aided design o f microwave circuits. Dedham, MA: ARTECH HOUSE, INC., 1981. [2] T. Itoh, Numerical techniques fo r microwave and millimeter-wave passive structures. New York: John Wiley & Sons, 1989. [3] K. C. G upta, R. Garg, and I. J. Bahl, M icrostrip lines and slotlines. Dedham, MA: ARTECH HOUSE, INC., 1979. [4] A. R. Mitchell and D. F. Griffiths, The fin ite difference method in partial difference equations. Chichester: John Wiley & Sons, 1980. [5] P. P. Silvester and R. L. Ferrari, Finite elements fo r electrical engineers, 2nd ed. Cambridge: CAMBRIDGE UNIVERSITY PRESS, 1990. [6] R. F. Harrington, Field computation by m om ent methods. New York: The M acmillan Co., 1968. [7] K. S. Kunz and R. J. Luebbers, The finite difference time domain method fo r electromagnetics. Boca R aton : CRC Press, 1993. [8] X. Zhang and K. K. Mei, “Time-domain finite difference approach to the calculation of the frequency-dependent characteristics of microstrip disconti nuities,” IE E E Trans. Microwave Theory Tech., vol. 36, pp. 1775-1787, Dec. 1988. [9] D. M. Sheen, S. M. Ali, M. D. Abouzahra, and J. A. Kong, “Application of the three-dimensional finite-difference tim e-dom ain method to the analysis of planar m icrostrip circuits,” IE E E Trans. Microwave Theory Tech., vol. 38, pp. 849-857, Jul. 1990. [10] R. F. Harrington, Time-harmonic electromagnetic fields. New York: McGraw-hill book Co., INC. , 1961. [11] W. C. Chew, Waves and fields in inhomogeneous media. New York: Van Nostrand Reinhold, 1990. [12] A. Sommerfeld, Partial differential equations in Physics. New York: Aca demic Press, 1949. [13] Y. Rahm at-Sam ii, R. M ittra, and P. Parham i, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics, vol. 1, pp. 1-28, Jan. 1981. 118 with permission of the copyright owner. Further reproduction prohibited without permission. [14] J. R. Mosig, “A rbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation.” IE E E Trans. Microwave Theory Tech., vol. 36. pp. 314-323, Feb. 1988. [15] S. M. 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