# Equivalent boundary conditions of strip gratings and their application to microwave filters

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EQUIVALENT BOUNDARY CONDITIONS OF STRIP GRATINGS AND THEIR APPLICATION TO MICROWAVE FILTERS by Tungyi Wu B.S.C.E., National Chiao-Tung University, 1988 M.S.E.E., University of Massachusetts, 1992 A thesis submitted to the Faculty o f the Graduate School of the University of Colorado in partial fulfillment o f the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9709547 UMI Microform 9709547 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. This thesis for the Doctor of Philosophy degree by Tungyi Wu has been approved for the Department of Electrical and Computer Engineering by Edward F. Kuester Zoya Popovic Date 8 M e u e J L l< i ’? 6 L Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j Wu, Tungyi (Ph.D., Electrical Engineering) Equivalent Boundary Conditions of Strip Gratings and Their Application to Microwave Filters Thesis directed by Professor Edward F. Kuester This thesis develops the equivalent boundary conditions (EBCs) for a periodic metallic strip grating which is located between two media with uniform phase shift between adjacent strips. A method of homogenization based on the technique of multiple scales is employed. The derived EBCs modify previous results which didn’t take into account the required phase shift and are suitable for two different media on either side of the grating. Based on the developed EBCs, the propagation of surface waves along a periodic strip grating on grounded dielectric slab is then investigated in an efficient and accurate approach. Propagation along a grating with a uniform phase shift between strips and at an oblique angle with respect to the strips is assumed. The determination o f propagation characteristics o f metal gratings on grounded substrates then becomes analytically simple and computationally fast compared to a numerical approach. Finally, dispersion equations o f periodic grating elements formed by truncating an infinite array of strips, such as microstrip meander lines, comb lines, and hairpin lines, are investigated and show the filtering properties. Then a class of compact grating filters is designed. To efficiently compute the responses of grating \ 1 j_ _ _ _ _ _ _ _ _ _ _ _ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. filters, an equivalent transmission line (ETL) model based on EBCs is employed. The behaviors o f filters are also verified by experiment and compare well. For a meander line loaded with lumped elements, an interesting property of bandwidthtuned exists which can find application in voltage-tunable microwave filters. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would first like to thank my advisor, Prof. Edward F. Kuester, for his supervision and encouragement during the entire course of my thesis research. Professor Kuester provided the important ingredients that helped me finish this work during my three years at the University of Colorado. I would also like to thank the members o f my thesis committee, K. C. Gupta, Zoya Popovic, Melinda Piket-May, and Arlan Ramsay, for their valuable suggestions about this thesis. I would also like to pay many thanks to several people outside the University o f Colorado. Dr. Ming H. Chen, President of Victory Industrial Corp. in Taiwan, who offered me the interships for several summers and winters. My loving parents and families supported and encouraged me to make all things possible in my life. Special thanks to my wife Suming for all the love and infinite patience in the past years. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONTENTS CHAPTER 1 INTRODUCTION 2 1 1.1 Thesis Goals 1 1.2 Thesis Overview 3 EQUIVALENT BOUNDARY CONDITIONS 5 2.1 Introduction 5 2.2 The Geometry and Assumptions 6 2.3 Maxwell’s Equations and Expansion o f the Scattered Fields 10 2.4 Boundary Conditions 13 2.5 The Zeroth Order Fields 16 2.5.1 The Zeroth Order Magnetic Fields 17 2.5.2 The Surface Current Density 25 2.5.3 The Odd Current Density 28 2.5.4 The Even Current Density 29 2.5.5 The Zeroth Order Electric Fields 37 2.5.6 38 2.6 Boundary-layer Grating Voltages and Currents The First Order Fields ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 2.7 Equivalent Boundary Conditions (EBCs) 3 2.7.1 EBCs - Electric Field 44 2.7.2 EBCs - Magnetic Field 46 2.8 Results and Conclusions 51 SURFACE WAVES ALONG A METALLIC STRIP GRATING 55 3.1 Introduction 55 3.2 Determination of the Propagation Characteristics 57 3.2.1 z-directed Propagation Constant Pvs. Phase Shift 0 57 3.2.2 Propagation Constant k j vs.Propagation Angle (J) 67 3.2.3 Characteristic Impedance Zc 69 3.3 Numerical Results 4 43 73 3.3.1 Comparison of 3 with Weiss’ Model 74 3.3.2 Comparison of k t with Bellamine’s Result 76 3.3.3 Comparison of Zc with Weiss’ Model 76 3.4 Conclusions 79 CHARACTERISTICS OF MICROSTRIP GRATING ELEMENTS 80 4.1 Introduction 80 4.2 Dispersion Equations 81 4.2.1 Meander Lines 81 4.2.2 Hairpin Lines 87 4.2.3 Comb Lines 90 vii L. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 4.4 4.5 5 Image Impedance Zim 93 4.3.1 Meander Lines 93 4.3.2 Hairpin Lines 97 4.3.3 Comb Lines 99 Numerical Results and Prediction o f Stopband and Passband 99 4.4.1 Meander Lines 99 4.4.2 Hairpin Lines 108 4.4.3 Comb Lines 112 Conclusions MICROSTRIP GRATING FILTERS 118 5.1 Introduction 118 5.2 Meander-line Bandreject Filters 119 5.2.1 6 117 Equivalent Transmission Line (ETL) Model: Normal Modes 119 5.2.2 Dual-mode ETL Model: Complex Wave Modes 122 5.2.3 Design of a Microstrip Meander-line Bandreject Filter 126 5.3 Microstrip Comb-line Bandreject Filters 130 5.4 Microstrip Hairpin-line Bandpass Filters 133 5.5 Tunable Microstrip Meander-line Bandreject Filters 136 5.6 Conclusions 143 SUMMARY AND FUTURE WORK viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 BIBLIOGRAPHY 148 APPENDIX A Evaluation o f Equation (2.87) 158 B Evaluation o f Equation (2.125) 161 C Evaluation o f Identities 164 D Grating Voltages and Currents 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FIGURES FIGURE 2.1 Geometry o f a periodic strip grating. 2.2 Cross-section of a strip grating on “fast-variable” coordinate. 3.1 A periodic array of strips loaded by a dielectric slab over a ground plane: (a) cross section; (b) top view. 3.2 7 24 56 Comparison o f P between quasi-TEM Green’s function method and homogenization EBCs method for even and odd modes. The parameters used are: kod = 3.325e - 3 , p / d = 1.632, a /p = 0.5588. and e r = 6.5. Weiss’ result (— ); present method (—). 3.3 75 Comparison of normalized propagation constant versus propagation angle. The parameters used are: kod = 2.0944e - 3, p / d = 1.86, a/p = 0.07, and e r = 9.6. Bellamine’s result ( - •); Crampagne’s result ( - -); present method (—). 3.4 77 Comparison o f Zc between quasi-TEM Green’s function method and homogenization EBCs method for even and odd modes. The parameters used are the same as those in Figure 3.2. Weiss’ result (— ); present method (—). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 4.1 Top view o f a microstrip meander-line structure consisting of an infinite array o f strips. 4.2 82 Top view o f a microstrip meander-line structure loaeded with series impedances Zi and Z i . 83 4.3 Hairpin-line structure with a defined unit cell. 88 4.4 (a) A comb-line structure loaded by reactances; (b) a unit cell of (a). 91 4.5 Comparison o f dispersion diagram (backward-wave) of a meander line between homogenization method ( - - ) and Crampagne’s result (—) with a = 0.13mm, b = 0.8mm, d = 1.0mm, L = 18.36mm, and e r —9.6. 4.6 101 Dispersion diagram (backward-wave) o f a meander-line structure with p = 1.2mm, d = 1.012mm, L - 18.36mm, and e r = 10.2 for ajb = 2.0 (—), 5.0 (• •), 1.0 (— ), and 0.5 ( - •). 4.7 102 Calculated image impedances (forward-wave) of a meander line with p - 1.2mm, d - 1.012mm, L = 18.36mm, and e r = 10.2 for ajb = 2.0 (—), 5.0 (• •), 1.0 (— ), and 0.5 ( - •)- 4.8 103 Dispersion diagram (backward-wave) of a meander-line structure with a = 0.8mm, b = 0.4mm, d - 1.016mm, and z r = 10.2 for 4.9 L = 18.9mm (—), 18.36mm (• •), 17.5mm (— ), and 19.5mm ( - •). 105 Example o f a diode-loaded meander-line structure. 106 4.10 Dispersion diagram (backward-wave) o f a loaded meander line XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for Ct = 1 p F (—), 15 p F (• •), 5 p F (— ), and 0.5 p F ( - •) with a = 0.8mm, b = 0.4m m , L = 19mm, d = 1.016mm, and e r = 10.2. 107 4.11 Normalized group velocity o f a meander-line structure with a = 0.13 m m , b = 0.8mm, d = 1.0mm, L = 18.36mm, and e r = 9.6. 109 4.12 Dispersion diagram o f a hairpin line for c/ = 1.27mm, L = 18.36mm , and e r = 10.2in terms o f a and b . 110 4.13 Dispersion diagram o f a hairpin line with a = 0.8mm, b = 0.4mm , d = 1.27mm, and e r = 10.2 for L = 18.9mm (— ), 18.36mm (—) , 18.8mm (• •), and19.2mm ( - •)• 111 4.14 Image Impedance of a hairpin line for p = 1.2mm and d = 1.27mm with (a) L = 18.66mm and a /p = 2/3; (b) s r = 10.2 and a/p = 2/3. 113 4.15 Dispersion diagram of a comb-line structure for a = 0.8mm, b = 0.4m m , L = 18.36mm, d = 1.016mm, and e r = 10.2. Equivalent reactances: Ls = 0.7239/?// and Cs = 0.0112//F with Cp = 0 (— , C s = 0 ), 0.03p F ( - •), 0.3//F ( - -), and \.0pF (” ). 115 4.16 Calculated image impedances of a comb-line structure in terms of a and b with p = 1.2mm, L = 18.36mm, d = 1.016mm, and e r = 10.2: a/b = 2.0 (—), 1.0 (• •), 0.5 ( - - ) , and 0.2 ( - •)• xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 5.1 (a) Top view o f a microstrip meander-line filter, (b) Equivalent transmission line model of (a). 5.2 120 (a) Dual-mode ETL model for a meander-line filter, (b) Admittance-matrix representation of (a). 5.3 124 Complex propagation constant y of a microstrip meander-line filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, Z, = 18.36 m m , and e r = 10.2: y in passbands (—) and P5 ( - •) and - a s (• •) in stopband. 5.4 127 Measured (• •) and computed (—) responses of a microstrip meander -line bandreject filter with a = 0.8mm, b = 0.4m m , d = 1.016mm, w = 30mm, L = 18.36mm, and e r = 10.2. 128 5.5 An example o f a microstrip comb-line bandreject filter. 130 5.6 Measured (• •) and calculated (—) responses o f a microstrip combline filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36 5.7 mm, w= 14.4mm, and s r = 10.2. 131 An example o f a microstrip hairpin-line bandpass filter. 133 5.8 Measured insertion loss of a microstrip hairpin-line bandpass filter for a - 0.8mm, b - 0.4mm, d = 1.27mm, L = 18.36mm, w = 31.2mm, and e r = 10.2. 5.9 134 Calculated responses of a microstrip hairpin-line bandpass filter with a = 0.15mm, b = 2.0mm, d = 0.25mm, L = 18.36mm, xm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w = 43mm, and e r = 18. 5.10 An example of a tunable microstrip meander-line bandreject filter with loaded capacitances. 5.11 Dispersion diagram (backward-wave) o f a loaded meander line for C* =1.5 p F (— ), 3.3 p F (• •), 4.8 p F ( - •), 10 p F ( - • • •) , and short link (—) with a = 3.5mm, b = 2.0mm, L = 100mm , d = 1.27mm, and e r = 10.2. 5.12 Measured responses o f a loaded meander-line bandreject filter for G =1-5 p F 3.3 p F ( - ) , 4.8 p F ( - •), 10 p F ( - • • •), and short link (• •) with a = 3.5mm, b = 2.0mm , L = 100mm, d = 1.27mm, w = 110m m , and e r = 10.2. 5.13 Calculated ( - -) and measured (—) 1^21| of a loaded meander-line filter with a = 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, w = 1lOmm and e r = 10.2 : (a) short link; (b) C j= 10 p F ; (c) G = 4 .8 p F \ (d) G =3.3 p F . 5.14 Calculated image impedances of a loaded meander line for Cs =1.5 p F ( - - ) , 3.3 p F (• •)» 4.8 p F (-•), 10 p F (------ ), and short link (—) with a = 3.5mm, b - 2.0m m , L = 100mm, d = 1.27mm, and e r = 10.2. 5.15 Measured transmission loss of a meander-line bandreject filter xiv Ii j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (— ) and hairpin-line bandpass filter (• •) with the same circuit dimensions: a = 2.5mm, b = 2m m , L = 50m m , d = 1.27mm , w = 90m m , and e r = 10.2. xv | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION 1.1 Thesis Goals The method of homogenization had been used to derive the equivalent boundary conditions in analyzing a strip grating antenna [11-12]. The use of homogenization method simplifies the antenna analysis and requires less computational memory compared to the analysis o f numerical method, yet the method employed in [11-12] is not quite general for most microwave circuits since the required phase shift between adjacent strips o f a periodic strip grating was assumed zero. In this dissertation, the equivalent or averaged boundary conditions (EBCs or ABCs) of a semi-infinite strip grating located at two different media are derived by using a modified method of homogenization based on the technique of multiple scales; the developed EBCs successfully modify the previous results [9, 1112] by considering a general phase shift between adjacent strips and are suitable for two different media on either side of the grating. To investigate the characteristics of surface-wave propagation on the grounded dielectric substrate covered by periodic strips for circuit application, the modified EBCs have been shown to make the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. analysis analytically simple and computationally fast compared to the results using numerical method [57-59], By making connections or disconnections between finger ends of periodic strips on grounded dielectric substrate, several compact planar grating elements are formed such as microstrip meander lines, hairpin lines, comb lines, and so on. Periodic strip grating structures can propagate slow-wave whose group velocity is less than the velocity of light and have pass bands and stop bands. By imposing the proper boundary conditions o f strip ends of the periodic grating elements, loaded or unloaded with lumped devices, the dispersion equations of microstrip grating elements are thus obtained and the calculated Brillouin diagrams show that they exist filtering properties similar to filters. Using an accurate and efficient model for metal gratings on grounded substrate based on the obtained EBCs, a new class of microwave grating filters is designed and their behaviors are also verified by experiment. Especially, when a meander line loaded with lumped elements, a promising bandwidth-tuned grating filter is constructed; this achievement enable us to design a voltage-tuned filter if nonlinear devices such as varactor diodes are loaded with a meander line. The impact of these grating filters promises to deliver broadband properties or highly selective behaviors and these techniques are applicable to more compact filters used in microwave and millimeter-wave integrated circuits. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 Thesis Overview This thesis contains six chapters. In chapter 2, the modified method of homogenization based on the technique of multiple scales is used to analyze a semi infinite periodic strip grating in a general manner. Then the equivalent boundary conditions (EBCs) for the average fields at a periodic array of perfectly conducting strips with a uniform phase shift between adjacent strips are developed. The firstorder EBCs which take into account the phase shift between strips are obtained under the assumption that the period length of the grating is much smaller than the wavelength o f incident wave such that the fields approach the average fields existing around the grating at a appreciably exceeding the period. Also the derived EBCs here generalize the previously obtained results. Chapter 3 deals with the propagation characteristics of surface waves along a grounded dielectric slab covered by periodic strip grating using the developed EBCs in chapter 2. Without loss of generality, the fields are assumed to have a uniform phase shift between adjacent strips and propagate at an oblique angle with respect to the strips. With the aid of EBCs, the propagation constant and characteristic impedance o f a strip grating on grounded substrate are obtained analytically in a simple and accurate manner compared to the results using numerical method in some other literatures. Chapter 4 presents the dispersion equations and image (iterative) impedances o f periodic grating elements loaded or unloaded with lumped devices, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i such as microstrip meander lines and hairpin lines, etc, formed by truncating an infinite array of strips. The calculated Brillouin diagrams predict that these grating elements have filtering properties and have the potential application in compact filter design. The calculated filtering characteristics of grating elements loaded with lumped devices are also presented; they find possible applications in electronically tunable filters. Chapter 5 compares the measured and calculated results of microstrip meander-line bandreject filters, hairpin-line bandpass filters, and comb-line bandreject filters. There the calculated responses are obtained by using the equivalent transmission line (ETL) model based on the obtained EBCs by considering normal-mode and complex-mode propagation along the grating structure. The measured results o f a microstrip meander-line filter loaded with lumped devices are also presented for the promising application in bandwidth-tuned filters. Finally, Chapter 6 make conclusions from the previous chapters and points out the suggestions and directions for future work. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 EQUIVALENT BOUNDARY CONDITIONS 2.1 Introduction This chapter develops equivalent boundary conditions (EBCs) of a periodic metal grating o f strips with uniform phase shift between adjacent strips using a modified method o f homogenization based on the technique of multiple scales. The development o f EBCs is useful for efficiently modelling the propagation characteristics o f fields on periodic slow-wave grating structures which form the basis of many microwave and millimeter-wave components. Several investigators have previously formulated equivalent boundary conditions for a strip grating assuming that there is no phase shift between conductors [1-4]. However, this assumption will not be valid for many microwave or millimeter-wave slow-wave structures [5-8, 26, 57-59], Ivanov [9-10] derived equivalent boundary conditions for an array o f strips or circular conductors excited by a field with a phase shift in free space using a quasi-static approximation, and these generalized the equivalent boundary conditions obtained by Vainshtein [4]. Although Ivanov's research was a pioneer work, it is not directly useful in application to microstrip or other substrate- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. based structures because it cannot describe a grating at the interface between different media. Such situations will be treated in this chapter. In this chapter, we will follow a modification o f DeLyser’s approach [1112] on homogenization analysis o f strip gratings with zero phase shift and obtain EBCs for a strip grating lying at the interface between two different media with nonzero phase shift. The detailed mathematical treatment o f the homogenization method has been discussed extensively in [13-20] and will not be repeated in this dissertation. 2.2 The Geometry and Assumptions Assume the periodic structure under analysis consisting of infinitely long and ideally conducting metallic ribbons of zero thickness in the z -direction and extending to infinity in the y -direction with period p , width b, and spacing a is located between two different media as shown in Figure 2.1, where the material properties are denoted by a “ + ” subscript for x > 0 and a “ subscript for x < 0. An assumption is made that the period length p of the structure is much smaller than the wavelength of incident waves. If incident waves exist in x > 0 plane, for example, they cause scattered fields everywhere. Thus the total fields can be represented as: 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t £ + = £ ? c + £+ ^ H+ = H T + Hs+ x > 0 and (2.1) « E l = EsH i = H t x < 0 t A J M w F mmm a Figure 2.1: Geometry o f a periodic strip grating. where the fields in equation (2.1) include the phase shift 0 occurring from the incident fields and -7t < 0 < 7t is assumed in the following derivation. The incident 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. waves in (2.1) can be represented as ~ e!+ c = E inc(r,Q)e~J®y' and owing to the dispersive homogenization theory [17-23], a more general incident waves can be handled as £ + c = ][inc(r,Q)e~jQy'dB where the variables r and y ' will be defined below. For a wave is incident on the metal strip, there exists a phase shift 9 between adjacent strips. Since the grid period p is much smaller than the wavelength of incident waves, the fields approach the average fields existing around the grating at a distance appreciably exceeding the period. Therefore a perturbation method called the technique o f multiple scales is applied [13-16] where the scattered fields are expanded in powers o f the period p . Assume the scattered fields are functions of a “fast” variable r ' and a “slow” variable r which are defined as: r = (x, y, z) . , r = rjp (2-2) where x , y , and z are the usual rectangular coordinates. Here we assume the “fast” variable is independent o f the “slow” variable temporarily [15], Thus the del vector operator is defined as: 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where we assume there is no field dependence on the “fast” variable z ' and this assumption will be used in the following derivation of EBCs. The MKS system of units and the time dependence eJat are used. Now expand the scattered fields with a phase shift in powers of p , Es ~ E ° i r X , y ’) e - j Qy' + P E l (r’x ' y ) e ~ JQy’ + 0(p2) Hs where e ~ ' e ~ jQy' (2-4) H ° ( r , x ' y ) e - J Qy' + P H l ( . r , x ' , y ' ) e - j Qy ' + o ( P 2) and e~^y \ i = 0,1,2 •••) are vector fields containing the non boundary layer fields (in captial letters) and boundary layer fields (in lower letters) which are defined as: El ~ E‘(r,x^ + e ' i r y y ) H 1~ (2.5) + V (r, x ' , y ' ) According to Floquet’s theorem [24], e i = 0 , 1,2 ■ ‘ and H l in (2-5) will correspond to the fundamental Floquet-Bloch mode, while e‘ and h l are the sum of all the higher Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. order modes. Also £ ' and f f 1 are not function of y ' because they are defined as the averaged fields of e ‘ and h ‘ over one period, which are E !(r,x') = f o E \ r , x ' y W —’ ( 2 .6 ) p l_ : H l (r ,x ’) = | aH , ( r , x ' , y ' W i = 0,1,2 - and equation (2.6) results in foe '( r , x \ y ') d y ' = 0 and £ h ‘ (r,x\y')<fy' = 0 i = 0 ,1 ,2 - (2.7) which means that the averaged boundary layer fields without e~jQy> dependence over one period are zero. Equation (2.7) is useful and important in our derivation of the EBCs and it also implies that e' (or p ) is periodic in y ' with a period of 1. 2.3 Maxwell’s Equations and Expansion of the Scattered Fields Apply the source-free Maxwell's equations to the scattered fields V x £ * = -j<a\iHs (2.8) V x f j s = y<ae£* and from equations (2.4) and (2.5), we have the scattered fields expanded in powers of p 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E? - [ E V . x O + e ^ r .x '.j O le - j0/ + + * ', / ) ] +o(p2) (2.9) H s ~ [H° (>.*') + A1V , x ' . y ’yie-fiy' + p[H l (r,*') + /»I(r >Jc'.yX k”^ ' +o(p2) We substitute equation (2.9) into Maxwell’s equations and then equate identical powers of p on both sides o f (2.8), and get P~l : p°: Vr - x (£ ° + e ° )+J/9 (£ ° +e°) x a y = 0 Vr x ( £ ° + e°) + Vr - x ( £ l + el) + J6( £ l + el) x a , = Pl■ + h°) ^210^ v r X (E l + e1) + V r- X ( £ 2 + e2) + 7 0( £ 2 + e2) x = - > H ( ^ l + Al) Now, take the average of the first equation of (2.10) over one period. W e have V r>x e° = 70 ay x e° (211) where equations (2.6) and (2.7) have been used. Again, take the average of the second equation of (2.10) and use in addition: 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j lQ(V r x & W = 0 and f^ (v r . x =0 / = 0,1,2 — (2. 12) Then we get Vr x £ ° + v r *x £ l + 70 £ l x gy = - j o (XH ° (2.13) Substituting equation (2.13) into the second equation of (2.10) yields (Vr - - y'0 a y ) x el = - J o H f ? - V r x e ° (2.14) Next, applying the operator (Vr«-JQ ay) • to equation (2.14) yields after some manipulation (V r ' -./Q o y M n /j0) = 0 (2.15) Similarly, following the same procedure for the third equation of (2.10), we have ( V r>-jQ ay ) h l = - V r t (2-16) Equations (2.11), (2.14), (2.15), and (2.16) are important for determining the boundary layer fields as will be shown later. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If we start from the second equation o f (2.8), the similar results to equations (2.11), (2.14), (2.15), and (2.16) can be obtained. Thus, gathering the results for boundary layer fields, we have: (Vr »- JQ ay ) x e° = 0 (2.17) (Vr ' - jQ ay) x /j° = 0 (Vr>—jQ ay) •(e e°) = 0 (2.18) (Vr - —jQ ay) •(n h°) - 0 ( W - jQ ay ) x e l = - Vr x e° (2.19) (Vr <- ; e ay) X A1= jme e° - Vr x h° ' (Vr <-y’0 ay)-el = - V r -e° ( 2 .20) (V r. - j B a y ) - h l = - V r h0 Note that when the phase shift 0 = 0, they reduce to DeLyser’s equations (2.20) (2.23) [12] as expected. 2.4 Boundary Conditions The boundary conditions at the plane o f the grating are: ax x(]E+C+ E+~ E - )|x=x'=0 - 0 ; ~E+C= E ‘+e~jQy 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.21) ax -[e+( £ ? C+ £ “1) - £ - ^= x'= 0 = P. (2.22) = [Ps(r) + P°s(r,r')\e-JQy' + p[Pls (r ) + P\( r ,r ') \e -j*y' + 0 ( p 2) ; H T ^ lf+ e -fly ' a * - [p + ( t f ? C+ 7 7 * ) ' P _ t f ^ x ' =o = 0 (2.23) a x x [ t f T + 7 & -7 F - ^x=x'=0 = (2.24) * [ j h r ) + f s (r,r')]e-JQy' +pCA(r) + j ls (r,r')]e-j* S + o (p 2) where P5 and j s are the surface charge density and surface current density which contain averaged and boundary-layer components have been expanded in powers of p . Then substitute equations (2.4) and (2.5) for E s and Jjs into equations (2.21) (2.24) and collect the like powers o f p to get the p ° components _ ,_0 -0 \ — r—i —0 —0 , ax x (e+ - e - )x=r--o - - ax x [£+ + E + ~ E - \ix=x'=0 (2.25) — r — 0 — 0l Clx'\£ + e+ E-e- Jjjc=ar'=0 - - a x - [e+(H!+ + E+) - s _ 0 30 ^ o0 )x=x'=o + /*? + ? (2.26) - (K ax x (fj+ h-j^x=x'=0 - r —zl — 0 —0 -,o . -0 1 (2.27) -0 - ~ ax x \jf+ + H +~ / / - ^x=x'=0 + J s + J 5 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _ r -0 ax-[V-+h+ ~ -0 ]jx=x'=0 (2.28) = - ax ■[ n + Q /V + h I) ~ H_ H - }x=x'=0 and the p l components ax x (e+ - e l -\ x=x >=q - - a x * [E?+ - E l- )x=x'=0 I ! (2.29) ax •[e + e+ - e _ e - ]|x=x'=0 (2.30) - - a x - [ e +£ i-e _ £ L ] |x = x ,=o + / >! + Py ax * (/i+ - /jl )|x=x'=0 (2.31) - - a x x ( t f i - H - ) |x=x'=0 + v i ax ■[M-+h+ ~ t1_ h - ]jx=x'=o (2.32) = ~ ax •O + h \ - H_H - ^x=x'=0 The boundary conditions (2.25) - (2.32) and equations (2.17) - (2.20) can be combined and reduced to the boundary-value problems for the zeroth order and first order boundary layer fields which will be discussed below. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.S The Zeroth O rder Fields Now, we have to rewrite equations (2.17) - (2.20) for easily solving the static electric and magnetic boundary-value problems. Let us define new functions e' and H' such that (2.33) i = 0,1,2 - Then equations (2.17) - (2.20) can be rewritten in terms of e' and h' as ' Vr 'x e °=0 (2.34) Vr -x h °=0 ' Vr '-ee°=0 (2.35) Vr .-Hh°=0 Vr ' x el = - y o n h 0 - Vr x e° (2.36) Vr »x 5 1 = yoee^ —Vr x ii0 Vr '- e l = - V r e (2.37) . V r - E l = - V r -E° 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equations (2.34) - (2.37) are similar to those obtained in DeLyser's results [11-12], We can follow the same process and assumptions as done in [11-12] to obtain the zeroth order fields. 2.5.1 The Zeroth O rd er Magnetic Fields From equation (2.34), we find as in [ 11] that e° = h2 = 0 (2.38) The zeroth order z -directed boundary layer fields are zero which imply from equation (2.25) that the z -components of the “slow” variable fields are continuous. Now, we need to solve for the magnetic field h°. From the second equation of (2.35), we define a vector potential A such that Uh° = Vr «x(Ae-fly") (2.39) where A = a z A z ( x ' , y ' ) is periodic in y ' with period 1. Because A is periodic in y ' with period 1, A can be expanded in a Fourier series as: a o(x') A z ( x ' , y ’) n=» n=Q0 n = ® = — - — + £ a a(x ')co s2n7ty, + £ 6 n(x')sin2m ty' 2 n=l n=l where 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.40) r I/2 a n = 2 1_^ Az ( x y ') cos 2 nny 'efy' n = 0,1,2 ' (2.41) fl/2 bn = 2 J-i/2 A z ( x 'j O s i n 2 n 7 t y '# ' n = 1,2 — Since /j° has zero average in one period with respect to y ' as mentioned previously, A also has zero average in one period. This leads to ao(x') = 0. Thus equation (2.40) becomes the following expression by multiplying e~JQy' on both sides of equation (2.40) n=oo n=ao A z {x’, y ' ) e ~jQy' = ^ aa(x')e -JQy' cos2mty’ + 5 ^ n ( x ')e"J0'y' sin2n;ry' n=l n=l = 2 Cn(x ') ey(2ll7l~ 0 ) y n=-oo a *° where Cn = J_V^ ( A z e - j Qy')e-->i2Tm _9)y'cfy' n*- (2.43) If taking the curl o f equation (2.39) with respect to the “fast” variable and using equation (2.34), we have V ^ x [ V r f x ( A e -ye/ ) ] = 0 (2.44) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or by a vector identity a2 « a1 { k z e - f l y ,) + -2- T { \ z e-fly') = Q dx'2 dy'2 (2.45) Substituting equation (2.42) into equation (2.45), we have d2 2 - c n( x ') - ( 2 n7t - 0 ) c n(*') = 0 dx'2 (2.46) whose solution is Cn(x ') = A ne(2n7t- 0)jf' + B ne - (2rOT- 0)jf' n = ±l,±2,---- (2.47) where An and Bn are constants. Then the potential A z on either side o f the grid is given by equation (2.42) which is Az + e - W (2.48) = S Bne - (2lut “ 0) * V ( 2im - 0) / + A.a e _(2im +0)*'e--/(2mt + 0).y'] a=l for x ' > 0, and i 19 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A Z-e -J Qy' ~ ^ [ A n e ( 2n7t - 0 )Jr' ey(2nji - 0 ) / + B_n<?(2n* + 9 )* 'e -/(2im +0).y'] n=l for x ' < 0. From the continuity o f |dh° at x = 0 , the x -component of (2.39) together with equations (2.48) and (2.49) give the conditions for the constants A n and Bn : (2.50) where g n and f n are constants yet to be determined. Thus, the potential A z in equations (2.48) and (2.49) now becomes A z+ e-W n=0° = ^ ] [ - . / g ne “(2im - 0)*'e/(2n* ~ Q)y' + j f ae~(lnn +0)*'e- /( 2iHt +0 ) / ] n=l C2 51) for x ’ > 0, and P iz-e -W n=o° - j S ne(-2lm -Q)x'ej(2m -0 )y + y fQg(2njt +Q)x'e-j(2m + 0)/] n=l 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 52) for x ' < 0 . Then from equation (2.39), the zeroth order boundary layer fields without the phase dependence e ~jQy' can be solved as 1 n h i+ = ---- £[(2n7C + 0 )fn e _(2im +Q)x'e- j 2rmy' + n=l (2n7t - 0 )gne~(2lut - Q)Jc'ej 2imy'] (2.53) 1 n h%_ = — £ [ ( 2mt + 0 )fn e (2rot +0)^ 'e-y2njry' + n=l (2mt ~ 0 )gae(2mt ~Q)x 'e jinny'] and = — T [/(2n7t + 0 ) f ne -(2nn + 0)x'e -y2njt/ _ y(2njt - 0 )gne _(2nn -Q)x 'ejloxy'] i n=oo ifi "V- = — 2 ["-/(2niz + 0 )fn e (2mt +Q)X' e - j 2imy' + I*- n=l (2.54) j ( 2an - 0 )gne(2mt -®)x'e jinny'] which can be used to determine the surface current density on the strips. From equation (2.27), the ay and a2 components of the surface current density are 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J % + j% - [hy+ - hy_ + H y + + H y + - Hy_]^x =x'=0 * (2.55) J s y + j s y = ~[H z+ + H'z+ ~ H*z- ^x=r'=0 where h? = 0 is used. Let Kh = (H°y+ + H ‘y+ - H°y_ ^x=*'=0, and note that the transverse current density + yO j js identically zero as obtained in [11- 12]. Thus we have J sz+ J< L = K h + (hy+ - Ay_)|x=x'=0 (2.56) (H°z+ + H iz + - H ° Z-] x = x '= o = 0 Substituting (2.54) into equation (2.56), we obtain fsz = K h - J % + (— + — ) 5 f[/(2n7t + 0 ) f ne - / 2n n / - j ( 2 n n -Q )g neJ*my'] lt + j (2.57) It- n=l j n=oo = Kh ~ J% + (— + — ) ^ [ Gn cos 2n7ty' + Fnsin 2nny’] lI + I1- n=l where G a = y(2n7t + 0 ) f n - y ( 2n7i - 0 ) g n Fn = (2n7t + 0 ) f n + (2n7t - 0 )gn 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.58) Note that the right-hand side of equation (2.57) is the Fourier series representation o f j \^ which is periodic in y ' with period 1, so the Fourier coefficients are K h -J L ^ l^ lw w (2.59) (— + — )G „ = 2 j ^ n j l , ( y ’) c o s 2 im y ,d y ’ (— + — )F„ = 2 it M*— j % ( y ’) s m 2 n x y ’d y ' n = 1,2,3 Using the strip geometry on the “fast-variable” coordinate shown in Figure 2.2, the first equation o f (2.59) can be rewritten as (2.60) since J ^ + 7^ = 0 in the gaps. Then equation (2.60) becomes M (2.61) ' w 23 .. i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.2: Cross-section of a strip grating on “fast-variable” coordinate. The second equation o f (2.59) gives Gn = -b /ip [J°sz + fsz(y')]cos2My'cfy' 2(1 |i_ where u = ---and since H+ + H_ „ (2.62) is .y'-independent, the following identity has been used: \ ^ P/szcoslvm y'dy' = - ^0 J%cos2mzy'cfy' Jbf2 p Similarly, from the third equation of (2.59) we get 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.63) p®/2|> .n Fn = V-a]_m p Jsz(y ')v n 2 M y 'd y ' (2.64) To evaluate Kh, Fn and Gn explicitly, the zeroth order z -directed surface current density 2.5.2 must be evaluated. The Surface C urrent Density According to the boundary condition on the strips, the normal component of the magnetic flux density b equals to zero, i.e. (/»2 + # 2 + / & W = o = 0 on strip where hx can be found from equations (2.33) and (2.39) (2.65) as h°x = - i - ^ 7 A z - j Q A z) |i qy (2.66) Thus equation (2.65) becomes d ( - v , A Z- jQ A2)|x=x'=0 = °y + ^/x)|x=x'=0 1 1 — ~ on strip _ p0 D xi\x=x"=0 Obviously, the solution of equation (2.67) is 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.67) eP Az,x=x,=o = c iejQy' + (1 - ej Qy') where c\ is a constant to be determined and on strip (2.68) can be found from equations (2.51) and (2.52) as Aa„ x..o = "ft - j Z n ‘ J l m y + J n=l (2.69) n=oo = Z t(Sn + f n)sin lx™y’ + A f n “ 8n) cos 2n7^ ' l n=l Note that (fn + g ^ and (f n ~gn) can be found in terms of Fa 111(1 G n from equation (2.58), they are (f + s ) = 2mt Fa +jQ G n U n gnJ (2n7t + 0 )(2n7t - 0 ) (2JQ) re - a \ = - ^ F n - ^ n T tG n U n gn' (2n7t + 0 )(2n7t - 0 ) Substituting F n and G a in equations (2.62) and (2.64) into equation (2.70), then (2.69) becomes 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dO ^x/V=o — {cie-fiy' + — -^-(1 - eJQy')} V-a jQ 2k = 0 4 J-*/^ ♦a o o i i (n + 5 )(n -8 ) (2.71) 5=i (n + 8 )(n -5 ) n=oo y'S cos InKy' sin 2 nTty' - JJ-*/2p L V ^ o " ) “s Ly' T ( 7n +«5 v) ( n" -~5 ) : r ■»' where 5 = 0/2 7t. To solve equation (2.71), we separate into the even and odd parts: (2.72) Substituting (2.72) into equation (2.71) and after a little algebra, comparing the even and odd functions on both sides, we get pU 2k Dxi\x,=a Va jQ ~ pO + (ci~ ^xiV-o ) cosOy'} fi 5=1 n=oo - r '/ s o '^ z (n + 5 )(n -5 ) y'5cos2n7ty' cos 2n7ty " <fy" (n + 5 )(n -5 ) 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and pO 27C , a xi\x*=Q — { (/C l------r — )sm 0y } 0 =Jo 5=1 V (n + 5 )(n -5 ) <?■») fA/2P 0Of 'y ° n sin 2n?ty' sin 2n7g/" , J0 J s z W ) ^ (n + 5 )(n -8 ) ^ The even current density fj£ and odd current density fj£ will be found below. 2.5.3 The O dd C urrent Density Taking the derivative o f equation (2.74) with respect to y ', we have pO 2rc f £>x/|x’=o — KC1------X—) cos03/ } Va JQ Jo 5=1 p -w ) (n + S )(n -5 ) . Cbl2p ;Oo/ ..„xIl^ 0 n2cos2n7^, sin2n7ty"_t „ + Jn Jo Jsz (V ) 2 * 5=i T? TiTv sT y 5 ( n + 5 ) (n - 5 ) ^ and then subtracting equation (2.75) from equation (2.73) gives ^Xi\x'-n Min o. n 00 — = - \ bQ2Pfsz O'") £ c o s 2n7iy 'sin 2n;iy ’d y” 6/2 d = -JL a 1 n=oo n ^ ° n=ao n =oo y'SO'")r { £ s m 2 n jiO '+y") - £ s in 2 n x 0 ” - X W 2 n=l n=l 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.76) Using the formula j x sinnx = —cot— n=l 2 2 n=oo 0 < x < 7t (2.77) and after a little algebraic simplification, equation (2.76) yields nO ^xi\x'= 0 _ 1 rW p oo „ = IJ o ^ C v " )[c o tn C y ' + / ' ) - c o t 7 t ( y - y w 4 (2.78) = 1 f ' ^ ( T ) ------------------------ dy" 2 ■'° cos 2-izy' - cos 2xy " Note that equation (2.78) is exactly the same as DeLyser's equation (C.43) [12], thus we have the solution for y g (V > -— i ^a 2.5.4 : (2.79) Jsin 2——~ s in 2tty' V 2P The Even C urrent Density Substituting (2.79) into equation (2.73), we get an integral equation for Q J % + jg ) 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A * £ (? ■ )]■ E S=i r e Jo pO (n + 8 )(n -5 ) pO 2;c . ^ x/Ijc’sO , "xrV=o. _ ,, = — {— + (C1--------- ^ - ) c o s e y '} Va JQ _ 1*6 Jo / 2 (2.80) sinny"______ r^ ° cos2 nfly'sin2 m t y " „ »a I ■ z b* JsurV 2p surfly (n + 5 )(n -8 ) ^ Use the integral representation of a Legendre polynomial [25] f6*/2p cos(2n +1)/ 7C Atc, J —r = = = = = = d t —~ P n(cos ) • 2 blz ./sin r V 2/? ■ 2 , sm * 2 , , on (2.81) ^ where p n is a Legendre polynomial of order n. Then equation (2.80) becomes Jo 2i t r Q, o . [ e ia ^ +^ h=i (n + 5 )(n -S ) ,c pO l-c o s 9 y ' J °xi\x'=Q n ” cos2nny' _ _ ] _ _ ^ [g _ _ (2-82) .... s . (A)1 where ^ ( A ) = P n _ i(c o s A ) -/, n (cosA) = / >_n ( c o s A )-/)n(cosA) and A = bit/p. Letting the right-hand side of equation (2.82) equal to f ( y ' ) 7 an alternative representation o f (2.82) is 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or in integral form r t 2 / a + ^ c v " ) ] Si i cosf ° s 2-S " )f v Jo n ( n^+ 5 ‘)(n (2 84) = - 47t 2J0 Jjj 2f ( y ' \ ) d y ’\ d y '2 + A.\y' + A.2 where Ai and A2 are constants. Since the left-hand side of equation (2.84) is an even function, the right-hand side must be an even function too. Thus we have the condition o f Ai = 0. Now, apply the operator (d2ld y '2 + 4 n 25 2) on both sides o f equation (2.84), we have (2.85) where A 3 is a constant. Substituting fly ') which is the right-hand side o f equation (2.82) into equation (2.85), after some manipulation we get 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 271 e ^x»v=0 - f c 1+ — * , B xi\x'=o n^ f c o s 2 n 7 ^ ' 2, > - _ _ _ ( £ _ _ S l(4 ) I B»(. + « X n - « ) S- M ) * A i 2y (Aa n^ ° ^ n ( A ) n^ ° 5 n (A ) n=l n=l a=lH2 - 5 2 „ „ R§ (A') = P_§ (cos A') + P s (cos A'), P§ + X T~+ S n (2 '86) A r - c o s 2n7t/ } + A 3 n-4 Now, in Appendix A it is shown that y ° S n(A) _ k R 8 (A ’) n^l n 2 ~ 5 2 where 28 sin57c 1 52 is the Legendre function of noninteger order 5 , A = b n /p , and A' = a n / p . Furthermore, from (A.6) and (A.7) n=l nz n=l nz _ lim rV" ^ n ( ^ ) = c -> o l2 * n=l n 2 2 ~ •S'n(^) „ o n_ /-i 2rcos2n 7iy ] n=l n 2 - ? = 27t2/ 2 where L’Hospitai’s rule has been used twice. Thus equation (2.86) reduces to 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f'* ??\ (2.88) _2k \ia Cl j Dxi\x'=o r7t R8 (A') u 2sin57c 1 8 3 To determine A 3 in equation (2.89), we compare equation (2.89) with equation (2.82) by letting 8 -> 0, we find that A3 is equal to zero because lim R5 (A ')__ 1_ 5-*° 2sin§7E 28 _ lim rric/>_5 (cosA') 1 , , rn P 5 (cos A') 1 „ ......... = ( - - In — ” S-A') + ln - + 005 A ’) 2 2 2 2 = 0 Finally, we have the integral equation for (2 J% + j ^ ) : 0 0 n=l 11 (2 M ) = \ia li a = 2k, Cl 2sin87c ^ > '= 0 tc/? s(A ') 2rt 2sinS7t 8 1 8 Note that this integral equation is similar to equation (C.35) in [12]. The solution for equation (2.91) is 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 J%+j°sHy') - 8 ft , ii inrc;«2 A \ Hflln(sm - ^ y . o rlt jfe(A ') CI 2k 2sin87C 1„ 5 c o s tt/ (2.92) I . ,A ^ ~ Jsin2 — - s i n 2ny Next, we need to determine the constant ci in equation (2.92). The zeroth order surface current density is Substituting equations (2.79) and (2.92) into (2.93) gives ;0 f . . i \ _ ,0 JSZV / ~ ~ J sz | 2 fP sinity' I-= y]s-sin2ity' ^Xt|jC'=o 7tj?s (A') Ha lns 2sin5Tt 4JCC1 cos7ty' ; ; = ^ a lns ^Js-sin2Tiy’ (2.94) cos Tty' 5 J s - s m 2W where ■J= sin2 (A/2) and J% is constant over y ' as seen in (2.24). Integrating equation (2.94) with respect to y ' from - b / 2 p to b/2p and using equation (2.61), then we have 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Kh ___ ~4 k ci | 2J ^S.y-o tc /?§ (A') Ha ln s f^a ln 5 2 sin57t i 8 (2.95) so that 1 - ^ lnV L i y ^ /'*,°0r* /?g (A'> -4tc 2tc 2sin8it 5 (2.96) Substituting c\ into equation (2.94), the total zeroth order surface current density in the z -direction is thus [J°sz+jsz(y')]e-^y' = —1 A?® »mg ' +^ y j s - sin2 w _ co s^ ]e. j e y (2.97) y j s - sin2w The zeroth order boundary layer magnetic fields are shown in equations (2.53) and (2.54), where f n and gn can be expressed in terms of Fn and G n by equation (2.58), they are c _ Fn VGn n 2(2n7t + 0 ) = n (2.98) Fq+yGn 2(2nx - 0 ) then equations (2.53) and (2.54) becomes 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. h i = -5— Y r ^ — iS±ILe +(2rm + 0 )x 'e -y'2n7t/ + 1 H± n=l 2 F n ~f; / G ne y (2n7t - Q ) x ' e j2zmy'] + , n=o r p2 /,0 = _£L Y [ ! £ n l Z £ j L e +(2mt + 0 )x 'e -y2n n / + * H± S=i P S?) 2 —•/^ rte +(2mt -0)x'gy'2im /] 2 where Fn and G n can be represented in Legendre polynomials by substituting equations (2.97) and (2.94) into equations (2.62) and (2.64). And the integral representation of a Legendre polynomial is shown in (2.81). After some manipulation, we have Fn and Gn as Fn = -fi2,V , 0S„(A) (2. 100) Gn = ^ K n ( A ) where 7?n(A) = / >n _i(cosA) + / >n(cosA) = / >_I1(cosA) + />n(cosA). Note that when substituting (2.100) into equation (2.99), the zeroth order boundary layer magnetic fields are expressed in terms of Legendre polynomials. | 36 i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5.5 The Zeroth Order Electric Fields Following the similar procedures, we assume = -(Vr - - jQay)<& in the First equation of (2.17), where <t>(x',y') is periodic in y ' with period 1. Then we have the surface charge density on strip n 0 . « 0 / . . n _ o , r-0 si*1* / * s LV / ^ S O vi\x'—Q r ' 'J s - s i n 2ny' 47tea c2 *s 1 ^n,y cos7ty' i ■ -~ 'J s - s i n 2 ny' (2 . 101) 2J £ q £ y i\X'=o Ins R$ (A') _ cosny' 2 sin57t 5 J s-sin W 2 sin57t 5 where c i js 4tte a 2tc Then the total zeroth order surface charge density is [P? + P ? C v ') ] e '^ ' (2.103) - n * F° sin Tty' cosny' ~ l2 ea^yi\X'=o I — ----- + Ke I Je VJ - s in 2n y' y /s - s in ny' where ea = ^ ~ ~ and K e = [s +(£x++ E%+) ~ s -£ ? _ ]x=r'=o • Finally, the zeroth order boundary layer fields e0± and e0± are 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ± £ | - K n j ^ y L j i e :p(2njt + Q)X' e - j l i m y ' + eQ x* -— n=l 2 K n ~ =^ ne +(2im -Q )x 'e j2rm y'] 2 n=SO e° ey± _ n=l r (2.104) _ —■ 9 g+(2rot +0)x' e-/2imy' + 2 .L n t Z ^ f l e T(2nff - 0 )x'e J2imy'] where K„ = ^ - ( 2S „) 2s Q (2.105) where 0 n and <Pn are defined in (B.2). 2.5.6 Boundary-layer Grating Voltages and Currents From the zeroth order electric fields in (2.104) and magnetic fields in (2.99), the voltages and current on the surface of grating due to the zeroth order boundary layer fields can also be derived. The voltage at a strip center due to the component of the boundary-layer electric field (which we will call the boundarylayer voltage) is 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vh ^ : 2za i M A ”> + J ^ 4 sm 87i 1 [± - M 2 8tc ^ 1 ] ( 2 m 2 sin87t while the current flowing on a strip is pKh\x'=o 2sin(A 8) , JR^xi\x,=o lh = — T 1----- [Jt Rs (A )------- ^ _ 2 ] + -------- 1— S5 (A) 2* 5 Ho (2.107) The transverse voltage between adjacent strips is given by V&(e~jQ- 1), where JP sin(A'8 ) ^ Fe' M l ^ 5 ) £ lv = o (2.108) The voltage Vfe and current //, are evaluated at .y' = 0. The detailed mathematical discussions are shown in Appendix D. 2.6 The First Order Fields Equations (2.19) and (2.20) and the boundary conditions in section 3 give the boundary conditions for the first order fields, they are 39 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (Vr ' - j Q ay ) x e l = - Vr Xe° (Vr ' - y ’9 ay) - el = - Vr •e° ax x (e+ - ei)|x=x,=0 = ~ a x x \Jl + - £i]|x=x'=0 (2.109) ax ■(e +e+ - e -C-)jx=x'=o = ~ ax •(e+E+ ~ £ —£^- )|x=x'=0 + and (V r' - 7 0 ay) x a 1 = ycos e° - V r x ( V r.-T Q a jd - ^ - V ^ A 0 5x • (H + h \ - H_ h- )|x=x'=0 ( 2 . 110) = —ax'([i-+H+~ M-_//^-)jx=x'=0 Ox x (h+ - h i )jx=x'=0 = ~ a x x [//+ —//_ ]|x=x'=o + «/i + js Expand the first equation of (2.109) and (2.110) by using d/dz' = 0, e® 0, and tPz = 0, we have «*(“dy' 4 -y0 4) - ay^-4+& (tax'^4 ~^ t4+;e 4) obc ay' = a x ( ~ > p^x + ^ -e j) + a y ( - j ® l i h y & ' - j-e $ ) 3r + a* ( ^ - 4 - -|-e$) ay ax and 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission ( 2 . 11 1 ) a x (- z ~ ;h \ ay' - jQ h \) - a y ^ - j h l + a z { ~ - h ly - ^ - h lx +jQ h \ ) ox' dx' ' dy (2.112) = a x C/o e e%+ JJ- hy ) + a v O e e$ hx) + a z (£ - hx - 4~ A?) oz dz ay ox Equating ax and ay components, we have ^^7 4 -jQ 4 =-j®V>h°x +^-eO dy dz (2.113) ^ ~ h \ - jQ h\ = yoe e%+ dy dz and ^ —4 = > H A°+ -?-*? dx oz (2.114) A -lj\ = - j aee0+^-h x . dx dz Note that equation (2.114) gives the easy way to solve for e\ and h . Integrate (2.114) with respect to x ' from ±oo to x ’, we get £ (S 7e**)‘fc' = Ja v- + £ h<y+dx' + f dz for x' > 0, and 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.115) for x' < 0 where g0± and hy^ are shown in (2.104) and (2.99), then equations (2.115) and (2.116) become 2 e \t = -yea V {5 iLtZZlLe+(2n3t +0)x'e-j2mty' + ji=I 2n?t ^“0 SLl—Z f l e T(2n)i -Q)x' ej2rmy'\ 2n% - 9 ' (2.117) _ ny ° , j _ KnJ-y'Lgx x(2tOT +Q)X' -y2ntcy' + dz 2mt +0 ' j - ( K n - y L nx T(2nTC -Q)x'j2n n y’\ dz 2nrc - 0 ' ' Similarly, substitute (2.104) and (2.99) into the second equation of (2.114), we have 2 h \± = ±y<0 e± Y { 1 t£ l n + j/K llg+(2n7t + 0 )x ' e -j2nny' _ 2ntt +0 L n_t .yK n +(2mt-0 )x ' ej2nny'\ 2m i -0 ' (2.118) + _L y (A(Ln. A 0 * ) g+(2rm + 0 )x 'e -j2wty' + & 2n7i +0 ' ■ £ .(E n + /G nx x(2ik -0 )x ' y2m t/\ az 2nrc - 0 ' Then the first order fields in (2.117) and (2.118) can be employed to obtain the equivalent boundary conditions (EBCs) for the total z -directed fields. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7 Equivalent Boundary Conditions (EBCs) Evaluate (2.117) and (2.118) at the interface x ' = 0 , we obtain 24 V { Gn + U * .e-j2imy' + G n _./ F n y2imy'} ' o = “> ^ 2 n tt+ 0 2n7t - 0 ' (2.119) _ " y { — ( Kn +J-l?n) e -J2nny' + l . ( E }L~. J ^ ) e j2nny'\ dz 2mz + 0 dz 2n7t - 0 and (4 + h \- )|*'=o = - > e flny { - - jK n-e - j 2™y + k n -V l ^ a gy2rmy'\ 2n7t +0 2njt - 0 (2 . 120) L y 0/j_(Fn~ j & _ -n) g-y2rwty' + +■ / G n) ;2n ^ ,\ 2n7t + 0 ' d z K 2 mz - 0 ' ' where G „ ± y F „ = • ^ « „ ( A ) + y f l “li.,0S „(4 ) F„±yG „ = (2. 121) K „± y l„ = L n + yK n = 2ea R n (A ) ± y 4 > '- » s »(A) T ^ ^ n (A ) 2e a 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equations (2.119) and (2.120) will be shown to be the main contribution to the EBCs for a semi-infinite strip grating. 2.7.1 EBCs - Electric Field Take the ax component when expanding equation (2.13), we have dy where e\ 4t4 ~JQE\= -J° (2 -122) dz 7 + dy is independent of y ' and e® = -E?z, and e\ - - e \ on the strip. As we know e0 = 0, then equation (2. 122) becomes E°y i - j Q e \ = j(j)B0xi at x ' = 0 (2.123) or J E%\X'=o ~ ez|x'=o ® Bxi\X'=0 (2.124) where the incident fields must be considered. And ^ x,=0 can be found in (2.119) by inserting the constants in (2.121). Appendix B shows that e\\x'=o equals to 44 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. aj2n8y' p> jr ArA^ a + - ( - ^ ) ] f t ( A ,) + dz en 8sm07t (2.125) Substitute (2.125) into (2.124), we have e1 'r ^ “° OZ Ea 4tt5 R$ (A') (2.126) —_ r l — ■c 'Z |x '= 0 or U<»Kh \ia + -L ( (^X±y ) ] S 5 (A') -1 _ “ '~ " nr’a ' d z \ a E m -0 = --------------------- —-----------z|x-° 47t5 R$ (A') (2.127) where as before the p$ are the Legendre functions of order 5 for |S| < 1/2. The average field ( without the phase dependence e~ jQy' ) is given by [H - 12] Eav = (l?nC+ £?) + P E l (2.128) 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If we are interested in the equivalent boundary conditions for z -directed fields. Since the zeroth order fields are continuous, only the first order and higher order fields contribute the averaged fields of the EBCs. E «K - o= p £ * " o Therefore we have and K e ( y , z ) — [5 -t-(£r+ + £ * . ) —£ -£ ^ c _ ]r = x ’=0 = [e + £ j - z - E ^ ^x=x'=0 (2.129) K h (y .z ) = [(.Hy+ + H°y+)- H ° y _ ] x=x'=0 = [H $ - H } ^ = x '= 0 Then, the desired EBC for the z -directed electric field is Ez\x'=0 P S k (A') 47t5 R $(A') 2.7.2 . “ 1 a 1 ^ z a dz (2.130) 1 EBCs - M agnetic Field Now, we need to evaluate equation (2.120) in the gap to obtain the equivalent boundary condition for the z -directed magnetic field. Take the ay component of equation (2.31), we have J\y = - j ' s y - m l - H i V -o - i h l - h i (2.131) Taking the average value of equation (2.131) yields 46 i _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .._ _ _ _ _ _ _ _ _ _ _ _ ___ ____ Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 4 y = - ( f l l _ f li - V = 0 (2 l3 2 ) where we use j j\y d y ' = 0 and j^hldy' = 0 which are mentioned in (2.7). Thus equation (2.131) becomes -J/ sl y ~~W(hz l+ -A"z-J\x'=o 1 \ <2133> Substituting (2.120) into (2.133) yields t ?3 e~J2imy' eJ2imy' |i [ + 1<p" + i -Jsy= > e» n=« ° eS y2iwy' e -j2 sa q f 2n3C- 0 2n7t +0 n (2*134) & 2n7t - 0 a_ n^ ° r e~J2any' ^ dz h 2nn +0 2n7t +0 a eJ2imy 'ln 2nrt- 0 n where Q n and (pn are defined in terms of Legendre polynomials in (2.105) or 0 .2). As we know, j\y + j \ y = 0 in the gap, or 47 . _ L — Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /sy = - ( 4 . - h\_ Jjx’sO = - J sl y = ( H i - H\_ V=o (2-I35) And the zeroth order y-directed magnetic field need to be continuous in the gap, we have ( # y ++ Hy^ + hy+)]jr,=o = (Hy_ 'l' hy_ )|x'=o (2.136) Taking the average value of equation (2.136), we obtain that //^ + + 4 . - 4 - = ° which is equivalent to K h ~ 0 and hy+= hy_. From the boundary condition, the normal component o f electric flux density ~D must be continuous in the gap, this yields [s + (■£*■+ + £r+) —S - ■£*_ )x=x'=0 = (s —ex- —£ + er+ )lx=x' =0 1 1 = Ke in the gap Making use o f (2.135), equation (2.134) becomes 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.137) r- rO a p—jLvm pjinny' ^ x / | x ' •=l zot e~J2my' Y [ — ----- + - ---- —I© 2n7i +9 2n7i - 0 n gjtimy I\z « ej 2iwy' ^-yziuiy e -/2iBiy' - a K e 2 J - ---------------------2nrc+0 n=l 2n7 4 t -0 (2.138) n d , B° x i\ * - = o ^ (re - n ™ y , ej2 m y' > > '= ° '& (~ 7 r )1n4 l' { CD Ka 4k n+?~ n^ ° e /Z iH ty ' /frl n -5 Tn^-55 ~ ]• e -jin n y' ^ nr +o ^ r ]* " (4)} From the second equation o f (2.113) when evaluated on both sides of the grating, we have ^ 7 Aj+ - jQ h\+ = j o e +e§+ + J j / # + (2.139) 4dy'r f i - ~ JQ h\_ =y©e_e§_+oz Subtracting the first equation from the second equation in (2.139) yields h \j- m l- h \j (2.140) When evaluated in the gap, we have the following conditions 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! h \+ -h \_ = - ( H zl +- H \_ ) (2-141) - hy+~ hy_ = 0 Ke —(e - ex_ 6 +e x+ )|x=x'=0 Note that the right-hand side o f the first equation of (2.141) is independent of y ' . Substitute (2.141) into equation (2.140) yielding Q K e = -Q (H z1+~ h \_) in the gap (2.142) Then substitution of (2.142) into (2.138) with the aid of (C.7) and (C.8) yields ( h \+ ~ (2.143) R° —r in o IT® _ ^ f jal,x's^Mr ^5 (^) i - L/© Q^^/i\x'=0 a ( H y‘\x - Q dz 7t5 R$ (A) Thus as stated previously, the EBC for the magnetic field is mainly contributed by the first order field, i.e. ( H t - H 7 ) |*=x'=o = (2.144) ~ f i \ - ) |x=x'=o 50 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.8 Results and Conclusions From equations (2.130) and (2.144), we then have the EBCs for a semi- infinite metal strip grating with a phase shift 0 at the grating surface E^ [W ) ]U a (2.145) rr+ Hz where r r - _ P r ^ S (A) lr r- . 5 Hz [ _ - - - ][/© e a ^ v /y = o ~ ~ ( *6 £ s ( A) dz S5 (A') = £ -5 (cos A' ) - £5 (cos A'), 2 |i , (!_ A = b n /p , A' = a n / p , u = — ±— , a | i + + (i_ 11a )] £g(A ') = £ _ 5 (cosA') + £ 5 (cosA'), s +g_ £a = — -— , and £„• and Bxi denote 2 y total fields including the incident fields. The subscript “ i ” will be omitted in the following expressions. The obtained results will be compared by previous approximation [4, 10] for the strip grating in the air, i.e., |i+ = |i_ = |i0 e + = e_ = eo - From [4], the equivalent boundary condition can be expressed in terms of our notation as follows: E+ z + E~ = ./* o /o (5 )(tfJ; - H ~ ) + / q ( 5 ) £ ( £ ^ - E ~ ) (2.146) for CGS units, or transfer to MKS units as 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E z+ + E Z = j t o / 0( S ) J — ( / / t ~ H V ) + /0( 5 ) - |- ( £ + - E ~ ) Veo ' oz (2.147) where ko is the free-space wavenumber and /0(5) is defined in [10] which is (2I48) Note that Ez = E z = E z at x = 0 plane, then (2.147) becomes which agrees with the first equation o f our EBCs in (2.145) with (J.a = fi0 ^ s a = eo- Similarly, [4] also gives 3 + / l(5) — ( H i + H x ) dz H l - f G = -jk o li(5 \ + £ ? ) yn0 (2.150) where /j(5 ) is defined in [10] which is (2151) Also Ey = Ey = E y and = H x for n + = n_ = p Q. Thus we have 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 715 Pg (A) (2.152) dz which also agrees with the second equation of (2.145) with e a = eo and M-a = |a0. Therefore, we see that the derived EBCs using homogenization analysis to the first order are identical to Ivanov's results [10] which consider the fundamental mode only with quasi-static approximation. If we let the phase shift 0 equal to 0, i.e., 8 = 0 in (2.145), since lim P -8 (c o sA p -P g (cosAp 5_>0 8 [P_§ (cos A p + Pg (cos A p] = (cos A P - | - P g (COSAP] (2.153) = - In cos ^ (^ -) and lim Ps (cos A) - P_g (cos A) 5_>0 8 [Pg (cos A) + P_g (cos A)] = 58 (“ s A> - (“ s 4 >] (2.154) = In cos2 ( y ) where — p z ( u ) = ln -^ -^ and — p s(w) = -ln -^ -^ - have been used [25]. Then 68 6 2 68 6 2 the EBCs with zero phase shift become 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and A i 3 H t - H I = - In cos2 ( - ) [ > e aEy ------ — H x] K 2 n a dz (2.156) which are identical to DeLyser's results [11-12], The equivalent boundary conditions (EBCs) to the first order considering a uniform phase shift for a semi-infinite strip grating lying on different media have been derived based on the homogenization analysis. Comparison with Ivanov’s results [ 10] shows that our homogenization method solution is less cumbersome and more straightforward and our results generalize those of DeLyser [11-12] by taking into account phase shift between strips. Having obtained the EBCs, we will use them to investigate the propagation characteristics of the slow-wave grating structures on grounded substrates for planar microwave filter applications. This will be the subjects of next chapters. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 SURFACE WAVES ALONG A METALLIC STRIP GRATING 3.1 Introduction Periodic slow-wave structures have been well known for their transmission-line application such as filters, antennas, and other components at microwave and millimeterwave frequencies for many years [26-45], To analyze these periodic structures for circuit applications, the propagation characteristics of guided waves must be investigated. In this chapter, the propagation of surface waves on a semi-infinite periodic strip grating lying on a grounded dielectric slab (Figure 3.1) will be determined in an efficient and accurate approach. Several authors [46-59] have investigated the propagation of surface waves on the periodic structures using analytical or numerical methods. But most o f their work had assumed that the direction of propagation is perpendicular to or parallel to the strips or had additional restriction in practical application. Recently, Bellamine et al. [60] investigated the surface waves propagating at an oblique angle with respect to the strips over the periodic structure, as illustrated in Figure 3.1, where the equivalent boundary conditions (EBCs) based on the homogenization analysis derived in [11-12] were used to simplify the analysis. Although Bellamine’s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x=d x=0 m Ilf;; Q ound Plane (a) (b) Figure 3.1: A periodic array o f strips loaded by a dielectric slab over a ground plane: (a) cross section; (b) top view. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. analysis is analytically simple and computationally fast, its application is limited since the required phase shift over a period length was assumed zero in the derivation of the EBCs in [11-12]. In general, there exists a phase shift between adjacent strips in the application of several periodic slow-wave structures. Without loss o f generality, the fields along a grating are assumed to have a uniform phase shift 0 between adjacent strips and propagate at an oblique angle with respect to the strips, as illustrated in Figure 3.1(b). Then the modified EBCs derived in chapter 2 will be employed to investigate the propagation characteristics of surface waves along a strip grating. Comparisons will be made and show good results. 3.2 Determination of the Propagation Characteristics 3.2.1 z -directed Propagation Constant (3 vs. Phase Shift 6 Figure 3.1 shows the geometry of strip grating on a grounded dielectric slab where the strips extend to infinity in the z -direction; the width o f a slot in the grating is a and the width o f a strip is b . Based on the derivation of the equivalent boundary conditions in chapter 2, the grating period p is assumed much smaller in comparison with the wavelength of radiation, i.e., koP « 1, where ko is the free- space wavenumber and p is comparable to the substrate height, i.e., p / d will not be too large. Also, in the following discussion of the propagation of surface waves, we adopt the hypotheses made in [60] for the periodic structure of Figure 3.1. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The EBCs at a strip grating lying at the interface between two different media (Figure 3.1) derived in equations (2.145) can be rewritten as ( in MKS units and the time dependence eJat ) (3.1) where u = 2^ i H2 + *l, - Efl = ' g2 4~g l 2 S8 (A) = F*_5 (cos A) —P 8 (cos A ), A' = o k / p Rs (A) = p _5 (cos A) + p 8 (cos A ), , A = bit/p, the P 8 are Legendre functions o f noninteger order 5 which is defined as 5 = Q/2n, and the superscript I or II denotes the fields in dielectric region ( / ) or half-space region above the slab (II). Note that E z and H z shown in (3.1) do not contain the term e~jQyfP according to the derivation in Chapter 2. Assume the waves propagate in arbitrary direction on the surface o f the slab as shown in Figure 3.1(b), the fundamental Floquet-mode electric and magnetic fields can be denoted as E=E(x)e-jPze~jyy H=H(x)e~JVze-jyy (3.2) ; y =9/p 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I where 3 and y are the propagation constants in the z -direction (along the strips) and y-direction (transverse to the strips) respectively and the field quantities E and H are functions of x only. The higher-order Floquet modes (boundary-layer fields) concentrated near the grating surface can be ignored when applying boundary conditions at the ground plane and do not explicitly enter into the EBCs in (3.1). Then substitute equation (3.2) into the source-free Maxwell's equations, namely f V x E = —/cd \iH [ V xW = j a s E (3-3) in both the dielectric region ( / ) and the half-space region (II) in Figure 1. And “V” in equation (3.3) is the regular del vector operator. Using the following substitution dE dy ~jy E dH ■--jyH dy dE dz (3.4) ~3E dH ■ -J0 dz to expand (3.3), the tangential components of (3.3), and Hy. can be represented in terms o f Ex and Hx, we obtain 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E r ~ 72"[®Ja(^zY -5yP)H x~y(ayY + a z 3 ) — kj* dx — 1 d H Hr = -y [-ffle(azy - a y P ) E x - y ( a y Y + a z P )— -] kf dx (3.5) where k \ = p 2 +Y 2 = p 2 + (^ ) , and Ex as well as Hr satisfy region I ( 0 < x < d ) (£ + h2)H/ ° (3.6) h2 =co2 ^ o e i - ^ r and region II ( x ^ d ) (3.7) ; Re(q) > 0 where q must have a positive real part for a proper guided mode. Obviously, the forms of the solutions in region I are standing waves. Take into account the boundary condition at ground plane x - 0, the solutions for equation (3.6) are 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cos(hx) 1cos(hcf) x Hx = IIjSin(hx) x sin(luf) (38) and the forms o f solutions for equation (3.7) are exponentially decayed waves in xdirection, they are Ex= E 2e - ^ x- ^ (3.9) Hx= H 2e - <l(x- </) where E i, H i, E 2 ,and H2 are constants. Now, the boundary condition at interface x = d must be matched. By inserting equations (3.8) and (3.9) into equation (3.5), the tangential electric field on the slab are E f l w “ 7j [ a n ( 5zY - a y P ) H i + y ( a y y +az 3 )Eihtan(hcO] Kt l _ = 7rt© ^(5rY -5 y P )H 2 + j ( a y j + 5zP)E2q] (3l0) kT Equate the az and ay components in equation (3.10) by making E ^ a * = E r ^ * - , we have 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f cony H i+ y P h E ita n (h i/) = Q(iy H 2 + y ‘P q E 2 { -o (.iP H i+ y h Y E itan(ht/) = -<DjipH 2 + yqY E 2 ^ * Simplifying equation (3.11) results in the following conditions f Hi = h 2 (312) | q E 2 = hE[tan(ht/) Let us now substitute equation (3.2) into the EBC at x = d for the tangential E -field in equation (3.1), we have ] •O Va(Hy1e-JV* - Htye-Jfr) (3.13) + — | - ( e 2Ejc7 e a oz or (3.14) 0 |i ( H y ~Hy) where M-a = = Ho for nonmagnetic materials, s fl = -g l+ -g2, A' = a % f p , and the parameter I q(5) is defined as 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / (Sx 0 P r? -5 ( c o s A ')-P 5 (cosA') 2tiS P_8(cosA') + Pg(cosA') - j u s m (3,5) 2iz5l R5 (A t)1 From equations (3.5) and (3.12), we have Er and Hy at the interface x = d as Ez\x=d = 4 ( o f i y H i+y‘P qE 2) kT = 4-(ffl s 2 p E 2 + yqy H 2) (3.16) kT Hy\*=d- = Tjt® e iP E l-y h Y H ic o tM ] kf where Er and Hr in equations (3.8) and (3.9) are used. Then substitute equation (3.16) into equation (3.14), we have T [o HY Hi +ypq E 2] = kj 7 2 P s 2E 2 +yqy H 2 - a p e t E i + kf (3.17) jh y Hicot(hflO]- — ~ — (e2E2 - e {Ei)} E2 +6! Use condition (3.12) to simplify equation (3.17), after some arrangement we get I I 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Aronoy {1 + ^ p [ q + h cot(h^]}Hi (3.18) = -y'P Ei tan(hrf){h + -------- 2—][en cot(hd) - — ]} S ri+ e r2 q 2 where % = yj\x0/ e 0 is the intrinsic impedance in free space, e t = s r lso> e 2 = e r 2so> and ko is the free-space wavenumber. Similarly, substitute equation (3.2) into the equivalent boundary condition for the tangential H -field in equation (3.1) at x = d, we have W / e - J f r - Hle-JP* = ea E y e " ^ + — ^ ( B x e " ^ ) ] V-adz iz5 R§ (A) (3-19) or H? - Hi = 2 /, (6 )[-y *0.££1L|££2.E>,- yp HJ at x=d (3.20) where A = b n / p and ^ ( 5 ) is defined as /j(5 ) = P ^ - 5 (cos A ) - P s (cos A)j 27t5 P _5 (cos A) + P§ (cos A) P f 58 (A) 27t8 /? s (A ) With the aid o f equations (3.8) and (3.9), equation (3.5) gives |-t as 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.21) H zfr=«/+ “ T F H ® Y E 2 E 2 + 7 p q H 2] kT (3.22) = i [ - ® Y e 1Ei -yhP H icot(hJ)] kT Substitute equation (3.22) into (3.20) and use (3.5) for Ey and Hx, we obtain |j~Y E 1tan(iu0[srl cot(hrf) - - / i ( 8 )(er l + e r2)h] (3.23) = -y'P H i {q + h cot(hc0 + / t(5 )[2 k j - Ar§(e rl + e r2)]} where the conditions in (3.12) are used. Now, combine equations (3.18) and (3.23) to eliminate the common term Eitan(h</). For nontrivial solution of Hi, we obtain the dispersion equation for the surface wave propogation on the periodic strip grating (2 y 2(1 + M l [ q + h cot(hrf)]} {eMcot(kO 2. = - p * {q + h cot(lu/) + 1{(5 )[2 kj- {h + ^ [ * 5 2 e r l + s r2 q - /j<5 ) te ,,+ e r2)h} (erl + e r2)]} co«M ) - ^ S ] } q For simplicity, let gr2 = 1 and e r i - e r in Figure 3.1, then (3.24) becomes 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.24) k2y 2 {1 + / # ) [q + h 2 }{e r cot(hii) _ h _ /i(5 )(1 + g r)h} q = - p z {q + h c o t M + /i( 5 ) [ 2 A : f - ^ ( i+ e r )]} {h + /o ff). 2 (3.25) 2 fPh - -— —][er cot(hfO - - ] } 1+ E r q where some parameters are summarized as k l = P2 +Y2 , ; Y = 0 /p h i = kiQz r - k \ q2 = k2T - k l P p^sCA'), ; A' = an/p P r^sC^! (3.26) ; A = bn/p Dispersion equation (3.25) describes the z-directed propagation constant P versus the phase shift 0 when surface waves propagate along the infinite array of strips on a grounded dielectric slab. Now, let us investigate the limiting case when the phase shift 0 -> 0, i.e., 5 -> 0, then Iq and l\ in (3.26) are obtained with the aid of equations (2.153) and (2.154) as 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then equation (3.25) reduces to q + h cot(hrf) + 1’\[2 k\- - k \ (1 + e r)] = 0 (3-28) for TE mode, or i'n 2 Ict h + ^ r [* 0 - 2 l +er h r COt(hrf) ——] = 0 (3.29) Q for TM mode which are identical to those obtained in [60], This is quite evident because the EBCs in (3.1) reduce to DeLyser’s equivalent boundary condition [1112] when 9 - > 0 and the dispersion equation in (3.25) obviously reduces to Bellamine’s equation [60], 3.2.2 Propagation Constant k t vs* Propagation Angle <(> Equation (3.25) gives the propagation constant parallel to the strips, 3, versus phase shift 0 between adjacent strips. As seen in Figure 3.1(b), 3 is related to the propagation constant k f by an oblique angle 4>, so is y . Thus we introduce 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P = ^7'C0S<j) 0 , . . (3.30) Y s — = kj-sm^ P where <j> is the propagation angle of waves with respect to the strips. From the second equation o f (3.30), the phase shift 0 (or 5 ) can be reprsented as With (3.30) and (3.31), equation (3.25) can be rewritten for k j in terms of <j> as tan2(J){1 + 2 [q + hcot(h^)]}(er cot(bd) - —- / 1(5)(l + gr )h} q = -{q + hcot(h£/) + / l ( 5 ) [ 2 4 - ^ ( l + 6r)]}- (3.32) {h + ^ - [ k l - i ^ L ] [ 8r cot(hd) - -]} 2 l+er q where the parameters are now defined as h —j-Jkj'~SrkQ q= , 1 — ■] ^ 7 sm<t> /?§ (A ) / o (p ) = ';— '■ P ^ 7’Sin<j) R § ( A) R /JAr7’Sin(j) ------2tc , 5 = — ; A = b n /p , A’ = m / p 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.33) Now, equation (3.32) shows implicitly the propagation constant k r as a function of propagation angle <j) for the surface-wave propagation on a strip grating with an arbitrary angle. 3.2.3 Characteristic Impedance Z c The characteristic impedance Z c of a strip grating structure shown in Figure 3.1 is defined as Zc = y (3.34) where V is total voltage between grating surface and ground plane and I is total current flowing on one o f the strips in z -direction, this definition is similar to the characteristic impedance in parallel plate waveguide [61], The quantities V and / in (3.34) are both due to the boundary layer and non-boundary layer fields mentioned in Chapter 2. Then V is equal to V = V E + Ve fd where Ex = E\ „ (3.35) - e e ~ t i y in dielectric region and Ve is the voltage due to cos(h a) the zeroth order boundary-layer electric fields e% which is defined in (2.104). Since 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the boundary-layer fields decayed rapidly from the grating, taken identical to the V jl Ve in (3.35) could be obtained in (D.4) which depends on the non-boundary layer fields in equations (3.8) and (3.10), then p ie 0 e S - E e 'x Ve in (3.35) becomes J & * y /p ? ' = ---------- :---------— Ss(A'1 + 2ea 4sin57t .p E * x - d r i J 2 57t (3-36) J^y/p 2sin57c where e = e r so - For simplicity, take the strip located at >> = 0 in Figure 3.1 as reference, and omit the factor e~J^z hereafter. Then we have kr£ = - q £ 2/h 2 and from Ex and Ey in equation (3.10), we have K e\x=d = [zoEx - e Ex \ x = d - ^ o E i ~ ^ E \ Ey\x=d = 1 -y[-<d l^o P H i+ /y Eihtan(hrf)] kT (3.37) Substitute (3.37) into (3.36) with the aid of equations (3.12) and (3.18), then we have the total voltage V on the strip located at y = 0 (3.38) 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where ko Tlo hy {1 + y [ q + h cot(lu/)]} T= (3-39) At 2 P ( h q - f [*o - - J 1 ][q e rC 0 t(h < /) + h]} 2 1 +Br The total current / on the strip located at y = 0 is determined from integrating current density due to nonboundary- and boundary-layer magnetic fields over that strip, then I is ( omit the factor e ~ ^ z ) 1= J ! - H V * '* +\ y i „ - (3-40) ~ lH + Ih where current is due to non-boundary layer fields Hy and If, is due to zeroth order boundary layer magnetic fields hy . In calculating I H, equation (3.16) is used along with the aid o f equations (3.12) and (3.18), thus kr e o n + 3 i - ( ^ a P _ ! 3 .) e a m i ^ s S b M T h T Y where T is defined in equation (3.39) and If, is obtained in (D ll) which is 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3-41) in 27C ft(A ) - ^ 5 > ] + 5 s 5 (A) (3-42) Ho where K h\x=d - (Hy - Hy )]*=</ (3.43) 5*1*=* = (i0 Hxi^rf = Hx\x=d Substitute Hr and Hy into (3.42), after some manipulation we have lb = 2lt f t (A) - + ^ ~ f t ( A) 7 (3-44) E«P + VT ' (£ Th ® " ' T )cot(W)! (3.45) 5 where Therefore the total current on the strip located at y = 0 is ^ i M i i M & S)+ E m t l1 tR sW . ^ m ]+p j 2 S s W 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3-46) i I where Kh and T are defined in equations (3.45) and (3.39) respectively. Note that in the expressions of V and I in equations (3.38) and (3.46), there exists a common constant E i which will be canceled when calculating characteristic impedance Zc ■ By observing equations (3.38) and (3.46), computation of characteristic impedance Zc depends on the determination o f propagation constant (3 in the transcendental equation in (3.25). 3.3 Numerical Results This section presents results of z -directed propagation constant P versus phase shift 9 , propagation constant k j characteristic impedance Zc versus propagation angle <{>, and o f strip gratings when solving the transcendental equations (3.25) and (3.32). Although the dispersion equations (3.25) and (3.32) show that P (or k f ) ^ d Zc are frequency-dependent, once a small value of ko p is chosen and the assumption for the homogenization analysis o f the strip grating is satisfied, P (or k r ) and Zc can be treated to have a quasi-TEM character which means that they are frequency-independent in this way. To increase the speed o f computation and convergence, the Legendre function />§ in equations (3.25) and (3.32) are represented by a hypergeometric function [25] PS (cos£) = F(-8 ,5 +1; l;sin2| ) 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.47) to simplify computer computation and 5 in (3.47) could be a real or complex quantity. 3.3.1 Comparison of (3 with Weiss' Model The z -directed propagation constant P obtained in equation (3.25) is a function o f the phase shift 0 between adjacent strips. Now, we would like to compare P versus 0 based on homogenization method with Weiss’ quasi-TEM model for microstrip meander-line structure. In [57], each unit cell of the meanderline periodic structure is defined to contain two strips and there is a fixed phase shift cp between successive cells. Within each cell, the potential on the two strips can be either identical or 180° out of phase, they are called “even” mode for phase difference between strips o f cp/2 and “odd” mode for cp/2 + 7t phase difference. When compared to our derivation and notation for phase shift 0 between strips, 0 and 0 + 7t is equivalent to cp/2 in Weiss' definition [57] for even mode and cp/2 + n for odd mode, respectively. Figure 3.2 shows the normalized z-directed propagation constant versus phase shift per strip on the infinite array of parallel strips for even and odd modes when solving the transcendental equation (3.25). Note that Weiss' effective dielectric constant [57] is equivalent to our (J3/yfco) ^ t^ie definition o f the homogenization process. The parameters used to determine Figure 3.2 are: £o<^=3.325e-3, p / d =1.632, a j p =0.5588, and e r =6.5. Since k o p is chosen small enough, good 74 i I I I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Even Mode D e. r 3 fj- z0 . , .__________ . 50 , 'CO __________ _ 150 Phase Shift 8 (D ec ) Figure 3.2: Comparison o f (3 between quasi-TEM Green’s function method and homogenization EBCs method for even and odd modes. The parameters used are: k o d = 3 .3 2 5 e -3 , p / d = 1.632, a/p = 0.5588, and e r = 6.5. Weiss’ result ( - -); present method (—). 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. agreements are achieved between the homogenization EBCs method and Weiss' quasi-TEM Green's function approximation. Due to the mode symmetry, Figure 3.2 is symmetrical at 0 = 90° as expected. 3.3.2 Comparison of k j with Beilamine’s Result Bellamine has illustrated the normalized propagation constant hf/lco versus the propagation angle <j> in comparison with Crampagne’s results [58, 60], Good agreement was achieved only for the low propagation angle since Beilamine’s analysis did not take into account the phase shift between adjacent strips (i.e., 0->O ); in fact, 0 is related to propagation angle <j) by equation (3.30). Now, the dispersion equation in (3.32) has improved Bellamine’s situation. Figure 3.3 shows that equation (3.32) does improve Beilamine’s results when propagation angle greater than 45° and good agreement is achieved in comparison with Crampagne’s result [58, 60], The parameters used to determine Figure 3.3 are fco^=2.0944e-3, p / d = 1.86, a j p =0.07, and e r =9.6 where the assumptions of hop « I and p / d be finite are fulfilled. 3.3.3 Comparison of Z c with Weiss' Model The normal-mode characteristic impedance Z c of a grounded dielectric slab covered by an infinite array of metal strips is obtained by the ratio of voltage in equation (3.38) to current in equation (3.46). In those formulas, the z -directed propagation constant P in (3.25) has to be determinted first. Figure 3.4 shows good 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T 2 0 r— i- 0.00 O.’ O 0 .3 0 0.20 0 .4 0 3.50 Figure 3.3: Comparison o f normalized propagation constant versus propagation angle. The parameters used are: ko d = 2.0944e - 3, p / d = 1.86, a / p - 0.07, and s r = 9.6. Bellamine’s result ( - •); Crampagne’s result (— ); present method (-)- 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T Even Mode 20 Odd M ode *0i 20 - 0 50 '0 0 150 Phase Shift 0 (D eg ) Figure 3.4: Comparison of Zc between quasi-TEM Green’s function method and homogenization EBCs method for even and odd modes. The parameters used are the same as those in Figure 3.2. Weiss’ result (— ); present method (-)• 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. results of Zc in comparison with Weiss’ model [57] since Figure 3.2 gives good agreement for p . The parameters used to obtain Figure 3.4 are the same as those in Figure 3.2. 3.4 Conclusions The EBCs with a phase shift has been used to study the propagation characteristics o f surface waves along a strip grating on a grounded dielectric slab with finite thickness substrate under the assumption of ko p « 1- Unlike previous analyses, we take into account the phase shift between adjacent strips and assumes that the direction o f surface-wave propagation is at arbitrary angle with respect to strips on the grating. With the aid of the developed EBCs in Chapter 2, determination of eigenvalues of the dispersion equations for the propagation constants becomes analytically simple and computationally fast compared to the previous quasi-TEM Green’s function method [57-58], Comparisons have been made and show good results. The obtained propagation constant and characteristic impedance o f a semi-infinite strip grating on a grounded substrate are important in modelling the properties o f metal grating elements, such as meander-line structures, for compact microwave circuit elements design. This will be illustrated in the subsequent chapters. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 CHARACTERISTICS OF MICROSTRIP GRATING ELEMENTS 4.1 Introduction Chapter 4 will address the potential application of microstrip grating elements such as microstrip meander lines, hairpin lines, and comb lines for compact microwave components using the derived propagation characteristics of surface waves along a periodic array of strips discussed in chapter 3. Impose the proper boundary conditions for different grating element structures, the dispersion equations of microstrip grating elements are derived and show the filtering characteristics similar to filters. For grating elements loaded with nonlinear lumped devices, tunable filtering properties exist and show practical application in electronically tuned microwave filters. Also the image (iterative) impedances o f a semi-infinite periodic grating elements are defined and derived for efficiently modelling filter responses which will be discussed in Chapter 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Dispersion Equations 4.2.1 M eander Lines Consider a section of length L of an infinite array of strips on a grounded substrate shown in Figure 3.1, a microstrip meander-line is formed by connecting alternating ends of strips as illustrated in Figure 4.1. Without loss of generality, linear or nonlinear lumped elements can be loaded between adjacent strips, as illustrated in Figure 4.2, where the impedances Zi and Z2 could be inductive or capacitive. To apply the results o f propagation characteristics of metal gratings to meander-line structures, the previous assumptions made for periodic strips in Chapter 3 must be followed. We will show that this meander-line structure has bandreject properties, which indicate that this circuit element will be a bandstop filter. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.1: Top view of a microstrip meander-line structure consisting of an infinite array of strips. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z unit cell Figure 4.2: Top view of a microstrip meander-line structure loaded with series impedances Zl and Z l ■ As stated in Chapter 3, the propagation of surface waves in Figure 4.2 is in both z (along the strips) and y (transverse to strips) directions and the phase shift between adjacent strip is assumed to be 0 . In this case, the voltages and currents on strips 1 and 2 in Figure 4.2 can be denoted by 83 i I i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vi —[A+e J $ez + A - e J $ez —B+e j $ o z + B-e JP oz]e-i®/^ V2 ~ \_A+e~i P*2 + A —eJ $ez + B+e~J &oz —B~eJ Poz]e - /®/^ / l = {— [A+e~jVez - A - e j V e z] + — [ - B +e-JPoz - B - e j V o z]}eJQ/ 2 Zee (4 I ) Zco I 2 = {— [A+e-jVez - A - e j V e z] + — [B+e-JPoz + B-eJPoz]}e-JQ/ 2 Zee where A± Zco and B± are the amplitudes of the even- and odd-mode waves, respectively, for waves propagating in +z and - z directions. Zee, Z co, Pe , and P0 are the even- and odd-mode characteristic impedances and z -directed propagation constants for a strip grating, respectively. Assume the currents depicted in Figure 4.2 flow in the +z direction, the boundary conditions of the meander line are \ V \ ( L / 2 ) - V i ( L / 2 ) = Z \Ii( L/2) V2( - L / 2 ) - V 3( -L /2) = Z 2 I 3(-L/2) / 1(Z/2) = - / 2(L/2) (4.2) I l ( ~ L/2) = - / 3( - L/2) V 2(z) = V ^ e - j ™ . h ( z ) = I l( z )e - J 2Q where the last two equations of (4.2) are always true due to the iterative phase relationship. Applying equation (4.1) to boundary conditions in (4.2) gives a system of homogeneous equations for mode amplitudes A± and B±: 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4n An An An A 13 A 14 A+ A 14 A n A .- A 31 A 32 A 33 A 34 B+ A n A 42 A 43 An _ =0 (4.3) where the elements in the 4 x 4 matrix are given by A n = cos(0/2)e“/P e V 2 A 12 = -co s(0 /2 )e / PeV2 ^13 = -/nsin(0/2)e-y'PoV2 ^14 = —jr\ sin(0/2) eJ PoV 2 ^31 = [-TiieCOs(0/2) +y(2 -n ie)sin (0 /2 )]e^'P ez/ 2 ^32 = h i ecos(0/2)+y(2 +Tlle)sin(0/2)]e/Pe V 2 (4.4) ^33 = [Olio - 2)cos(0/2) + y rilosin(0/2)]e-y P0^ /2 A 34 = [(Tlio +2)cos(0/2)+yTilosin(0/2)]e/PoV 2 A n = [Tl2ccos(0/2)+y(2+Ti2e)sin(0/2)]ey Pei / 2 ^42 = [_Tl2ecos(0/ 2) + y(2 -r)2e)sin(0/ 2)]e_y P ei/2 -443 = C0l2o + 2)cos(0/2)+yTi2osin(0/2)]eyPoi / 2 -444 = where [Cn2o- 2)cos(0/2) + y ri2osin(0/2)]e-/Poi / 2 r\ = Z e e / Z c o , *11,2e = Z u /Z c e , and rji 2o = Z i,2 /Z c o . For nontrivial solutions o f (4.3), the determinant o f the 4 x 4 matrix in (4.3) must be equal to zero. After a lengthy algebraic and trigonometric simplification, we obtain the dispersion equation for a loaded meander-line (as illustrated in Figure 4.2) 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cnsin2 (9/2) tan(Pe L/2) - cos2 (9/2) cot(P0 L/2) - y'nio/2] • fa sin2 (9/2) cot(pe L/2) - cos2 (9/2) tan(p 0 L / 2 ) +jr \2o/2] -J “ 0 (^20 2 - , rrisin2(9/2) rllo)L . fn , x sm(PeZ,) (4.5) cos2(9/2)^ . rn ~ J sm (p0 I ) Two limiting cases should be noted. When the loaded impedances are equal, i.e., Zl = Z l = Zs in Figure 4.2, the dispersion equation (4.5) becomes Zc<?sin2(9/2)cot(Pe Z,/2) - Zcocos2(9/2)tan(P0 I /2 ) = - j Z s/ l Zcesin2(9/2)tan(PeZ ,/2)-Z cocos2(9/2)cot(P0 Z,/2) = jZ s / 2 which is identical to Dashenkov’s results [62] if only the series elements in his paper are considered. If Zs = 0 , which means that the parallel strips of the meadner-line structure are connected as short circuits as shown in Figure 4.1, then equations (4.5) and (4.6) become Zee , f tan(Pe L/2) tan(P0 L/2) — tan2(9 /2 )H Zco ' { cot(PeZ./2)cot(P0 Z,/2) (4-7) which has been shown in [57-58], with 0 from equation (4.7), to be equivalent to <j>/2. According to equation (4.6) or (4.7), there exist mathematically two dispersion equations in a meander-line structure, that is, there are two normal modes propagating along the structure. The upper and lower equations of (4.6) or (4.7) are 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. then referred to as forward-wave and backward-wave branches of the dispersion diagram, respectively. The forward-wave and backward-wave modes are defined by their directions of the group velocity (or energy velocity) and phase velocity. For forward-wave propagation, the group velocity and phase velocity are in the same direction, and for backward-wave in the opposite direction [87], Note that, as mentioned in Chapter 3, the even- and odd-mode grating characteristic impedances and propagation constants shown in equation (4.7) are functions o f phase shift 0 (or frequency). Obviously, at frequencies for which the arguments Pe £/2 and P0 L/2 in (4.7) are in different quadrants, the right-hand side of (4.7) must be negative. This requires that the phase shift 9 on the left-hand side of (4.7) be imaginary, i.e., the propagation along the meander line is cut off and stopband properties occur. This stopband phenomenon is applicable to bandstop filters. If a loaded impedance is considered in the dispersion equation as shown in equation (4.6), the location o f stopbands will shift, which means that the bandwidth of the stopbands is tunable by loading devices; this phenomenon is important and will be discussed numerically later. The existence and prediction of stopbands is essential and important for filter design purposes. Obviously, the bandwidth o f the bands will depend on the circuit geometry and impedances loading the strips. 4.2.2 Hairpin Lines By letting the impedances Zl - » °° and Z l = 0 in Figure 4.2, the structure then becomes a hairpin line or C-section line illustrated in Figure 4.3. Note 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that there are two strips in a unit cell which is defined from the center of one Csection to center of adjacent C-section line. This definition gives a reasonable way to find the image impedance of a hairpin line when modelled by transmission line theory and this will be discussed more in Chapter 5. Taking the limits of Zi * and Z l - 0 on both sides of equation (4.5) results in the dispersion equation for a hairpin line a 4 b 4- ► .,1 V, d V2 #2 #1 d V3 #3 unit cell Figure 4.3: Hairpin-line structure with a defined unit cell. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lim lim {[q sin2(0/2) tan(Pe 1/2) - cos2(0/2) cot(p0 L/2) - j q lo/2] z r « z 2-*° [Tisin2(0/2)cot(PeZ ./2)-cos2(0/2)tan(PoZ./2)+yrj2o/2] (4.8) j /•„ . rrisin2(0/2) cos2(0/2)11 n ~ T v l 2 o ~ n i 0)[ .--------- --------. . q ~7 x'JJ = 0 2 sm(PeL) sm ^L ) where ri20 = ZilZco and r|j0 = Zi/Zco • After simple arrangement, equation (4.8) becomes tan2(0/2) = - Zco tan(PeZ) Zee tan(P0 Z.) (4.9) which is identical to Crampagne’s dispersion equation for C-section periodic structure [59] where the phase shift between strips was defined as <p/2 which is equivalent to 0 in this thesis. Clearly from equation (4.9), at the frequencies for which arguments p eZ, and P0 L are in different quadrants, the right-hand side of equation (4.9) is positive and results in real phase shift 0 existing in the propagation band (passband). This phenomenon implies that this hairpin-line structure could be designed for a bandpass filter. In deriving the dispersion equation of (4.9) for a hairpin line (Figure 4.3), the end-effects o f strips are not considered, i.e., the coupling capacitance between strip ends is assumed to be small enough so that it can be ignored. On the other hand, the excess capacitance due to the open end discontinuity can be transformed into an 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equivalent length of small transmission line adding to the physical length L [63]. This approximation is essential and valid for the microwave frequency range. By looking at the dispersion equation (4.9), since the even- and odd mode propagation constant P and characteristic impedance Zc are symmetrical at 9 = 90° as calculated in chapter 3, then dispersion curve o f a hairpin line will also be symmetrical at 6 = 90° in the passband; this will be verified numerically later. 4.2.3 Comb Lines Figure 4.4 shows the general representation of a comb-line structure with reactive loads at the finger ends of the lines. Since only one strip is contained in the unit cell of a strip grating, the voltage and current on strip #1 of Figure 4.4 can be denoted as Vi = Ae~JPz + BeJPz (4.10) Zc where P and Zc are propagation constant in +z direction and characteristic impedance of strip grating, respectively. The boundary conditions for Figure 4.4(a) now are 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. #0 I #1 1 1 Io,V0 I V ° 1012 #2 I fl^V , 1 1 I*V2 l Figure 4.4: (a) A comb-line structure loaded by reactances; (b) a unit cell of (a). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Uoi + 1 l( V 2) = I \g + I C12 V0 ( - L / 2 ) - V l ( - L / 2 ) = j X l 0l (4.11) V X= V 0 e~JQ . I c l 2 = I c O i e ~ jQ where the last two equations o f (4.11) are true for the iterative phase assumption. Using KirchhofPs voltage and current laws on strips #1 and #2 of Figure 4.4, (4.11) can be reduced to the following conditions h ( i / 2) = - A Ki ( - Z./2) = ± - X2 + -k e-jB - lX«/0 - 0] n ( i / 2 ) X\ (4.12) jX h(-L/2) Substituting equation (4.10) into (4.12) gives a system of homogeneous equations for wave amplitudes A and B . For nontrival solutions of A and B , it gives the following dispersion equation for a comb line with nonzero reactance X \ [S i" 2 0 /2 ) + f 1 • [sin2 0 /2 ) + — 4 Z ctan(PZ.) --7^ — — 1 4ZcXi (4.13) sin 2 (6/ 2) -2Zcsin(2pZ,) For the limiting case of X \ —> °o and X 2 00 • (4.13) then becomes 92 Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. sin2 (0 / 2) = -^-tan(PZ-) 4 Zc (4.14) where we see that the reactance X in equation (4.14) can not be zero for proper solution of phase shift and for the consistency of unit on both sides of (4.14). This reactance would be the self-inductance due to the transmission line for practical reason. Equation (4.14) also shows that at frequencies for which tan(PL) is negative, then the phase shift 0 in the left-hand side of (4.14) is imaginary which cuts off the propagation and stopband phenomenon occurs. 4.3 Image Impedance Z/m 4.3.1 M eander Lines The image (iterative) impedance of a loaded meander line shown in Figure 4.2 is defined as Zim = ~ ~ IA (4.15) where V a and I a are voltage and current at point A o f Figure 4.2. This concept of image impedance o f (4.15) is derived based on the input impedance of an infinite array of strip conductors extending in the transverse direction (.y-direction) with connected alternating ends. However, as long as the periodic structure consists of a 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sufficient large finite number o f strips such that a uniform wave-propagating in the transverse direction is established, the mentioned input impedance can be considered equivalent to the image impedance for an infinite array of strips. The image impedance is sometimes referred to as iterative impedance since it repeats every period. Now according to Figure 4.2 with Zl = Z l = Z s , the image impedance defined in (4.15) can be written as 7- —-----Z d Lim IA _ Zs , Vi(tLI2) 2 _ Iii-L/2) eJPe^/2 A+ + e~j Pe^/2/f_ — e J B + + e~i Zs ^ [eJPe^/2A+—e~J Zee A^]-------[e/Po^/2 B+ + e~j Po^/2 BJ\ Zco (4.16) where V\ and I\ defined in equation (4.1) are used. To find out (4.16), the relation between wave amplitudes must be evaluated first. Then use the 4 set of equations in (4.3) to eliminate amplitudes B+ and B- , we obtain i± A(1 + e i P' L eJ Vol )[2t\ sin 2 (6/ 2) + 2 cos2(0 / 2)] -% (1 - eJ P«* ei Pql ) (eJ $ e L + eJ Po^)[2 cos2(0 / 2) - 2r|sin2(0 /2)]+ r\s (eJ PoL - eJ P « L) 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (41?) where % = Z s {Z c o • The lower equation of dispersion (4.6) for backward-wave mode can be expanded as (1 + eJ P* L e i Po L)[2t\ sin2(0/2) + 2 cos2(0/2)] - t\s (1 - eJ P*L ei Po i ) —- ( e i P*^ + e i Po^)[2cos2(0/2) —2 q sin 2(0 /2 )]—^ (e /P o ^ - - g 7 Pe ^) Equation (4.18) is then used to simplify (4.17), we obtain the following relation for wave amplitudes A± A+ = - A - for backward-wave (4.19) Similarly, by substituting the upper dispersion equation in (4.6) for the forwardwave branch in (4.17), we get another relationship A+ - A - for forward-wave (4.20) Similarly, by eliminating amplitudes A+ and A - from the homogeneous equations (4.3), we have B+ _ (1 + e i Pej ei Po^)[2t| sin2 (0/2) + 2 cos 2 (0 /2)] + % (ei P<L e i Pql - 1 ) B~ (ei Pe^ + e i Po^)[2 cos2 (0/2) - 2r) sin2 (0/2)] +r\s (e i^o^ - e i Pe^) (4.21) 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Again, use the dispersion equation for forward-wave and backward-wave branches in (4.6) to simplify (4.21), we obtain the following relation for wave amplitudes B ± for forw ard-w ave , , . for backward - wave (4.22) Employing the relationship o f wave amplitudes in (4.19), (4.20), and (4.22), after a lengthy algebraic and trigonometric simplification, the image impedance of a loaded meander line defined in (4.16) becomes - S‘-" ~ [Z ee tan(J3 e L f 2) + Zco cot(P 0 L /2 )] backward - wave (4 23) Zim — — —- [ Z e e cot(Pe L / 2 ) + Zco tan(P0 L /2 )] forward - wave For the case in Figure 4.1 where no series elements exist, i.e., Zi = Z l = Z s = 0 , we use the dispersion equation in (4.7) for zero series elements to simplify (4.23), and the image impedance in (4.23) becomes Zim \y tan(P0 Z,/2) =0 = J Z ee Z c o — =- I Zee Z c o V cot(p0Z /2) —-— cot(Pe£/2 ) „ „ for forw ard-w ave _ , for backw ard-w ave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is identical to Weiss’ meander-line image impedance [57]. 4.3.2 Hairpin Lines The voltages and currents on strip #1 and #2 in Figure 4.3 for a hairpin- line structure can also be represented by equation (4.1). Now in this case, the boundary conditions o f the structure are /l( L /2) = 0 I 2{Lf 2) = 0 / 2( - L / 2 ) + / 3(-L /2 ) = 0 (4.25) V 2( - L / 2 ) = V3( ' L / 2 ) r 3(z)=Fi(z)e- / 20 h ( z ) = h ( z ) e ~ J 2Q Substituting the expression of voltages and currents in equation (4.1) into boundary conditions in (4.25), we obtain a system o f 4 homogeneous equations and simplifying these 4 equations, the following relationship for wave amplitudes are obtained A+ = eJ$eL A A - _ y'sin(Q/2)r|eJ Pq V 2sin(P0Z,) B- ~ cos(9 /2)ej P«^/2 sin(P e L) B+ = - e J $ol B~ 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.26) Now, the definition o f image impedance shown in (4.15) can be adapted for a hairpin-line structure. Since a unit cell o f a hairpin-line structure is defined from center of one C-section to center of adjacent C-section such that the image impedance of this structure can be well defined. Thus the image impedance of the hairpin line shown in Figure 4.3 is _ VA V x (-L !2) Zm~TA - W w ) ei V'L/ 2 A + + e - j P* V 2 A .- - e i P° L/ 2 B+ + e~J = — [ei Pe A + —e~i Pe V2 Zee V2B- (4-27) [gj P0 £/2 B++e~i Po^/2 /j_] Zco where Fj and /[ are depicted in Figure 4.3 and expressed in equation (4. 1). Use conditions (4.26) and hairpin-line dispersion equation in (4.9) to simplify equation (4.27), we then obtain ^ _ Z e e tan(0/2) £im —------- —— ~~— tan(PgZ) —Zco (4.28) tan(0/2)tan(PoI ) —Z e e Zco tan(Pe Z,)tan(P0Z,) where the third expression is obtain by taking square root of product of the first and second expressions. As stated previously, if PeL and P 0 L are in different 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. quadrants and the phase shift 8 is real in the passband region, then the image impedance shown in (4.28) is real in the propagation band, which is expected. 4.3.3 Comb Lines From Figure 4.4, the image impedance o f a comb-tine structure can be defined as ; V o i - V \ ( r L [ 2 ) + j — lQ[ '01 (4.29) 2 where Foi and / qi are depicted in Figure 4.4. Using KirchhofFs voltage and current laws on the unit cell o f Figure 4.4(b), equation (4.29) becomes Zim = -~cot (9/2) (4.30) Note that this image impedance depends on the reactance loaded on the series line which connects the parallel strips, and reactance X could not be zero for reasonable image impedance. 4.4 Numerical Results and Prediction of Stopband and Passband 4.4.1 Meander Lines The dispersion equation of unloaded meander-line shown in (4.7) will be used to calculate the important 0 - / diagram (Brillouin diagram) and predict the 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. filtering characteristics of the structure. Figure 4.5 shows the comparison of prediction of stopband and dispersion diagram o f backward-wave branch for an unloaded meander-line between homogenization EBCs method and Crampagne’s results [58]. The frequency range between the extremes of a 0 - / diagram is called a passband since the phase shift is real and a frequency range between two passbands is a stopband. These two results agree well as expected since the characteristic impedance Zc and z -directed propagation constant (J of the periodic strip grating were numerically compared well as shown in Chapter 3. Figure 4.6 shows the dispersion diagram (backward-wave) of an unloaded meander-line structure in terms of slot width a and strip width 6 with p - 1.2mm, L = 18.36mm, d = 1.016mm, and g r = 10.2. We see that the bandwidth and the diagram curves do not change much when a/6 is adjusted. Even though, the image impedance will change significantly with different a /6 . Figure 4.7 shows the calculated image impedances of a meander line (forward-wave) where the circuit parameters used are the same as those in Figure 4.6 and we see that the image impedance goes to either zero or infinity at the edges of stopband; this is obvious since the propagation approaches to cut off near the edges o f the stopband. The selection of image impedances is important to this class of grating filter design since they correspond to the “equivalent” characteristic impedances of the structure when modelled by transmisssion line theory and this willl be explained more in the next chapter. The dispersion diagram in terms of the strip length L with fixed slot and strip width is 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2QG;— 50 x 2 4 F re q u e n c y (GHz) Figure 4.5: Comparison o f dispersion diagram (backward-wave) of a meander line between homogenization method ( - - ) and Crampagne’s result (—) with a - 0.13mm, b = 0.8mm, d = 1.0mm, L = 18.36mm, and e r = 9.6. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200- X • 50 50 -L 2 F r e c u e n c y (GHz) Figure 4.6: Dispersion diagram (backward-wave) of a meander-line structure with p = 1.2mm, d = 1.012mm, L = 18.36mm, and e r = 10.2 for a[b = 2.0 (—X 5.0 (• •), 1-0 (— ), and 0.5 ( - •). 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I I (Olitna) t Imcnjo Impedance r 00 t 0 1 2 3 4 - 5 6 ►Vecuency (G h z ) Figure 4.7: Calculated image impedances (forward-wave) o f a meander line with p = 1.2m m , d - 1.012mm, L = 18.36mm, and e r = 10.2 for a/b = 2.0 (— ), 5.0 (• •), 1.0 ( - -), and 0.5 ( - •)• 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shown in Figure 4.8 for backward-wave branch where we see that the location of stopband varies with the strip length. It is expected since the resonant frequency of a single strip varies with the length of strip. For an unloaded meander-line structure, the physical strip length L is approximately equal to the guided half-wavelength in microstrip and this will be shown experimentally later. The loaded series impedances shown in Figure 4.2 can be replaced by identical nonlinear devices such as varactor diodes, as illustrated in Figure 4.9, where the DC bias circuit is not shown and the junction capacitance Cj of the diodes can be tuned by appling DC voltage. Figure 4.10 shows the backward-wave branch dispersion characteristics in terms of total capacitance Ct of a device-loaded meander-line structure using equation (4.6), where Ct is the total capacitance loaded between strips. We see from Figure 4.10 that, for different capacitance Ct near the stopband, the first edge does not change too much, but the second stopband edge shifts as the loaded capacitance is changed. Basically, the bandwidth of the stopband of this device-loaded meamder-line structure is electronically tunable when the grating element is loaded with nonlinear lumped devices and this phenomenon is promising for tunable compact filter design. 104 i j. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T 2CG— CO 50 0 1 2 3 F r e q u e n c y (GHz) 4 5 5 Figure 4.8: Dispersion diagram (backward-wave) of a meander-line structure with a = 0.8mm, b - 0.4mm, d = 1.016mm, and z r = 10.2 for L = 18.9mm (—), 18.36mm (• •), 17.5mm ( - - ) , and 19.5mm ( - •). I 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.9: Example of a diode-loaded meander-line structure. To show that a periodic structure such as a meander line is a class of slow-wave structures, the group velocity vg must be examined. With the aid of the Brillouin diagram ( 0 - / diagram) shown in Figure 4.5, group velocity Vg is defined as 3(9 /p) da 2np% (4'31) where Q/p is the “equivalent” propagation constant along the periodic system in y direction for fundamental mode as mentioned in Chapter 3, / is the frequency, and df/dQ is the inverse o f slope of the Brillouin diagram shown in Figure 4.5. To 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2C0 — 0 3 2 O f r e q u e n c y (GHz) Figure 4.10: Dispersion diagram (backward-wave) o f a loaded meander line for Ct= 1.0 p F (—), 15 p F (• •), 5 p F and 0.5 p F (-•) with a = 0.8mm, b = 0.4mm, L = 19mm, d = 1.016mm, and e r = 10.2. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. evaluate the slope of the dispersion curve, one can fit the curve using mathematical functions or polynomials with high accuracy and take the derivative directly from the fitted curve. The absolute value of normalized group velocity to the light speed in free space vq versus frequency for the meander-line structure with the same parameters as those in Figure 4.5 is shown in Figure 11, where the dispersion diagram o f Figure 4.5 is fitted using 10th degree polynomial. We see that the group velocity o f this meander-line structure is much smaller than the speed of light in free space as expected for slow-wave structure. Also near the edges of stopband, group velocity approaches to zero which indicates that the structure carries no power in stopband region. 4.4.2 H airpin Lines The prediction of passband characteristics or 0 - / diagram of a hairpin line can be calculated by using equation (4.9). Figure 4.12 shows the calculated dispersion diagram of a hairpin-line structure in terms of slot width a and strip width b . These dispersion curves are symmetrical at 0 = 90° as predicted by (4.9) in section 4.2.2. We also see that the passband characteristics do not change too much for different values of strip and slot width except for the tightly coupled case of a = 0.2mm and b = 1.8mm. In this case, the dispersion curve seems degenerate with little bandwidth and the group velocity (corresponding to inverse of the slope of the curve) approaches to zero resulting in little power flowing through the structure. Figure 4.13 shows that the location of passband in terms of the length of strip; this 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ |OA/f>A| 0 1 2 2 rreauency ^ 5 5 (G H z) Figure 4.11: Normalized group velocity of a meander-line structure with a = 0.13mm, b = 0.8mm, d = 1.0mm, L = 18.36mm, and s r = 9.6. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2C 0- T a a t/ a a = 02mm /> = 18mm - 10mm h - 05mm = 0 8mm h = 0 4mm - 06mm h = 0 3mm = 0 2mm h = 0 4mm 50- 02.0 2.5 3.0 7 F r e q u e n c y (GHz) Figure 4.12: Dispersion diagram of a hairpin line for d = 1.27mm, L = 18.36mm, and e r = 10.2 in terms of a and b . 110 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ v •co - Phose Shill (Di.fj ) \ 50 or 1.0 1.5 2 .0 2.5 3 .0 3.5 F r e c - e r i c y (GHz) Figure 4.13: Dispersion diagram of a hairpin line with a = 0.8mm, b = 0.4mm, d = 1.27mm, and e r = 10.2 for L - 18.9mm 18.36mm (—), 18.8mm (• •) , and 19.2mm ( - •)• 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. phenomenon is obvious and expected as mentioned in the case of meander-line structure. Figure 4.14 shows the calculated image impedances for various er and L using equation (4.28). We see that the image impedance becomes sharply large at the edges of passband for each case and for higher dielectric constant, the image impedance of passband decreases and the location of passband shifts toward lower frequency. From Figure 4.14(b) we also see that as strip length L increases, the bandwidth of passband decreases; however, the fractional bandwidth is the same. The variation of L doesn’t change the impedance level as expected. Although Figure 4.14 does not show good circuit dimensions for filter design purpose since the image impedances are quite high, it provides an important information for a hairpin-line structure. 4.4.3 Comb Lines For the comb-line structure illustrated in Figure 4.4, if no external elements are loaded with the structure, then X j could represent the reactance due to open-end discontinuity effect which can be represented by an excess capacitance Cp [63] to ground, X \ could be the reactance due to coupling capacitance Cs between adjacent strips which can be found in Getsinger’s paper using the concept of negative inductance in quasi-TEM transmission line [65], and X is the reactance due to the self-inductance Ls produced by the short transmission line connected between two parallel strips and can be modelled by 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ ik-.CC (a) L - 18 6 6 m m ■ 18 0 m m • 19 5m m ■ 21 0mm ■ 22 5m m - -re c u e rc y (GHz) | K , ' n __ \ 5CG r ~ (b) ; Er= 102 96 11 9 130 150 o2.0 2.5 " re c u e rc y (GHz) 3.0 Figure 4.14: Image impedance of a hairpin line for p = 1.2mm and d = 1.27mm with (a) L = 18.66mm and ajp = 2/3; (b) e r - 10.2 and ajp = 2/3. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I L s — A/ Zo -J&e (4.32) CO where A/ is the length of the short transmission line which is equal to the period of the structure, Zo and ee represent the characteristic impedance of the short transmission line A/ and effective dielectric constant of the substrate, respectively, and co is the speed of light in free space. Then the reactances shown in Figure 4.4 can be represented as X —co Ls (4.33) X i = - l / o Cs Xi = - l/ o C P Now, use the structure parameters o f a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, and g r = 10.2, then the mentioned reactances are calculated as: Cp = 0.02p F , C s = 0 01 \ 2p F , and L s = 0.1229nH. The dispersion diagram of the comb line with the above computed equivalent reactances are shown in dash-dot line in Figure 4.15. Also, if the end-loading capacitance Cp is replaced by an identical varactor diode whose capacitance is tunable by applied DC voltage, then the location of stopband will be tuned electronically as shown in Figure 4.15. Thus an electronically tunable comb-line filter suggested in [66, 67] can be constructed in microstrip. Figure 4.16 shows the calculated image impedance of a comb-line 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2QC- • 00 — < u m 50- 0 o 7 2 Frequency (G H z) Figure 4.15: Dispersion diagram o f a comb-line structure for a = 0.8mm, b = 0.4mm, L = 18.36mm, d = l.016m m , and e r = 10.2. Equivalent reactances: Ls = 0.7239nH and Cs = 0.0112p F with Cp = 0 (—, Cs = 0), 0.03p F ( - -), 03 p F and 1.0p F (• •)• 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i) u 1cr> ) c 50 0 2 3 4 5 “ r e q u e n c y (G h z) Figure 4.16: Calculated image impedances o f a comb-line structure in terms of a and b with p = 1.2mm, L = 18.36mm, d = 1.016mm, and s r = 10.2: a/b = 2 (—), 1.0 (• •), 0.5 (— ), and 0.2 ( - •)■ 116 Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. structure in terms o f a and b using equation (4.30) where the equivalent reactances are considered. The calculated image impedances approach to either zero or infinity at the edges o f stopband; this phenomenon is quite obvious or can be understood by checking the expression of image impedance in (4.30). 4.5 Conclusions Dispersion equations of microstrip periodic grating elements such as microstrip meander lines, hairpin lines, and comb lines are derived analytically in this chapter. With the aid of propagation characteristics o f surface waves along periodic strip gratings, the filtering properties of the grating elements are numerically predicted which provide the important information to microstrip grating filter design. The image impedances o f the grating elements are also defined and derived. Since image impedance corresponds to the “equivalent” characteristic impedance o f grating elements when modelled by equivalent transmission line theory, the selection of circuit parameters must be careful to provide proper impedance level for impedance-matching purpose. The assumptions made in previous chapter that the substrate height has to be finite and the grating period p must be much smaller than the wavelengh of radiation should be obeyed for all the design. The design of the grating filters will be discussed in next chapter. 117 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 MICROSTRIP GRATING FILTERS 5.1 Introduction To efficiently and accurately compute the responses of microstrip grating elements as microwave filter applications, the grating filters will be modelled by an equivalent transmission line (ETL) with a defined propagation constant and characteristic impedance based on the structure. The results of propagation characteristics of strip gratings on grounded substrates derived from the EBCs will be used to model the ETLs. Then the performances of unloaded and loaded microstrip meander-line bandreject filters, hairpin-line bandpass filters, and combline bandreject filters are calculated using the ETL model based on the normal and complex modes theory. For the grating elements loaded with lumped devices, they find potential applications in bandwidth-tuned filters. The calculated responses are also compared with the measured results. The development of this class of grating filters promises to deliver broadband properties or highly selective behaviors of compact filters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Meander-line Bandreject Filters 5.2.1 Equivalent Transmission Line (ETL) Model: Normal Modes Consider a section o f microstrip meander-line structure with equivalent length o f w in the transverse direction ( y -direction) and the structure is connected with nominal 50- Cl microstrip lines on both input and output ports as shown in Figure 5.1, where w is equal to the number of strips times the period length p . To represent this meander-line structure from the standpoint of circuit, an equivalent transmission line (ETL), as illustrated in Figure 5.1(b), with propagation constant y = 9 Ip in transverse direction as mentioned in Chapter 3 and characteristic impedance replaced by Zim which is defined in Chapter 4 as an image (iterative) impedance for semi-infinite periodic structure will be employed. The use of Q/p as the propagation constant of ETL is quite obvious as mentioned in Chapter 2 and the characteristic impedance of the ETL can be modelled by Zim without doubt according to the defintion o f image impedance for symmetrical two-port networks [61]. Then the transmission or ABCD matrix of the ETL o f Figure 5.1(b) is n [ cos(wy) y Z/m sin(HY) sin(ivy) cos(vvy) Zim and return loss S n and insertion loss S 21 (5.1) this transmission line can be obtained by transforming transmission matrix o f (5.1) into scattering matrix, which is 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. port #1 ■ port #2 ) -W (a) w z* y=Q/p i ------------------------------------• (b) Figure 5.1: (a) Top view of a microstrip meander-line filter, (b) Equivalent transmission line model of (a). 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 co s(w y ) + j ( Z i m / Z o + Z o/Z /m )sin(w Y ) (5.2) 5 i 1 = j - ^ i Z i m / Z o - Zo/Zim)sin(wy) where Zo is the characteristic impedance o f microstrip line connected on both ports of the meander line and is set to 50 £2 normally. As mentioned in the meander-line dispersion diagram in Chapt 4, y =Q/p in equation (5.2) is a real quantity for passbands and is imaginary for stopband. If a Bloch wave propagating in +y direction (see Figure 4.1) is assumed, we can define the equivalent propagation constant o f ETL in more general way (wave propagation with exp(-jyy) dependence is assumed) y =Q/p passband (5 .3 ) =Q /p= $s + jas stopband where 0 and 0 are real and complex, respectively. The second equation of (5.3) turns out that (3^ and a s are resultant phase constant (positive) and attenuation constant (negative) o f ETL at stopband; this complex representation of resultant propagation constant due to the mode coupling will be discussed more later. Note that this proposed ETL model only holds for a normal mode propagation since the interaction or coupling between two normal modes existing in 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. meander-line structures is not considered. The importance of the coupling between two different propagation modes will be discussed in next section. The derived equations of finding S-parameters of ETL shown in (5.2) can also be applied to hairpin lines and comb lines since only a single normal mode exists in those structures. 5.2.2 Dual-mode ETL Model: Complex Wave Modes As derived in equation (4.6) or (4.7), there are two normal modes existing in meander-line structures. We now must take into account the interaction between these two modes; the interaction o f normal modes of propagation is referred as complex wave modes. Complex waves are the modes having complex propagation constants in spite o f the assumption that the structures are lossless. Complex wave modes in lossless systems were first observed in a waveguide with anisotropic impedance walls by Miller [68]. Later, the existence and properties of complex waves in lossless waveguiding structures, e.g., dielectric-loaded circular waveguides, shielded microstrip lines, and finlines, has been investigated by several authors [6985] for the analysis o f discontinuity problems o f the structures. Investigations showed that complex wave modes constitute essential parts o f mode spectrum (complex propagation constant). In addition, in coupled lossless transmission lines and periodic structures, complex wave propagation was predicted to occur in a certain frequency band and carry no power flow [83-85]; this peculiar feature is characteristic o f wave filters. It has been mentioned in [85] that waves with complex 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. propagation constants can only occur in structures which exist at least two normal modes. For our meander-line structure, there exist two normal modes propagation as derived in dispersion equations of (4.6) and (4.7). Then the coupling o f the backward waves and forward waves constitutes complex wave modes and gives rise to the stopbands. As pointed out in [77], ignoring the effects of complex wave modes in certain frequency band will lead to erroneous solutions. So we will apply the concept o f complex wave modes to the previous microstrip meander-line bandreject filter and modify the normal-mode ETL model. It has been shown that complex waves always exit in pair [72, 76], so there are a set o f four possible complex waves resulting in standing and evanescent waves along the meander-line structure. To consider the coupling of these two modes using a circuit model, the concept of ETL model shown in Figure 5.1(b) still holds; the difference now is, we use two ETLs connected in parallel representing forward- and backward-wave mode as shown in Figure 5.2 where y b , Z bm , y A and Z{m are equivalent complex propagation constants and image impedances for backward- and forward-wave mode, respectively. This concept was first introduced by Clarricoats [86] in 1963 by using homogeneous and inhomogeneous waveguides coupled with forward and backward waves for a frequency-selector application. According to complex mode theory [72, 76, 85], y b and y f are complex conjugate with opposite sign in such a way that there is no real power being carried by these Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Forward-wave mode, Z^, Backward-wave mode, yf y1 (a) [Y*] •------------C- ---- 1 [Y 6] (b) Figure 5.2: (a) Dual-mode ETL model for a meander-line filter, (b) Admittancematrix representation of (a). 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. two modes. Thus these waves at stopband in the system contain one of the forms in y -dependence exp(±y Psy)exp(a.<r>') exp(±yp^)exp(-a5>0 (5.4) where Ps = Re(y) and a s = Im(y) < 0. Since these two ETLs are connected in parallel, we may use admittance-matrix to represent each section of the dual-mode ETL, as illustrated in Figure 5.2(b), and find the scattering matrix. Then transform the transmission matrix shown in (5.1) for each section of the ETL of Figure 5.2(a) into the admittance-matrix, we have Db’f Bb' f Cb' f - A b' f D b' f B b' f -1 Bb' f Ab' f _Bb' f Bb’f (5.5) where A, B, C, and D are shown in equation (5.1) with the superscripts b and / standing for backward- and forward-wave modes. Then the overall admittancematrix for the dual-mode ETL is [K ]=[r*]+[r/] (5.6) 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and convert admittance-matrix into scattering-matrix, we have the S-parameters representation o f the dual-mode ETL P l = - ( P 1 + I 'o M r 'd H - I'o W (5.7) where [/] is the unit diagonal matrix and Y q = I/Z q . 5.2.3 Design of a Microstrip Meander-line Bandreject Filter Based on the previously developed ETL models for normal and complex modes, a microstrip meander-line bandreject filter shown in Figure 5.1(a) with parameters o f a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, w = 30mm, and e r = 10.2 will be designed. Figure 5.3 shows the calculated complex propagation constant versus frequency of a microstrip meander-line filter using ETL models. At stopband region, we see that the phase constant (3 * ) is small and merges with the edges o f passbands and minus of attenuation constant ( - a s ) is maximum at the center of stopband and goes to zero at the edges of stopband as expected. Figure 5.4 shows the measured [Sul and |^2i| of a microstrip meanderline bandreject filter. The calculated results without considering loss factors are obtained from the developed ETL models; the parameters used are the same as those in Figure 5.3. We see that the measured and calculated results agree well except the frequency shift o f about 0.2 GHz; this discrepancy of frequency shift at stopband is due to the strip width of short transmission line connected between two parallel 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 30 25 2C \ \ stopband retpon i 2 3 - 5 5 Frequency (GHz) Figure 5.3: Complex propagation constant y of a microstrip meander-line filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, and e r = 10.2: y in passbands (—) and Ps ( - •) and - a * (• •) in stopband. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2G -5 0- 0 - 1 2 3 F-ecuency (GHz) 4 - 5 20- rs A -ac- j - 100- 0 1 2 3 4 5 Frequency (GHz) Figure 5.4: Measured (• •) and computed (— ) responses of a microstrip meander -line bandreject filter with a = 0.8mm, b = 0.4m m , d = 1.016mm, w = 30mm, L = 18.36mm, and e r = 10.2. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strips. This phenomena was experimentally predicted by Vesslkov et al. [8] in calculating transmission loss o f a meander-line structure at stopband. Their experimental curves show that the strip width of the short link not only affects the transmission loss but also shifts the center frequency of stopband o f a meander-line structure. This can be explained that the equivalent circuit due to the strip width and discontinuities of strip ends between two parallel strips varies the transmission characteristics of the structure. To improve this frequency discrepancy, the equivalent circuit between two parallel strips should be modelled correctly. In this filter structure, the strip length L is approximately equal to half guided-wavelength computed from the center frequency of stopband and the measured fractional bandwidth is about 25%. Compared to the conventional quarter-wave coupled line filters, this class of grating filter provides both wider bandwidth and compact design. The maximum measured insertion loss of this filter at passbands is about 2 dB mainly due to the conductor loss since the transmission loss of the whole structure is calculated about 1.7 dB at 2 GHz [61], The average return loss is about 10 dB in the first passsband; To improve the performance of the return loss, the image impedance of the structure (or characteristic impedance o f ETL) should be matched, i.e., the dimensions o f this meander-line filter must be adjusted such that the image (iterative) impedance of filter is close to the microstrip characteristic impedances (50 £2) connected on both ports, as illustrated in Figure 4.7. Some other loss may be due to the step junction between microstrip line and meander-line 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I structure, this type of discontinuities should be modelled properly. Also from the measured insertion loss o f stopband in Figure 5.4, we see that the cut-off slopes are relatively steep, better than 0.07 dB/MHz. Although the meander-line bandreject filter shown in Figure 5.4 is not optimumly designed, it provides the potential application in compact microwave filter design with sharp skirts in i| and with less complicated computation by using the accurate and efficient ETL models. 5.3 M icrostrip Comb-line Bandreject Filters Figure 5.5: An example of a microstrip comb-line bandreject filter. Figure 5.5 shows an example of a comb-line bandreject filter in microstrip. The measured and calculated return loss and insertion loss of a comb-line bandreject filter are shown in Figure 5.6. The calculated results are based on the ETL model mentioned in section 5.2.1 for a single normal mode. The circuit parameters used to achieve Figure 5.6 are: a = 0.8mm, b = 0.4mm, L = 18.36mm, 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - • o — -20 A i - JG i ‘I -5 0 0 Frequency (GHz) - 30 r uu -4 -5 0 — 0 Frequency (GHz) Figure 5.6: Measured (• •) and calculated (—) responses o f a microstrip comb-line filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, w = 14.4mm, and, e r = 10.2. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w - 14.4mm (12strips), d = 1.016mm, and e r = 10.2. Based on the circuit geometry of Figure 4.4, the equivalent reactances are calculated in Chapter 4 as: Cp = 0.03pF, Cs = O.Q\\2pF, and Ls = 0.7239nH where Cp is the excess capacitance due to the open-end discontinuity and Cs is the coupling capacitance between parallel strips. To match the measured results at lower frequency, Ls is chosen as 0.9049n H in obtaining Figure 5.6. As illustrated in Figure 5.6, the calculated results are quite close to the measurement except at higher frequency edge. The return loss at the first passband is not quite good since the circuit parameters were not optimumly chosen such that the image impedance is about one half o f the characteristic impedance of the microstrip line (50 ) connected on both ports, as predicted numerically in Figure 4.16, and this contributes a severe impedance mismatch. For this type o f filter, equivalent reactances due to the side- and endeffects are more sensitive to the reponses since the dispersion characteristics of comb lines are strongly dependent on the circuit model as shown in (4.13). For practical reason, an electronically bandwidth-tuned filter can be constructed by loading varactor diodes at the end o f strips, as suggested in [66, 67]; an example of tunable dispersion curve o f a comb-line structure is shown in Figure 4.15. 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 M icrostrip Hairpin-line Bandpass Filters Figure 5.7: An example o f a microstrip hairpin-line bandpass filter. Figure 5.7 shows an example of microstrip hairpin-line bandpass filter. The measured insertion loss of fabricated hairpin-line filter is shown in Figure 5.8 with circuit parameters o f a = 0.8mm, b = 0.4mm, d = 1.27mm, L - 18.36mm, w = 31.2mm (26 strips), and e r = 10.2. We see that the average insertion loss at passband is about 5 dB which is not quite good; this is expected since the calculated image impedance at passband with the above circuit parameters is quite high as shown in Figure 4.14 (solid line) where the equivalent length o f about 0.3mm due to open-end discontinuities [63] is considered. This high image impedance will result in a severe impedance mismatch at passband. However, the edge frequencies of passband are well predicted. In addition, this bandpass filter provides sharp attenuation rate at the edges which is of importance in practical filter design. 133 .1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REEF" —t>0 . 0 d B 1 0 .0 dB/ START STO P 1.000000000 4.000000000 GHz GHz Figure 5.8: Measured insertion loss of a microstrip hairpin-line bandpass filter for a = 0.8mm, b = 0.4mm , d = 1.27mm, L = 18.36mm, w = 3 1.2mm, and e r = 10.2. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • 'G — :n £ -20 u — - 3 0 - L -40 — 2.0 2. 5 F r e q u e n c y (GHe) Figure 5.9: Calculated responses of a microstrip hairpin-line bandpass filter with a - 0.15mm, b = 2.0mm, d = 0.25mm, L = 18.36mm, w = 43mm, and e r = 18. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.9 shows the computed return loss and insertion loss of ahairpinline bandpass filter using ETL model of (5.2) with a = 0.15mm, b = 2.0mm, L = 18.36mm, d = 0.25mm, w = 43mm (20 strips), and s r = 18. Since this set of circuit parameters provides good image impedance at passband region, the calculated responses show better insertion loss at passband and sharp skirts at the edges. 5.5 Tunable Microstrip Meander-line Bandreject Filters Cs H h Cs Cs Cs Cs Hh Hh HH HH H HH HH HH HH 2Cs Cs Cs Cs Cs 4 2Cs Figure 5.10: An example of a tunable microstrip meander-line bandreject filter with loaded capacitances. Figure 5.10 shows a meander-line structure loaded with lumped capacitances in microstrip form. As shown in Figure 4.10, this kind of loaded meander-line has bandwidth-tuned property. To demonstrate the phenomenon, a 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. meander line structure loaded with chip capacitances is fabricated with a = 3.5mm, b = 2.0mm, d = 1.27mm, L = 100mm, w = 110mm (20 strips), and e r - 10.2. Figure 5.11 shows the calculated periodic dispersion diagram for this structure using the lower equation o f (4.6). We see that the bandwidth of stopband increases as frequency goes up; this phenomenon will be verified by experiment. Figure 5.12 shows the measured insertion loss and return loss of the capacitance-loaded meander-line bandreject filter, where we see that the lower edges of each stopband stay almost close for all cases and the location of stopband and bandwidth are tunable as the loaded capacitance is varied; both phenomena are correctly predicted in the dispersion curves o f Figure 5.11. So basically speaking, this kind of meanderline filter is bandwidth-tuned. This structure can be extended to load nonlinear devices such as varactor diodes to meander lines, as illustrated in Figure 4.9 with proper bias circuit, for voltage-tuned filter applications. The calculated and measured IS21I are shown in Figure 5.13 where the measured insertion losses in passband increase for lower value of capacitance and the losses are mainly due to the conductor loss and impedance-mismatched loss as shown Figure 5.14. To correctly compute the responses, the loss factors and the model of chip capacitances should be taken into account, especially for higher frequency. Another promising application suggested by Onoh et al. [64] is loading PIN diodes into meander lines to control the passband and stopband properties for multiplexer application. In our meander-line structure shown in Figure 4.2, if PIN 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2:0 • — r i ’ \ 1 ■A * 52 — - V , ! \ - , ;! ■ ' - I — \ - \ , - \ ( V \ \ V : w\ •\ v„ ■ v . ; '■ V - . \ / . ' . ■■ F : : 50 — : : ■ : n i f *' - f \ / r ■ •\ V - 0— - • ■ F /; /' /.. /' • /" i 0. 5 / ' : r ’ 0 -'?cue / ' .5 - c y (GHz) A . ; i ‘ - ■i i 2.3 2: Figure 5.11: Dispersion diagram (backward-wave) o f a loaded meander line for Cs =1.5 p F 3.3 p F (• •), 4.8 p F ( - •), 10.0 p F ( - • • •), and short link (—) with a = 3.5mm, b - 2.0mm, L = 100mm, d = 1.27mm, and e r = 10.2. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.12: Measured responses of a loaded meander-line bandreject filter for Cj=l-5 p F (— ), 3.3 p F (—), 4.8 p F ( - •), 10 p F ( - • • •), and short link (• •) with a = 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, w = 110mm, and e r = 10.2 . 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fa) -20 - C. 0 .5 ' 0 " '■ e c je r’c y (G h z ) 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i i t (c) "I ".'I | I 1 ;|j ; y,r ■< -2C ^ e c - e r c v ''GHz, i Or A (d) ; _ i'M i - i •i. 1-'Jn f aw F 0.0 0.5 ^ 'ecuencv (GHz) Figure 5.13: Calculated (— ) and measured (—) 1^21| of a loaded meander-line filter with a - 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, w = 110mm, and e r = 10.2 : (a) short link; (b) Cj=10 p F \ (c) Cy=48 p F \ (d) G =3.3 p F . 141 ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I.'irnl ( wt . i n: . V 1 0.5 VC '.5 -'ecu e-cy 'G Hz; Figure 5.14: Calculated image impedances of a loaded meander line for Cr=l-5 p F ( - -), 3.3 p F (• •), 4.8 p F (- •), 10 p F ( - • • •), and short link (—) with a = 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, and e r = 10.2. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. diodes are loaded on the upper side of structure (replace Z i) and Z i is replaced by a short link, then this structure becomes a meander-line bandreject filter when the diode is “on” and becomes a hairpin-line bandpass filter when the diode is “o ff’. Thus a two-staged filter with the same bandwidth in one circuit can be constructed. Figure 5.15 demonstrates the measured transmission loss of a meander-line and hairpin-line filter with the same circuit dimensions. We see that the passband and stopband of the structures can be switched with the same bandwidth by controling Z i in Figure 4.2. This idea can be realized by loading PIN diodes into the structure in future work according to the good measured results of Figure 5.15. 5.6 Conclusions This chapter develops accurate and efficient models for microstrip grating elements such as meander lines, hairpin lines, and comb lines for compact filter applications. The computed results are verified by experiment and compared well. Especially, a loaded meander-line structure provides application in voltagetuned filters. These grating filters promise to deliver sharp attenuation rate at stopband or passband edges which is necessary in practical design. The only thing needed to be improved is the loss factor. To well design the grating filters, the most important thing is that one should select the optimum circuit parameters to provide proper image impedance to match the characteristic impedance on both ports such that the impedance-mismatched loss can be minimized. To be more accurate, the 143 Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. h,/11 " r e c u e r c y (GHz) Figure 5.15: Measured transmission loss of a meander-line bandreject filter (—) and hairpin-line bandpass filter (• •) with the same circuit dimensions: a = 2.5mm , b = 2.0m m , L = 5 0 m m , d = 1.27mm, w = 9 0 m m , and e r = 10.2. I 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. model o f the side edge effect should be considered seriously. The overall performances of filters presented in this chapter are able to be further improved by optimum design. All the measured results we show in this chapter are performed on HP 8510 using full two-port calibration in coaxial connectors. For accurate measurement in microstrip circuits, the TRL calibration should be used by taking into account the effects on both ports. 145 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 SUMMARY AND FUTURE WORK The main contribution in this dissertation is the development of equivalent boundary conditions (EBCs) of a periodic strip grating lying at the interface between two media. The obtained EBCs taking into account the phase shift between adjacent strips are derived from the modified method of homogenization based on the technique of multiple scales. This technique is employed to expand the scattered fields with a phase shift from a strip grating in powers of grating period p and this process leads to solving the static electric and magnetic boundary-value problems to obtain EBCs. The derived EBCs with a phase shift are then used to investigate the propagation characteristics o f surface waves along a periodic strip grating on a grounded substrate. By using EBCs, the analysis of the periodic structure becomes less complicated and more efficient compared to the numerical method used in [5759]. Having the propagation characteristics of metal gratings on grounded substrates, planar grating elements loaded or unloaded with lumped devices are then investigated and the results show the stopband or passband properties which are characteristics o f periodic structures. Then grating filters are designed by using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equivalent transmission line (ETL) models and verified experimently. The impact of these compact grating filters is to promise to deliver either broadband properties or highly selective behaviors. In this analysis, the key limitations are p /d will not be too large and the period length p of the grating is much smaller than the wavelength in free space due to the homogenization process. In all the design through this thesis, the loss factor and the model of side edge effect o f the structure are not considered. This will be very important in practical design and could be the future subject. Finally, it should be very possible to extend the grating filters to other prominent transmission-line structures, such as coplanar waveguide (CPW), stripline, and multi-layer microstrip. For some other grating elements application for planar microwave circuits, such as switches and pulse shapers, the developed EBCs need to be used to efficiently model metal grating on substrates and this will be an interesting subject. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I BIBLIOGRAPHY [1] S. Ye. Bankov and I. V. Levchenko, “Equivalent boundary conditions for a closely spaced ribbon grating at the interface of two media,” Soviet Jn. Comm. Tech. and Electron., vol. 34, no. 5, pp. 67-72, 1989. [2] A. I. Adonina and V. V. Shcherbak, “Equivalent boundary conditions at a metal grating situated between two magnetic material,” Soviet Phys. - Tech. Phys., vol. 9, pp. 261-263, 1964. [3] B. Ia. Moizhes, “On the theory of electromagnetic wave propagation in a helix,” Soviet Phys. - Tech. Phys., vol. 3, pp. 1196-1201, 1958. [4] L. A. Vainshtein, “On the electrodynamic theory of grids,” in High-Power Electronics. Oxford: Pergamon Press, pp. 14-48, 1966. [5] J. T. Bolljahn and G. L. Matthaei, “A study of the phase and filter properties of arrays o f parallel conductors between ground planes,” Proc. IRE, vol. 50, pp. 299-311, March 1962. [6] P. N. Butcher, “The coupling impedance of tape structures,” Proc. IEE, vol. 104, pt. B, pp. 177-187, March 1970. [7] R. M. 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Phys., vol. 14, pp. 350-354, 1969. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [47] V. N. Ivanov, “Surface waves in a semi-infinite ribbon array,” Radio Eng. Electron. Phys., vol. 14, pp. 1520-1525, 1969. [48] L. G. Naryshkina and M. E. Gertsenshtein, “Slow waves in an anisotropically conducting plane lying in a dielectric,” Radiophys. Quantum Electron., vol. 10, pp. 45-48, 1967. [49] V. N. Zlunitsyna, “A transmission line structure consisting of a metallic strip grid embedded in a screened dielectric,” Radiophys. Quantum Electron., vol. 11, pp. 352-354, 1968. [50] L. G. Naryshkina, “Effect o f the gap between a slow-wave array and a dielectric on the propagation of surface waves,” Radiophys. Quantum Electron., vol. 12, pp. 483-485, 1969. [51] J. Jacobsen, “Analytical, numerical, and experimental investigation of guided waves on a periodically strip-loaded dielectric slab,” IEEE Trans. Antennas Propagat., vol. Ap-18, pp. 379-388, May 1970. [52] R. A. Sigelmann, “Surface waves on a grounded dielectric slab covered by a periodically slotted conducting plane,” IEEE Trans. Antennas Propagat., vol. Ap-15, pp. 672-676, Sept. 1967. [53] V. Rizzoli and A. Lipparini, “Bloch-wave analysis of stripline- and microstriparray slow-wave structures,” IEEE Trans. Microwave Theory Tech., vol. MTT29, pp. 143-150, February 1981. [54] V. N. Ivanov and V. D. Ivanova, “TEM-approximation in the theory of strip slow-wave systems,” Elektronnaya Tehknika, ser. Elektronika SVCh, No. 9, pp. 118-121, 1977. [55] R. R. Yurgenson, “On the modelling of microstrip meander lines on ferrite substrates,” Radiotekh Elektron., vol. 24, pp. 397-401, 1979. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [56] V. D. Zosim and R. R. Yurgenson, “Calculation of dispersion in microstrip slow-wave systems o f the meander and comb types on a magnetodielectric substrate,” Antenny, vyp. 28. Moscow: Svyaz’, pp. 114-121, 1980. [57] J. A. Weiss, “Dispersion and field analysis of a microstrip meander-line slowwave structure,” IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 1194-1201, Dec. 1974. [58] R_ Crampagne and M. Ahmadpanah, “Meander and interdigital lines as periodic slow-wave structures. Parts I and II,” Int. J. Electron., vol. 43, pp. 1932, 33-39, 1977. [59] R. Crampagne and M. Ahmadpanah, “Interesting microwave filtering properties of C-section periodic structure,” Electronics Letters, vol. 13, no. 7, pp. 199-201, 1977. [60] F. H. Bellamine and E. F. Kuester, “Guided waves along a metal grating on the surface of a grounded dielectric slab,” IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 1190-1197, July 1994. [61] D. M. Pozar, Microwve Engineering, Addison-Wesley Publishing Company, Inc., 1990. [62] V. M. Dashenkov, “The dispersion and coupling impedance o f a periodic multistage system o f coupled lines,” Radio Electron. Commun. Syst., vol. 20, pp. 57-64, 1977. [63] R. Garg and I. J. Bahl, “Microstrip discontinuities,” Int. J. Electron., vol. 45, pp. 81-87, 1978. [64] G. N. Onoh and H. C. Inyiama, “A new class of microwave filters for multiplexing applications,” Modelling, Simulation & Control, vol. 36, pp. 122, 1991. 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [65] W. J. Getsinger, “End-effects in quasi-TEM transmission lines,” IEEE Tram. Microwave Theory Tech., vol. MTT-41, pp. 666-672, April 1993. [66] M. A. R. Gunston and B. F. Nicholson, “Interdigital and comb-line filters,” M arconi Review, vol. 28, pp. 133-147, 1965. [67] V. K. Tripathi and G. I. Haddan, “Varactor-loaded ladder lines,” 7th International Conference on Microwave and Optical Generation and Amplification, pp. 594-599, 1968. [68] M. A. Miller, “Electromagnetic waves propagation over a plane surface with anisotropic impedance boundary conditions,” Doklady A N SSSR, vol. 87, pp. 571-574, 1952. [69] P. J. B. Clarricoats and K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electronics Letters, vol. 1, no. 5, pp. 145-146, 1965. [70] P. J. B. Clarricoats and B. C. Taylor, “Evanescent and propagating modes of dielectric-loaded circular waveguide,” Proc. IEE, vol. I l l , pp. 1951-1956, December 1964. [71] J. D. Rhodes, “General constraints on propagation characteristics of electromagnetic waves in uniform inhomogeneous waveguides,” Proc. IEE, vol. 118, pp. 849-856, July 1971. [72] S. B. Rayevskiy, “Some properties of complex waves in a double-layer, circular, shielded waveguide,” Radio Eng. Electron. Phys., vol. 21, pp. 36-39, 1976. [73] T. F. Jablonski, “Complex modes in open lossless dielectric waveguides,” J. Opt. Soc. Am., vol. 11, pp. 1272-1282, April 1994. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [74] G. I. Veselov, A. V. Gureev, and V. Yu. Soldatkin, “Diffraction characteristics of dielectric slug inside circular waveguide,” Radiophys. Quantum Electron., vol. 27, pp. 980-985, 1984. [75] G. I. Veselov and A. V. Gureev, “Features of the diffraction of electromagnetic waves in partially filled waveguides with a complex spectrum,” Radiophys. Quantum Electron., vol. 27, pp. 232-236, 1984. [76] A. S. Omar and K. Schunemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 268-275, March 1987. [77] A. S. Omar and K. Schunemann, “The effect of complex modes at finline discontinuities,” IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 1508-1514, December 1986. [78] C.-K. C. Tzuang and J.-M. Lin, “On the mode-coupling formation of complex modes in a nonreciprocal finline,” I FEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1400-1408, August 1993. [79] A. Magos, “Singular frequencies in wave guides with inhomogeneous dielectric,” Int. J. o f Applied Electromagnetics in Materials 4, pp. 189-196, 1994. [80] P. E. Krasnushkin, “Conversion of normal waves in periodic and smooth lossless waveguides,” Radio Eng. Electron. Phys., vol. 19, pp. 1-12, 1974. [81] A. S. Omar and K. Schunemann, “Formulation of the singular integral equation technique for planar transmission lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 1313-1322, December 1985. [82] M. Mrozowski and J. Mazur, “Matrix theory approach to complex waves,” IEEE Trans. Microwave Theory Tech., vol. MTT-40, pp. 781-785, April 1992. 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [83] W.-X. Huang and T. Itoh, “Complex modes in lossless shielded microstrip lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 163-165, January 1988. [84] J. R. Pierce, “Coupling of modes o f propagation,” J. Appl. Phys., vol. 25, no. 2, pp. 179-183, Feb. 1954. [85] A. M. Belyantsev and A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys., vol. 9, pp. 980-988, 1964. [86] P. J. B. Clarricoats, “Circular-waveguide backward-wave structures,” Proc. IEE, vol. 110, pp. 261-268, February 1963. [87] L. Brillouin, Wave Propagation in Periodic Structure, Dover Publications Inc., 1953. 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A EVALUATION OF EQUATION (2.87) The goal o f Appendix A is to evaluate the expression shown in equation (2.87). Ivanov showed that [10] n=c» 2 —/2nJtv' / ’n ( « » 4 )[ 5 -/2 (n + l)n v ' «. — + i --------- —] = ey 2 S » y '_ _ _ ^ _ s (cosA ') n+5 n + 1 -5 smott (A .l) where 0 < j27ty'| < A, A' = 7t - A, and 6 is not an integer. Taking the real part of both sides o f (A. 1), we have cos 2n8y' —- — P_g (cos A') sinSre ^ / A\rCOs2n7tv' cos2(n + l)7ty', = S P n(cos A)[------ f - + V i ] £0 n +5 n + 1 -5 r A.cos2n7iy' = £ P n(cos A) n=0 n+0 IS ° ^ 0052(0 + 1)79/' J P n(cosA) ^ >* n=0 n + 1 -5 Let n -+ n - 1 in the second term of right-hand side of (A.2), we get Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A-2) I cos27t5y'—- — />_§(cos A') sin 87c o r * \ c° s 2 n it/ n / A,c o s 2 n7cy' = X P n (c°sA )c. + Z P n-lC cosA ) - fn=0 11 + 5 n=l (A.3) 11-5 or "v n t 2 ; Pa n=l ...cos2miy' (COS A ) ---------— i - + 2 ..c o s 2n7ty' / ^ ( c o s A ) ----------- n+8 = 7rcos27t5y' n -5 , . . (A.4) I r - r - ^ - P - s i c o s A ') - sino7t 0 By letting 5 —> -5 in (A.4), we have V °n r ANcos2n7ty' A.c o s 2n7ty' 2 ; P n (cos A)------- ■~—Jr 2 P n _ 1(cos A) -~— n=l n -8 n^i n+5 (A-5) 7t cos 27c5y' „ . . 1 = ----- — — i»5(cosA') + r -sino7t 0 Then subtracting (A. 5) from (A.4) yields ny ° *^n(A)cos2n7ty' _ tc cos 2tt8y' R§ (A') n2 - 8 2 25 sin 87c 1 52 Let y ’ = 0 in (A.6), we then obtain the identity shown in equation (2.87) 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.6) n=oo y S n ( A ) _ 7C/?g(A>) n=l n2 -8 2 25 sinSre 1 (A.7) 52 where Sn(A) = P n _ l ( c o s A ) - P a(cosA), /?5 (A') = P_5 (cosA') + P 8 (cosA'), A = bit/p, and A' = aiz/p. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B EVALUATION OF EQUATION (2.125) Appendix B finds the first-order z -component boundary-layer electric field on the surface o f grating, as shown in equation (2.125) for Substituting (2. 121) into equation (2.119), we then have i -2 V - 0 = U® d K „ It ? 3 <?-y2n7iy' „/2imy' + a 11— ) 1 Z [ + 2 S T ¥ 19 » (B .l) r no . dr) " r - e - J ' 2n*y' eJ2imy \ where S n = T ^ n - l ( cos A ) + F’nCcos A)] = ^ -fln (A ) 2 2 cpn = P n( c o s A ) - P n _ 1(cosA) = - 5 n (A) (B.2) ; & = bn/p Now, we have to evaluate the summation terms in (B .l) first. Substitute in (B.2) into (B.l), the summation terms become Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and ny ° r ^ — — gy2n7ly'n 2nit+0 + 2n7C-0 1 n^ ° g-y'2njty' {Z ey2n7tv' =— [^n(cos A)------— +Pn- i( cos A)------— ]+ 4tt n=i n+5 n -5 ejinny' ^2° (B.3) p-jinny' Z t^n(cosA)------— + /3n_ 1(cosA)----- — “ l n -5 n+5 ]} and ejinny' ny ° - e -y2n7ty' £i 2n7t +0 + 2n7t -0 1 I\z ? =— { Z 2k n ey2n7ry' p-jinny' I A ( c° s A )------ — +/>n_ l(c o s A )------ — - ] - n -5 ^ n+5 (B.4) “2^° e-jinny' ej2nny' X [PnCcosA)------— — + P n_ l (c o sA )----- — ]} n=i n+5 n—5 where 5 = 9 /2 tc. We have to make use of (A.1) to obtain (B.3) and (B.4). After regrouping the terms in the summation in (A.1), (A.1) can be rewritten as n_z?° e -jin n y' pjln n y’ n -« £ P n(cosA)-------— + £ P n _!(cosA ) — n=i n+8 n=1 n -5 n ejin n y’ i = — -------- P _ 5 ( cosA ' ) - sin57t ^ 5 Also replace 5 by 5 +1 in (A.1), with a little rearrangement we have 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nJZ? g -jlr a iy ' n=°° p jltm y ' £ P n _i(cos A) — + £ Pn (cosA ) —n-, n+6 n-8 k 1 eJ'2my ' = - = ^ - * - ' (“ s 4 ') + r Substitute (B.5) and (B.6) into (B.3) and (B.4), we obtain n“f° p -jinn y' pj2nvy' p j’& ny' and It2 ° -e-jln xy' eJlnxy' 1 1 trpj^Snty' S [ - - ‘ J . - + * ----- „]<Pa = i [ i - S ^ - / - « S(A')] £=l 2n7t +6 2n7t - 0 7t 5 2 sin57t (B.8) Then substitution o f (B.7) and (B.8) into (B .l) yields (2.125), - eer|*'=o l pjln 8 y' K W a ^ ) W I A ' )+- n r) n la (B.9) 1 ireJ'2n8y' 5 2 smote where identities P _ s _ 1(cosA ') = />5 (cosA ') and P _ 8 ( c o sA ') = P s_ i(co sA ') are used. 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C EVALUATION OF IDENTITIES Appendix C shows two important identities which are helpful to evalute equation (2.138). Ivanov [10] gave the following identity ^ n( c o s A ) [ ^ ^ + ^ n=0 ^ ] =0 A <| / | < (C l) Multiply both sides by e'Jt5 and integrate this equation with respect to t from 2izy' to 7t, we obtain another identity a-f° e -j2n(n+8)y' £ Pn(cosA)[S n—o n"° gj2n (n + l-5)y' -------- 1------n+5 n + 1 -5 (C.2) = n=0 Z ° (- 0 " ^ i.( cosA) [nr+~oT *nr+r ri T —o^ " - /"6 Use the following formula [ 10] ! _ ] ( _ i ) « = _ J L - f 6 (cosA) "If />n(c o s A )[-!------------5 -n n + 1+5 sinSre Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.3) By making 5 —>-5 in (C.3) and substituting into (C.2), then (C.2) becomes n^ ° , , X /»n(cosA)[ i^O e -/2rc(n+5)y' ej2 n (n + \-§ )y' — ----------------- — — ] n+5 n + I-5 (C.4) 7t e = - — — /*_5 (cos A) smS7t After a little rearrangement, (C.4) yields ay P n.-l(cos A) ej2my' _ y ° ^ n (cos A) g-y?nny» n^l n -5 £i n+5 T i 7t ej^ny'-7t)5 (C.5) ; A < |2jty '|< it Also replace 5 by 5 +1 in (C.4), we obtain the identity y 0 /!n-l(£ps A) ^ _y2n^y' _ ny n=l n+5 P n(cos A) / 2my* n -5 - i 7t e j(2ny' ~n ^ = T~ + ------ r 1 -------P 5 (cos A) 5 sm 57t (C.6) ; A< 2 7 t/< 7 t where p_g _ ^ co sA ) = PgtcosA ) has been used in obtaining (C.5) and (C.6). Adding and subtracting o f (C.5) and (C.6) respectively, we obtain the following important identities 165 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e -jinn y' I ejinny' ; l _ _ +£ ( ~ „j(lny'-n)S C . 7 ) n=l and n=oo e jinny' e -jinny' 2 n p lO ^ y '-^ I t <c -8> n=l where S n , S5 , R n , and R$ are shown in Appendix A or B. 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX D GRATING VOLTAGES AND CURRENTS In this Appendix, the voltages and current at the surface of grating due to the zeroth order boundary layer fields are evaluated; two grating voltages will be defined. First, consider the voltage difference between the strip grating and -oo due to the vertical component o f the boundary layer electric fields. We will call this the boundary-layer voltage. From (2.104), the boundary-layer voltage at the strip grating with respect to —oo, V h , which is function of y ' is jri i znk +0 znrc - o ifr) 2 [ 2mt +0 2n7i - 0 With K n and L n given in (2.105), then (D .l) becomes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pK„\ . „ n=co -jin n y' jin n y’ S — - +f — - » .+ 2ntt+ 0 2n7t - 0 2 sa zrO J = o n=® r- e ~j2 m y ' e j 2 m y ' , —------- / [— + - -------- ]cp 2 L 2n7t + e 2 n r t - 0 JVn ( ° - 2) p E yi\x' where the summation terms in (D.2) have been obtained in (B.7) and (B.8). Equation (D.2) then becomes v b, 0”) J K ^ 2 ea (D.3, 4 smo7t 2 57t 2sm57t At y ' = 0 , (D.3) becomes = 2sa 4 sin57t 2 07t 2 sinS7t ( d .4) The current If, flowing in z -direction on the strip centered at y ' = 0 due to zeroth order boundary layer magnetic field can be obtained from integrating the boundary layer current density over the strip, i.e. h = I ! i “ hy_]e~jQy/Pdy on grating 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.5) From equation (2.99), hy on the grating is (D6) h ^ = — 2 ][ —n +-j F n e~J2n7zy/p+ Gn a=l 2 2 where Fn and Ln are defined in (2.100). Substitute ( D .6 ) into (D .5 ), after some manipulation we have (assume _ P K h \ x ’=o = |Iq ) V f . sinQi-SjA ^ sinpi+SjA^ 2it n -5 n+5 (D.7) j p ( ^xi|jr'=0) sin(n+5 )A sin(n-5 )A, . . . ---------------- 2 - l---------- ;----------------; — Jon(A) Ho* n=l n+8 n -5 The summation terms in (D.7) can be evaluated by the following identities. From (C.4), after a little arrangement in (C.4) we have g - j l n (n + 5)y £ i>n (cosA) ------ — n +5 n=« £ P n -i(cosA ) i£l ej2n(n-5)y' ---n"5 Tte- ^ 6 n , ., e ' W = . g P - s (cos A ) ----smSrt 6 Taking the imaginary part of (D .8 ) yields 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.8) ”f [/>.(«. ttfZ& ULpX. + n=i ri»2.c-<y n +o n -5 _ . .. sin(27c8.y') = n P _5 (cos A ) — — (D.9) By letting 8 -> -6 in (D.9) we have V °rD ( AxSin27c(n-8)y .. sin27t(n+5)y' 2 t^n(C0SA) -V + ^n-l(C0S A) ] nTj n -5 n+5 (D.10) = x/>5 ( c o s A ) - ^ ^ Substitute (D.9) and (D.10) by replacing y ' by b/2p into (D .7 ), the current on the center strip due to zeroth order boundary layer magnetic field is obtained as lh = PKh\X’ = o 2sin(A5) 1 JP B xi^o T [« Rs (A) --------------- + ---------1— S5 (A) lK 5 Ji0 (D. 11) where ' ^ k = o = ( //? + + ^ + - ^ ? - V = ° (D .12) 4 y =0 = H o ( ^ ++ ^ + V=o = Note that If, obtained in (D .ll) is equal to the integration o f the boundary-layer surface current density shown in (2.97) with respect to y from -6 /2 to 6/2. 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, consider the voltage difference due to the transverse component o f the zeroth order boundary layer electric fields. Let Vi be the voltage on the strip at y ' = 0 and Vie~J® is the voltage on the strip to the right o f y ' - 0 due to the boundary layer electric fields, then we have the voltage difference between these two strips Ve(e~JQ - 1 ) = where J 1bj2ey+e~-iQy'd y 011 the strip (D. 13) is continuous in the gaps and zero on the strips and the fast variable y ' is replaced by y [ p . Then substitute e® in equation (2.104) into (D.13), we have Ve(e~JQ - 1) = J~ ^ a e - j (2tat+Q)y/Pdy+ 2 (D. 14) r bl2ej(2m -Q)y/pdy] *b/2 where the integrations in (D. 14) are carried out as ra+b/2 ... . -y'(n+5)(27t-A)_ -y'(n+S)A A= j e -y(2nji +Q)y/P(fy = j p t -------------------- e-----------Jb/2 r 2mt +0 ° r a+bn / ^ (n -S )(2 7 i-A )_ ^ (n -S )A B= I eJ(2m ~Q)ylPdy = - j p —----------------------------I** * Jy 2 hk - 0 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.15) Then (D. 14) becomes U=«J T r X e - f l - l ) * Z [ ± ? - ( A + B) + ^ - ( 3 - A ) l n=l 2 2 (D.16) where ■R« * \ e -yn(27i-A) ^ n (2 * -A ) /nA -ynA A + B = jp { e -J 5 V n - V [ - -----------------] + e-yA5 |- _ £ ------------ e-------- 2n?t +0 2n7C- 0 2n7t - 0 2nrc +0 •so* a\ eyn(27l-A) e- M 2n~ V .AS J * * e~jnA B - A = -jp{e~J?>(27t ~A) [ + ] - e~J--- [— ------- + —------- ]} 2n7t - 0 2 nrc +0 2n7t - 0 2ntt +0 (D.17) and L n and K n are shown in (2.105). Substitute (D.8), L n , and K n into (D.16) with the aid of (B.7) and (B.8), after some manipulation we have ,-y'A5 _ ^/A5 g-ye 1 + £%r J ] k8 (D.18) 11=50 /X 2 n=l -J ± ( B - A ) = o 2 Then (D. 16) becomes ,-y'A5 _ gj'AS e -ye 7t6 ] 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.19) or the voltage on the strip located at y ' - 0 due to the zeroth order boundary-layer electric field e£ is 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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