Electromagnetic simulation and analysis for novel filter design

... convergence and accuracy have been achieved through our method The MoM algorithm is next used to design and analyze novel bandpass filters Several miniaturization techniques for bandpass filter design. .. made of filter structures, there is a need to investigate miniaturization of the filters Filters can be categorized into bandpass filters, bandstop filters, lowpass filter and highpass filters... Various novel bandpass filter designs are investigated in Chapter As the design of compact, lowloss and good performance bandpass filter has attracted a lot of attention recently, all the filters

Acknowledgement I would like to express my most sincere gratitude to my supervisors Associate Professor Ooi Ban Leong, Professor Prof. Leong Mook Seng and Dr. Guo Li Hui for their invaluable guidance, suggestions, and constructive criticisms during the investigation of the project and the preparation of this thesis. At the same time, I would like to thank my friends and colleagues from our microwave group and Institute of Microelectronics (IME), Singapore for their support and kind assistance. Finally, I would like to express my appreciation to my family and friends, from whom I received much encouragement in the Ph.D. process. i Table of Contents Acknowledgement ...........................................................................................................i Table of Contents............................................................................................................ii List of Figures................................................................................................................vi List of Tables .................................................................................................................xi List of Symbols.............................................................................................................xii Summary ..................................................................................................................... xiii Chapter 1 Introduction.................................................................................................1 1.1 Literature Review ...........................................................................................1 1.2 Scope of Work ..............................................................................................10 1.3 Achievements and Contributions..................................................................11 1.3.1 List of Journal Publications .................................................................. 12 1.3.2 List of Conference Publications............................................................ 13 Chapter 2 Bi-complex Algebra and Quaternion Electromagnetics ...........................15 2.1 Introduction...................................................................................................15 2.2 Network Quaternion Integral Equation Formulation....................................16 2.3 Mixed Potential Integral Equation Formulation ...........................................31 Chapter 3 A Useful Multilayered Microstrip Pole Extraction Technique.................41 3.1 Introduction...................................................................................................41 3.2 Pole Extraction for Two Layered Structures ................................................44 3.2.1 Introduction........................................................................................... 44 3.2.2 Two-layered Microstrip Geometry in TM Mode.................................. 44 3.2.3 Two-layered Microstrip Geometry in TE Mode................................... 50 3.2.4 Numerical Results and Discussions...................................................... 53 3.3 Extraction of Poles for N-layered Microstrip Geometry ..............................58 ii 3.3.1 Extraction of Surface Wave Poles ........................................................ 58 3.3.2 Extraction of Leaky Wave Poles for N-layered Microstrip Geometry . 65 3.3.3 Extraction of Lossy Improper Poles for N-layered Microstrip Geometry 68 3.3.4 Extraction Algorithm ............................................................................ 68 3.3.5 Numerical Results and Discussions...................................................... 70 Chapter 4 Bandpass Filters Miniaturization ..............................................................76 4.1 Introduction...................................................................................................76 4.1.1 Method of Moment with developed DCIM algorithm ......................... 76 4.1.2 Bandpass filters Miniaturization........................................................... 82 4.2 Basic Structures for Filter Design.................................................................84 4.2.1 Resonator .............................................................................................. 84 4.2.2 Basic Types of Microstrip Bandpass Filters......................................... 86 4.3 4.2.2.1 End-coupled, Half-wavelength Resonator Filters............................. 86 4.2.2.2 Parallel-coupled, Half-wavelength Resonator Filters....................... 87 4.2.2.3 Hairpin-line Bandpass Filters ........................................................... 87 4.2.2.4 Interdigital Bandpass Filters ............................................................. 88 4.2.2.5 Stepped Impedance Resonator (SIR) Filters..................................... 89 4.2.2.6 Dual Mode Resonator Filters............................................................ 89 4.2.2.7 Other Types of Bandpass Filters....................................................... 90 Bandpass Filter Miniaturization for Microstrip Structures...........................91 4.3.1 Bandpass Filter Miniaturization Using PBG ........................................ 91 4.3.1.1 Design Structure of a SIR Bandpass Filter ....................................... 93 4.3.1.2 Design Structure of a SIR Bandpass Filter using PBG..................... 95 4.3.1.3 Experimental Results and Discussions ........................................... 100 iii 4.3.2 New Planar Filter without Using Cross-Coupled Effect .................... 103 4.3.2.1 Intermediate, Dual Mode Microstrip Resonator............................. 104 4.3.2.2 The Proposed Filters ....................................................................... 109 4.3.2.3 Numerical and Experimental Results ............................................. 110 4.3.3 Miniaturized Open-Loop Resonator with Wide Frequency Perturbation 116 4.3.3.1 Preliminary Analysis and Design ................................................... 117 4.3.3.2 Novel Resonator Implementation ................................................... 125 4.3.3.3 Novel Resonator Design ................................................................. 131 4.4 Coplanar Filter Miniaturization Using Advanced Technologies ................134 4.4.1 Novel Miniaturized Coplanar Filter Design Using WTT ................... 134 4.4.1.1 Introduction..................................................................................... 134 4.4.1.2 Fabrication technology ................................................................... 136 4.4.1.3 Coplanar Bandpass Filter Design ................................................... 139 4.4.1.4 Experimental Results and Discussions ........................................... 151 4.4.2 A Modified Miniaturized LTCC Hairpin-combline Resonator .......... 155 4.4.2.1 Introduction..................................................................................... 155 4.4.2.2 LTCC Materials and Structures Used in This Thesis ..................... 155 4.4.2.3 Preliminary Analysis and Design ................................................... 157 4.4.2.4 Two Proposed Designs Using LTCC ............................................. 166 4.4.2.5 Experimental Results and Discussions ........................................... 171 Chapter 5 Conclusions and Future Work ................................................................177 5.1 Conclusions.................................................................................................177 5.2 Recommendations for Future Work ...........................................................180 APPENDIX.................................................................................................................182 iv Definition of Quaternions .......................................................................................182 Some Properties of Bi-complex Numbers ..............................................................183 Non-commutative multiplication of Quaternions............................................... 183 Associativity of Multiplication of Quaternions .................................................. 184 Conjugation of Quaternions................................................................................ 184 Quaternion Fourier Transform................................................................................185 Basic Quaternion Electromagnetics........................................................................187 Source-free, Homogeneous, Lossless TEM Case............................................... 187 Source-free, Inhomogeneous, Lossless TEM Case ............................................ 191 References...................................................................................................................196 v List of Figures Fig. 1 Rotation of the axis.........................................................................................22 Fig. 2 Transmission line models ...............................................................................26 Fig. 3 Initial guess evaluation. ..................................................................................47 Fig. 4 Two-layered microstrip topology ...................................................................48 Fig. 5 A numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . (c)-(d) For the second TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 ....................................55 Fig. 6 A numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 . (c)-(d) For the first TE root of the two-layered microstrip geometry for the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . .......................................56 Fig. 7 A numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations. (a)-(b) For the second TE root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 .(c)-(d) For the first TE root of the two-layered microstrip geometry under the case of ε r1 < ε r 2 and . k o ε r1 < k ρ ≤ k o ε r 2 .....................57 Fig. 8 N-layered microstrip topology........................................................................59 Fig. 9 Initial guess evaluation. ..................................................................................64 Fig. 10 Graphic solution for TM and TE surface and leaky wave modes, where the x semicircles represent ± B 2 − x 2 . (a) TM: ± B 2 − x 2 = tan ( x ) . (b) TE: εr ± B 2 − x 2 = − x cot ( x ) .........................................................................................67 Fig. 11 A numerical comparisons of the two methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the fourlayered microstrip geometry under the case of ε r1 < ε r 2 < ε r 3 < ε r 4 and k o ≤ k ρ ≤ k o ε r1 ..................................................................................................71 Fig. 12 Schematic of a RWG edge element and the dipole interpretation................77 vi Fig. 13 Barycentric subdivision of the primary triangle. The triangle’s midpoint is shown by a white circle. .......................................................................................78 Fig. 14 Photograph of the bandpass filter with SIR..................................................81 Fig. 15 Comparison of measured and simulated results ...........................................82 Fig. 16 Distributed line resonators............................................................................84 Fig. 17 Ring resonator ..............................................................................................85 Fig. 18 Patch resonator .............................................................................................86 Fig. 19 General configuration of end-coupled microstrip bandpass filter ................87 Fig. 20 General structure of parallel (edge)-coupled microstrip bandpass filter ......87 Fig. 21 Layout of a hairpin-line microstrip bandpass filter ......................................88 Fig. 22 General configuration of interdigital bandpass filter....................................89 Fig. 23 Bandpass filter with SIR...............................................................................93 Fig. 24 Structural variation of λg / 2 type SIRs. ......................................................94 Fig. 25 Bandpass filter with SIR and PBG ...............................................................96 Fig. 26 Dimension of one unit PBG structure ..........................................................96 Fig. 27 An EBG unit .................................................................................................97 Fig. 28 Lossless equivalent circuit............................................................................97 Fig. 29 Equivalent circuit of an EBG unit ................................................................97 Fig. 30 Photographs of the proposed bandpass filters ............................................101 Fig. 31 Simulated (MoM) and measured results for the filter with SIR.................101 Fig. 32 Simulated (MoM) and measured results for the filter with SIR and PBG .102 Fig. 33 A typical measured response of a squared open-loop resonator ................104 Fig. 34 Comparison of shifted feed lines ................................................................106 Fig. 35 Equivalent circuit for the parallel direct feed structure ..............................106 Fig. 36 Physical dimensions of the proposed structure ..........................................111 vii Fig. 37 A photograph of the prototype ...................................................................112 Fig. 38 (a) Comparison of S11 on the effect of the meander loop with and without the presence of the open-circuit stub; (b) Comparison of S21 on the effect of the meander loop with and without the presence of the open-circuit stub ...............113 Fig. 39 Comparison between the simulated (MoM) and measured response of the proposed filter .....................................................................................................114 Fig. 40 The proposed miniaturized resonator with its physical dimension ............115 Fig. 41 The measured and simulated (MoM) responses of the structure in Fig. 40115 Fig. 42 Preliminary structure ..................................................................................118 Fig. 43 Simulated responses of the preliminary structure. .....................................118 Fig. 44 Equivalent circuit of the preliminary structure...........................................119 Fig. 45 (a) Even-mode and (b) odd-mode equivalent circuit..................................119 Fig. 46 Lumped element equivalent circuit for even- and odd-mode.....................120 Fig. 47 Various resonator implementations and their S21 responses. .....................127 Fig. 48 The dimension of the preliminary structure. ..............................................128 Fig. 49 (a) Comparison of measured and simulated (MoM) results and (b) A closedup view of the response. .....................................................................................129 Fig. 50 Comparison of S-parameters with ε r = 9.95 , ε r = 10.2 and ε r = 10.45 .....130 Fig. 51 A novel elliptic filter. .................................................................................132 Fig. 52 Photograph showing (i) the fabricated prototype (parallel feed): area = 49.7 mm2 (ii) the fabricated prototype (orthogonal feed): area = 60.84 mm2, (iii) the loop resonator [167]: area = 144 mm2. All filters are designed centered at 2.5 GHz. ....................................................................................................................132 Fig. 53 Comparison of the simulated (MoM) and measured responses for Fig. 52 (ii) ............................................................................................................................133 Fig. 54 Frequency response for the loop resonator for Fig. 52 (iii)........................133 Fig. 55 Substrate structures using WTT .................................................................137 Fig. 56 WTT process flow ......................................................................................138 Fig. 57 Layout of a CPW resonator ........................................................................139 viii Fig. 58 Proposed filter design in reference [181] ...................................................142 Fig. 59 CPW resonator with additional grounded meander lines ...........................143 Fig. 60 Equivalent circuit of the CPW resonator with additional grounded meander lines.....................................................................................................................144 Fig. 61 Equivalent circuit of the meander lines ......................................................145 Fig. 62 Novel bandpass filter design using WTT ...................................................149 Fig. 63 Proposed filter structure and simulated results...........................................151 Fig. 64 Photograph of the wafer .............................................................................152 Fig. 65 Photograph of the coplanar bandpass filter ................................................153 Fig. 66 Simulated (IE3D) and measured results .....................................................153 Fig. 67 Cross section view of the LTCC substrate .................................................156 Fig. 68 Preliminary structure I. All the other lines and space is 0.1 mm................158 Fig. 69 Simulated (IE3D) results for preliminary structure I for Fig. 68 ...............159 Fig. 70 Approximated equivalent circuit for preliminary structure for Fig. 68......159 Fig. 71 Preliminary structure II...............................................................................165 Fig. 72 Simulated (IE3D) results for preliminary structure II ................................166 Fig. 73 Proposed coplanar bandpass filter ..............................................................168 Fig. 74 Simulated (IE3D) results for the structure in Fig. 73 .................................168 Fig. 75 Dimensions of the proposed two-layered filter design...............................170 Fig. 76 Simulated (IE3D) results for the proposed two-layered design .................171 Fig. 77 Photographs for the proposed designs: (i) Single-layered CPW bandpass filter as shown in Fig. 73, (ii) Two-layered CPW bandpass filter with via as shown in..............................................................................................................173 Fig. 78 Simulated (IE3D) and measured results for Design (i) in Fig. 77..............173 Fig. 79 Comparison of S-parameters with ε r = 5.7 , ε r = 5.9 and ε r = 6.1 ............174 Fig. 80 Simulated (IE3D) and measured results for Design (ii) in Fig. 77.............175 ix Fig. 81 Comparison of S-parameters with ε r = 5.7 , ε r = 5.9 and ε r = 6.1 ............176 Fig. 82 Properties of the three fundamental elements ............................................182 x List of Tables Table 1 Equations for the transmission line model...................................................27 Table 2 Two sets of Maxwell equations for electric current-source free and magnetic current-source free ................................................................................31 Table 3 Two possible forms for both G A and G M in the ( u, v ) plane.....................35 Table 4 Comparison of the proposed approach and the Davidenko’s method [126] for leaky-wave poles extraction in TM mode. The parameters adopted are ε r1 = 15 , ε r 2 = 59 , ε r 3 = 37.5 , h1ko = 0.8 , h2 ko = 0.9 , h3 ko = 0.95 , and ε r1 ko < k ρ < ε r 3 ko . ...........................................................................................72 Table 5 Comparison of the proposed approach and the Davidenko’s method [126] for lossy improper poles extraction in TM mode. The parameters adopted are ε r1 = 35.5 + i0.1 , ε r 2 = 37.5 + i0.1 , ε r 3 = 59 + i0.1 , h1ko = 0.4 , h2 ko = 0.4 , and h3 ko = 0.5 ..............................................................................................................74 Table 6 Comparison of the proposed approach and the Davidenko’s method [126] lossy improper poles extraction in TM mode. The parameters adopted are ε r1 = 40 + i 25 , ε r 2 = 75 + i 25 , ε r 3 = 40 + i 25 , ε r 4 = 80 + i 25 , h1ko = 0.4 , h2 ko = 0.45 , h3 ko = 0.4 , and h4 ko = 0.45 . .............................................................75 Table 7 Material properties for LTCC used in this section ....................................157 xi List of Symbols εo Permittivity of free space ( 8.854 × 10−12 F/m) µo Permeability of free space ( 4π × 10 −7 H/m) η Free space wave impedance ( Ω ) ko Free space wave number (rad/m) λ Wavelength (m) E Electric field (V/m) H Magnetic field (A/m) A Magnetic vector potential (Wb/m) J Electric current density (A/m2) xii Summary Various numerical techniques have been developed to efficiently and accurately calculate the fields of a layered medium. Traditionally the electric and magnetic fields are derived separately. With the concept of quaternion algebra, a novel and compact quaternion formulation is derived for EM analysis. The resultant formulation is subsequently utilized in the Method of Moments (MoM), which is a powerful technique to analyze planar passive circuits. The MoM used herein requires the Green’s function in the spatial domain. The closed-form Green’s function can be obtained using the Discrete Complex Image Method (DCIM), which needs the accurate evaluation of the poles of the Green’s function in the spectral domain. The extraction of the poles of the Green’s function is one of the bottlenecks for the analysis of multilayered structures due to the complexity of the multilayered Green’s function. Although several methods have been introduced for pole extraction, they are not efficient and accurate enough. Thus, a fast, stable and efficient multilayered pole extraction technique for fast evaluation of DCIM is introduced in this thesis. Suitable initial guesses, good convergence and accuracy have been achieved through our method. The MoM algorithm is next used to design and analyze novel bandpass filters. Several miniaturization techniques for bandpass filter design have been explored for the first time. These include the Photonic Bandgap (PBG) structure, dual mode resonator without cross coupling effect, novel miniaturized open-loop resonator with wide frequency perturbation, novel coplanar filter design with Wafer Transfer Technology (WTT) and a modified miniaturized LTCC hairpin-combline resonator. xiii Chapter 1 Introduction 1.1 Literature Review The electromagnetic field computation in layered media plays a crucial role in certain applications, such as geophysical prospecting [1]–[3], remote sensing [4], wave propagation [5]-[6], and microstrip circuits and antennas [7]–[9]. Researchers have developed many computationally efficient and accurate numerical techniques for the field computation of layered media. To name a few, these include the method of moments (MoM) and its variants [10], the finite-element method (FEM) [11], and the finite-difference time-domain (FDTD) method [12]. In general, the FEM and the FDTD method solve differential equations, whereas MoM deals with the integral equations. MoM is a more powerful and efficient method for solving problems with simple planar geometry compared to FEM and FDTD. FEM [13] and FDTD [14] are more suitable for arbitrarily shaped, inhomogeneously filled and anisotropic scatterers, but are not particularly efficient at modelling the finite source region, especially when it is necessary to describe it in details. In here, the source cannot be easily described as dirac function. For solving planar layered microstrip and coplanar waveguide (CPW) structures, the MoM is regarded as one of the most popular techniques [15]-[16]. For the MoM formulation, an integral equation describing the electromagnetic problem can be formulated as the mixed potential integral equation (MPIE) or the electric field integral equation (EFIE). These integral equations require the evaluation of problemdependent Green’s functions, which can be assumed to have the forms of the vector 1 and scalar potentials based on MPIE formulation or the electric fields based on EFIE formulation. In general, the MPIE equation provides a less singular kernel compared with the EFIE equation for analyzing multilayered structure. The MoM analysis can be carried out either in the spectral domain [17]-[18] or in the spatial domain [19]–[22]. In the spectral domain formulation, the MoM matrix elements involve two-dimensional (2-D) integrals of complex, oscillatory, and slowconverging functions over an infinite domain [23]. Therefore, the numerical evaluation of these elements is quite time consuming, rendering the technique computationally inefficient. Although acceleration techniques and approximations can improve the computational efficiency of the spectral-domain MoM, they may impose some restrictions. The basis and testing functions are restricted to those that have analytical Fourier transforms, such as the rooftop [24] and piecewise sinusoidal basis functions [25]. On the other hand, the application of the spatial-domain MoM to the mixedpotential integral equation (MPIE) requires the evaluation of Green’s functions in the spatial domain [15]. The spatial-domain Green’s functions can be obtained from their spectral domain counterparts, which can be derived analytically for planar multilayered media, via the Hankel transformation, also called Sommerfeld integral [26]- [27]. Since the kernel of the transformation is the Bessel function of the first kind and the function to be transformed is the spectral-domain Green’s function, the integrand is often an oscillatory and slow-converging function. Therefore, the calculation of the spatial domain Green’s function, i.e., the numerical implementation of the Hankel transformation, is the main computational bottleneck of the spatialdomain MoM. The spatial-domain MoM has no restriction on the basis and testing 2 functions [28]. In this thesis, a spatial-domain MoM is used to analyze planar structures. In modeling microstrip structures using the MoM, much effort has been devoted to the computation of the Green’s functions because the computation of Sommerfeld integrals (SIs) is very time consuming. There are three methods that can be found in the literature for the evaluation of these SIs: (1) numerical integration method; (2) asymptotic method; (3) discrete complex image method. The numerical integration method in [29]-[30] is suitable only when the field points are very close to the source points. Generally, this method requires the largest computation time because the integrands are oscillatory. The asymptotic method [31] is the fastest, but it is also the least accurate, especially when the field points are close to the source points. They are also complicated and cannot be directly used for multilayered microstrip structures. To address these limitations, a method based on the Sommerfeld identity, called the discrete image method (DCIM) [33]–[48], has been developed. The DCIM approximates the spectral-domain Green’s functions in terms of complex exponentials and casts the integral representation into closed-form expressions via an integral identity, namely the Sommerfeld identity [49]. The DCIM divides the Green’s function into three main parts [34]. They are namely: (1) extraction of the quasi-static terms dominating in the near-field region, (2) contributions of the surface wave poles (SWP) dominating in the far-field region of the substrate surface, and their extraction, and (3) the remaining terms, which are related to leaky waves and are very important in the intermediate field region. The quasi-static contribution is inverse transformed analytically through the Sommerfeld identity. The 3 surface wave contribution comes from the poles in the spectral Green’s functions. The surface wave poles are extracted, and subsequently the contribution of surface waves is evaluated through the residue calculus. The remaining portion in the spectral representation is expanded into a series of exponentials using either the Prony [34] or generalized pencil-of-functions method (GPOF) methods [38], [50]. Conventionally, both the vector magnetic potential A and the vector electric potential F are evaluated separately through the mixed potential formulation. To achieve more compactness in the derivation, a novel quaternion mixed potential formulation is presented in the thesis. In reference [51], concepts of quaternion Fourier transform (QFT), quaternion convolution (QCV), and quaternion correlation, which are based on quaternion algebra, have been found useful in the solution of Maxwell’s equations. It has been shown that the derivation of regular TEM waves in homogeneous medium can be given in a very compact way, yielding all possible polarization cases simultaneously. In this thesis, a new way of representing the dyadic Green’s function in a compact form has been introduced based on the concepts of quaternion algebra in [51]. Several issues need to be considered when applying DCIM method for multilayered structures. One of them is the difficulty of extracting the surface wave poles for the multilayered structures. In applying DCIM to general multilayered media, there is a lack of a reliable procedure for the extraction of surface wave components [41]. In [52] and [53], a Newton–Raphson root-searching procedure is applied to find the TE- and TM-mode surface wave poles of a grounded dielectric slab. However, due to the existence of branch cuts, the initial values close to the roots are needed. In this 4 connection, reliable and suitable initial values need to be investigated for the grounded dielectric slab, but such initial values for general multilayered media are not available. In [54] a contour integral method is used for seeking the zeros of an analytical function, which are the surface wave poles of three-layered media grounded both at the top and bottom. In [55] the same method is applied to a three-layered medium which is grounded only at the bottom by eliminating the branch points associated with the semiinfinite top layer through a variable transform. However, this method requires evaluation of the first-order derivative of the function, the zeros of which are to be sought, which is difficult to derive for general multilayered media. When the number of zeros in the contour is greater than four, the contour should be replaced by several smaller contours so that each contains no more than four zeros. This process is generally rather tedious. In [56], a two-stage root-searching procedure is proposed for seeking the roots of these denominators. The residuals associated with the surface wave poles are calculated through contour integration. To find all the roots automatically, the interval between the minimum and maximum wave number of all media is divided uniformly into a number of sections. For each section, the golden search procedure is first applied to find the minimum of F ( k ρ ) [56], which is then used as the starting point for the Newton– Raphson procedure. In general, this method is not very efficient since it cannot find the total number of the roots automatically and it has difficulties to seek the initial values for the surface wave poles. Also, this method suffers from local minimum termination. In [57], an efficient and robust iterative algorithm is introduced based on contraction mapping, which can locate all the proper and improper solutions of the characteristics equations of the grounded dielectric slab. However, this method will become extremely complex when it is extended to multilayered structures and the functional form, which is crucial for 5 the accurate evaluation of the overall spatial Green’s function, is difficult to find. As a result, there is a need to devise an efficient, accurate and fast poles extraction algorithm for multilayered structures. Our earlier success in deriving fast but efficient pole extraction procedure for a single-layered structure [58] provided a good starting point for deriving a generalized approach for multilayered microstrip pole extraction. In this thesis, a general and fast algorithm is introduced for surface wave, leaky wave and lossy pole extraction for multilayered structures. Another difficulty associated with the DCIM method is the approximation method for the complex image terms. The original derivation of the closed-form Green’s functions, as proposed in [34], employed the original Prony method for complex image terms. It was limited in use to thick and single layer structures, which was due to inadequacy of the original Prony method. This problem was eliminated by employing the least squares Prony method [35]. Then the approximation was further improved by using the generalized pencil-of-functions method (GPOF) [38], which is less noise sensitive and more robust compared to the Prony methods. However, the algorithm for the exponential approximation is still computationally expensive, because Prony’s methods and the GPOF method require uniform sampling of the function to be approximated along the range of approximation. This, in turn, makes it necessary to take a large number of samples for functions with local oscillations and fast variations, like spectral-domain Green’s functions in general, rendering the algorithm computationally expensive and not robust. Recently, a two-level approach that requires piecewise uniform sampling has been introduced to eliminate this problem, and is demonstrated to be much more efficient and robust [40]. Hence, the spatial- 6 domain closed-form Green’s functions can be employed efficiently in the solution of MPIE for planar, multilayered geometries. With the appropriate spatial-domain Green’s function and boundary conditions, the surface current of a specified problem can be calculated. Based on the appropriate integral equation, we need to expand the unknown function in terms of some known basis functions with unknown coefficients. The preferred choice of basis functions is the one developed by Rao et al. [59], which is now commonly known as the Rao– Wilton–Glisson (RWG) basis function. This basis function provides a great capability to model arbitrarily shaped microstrip structures. The boundary conditions are then implemented in an integral sense through the testing procedure. The integral equation is next transformed into a matrix equation, whose entries are double integrals for the general 2-D geometries. However, for planar 2-D geometries, the MoM matrix entries can be reduced to single integrals by transforming the convolution integrals onto the basis and testing functions and by evaluating the resulting integrals analytically [60]. In either domain, for moderate-size geometries the computational efficiency of the MoM lies in the evaluation of the MoM matrix entries. In the spectral-domain application of the MoM, since the Green’s functions are known in closed form, the matrix entries become single integrals over an infinite domain. For a geometry requiring a large number of unknowns, the matrix solution time dominates the overall performance of the technique, and therefore, the efficiency of the method is defined by the efficiency of the linear system solver [61]. The MoM is a powerful technique to analyze planar passive circuits, such as antennas and filters. With the widespread evolution of wireless communications, 7 miniaturization for personal communications equipment has become one of the most fundamental requirements. Since many of the passive circuits are made of filter structures, there is a need to investigate miniaturization of the filters. Filters can be categorized into bandpass filters, bandstop filters, lowpass filter and highpass filters. Among them, the bandpass filter plays a pivotal role in wireless communications systems such as satellite and mobile communications systems. In general, bandpass filters can be designed based on single- or multiple-resonator structures. Microstrip resonators for filter designs may be classified as lumped-element or quasilumped-element resonators, distributed line resonators or patch resonators. Conventional bandpass filters include stepped-impedance filters [62]-[66], open-stub filters [67]-[68], semi-lumped element filters [69], end- [70] and parallel-coupled [71][72] half-wavelength resonator filters, hairpin-line filters [73]-[75], interdigital [76][79], combline filters [80]-[81], pseudo-combline filters, and stub-line filters [82]. Recently some research has been conducted on designing compact, low-loss bandpass filters with good performance. In view of this, we will focus on miniaturization of microstrip and coplanar bandpass filters in this thesis. To minimize the size of the bandpass filter, several structures have been demonstrated including the steppedimpedance resonators (SIR) [62]-[66], the dual mode resonators [83]- [85], meandered open loop resonators [86]-[88], and the photonic bandgap structures [89]. Conventionally, filters using uniform impedance resonators (UIRs) [91] were first used in microwave communication systems. They suffer from poor harmonic suppression. To alleviate the problem, the stepped impedance resonator (SIR) was developed to solve the problem. Stepped impedance resonators (SIR) are composed of 8 transmission lines with different characteristic impedances. They provide an effective way to minimize circuit space and push spurious resonant frequencies away from the passband [62]. Dual-mode resonators are one of the most effective means for miniaturizing a bandpass filter (BPF). Dual-mode [90] means that two degenerate resonant modes are excited by asymmetric feed lines and by the addition of notches or stubs on the ring or patch resonators. Therefore, the perturbed resonator may be used as a doubly-tuned resonant circuit. Consequently, the number of resonators for a n -degree filter can be reduced by half, and the overall size of the filter can be compacted [91]-[92]. A dualmode square patch resonator has been used to build Chebyshev and Elliptic filters [93]. Conventional square patches suffer from large size. As a result of this, several novel dual-mode structures have been introduced in this thesis for filter miniaturization. Photonic bandgap (PBG) structures are specific periodic structures artificially created in materials such as metals or substrates to influence or even change the electromagnetic properties of materials. These PBG structures can be integrated in many microstrip filters so as to increase the maximum attenuation at the stopband, suppress the spurious transmission and reduce the overall circuit area. For filter implementation, the available technologies include the microstrip line and coplanar waveguide (CPW). Microstrip bandpass filters are finding wide range of applications in many RF/microwave circuits and systems owing to their low-cost, light weight and ease of integration with other components on printed circuit boards (PCBs). However, ground connections in microstrip require either a metallised via hole 9 through the substrate or else a wrap-around metallization at the edge of the substrate. Both of these techniques increase the meanufacturing complexity and hence the cost. Unlike microstrip or stripline, coplanar waveguide has the ready access to the ground plane on the topside of the substrate and permits easy parallel and series insertion of circuit components. Its circuit parameters are also less sensitive to the substrate thickness, but sensitive to other parameters. For filter miniaturization, the available techniques include using slow wave structures [94], meander line structures [95]- [96], or slot-loaded CPW structures [97]- [98]. Besides the above-mentioned novel structures for filter design, there are also many advanced materials and technologies currently available for filter fabrication. They include the high-temperature superconductors (HTS) [99]-[102], ferroelectrics, micromachining or microelectromechanical systems (MEMS) [103]-[105], hybrid or monolithic microwave integrated circuits (MMIC) [106]- [107], active filters, photonic bandgap (PBG) materials/structures, low-temperature cofired ceramics (LTCC) [108][109] and wafer transfer technology (WTT). These technologies have stimulated the rapid development of new bandpass filters. In this thesis, we will introduce several novel structures of bandpass filters for microstrip structure and CPW using conventional and advanced fabrication technologies. 1.2 Scope of Work In Chapter 1, an introduction to the history and current research concerning EM modeling for multilayered structures is presented. Different numerical techniques have been investigated to efficiently and accurately calculate the fields of layered medium. It is noted that the Method of Moments is a powerful technique to analyze planar passive circuits. In general, MoM requires the Green’s function in spatial and spectral 10 domains. In this research, the development of closed-form spatial-domain Green’s function are investigated using EM simulation. Various techniques of filter miniaturization are introduced. Chapter 2 introduces a novel and compact quaternion analysis for EM analysis. Normally, the electric and magnetic fields are derived separately. With the concept of quaternion algebra, it is possible to give a compact formulation for DCIM analysis. A multilayered pole extraction technique for fast evaluation of DCIM is introduced in Chapter 3. Extracting the poles of the Green’s function is one of the difficulties for the analysis of multilayered structures due to the complexity of the multilayered Green’s function. Although several authors provide some techniques for pole extraction, these methods are not very efficient and accurate. Thus, there is a need to investigate a new general and fast algorithm for pole extraction in multilayered structures. MoM can be used to design and analyze different planar circuits. Various novel bandpass filter designs are investigated in Chapter 4. As the design of compact, lowloss and good performance bandpass filter has attracted a lot of attention recently, all the filters introduced in this chapter focus on miniaturization, which provide low cost and good performance. Several advanced design and fabrication techniques are applied for further miniaturization of bandpass filters compared to the conventional PCB technology. 1.3 Achievements and Contributions As a result of the research work, the following contributions have been achieved: 11 a) A novel and compact quaternion analysis has been derived. Without separately deriving the electric and magnetic field, the quaternion MPIE provides a faster formulation method for the DCIM analysis. b) A multilayer pole extraction technique for fast evaluation of DCIM has been developed during the project work. Suitable initial guesses and good functional expressions with fast convergence have first been derived. Compared to the conventional methods, good convergence and accuracy are achieved. The method is noted to be fast and stable, and does not suffer from local minimum termination. c) Several miniaturization techniques for bandpass filter design have been explored for the first time. These includes the PBG structure, dual mode resonator without cross coupling effect, novel miniaturized open-loop resonator with wide frequency perturbation, novel coplanar filter design with Wafer Transfer Technology (WTT) and miniaturized two-layered LTCC coplanar filter using modified hairpin-combline resonator. As a result of these investigations, the following publications have been produced: 1.3.1 List of Journal Publications z Y. Wang and B. L. Ooi, “Useful multilayered microstrip pole extraction technique”, IEE Proceedings Microwaves, Antennas and Propagation, Vol. 152, Issue 3, pp. 149 – 154, June 2005. z Y. Wang, B. L. Ooi, M. S. Leong, “Efficient and fast approach for surface wave poles extraction in two-layered microstrip geometry”, Microwave and Optical Technology Letters, pp. 253-258, May 2004. z H. Y. Fong, Y. Wang, B. L. Ooi, “A novel microstrip loop filter”, Microwave and 12 Optical Technology Letters, Volume 47, Issue 3, pp. 279-281, Nov 2005. z B. L. Ooi, D. X. Xu, Y. Wang, B. Chen and M. S. Leong , “A novel LTCC power combiner”, Microwave and Optical Technology Letters, Vol. 42, Issue 3, pp. 255257, Aug 2004. z B. L. Ooi, Y. Wang, and H. Y. Fong, “Experimental Study of New Planar Filter without Using Cross-Coupled Effect”, IEE Proceedings Microwaves, Antennas and Propagation, Vol. 153, No 3, pp. 226 – 230, June 2006. z Ban-Leong Ooi and Ying Wang, “Novel Miniaturized Open-Square-Loop Resonator with Inner Split Rings Loading”, IEEE Trans. on Microwave Theory and Techniques, Vol. 54. No. 7, pp. 3098-3103, July 2006. z Ying Wang, Ban Leong Ooi, Li Hui Guo and Mook Seng Leong, “Implementation of Novel Millimeter-Wave Filter Using Wafer Transfer Technology”, Accepted by IEEE Electronics Letters. 1.3.2 List of Conference Publications z Ying Wang, B. L. Ooi and M. S. Leong, “Fast and efficient approach N-layered microstrip poles extraction”, International Union of Radio Science National Radio Science Meeting, (URSI) Colorado, USA, Jan 5-8, 2004. z Ying Wang, B. L. Ooi and M. S. Leong, “Novel multi-layered poles’ extraction technique”, Progress in Electromagnetics Research Symposium, Pisa, Italy, 2831 March 2004. z Ying Wang, B. L. Ooi and M. S. Leong, “Comparison of various methods for poles' extraction in microstrip problem”, International Symposium on Antennas and Propagation, Sendai, Japan, Aug 17-21, 2004. (Invited) 13 z Y. Wang, B. L. Ooi, Y. J. Fan, M. S. Leong, Y. Q. Zhang and L. H. Guo, “Implementation of a novel millimeter-wave filter using wafer transfer technology”, submitted to 2006 IEEE Radio Frequency Integrated Circuits Symposium (RFIC2006), San Francisco, California on June 11-13, 2006. z Y. Wang, B. L. Ooi, Albert Lu, K. M. Chua, and M. S. Leong, “A modified miniaturized LTCC hairpin-combline resonator”, submitted to 2006 IEEE Radio Frequency Integrated Circuits Symposium (RFIC-2006), San Francisco, California on June 11-13, 2006. 14 Chapter 2 Bi-complex Algebra and Quaternion Electromagnetics 2.1 Introduction The recently developed concepts of quaternion Fourier transform (QFT), quaternion convolution (QCV), and quaternion correlation, which are based on quaternion algebra, have been found to be useful for color image processing [110]-[114]. This concept has also been found useful in the solution of Maxwell’s equations [51]. It has been shown that the derivation of regular TEM waves in homogeneous media can be given in a very compact way, yielding all possible polarization cases simultaneously using quaternion algebra. From a physical point of view, the implication is that the electric and magnetic fields are not really separate entities, but they merely constitute components of a composite field with bicomplex mathematical nature, satisfying only one (instead of two) Maxwell-like equation. In all these cases, the number of the unknown quantities has been reduced by half, compared to conventional techniques [51]. EFIE and MFIE are traditionally derived separately. In this chapter, based on the concept in [51], a compact and elegant way has been introduced to derive both the EFIE and MIFE simultaneously. A dyadic Green’s function in spectral domain based on the quaternion Fourier transform [115] has been derived to theoretically verify the validity of this method. The adopted approach is an extension of the work found in [115] and differs greatly from [115] through the dyadic analysis. The efficiency of this analysis is that the first order bicomplex differential equation can model wave 15 propagation in two directions, as compared to the conventional second order complex equations [51]. Following the approach in [115] and [51], this chapter will introduce a novel and compact quaternion analysis in spectral domain has been derived. Without separately deriving the electric and magnetic field, the quaternion MPIE provides a faster formulation method for the DCIM analysis. 2.2 Network Quaternion Integral Equation Formulation Based on the detailed derivation shown in Appendix, with reference of [51] and applying the concept of quaternion Fourier transform in [115], we extend the concept of quaternion analysis into transmission line models in spectral domain for the first time. In the next two sections, the derivation of the Green’s function in spectral would be used for poles extraction and MoM algorithm in the subsequent chapters. Consider a uniaxially anisotropic medium, where the general complex-valued permeability and permittivity dyadics are given as ( ) µ = xx + y y µt + z zµ z = I t µt + z zµ z and ε = I t ε t + z zε z . Let E = ηo 2 ( F + F ) and H = + ( i F+ − F 2 ηo (2.1) ). From Maxwell’s equations: ∇ × E = − jωµo µ • H − M , 16 ⎡ d ( Fz+ + Fz ) d ( Fy+ + Fy ) ⎤ ⎢ ⎥ − dy dz ⎢ ⎥ ⎢ ⎥ + + d ( Fx + Fx ) d ( Fz + Fz ) ⎥ 2M ⎢ , ⇒⎢ − = ijko µ • F + − F − ⎥ dz dx ηO ⎢ ⎥ + + ⎢ d ( Fy + Fy ) d ( Fx + Fx ) ⎥ − ⎢ ⎥ dx dy ⎢⎣ ⎥⎦ ( ) (2.2) ∇ × H = jωε oε • E + J , and ( ) ⎡ d Fz+ − Fz d ( Fy+ − Fy ) ⎤ ⎢ ⎥ − dy dz ⎢ ⎥ ⎢ ⎥ + + ⎢ d ( Fx − Fx ) d ( Fz − Fz ) ⎥ + ⇒i⎢ − ⎥ = jkoε • F + F + 2 ηo J , dz dx ⎢ ⎥ + + ⎢ d ( Fy − Fy ) d ( Fx − Fx ) ⎥ − ⎢ ⎥ dx dy ⎢⎣ ⎥⎦ ( ) (2.3) To easily remove the exponential term in the Fourier transform integral, we apply the right-side Fourier transform pair, namely, ( ) ∫ ∫ f (r )e ( ~ F kρ = ∞ ∞ j kx x+k y y ) dxdy , (2.4) − ∞− ∞ f (r ) = ∞ ∞ 1 4π 2 ∫ ∫ F (k ~ x , k y )e ( − j kx x+k y y ) dk x dk y , (2.5) − ∞− ∞ and therefore, d 1 if ( r ) ) = ( dx 4π 2 ∞ ∞ ∂ 1 − j(kx x+ k y y ) ⎤ dk dk = − 2 ∫−∞ −∞∫ ∂x ⎡⎢⎣iF k ρ e ⎥⎦ x y 4π ( ) ∞ ∞ ∫ ∫ iF ( k ρ ) jk e x −∞ −∞ ( − j kx x+ k y y ) dk x dk y . (2.6) Thus, from eqn. (2.2), we have 17 ( ) ⎡ d Fy+ + Fy ⎤ + ⎢ − Fz + Fz jk y − ⎥ dz ⎢ ⎥ ⎢ d F+ + F ⎥ 2M x x + ⎢ ⎥ ⇒ + Fz + Fz jk x = ijko µ • F + − F − , ⎢ ⎥ dz ηO ⎢ ⎥ + + ⎢ − Fy + Fy jk x + Fx + Fx jk y ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ( ) ( ) ( ( ) ) ( ) ( ) ( ) ⎡ d Fy+ + Fy ⎤ + ⎢ − j Fz + Fz k y − ⎥ dz ⎢ ⎥ ⎢ d F+ + F ⎥ 2M x x + ⎢ ⇒ + j Fz + Fz k x ⎥ = − jko µ • i F + − F − , ⎢ ⎥ dz ηO ⎢ ⎥ + + ⎢ − j Fy + Fy k x + j Fx + Fx k y ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ( ( ) ) ( ) (2.7) ( ( ) ( ) (2.8) ) and from eqn. (2.3) ( ) ⎡ d Fy+ − Fy ⎤ + ⎢ − Fz − Fz jk y − ⎥ dz ⎢ ⎥ ⎢ d F+ − F ⎥ x x + ⎢ ⇒i + Fz − Fz jk x ⎥ = jkoε • F + + F + 2 ηo J ⎢ ⎥ dz ⎢ ⎥ + + ⎢ − Fy − Fy jk x + Fx − Fx jk y ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ , ( ) ( ) ( ( ) ( ) ( ) ) ( ) ⎡ d Fy+ − Fy ⎤ + ⎢ −ijk y Fz+ − Fz − i ⎥ dz ⎢ ⎥ ⎢ d F+ − F ⎥ + x x + ⎢ ⎥ = jk ε • F + + F + 2 η J , ⇒ i + ijk x Fz − Fz o o ⎢ ⎥ dz ⎢ ⎥ + + ⎢ −ij Fy+ − Fy k x + ij Fx+ − Fx k y ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ ( ) ( ) ( ) ( ) ( ) ) ( ( ) ( ( ) ) ⎡ d Fy+ − Fy ⎤ ⎢ − jk y i Fz+ − Fz − ⎥ dz ⎢ ⎥ ⎢ d F+ − F ⎥ x x + ⎢ ⇒ i + jk x i Fz − Fz ⎥ = jkoε • F + + F + 2 ηo J . ⎢ ⎥ dz ⎢ ⎥ + + ⎢ − ji Fy − Fy k x + ji Fx − Fx k y ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ ( ) ( ( ) ) (2.9) ( ) (2.10) (2.11) 18 Extracting the last eqn. from eqns. (2.8) and (2.11), ( ( ⎡ ko µ z i Fz+ − Fz ⎢ ⇒ ⎢k ε j F + + F z z ⎣ o z ⎡ 2 jM ) ⎥⎤ ⎢ ( F + F ) k − ( F + F ) k + η =⎢ ⎥ )⎦ ⎢ ji ( F − F ) k − ji ( F − F ) k − 2 η + y y + x x z x y o + x ⎣⎢ x + y y y x ⎤ ⎥ ⎥, ⎥ o Jz ⎦ ⎥ (2.12) Putting the last eqn of (2.12) into eqn. (2.8), ( ) ( ) ( ) ⎤⎥ ⎡ − ⎡ jik F + − F − jik F + − F − 2 η J ⎤ k d Fy+ + Fy x x y y o z⎦ y ⎢ ⎣ y x − ⎢ ko ε z dz ⇒⎢ + + ⎡ jik y Fx − Fx − jik x Fy − Fy − 2 ηo J z ⎤ jk x ⎢ d Fx+ + Fx ⎦ ⎢ +⎣ dz k oε z ⎢⎣ ( ( ) ) ( ) ⎥ ⎡ Fx+ − Fx ⎤ 2 ⎡M x ⎤ ⎥ = − jko µt i ⎢ + ⎥− ⎢ ⎥ ηo ⎢⎣ M y ⎥⎦ ⎢⎣ Fy − Fy ⎥⎦ ⎥ ⎥ ⎥⎦ (2.13) ( ) ( ) ( ) ( ) ( ) ( ) + + ⎡⎡ ⎤ ⎤ ⎡ d Fy+ + Fy ⎤ ⎢ ⎣ − jk y i Fx − Fx + jk x i Fy − Fy + 2 ηo J z ⎦ k y ⎥ ⎢ ⎥ ⎢ ⎥ koε z dz ⎢ ⎥=⎢ ⎥ ⇒ + ⎢d F+ + F ⎥ ⎢⎡ + + ⎤k ⎥ 2 η − − + − + jk i F F jk i F F J x x ⎥ y x x x y y o z⎥ x ⎢ (2.14) ⎦ ⎥ ⎢ ⎣⎢ ⎢⎣ ⎥ dz ⎦ ⎢ ⎥ koε z ⎣ ⎦ + ⎡ − Fx − Fx ⎤ 2 ⎡−M x ⎤ ⎥− − jko µt i ⎢ ⎢ ⎥ ηo ⎢⎣ M y ⎥⎦ ⎢⎣ Fy+ − Fy ⎥⎦ ( ( ) ( ) ) ⎡ d Fy+ + Fy ⎤ + + 2 ⎢ ⎥ ⎡ ⎤ 1 ⎢ − jk y i Fx − Fx + jk x k y i Fy − Fy + 2k y ηo J z ⎥ dz ⎢ ⎥ ⇒ = ⎢ ⎥ koε z ⎢ − jk k i F + − F + jk 2i F + − F + 2k η J ⎥ + y x x x x y y x o z⎦ ⎢ d Fx + Fx ⎥ ⎣ , ⎢⎣ ⎥ dz ⎦ ( ) ( ( ( ) ) ) ( ⎡ − Fx+ − Fx ⎢ − jko µt i ⎢⎣ Fy+ − Fy ( ) ( ) ⎡d F+ + F ⎤ x x ⎢ ⎥ + ⎢ ⎥ − jkt2 ⎡ i Fy − Fy dz ⎢ ⇒⎢ ⎥= ⎢ d Fy+ + Fy ⎥ koε t ⎢⎣ −i Fx+ − Fx ⎢ ⎥ ⎥⎦ dz ⎣⎢ jυ e + ko ε t ( ( ⎡ k x2 ⎢ ⎢⎣ k y k x )⎥⎤ − ⎥⎦ 2 ⎡−M x ⎤ ⎢ ⎥ ηo ⎣⎢ M y ⎦⎥ ) ⎤⎥ )⎥⎦ ( ( (2.15) (2.16) ) ⎤⎥ 2 η J + )⎥⎦ k ε + k y k x ⎤ ⎡ i Fy − Fy ⎢ ⎥ k y2 ⎥⎦ ⎢ −i F + − F x x ⎣ o o z z ⎡ kx ⎤ 2 ⎡ My ⎤ ⎢ ⎥ ⎢k ⎥ − ηo ⎣ − M x ⎦ ⎣ y⎦ 19 ⇒ ( d Ft + + Ft ) = − j (k ko ε t dz 2 t ) ( 2 ηo J z k ρ 2 − υ e k ρ k ρ • ⎡i F + − F × z ⎤ + − M t × zˆ ,(2.17) ⎥⎦ ⎣⎢ koε z ηo ) ~ ⎡ F~x ⎤ e ε t where Ft = ⎢ ~ ⎥ , υ = , k t = k o µ t ε t , k ρ = k y xˆ + k y yˆ , and k o = ω µ o ε o . εz ⎢⎣ Fy ⎥⎦ Similarly, putting the first eqn. of (2.12) into eqn. (2.11), ⎡ ⎡ 2 jM z ⎤ ⎢ − jk y ⎢ − k y Fx+ + Fx + k x Fy+ + Fy + ⎥ ηo ⎥⎦ d Fy+ − Fy ⎢ ⎢⎣ −i ⎢ ko µ z dz ⎢ ⇒ ⎢ ⎡ 2 jM z ⎤ ⎢ jk x ⎢ − k y Fx+ + Fx + k x Fy+ + Fy + ⎥ + ⎢ d Fx − Fx ηo ⎦⎥ ⎢ ⎣ + ⎢i dz ko µ z ⎣⎢ ( ( ) ) ( ) ( ) ( ( ) ) ⎤ ⎥ ⎥ ⎥ ⎡ + ⎤ ⎥ = jk ε ⎢ Fx + Fx ⎥ + 2 η ⎡ J x ⎤ ⎥ o t o ⎢ + ⎥ ⎢⎣ Fy + Fy ⎥⎦ ⎣Jz ⎦ ⎥ ⎥ ⎥ ⎦⎥ (2.18) ⎡⎡ 2 + 2k y M z ⎤ ⎤ ⎢ ⎢ jk y Fx + Fx − jk x k y Fy+ + Fy + ⎥⎥ ηo ⎦⎥ ⎥ ⎡ d Fy+ − Fy ⎤ ⎢ ⎣⎢ ⎢i ⎥ ⎢ ⎥ ko µ z dz ⎢ ⎥ ⎢ ⎥ ⇒ = ⎢ d F+ − F ⎥ ⎢⎡ ⎤⎥ x x ⎥ ⎢i ⎢ ⎢ jk y k x Fx+ + Fx − jk x2 Fy+ + Fy + 2k x M z ⎥ ⎥ ⎢⎣ ⎥⎦ ⎢ ⎢ ηo ⎥⎦ ⎥ dz ⎣ ⎢ ⎥ ko µ z ⎢⎣ ⎥⎦ ( ) ( ) ( ) ( ) ( ) ( ) ( ⎡ − Fx+ + Fx + jkoε t ⎢ ⎢⎣ Fy+ + Fy ⇒i ( d Ft + − Ft dz ) = jk ⎡ Fy+ + Fy ⎢ ko µt ⎢ − Fx+ + Fx ⎣ 2 t ( ) ⎤ jυ µ ⎥− ⎥⎦ ko µt ⎡ k x2 ⎢ ⎢⎣ k x k y )⎤⎥ + 2 ⎥⎦ + k x k y ⎤ ⎡ Fy + Fy ⎢ ⎥ k y2 ⎥⎦ ⎢ − Fx+ + Fx ⎣ ( ) (2.19) ⎡− J x ⎤ ⎥ ⎣⎢ J y ⎦⎥ ηo ⎢ ⎤ ⎡J ⎤ 2M z ⎥ + 2 ηo ⎢ y ⎥ + η o ko µ z ⎥⎦ ⎣− J x ⎦ ⎡ kx ⎤ ⎢k ⎥ ⎣ y⎦ (2.20) ⇒i ( d Ft + − Ft dz ) = − j (k k o µt 2 t ( ) 2M z kρ − υ h kρ kρ • ⎡ z × F + + F ⎤ + − 2 ηo zˆ × J t , (2.21) ⎢⎣ ⎥⎦ η o ko µ z ) 20 ηo where υ = µt µ z . Using E = h 2 (F + F ), H = + ( i F+ − F 2 ηo ) , eqns (2.17) and (2.21) respectively becomes ⎧ dE Jk 1 kt2 − v e k ρ k ρ • ⎡⎢ H t × z ⎤⎥ + z ρ − M × z , ⎪ t = ⎣ ⎦ ωε oε z jωε oε t ⎪ dz ⇒⎨ ⎪ dH t 1 h 2 ⎡ z × E ⎤ + M z kρ − z × J . k v k k = − • t t⎥ ρ ρ ⎪ dz ⎢⎣ ⎦ ωµo µ z jωµo µt ⎩ ( ) ( (2.22) ) Instead of using eqn. (2.22), in general, we can combine eqns. (2.17) and (2.22) F ( through Q = t + + Ft 2 ) + i i(F t ( + − Ft ) = F to yield t 2 ) ( ) iM z k ρ dFt M ×z ij −j kt2 − υ e k ρ k ρ • ⎡i F + − F × z ⎤ − kt2 − υ h k ρ k ρ • ⎡ zˆ × F + + F ⎤ − t = + ⎥⎦ 2ko µt ⎣⎢ ⎣⎢ ⎦⎥ dz 2koε t ηo ηo ko µ z ( −i η o z × J t + ) ( ) ηo J z k ρ . koε z (2.23) To simplify the computation, we perform a rotation of the axis through ⎡u ⎤ 1 ⎡ k x ⎢ ⎥= ⎢ ⎢⎣ v ⎥⎦ k ρ ⎣ −k y k y ⎤ ⎡ xˆ ⎤ , k x ⎦⎥ ⎣⎢ yˆ ⎦⎥ (2.24) where k ρ = k x2 + k y2 . The inverse of this rotation is given as ⎡ x ⎤ 1 ⎡ kx ⎢ ⎥= ⎢ ⎢⎣ y ⎥⎦ k ρ ⎣ k y −k y ⎤ ⎡uˆ ⎤ . k x ⎦⎥ ⎣⎢ vˆ ⎦⎥ (2.25) 21 (k ) y zˆ × k ρ ky kρ yˆ vˆ uˆ ξ xˆ kx ( kx ) (a) yˆ Ey vˆ uˆ E Eu H Ev ξ zˆ Ex xˆ (b) Fig. 1 Rotation of the axis Therefore any electric and magnetic current in the ( x, y ) plane will become ⎡ M u ⎤ 1 ⎡ kx ⎢ ⎥= ⎢ ⎣ M v ⎦ kρ ⎣−k y ⎡ Ju ⎤ 1 ⎡ kx ⎢ ⎥= ⎢ ⎣ J v ⎦ kρ ⎣ −k y k y ⎤ ⎡ M x ⎤ 1 ⎡ kx M x + k y M y ⎤ ⎢ ⎥= ⎢ ⎥, k x ⎥⎦ ⎢⎣ M y ⎥⎦ k ρ ⎢⎣ − k y M x + k x M y ⎥⎦ k y ⎤ ⎡ J x ⎤ 1 ⎡ kx J x + k y J y ⎤ ⎢ ⎥= ⎢ ⎥. k x ⎥⎦ ⎣⎢ J y ⎦⎥ k ρ ⎢⎣ − k y J x + k x J y ⎦⎥ (2.26) (2.27) To solve Eqns. (2.17) and (2.21), let Ft in the ( u, v ) plane be defined as ⎛ Ve ⎞ ⎛ Vh ⎞ Ft = ⎜ − i ηo I h ⎟ u + ⎜ + i ηo I e ⎟ v = Q1 u + Q2 v , ⎜ η ⎟ ⎜ η ⎟ ⎝ o ⎠ ⎝ o ⎠ (2.28) we have 22 dFt ⎛ 1 dV e dQ dQ dI h ⎞ ⎛ 1 dV h dI e ⎞ η =⎜ − i ηo + + u i ⎟ ⎜ ⎟v = 1 u + 2 v o ⎜ ⎟ ⎜ ⎟ dz ⎝ ηo dz dz ⎠ ⎝ ηo dz dz ⎠ dz dz (2.29) In the ( x, y ) plane, we have ⎡ ⎢ −k y ⎤ ⎢ k x ⎥⎦ ⎢ ⎢ ⎢⎣ ⎡ F ⎤ 1 ⎡ kx Ft = ⎢ x ⎥ = ⎢ ⎣⎢ Fy ⎦⎥ k ρ ⎣ k y ⎤ − i ηo I h ⎥ ηo ⎥= 1 h ⎥ k ρ ηo V + i ηo I e ⎥ ηo ⎥⎦ Ve ⎡kx ⎢k ⎣ y − k y ⎤ ⎡V e ⎤ i ηo + k x ⎥⎦ ⎢⎣V h ⎥⎦ kρ ⎡kx ⎢k ⎣ y −k y ⎤ ⎡− I h ⎤ , k x ⎥⎦ ⎢⎣ I e ⎥⎦ (2.30) ( ( ) ⎥⎤ 2 η = )⎥⎦ k ⎡ i Fy+ − Fy i Ft − Ft × zˆ = ⎢ ⎢ −i F + − F x x ⎣ ( ) + ( ⎡ − F + Fy zˆ × Ft + + Ft = ⎢ + ⎣⎢ Fx + Fx ( ) + y o ρ )⎥⎤ = ⎦⎥ ⎡ ky ⎢ −k ⎣ x k x ⎤ ⎡ − I h ⎤ 2 ηo = k y ⎥⎦ ⎢⎣ I e ⎥⎦ kρ ⎡ kx ⎢k ⎣ y ky ⎤ ⎡ I e ⎤ , − k x ⎥⎦ ⎢⎣ − I h ⎥⎦ (2.31) ⎡ −k y ⎢k ηo ⎣ x −k x ⎤ ⎡V ⎤ , −k y ⎥⎦ ⎢⎣V h ⎥⎦ 2 kρ e (2.32) and from Eqn. (2.12), namely, ( ( ⎡ ko µ z i Fz+ − Fz ⎢ ⎢k ε j F + + F z z ⎣ o z ( ) ⎡ 2 jM ) ⎤⎥ ⎢ ( F + F ) k − ( F + F ) k + η =⎢ ⎥ )⎦ ⎢ ji ( F − F ) k − ji ( F − F ) k − 2 η + y y + x x z x y o + x ⎢⎣ ( x + y y ) ( y x ) ⎤ ⎥ ⎥, ⎥ o Jz ⎥ ⎦ ( ) (2.33) + + i Fy+ + Fy k x − i Fx+ + Fx k y 2ijM z 1 ⎡ i Fx − Fx k y − i Fy − Fy k x 2 j ηo Jz + ⇒ Fz = ⎢ + + k oε z koε z ko µ z 2⎢ ko µ z η o ⎣ ⎤ ⎥ ⎥ ⎦ (2.34) 1 ⇒ Fz = kρ ⎧⎪ − ηo k y ( k x I h + k y I e ) − ηo k x ( − k y I h + k x I e ) ik x ( k yV e + k xV h ) − ik y ( k xV e − k yV h ) ⎫⎪ + ⎨ ⎬ k oε z ko µ z η o ⎪⎩ ⎪⎭ + j ηo J z koε z + ijM z ko µ z η o (2.35) ⇒ Fz = k ρ ⎧⎪ ηo e iV h I − + ⎨ ko ⎩⎪ ε z µ z ηo ⎫⎪ j ηo ijM z Jz + . ⎬+ ε k µ η k o z o z o ⎭⎪ (2.36) Thus, 23 ⎛ Ve ⎞ ⎛ Vh ⎞ ⎛ k η ik ρV h j ηo ijM z F =⎜ − i ηo I h ⎟ u + ⎜ + i ηo I e ⎟ v + ⎜ − ρ o I e + + Jz + ⎜ η ⎟ ⎜ η ⎟ ⎜ k oε z koε z ko µ z η o ko µ z ηo ⎝ o ⎠ ⎝ o ⎠ ⎝ ⎞ ⎟z , ⎟ ⎠ (2.37) Therefore, from eqn. (2.23), namely, dFt −j kt2 − υ e k ρ k ρ • = dz 2koε t ( ) ⎡⎢⎣i ( F + ) ( ij k 2 − υ h kρ kρ • − F × z⎤ − ⎥⎦ 2ko µt t ) ⎡⎢⎣ z × ( F + ) +F ⎤ ⎥⎦ (2.38) η Jk M ×z − t + − i ηo z × J t + o z ρ , k oε z ηo η o ko µ z iM z k ρ we have in the ( u, v ) plane ⎡ dQ1 ⎤ ⎢ ⎥ −j η ⎢ dz ⎥ = 2 o ⎢ dQ2 ⎥ k ρ k oε t ⎢ ⎥ ⎣ dz ⎦ − + ⎡ kx ⎢−k ⎣ y k y ⎤ ⎡ kt2 − υ e k x2 ⎢ k x ⎥⎦ ⎢⎣ −υ e k y k x ⎡ kx ⎢ 2 k ρ ko µt η o ⎣ − k y ij 1 kρ ⎡ kx ⎢ ⎣−k y Using Q1 = Ve ηo k y ⎤ ⎡ kt2 − υ h k x2 ⎢ k x ⎥⎦ ⎣⎢ −υ h k x k y ky ⎤ ⎡ I e ⎤ − k x ⎥⎦ ⎢⎣ − I h ⎥⎦ −υ h k x k y ⎤ ⎡ − k y ⎥⎢ kt2 − υ h k y2 ⎥⎦ ⎣ k x ⎧ 1 ⎡−M y ⎤ iM z ⎪ ⎢ ⎥+ η o ko µ z k y ⎤ ⎪ ηo ⎣ M x ⎦ ⎨ ⎥ kx ⎦ ⎪ η J ⎡ kx ⎤ o z ⎪+ k ε ⎢ k ⎥ o z ⎣ y⎦ ⎩ ⎡ dQ1 ⎤ ⎢ ⎥ − j η ⎡k 2 − υ e k 2 o t ρ ⎢ dz ⎥ = ⎢ koε t ⎣ 0 ⎢ dQ2 ⎥ ⎢ ⎥ ⎣ dz ⎦ + −υ e k y k x ⎤ ⎡ k x ⎥⎢ kt2 − υ e k y2 ⎥⎦ ⎣ k y − k x ⎤ ⎡V e ⎤ (2.39) − k y ⎥⎦ ⎢⎣V h ⎥⎦ ⎡ J y ⎤⎫ ⎡ kx ⎤ ⎥⎪ ⎢ k ⎥ + i ηo ⎢ ⎣ y⎦ ⎣− J x ⎦ ⎪ ⎬, ⎪ ⎪ ⎭ ⎡0 0 ⎤ ⎡ Ie ⎤ ij − ⎢ 2⎥⎢ h⎥ − kt ⎦ ⎣ − I ⎦ ko µt ηo ⎣ kt2 − kt2 + υ h k ρ2 ⎤ ⎡V e ⎤ ⎥⎢ h⎥ 0 ⎦ ⎣V ⎦ (2.40) ⎡ J v ⎤ ⎧⎪ iM z η J ⎫⎪ ⎡ k ⎤ 1 ⎡−M v ⎤ + o z ⎬⎢ ρ ⎥, ⎢ ⎥ + i ηo ⎢ ⎥+⎨ koε z ⎪⎭ ⎣ 0 ⎦ ηo ⎣ M u ⎦ ⎣ − J u ⎦ ⎪⎩ ηo ko µ z − i ηo I h , Q2 = Vh ηo + i ηo I e , 24 ⎡ d ⎛ Ve ⎞⎤ − i ηo I h ⎟ ⎥ ⎢ ⎜ ⎟⎥ ⎢ dz ⎜⎝ ηo ⎠ ⎢ ⎥ ⎞ ⎢ d ⎛ Vh ⎥ + i ηo I e ⎟ ⎥ ⎢ ⎜⎜ ⎟ dz (2.41) ⎠ ⎦⎥ ⎣⎢ ⎝ ηo ⎡ − j ( kt2 − υ e k ρ2 ) ηo I e ij ( kt2 − υ h k ρ2 )V h ⎤ ⎡ M ik M k η J ⎤ ⎢ ⎥ ⎢ − v + i ηo J v + ρ z + ρ o z ⎥ + εt koε z ⎥ µt η o η o ko µ z ⎥ ⎢ ηo 1 ⎢ = ⎢ +⎢ ⎥ ⎥ ko ⎢ Mu − jkt2 ηo I h ijkt2V e ⎥ ⎢ ⎥ − i ηo J u − ⎢ ⎥ ⎢ ⎥ η εt µt η o o ⎦ ⎣⎢ ⎦⎥ ⎣ Equating the bi-real and bi-imaginary parts, we obtain a set of transmission line equations, namely ⎧ dV e − j ( kt2 − υ e k ρ2 )ηo I e kη J e e e e ⎪ = − jk z Z I + v = − Mv + ρ o z , koε t koε z ⎪ dz ⎪ h − j ( kt2 − τ h k ρ2 ) V h k M ⎪ dI h h h h − Jv − ρ z , ⎪⎪ dz = − jk z Y V + i = koηo µt koηo µ z ⇒⎨ ⎪ − jkt2ηo I h dV h h h h h jk Z I v = − + = + Mu , z ⎪ dz k ε o t ⎪ ⎪ jkt2V e dI e e e e e = − jk z Y V + i = − − Ju , ⎪ dz koηo µt ⎪⎩ where k ze = kt2 − υ e k ρ2 , k zh = kt2 − υ h k ρ2 , k t = k o µ t ε t , Z e = Zh = (2.42) 1 ηo kZe = Y e koε t and 1 koη o µt . The square root branch of k ze and k zh is specified by the condition = h h Y kz that −π ≤ arg ( k zp ) ≤ 0, ∀p = e, h. (2.43) The voltage and current sources are respectively given as ve = −M v + k ρηo J z k oε z , (2.44) 25 ih = − Jv − kρ M z koηo µ z , (2.45) vh = M u , (2.46) ie = − Ju . (2.47) The subscripts e and h represent the fields that are respectively the TM and TE wave. Grouping eqn. (2.42), we arrive at a set of transmission line equations, namely: dV p = − jk zp Z p I p + v p , dz (2.48) dI p = − jk zpY pV p + i p , dz (2.49) where p = e, h . To solve eqn. (2.42), consider the following transmission lines. I iP k zp k zp δ ( z − z′) Zp Zp z = z′ −∞ ← z z→∞ (a) − k zp + δ ( z − z′) Zp −∞ ← z Fig. 2 dVi p = − jk zp Z p I iP , dz Vi P z I vP k zp VvP Zp z = z′ (b ) z→∞ z Transmission line models (2.50) dVv p = − jk zp Z p I vP + δ ( z − z′ ) , (2.54) dz 26 dI ip = − jk zpY pVi p + δ ( z − z ′ ) , dz (2.55) Multiply both eqns. by v p , Multiply both eqns. by i p , ip dI vp = − jk zpY pVv p , dz (2.51) dVi p = − jk zp Z p I iP i p , dz (2.52) vp dVv p = − jk zp Z p I vP v p + v pδ ( z − z′ ) , dz dI ip = − jk zpY pVi p i p + i pδ ( z − z′ ) , i dz (2.56) p dI vp v = − jk zpY pVv p v p , dz p (2.53) Table 1 (2.57) Equations for the transmission line model Summing along similar rows and integrate along z, we have ⎧d p p p P P p p p p P p ⎪ dz ∫ ( i Vi + v Vv ) dz ′ = − jk z Z ∫ ( i I i + v I v ) dz′ + v , ⎪ z ⇒⎨ z ⎪ d ( i p I p + v p I P ) dz ′ = − jk PY p ( i pV p + v pV P ) dz ′ + i p . i v z v ∫z i ⎪⎩ dz ∫z (2.58) Comparing these eqns with eqns. (2.42)(b) and (2.42) (c), we have V P = ∫ ( i pVi p + v pVvP ) dz ′ (2.59) z I p = ∫ ( i p I ip + v p I vP ) dz ′ (2.60) z Upon multiplying eqns. (2.50) to (2.54) with I vp , I ip , Vv p , and Vi p respectively, and subtracting the sum of eqns. (2.50) and (2.51) with the sum of eqns. (2.54) and (2.55), we have 2 ⎛ dV p dI P dI p dV P ⎞ ℜ = ∫ ⎜ I vP i + Vi p v − VvP i − I ip v ⎟dz dz dz dz dz ⎠ z1 ⎝ z = ( I vPVi p − VvP I ip ) z2 z1 (2.61) ( ) = ∫ − jk zP ( Z p I vP I ip + Y pVi pVvP − Y pVvPVi p − Z p I ip I vP ) + −vˆ p I ip − iˆ pVvP dz. z At the terminal end of the transmission line, we have 27 Vi p ( z = zL ) = Z L I iP ( z = zL ) , (2.62) Vvp ( z = zL ) = Z L I vp ( z = zL ) . (2.63) Thus, ℜ = 0 . If vˆ p = iˆ p = δ ( z − z′ ) , Vvp ( z | z′ ) = − I ip ( z′ | z ) or I ip ( z | z′ ) = −Vvp ( z′ | z ) . This condition is termed as the reciprocity theorem. By symmetry, we have Vi P ( z | z ′ ) = Vi p ( z′ | z ) and I vp ( z | z′ ) = I vp ( z′ | z ) . Therefore, from eqn. (2.37), i.e. ⎛ Ve ⎞ ⎛ Vh ⎞ ⎛ k ρ ηo e ik ρV h j ηo ijM z h e F =⎜ − i ηo I ⎟ u + ⎜ + i ηo I ⎟ v + ⎜ − I + + Jz + ⎜ η ⎟ ⎜ η ⎟ ⎜ k oε z koε z ko µ z η o ko µ z ηo ⎝ o ⎠ ⎝ o ⎠ ⎝ ⎞ ⎟z ⎟ ⎠ (2.64), we have F= 1 ∫ (i V e ηo + − + e i z 1 ηo z ∫ (i V h z k oε z koε z h i k ρ ηo j ηo + v eVve ) dz ′u − i ηo ∫ ( i h I ih + v h I vh ) dz ′uˆ + v hVvh ) dz ′v + i ηo ∫ ( i e I ie + v e I ve ) dz ′v z ∫ (i I e e i +v I e e v z Jz z + ) dz′ z + k µ ik ρ o ijM z ko µ z η o z ∫ (i V h ηo i h +v V h h v ) dz′ z (2.65) z z. Using eqn. (2.44) to (2.47), 28 ⎛ ⎞ k ρηoVve u z e e ˆ ⎜ ⎟ dz ′ − • + • − • F= uuV J u J z V uv M v i u z v v ⎟ koε z′ ηo ∫z ⎜⎝ ⎠ 1 ⎛ h ⎞ k ρ I ih u z + i ηo ∫ ⎜ I i uv • J v v + • M z z − I vh uu • M u u ⎟ dz ′ ⎜ ⎟ koηo µ z′ z⎝ ⎠ h ⎞ k ρVi vz 1 ⎛ h h ′ ⎜ V vv J v M z V vu M u − • + • − • i v z v u ⎟ dz ⎟ koηo µ z′ ηo ∫z ⎜⎝ ⎠ e ⎛ ⎞ k η I vz o − i ηo ∫ ⎜ I ie vu • J u u + I ve vv • M v v − ρ v • J z z ⎟ dz ′ ⎜ ⎟ koε z′ z⎝ ⎠ e ⎞ k η ⎛ k η I zz + ρ o ∫ ⎜ I ie zu • J u u − ρ o v • J z z + I ve zv • M v v ⎟ dz ′ ⎟ koε z z ⎜⎝ koε z′ ⎠ h ⎛ h ⎞ ik ρ k ρVi z z h ′ ⎜ • − • V zv J v M z V zu M u − • + z v u ⎟ dz v ∫ i ⎟ koηo µ z′ ko µ z ηo z ⎜⎝ ⎠ +∫ z j ηo ij δ ( z − z ′ ) z z • J z zdz ′ + ∫ δ ( z − z ′ ) z z •M z zdz ′. koε z′ z ko µ z′ η o Comparing with F = ∫ 1 ηo (G EJ ) (2.66) ) ( • J + G EM • M dz′ + i ηo ∫ G HJ • J + G HM • M dz′ , we have the spectral dyadic Green’s function (DGF): G EJ = −uuVi e − vvVi h + u z ⎡ k 2η 2 I e jη δ ( z − z ′ ) ⎤ k ρηoVve k η Ie + zu ρ o i − z z ⎢ ρ2 o v − o ⎥ ,(2.67) koε z′ koε z koε z′ ⎢⎣ ko ε z ε z′ ⎥⎦ G EM = −uvVve + vuVvh + zv G HJ = uvI ih − vuI ie − zv G HM = −uuI − vvI + h v e v k ρηo I ve k oε z k ρVi h koηo µ z ( − vz + vz k ρ Vvh zu µ z + u zI ih µ z′ koηo k ρVi h koηo µ z′ , (2.68) k ρηo I ve , koε z′ ) − zz k ρ2Vi h k η µ z µ z′ 2 2 o o (2.69) + zz jδ ( z − z ′ ) .(2.70) koηo µ z′ where the subscript prime indicates source coordinate. These Green’s functions are related to the electric and magnetic fields through 29 ) ( E = ∫ G EJ • J + G EM • M dz ′ , (2.71) ) ( H = ∫ G HJ • J + G HM • M dz ′ . (2.72) The spatial Green’s function can be obtained through ( ) G PQ = ℑ−1 G PQ ( k ρ ; z | z′ ) = 1 4π ∫ξ η∫ G ( kρ ; z | z′) e PQ 2 − j (ξ x +η y ) dξ dη , (2.73) where G PQ is the dyadic Green’s function relating a P-type fields at r and Q-type currents at r ′ . After the transformation from Cartesian into cylindrical coordinate given by ξ1 = k ρ cos β , ξ 2 = k ρ sin β , x − x′ = r cos φ , y − y′ = r sin φ , a Sommerfeld type integral ⎧ sin ⎫ n sin ℑ−1 ⎨ nζ f ( k ρ ) ⎬ = ( − j ) nϕ S n f ( k ρ ) , n=0,1,2, cos ⎩cos ⎭ { { } where S n f ( k ρ ) = 1 2π } (2.74) ∞ ∫ f ( kρ ) J ( kρ r ) kρ dkρ , is obtained. n 0 Typical useful expressions are given below: ( ) ( ) ℑ−1 f ( k ρ ) = So f ( k ρ ) , (2.75) ( ) ( ) (2.76) ( ) ( ) (2.77) ℑ−1 jk x f ( k ρ ) = cos φ S1 f ( k ρ ) , ℑ−1 jk y f ( k ρ ) = sin φ S1 f ( k ρ ) , ( ) )} (2.78) ( ) 12 {cos 2φ S ( f ( k )) + S ( k f ( k ))} , (2.79) ℑ−1 k x2 f ( k ρ ) = − ℑ−1 k y2 f ( k ρ ) = { ( ) ( 1 cos 2φ S 2 f ( k ρ ) − So k ρ2 f ( k ρ ) , 2 2 ρ 2 o ρ ρ 30 ( ) ( ) 1 ℑ−1 k y k x f ( k ρ ) = − sin 2φ S 2 f ( k ρ ) . 2 (2.80) 2.3 Mixed Potential Integral Equation Formulation Consider the two set of Maxwell equations for electric current-source free and magnetic current-source free as shown in Table 2: Electric current-source free ∇ × E1 = − jωµo µ • H1 − M ∇ × H1 = jωε oε • E1 , ( ) ∇ • µ o µ • H1 = ρ m , ( ) ∇ • ε oε • E1 = 0 , Table 2 Magnetic current-source free (2.81) ∇ × E2 = − jωµo µ • H 2 , (2.82) ∇ × H 2 = jωε oε • E2 + J , (2.86) ( ) (2.83) ∇ • µo µ • H 2 = 0 , (2.84) ∇ • ε oε • E2 = ρe , ( ) (2.85) (2.87) (2.88) Two sets of Maxwell equations for electric current-source free and magnetic current-source free By taking the superposition of the two cases, the general Maxwell’s equations can be arrived. From the curl identity, i.e. ∇ • ∇× Y = 0 , we have from eqns. (2.84) and (2.87), µo µ • H 2 = ∇ × Ae , (2.89) ε oε • E1 = −∇ × Am . (2.90) Using these in eqns. (2.82) and (2.85), we have ( ) (2.91) ∇ × E2 + jω Ae = 0 . ( ) (2.92) H1 = − jω Am − ∇φm , (2.93) ∇ × H1 + jω Am = 0 , Therefore, 31 E2 = − jω Ae − ∇φe . (2.94) Grouping these equations with quaternion, we have F= ⎞ ⎛ ⎞ 1 ⎛ 1 −1 1 −1 m e m µ • ∇ × Ae ⎟ (2.95) ⎜ − ε • ∇ × A − jω A − ∇φe ⎟ + i ηo ⎜ − jω A − ∇φm + µo ηo ⎝ ε o ⎠ ⎝ ⎠. Expanding eqns. (2.81) and (2.86) with eqn. (2.89) and eqn. (2.90), and assuming that ε and µ are not functions of position, we have e ⎡⎛ 1 1 ⎞ e ⎤ 2⎛ A ⎞ − − ∇ A ⎢⎜ ⎜ ⎟ ⎟ ⎥ ⎢⎣⎝ µ z µt ⎠ ⎥⎦ ⎝ µz ⎠ ⎡ dφ ⎤ ⎤ ⎤ ⎢ εt e ⎥ ⎡ dx ⎥ ⎢ ⎢ ⎥ µo J x ⎥ 0 ⎥ e ⎢ ⎥ ⎢ dφ ⎥ 0 ⎥ • A − jωε o µo ⎢ ε t e ⎥ + ⎢ µo J y ⎥ , dy ⎥ ⎢ ⎢ µt ε z ⎥⎥ µµJ ⎥ ⎢ µt ε z dφe ⎥ ⎢ o t z ⎥ µ z ⎦⎥ ⎢ ⎥ ⎣⎢ µ z ⎥⎦ ⎣ µ z dz ⎦ ( ( ∇ ∇ • µt−1µ z−1µ • Ae ⎡ ⎢ε ⎢ t 2 = ko ⎢ 0 ⎢ ⎢0 ⎣⎢ ( ( 0 εt 0 ∇ ∇• ε ε ε • A ⎡ ⎢µ ⎢ t = ko2 ⎢ 0 ⎢ ⎢0 ⎢⎣ −1 −1 t z 0 µt 0 )) m + )) d2 dz 2 d2 + 2 dz (2.96) m ⎡⎛ 1 1 ⎞ m ⎤ 2⎛ A ⎞ ⎢⎜ − ⎟ A ⎥ − ∇ ⎜ ⎟ ⎝ εz ⎠ ⎣⎢⎝ ε z ε t ⎠ ⎦⎥ ⎡ dφ ⎤ ⎤ ⎤ ⎢ µt m ⎥ ⎡ dx ⎥ ⎢ ⎢ ⎥ 0 εoM x ⎥ ⎥ m ⎢ ⎥ ⎢ ⎥ dφ 0 ⎥ • A − jωε o µo ⎢ µt m ⎥ + ⎢ ε o M y ⎥ . dy ⎥ ⎢ ⎢ ε t µ z ⎥⎥ ε oε t M z ⎥⎥ ⎢ ⎢ µ zε t dφm ⎥ ε z ⎥⎦ ⎢ ⎥ ⎢⎣ ε z ⎥⎦ ⎣ ε z dz ⎦ (2.97) Using Lorentz conditions, namely, ⎪⎧ ( ( µ z ⎨∇ ∇ • µt−1µ z−1µ • Ae ⎪⎩ )) ⎡ ⎢ µ zε t ⎢ d 2 ⎡⎛ 1 1 ⎞ e ⎤ ⎪⎫ ⎢ + 2 ⎢⎜ − ⎟ A ⎥ ⎬ = − jωε o µo ⎢ µ zε t dz ⎣⎢⎝ µ z µt ⎠ ⎦⎥ ⎪⎭ ⎢ ⎢ ⎢⎣ µt ε z dφe ⎤ dx ⎥⎥ dφe ⎥ , (2.98) dy ⎥ ⎥ dφe ⎥ dz ⎥⎦ 32 ⎪⎧ ( ( ε z ⎨∇ ∇ • ε t−1ε z−1ε • Am ⎪⎩ )) ⎡ ⎢ µt ε z ⎢ d 2 ⎡⎛ 1 1 ⎞ m ⎤ ⎪⎫ ⎢ + 2 ⎢⎜ − ⎟ A ⎥ ⎬ = − jωε o µo ⎢ µt ε z dz ⎣⎢⎝ ε z ε t ⎠ ⎦⎥ ⎪⎭ ⎢ ⎢ ⎢⎣ µ zε t dφm ⎤ dx ⎥⎥ dφm ⎥ , (2.99) dy ⎥ ⎥ dφm ⎥ dz ⎥⎦ in eqns. (2.96) and (2.97), we obtain the Helmholtz vector wave equations: ⎡ε z µt ∇ A + k ⎢⎢ 0 ⎢⎣ 0 2 2 o m ⎡ µ zε t ∇ A + k ⎢⎢ 0 ⎢⎣ 0 2 e 2 o 0 ε z µt 0 0 µ zε t 0 0 ⎤ 0 ⎥⎥ • Am = −ε oε p • M , ε t µ z ⎥⎦ (2.100) 0 ⎤ 0 ⎥⎥ • Ae = − µo µ p • J , µt ε z ⎥⎦ (2.101) where ⎡µz µ p = ⎢⎢ 0 ⎢⎣ 0 ⎡ε z ε p = ⎢⎢ 0 ⎢⎣ 0 0 µz 0 0 εz 0 0⎤ 0 ⎥⎥ , µt ⎥⎦ 0⎤ 0 ⎥⎥ . ε t ⎥⎦ (2.102) (2.103) In here, we are assuming the relative permeability and permittivity are independent of positions. To solve eqns. (2.102) and (2.103), we assume the existence of following un-normalized DGFs, ⎡ε z µt ∇ G + k ⎢⎢ 0 ⎢⎣ 0 2 M 2 o ⎡ε z µt ∇ G + k ⎢⎢ 0 ⎢⎣ 0 2 e 2 o 0 ε z µt 0 0 ε z µt 0 0 ⎤ 0 ⎥⎥ • G M = −ε pδ ( r − r ′ ) , ε t µ z ⎥⎦ 0 ⎤ 0 ⎥⎥ • G e = − µ pδ ( r − r ′ ) . ε t µ z ⎥⎦ (2.104) (2.105) 33 Through the dot product of eqns. (2.104) and (2.105) respectively with ε o M and µo J , and followed by an integration within its spatial support, we have ε o∇ 2 ∫ G M • Mds′ + ε o ko2ε p • µ • ∫ G M • Mds′ = −ε o ∫ ε pδ ( r − r ′ ) • Mds′ = −ε oε • M , s s s (2.106) µo∇ 2 ∫ G A • Jds′ + ko2 µoε • µ p • ∫ G A • Jds′ = − µo ∫ µ pδ ( r − r ′ ) • Jds′ = − µo µ • J . s s s (2.107) Comparing eqns. (2.106) and (2.107) with eqns. (2.102) and (2.103), we have Am = ε o ∫ G M • Mds′ , (2.108) Ae = µo ∫ G A • Jds′ . (2.109) s s Using eqns. (2.89), (2.90), (2.71), (2.72), (2.108) and (2.109), we have µo µ • H 2 = µo µ • ∫ G HJ • J s ds′ = ∇ × µo ∫ G A • J s ds′ ⇒ µ • G HJ = ∇ × G A , (2.110) s s ε oε • H 2 = ε oε • ∫ G EM • M s ds′ = ∇ × ε o ∫ G M • M s ds′ ⇒ ε • G EM = ∇ × G M . (2.111) s s Both eqns. (2.110) and (2.111) does not uniquely specify G A and G M , making different formulation possible. Two possible forms for both G A and G M in the ( u, v ) plane are given in Table 3: Form of G A ⎡GuuA ⎢ A ⎢ Guv ⎢ 0 ⎣ GuvA A vv G 0 0 ⎤ ⎥ 0 ⎥ GzzA ⎥⎦ Form of G M ⎡GuuM ⎢ M ⎢Guv ⎢ 0 ⎣ GuvM M vv G 0 0 ⎤ ⎥ 0 ⎥ GzzM ⎥⎦ 34 ⎡ GvvA ⎢ ⎢ 0 ⎢GzuA ⎣ Table 3 0 ⎤ ⎥ 0 ⎥ GzzA ⎥⎦ 0 A vv G 0 ⎡GvvM ⎢ ⎢ 0 ⎢GzuM ⎣ 0 M vv G 0 0 ⎤ ⎥ 0 ⎥ GzzM ⎥⎦ Two possible forms for both G A and G M in the ( u, v ) plane For a horizontally Hertzian dipole over a dielectric space, two components of the vector potential are required to satisfy the boundary conditions at the interfaces. Traditionally, the z component has been selected in addition to the v component (see second row). Alternatively, one may postulate the u component of the vector potential to accompany the v component (see first row). For convenience, consider initially the second row of G A . In the spectral domain, it becomes ⎡ GvvA ⎢ GA = ⎢ 0 ⎢GzuA ⎣ 0 GvvA 0 0 ⎤ ⎥ 0 ⎥ GzzA ⎥⎦ (2.112) . By projecting to the ( x, y ) plane through eqn. (2.24), we have G A = Gvv uu + Gvv vv + Gzu zu + Gzz z z ⎡ k x2 1 ⎢ = Gvv 2 ⎢ k x k y kρ ⎢ 0 ⎣ ⎡ k y2 0⎤ −k x k y 1 ⎢ ⎥ 0 ⎥ + Gvv 2 ⎢ −k x k y k x2 kρ ⎢ 0 0 0 ⎥⎦ 0 ⎣ k k = Gvv xx + Gvv y y + x Gzu z x + y Gzu z y + Gzz z z kρ kρ kx k y k y2 0⎤ ⎛k k ⎞ ⎥ 0 ⎥ + Gzu z ⎜ x x + y y ⎟ + Gzz z z ⎜k k ρ ⎟⎠ ⎝ ρ 0 ⎥⎦ , (2.113) which clearly indicates that horizontal and vertical components of the vector potential are involved for a horizontal current source. Let G = ε • G EM + i µ • G HJ and then expressing it in spectral domain, we have 35 () ) ( ( ℑ−1 G = ℑ−1 ε • G EM + i µ • G HJ = ∇ × ℑ−1 G M + iG A d ⎞ d ⎞ ⎛ ⎛ = ⎜ − jk ρ u + z ⎟ × G M + i ⎜ − jk ρ u + z ⎟ × G A dz ⎠ dz ⎠ ⎝ ⎝ ) . (2.114) Equating eqn. (2.114) with eqns. (2.67) to (2.70) gives ⎡ ε z k ρηo I ve ε t k ρVi h ⎤ e h − vz ⎢ −uvε tVv + vuε tVv + zv ⎥ koε z koηo µ z′ ⎦⎥ ⎣⎢ ⎡ µ z k ρVi h µt k ρηo I ve ⎤ h e +i ⎢uvµt I i − vuµt I i − zv + vz ⎥ koηo µ z koε z′ ⎥⎦ ⎢⎣ . (2.115) ⎡⎛ ⎞⎤ ⎛ dGvvM dGvvA ⎞ dGvvM ⎞ ⎛ dGvvA M − jk ρ GzuA ⎟ ⎥ vu = −⎜ −i ⎟ uv + ⎢⎜ jk ρ Gzu + ⎟−i⎜ dz ⎠ dz ⎠ ⎝ dz ⎠ ⎦⎥ ⎝ dz ⎣⎢⎝ ( ) ( ) − jk ρ GvvM − iGvvA zv + jk ρ GzzM − iGzzA vz Equating the components of the bi-real and bi-imaginary parts yields ε tVi h , koηo µ z′ (2.116) − j µtηo I ve G = , koε z′ (2.117) GzzM = j A zz GvvM = j ηo I ve ko , (2.118) − jVi h G = , koηo (2.119) A vv ⎛ dGvvM dGvvA i − ⎜ dz ⎝ dz ⎞ e h ⎟ = ε tVv − iµt I i , ⎠ ⎡⎛ ⎞⎤ dGvvM ⎞ ⎛ dGvvA M − jk ρ GzuA ⎟ ⎥ = ε tVvh − i µt I ie . ⎢⎜ jk ρ Gzu + ⎟−i⎜ dz ⎠ ⎝ dz ⎢⎣⎝ ⎠ ⎥⎦ (2.120) (2.121) Putting eqn. (2.120) in eqn. (2.121), we have G M zu = jε t (Vve − Vvh ) kρ , (2.122) 36 j µt ( I ie − I ih ) GzuA = kρ . (2.123) Therefore, ⎡ − jV h i ⎢ koηo ⎢ ⎢ 0 GA = ⎢ ⎢ ⎢ ⎢ j µt ( I ie − I ih ) ⎢ kρ ⎢⎣ ⎡ jηo I ve ⎢ ko ⎢ ⎢ GM = ⎢ 0 ⎢ ⎢ ⎢ jε t (Vi e − Vi h ) ⎢ kρ ⎣⎢ 0 ⎤ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ − j µtηo I ve ⎥ ⎥ koε z′ ⎥⎦ , 0 − jVi h koηo j µt ( I ie − I ih ) kρ 0 jηo I ve ko jε t (Vi e − Vi h ) kρ (2.124), ⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ jε tVi h ⎥ ⎥ koηo µ z′ ⎦⎥ . 0 (2.125) From the Lorentz condition given below, ⎧⎪ ( ( µ z ⎨∇ ∇ • µt−1µ z−1µ • Ae ⎩⎪ ⎧⎪ ( ( ε z ⎨∇ ∇ • ε t−1ε z−1ε • Am ⎩⎪ ⎡ ⎢ µ zε t ⎢ d 2 ⎡⎛ µ z ⎞ e ⎤ ⎫⎪ + 2 ⎢⎜ 1 − ⎟ A ⎥ ⎬ = − jωε o µo ⎢⎢ µ zε t dz ⎢⎣⎝ µt ⎠ ⎥⎦ ⎭⎪ ⎢ ⎢ ⎢⎣ µt ε z dφe ⎤ dx ⎥⎥ dφe ⎥ , dy ⎥ ⎥ dφe ⎥ dz ⎥⎦ (2.126) ⎡ ⎢ µt ε z ⎢ d 2 ⎡⎛ ε z ⎞ m ⎤ ⎫⎪ + 2 ⎢⎜1 − ⎟ A ⎥ ⎬ = − jωε o µo ⎢⎢ µt ε z dz ⎢⎣⎝ ε t ⎠ ⎥⎦ ⎭⎪ ⎢ ⎢ ⎢⎣ µ zε t dφm ⎤ dx ⎥⎥ dφm ⎥ , dy ⎥ ⎥ dφm ⎥ dz ⎥⎦ (2.127) )) )) the scalar potentials can be obtained from the last auxiliary equations (2.126) and (2.127), i.e. 37 ⎫⎪ d ⎧⎪ d ⎡⎛ 1 1 ⎞ e ⎤ −1 −1 e ⎢⎜ − ⎟ A ⎥ + jωε o µo µt ε zφe ⎬ = 0 , ⎨ µ z ∇ • µt µ z µ • A + µ z dz ⎩⎪ dz ⎢⎣⎝ µ z µt ⎠ ⎥⎦ ⎭⎪ (2.128) ⎫⎪ d ⎧⎪ d ⎡⎛ 1 1 ⎞ m ⎤ −1 −1 m ⎢⎜ − ⎟ A ⎥ + jωε o µo µ zε tφm ⎬ = 0 . ⎨ε z ∇ • ε t ε z ε • A + ε z dz ⎩⎪ dz ⎢⎣⎝ ε z ε t ⎠ ⎥⎦ ⎭⎪ (2.129) ( ) ( ) Therefore, ⎛ Ae ⎞ ⎟ + jωε o µo µt ε zφe = 0 , ⎝ µz ⎠ (2.130) ⎛ Am ⎞ ⎟ + jωε o µo µ zε tφm = 0. ⎝ ε z′ ⎠ (2.131) µz∇ • ⎜ ε z∇ • ⎜ To arrive at the mixed-potential form, we postulate that ε t−1∇ • ( µt−1µ z−1µ • G A ) = −∇′K AΦ + C AΦ zˆ , (2.132) µt−1∇ • ( ε t−1ε z−1ε • G M ) = −∇′K MΦ + CMΦ zˆ , (2.133) where K Φ is a scalar potential kernel and C Φ is the correction factor, which arises in general when both horizontal and vertical current components are present. Grouping eqns (2.132) and (2.133) with quaternion, we have µt−1∇ • ( ε t−1ε z−1ε • G M ) + iε t−1∇ • ( µt−1µ z−1µ • G A ) = −∇′K MΦ + CMΦ z − i∇′K AΦ + iC AΦ z . (2.134) Expressing eqn. (2.134) in spectral domain, i.e. replacing ∇ = − jk ρ uˆ + zˆ ∇′ = jk ρ uˆ + zˆ d and dz d , we have dz ′ 38 d ⎞ ⎛ −1 −1 M M M M ⎟ • ε t ε z ⎡⎣ε t Gvv uu + ε t Gvv vv + zuε z Gzu + z zε z Gzz ⎤⎦ + dz ⎠ ⎝ d ⎞ ⎛ iε t−1 ⎜ − jk ρ u + z ⎟ • µt−1µ z−1 ⎡ µt GvvA uu + µt GvvA vv + zuε z GzuA + z zµ z GzzA ⎤ ⎣ ⎦ dz ⎠ ⎝ d ⎞ Φ d ⎞ Φ ⎛ ⎛ Φ K M + CMΦ z − i ⎜ jk ρ u + z = − ⎜ jk ρ u + z ⎟ ⎟ K A + iC A z , ′ ′ dz ⎠ dz ⎠ ⎝ ⎝ ( ( ) ) (2.135) ⎛ dG M dG M ⎞ ⇒ µt−1 ⎜ − jk ρ ε z−1GvvM u + ε t−1 zu u + ε t−1 zz z ⎟ dz dz ⎠ ⎝ ⎛ dG A dGzzA ⎞ + iε t−1 ⎜ − jk ρ µ z−1GvvA u + µt−1 zu u + µt−1 z⎟ dz dz ⎠ ⎝ dK Φ dK AΦ = − jk ρ K MΦ u − z M + CMΦ z − ijk ρ K AΦ u − zi + iC AΦ z dz ′ dz ′ . (2.136) µt−1 ⎜ − jk ρ u + z Substituting eqns. (2.116) to (2.121) in eqn. (2.136) and equating the bi-real and biimaginary terms, we have e h ηo I ve 1 d (Vv − Vv ) , K = j − µt ε z′ ko k ρ2 µt dz (2.137) e h 1 d ( Ii − Ii ) , K =− − koηoε t µ z′ k ρ2ε t dz (2.138) Φ M Φ A CMΦ = jVi h dVi h dK MΦ , + koηo µt µ z′ dz dz ′ 1 − jηo dI ve dK AΦ + C = . koε t ε z′ dz dz ′ Φ A (2.139) (2.140) Using eqns. (2.50), (2.51), (2.54), (2.55) i.e. ⎧ dVi p = − jk zp Z p I iP , ⎪ dz ⎪ ⎪ dVvp p p P p p P p ⎪⎪ dz = − jk z Z I v + δ ( z − z ′ ) = − jk z Z I v + vˆ , ⎨ p ⎪ dI i = − jk pY pV p + δ ( z − z ′ ) = − jk pY pV p + iˆ p , z i z i ⎪ dz ⎪ dI vp ⎪ = − jk zpY pVv p , ⎪⎩ dz (2.141) 39 where k = k − υ k ρ , k = k − υ k ρ , k t = k o µ t ε t e z Zh = 2 t e 2 h z 2 t h 2 1 ηo kZe , Z = e = Y koε t e and 1 koη o µt , = Yh k zh For the reciprocity property and the symmetry property, we have ko2 µt ε t − ε t k ρ2 ε z′ )ηo e ⎞ jkoηo e h ( jηo I ve j ⎛ h ⎜k η µ I − − K = I v ⎟ = 2 ( I v − I v ) , (2.142) ⎟ µt ε z′ ko k ρ2 µt ⎜ o o t v k oε t kρ ⎝ ⎠ Φ M K AΦ = − jVi h koηoε t µ z′ − h 2 2 e h j ⎛ Vi ( ko µt ε t − k ρ µt µ z ) koε tVi e ⎞ jko (Vi − Vi ) ⎜ ⎟ − = , k ρ2ε t ⎜ koηo µt ηo ⎟ ηo k ρ2 ⎝ ⎠ − jI ih jkoηo ⎛ koε t′ e + 2 ⎜− j C = V + k ρ ⎜⎝ µ z′ ηo i Φ M ⎛ 2 µt k ρ2 ⎞ Vi h j ⎜ ko ε t′µt − ⎟ ⎜ µ z′ ⎟⎠ koηo µt ⎝ ⎛k ε ⎞ −Vve 1 ⎛ C = + 2 ⎜ ko2 µt′ε t − k ρ2 t ⎟ I ie − ⎜ o ⎜k ε z′ k ρ ε t ⎝ ε z′ ⎠ ⎝ ρ Φ A (2.143) ⎞ Vvh − jI ih ko2ε t′ e + 2 (Vi − Vi h ) ⎟= ⎟ kρ µ z′ ⎠ ,(2.144) 2 ⎞ ko2 µt′ e ko2 µt′ h h h e ⎟⎟ µt I i = 2 ( I i − I i ) = 2 (Vv − Vv ) . kρ kρ ⎠ (2.145) In the eqn. (2.145), we have used Vv p ( z | z′ ) = − I ip ( z′ | z ) . We have achieved a new quaternion mixed potential formulation which is much more compact than the conventional mixed potential formulation. This leads to a new way of representing the dyadic Green’s function in a compact form. This new quaternion mixed potential formulation is much compact in terms of its derivation as both E & H are not separately evaluated. 40 Chapter 3 A Useful Multilayered Microstrip Pole Extraction Technique 3.1 Introduction In Chapter 2, we have provided an alternative way of deriving Green’s function in spectral domain. In this chapter, the poles of the Green’s function in spectral domain were extracted from the eqn. (2.124) and (2.125). In general, planar structures can be analyzed using the Method-of-moments (MoM), which requires the accurate evaluation of Green’s function. To calculate the Green’s functions efficiently, discrete complex image method (DCIM) can be used and the precise information on the pole locations of the Green’s function in spectral domain is thus needed. In this chapter, a fast and accurate method of pole extraction is introduced based on the derivation of the Green’s function obtained in Chapter 2. Method-of-moments (MoM) is known to be the most suitable numerical algorithm for the rigorous analysis of multilayered printed structures of small to medium sizes (in terms of wavelength) compared to other techniques such as finite elements and finitedifference time domain methods [117]–[120]. The application of the MoM for the solution of integral equations, either in spectral or spatial domain, requires the knowledge of the Green’s functions in the corresponding domain. The Green’s functions for multilayered media are traditionally represented by the Sommerfeld integrals in the spatial domain, and are obtained as closed-form expressions in the spectral domain. These representations of the Green’s functions are not computationally efficient to use in conjunction with the MoM. This is mainly due to 41 the oscillatory nature of the Sommerfeld integrals in the spatial domain, and due to slow-decaying nature of the spectral-domain Green’s functions [32]. To calculate the Green’s functions efficiently, discrete complex image method (DCIM) [33]–[48] has been introduced to approximate the spectral-domain Green’s functions in terms of complex exponentials and cast the integral representation into closed-form expressions via an integral identity, namely the Sommerfeld identity [49]. DCIM divides the Green’s function into three parts. They are the extraction of the quasi-static terms, the contributions of the surface wave poles (SWP) and the remaining terms. The quasistatic contribution is inverse transformed analytically using the Sommerfeld identity. The surface wave contribution comes from the poles in the spectral Green’s functions; these are removed and evaluated using residue calculus. The remaining portion in the spectral representation is expanded into a series of exponentials using either the Prony or GPOF methods. The precise pole locations of the Green’s function are very important for the accurate evaluation of the closed-form spatial-domain Green’s function for microstrip problems using DCIM [34], [121]-[122]. The poles of the Green’s function for the microstrip geometry correspond to the surface- and leaky-wave modes. In most of the analysis, the precise information on the pole locations is needed for the accurate evaluation of the residue terms used in the complex image method [34], [122]. In the spectraldomain method, the poles also play a crucial role in the accurate evaluation of the Sommerfeld Integral [123]. The extracted poles also have its importance in leaky wave antenna design. 42 With the introduction of new technologies such as the low-temperature cofired ceramics (LTCC), meta-materials and multilayered PCBs, it has become increasingly necessary to look into the extraction of poles in multilayered microstrip geometry. So far, most of the proposed techniques of pole extraction [52]-[53], [57]-[58], [123][124] have been confined to a single-layered, dielectric microstrip geometry. We have shown that it is possible to obtain the locations of these poles through the functional analysis such as “contraction mapping” [57]. However, the extension of this method to multilayered structures is rather complex and impractical. The proposed solution in [57] did not address two important issues namely, the: (a) speed of convergence of the derived functional expression, and the (b) initial guess needed for initializing the contraction mapping algorithm. There can be many functional forms that satisfy the contraction mapping criteria, and each form can have its own speed of convergence. The search for a correct, convergent operator as used in [57] will become extremely difficult when it is extended to multilayered structures. An intelligent starting point will surely result in fast convergence when the contraction mapping is adopted. However, unless the user has the knowledge on the shape of the derived functional expression, which is usually transcendental in nature, one cannot have a fully automatic procedure for arbitrarily dielectric layered pole extraction. Moreover, it is noted that with an increased number of dielectric layers, different functional forms would be required and the quest for a generalized, unified method would thus be lost. Our earlier success in deriving fast but efficient pole extraction for single-layered [58] and double-layered [125] dielectric microstrip problem provided a good starting point 43 for deriving a generalized approach for multilayered microstrip pole extraction. We have been able to derive novel functional expressions, which have quadratic convergence, for finding the roots of DTE and DTM [58] in a single-layered microstrip structure and [125] in a double-layered microstrip problem. In this chapter, we will illustrate how these functional expressions, coupled with the derived initial values given in [58], [125], can be extended to solve the multilayered microstrip problem. Instead of using the real analysis for our explanation, an algorithmic approach will be adopted in explaining the proposed method. In Section 3.2, we will start with the pole extraction for two-layered structures as shown in [125]. In the next section, this algorithm is improved and developed for multilayered structures. 3.2 Pole Extraction for Two Layered Structures 3.2.1 Introduction The surface wave poles of a two-layered structure are extracted using the following algorithm shown in this section. These poles actually correspond to the intersection of a circle and a tangent or cotangent curve in the first quadrant for our proposed method. 3.2.2 Two-layered Microstrip Geometry in TM Mode In a two-layered microstrip problem, with different dielectric permittivity, several cases of characteristic equations can be derived after tedious manipulation. These expressions are given as follows: For ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 , DTM = ε r 2 ∆ o2 − x 2 − x xε r1h1 tan ( x ) + yε r 2 h2 tan ( y ) = 0, xε r1h1 − yε r 2 h2 tan ( x ) tan ( y ) (3.1) 44 For ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 , DTM = ε r 2 ∆ o2 − x 2 − x xh1ε r1 tan ( x ) − jyh2ε r 2 tanh ( jy ) = 0, xh1ε r1 + jyh2ε r 2 tanh ( jy ) tan ( x ) (3.2) For ε r1 < ε r 2 and k o ε r 2 < k ρ < ∞ or ε r1 > ε r 2 and k o ε r1 < k ρ < ∞ , DTM = ε r 2 ∆ o2 − x 2 + jx xε r1h1 tanh ( jx ) + yε r 2 h2 tanh ( jy ) = 0, xε r1h1 + h2 yε r 2 tanh ( jy ) tanh ( jx ) (3.3) For ε r1 > ε r 2 and k o ≤ k ρ ≤ k o ε r 2 , xε r1h1 tan ( x ) + yh2ε r 2 tan ( y ) = 0, xε r1h1 − ε r 2 yh2 tan ( y ) tan ( x ) DTM = ε r 2 ∆ o2 − x 2 − x (3.4) For ε r1 > ε r 2 and k o ε r 2 ≤ k ρ ≤ k o ε r1 , DTM = ε r 2 ∆ o2 − x 2 − x ε r 2 yh2 tan ( y ) − jxε r1h1 tanh ( jx ) = 0, xh1ε r1 + jyε r 2 h2 tan ( y ) tanh ( jx ) (3.5) For ε r1 = ε r 2 and k o ≤ k ρ ≤ k o ε r1 (single-layered case), DTM = ε r1 ∆2o − x 2 − x tan( x ) = 0, (3.6) where the reduced variables are defined as x = h2 ε r 2 k o2 − k ρ2 , (3.7) ∆ i = k o h2 ε r 2 − ε ri , ∀i = 0,1 (3.8) ε r 0 = 1, y = h1 x 2 − ∆21 h2 , (3.9) (3.10) and j = − 1 . Since all the equations (3.1) to (3.6) have similar forms, for brevity, only equations (3.1) and (3.2) will be discussed in this section. Similar procedures can be applied to the rest of the equations. 45 Consider equation (3.1), if we let tan(ξ) = ε r 2 h2 y tan( y ) xh1ε r1 , (3.11) we obtain from equation (3.1) DTM = ε r 2 ∆2o − x 2 − x tan(ξ + x ) = 0. (3.12) This equation is in the same form as what we have adopted in [58] and some minor modifications on the arguments are needed. These expressions are given below: DTM ⎧ if k = ( N − 1) , kπ < ∆′ < ( 2k + 1) π 2, ∆oε r 2 ε r22 + tan 2 (ξ + x ) − x, ⎪ ⎪ 2 ⎡ε =⎨ ∆ x ) − 1 − tan (ξ ) ⎤ −1 ⎢ r 2 ( o ⎥ + kπ − x, elsewhere for k = 0,1, 2,…, N , ⎪tan 2 ⎢ ⎥ ⎪ x + ∆ − 1 ε tan ξ 1 ( ) ( ) r2 o ⎣ ⎦ ⎩ (3.13) where ( ( ) ) ∆ ′ = (ξ + x ) x = ∆ = ∆ o + tan −1 ε r 2 h2 ∆2o − ∆21 tan h1 ∆2o − ∆21 h2 ∆ o ε r1 h1 . (3.14) o The Newton-Raphson method is subsequently applied to solve both equations (3.13) with the initial values evaluated from ⎧ ⎡ ⎛ ⎪bk − ( bk − ak ) cos2 ⎢tan−1 ⎜ ⎜ ⎪ ⎢ ⎪ ⎝ ⎣ xk = ⎨ ⎡ −1 ⎛ ⎪ 2 ⎪ ∆ − ( ∆ − ak ) cos ⎢tan ⎜⎜ ⎢⎣ ⎪⎩ ⎝ ⎡ b −a ⎤ ⎡ ∆ − ak ⎤ ∆2 − ak2 ⎞⎤ −1 k k ⎟⎥ , if cos−1 ⎢ ⎥ > tan ⎢ 2 2 ⎥ bk − ak ⎟⎥ b a − ⎣ k k⎦ ⎠⎦ ⎣⎢ ∆ − ak ⎦⎥ ∆ + ak ∆ − ak ⎞⎤ ⎟⎟⎥ , ⎠⎥⎦ ⎡ b −a ⎤ ⎡ ∆ − ak ⎤ −1 k k if cos−1 ⎢ ⎥ < tan ⎢ 2 2 ⎥ ⎢⎣ ∆ − ak ⎥⎦ ⎣ bk − ak ⎦ (3.15) where k = 0,1,2, … , N , N is the number of DTM or DTE roots, ak and bk denote respectively the location of the zeros and infinities of either the tan ( x + ξ ) curve in equation (3.12) or cot ( x ) curve in equation (3.27). Fig. 3 shows how equation (3.15) is obtained from the perpendicular projection from point (a k ,0 ) to the line joining points 46 (a , k ∆2o − a k2 ) and (b ,0) . For TM surface modes, k N < 1 + ∆ π whereas for TE surface modes, N < 0.5 + ∆ π . Although the proposed method is not limited to the cases to be demonstrated, for simplicity, we will confine our explanation to those cases where the bottom layer is the ground and the top layer is air (See Fig. 4). For convenience, we shall assume that metal is found only on the top layer for all the cases to be studied. ∆2o − a 22 a1 b1 a2 Fig. 3 b2 Initial guess evaluation. 47 εo h2 εr 2 ε r1 h1 Fig. 4 Two-layered microstrip topology The locations for the zeros and singularities (poles) of tan ( x + ξ ) as used in equation (3.15) can be approximated through the locations of zeros and singularities of tan( x + y ) . Similar to equation (3.1), if we let tan(ζ ) = jyh2 ε r 2 tanh( jy ) (ε r1h1 x ), (3.16) we obtain from equation (3.2) DTM = ε r 2 ∆2o − x 2 − x tan( x − ζ ) = 0. (3.17) The required modifications for equations (3.13) are given as follows: DTM ⎧ if k = ( N − 1) , kπ < ∆′ < ( 2k + 1) π 2, ∆oε r 2 ε r22 + tan 2 ( x − ζ ) − x, ⎪ ⎪ 2 ⎡ε =⎨ ∆ x ) − 1 + tan (ζ ) ⎤ −1 ⎢ r 2 ( o ⎥ + kπ − x, elsewhere for k = 0,1, 2,…, N , ⎪tan ⎢1 − ε tan (ζ ) ( ∆ x )2 − 1 ⎥ ⎪ r2 o ⎣ ⎦ ⎩ 48 (3.18) and ( ( ) ) ∆ ′′ = ( x − ζ ) x = ∆ = ∆ o − tan −1 ε r 2 h2 ∆21 − ∆2o tanh h1 ∆21 − ∆2o h2 ∆ o ε r1 h1 . o (3.19) The locations of the zeros and sigularities used in equation (3.15) can respectively be determined by ( ) ⎧ ⎡ ⎛ ∆1 + kπ ⎞ ⎤ 2 ⎪ ∆1 − 2∆ ( ∆ − kπ ) − ∆12 − ( kπ ) cos ⎢ tan −1 ⎜⎜ ⎟⎥ , ∆1 − kπ ⎟⎠ ⎦⎥ ⎪ ⎢ ⎝ ⎣ ⎪ ak = ⎨ 2 2 ⎡ 2 ⎛ ⎞ ⎪ π ⎛ 2 2 2 ⎛π ⎞ 2 ⎢ −1 ⎜ ∆1 − ( kπ ) ⎪ kπ + 2 − ⎜⎜ ∆1 − ( kπ ) + ⎜ 2 ⎟ − ∆1 − ( kπ ) ⎟⎟ cos ⎢ tan ⎜ π /2 ⎝ ⎠ ⎪⎩ ⎝ ⎠ ⎝ ⎣ ⎛⎢∆ ⎥ ⎞ if k = ⎜ ⎢ 1 + 1⎥ − 1⎟ , kπ < ∆1 < ( 2k + 1) π /2 ⎦ ⎠ ⎝⎣ π ⎞⎤ ⎟ ⎥ , elsewhere for k = 0,1, 2,... ⎢ ∆1 + 1⎥ ⎢⎣ π ⎥⎦ ⎟⎥ ⎠⎦ (3.20) and 2 ⎧ ⎛ ⎛ ⎜ ⎜ ∆ 2 − ⎛⎜ ⎛ k + 1 ⎞ π ⎞⎟ ⎪ ⎟ ⎜ 2 ⎤ 2 2 1 ⎡ ⎜ ⎜ ⎜ ⎪ 2 ⎠ ⎟⎠ ⎛⎛ ⎛ ⎛⎛ 1⎞ ⎞ 1⎞ ⎞ 1⎞ ⎞ ⎛ 2 2 ⎝⎝ −1 ⎪ ∆ 1 − ⎢ ∆ 1 − ⎜⎜ ⎜ k + ⎟ π ⎟⎟ + ⎜⎜ ∆ 1 − ⎜ k + ⎟ π ⎟⎟ − ∆ 1 − ⎜⎜ ⎜ k + ⎟ π ⎟⎟ ⎥ cos ⎜ tan ⎜ 1⎞ ⎥ ⎢ 2 2 2 ⎛ ⎜ ⎠ ⎠ ⎝ ⎠ ⎠ ⎠ ⎠ ⎜ ⎝⎝ ⎝ ⎝⎝ ⎪ ∆1 − ⎜ k + ⎟π ⎦ ⎣ ⎜ ⎪ 2⎠ ⎝ ⎜⎜ ⎜ ⎝ ⎪ ⎝ ⎪ ifif k = ⎛⎜ ⎢ ∆ 1 + 1⎥ − 1 ⎞⎟ & k π < ∆ < (2 k + 1 )π 2 , ⎪ 1 ⎜⎢ π ⎟ ⎥ ⎪ ⎦ ⎝⎣ ⎠ bk = ⎨ 2 ⎞⎞ ⎛ ⎛ ⎪ ⎜ ⎜ ∆ 2 − ⎛⎜ ⎛ k − 1 ⎞ π ⎞⎟ ⎟ ⎟ ⎜ ⎟ ⎟ ⎟⎟ 2 ⎤ 2 ⎪ 2 1 ⎡ ⎜ ⎜ ⎜ 2⎠ ⎠ ⎛⎛ ⎛⎛ 1⎞ ⎞ 1⎞ ⎞ ⎛π⎞ ⎝⎝ ⎪ ⎟ ⎟, k π − ⎢ ∆ 21 − ⎜⎜ ⎜ k − ⎟ π ⎟⎟ + ⎜ ⎟ − ∆ 21 − ⎜⎜ ⎜ k − ⎟ π ⎟⎟ ⎥ cos ⎜ tan − 1 ⎜ ⎪ ⎢ π 2 2⎠ ⎠ 2⎠ ⎠ ⎥ ⎜ ⎝2⎠ ⎝ ⎝ ⎜ ⎟⎟ ⎝ ⎝ ⎣ ⎦ ⎪ ⎜ ⎜⎜ ⎟⎟ ⎟⎟ ⎜ ⎪ ⎝ ⎠⎠ ⎝ ⎪ ⎛⎢∆ ⎞ ⎥ ⎪ elsewhere for k = 0 ,1, 2 , … , ⎜⎜ ⎢ + 1⎥ − 1 ⎟⎟. elsewhere for ⎪⎩ ⎦ ⎝⎣π ⎠ ⎞⎞ ⎟⎟ ⎟⎟ ⎟ ⎟, ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠ (3.21) The symbol ⎣•⎦ denotes the supremum of that enclosed by the bracket. Fig. 3 provides an illustration of how ak and bk are derived from the circular arc from the point (a k ,0 ) ( ) to the line joining points a k , ∆2o − a k2 and (bk ,0) . 49 Similar method can be applied for the TM improper pole extraction. For brevity, they are not shown here. 3.2.3 Two-layered Microstrip Geometry in TE Mode For the TE mode of a two-layered microstrip problem, the characteristic equations are given as follows: For ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 , DTE = ∆ o2 − x 2 + x yh2 − xh1 tan ( x ) tan ( y ) xh1 tan ( y ) + yh2 tan ( x ) = 0, (3.22) For ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 , DTE = ∆ o2 − x 2 + x jyh2 − xh1 tanh ( jy ) tan ( x ) = 0, xh1 tanh ( jy ) + jyh2 tan ( x ) (3.23) For ε r1 < ε r 2 and k o ε r 2 < k ρ < ∞ or ε r1 > ε r 2 and k o ε r1 < k ρ < ∞ , DTE = ∆ o2 − x 2 + jx yh2 + xh1 tanh ( jy ) tanh ( jx ) = 0, xh1 tanh ( jy ) + yh2 tanh ( jx ) (3.24) For ε r1 > ε r 2 and k o ≤ k ρ ≤ k o ε r 2 , DTE = ∆ o2 − x 2 + x yh2 − xh1 tan ( x ) tan ( y ) xh1 tan ( y ) + yh2 tan ( x ) = 0, (3.25) For ε r1 > ε r 2 and k o ε r 2 ≤ k ρ ≤ k o ε r1 , DTE = ∆ o2 − x 2 + x jyh2 − xh1 tan ( y ) tanh ( jx ) = 0, jxh1 tan ( y ) + yh2 tanh ( jx ) (3.26) For ε r1 = ε r 2 and k o ≤ k ρ ≤ k o ε r1 (single-layered case), DTM = ∆ o2 − x 2 + x cot ( x ) = 0, (3.27) 50 where the reduced variables are defined as x = h2 ε r 2 k o2 − k ρ2 , (3.28) ∆ i = k o h2 ε r 2 − ε ri , ∀i = 0,1 (3.29) ε r 0 = 1, (3.30) y = h1 x 2 − ∆21 h2 , (3.31) Using the same approach as above, the TE surface wave modes characteristic equations can be derived for the case when ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . This expression is given as follows: DTE = ∆2o − x 2 + x cot( x + ς ) = 0, (3.32) where tan (ς ) = xh1 tan ( y ). yh2 (3.33) The required modification on equation (3.13) is thus DTE ⎧ ∆ o 1 + cot 2 (x + ς ) − x, ifif k = N & (2 N + 1)π 2 < ∆ ′′′ < Nπ, ⎪⎪ = ⎨ −1 ⎛⎜ x ∆2o − x 2 + tan (ς ) ⎞⎟ − x + kπ, elsewhere for k = 1,2,… , N , elsewhere for ⎪tan ⎜ 2 2 ⎟ ( ) ς ∆ − − x tan x 1 ⎪⎩ o ⎝ ⎠ (3.34) where ∆ ′′′ = ( x + ς ) x = ∆ ⎛ ⎛ h ∆2 − ∆2 ∆o 1 ⎜ = ∆ o + tan tan⎜ 1 o ⎜ 2 2 ⎜ h2 ⎝ ⎝ ∆ o − ∆1 −1 o ⎞⎞ ⎟ ⎟. ⎟⎟ ⎠⎠ (3.35) 51 Similar to TM mode, the locations for the zeros and singularities of cot ( x + ς ) as used in equation (3.15) can be approximated through the locations of zeros and singularities of cot ( x + y ) . When ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 , and this is given as follows: DTE = ∆ o2 − x 2 + x cot ( x + δ ) = 0, (3.36) xh1 tanh ( jy ) . jyh2 (3.37) where tan (δ ) = The required modification on equation (3.13) is given as ⎧ ifif k = N & ( 2 N + 1) π 2 < ∆ '''' < Nπ , ∆ o 1 + cot 2 ( x + δ ) − x, ⎪ ⎪ DTE = ⎨ ⎛ x ∆ 2 − x 2 + tan (δ ) ⎞ o −1 ⎜ ⎟ − x + kπ , tan elsewhere for k = 1, 2,… , N , elsewhere for ⎪ 2 2 ⎜ ⎟ tan δ 1 ∆ − − x x ( ) o ⎪⎩ ⎝ ⎠ (3.38) where ⎛ ⎛ h ∆2 − ∆2 ∆o 0 ∆ '''' = ( x + δ ) x =∆ = ∆ o + tan −1 ⎜ tanh ⎜ 1 1 2 2 o ⎜ h ⎜ ∆1 − ∆ 0 2 ⎝ ⎝ ⎞⎞ ⎟ ⎟. ⎟⎟ ⎠⎠ (3.39) The locations of the zeros and singularities used in equation (3.15) can respectively be determined using equation (3.20) and (3.21). Similar method can be applied for the extraction of improper TE mode poles. 52 3.2.4 Numerical Results and Discussions In the previous section, we have introduced a new algorithm for extracting the poles for a two-layered structure. This method presents good initial guess and good function expression with fast convergence. Initial guess is very important for poles extraction, our automatic algorithm for searching initial guess is much more convenient and fast compared to the other methods mentioned in the introcution. And in order to compare the convergence of our proposed functional expression and the conventional functional expression, some numerical results are shown below. The permittivity for the various microstrip configurations was selected as ε r1 = 5.9, ε r 2 = 10.9, ε r 3 = 12.8, and ε r 4 = 17.3. The adopted dielectric thickness are ko h1 = 0.4 , ko h2 = 0.4 and ko h3 = 0.4 . Based on the above algorithm, a Matlab program was developed. Fig. 5 (a) to (d) shows the numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations for the evaluation of the first and second TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . Similarly, Fig. 6(a) and 4(b) depicts the numerical comparison for the first TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 , whereas Fig. 6(c) and (d) gives the first TE surface mode of the two-layered microstrip geometry for the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . Fig. 7(a) and 5(b) give the numerical comparisons of the second TE root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 , whereas Fig. 7(c) and 5(d) depict the first TE root of the two53 layered microstrip geometry under the case of ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 . All the methods shown in Fig. 5 to Fig. 7 are subject to the same constraints, namely, (a) the function variation, df = 1 − dx = 1 − f n +1 < 10− 9 , (b) the step size variation, fn xn +1 < 10−9 dx, and (c) the same initial guesses are used. As shown from the xn figure, the conventional functional expression can not converge sometimes which leads to failure of finding the actual roots of the Greens’ function. As noted from all the figures, our proposed algorithm is better than all the various classical methods in terms of accuracy and number of iterations. As shown in Fig. 5(a), Fig. 5(c), Fig. 6(a), Fig. 6(c), Fig. 7(a) and Fig. 7(c), the Newton-Raphson method on classical equation can at times diverge. On average, an accuracy of 10 −16 and with a maximum of 4 iterations are achieved for the determination of each root under our proposed new method. A unified approach for fast evaluation of the pole locations in two-layered microstrip technology has been presented in this section. Using a simple re-organization of the classical equations, together with the Newton-Raphson algorithm, a quadratic convergence is observed for the two-layered microstrip pole extraction. For the first time, good initial values have also been proposed. 54 (a) (c) Fig. 5 (b) (d) A numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . (c)-(d) For the second TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . 55 Bisection Brent Newton New method (a) (b) Brent Newton Bisection New method (c) Fig. 6 (d) A numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the two-layered microstrip geometry under the case ε r1 < ε r 2 and k o ε r1 < k ρ ≤ k o ε r 2 . (c)-(d) For the first TE root of the two-layered microstrip geometry for the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 . 56 Fig. 7 (a) (b) (c) (d) A numerical comparisons of the various classical methods in terms of the residue of the function and the number of iterations. (a)-(b) For the second TE root of the twolayered microstrip geometry under the case ε r1 < ε r 2 and k o ≤ k ρ ≤ k o ε r1 .(c)-(d) For the first TE root of the two-layered microstrip geometry under the case of ε r1 < ε r 2 and . k o ε r1 < k ρ ≤ k o ε r 2 57 3.3 Extraction of Poles for N-layered Microstrip Geometry Based on the formulation introduced in Section 3.2, this algorithm is improved and extended for N-layered structures. In this section, a fast, accurate and generalized poles’ extraction algorithm for N-layered microstrip geometry is introduced. 3.3.1 Extraction of Surface Wave Poles For the ease of explanation, it is assumed that there is a ground layer on the 0th-layer and the metal layers are all infinitely thin (See Fig. 8). It should be noted that the proposed methodology is not limited to these assumptions, since the same functional expressions and initial guesses derived for all the various types of modes can still be applied in more general problems. One needs only to change A(′1) and B(′1) of the various characteristic equations for the case of surface wave poles, leaky-wave poles and lossy improper mode poles in order to cater for the changes in boundary conditions. The parameters A(′1) and B(′1) are explained in the following section. 58 Metallic conductors εo ε rN hN hN −1 ε r ( N −1) hN − 2 ε r ( N −2 ) h3 εr3 h2 εr 2 ε r1 h1 Fig. 8 (i) N-layered microstrip topology Functional Expression Derivation For the TM mode, the required recursive characteristic equation for multilayered substrate is given as DTM = ± k ρ2 − ko2 − kz( N ) ε r( N ) ( ) tan k z ( N ) h( N ) − jA( N −1) , (3.40) where N denotes the total number of layers as shown in Fig. 8 and k z ( N ) = ε r ( N ) ko2 − k ρ2 , ( ) tanh A( N −1) = k z ( N −1) ε r ( N ) ε r ( N −1) k z ( N ) ( ) tanh A(1) = (3.41) ( ) tanh A( N − 2) + jk z ( N −1) h( N −1) , k z (1)ε r ( 2) k z ( 2 )ε r (1) ( ) tanh jk z (1) h(1) . (3.42) (3.43) 59 The “ + ”sign before the square root operator in Eqn. (3.40) stands for the proper surface wave pole whereas “ − ” sign is for the improper or leaky wave pole. We will focus on the proper pole extraction algorithm in this section and discuss the improper pole extraction algorithm in subsequent sections. Using the reduced variables x = h( N ) ε r ( N ) ko2 − k ρ2 = y( N ) , (3.44) ∆i −1 = ko h( N ) ε r ( N ) − ε r ( i −1) , ∀i = 1, 2,… , N , (3.45) ε r ( 0) = 1. (3.46) The above TM mode characteristic equation becomes ∆ o2 − x 2 − x ( ) tan x − jA(' N −1) = 0, (3.47) ( ) (3.48) ( ) (3.49) y( i ) = h(i ) x 2 − ∆ (2i ) h( N ) , ∀i = 1, 2,… , N . (3.50) ε r( N ) where ( ) tanh A(' N −1) = y( N −1) ε r ( N ) h( N ) y( N ) ε r ( N −1) h( N −1) ( ) tanh A(' N − 2) + jy( N −1) , tanh A('1) = y(1) ε r ( 2) h( 2) y( 2) ε r (1) h(1) tanh jy(1) , In here, the proposed functional expressions necessary for the efficient extraction of the surface wave poles are given as 60 ( ) ) ⎧ if p = ( M −1) , ( M −1) π bM and m = 1,2 ...M -1 , (3.61) if ∆o > bM and m = M where m = 1, 2,… , M , M is the number of DTM or DTE roots, am and bm denote respectively x ε r( N ) ( the location of the ) zeros and singularities ( of either the ) tan x − jA(' N −1) curve in equation (3.47) or x cot x − jB(' N −1) curve in equation (14). Fig. 9 shows how the initial guess, xm , in equation (3.61) is obtained from the projection of the point Cm onto the horizontal axis. The angle, θ m , where m = 1, 2,… , M in Fig. 9 is easily evaluated from the respective right-angle triangle in each quadrant. Using the property of right-angled triangle of each quadrant, the initial guess, xm , can thus be evaluated through simple trigonometry function as in Fig. 9. The number of TM surface wave modes can be determined by using M TM < 1 + ∆ˆ TM π whereas for the number of TE surface wave modes can be determined from M TE < 0.5 + ∆ˆ TE π . 63 ⎧ ⎛ ⎪ tan −1 ⎜ ⎜ ⎪⎪ ⎝ θm = ⎨ ⎪ −1 ⎛ ⎪ tan ⎜ ⎜ ⎪⎩ ⎝ ∆ o2 − am2 ⎞ ⎟ else where bm − am ⎟ ⎠ ∆ o2 − am2 ⎞ ⎟ for the last root and ∆ o > bM ∆ o − am ⎟ ⎠ ∆ o2 − a12 ∆ o2 − a22 ∆ o2 − a32 θ1 θ2 C1 b1 a1 θ3 C2 x1 Fig. 9 C3 b2 a2 x2 b3 a3 x3 ∆o Initial guess evaluation. 64 3.3.2 Extraction of Leaky Wave Poles for N-layered Microstrip Geometry The extraction of the leaky-wave poles is given in the following section. (i) Functional Expression Derivation For the TM leaky-wave mode, the characteristic equation becomes − ∆ o2 − x 2 − ( x ) tan x − jA(' N −1) = 0, ε r( N ) (3.62) Similarly, for the TE leaky-wave mode, the characteristic equation for extracting the poles becomes ( ) − ∆ o2 − x 2 + x cot x − jB(' N −1) = 0, (3.63) The required functional expression for the TM leaky-wave mode can be expressed as ( ( ∆ o2 − x 2 1 + tan ( x ) tan jA(' N −1) )) + ε x ( tan ( x ) − tan ( jA( ) )) = 0, ( ) ' N −1 (3.64) r N whereas the required functional expression for the TE leaky-wave mode is expressed as ( ( − ∆ o2 − x 2 tan ( x ) − tan jB(' N −1) (ii) )) + x (1 + tan ( x ) tan ( jB( ) )) = 0. ' N −1 (3.65) Initial Guess The locations of the zeros and infinites of x ε r( N ) ( ) ( tan x − jA(' N −1) and x cot x − jB(' N −1) ) of equations (3.62) and (3.63) respectively are obtained by using the same method as described in Section 3.3.1. From Fig. 10, we notice that as P reaches a value Po , such that the semicircle in Fig. 10 is tangent to the “tan” function, the two leaky wave poles merge to a single pole. Thus, special check is necessary to ensure the correct number 65 of poles is extracted. The initial guess for this case is obtained from the following steps: Step 1: Assume two leaky wave poles in the last quadrant, where xleft is on the left side of Po and xright is on the right side of Po . Step 2: Use similar method from previous section to obtain the initial guess for all the roots on the left side of Po , x1 , x2 , xM −1and xleft . Step 3: The initial guess for the last root is obtained using xright = xleft + bM − aM . Step 4: Using the derived initial guess, we apply the well-known Halley’s method [127] to the functional expressions. As shown from Fig. 10, there are three possibilities in the last quadrant of tangent curve. They are, namely, no root, 1 root and 2 roots. z If xleft ≠ xright and the function variation df = 1 − f n +1 < 10− 9 , the last fn quadrant has two roots. z If xleft = xright and the function variation df = 1 − f n +1 < 10− 9 , the last fn quadrant has one root. z If the algorithm cannot find any root within 100 iterations, this implies that there is no root in the last quadrant. 66 Po (a) Po (b) Fig. 10 Graphic solution for TM and TE surface and leaky wave modes, where the x semicircles represent ± B 2 − x 2 . (a) TM: ± B 2 − x 2 = tan ( x ) . (b) TE: εr ± B 2 − x 2 = − x cot ( x ) 67 3.3.3 Extraction of Lossy Improper Poles for N-layered Microstrip Geometry For lossy substrate, we have ε r = ε rr + iε ri . The extraction of the lossy improper poles is described in the following section. (i) Functional Expression Derivation The required functional expression for the lossy improper TM mode is given as ( ( ∆ o2 − x 2 1 + tan ( x ) tan jA(' N −1) )) − ε x ( tan ( x ) − tan ( jA( ) )) = 0, ( ) ' N −1 (3.66) r N whereas the required functional expression for the lossy improper TE mode is expressed as ( ( ∆ o2 − x 2 tan ( x ) − tan jB(' N −1) (ii) )) + x (1 + tan ( x ) tan ( jB( ) )) = 0. ' N −1 (3.67) Initial Guess The initial guess for this case is obtained from the following steps: Step 1: Assume initially that the substrate is lossless and let ε r = ε rr and ε ri = 0. Step 2: Use the exact same method as outline in section 3.3.1 to obtain the initial guess of xini _ lossy . Step 3: Using the initial guess xini _ lossy , we apply the well-known Halley’s method [127] to the functional expressions to extract the roots. 3.3.4 Extraction Algorithm 68 With all the functional expressions and initial guesses for the surface wave modes, leaky wave modes and lossy improper modes been derived, we next apply the wellknown Halley’s method [127] for the pole extraction. The Halley’s method is given as x n +1 = x n − 2 f ( x n ) f ′( x n ) 2 f ′( x n ) − f ( x n ) f ′′( x n ) 2 . (3.68) The single prime in equation (3.68) denotes the first derivative and the double prime indicates the second derivative. 69 3.3.5 Numerical Results and Discussions Based on the above algorithm, a Matlab program running on Pentium 4 platform has been written to extract all the various types of poles. (i) Surface Wave Poles The selected permittivity for the various microstrip configurations were selected as ε r1 = 5.9, ε r 2 = 10.9, ε r 3 = 12.8, and ε r 4 = 17.3. The adopted dielectric thickness are h1ko = 0.4, h2 ko = 0.4, h3 ko = 0.4, and h4 ko = 0.4 . Our algorithm with the new functional expressions is compared with the Davidenko’s method [126] with the original characteristic equations. Fig. 11 (a) and Fig. 11 (b) show the numerical comparisons of the two methods in terms of the residue of the function and the number of iterations for the evaluation of the first and second TM root of the four-layered microstrip geometry under the case ε r1 < ε r 2 < ε r 3 < ε r 4 and k o ≤ k ρ ≤ k o ε r1 . With the exception of the functional expression, the results obtained in Fig. 4 are subject to the same constraints, namely, (a) the function variation, df = 1 − step size variation, dx = 1 − f n +1 < 10− 9 , (b) the fn xn +1 < 10− 9 , and (c) the same initial guesses are used. As xn shown in Fig. 11 (a) and Fig. 11 (b), our proposed algorithm achieves better results in terms of accuracy and number of iterations compared to the conventional functional expression. On average, an accuracy of 10 −16 after an average of 6 iterations is achieved for the determination of all of roots under our proposed new method. As 70 shown from the figure, the conventional method can not converge sometimes which leads to failure of finding the actual roots of the Greens’ function. (a) Fig. 11 (b) A numerical comparisons of the two methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the four-layered microstrip geometry under the case of ε r1 < ε r 2 < ε r 3 < ε r 4 and k o ≤ k ρ ≤ k o ε r1 . 71 (ii) Leaky Wave Poles To validate our method for the leaky-wave pole extraction, a three-layered structure is first selected for comparison. The respective comparison results of using Halley’s method with the new functional expressions for leaky wave poles extraction and Davidenko’s method with classical equations [126] are shown in Table 4. These two methods are using same initial guess and the conventional function expression fails to converge as shown in the shadow area. Halley’s method with Eqn. Davidenko's method with Eqn. (3.64) (3.62) No. of Iterations 6 100 1st root 1.583226666 -1.424774824 df 10 −32 -0.948361489 dx 10 −32 481.9477339 2nd root 3.575614406 3.575614406 df 10 −32 10 −32 dx 10 −32 10 −32 Leaky Poles Table 4 Comparison of the proposed approach and the Davidenko’s method [126] for leaky-wave poles extraction in TM mode. The parameters adopted are ε r1 = 15 , ε r 2 = 59 , ε r 3 = 37.5 , h1ko = 0.8 , h2 ko = 0.9 , h3ko = 0.95 , and ε r1 ko < k ρ < ε r 3 ko . As noted from the tables, our proposed method truly outperforms the Davidenko’s method with classical equations [126] for all cases in terms of accuracy and the number of iterations required. In addition, it is noted that for some cases in Table 4 72 (indicated by the shaded region), the Davidenko’s method, at times, fails to converge to the proper roots. On average, an accuracy of 10 −32 is obtained for our newly proposed method. (iii) Lossy Improper Poles To validate the proposed approach for the lossy improper pole extraction, a threelayered structure is first selected. Table 5 presents the comparison results between the proposed method and Davidenko’s method [126]. To clearly illustrate the advantages of our proposed method, the values of dielectric permittivities are selected in a random order. To test the algorithm robustness, we have selected a very highly lossy dielectric of ε r1 = 40 + i 25 , ε r 2 = 75 + i 25 , ε r 3 = 40 + i 25 , ε r 4 = 80 + i 25 , h1ko = 0.4 , h2 ko = 0.45 , h3 ko = 0.4 , and h4 ko = 0.45 for our comparison and Table 6 gives the comparison results. As noted from the tables, our proposed method can indeed extract the roots accurately and within a minimum number of 14 iterations. It is noted that the Davidenko’s method using classical function expression can at times give wrong results (as indicated by the shaded box). In this Chapter, a unified approach for the fast evaluation of the locations of the surface wave poles, leaky wave poles and lossy improper poles in multilayered microstrip technology has been presented for the first time. Using a simple reorganization of the classical characteristic equations, together with the Halley’s method, a third order convergence is observed for the multilayered microstrip pole extraction. For the first time, reliable initial values have also been proposed. 73 Halley’s method with Eqn. Davidenko's method with (3.66) Eqn.(3.47) 14 100 1.01182227081261 - 1.23730715979912 + 0.00043924163301i 0.31064812216076i df 10 −32 -3.23468912 dx 10 −32 5.752047286 2.97282693490741 + 2.97282693490741 + 0.01579544532283i 0.01579544532283i df 10 −32 10 −32 dx 10 −32 10 −32 6.38008671592520 + 6.13059487401265 + 0.00807064377279i 0.01884096006964i df 10 −32 2.309324782 dx 10 −32 2.623262689 Lossy poles No. of Iterations 1st root nd 2 root rd 3 root Table 5 Comparison of the proposed approach and the Davidenko’s method [126] for lossy improper poles extraction in TM mode. The parameters adopted ε r1 = 35.5 + i0.1 , ε r 2 = 37.5 + i0.1 , ε r 3 = 59 + i0.1 , h1ko = 0.4 , are h2 ko = 0.4 , and h3 ko = 0.5 . 74 Halley’s method with Eqn. Davidenko's method with (3.66) Eqn. .(3.47) 14 100 2.46919934276706 - 2.46919934276706 - 0.07871013515017i 0.07871013515017i df 10 −32 10 −32 dx 10 −32 10 −32 3.60204381679136 + 3.60204381679136 + 0.02588984853330i 0.02588984853330i df 10 −32 -0.04405944499115 dx 10 −32 10 −32 4.96220186733286 + 4.96220186733286 + 0.04776474037665i 0.04776474037665i df 10 −32 10 −32 dx 10 −32 10 −32 Lossy Poles No. of Iterations st 1 root nd 2 root 3rd root Table 6 Comparison of the proposed approach and the Davidenko’s method [126] lossy improper poles extraction in TM mode. The parameters adopted are ε r1 = 40 + i 25 , ε r 2 = 75 + i 25 , ε r 3 = 40 + i 25 , ε r 4 = 80 + i 25 , h1ko = 0.4 , h2 ko = 0.45 , h3 ko = 0.4 , and h4 ko = 0.45 . 75 Chapter 4 Bandpass Filters Miniaturization 4.1 Introduction 4.1.1 Method of Moment with developed DCIM algorithm With the location of the poles of the Green’s function obtained in Chapter 3 and based on the dyadic Green’s function in spectral domain derived in Chapter 2, a Matlab program was written to analyze the planar structures using the Method of Moments. We can evaluate the Green’s function accurately using DCIM with the location of the poles. The Green’s function is then applied to MoM to analyze planar structures such as antennas and filters. In this section, a brief introduction of MoM is given and a simple microstrip bandpass filter is designed, fabricated and measured for comparison. The precise location of the poles’ of the Green’s function was used in the DCIM method for accurate evaluating the Green’s function in spatial domain. The DCIM divides the Green’s function into three main parts [34]. They are extraction of the quasi-static terms, contributions of the surface wave poles (SWP) which are obtained in the previous sections, and the remaining portion. The quasi-static contribution is inverse transformed analytically through the Sommerfeld identity. The surface wave contribution comes from the poles in the spectral Green’s functions. The surface wave poles are extracted first, subsequently the contribution of surface wave is evaluated through the residue calculus. The remaining portion in the spectral representation is expanded into a series of exponentials using either the Prony or GPOF methods. 76 With the appropriate Green’s function and boundary conditions, the surface current of a specified problem can be calculated. With an appropriate integral equation, we need to expand the unknown function in terms of known basis functions with unknown coefficients. The basis function for MoM in this thesis is chosen to be RWG (RaoWilton-Glisson) basis function. The surface of the metal filter is divided into separate triangles. Each pair of triangles, having a common edge, constitutes the corresponding RWG edge element as shown in Fig. 12. One of the triangles has a plus sign and the other a minus sign. A vector function (or basis function) ⎧( l / 2 A+ ) ρ + ( r ) , ⎪ − ⎪ f ( r ) = ⎨( l / 2 A− ) ρ ( r ) , ⎪ ⎪0, ⎩ r in T + r in T − (4.1) otherwise is assigned to the edge element. Here l is the edge length and A± is the area of ± + triangleT ± . Vectors ρ are shown in Fig. 12. Vector ρ connects the free vertex of the − plus triangle to the observation point r . Vector ρ connects observation point to the free vertex of the minus triangle. ρ l − T− d T+ ρ + + C rC r − r O Fig. 12 Schematic of a RWG edge element and the dipole interpretation 77 The surface electric current on the metal surface (a vector) is a sum of the contributions of eqn. (4.1) over all edge elements, with unknown coefficients. These coefficients are found from the moment equations. The moment equations are a linear system of equations with the impedance matrix Z . There are a number of methods for the calculation of the impedance matrix [59], [128]-[131]. These methods employ different ways of integrating over surface triangles. Analytical approaches (line integrals and potential integrals) are accurate and fast but require extensive preliminary mathematical work. An alternative is to use a numerical integration over a triangle [131]. If the quadrature points do not coincide with the triangle’s midpoint, no separate calculations for the diagonal elements of the impedance matrix are necessary. All elements of the impedance matrix can be calculated straightforwardly, using the same formula. Fig. 13 Barycentric subdivision of the primary triangle. The triangle’s midpoint is shown by a white circle. Fig. 13 shows the so-called barycentric subdivision of an arbitrary triangle [132]. Any primary triangle can be divided into 9 equally small subtriangles by the use of the 78 “one-third” rule. Further we assume that the integrand is constant within each small triangle. Then the integral of a function g over the primary triangle Tm is equal to ∫ g ( r )dS = Tm Am 9 9 ∑ g (r ) (4.2) c k k =1 where m represents m -th primary triangle and points rkc , k = 1,...9 are the midpoints of nine subtriangles shown in Fig. 13 by black circle. Am is the area of the primary triangle. The impedance matrix determines electromagnetic interaction between different edge elements. If the edge elements m and n are treated as small but finite electric dipoles, the matrix element Z mn describes the contribution of dipole n (through the radiated field) to the electric current of dipole m , and vice versa. The size of the impedance matrix is equal to the number of edge elements. The impedance matrix of the electric field integral equation is given by ( ) + − + Z mn = lm ⎡ jω A mn ⋅ ρ mC / 2 + A −mn ⋅ ρ mC / 2 + Φ −mn − Φ +mn ⎤ ⎣ ⎦ (4.3) where index m and n correspond to two edge elements, the term with the bracket ( ⋅) C± denotes the dot product. lm is the edge length of element m . ρ m are vectors between ± the free vertex point, vm± , and the centroid point rmC , of the two triangles Tm± of the C+ edge element m , respectively. ρ m C− whereas ρ m is directed away from the vertex of triangle Tm+ , is directed toward the vertex of triangle Tm− . The expressions for vector A and the scalar Φ can be found in [59]. 79 The surface current density on a surface S of the plate or on other perfectly electrically conducting (PEC) structures is given by an expansion into RWG basis functions over M edge elements [59]. The current coefficients I m can be obtained by solving the impedance equation Z ⋅I =V (4.4) where V is a voltage excitation vector. To demonstrate the accuracy and reliability of our DCIM algorithm with the new pole extraction method in the previous chapter, a bandpass filter using the Stepped Impedance Resonator (SIR) was designed and fabricated, which is shown in Fig. 14. The filter is fabricated on a substrate with dielectric permittivity of ε r = 10 and thickness of 62 mils. As shown in Fig. 15, the simulated and measured centre frequency of the filter obtained is around 2.65 GHz. Relatively good agreement is observed between the results obtained using IE3D and the MoM alrogithm. The discrepancy is attributed to the different ways of approximating the Green’s function, the meshing of the filter structure, the zero metal thickness assumption in the MoM code. In general, the limitation of the simulation software may lead to discrepancy between the simulated and the measured results. Both our MoM code and the IE3D assume infinite ground and substrate. The MoM code, on the other hand, assumes zero metal thickness and lossless substrate. The variation of the substrate, i.e. permittivity and thickness, can cause the difference between the actual results and the expected data. There also exist fabrication tolerance and measurement errors which cause the discrepancy between simulation and measurement, and these include the variation of the width and the length of the microstrip lines, the soldering between SMA 80 connectors and the circuit, the measurement calibration while using the network analyzer. Fig. 14 Photograph of the bandpass filter with SIR (a) Comparison of measured and simulated S11 using MoM and IE3D 81 (b) Comparison of measured and simulated S21 using MoM and IE3D Fig. 15 Comparison of measured and simulated results 4.1.2 Bandpass filters Miniaturization With the developed MoM algorithm introduced in the previous section, different types of miniaturized microstrip bandpass filters were designed in this thesis. Nowadays, microwave filters are widely used in telecommunication systems such as the cellular base-station transceiver, code-division multiple access (CDMA), the personal communication system (PCS), and Bluetooth. A lot of research has been conducted on designing small-size, compact, low-loss bandpass filters. Planar filters are currently a popular structure because they can be fabricated using printed circuit technology and are suitable for commercial applications due to their small size and lower fabrication cost. In section 4.2, some basic theory for filter design is introduced. Following this in section 4.3, we focus on novel bandpass filter designs for microstrip structures 82 fabricated using printed circuit board (PCB) technology. These filter structures are simulated and analyzed using MoM introduced in Chapter 3. We commence with the simple bandpass filter using the Stepped Impedance Resonator (SIR). This filter is miniaturized using the photonic bandgap (PBG) structure. Secondly, as the size reduction using PBG structure is not satisfactory as compared to the original SIR design, two novel microstrip dual mode filters, with the basic configuration of a square open-loop but with two added gaps at the off-diagonal corners, are proposed. The proposed filters are much smaller in terms of corresponding wavelength compared to the traditional cross-coupled open-loop resonators. Thirdly, to further reduce the size of the microstrip filters, we propose an alternate way of producing a size reduction of an open-loop squared filter with wide frequency perturbation. In Section 4.4, we investigate some advanced fabrication technologies, such as wafer transfer technology (WTT) and Low-temperature Cofired Ceramic (LTCC) technology. Instead of exploring filter miniaturization for microstrip structures, we will look into size reduction for coplanar structures. The simulation tool in this section is IE3D, since the MoM code is more suitable for planar microstrip structure. A novel and compact coplanar bandpass filter using two resonators is designed and fabricated using the WTT for millimeter wave applications. To further reduce the filter size in terms of wavelength, we propose an alternative way of filter miniaturization using a modified hairpin-combline resonator, which leads to the effective circuit dimension to be less than λ /17 in terms of wavelength. 83 4.2 Basic Structures for Filter Design 4.2.1 Resonator A resonator can be defined as the structure which contains at least one oscillating electromagnetic field. There are many forms of resonators, such as lumped-element or quasi-lumped element resonators, distributed line resonators or patch resonators. Microstrip, stripline or coplanar waveguide structure can be used for the design of resonators. In this section, some basic types of resonators are given. For simplicity, we will focus on the microstrip resonators. Fig. 16 shows the distributed line resonators. Fig. 16 (a) and (b) show the quarterwavelength resonators, whereas Fig. 16 (c) depicts the half-wavelength resonator. Quarter-wavelength resonators can also resonate at other higher frequencies when f ≈ ( 2n − 1) f o for n = 2,3 ⋅ ⋅⋅ . Half-wavelength resonators can also resonate at f ≈ nf o for n = 2, 3 ⋅⋅⋅ . These kind of resonators can be shaped into many different configurations for filter implementations, such as the open-loop resonator [87]. λgo λgo 4 4 λgo 2 (a) (b) Fig. 16 (c) Distributed line resonators 84 Another type of distributed line resonator is the ring resonator [133] shown in Fig. 17. Its fundamental frequency f o can be obtained using 2π r ≈ λgo , where r is the median radius of the ring and λgo is the guided wavelength. The higher resonant modes occur at f ≈ nf o for n = 2, 3 ⋅⋅⋅ . A resonance can occur in either of two orthogonal coordinates due to its symmetrical geometry. It can support a pair of degenerate modes that have the same resonant frequencies but orthogonal field distributions. This feature can be utilized to design dual-mode filters. Similarly, it is possible to construct this type of line resonator using different configurations, such as square and meander loops [134]-[135]. r λgo ≈ 2π r Fig. 17 Ring resonator Patch resonators can also be used for the design of filters. Their advantages include lower conductor losses compared to narrow microstrip line resonators and increased power handling capability [136]–[137]. Their disadvantage is their strong radiation, which can be minimized with enclosed metal housing. Patch resonators usually have a larger size compared to others kinds of resonators. The resonant patches may have different shapes, such as circular in Fig. 18(a) and triangular in Fig. 18(b). These microstrip patch resonators can be analyzed as waveguide cavities with magnetic walls on the sides. 85 (a) (b) Fig. 18 Patch resonator 4.2.2 Basic Types of Microstrip Bandpass Filters Conventional microstrip bandpass filters include stepped-impedance filters, open-stub filters, semi-lumped element filters, end- and parallel-coupled half-wavelength or quarter-wavelength resonator filters, hairpin-line filters, interdigital and combline filters, pseudocombline filters, and stub-line filters. They are widely used in many RF/microwave applications. In this section, the designs of some of these filters will be briefly introduced. 4.2.2.1 End-coupled, Half-wavelength Resonator Filters An example of end-coupled microstrip bandpass filter is shown in Fig. 19. The length of each open-ended microstrip resonator is approximately half-wavelength, λgo / 2 , at the centre frequency f o of the bandpass filter. The capacitive coupling between the two resonators is through the gap between the two adjacent open ends. The gap can be represented by inverters. These J-inverters tend to reflect high impedance level to the end of each resonator, which causes the resonators to exhibit a shunt-type resonance [138]. 86 l1 l2 Z o ,θ2 Z o , θ1 Fig. 19 ln Z o ,θn General configuration of end-coupled microstrip bandpass filter 4.2.2.2 Parallel-coupled, Half-wavelength Resonator Filters Fig. 20 shows the general structure of parallel-coupled or edge-coupled microstrip bandpass filters using half-wavelength resonators. The adjacent resonators are parallel to each other along half of their length, which gives larger coupling for a given spacing between resonators compared to end-coupled resonators. This structure can be used for designing bandpass filters with wider bandwidth as compared to the endcoupled microstrip filters. l1 l2 l3 l4 Z1 Z2 Z3 Fig. 20 Z4 ln −1 ln Z n −1 Zn General structure of parallel (edge)-coupled microstrip bandpass filter 4.2.2.3 Hairpin-line Bandpass Filters Hairpin-line bandpass filters can be obtained by folding the resonators of parallelcoupled, half-wavelength resonator filters into a “U” shape. The “U” shape resonator 87 is the so-called hairpin resonator. When folding the resonators, it is necessary to take into account the reduction of the coupled-line lengths, which reduces the coupling between resonators. If the two arms of each hairpin resonator are close to each other, they function as a pair of coupled line, which will affect the coupling as well. Fig. 21 Layout of a hairpin-line microstrip bandpass filter 4.2.2.4 Interdigital Bandpass Filters A type of interdigital bandpass filter is shown in Fig. 22, which consists of an array of n TEM-mode or quasi-TEM-mode transmission line resonators, each of which has an electrical length of 90° at the centre frequency and is short-circuited at one end and open-circuited at the other end with alternative orientation. Generally the physical dimensions of the resonators can be different in lengths and widths. Coupling is achieved through the fields fringing between adjacent resonators. This kind of filter requires the grounding microstrip resonators, which can be accomplised using via holes. Since the resonators are quarter-wavelength long, the second passband of the filter is centered at about three times the centre frequency f o . 88 Fig. 22 General configuration of interdigital bandpass filter 4.2.2.5 Stepped Impedance Resonator (SIR) Filters Filters with uniform impedance resonators (UIRs) [91] suffer from poor harmonic suppression. To alleviate the problem, the stepped impedance resonator (SIR) was developed to solve the problem. Stepped impedance resonators (SIR) are composed of transmission lines with different characteristic impedances. They provide an effective way to minimize circuit space and push spurious resonant frequencies away from the passband [62]. SIRs also provide a wide degree of freedom in structure and design and a wide range of applicable frequency through the use of various types of transmission lines (coaxial, stripline, microstrip, coplanar). The resonant frequencies of SIRs can be easily altered by tuning its geometric parameters. 4.2.2.6 Dual Mode Resonator Filters Dual-mode resonators [90]-[92] have been widely used to realize many RF/microwave fitlers. A main feature and advantage of this type of resonator lies in the fact that each of dual-mode resonators can used as a doubly tuned resonant circuit, and therefore the 89 number of resoantors required for a n-degree filter is reduced by half, resulting in a compact filter configuration. For example, the field distributions of the two modes of a microstrip square patch resonator are orthogonal to each other. In order to couple them, some perturbantion to the symmetry of the caviry is needed, and the two coupled degenerate modes function as two coupled resonators. A microstrip dual-mode resonator is not necessarily square in shape, but usually has a two-dimentional symmetry. A small perturbation can be applied to each dual mode resonator at a location that is assumed at a 45o offset from its two orthogonal modes. For example, a small notch or a small cut is used to disturb the disk and square patch resonators, while a small patch is added to the ring, square loop, and meander loop resonantors, respectively. 4.2.2.7 Other Types of Bandpass Filters There are some other types of bandpass filters, such as combline filters and pseudocombline filters, which comprises of an array of coupled resonators. Stub bandpass filters include λg 0 / 4 short-circuited stubs or λg 0 / 4 open-circuited stubs. A lot of research has been conducted on designing compact, low-loss and good performance bandpass filters. In the next two sections, we will introduce several novel compact bandpass filters for microstrip structures and coplanar waveguide structures respectively. 90 4.3 Bandpass Filter Miniaturization for Microstrip Structures Among the various planar filter configurations reported in the literature, the microstrip open-loop ring resonator [135], [139]-[141] has attracted much attention due to its capability of generating elliptic or quasi-elliptic narrow resonances at specific frequencies. In these designs, narrowband filter performance is always realized through cross-coupled effect, and usually, more than one square open-loop ring resonator is required. With these cross-coupled open-loop resonators, the required estate area is increased. In this section, we will focus on miniaturization of microstrip bandpass filter using open-loop ring resonators. Firstly, we will use PBG structures to reduce the circuit size with three open-loop ring resonators. Then, several novel microstrip bandpass filters are introduced for further size reduction. All the microstrip structures were analyzed using the MoM algorithm introduced above. 4.3.1 Bandpass Filter Miniaturization Using PBG Filter miniaturization is important in many applications. The concepts of the Stepped Impedance Resonator (SIR) [62] and Photonic Band-Gap structures (PBG) have the capability of shortening resonator length, thus reducing the overall circuit area without affecting the circuit’s performance. In general, PBG structures are specific periodic structures artificially created in materials such as metals or substrates to influence or even change the electromagnetic properties of materials. For example, certain bands of frequencies cannot propagate [142]. Novel PBG structures have shown great potential applications in antennas, 91 filters, and other devices [143]–[159]. Application of PBG to microwave circuits has many advantages, one of which is the slow wave property that makes it possible to miniaturize the circuit size and reduce the circuit cost [160]. PBG structures can be realized by using metallic, dielectric, ferromagnetic, or ferroelectric implants. Dielectric PBG structures for microstrip circuits [147]–[148] require a periodic pattern through the substrate of microstrip. Many types of such PBG structures may be created, such as square lattice with square hole, triangular lattice with square hole, and honeycomb lattice with circular or square hole [151]. These structures possess distinct stopbands for quasi-TEM wave propagation in microstrip lines. An alternative approach to implement PBG structures for microstrip-based applications is to introduce some periodic patterns on the ground plane of the microstrip. This led to the development of the so-called uniplanar PBG [151]. Numerous periodic patterns can easily be created by printed-circuit techniques such as etching. In this case, no drilling through the substrate is required. The fabrication process is greatly simplified compared to the dielectric PBG. This type of PBG structure can generally achieve larger stopbands than the dielectric PBG structure based on drilling holes on the substrate [144]. The disadvantage of this PBG structure is that the etched ground plane must be far enough from any metal plate in order to keep etched patterns in function. This may cause packaging problem and realization of MMICs [145]. Unlike having a PBG structure in the ground, a bandpass filter with periodic microstrip signal lines does not suffer from such packaging problems. In [145] and 92 [146], microstrip bandpass filters with sinusoidal variation of characteristic impedance are introduced. In [152], a meandered pattern of PBG on microstrip line is introduced for harmonic suppression. In this section, a microstrip PBG structure, shaped like “王”, has been utilized in a conventional bandpass filter using open-loop stepped-impedance resonators to realize circuit miniaturization without etching the ground plane. 4.3.1.1 Design Structure of a SIR Bandpass Filter We start with a simple three-stage bandpass filter introduced in P.140 of [62]. As stated in [62], this BPF using folded line SIR shown in Fig. 23 is not suitable for a wide-band application due to structural restrictions making it difficult to obtain a large I/O coupling, a narrow-band filter structure with a relative bandwidth of several percent can be achieved, providing advantages of miniaturization. 1.183 mm y 0.3 mm Port 1 x Port 2 3.2852 mm Fig. 23 6.59 mm 9.722 mm Bandpass filter with SIR The SIR resonator shown in Fig. 23 can be obtained by folding a basic SIR as shown in Fig. 24(a). Z 2 ,θ 2 Z1 , 2θ1 Z 2 ,θ 2 (a) Basic SIR 93 (b) An equivalent circuit of the basic SIR (c) Folded-line SIR Fig. 24 Structural variation of λg / 2 type SIRs. To analyze the basic structure in Fig. 24(a), the input admittance Yi seen from an open-end must be obtained, Yi = jY2 2 ( Rz tan (θ1 ) + tan (θ 2 ) ) ( Rz − tan (θ1 ) tan (θ 2 ) ) Rz (1 − tan 2 (θ1 ) ) (1 − tan 2 (θ 2 ) ) − 2 (1 − Rz2 ) tan (θ1 ) tan (θ 2 ) . (4.5) Resonance conditions are obtained by taking Yi = 0 , thus giving Rz = Z2 = tan (θ1 ) tan (θ 2 ) . Z1 (4.6) The design procedure for this three-stage BPF shown in Fig. 23 is given below: 1. A microstrip bandpass filter is designed to meet the specifications: Center frequency 2.65 GHz Passband width 5-10% Passband return loss 130 MPa Material properties for LTCC used in this section 4.4.2.3 Preliminary Analysis and Design As shown in [108]-[109], compact bandpass filters with 6 to 8 metal layers can be realized using LTCC technology. Although the circuit area is considered to be very compact in terms of wavelength, the volume of the circuit is still quite big. In this section, two novel CPW bandpass filters using a modified miniaturized hairpincombline resonator, which have compact circuit area and volume are presented. One of them is a single-layered CPW bandpass filter and the other is two-layered CPW bandpass filter. The design topology is suitable for any available fabrication technologies, such as PCB, LTCC or WTT. However, in order to increase the quality of the microstrip lines, the proposed designs are constructed using LTCC technology. Before looking at the proposed novel miniaturized two-layered coplanar bandpass filter, two preliminary structures are first investigated. The basic layout of our preliminary integration of the hairpin and combline interdigital filter is illustrated in Fig. 68. As shown in Fig. 68, the input coplanar feed lines are proximity coupled to the 157 hairpin-cum-combline resonator. The hairpin-cum-combline resonator is shorted to the coplanar waveguide side walls. The proposed LTCC CPW filter is simulated with a six-layered Ferro A6 LTCC substrate of permittivity 5.9. Each layered has a thickness of 0.1905 mm. The proposed preliminary coplanar resonator is designed using IE3D and the center frequency is 2.5 GHz. The simulated response of the proposed preliminary structure is illustrated in Fig. 69. As noted from Fig. 69, the proposed quasi-elliptic filter has two poles, which is clearly visible at 2.5 GHz and 2.75 GHz and two transmission zeros (attenuation poles), which is located at 2.25 GHz and 2.8 GHz. The quasi-elliptic filter also has a good harmonic suppression at 5 GHz. Fig. 68 Preliminary structure I. All the other lines and space is 0.1 mm 158 Fig. 69 Fig. 70 Simulated (IE3D) results for preliminary structure I for Fig. 68 Approximated equivalent circuit for preliminary structure for Fig. 68 159 The modified hairpin-combline resonator used in Fig. 68 can be approximated and analyzed using the model of three coupled lines with the same characteristic impedance, as shown in Fig. 70. This structure can be analyzed using a triple coupled line formulation derived in section 4.3.3, which is shown below: ⎡i1 ⎤ ⎡Y11 ⎢ ⎥ ⎢ ⎢i2 ⎥ ⎢Y21 ⎢i3 ⎥ ⎢Y31 ⎢ ⎥=⎢ ⎢i4 ⎥ ⎢Y41 ⎢i ⎥ ⎢Y ⎢ 5 ⎥ ⎢ 51 ⎣⎢i6 ⎦⎥ ⎣⎢Y61 Y12 Y13 Y14 Y15 Y22 Y23 Y24 Y25 Y32 Y33 Y34 Y35 Y42 Y43 Y44 Y45 Y52 Y53 Y54 Y55 Y62 Y63 Y64 Y65 ⎡Y11 Y12 ⎢ ⎢Y12 Y22 ⎢Y Y = ⎢ 13 12 ⎢Y14 Y15 ⎢Y Y ⎢ 15 14 ⎢⎣Y16 Y15 Y13 Y14 Y15 Y12 Y15 Y11 Y16 Y14 Y15 Y16 Y11 Y12 Y15 Y12 Y22 Y14 Y12 Y13 Y16 ⎤ ⎡V1 ⎤ ⎥⎢ ⎥ Y26 ⎥ ⎢V2 ⎥ Y36 ⎥ ⎢V3 ⎥ ⎥⎢ ⎥ Y46 ⎥ ⎢V4 ⎥ Y56 ⎥ ⎢V5 ⎥ ⎥⎢ ⎥ Y66 ⎦⎥ ⎣⎢V6 ⎦⎥ Y16 ⎤ ⎡V1 ⎤ ⎥⎢ ⎥ Y15 ⎥ ⎢V2 ⎥ Y14 ⎥ ⎢V3 ⎥ ⎥ ⎢ ⎥, Y13 ⎥ ⎢V4 ⎥ Y12 ⎥ ⎢V5 ⎥ ⎥⎢ ⎥ Y11 ⎥⎦ ⎢⎣V6 ⎥⎦ (4.101) where Y11 = 1 jω L1 + jω C1m + C2 m + C1 p , ( ) (4.102) Y12 = − jωC1m − M 1 ( jω L ) , (4.103) Y13 = − jωC2 m − M 2 ( jω L ) , (4.104) 2 1 2 1 Y14 = −1 ( jω L1 ) , (4.105) Y15 = M 1 ( jω L ) , (4.106) Y16 = M 2 ( jω L ) , (4.107) 2 1 2 1 ( ) Y22 = 1 ( jω L1 ) + jω C1m + 2C1 p , (4.108) im and Vm are respectively the terminal currents and voltages for the network inside 160 box with dashed lines. The L and C in equations (4.102)-(4.108) are shown in Fig. 46 in section 4.3.3. Firstly, we look at the box on the left-hand side as shown in Fig. 70, which consists port 1 to port 6. Since port 1 and port 2 are connected and port 3 is connected to the ground, we have V3 = 0,V1 = V2 and i1 = −i2 . Thus, Y11V1 + Y12V1 + Y14V4 + Y15V5 + Y16V6 = − (Y21V1 + Y22V1 + Y24V4 + Y25V5 + Y26V6 ) , (4.109) → V1 (Y11 + Y12 + Y21 + Y22 ) = − (Y14 + Y24 ) V4 − (Y15 + Y25 ) V5 − (Y16 + Y26 ) V6 , (4.110) → V1 = 1 ⎡ − (Y + Y ) V − (Y + Y ) V − (Y + Y ) V ⎤ . (4.111) (Y11 + Y12 + Y21 + Y22 ) ⎣ 14 24 4 15 25 5 16 26 6 ⎦ Let ∆ = Y11 + Y12 + Y21 + Y22 , then ⎛ ⎞ (Y41 + Y42 ) (Y + Y ) (Y + Y ) (Y14 + Y24 ) Y45 − 41 42 (Y15 + Y25 ) Y46 − 41 42 (Y16 + Y26 ) ⎟ ⎜ Y44 − ∆ ∆ ∆ ⎟ ⎛V ⎞ ⎛ i4 ⎞ ⎜ ⎜ ⎟⎜ 4 ⎟ Y51 + Y52 ) Y51 + Y52 ) Y51 + Y52 ) ( ( ( ⎜ ⎟ (Y14 + Y24 ) Y55 − (Y15 + Y25 ) Y56 − (Y16 + Y26 ) ⎟ ⎜ V5 ⎟ ⎜ i5 ⎟ = ⎜ Y54 − ∆ ∆ ∆ ⎟⎜ ⎟ ⎜i ⎟ ⎜ ⎝ 6⎠ ⎜ ⎟ ⎝ V6 ⎠ (4.112) Y61 + Y62 ) Y61 + Y62 ) Y61 + Y62 ) ( ( ( ⎜ Y64 − (Y14 + Y24 ) Y65 − (Y15 + Y25 ) Y66 − (Y16 + Y26 ) ⎟⎟ ⎜ ∆ ∆ ∆ ⎝ ⎠ V A A A ⎛ 11 12 13 ⎞ ⎛ 4 ⎞ ⎜ ⎟⎜ ⎟ = ⎜ A21 A22 A23 ⎟ ⎜ V5 ⎟ . ⎜A ⎟⎜ ⎟ ⎝ 31 A32 A33 ⎠ ⎝ V6 ⎠ Similarly, for the network in the right box, we have V10 = 0,V11 = V12 and i11 = −i12 . Then 161 ⎡ i7 ⎤ ⎡ P11 ⎢ ⎥ ⎢ ⎢ i8 ⎥ ⎢ P21 ⎢ i9 ⎥ ⎢ P31 ⎢ ⎥=⎢ ⎢i10 ⎥ ⎢ P41 ⎢i ⎥ ⎢ P ⎢ 11 ⎥ ⎢ 51 ⎢⎣i12 ⎥⎦ ⎢⎣ P61 ⎡ P11 ⎢ ⎢ P12 ⎢P = ⎢ 13 ⎢ P14 ⎢P ⎢ 15 ⎣⎢ P16 P12 P22 P13 P23 P14 P24 P15 P25 P32 P33 P34 P35 P42 P52 P43 P53 P44 P54 P45 P55 P62 P63 P64 P65 P12 P13 P14 P15 P22 P12 P12 P11 P15 P16 P14 P15 P15 P16 P11 P12 P14 P15 P15 P14 P12 P13 P22 P12 P16 ⎤ ⎡V7 ⎤ ⎥⎢ ⎥ P26 ⎥ ⎢V8 ⎥ P36 ⎥ ⎢V9 ⎥ ⎥⎢ ⎥ P46 ⎥ ⎢V10 ⎥ P56 ⎥ ⎢V11 ⎥ ⎥⎢ ⎥ P66 ⎥⎦ ⎢⎣V12 ⎥⎦ P16 ⎤ ⎡V7 ⎤ ⎥⎢ ⎥ P15 ⎥ ⎢V8 ⎥ P14 ⎥ ⎢V9 ⎥ ⎥ ⎢ ⎥, P13 ⎥ ⎢V10 ⎥ P12 ⎥ ⎢V11 ⎥ ⎥⎢ ⎥ P11 ⎦⎥ ⎣⎢V12 ⎦⎥ (4.113) P51V7 + P52V8 + P53V9 + P55V11 + P56V11 = − ( P61V7 + P62V8 + P63V9 + P65V11 + P66V11 ) ,(4.114) → V11 ( P55 + P56 + P65 + P66 ) = − ( P51 + P61 ) V7 − ( P52 + P62 ) V8 − ( P53 + P63 ) V9 , (4.115) → V11 = 1 ⎡ − ( P + P ) V − ( P + P ) V − ( P + P ) V ⎤ .(4.116) ( P55 + P56 + P65 + P66 ) ⎣ 51 61 7 52 62 8 53 63 9 ⎦ Let ∆ ' = P55 + P56 + P65 + P66 , then ⎛ ⎞ ( P15 + P16 ) (P + P ) (P + P ) ( P51 + P61 ) P12 − 15 ' 16 ( P52 + P62 ) P13 − 15 ' 16 ( P53 + P63 ) ⎟ ⎜ P11 − ' ∆ ∆ ∆ ⎟ ⎛V ⎞ ⎛ i7 ⎞ ⎜ ⎜ ⎟⎜ 7 ⎟ P P P P P + + ( 25 26 ) ( ( ⎜ ⎟ 25 26 ) 25 + P26 ) i P P P P P P P P P = − + − + − + ⎜ ( 51 61 ) 22 ( 52 62 ) 23 ( 53 63 ) ⎟ ⎜ V8 ⎟ 21 ⎜8⎟ ∆' ∆' ∆' ⎟⎜ ⎟ ⎜i ⎟ ⎜ ⎝ 9⎠ ⎜ ⎟ ⎝ V9 ⎠ (4.117) P35 + P36 ) P35 + P36 ) P35 + P36 ) ( ( ( ⎜ P31 − ( P51 + P61 ) P32 − ( P52 + P62 ) P33 − ( P53 + P63 ) ⎟⎟ ⎜ ∆' ∆' ∆' ⎠ ⎝ ⎛ B11 B12 B13 ⎞ ⎛ V7 ⎞ ⎜ ⎟⎜ ⎟ = ⎜ B21 B22 B23 ⎟ ⎜ V8 ⎟ . ⎜B ⎟⎜ ⎟ ⎝ 31 B32 B33 ⎠ ⎝ V9 ⎠ From the equivalent circuit, it is obvious that V5 = V8 and i5 = −i8 , we have A21V4 + A22V5 + A23V6 = − ( B21V7 + B22V5 + B23V9 ) , (4.118) → ( A22 + B22 ) V5 = − A21V4 − A23V6 − B21V7 − B23V9 , (4.119) → V5 = 1 (− A V − A V − B V − B V ) . A + ( 22 B22 ) 21 4 23 6 21 7 23 9 (4.120) Let ∆ '' = A22 + B22 , then 162 A12 A21 ⎛ ⎜ A11 − ∆ '' ⎜ ⎛ i4 ⎞ ⎜ A A ⎜ ⎟ ⎜ A31 − 32 '' 21 i ∆ ⎜ 6⎟=⎜ ⎜ i7 ⎟ ⎜ B12 A21 ⎜⎜ ⎟⎟ ⎜ − ∆ '' ⎝ i9 ⎠ ⎜ ⎜⎜ − B32 A21 ∆ '' ⎝ A12 A23 ∆ '' A A A33 − 32 '' 23 ∆ B12 A23 − ∆ '' B A − 32 '' 23 ∆ A13 − B21 A12 ∆ '' B A − 21 '' 32 ∆ B12 B21 B11 − ∆ '' B B B31 − 32 '' 21 ∆ − B23 A12 ∆ '' B A − 23 '' 32 ∆ B12 B23 B13 − ∆ '' B B B33 − 32 '' 23 ∆ − ⎞ ⎟ ⎟ V ⎟ ⎛⎜ 4 ⎞⎟ ⎟ ⎜ V6 ⎟ ⎟⎜ ⎟ . ⎟ ⎜ V7 ⎟ ⎟ ⎜V ⎟ ⎟⎝ 9 ⎠ ⎟⎟ ⎠ (4.121) Then from the equivalent circuit, i4 + i7 = i13 i6 + i9 = i14 V13 = V4 = V7 . (4.122) V14 = V6 = V9 Thus ⎛ A − ⎛ i13 ⎞ ⎜ 11 = ⎜ ⎜ ⎟ ⎝ i14 ⎠ ⎜ A − ⎜ 31 ⎝ A12 A21 B12 A21 − ∆ '' ∆ '' A32 A21 B32 A21 − ∆ '' ∆ '' A12 A23 B12 A23 − ∆ '' ∆ '' A A B A A33 − 32 '' 23 − 32 '' 23 ∆ ∆ A13 − B21 A12 B B + B11 − 12 '' 21 ∆ '' ∆ B21 A32 B32 B21 − + B31 − ∆ '' ∆ '' − B23 A12 B B ⎞ ⎛ V4 ⎞ + B13 − 12 '' 23 ⎟ ⎜ ⎟ , '' V6 ∆ ∆ ⎟⎜ ⎟ B23 A32 B32 B23 ⎟ ⎜ V7 ⎟ − + B33 − ⎟⎜ ⎟ ∆ '' ∆ '' ⎠ ⎜⎝ V9 ⎟⎠ − (4.123) A A B A B A B B ⎛ A − 12 '' 21 − 12 '' 21 − 21 '' 12 + B11 − 12 '' 21 ⎛ i13 ⎞ ⎜ 11 ∆ ∆ ∆ ∆ ⎜ ⎟=⎜ ⎝ i14 ⎠ ⎜ A − A32 A21 − B32 A21 − B21 A32 + B − B32 B21 ⎜ 31 31 ∆ '' ∆ '' ∆ '' ∆ '' ⎝ ⎛ Y ' Y12' ⎞ ⎛ V13 ⎞ = ⎜ 11' ⎟ ' ⎟⎜ ⎝ Y21 Y22 ⎠ ⎝ V14 ⎠ A12 A23 B12 A23 B23 A12 B B ⎞ − − + B13 − 12 '' 23 ⎟ '' '' '' ⎛ V13 ⎞ ∆ ∆ ∆ ∆ ⎟⎜ ⎟ A A B A B A B B V . A33 − 32 '' 23 − 32 '' 23 − 23 '' 32 + B33 − 32 '' 23 ⎟⎟ ⎝ 14 ⎠ ∆ ∆ ∆ ∆ ⎠ A13 − (4.124) Since the two networks in the two dashed boxes are exactly the same and the three coupled lines are assumed to have the same characteristic impedance, the S-parameter for the overall two port network (port 13 and port 14) can be simplified and expressed as 163 ⎛ S11 ⎜ ⎝ S 21 ⎛ (1 − Y11' )(1 + Y22' ) + Y12' Y21' ⎜ ' ' ' ' S12 ⎞ ⎜ (1 + Y11 )(1 + Y22 ) − Y12Y21 ⎟=⎜ S 22 ⎠ ⎜ −2Y21' ⎜ ⎜ (1 + Y11' )(1 + Y22' ) − Y12' Y21' ⎝ ⎞ −2Y12' ⎟ (1 + Y11' )(1 + Y22' ) − Y12' Y21' ⎟ ⎟ (1 + Y11' )(1 − Y22' ) + Y12' Y21' ⎟ ⎟ (1 + Y11' )(1 + Y22' ) − Y12' Y21' ⎟⎠ (4.125) Let W1 = (Y14 + Y15 )(Y16 + Y15 ) W2 = Y12 − W3 = (4.126) Y11 + 2Y12 + Y22 (Y14 + Y15 ) 2 (4.127) Y11 + 2Y12 + Y22 W22 + 2 (Y12 − W1 ) W2 + (Y12 − W1 ) 2Y22 − 2 (Y14 + Y15 ) (4.128) 2 Y11 + 2Y12 + Y22 (Y + Y ) + (Y16 + Y15 ) − 14 15 2 W4 = 2Y11 2 Y11 + 2Y12 + Y22 2 − W3 (4.129) We have S11 = S 22 (1 − W ) + ( 2Y = S12 = S 21 2 4 13 − 2W1 − W3 ) 2 (1 + W4 ) − ( 2Y13 − 2W1 − W3 ) 2 −4Y13 + 4W1 + 2W3 (1 + W4 ) − ( 2Y13 − 2W1 − W3 ) 2 2 2 (4.130) (4.131) Based on the analysis of this modified hairpin-combline resonator, a simple two-pole filter is designed as shown in Fig. 68. However, as shown in Fig. 68, the proposed filter occupies quite a substantial area along the vertical direction, which makes the design very undesirable for LTCC implementation. 164 To achieve a more compact design in both the horizontal and vertical directions, we extend the meander line outwards to form a square loop as shown in Fig. 71. In doing so, the vertical direction is reduced in length and the occupied area would be reduced. The simulated response of the filter structure is given in Fig. 72. As observed from Fig. 72, with a broadening of the meander line to a square loop, the outer pole is shifted to a higher frequency at 3 GHz. Fig. 71 Preliminary structure II 165 Fig. 72 Simulated (IE3D) results for preliminary structure II 4.4.2.4 Two Proposed Designs Using LTCC The design procedure for this BPF shown given below: 1. A coplanar bandpass filter is designed to meet the specifications: Center frequency 2.5 GHz Passband return loss 100 Ω ). 4. To achieve a more compact design in both the horizontal and vertical directions, we extend the meander line outwards to form a square loop as shown in Fig. 71. 5. In order to move the outer pole to 2.5 GHz, the proposed filter needs to be loaded with a capacitor. In order to merge the two split resonant peaks together, we introduce additional triangular protrusions between the meander lines as illustrated in Fig. 73. 6. Fine tune the structure to improve the matching. The physical dimensions of the bandpass filter given in Fig. 73 are the same as those provided in Fig. 71, but with the additional triangular protrusions. The simulated results of the filter are depicted in Fig. 74. This proposed single-layered structure is compact in size, and has a good stop-band performance and low insertion loss of approximately 2 dB. The effective circuit area for this design is 6.54 mm × 8.564 mm , and the guided wavelength for this case is approximately λe = 68 mm . 167 2.78 mm Fig. 73 Fig. 74 Proposed coplanar bandpass filter Simulated (IE3D) results for the structure in Fig. 73 To further reduce the filter size and to fully utilize the LTCC multilayered capability, a two-layered structure with a single via-hole is introduced in Fig. 75. As mentioned 168 above, additional capacitance is needed in order to merge the two resonant peaks. Instead of using a triangular protrusion, the additional capacitance is realized through the coupling between 1st layer and 2nd layer in the second proposed design. The overall circuit area is reduced due to open meandered stubs in the 2nd layer, as shown in Fig. 75. The connection of the 1st layer and the 2nd layer is realized using a via hole. Quasielliptic filter response is thus obtained as shown in Fig. 76. The effective circuit volume for this design is now reduced to 3.524 mm × 3.712 mm × 0.1905 mm in x-, yand z- direction accordingly. It is noted that the volume of this design is much smaller compared to other LTCC bandpass filters designs, which normally consist 6 to 10 metal layers [108]-[109]. 169 Fig. 75 Dimensions of the proposed two-layered filter design 170 Fig. 76 Simulated (IE3D) results for the proposed two-layered design 4.4.2.5 Experimental Results and Discussions The two proposed CPW bandpass filters were fabricated using an in-house six-layered Ferro A6 LTCC technology. The relative permittivity of the substrate is 5.9 and the thickness of each layer is 0.1905 mm. The photographs of the two proposed designs are given in Fig. 77. The responses of these filters were measured using HP-8510C vector network analyzer. The simulated and measured results for the single-layered structure are compared in Fig. 78. This bandpass filter was designed to have a centre frequency of 2.46 GHz with a minimum insertion loss of 1.8 dB, a return loss of 27.2 dB at centre frequency and a fractional 3-dB bandwidth of 5.4%. From the measured results shown in Fig. 78, 171 the center frequency is slightly shifted to 2.39 GHz and the insertion loss at centre frequency is 4.12 dB and the return loss is around 26 dB. The simulated and measured results for the two-layered structure are compared in Fig. 80. This filter is designed to have a centre frequency at 2.45 GHz, which has a minimum insertion loss of 0.375 dB, a return loss of 25dB at centre frequency and a fractional 3-dB bandwidth of 30.6%. From the measured results shown in Fig. 78, the center frequency is shifted to 2.2 GHz. The measured insertion loss is 1.2 dB and the measured return loss is around 25 dB. It is also noted that the 3-dB bandwidth for the two-layered filter is increased to 30.6% as compared to single-layered filter, which is given as 5.4%. The possible reasons of discrepancy between simulated and measures results have been discussed in detail in the previous chapter. For single-layered structure, the effect of variation of the permittivity is shown in Fig. 79. A frequency shift of 70 MHz is observed at two extreme cases, i.e. ε r = 5.7 and ε r = 6.1 . For two-layered structure, the effect of variation of the permittivity is shown in Fig. 81. A frequency shift of 60 MHz is observed at two extreme cases, i.e. ε r = 5.7 and ε r = 6.1 . Instead of using lumped elements for LTCC filter implementation, two novel and miniaturized LTCC CPW filters were realized by using a sequence of capacitance loading. For the single-layered filter, an insertion loss of 4.12 dB and a return loss of 26 dB were achieved. Similarly, for the two-layered filter, an insertion loss of 1.2 dB and a return loss of 25 dB were obtained. 172 (i) (ii) Fig. 77 Photographs for the proposed designs: (i) Single-layered CPW bandpass filter as shown in Fig. 73, (ii) Two-layered CPW bandpass filter with via as shown in Fig. 78 Simulated (IE3D) and measured results for Design (i) in Fig. 77 173 (a) Comparison of S11 (b) Comparison of S21 Fig. 79 Comparison of S-parameters with ε r = 5.7 , ε r = 5.9 and ε r = 6.1 174 Fig. 80 Simulated (IE3D) and measured results for Design (ii) in Fig. 77 (a) Comparison of S11 175 (b) Comparison of S21 Fig. 81 Comparison of S-parameters with ε r = 5.7 , ε r = 5.9 and ε r = 6.1 176 Chapter 5 Conclusions and Future Work 5.1 Conclusions In the first chapter, an introduction to the history and current research concerning multilayered EM modeling was presented. Different numerical techniques, which are used to efficiently and accurately calculate the fields of a layered medium were compared. The problems related to MoM with DCIM algorithm were discussed in the first chapter. Passive circuits, such as filters, can be designed and simulated using the techniques described in the thesis. Work was conducted on designing compact, lowloss and high performance bandpass filter. In view of this, various ways of filter miniaturization techniques were introduced. A novel and compact quaternion formulation for EM analysis was derived. Normally, the electric and magnetic fields are derived separately. With the concept of quaternion algebra, it is possible to give a compact formulation for DCIM analysis, which is shown in Chapter 2. A multilayered pole extraction technique for fast evaluation of DCIM was introduced in Chapter 3. Extracting the poles of the Green’s function is one of the difficulties for the analysis of multilayered structures due to the complexity of the multilayered Green’s function. Although several authors provide some techniques for pole extraction, these are not very efficient and accurate. Thus, there is a need to investigate a new general and fast algorithm for pole extraction for multilayered structures. An efficient, fast and stable multilayered pole extraction algorithm was introduced in this chapter for fast evaluation of DCIM. Good initial guesses were derived and good 177 convergence and accuracy were achieved through this technique. This method is noted to be fast and stable, and does not suffer from local minimum termination. MoM can be used to design and analyze different planar circuits. In this thesis, various novel bandpass filter designs were investigated in Chapter 4. All the filters introduced in this chapter focused on miniaturization, which provides low cost and good performance. Several advanced design and fabrication techniques were applied for further miniaturization of bandpass filters compared to the conventional PCB technology. They were: • Filter with PBG structure. PBG was used together with the concept of SIR to realize circuit miniaturization without etching the ground plane. The circuit area of the resonator with PBG was reduced to 44.38% of the size of the resonator without PBG. • Dual mode resonator without cross coupling effect. Two novel microstrip dual mode filters based on the basic configuration of a square open-loop but with two added gaps at the off-diagonal corners, were proposed. The proposed filters do not require the use of cross-coupled effect to realize the narrowband filter response. As a result, the proposed filters have the advantages of small-size, low cost and easy fabrication. By applying the meandering technique at the two vertical sides of the loop, the magnetic field coupling in the proposed filters was enhanced, thus leading to size reduction. • Novel miniaturized open-loop resonator with wide frequency perturbation. Instead of using lumped element components or adding a mixed dielectric to the design, which will eventually increase the overall cost of production, an alternate way of producing a size reduction of an open-loop squared filter was explored. 178 Miniaturization was achieved by first designing the respective filter at a higher frequency, and through loading, the desired filter response at a lower frequency was achieved. This method involves inserting multiple circular rings within the open-loop square resonator. The use of these multiple rings allows a very wide frequency perturbation and eventually allowing significant size reduction as compared to the meander loop resonator or the hairpin resonator. • Novel coplanar filter design with Wafer Transfer Technology (WTT). Wafertransfer technology (WTT) was originally applied to transfer RF components from a silicon wafer to an opaque plastic substrate (FR-4). As the substrate loss of FR-4 is high at high frequencies, this technology is improved to transfer a novel design of coplanar bandpass filter from a Si to RO4003. This bandpass filter, which consists of two CPW resonators with additional grounded meander lines, has compact size and good harmonic suppression. • Miniaturized single-layered and two-layered coplanar filters using LTCC technology. We integrated both the hairpin resonator and the combline interdigital filter together into a novel single, small-size narrowband filter. Novel techniques of loading the resultant resonator so as to achieve miniaturization were attempted. 179 5.2 Recommendations for Future Work As an extension of the quaternion analysis, there is a need to investigate the discrete quaternion Fourier Transform for completeness. Since there exists a discrete Fourier transform, it becomes natural to investigate the discrete quaternion Fourier Transform. Moreover, it is found in the earlier chapter that the presented quaternion electromagnetic analysis has similar form as the S-parameter as shown in eqn. (2.119). As such, one can investigate the possibility on the concept of bi-complex S-parameter analysis. The complex image approach for the Green’s function of a mixed-potential integral equation formulation is limited to low values of ko ρ (in this context ko ρ = 2πρ / λo , where ρ is the distance between the source and the field points of the Green’s function and λo is the free space wavelength). This leads to a clear limitation for problems of large dimension or high frequency where this limit is easily exceeded. Finding a better approach to evaluate the Green’s function for large values of ko ρ is therefore desirable. Also in Chapter 2, we have applied Halley’s method for pole extraction algorithm. This method is not good enough as it requires the evaluation of the 2nd order derivatives of the desired function. A more promising method would be the curvilinear Newton’s method for solving the transcendental poles’ equations. In this thesis, only narrowband filters were investigated. Wideband filters also play a crucial role in wireless communications as such there is a need to investigate 180 miniaturization of broadband pass filters. There is also a need to look into higher order filter network for superior performance and increase of the payload. With regard to Wafer Transfer Technology, the loss at the interfaces is noted to be high. A novel way of interfacing the different layers needs to be considered to improve the performance. 181 APPENDIX Definition of Quaternions The concept of the quaternion was introduced by Hamilton in 1843 [116]. It is the generalization of a complex number. A complex number has two components: the real and the imaginary part, i.e. z = a + bi , (6.1) where i = −1 , a and b are arbitrary real numbers. The quaternion, however, has four components, i.e., one real part and three imaginary parts: z = a + bi + cj + dk , (6.2) i 2 = −1, j 2 = −1, k 2 = −1, (6.3) where The properties of the three fundamental elements (i, j, k) can be illustrated as shown in Fig. 82 and they forms a cyclic set. This figure shows ij = k , ji = − k , (6.4) jk = i, kj = −i (6.5) ki = j , ik = − j. (6.6) i j k Fig. 82 Properties of the three fundamental elements 182 Quaternions can be conceptualized as ordered pairs of complex numbers, called bicomplex number. For example, let u = a + jb, v = c + jd , a, b, c, d ∈ R . (6.7) If w = u + iv , we have w = u + iv = ( a + jb ) + i ( c + jd ) = a + jb + ic + ijd = a + jb + ic + kd (6.8) where u is called the bi-real part and v is called the bi-imaginary part. Some Properties of Bi-complex Numbers Non-commutative multiplication of Quaternions Let’s assume u = a1 + b1i + c1 j + d1k , v = a2 + b2i + c2 j + d 2 k , then uv = ( a1 + b1i + c1 j + d1k )( a2 + b2i + c2 j + d 2 k ) = a1a2 + b1b2i 2 + c1c2 j 2 + d1d 2 k 2 + ( a1b2 + b1a2 ) i + ( a1c2 + a2 c1 ) j + ( a1d 2 + a2 d1 ) k + b1c2ij + b1d 2ik + c1b2 ji + c1d 2 jk + d1b2 ki + d1c2 kj .(6.9) = ( a1a2 − b1b2 − c1c2 − d1d 2 ) + ( a1b2 + b1a2 + c1d 2 − d1c2 ) i + ( a1c2 + a2 c1 + d1b2 − b1d 2 ) j + ( a1d 2 + a2 d1 + b1c2 − c1b2 ) k vu = ( a2 + b2i + c2 j + d 2 k )( a1 + b1i + c1 j + d1k ) = a1a2 + b1b2i 2 + c1c2 j 2 + d1d 2 k 2 + ( a1b2 + b1a2 ) i + ( a1c2 + a2 c1 ) j + ( a1d 2 + a2 d1 ) k + b1c2 ji + b1d 2 ki + c1b2ij + c1d 2 kj + d1b2ik + d1c2 jk .(6.10) = ( a1a2 − b1b2 − c1c2 − d1d 2 ) + ( a1b2 + b1a2 − c1d 2 + d1c2 ) i + ( a1c2 + a2 c1 − d1b2 + b1d 2 ) j + ( a1d 2 + a2 d1 − b1c2 + c1b2 ) k ⇒ uv ≠ vu . Thus, it is proven that multiplication of quaternions is not commutative. 183 Associativity of Multiplication of Quaternions The associativity of multiplication of quaternions can be expressed as follows (q1 q2 )q3 = q1 (q 2 q3 ) , (6.11) where q n = a n + bn i + c n j + d n k , where n = 1,2,3. Conjugation of Quaternions For bi-complex numbers, examples of different kinds of conjugations are shown in the following: (i) Conjugation with respect to j: Let w = u + iv, u = a + bj , v = c + dj ⇒ w∗ = u ∗ + iv∗ ⇒ w∗ = a − bj + i ( c − dj ) ⇒ w* = a − bj + ci + dk (ii) . (6.12) Conjugation with respect to i: Let w = u + iv, u = a + bj , v = c + dj ⇒ w+ = u + − v +i ⇒ w+ = u − vi = a + bj − (c + dj )i ⇒ w+ = a + bj − ci + dk . (6.13) If p, q ∈ set of bicomplex number, B , then jq = q + j , ( pq )+ = p+q+ , ( pq )∗ = (iii) (6.14) (6.15) p*q* . (6.16) Conjugation with respect to i, j, k: Let q = q + bi + cj + dk ∈ Q ⇒ q = a − bi − cj − dk Then, we have qq = q = (a + bi + cj + dk )(a − bi − cj − dk ) = a 2 + b 2 + c 2 + d 2 2 qq + ≠ q 2 , qq ∗ ≠ q 2 . (6.17) (6.18) 184 Quaternion Fourier Transform There are at least three types of Fourier Transform for quaternions, namely, two-side, left-side and right-side Fourier Transform [115]. Suppose they are two unit quaternions, namely, µ1 = µ1,i i + µ1, j j + µ1,k k , (6.19) µ 2 = µ 2,i i + µ 2, j j + µ 2,k k , (6.20) which are orthogonal to each other. Therefore, we have 2 2 (6.21) µ1,i µ 2,i + µ1, j µ 2, j + µ1, k µ 2, k = 0 (6.22) µ1 = µ1,2i + µ1,2 j + µ1,2k = µ2 = µ 2,2 i + µ2,2 j + µ2,2 k = 1 (A) Type 1 QFT (Two-side): ~ H q1 (ω , v ) = ∞ ∞ ∫ ∫e µ1ωx h( x, y )e µ1vy dxdy (6.23) − ∞− ∞ where µ1 and µ2 are two unit, pure orthogonal quaternions (i.e. the quaternions with unit magnitude having no real part.). (B) Type 2 QFT (Left-side) The type 2 QFT is given as ~ H q 2 (ω , v ) = ∞ ∞ ∫ ∫e µ1 (ωx + vy ) h( x, y )dxdy , (6.24) − ∞− ∞ (C) Type 3 QFT (Right-side) 185 The type 3 QFT is given as ~ H q 3 (ω , v ) = ∞ ∞ ∫ ∫ h(x, y )e µ1 (ωx + vy ) dxdy . (6.25) − ∞− ∞ Their inverse quaternion Fourier transform (IQFT) are expressed as: (A) Type 1 IQFT (Two-side): h ( x, y ) = ℑ −1 ∞ ∞ 1 ( H (ω, v ) ) = 4π ∫ ∫ e q1 − µ1ω x 2 H q1 (ω , v )e − µ1vy dω dv , (6.26) H q 2 (ω , v )d ω dv , (6.27) −∞ −∞ (B) Type 2 IQFT (Left-side) ( ) h ( x, y ) = ℑ−1 H q 2 (ω , v ) = (C) ∞ ∞ 1 4π 2 ∫ ∫e − µ1 (ω x + vy ) −∞ −∞ Type 3 IQFT (right-side) ( ) h ( x, y ) = ℑ−1 H q 3 (ω , v ) = ∞ ∞ 1 4π 2 µ (ω ∫ ∫ H (ω , v ) e − q3 1 x + vy ) dω dv . (6.28) −∞ −∞ 186 Basic Quaternion Electromagnetics Source-free, Homogeneous, Lossless TEM Case Consider the homogeneous, source-free, isotropic medium where the Maxwell’s equations are given namely as: ∇ × E = − jωµH = − jkηH , ∇ × H = J + jωεE = (σ + jωε )E = jωεE = j where k = ω µε , η = (6.29) k η E, (6.30) ∇ • H = 0, (6.31) ∇ • εE = ρ . (6.32) µ and e jωt time variation is assumed. ε Let bicomplex vector field F be defined as F= 1 η E + i ηH . (6.33) Re-arrange eqns. (6.29) & (6.30) and multiply eqn (6.30) with i on the left, we have ∇× E η = − jk η H , i η ∇ × H = ij k η E, (6.34) (6.35) Adding eqns. (6.34) and (6.35), we obtain ∇ × F = ijkF + . (6.36) Similarly, multiply eqn. (6.31) with i η and divide eqn. (6.32) with η , we achieve ∇•F = ρ . ε η (6.37) 187 As mentioned above, F has the following properties: ∇ × F = ijkF + , (6.38) ρ . ε η (6.39) ∇•F = Maxwell equations can be simplified into two equations relating to a bi-complex function F . Both equations are valid for all coordinate systems. Consider a general TEM wave propagating in the z direction, (i.e. a vanishing zcomponent) and let the auxiliary quaternion Q y = iFy . Using Eqn. (6.35), we have ∇ × F = ijkF + ⎛∂ ∂ ⎞ ∂ ⇒ ⎜⎜ xˆ + yˆ + z ⎟⎟ × (Fx xˆ + Fy yˆ ) = ijk (Fx+ xˆ + Fy+ yˆ ) ∂z ⎠ ∂y ⎝ ∂x ⎛∂ ∂ ⎞ ∂ + ⇒ ⎜⎜ xˆ + yˆ + z ⎟⎟ × (Fx xˆ − iQy yˆ ) = ijk (Fx+ xˆ + iQy+ yˆ ) = ijk (Fx xˆ − iQy yˆ ) ∂z ⎠ ∂y ⎝ ∂x ⎧ ∂Q y+ = jkFx ⎪ z ∂ ⎪ ⎪ ∂Fx = jkQ y+ ⇒⎨ ⎪ ∂z ⎪ ∂Fy − ∂Fx = 0 ⎪ ∂x ∂y ⎩ . , (6.40) The first and second equations are used to find the solution, whereas the last equation is used as an additional condition to restrict the final possible solution. Adding and subtracting the first two equations by parts to yield the equivalent pair ∂ ( Fx + Q y+ ) = jk (Fx + Q y+ ) , ∂z (6.41) 188 ∂ ( Fx − Q y+ ) = − jk (Fx − Q y+ ). ∂z (6.42) Multiply eqn (6.42) by i and sum it with eqn. (6.41), [( )] [( )] ∂ Fx + Q y+ + i Fx − Q y+ = jk Fx + Q y+ − ijk Fx − Q y+ = jk Fx + Q y+ + i Fx − Q y+ .(6.43) ∂z ) ( ( ) ( ) ) ( Let ( ) ( ) P( z ) = Fx + Q y+ + i Fx − Q y+ , ⇒ ∂P ( z ) = jkP ( z ) . ∂z (6.44) Multiply eqn. (6.44) with e − jkz , we have e − jkz ∂P( z ) − jkz ∂ − jkz − e jkP (z ) = 0 ⇒ e P(z ) = 0 ∂z ∂z , [ ⇒ e − jkz P( z ) = u + iv where u = ( 2 η ] [C1 − iC 2 ] and v = ) ( 2 η (6.45) [D1 + iD2 ] (for conformity) ) 2 ⎧ − jkz [C1 − iC 2 ] ⎧⎪ Fx + Q y+ = 2 e jkz [C1 − iC 2 ] Fx + Q y+ = ⎪e η η ⎪ ⎪ ⇒⎨ ⇒⎨ 2 2 ⎪ie jkz Fx − Q y+ = i[D1 + iD2 ] ⎪ Fx − Q y+ = e − jkz [D1 + iD2 ] ⎪⎩ ⎪⎩ η η ( ) ( (6.46) ) ⎧ Ex ⎛ Ey ⎞ E iE y + i η H x − i⎜ −i ηH y ⎟ = x +i ηHx − − ηH y = ⎪ ⎜ η ⎟ η η ⎪ η ⎝ ⎠ ⇒⎨ ⎛ ⎞ E iE y E E y ⎪ x + i η H + i⎜ −i ηH y ⎟ = x +i ηHx + + ηH y = x ⎪ η ⎜ η ⎟ η η ⎝ ⎠ ⎩ 2 η 2 η e jkz (C1 − iC2 ) (6.47) e − jkz (D1 + iD2 ) . Equating the bi-real and bi-imaginary parts of eqn. (6.47), and summing the respective eqns, [ ] 2 jkz ⎧ 2E x = e C1 + e − jkz D1 ⎧ E x = e jkz C1 + e − jkz D1 ⎪ η η ⎪ ⎪ 1 ⇒⎨ ⇒⎨ H x = − e − jkz C 2 + e jkz D2 2 jkz − jkz ⎪2 η H x = ⎪⎩ − e C 2 + e D2 η ⎪⎩ η [ ] ( ) . (6.48) Similarly, equating the bi-real and bi-imaginary parts of eqn. (6.47), and subtracting the respective eqns, 189 ⎧ 2E y 2 − jkz e C 2 + e jkz D2 = ⎧ E y = (e − jkz C 2 + e jkz D2 ) ⎪ ⎪ ⎪ η η . ⇒⎨ ⇒⎨ 1 − jkz jkz ( ) H e D e C 2 = − − jkz jkz y 1 1 ⎪ ⎪2 η H y = e D1 − e C1 η ⎩ ⎪⎩ η [ ] [ ] (6.49) Therefore, E = (e jkz C1 + e − jkz D1 )xˆ + (e − jkz C 2 + e jkz D2 )yˆ , H= (− e η 1 − jkz C 2 + e jkz D2 )xˆ + (e η 1 − jkz D1 − e jkz C1 )yˆ . (6.50) (6.51) The last eqns of (6.40) confirms that C1 , C 2 , D1 , and D2 are constants. For a +z-directed wave, if C1 = D2 = 0 , we will have (i) linear polarization in x if C 2 = 0 , (ii) linear polarization in y if D1 = 0 , (iii) circular right-handed polarization if D1 = jC 2 , (iv) circular left-handed polarization if D1 = − jC 2 , and (v) elliptic for the general case. The efficiency of this analysis is that the first order bi-complex differential equation can model wave propagation in two directions, as compared to the conventional second order complex equations. It should be noted that ∂ − jkz ∂ ⎡⎣e P ( z ) ⎤⎦ = 0 is not the same as ⎡⎣ P ( z ) e − jkz ⎤⎦ = 0 as the ∂z ∂z first differential equation has a solution of P ( z ) = e jkz Po whereas the latter has a solution of P ( z ) = Po e jkz . This is because the product is not commutative. 190 Source-free, Inhomogeneous, Lossless TEM Case For the inhomogeneous, source-free medium, where the medium characteristics depend on the location, k and η will be functions of co-ordinates. Maxwell Equations can be expressed as ∇ × E = − jkηH , ∇× H = j k η (6.52) E, (6.53) ∇ • H = 0, (6.54) ∇ • εE = ρ . (6.55) Let F= F+ = 1 η 1 η E + i η H , and (6.56) E −i ηH , (6.57) Therefore, E= η 2 (F + F ) and H = + −i (F − F ) = + 2 η i 2 η (F + −F ) , (6.58) Using eqn. (6.52), ⎛F + F+ ⇒ ∇ × η ⎜⎜ ⎝ 2 ⎛F+ − F ⎞ ⎞ ⎞ ⎛ + ⎟ = ijk η ⎜ F − F ⎟ ⎟ = − jkηi⎜ ⎟ ⎜ 2 ⎟ ⎜ 2 η ⎟ ⎠ ⎠ ⎝ ⎝ ⎠ , (6.59) Using eqn. (6.53), ⎛F+ −F ⎞ ⎛F+ + F ⎞ k jk ⎜ ⎟ ⎟= ⇒ ∇×i η ⎜⎜ F+ + F , = j ⎟ ⎜ 2 η ⎟ η ⎝ 2 ⎠ 2 η ⎝ ⎠ ( ⎛F −F+ ⇒ ∇×⎜ ⎜ 2 η ⎝ ⎞ ⎟ = ijk F + + F , ⎟ 2 η ⎠ ( ) ) (6.60) (6.61) 191 1 Multiply eqn. (6.59) by η ⎛ F + F+ ∇ × η ⎜⎜ η ⎝ 2 1 ,⇒ ⎛F −F+ Multiply eqn. (6.61) by η , ⇒ η ∇ × ⎜ ⎜ 2 η ⎝ ⎞ ⎛ F+ − F ⎞ ⎟ = ijk ⎜ ⎟ ⎟ ⎜ 2 ⎟, ⎠ ⎝ ⎠ (6.62) ⎞ ijk + ⎟= F +F , ⎟ 2 ⎠ ( ) (6.63) Summing eqns (6.63) from (6.62), ⎛F + F+ ∇ × η ⎜⎜ η ⎝ 2 1 ⇒ ⎛F −F+ ⎞ ⎟ + η∇ × ⎜ ⎟ ⎜ 2 η ⎠ ⎝ ⎞ ⎛ + ⎞ ⎟ = ijk ⎜ F − F ⎟ + ijk F + + F = ijkF + (6.64) ⎜ 2 ⎟ 2 ⎟ ⎝ ⎠ ⎠ , ( ) ( ) Using ∇ × ΨA = ∇Ψ × A + Ψ∇ × A , ⇒ ⎛F + F+ ∇ η × ⎜⎜ η ⎝ 2 1 ⇒ ∇× F + ⎛ F + F+ ⎞ ⎟ + ∇×⎜ ⎜ 2 ⎟ ⎝ ⎠ ⎡⎛ F + F + 1 ⎡ ⎢∇ η × ⎢⎜⎜ η ⎢⎣ ⎢⎣⎝ 2 ⎛F −F+ ⎞ ⎟ + ∇×⎜ ⎜ 2 ⎟ ⎝ ⎠ ⎞ ⎛ F − F+ ⎟−⎜ ⎟ ⎜ 2 ⎠ ⎝ [∇ η ⎞ ⎟ = ijkF + ,(6.65) ⎟ ⎠ ⎞⎤ ⎤ 1 ⎟⎥ ⎥ = ∇ × F + ∇ η × F + = ijkF + ⎟ η ⎠⎥⎦ ⎥⎦ ,(6.66) [ ] ] 1 ⇒ ∇ × F = ijkF + − ⎛ F − F+ ⎞ ⎟ −∇ η ×⎜ ⎟ ⎜ 2 η ⎠ ⎝ η ×F+ . (6.67) − F = 0, ) (6.68) (F + F ) = ρ . (6.69) Similarly, for eqns. (6.54) and (6.55), ∇•H = ∇• i 2 η ∇ • εE = ∇ • (F ε η 2 + + ( ) Multiply eqn. (6.68) by η and eqn. (6.69) by 1 ε η , and after taking the sum of the resultant eqns, ⇒ η∇ • ( ) (F + −F 2 η ( )+ 1 ε η ) ∇• ε η 2 (F + F ) = + ) ( ) ρ , ε η ( (6.70) ) ⎛ 1 ⎞ F+ −F ρ , (6.71) F+ −F 1 F+ + F F+ + F ⎟• +∇• + ∇ε η • +∇• = ⇒ η ∇⎜ ⎜ η⎟ 2 2 2 2 ε η ε η ⎠ ⎝ ( 192 ( ) ) ( ) ⎛ 1 ⎞ F+ − F 1 F+ + F ρ ⎟• . ⇒ ∇ • F + = − η ∇⎜ − ∇ε η • + ⎜ η⎟ 2 2 ε η ε η ⎝ ⎠ ( (6.72) Consider a generalized TEM wave propagating in z-direction in an inhomogeneous medium and let a= ∇ η η . Expanding eqn. (6.67), i.e. ∇ × F = ijkF + − ∂Fy ∂x − [∇ η 1 ] η ×F+ , + az Fy+ = −ijkFx+ , (6.74) ∂Fx + a z Fx+ = ijkFy+ , ∂z (6.75) ∂z ∂Fy (6.73) ∂Fx + a x Fy+ − a y Fx+ = ijkFz+ = 0 . ∂y (6.76) Let Q y = iF y . Using eqns. (6.74) and (6.75), −i∂Qy ∂z + az iQy+ = −ijkFx+ ⇒ ∂Qy ∂z = az Qy+ + jkFx+ , ∂Fx ∂F + az Fx+ = ijkFy+ = − jkiFy+ = jkQy+ ⇒ x = − az Fx+ + jkQy+ . ∂z ∂z (6.77) (6.78) Taking the conjugation with respect to i for eqn. (6.77), ⇒ ∂Qy+ ∂z = az ( Qy+ ) + jkFx . + (6.79) Let U = Fx + Q y+ , W = Fx − Q y+ and summing and dividing eqns. (6.79) and (6.78), ∂ ⎡ U ⎤ ⎡ jk = ∂z ⎢⎣W + ⎥⎦ ⎢⎣ −az − az ⎤ ⎡ U ⎤ . − jk ⎥⎦ ⎢⎣W + ⎥⎦ (6.80) If k and az do not depend on z, the solution can be solved as follows: 193 ∂2 ∂z 2 −az ⎤ ∂ ⎡ U ⎤ ⎡ jk = − jk ⎦⎥ ∂z ⎣⎢W + ⎦⎥ ⎣⎢ − az ⎡ U ⎤ ⎡ jk ⎢ +⎥ = ⎢ ⎣W ⎦ ⎣ − az −az ⎤ ⎡ jk − jk ⎦⎥ ⎣⎢ − az −az ⎤ ⎡ U ⎤ ⎡ az2 − k 2 =⎢ − jk ⎦⎥ ⎣⎢W + ⎦⎥ ⎣ 0 0 ⎤⎡ U ⎤ ⎥⎢ ⎥ a − k 2 ⎦ ⎣W + ⎦ 2 z . (6.81) Solving the first row of eqn. (6.81), we have U = e λ1z A + e λ2 z B , where λ1,2 = ± az2 − k 2 . (6.82) Putting it into first row of eqn. (6.80), W+ = 1 1 ⎡⎣( jk − λ1 ) eλ1z A + ( jk − λ2 ) eλ2 z B ⎤⎦ ⇒ W = ⎡⎣( jk − λ1 ) eλ1z A+ + ( jk − λ2 ) e λ2 z B + ⎤⎦ . az az (6.83) Therefore, U = Fx + Qy+ = eλ1z A + eλ2 z B , W = Fx − Qy+ = ⇒ Fx = (6.84) 1 ⎡⎣( jk − λ1 ) eλ1z A+ + ( jk − λ2 ) eλ2 z B + ⎤⎦ , az (6.85) 1 ⎡ ( jk − λ1 ) eλ1z A+ + ( jk − λ2 ) eλ2 z B + + az eλ1z A + eλ1z B ⎤⎦ , ⎣ 2az ( ⇒ Qy+ = ) (6.86) 1 ⎡ az eλ1z A + eλ2 z B − ( jk − λ1 ) eλ1z A+ − ( jk − λ2 ) eλ2 z B + ⎦⎤ . 2az ⎣ ( ) Solving for the electric and magnetic fields by putting F = 1 η (6.87) E + i η H and Q y = iF y into eqns. (6.86) and (6.87), ⎡ 1 ⎤ ⎡ 1 ⎤ Ex = η eλ1z K ⎢1 + ( jk − λ1 ) ⎥ + η eλ2 z R ⎢1 + ( jk − λ2 ) ⎥ , ⎣ az ⎦ ⎣ az ⎦ Hx = * ⎡ 1 ⎤ 1 λ2* z ⎡ ⎤ 1 eλ1 z L ⎢1 + ( jk + λ1* ) ⎥ + e S ⎢1 + ( jk + λ2* ) ⎥ , η η ⎣ az ⎦ ⎣ az ⎦ 1 (6.88) (6.89) 194 * * ⎡ 1 ⎤ ⎡ 1 ⎤ E y = − η eλ1 z L ⎢1 − ( jk + λ1* ) ⎥ − η eλ2 z S ⎢1 − ( jk + λ2* ) ⎥ , ⎣ az ⎦ ⎣ az ⎦ Hy = − ⎡ 1 ⎤ 1 λ2 z ⎡ 1 ⎤ eλ1z K ⎢1 − ( jk − λ1 ) ⎥ − e R ⎢1 − ( jk − λ2 ) ⎥ , η η ⎣ az ⎦ ⎣ az ⎦ 1 (6.90) (6.91) where K, L, R, S are, in principle, arbitrary complex functions of x and y. These functions are explicitly determined by enforcing eqn. (6.76). 195 References [1] A. Karlsson and G. Kristensson, “Electromagnetic scattering from subterranean obstacles in a stratified ground,” Radio Sci., Vol. 18, pp. 345–356, May/June 1983. [2] P. E. Wannamaker, G. W. Hohmann, and W. A. SanFilipo, “Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations,” Geophys., Vol. 49, pp. 60–74, Jan 1984. [3] R. H. Hardman and L. C. 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Champ, “Thick-film technology offers high packaging density,” Microwaves & RF, Vol. 38, Issue 2, pp. 77–86, Feb 1999. 222 [...]... techniques for pole extraction, these methods are not very efficient and accurate Thus, there is a need to investigate a new general and fast algorithm for pole extraction in multilayered structures MoM can be used to design and analyze different planar circuits Various novel bandpass filter designs are investigated in Chapter 4 As the design of compact, lowloss and good performance bandpass filter has... filters [69], end- [70] and parallel-coupled [71][72] half-wavelength resonator filters, hairpin-line filters [73]-[75], interdigital [76][79], combline filters [80]-[81], pseudo-combline filters, and stub-line filters [82] Recently some research has been conducted on designing compact, low-loss bandpass filters with good performance In view of this, we will focus on miniaturization of microstrip and. .. communications, 7 miniaturization for personal communications equipment has become one of the most fundamental requirements Since many of the passive circuits are made of filter structures, there is a need to investigate miniaturization of the filters Filters can be categorized into bandpass filters, bandstop filters, lowpass filter and highpass filters Among them, the bandpass filter plays a pivotal role... a novel and compact quaternion analysis for EM analysis Normally, the electric and magnetic fields are derived separately With the concept of quaternion algebra, it is possible to give a compact formulation for DCIM analysis A multilayered pole extraction technique for fast evaluation of DCIM is introduced in Chapter 3 Extracting the poles of the Green’s function is one of the difficulties for the analysis. .. [108][109] and wafer transfer technology (WTT) These technologies have stimulated the rapid development of new bandpass filters In this thesis, we will introduce several novel structures of bandpass filters for microstrip structure and CPW using conventional and advanced fabrication technologies 1.2 Scope of Work In Chapter 1, an introduction to the history and current research concerning EM modeling for. .. as satellite and mobile communications systems In general, bandpass filters can be designed based on single- or multiple-resonator structures Microstrip resonators for filter designs may be classified as lumped-element or quasilumped-element resonators, distributed line resonators or patch resonators Conventional bandpass filters include stepped-impedance filters [62]-[66], open-stub filters [67]-[68],... for pole extraction, they are not efficient and accurate enough Thus, a fast, stable and efficient multilayered pole extraction technique for fast evaluation of DCIM is introduced in this thesis Suitable initial guesses, good convergence and accuracy have been achieved through our method The MoM algorithm is next used to design and analyze novel bandpass filters Several miniaturization techniques for. .. all the filters introduced in this chapter focus on miniaturization, which provide low cost and good performance Several advanced design and fabrication techniques are applied for further miniaturization of bandpass filters compared to the conventional PCB technology 1.3 Achievements and Contributions As a result of the research work, the following contributions have been achieved: 11 a) A novel and compact... convergence and accuracy are achieved The method is noted to be fast and stable, and does not suffer from local minimum termination c) Several miniaturization techniques for bandpass filter design have been explored for the first time These includes the PBG structure, dual mode resonator without cross coupling effect, novel miniaturized open-loop resonator with wide frequency perturbation, novel coplanar filter. .. resonators for a n -degree filter can be reduced by half, and the overall size of the filter can be compacted [91]-[92] A dualmode square patch resonator has been used to build Chebyshev and Elliptic filters [93] Conventional square patches suffer from large size As a result of this, several novel dual-mode structures have been introduced in this thesis for filter miniaturization Photonic bandgap (PBG)