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Static and Dispersion Analysis of Strip-Like Structures EE 694 Seminar Report on Static and Dispersion Analysis of Strip-like Structures Submitted by Ravindra.S.Kashyap 06307923 [email protected] Under the guidance of Prof. Girish Kumar Dept. of Electrical Engineering Indian Institute of Technology Bombay 1 CONTENTS I Introduction 2 II Static analysis 2 II-A The Conformal Mapping method . 3 II-B Variational approach . 4 II-C Finite Difference Method (FDM) . 6 II-D Losses . 8 III Dispersion analysis 9 III-A Spectral Domain Analysis . 10 III-B Method of Lines . 10 IV New Techniques for Microwave CAD 10 V Static Analysis for Various Microstrip line Configurations 11 V-A Tri-plateR . 11 V-B Microstrip Lines . 12 V-C Coupled Microstrip line . 14 V-D Suspended Microstrip Lines . 15 V-E Inverted Microstrip lines . 16 V-F Slot Lines . 16 V-G Coplanar Waveguide . 17 VI Dispersion Analysis for Various Microstrip line Configurations 17 VI-A Microstrip lines . 17 VI-B Coupled Microstrip Lines . 18 VI-C Suspended Microstrip lines . 19 VI-D Inverted Microstrip Line . 19 VII Conclusions 19 Appendix 19 References 20 2 Static and Dispersion Analysis of Strip-like Structures Ravindra.S.Kashyap (06307923) Abstract— Different models for the static and dispersion velocity (vp). These three parameters are related to the analysis of strip-like structures will be studied. We will be Capacitance (C) of the structure [2]. We know that studying, in particular, Tri-plate stripline, Micro stripline, the wave propagates through a medium with velocity p Coupled micro stripline, Inverted and suspended micro v = 1= ¹². If the medium is not free space but a striplines, Slot lines, and Coplanar waveguide. We delve uniform dielectric, then the velocity of propagation is into the analysis of the losses involved in some of these transmission structures. We will also look at the design given by (1). equations for these transmission structures p v = v0= ²r (1) p where v0 = 1= ¹0²0 is the free-space velocity and ²r I. INTRODUCTION is the relative permit Millimeter waves are extensively being used for Radar V V 1 and Wireless communication. Wireless communication Z = 0 = 0 = (2) includes wireless computer networks, voice and data net- I0 Qv Cv works etc., Transmission lines forms an integral part of where C is the capacitance between the conductors these Microwave Integrated Circuits (MICs) and Mono- per unit length. The electrostatic capacitance used in (43) lithic Microwave Integrated Circuits (MMICs). One ma- is independant of the operating frequency and depends jor constraint on the transmission lines for their use in only on the static field configuration in the transmission MICs is that they have to be planar. Microstrip lines, slot line [3]. If the substrate is thin in terms of wavelength lines and coplanar structures are used as the fundamental and the strip width is also narrow compared to the blocks in building these circuits. All these structures wavelength, and high dielectric substrates are used, then are planar, their characteristics being controlled by their static analysis itslef can be enough [4], [5].Substituting dimensions in that plane. the dielectric by air (²r = 1) in (43), we have Over the years various static and dispersion mod- 1 els have been developed for the analysis of striplines. Za = (3) C v Various models for the analysis of these structures has a a where C is the capacitance per unit length of the been consolidated here. Different methods adopted for a transmission line with dielectric replaced by air. Dividing Static and dispersion analysis of microstrips are shown (3) by (43) and putting ² = C=C we get in Fig 1 (next page). We shall begin with a brief r a introduction on the classic methods employed for the Z Z = p a (4) static and dispersion analysis of the various strip line ²r configurations. We shall then proceed with the static and [2]Open transmission lines like Microstrip lines are dispersion analysis of the individual striplines mentioned examples of mixed dielectric problem and as such cant above. We shall conclude with a section on the various support TEM waves. To aid in the analysis of these design equations that can be used during the stripline structures, a new quantity called ’effective dielectric con- design process. stant’is defined under quasi-static approximation. This quantity is defined as II. STATIC ANALYSIS C Static analysis of striplines involves the analysis of ²eff = (5) C the transmission structure at frequency, f = 0. The a The expressions for the phase velocity and character- analysis is carried out to find the vital parameters of the istic impedance Z then follows transmission lines viz., Characterisitc impedance(Z ), 0 p Effective dielectric permittivity (²eff ), and the phase v = va= ²eff (6) 3 Fig. 1. Various techniques for Microstrip Analysis [1] criterion for analyticity is that: if u(x; y) and v(x; y) p @u @v Z = Za= ²eff (7) satisfy the Cauchy-Reimann equations, @x = @y and @u = ¡ @v at all points in domain D, then the function Propagation constants calculated using (5) - (7) give @y @x results accurate enough for most of the practical cases f(z) = u(x; y) + jv(x; y) is analytic everywhere in D [2]. provided u(x; y) and v(x; y) are continious and has first- The electrostatic capacitance is found by the solution order partial derivatives. of a two dimensional Laplace Equation The reason we can apply conformal mapping to solve Laplace equations follow directly from a result in Com- @2Á @2Á plex analysis, which states that: If f is an analytic + = 0 (8) 0 @x2 @y2 function that maps from a domain D to D and if 0 The solution of (8) would then give us the field within W is harmonic in D , then the real valued function the structure. The total charge can be found out by using w(x; y) = W (f(z) is harmonic in in D [6]. Gauss law ZZ Q = ²0²r E ¢ dS (9) The integration in (9) being carried out over the entire surface of the transmission structure. The determination of the characteristic impedance Z0 proceeds with first finding the capacitance C of the transmission line and then using that in (5) to find the effective dielectric constant and then putting it into (6) and (7) to find out phase velocity and characteristic impedance respectively. We will now discuss the theoretical aspects of the Fig. 2. A system of curves static analysis techniques namely, The Conformal map- ping method, The Variational method, and The Finite [3], [2] Consider the solution of a two dimensional 2 Difference method. laplace equation r Á = 0 for the system of conductors shown in Fig 2 with the boundary conditions as Á = Á1 on S and Á = Á on S . The principle behind conformal A. The Conformal Mapping method 1 2 2 mapping approach is to transform the given system of [6]A mapping w = f(z) defined on a domain D is conductors to a different complex plane where it may be called conformal at z = z0 if the angle between any two easier to solve the given laplace equation. This technique curves in D intersecting at z0 is preserved by f. Such a gives an upper bound on Z0. Let us consider a conformal mapping is known as Conformal mapping (a.k.a Angle- transformation W given by (10) preserving mapping). If f(z) is analytic in domain D 0 and f (z0) 6= 0, then f is conformal at z = z0. The W = F (z) = F (x + jy) = u + jv (10) 4 where, The energy stored in the electrostatic field is given by u = u(x; y) ZZ " ¶ ¶ # 1 @Á 2 @Á 2 We = + dxdy and 2² @x @y ZZ v = v(x; y) 1 = j@ Áj2 dxdy 2² t Assuming a Transverse Electromagnetic Mode (TEM), ZZ " ¶ ¶ # 1 1 @Á 2 1 @Á 2 we shall define the gradient operator as rt = 2 + 2 h1dxh2dy 2² h1 @u h2 @v @u ax @u ay ZZ " ¶ ¶ # rtu = + (11) 1 @Á 2 @Á 2 @x h1 @y h2 = + dudv 2² @u @v and 2 @v ax @v ay = 1=2C(Á2 ¡ Á1) (14) rtv = + (12) @x h1 @y h2 The last equation shows that capacitance C is same in The scale factors h1 and h2 are given by the conformal mapped domain. ¶ ¶ Situations may sometimes arise where a particular 1 @u 2 @u 2 type of a stripline can be considered as a polygon. 2 = + h1 @x @y This polygon can then be conformal mapped to the ¶2 ¶2 upper half of a plane by the use of Schwarz-Christoffel 1 @v @v Transformation. The transformation is formulated as [6]: 2 = + h2 @x @y Let f(z) be a function that is analytic in the plane y > 0 ¯ ¯ having the derivative 1 1 1 ¯dW ¯2 = = = ¯ ¯ 0 2 2 2 ¯ ¯ f (z) = A(z¡x )®1=¼¡1(z¡x )®2=¼¡1 ¢ ¢ ¢ (z¡x )®n=¼¡1 h1 h2 h dz 1 2 n (15) where the last step directly follows from the Cauchy- where x1 < x2 < ¢ ¢ ¢ < xn and each ®i satisfies Reimann equations. Laplace equations in uv plane are 0 < ®i < 2¼. Then f(z) maps the upper half of the given by plane y > 0 to a polygon with its interior angles 2 2 ®1; ®2; ¢ ¢ ¢ ; ®n Successive applications of this transfor- @ h2 @Á @ h1 @Á @ Á @ + = + = 0 (13) mations would sometimes be needed to end up with the @u h @u @v h @v @u2 @v2 1 2 desired configuration. Detailed explanation on Schwarz- The above result shows that the potential function Á Christoffel Transformation and the method to obtain the satisfies the same Laplace equations in uv co-ordinate functions for such a mapping is outlined in [6].The steps systems.
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