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Conductor Loss, Dielectric Loss, Multilayer Coplanar Waveguide International Journal of Electromagnetics and Applications 2012, 2(6): 174-181 DOI: 10.5923/j.ijea.20120206.07 Loss Computation of Multilayer Coplanar Waveguide using Single Layer Reduction Method A. K. Verma, Paramjeet Singh* Department of Electronic Science University of Delhi South Campus, New Delhi 110021, India Abstract In this paper, the quasi-static spectral domain approach (SDA) and single layer reduction (SLR) method applicable for multilayer coplanar waveguide (CPW) on isotropic dielectric substrate that incorporates two layer model of conductor thickness is used to compute the impedance, effective dielectric constant, dielectric loss and conductor loss. The transverse transmission line (TTL) technique is used to find the Green’s function for the multilayer CPW in Fourier domain. Die lectric loss of multilayer coplanar waveguide is computed by converting multilayer CPW structure into equivalent single layer CPW using SLR method. Perturbation method is used to compute the conductor loss. The effect of finite conductor thickness is analysed on the impedance, effective dielectric constant, dielectric loss and conductor loss. The present formulation also accounts for low frequency dispersion in computation of quasi-static effective dielectric constant and characteristic impedance. Ke ywo rds Conductor Loss, Dielectric Loss, Multilayer Coplanar Waveguide, Quasi-Static Spectral Domain Approach, Single Layer Reduction Method, Transverse Transmission Line Technique the impedance, effective relative permittivity, dielectric loss 1. Introduction and conductor loss of multilayer CPW structures using quasi-static SDA method on isotropic dielectric substrate. There has been a growing interest in different Dispersion characteristics of multilayer CPW is also configurations of coplanar waveguide (CPW) transmission analysed using single layer reduction (SLR) method. line such as conductor backed CPW, CPW with finite ground width, shielded multilayer CPW, elevated or suspended CPW, asymmetric CPW etc. The CPW offers many 2. Coplanar Waveguide Analysis advantages over microstrip line such as easy connection of The conventional CPW consist of centred conductor both series and shunt components without drilling holes between two infinite ground planes with all on the same through the dielectric substrate, less dispersive and provides plane. The height of substrate is also infinite. But in practical ease of controlling the characteristics of line by changing the case, both the width of the ground planes and height of strip and slot width. Therefore, CPW transmission lines are substrate are finite. The cross section structure of shielded widely used in the design of components and interconnect in multilayer coplanar waveguide of finite conductor thickness modern microwave circuit design. The accurate modelling of under investigation using the quasi-static SDA formu lation transmission lines is important in the design and is shown in Figure. 1. development of modern integrated circuits. Most research In this paper an efficient method for calculating the line done in literature on CPW was based under the assumption capacitance of shielded multilayer CPW of finite conductor of zero conductor thickness using the conformal mapping thickness is presented. The effect of conductor thickness is and spectral domain approach (SDA) method. But in accounted by accommodating the two layer strip conductor practical cases, the conductor thickness can not be zero and model that is suggested for a microstrip line[4]. The potential affects the impedance, effective dielectric constant and and charge distribution basis function on the strips is losses etc. Several research papers have been presented to required to compute the line capacitance. The potential on characterize planar transmission line structure with finite the strips is expressed in terms of charge distribution on the metallization thickness[1-13]. upper and lower strips. The unknown charge distribution This paper presents the effects of finite strip thickness on function can be expanded in term of linear combination of basis function. The total potential in the range[0, L] is equal * Corresponding author: [email protected] (Paramjeet Singh) to the sum of the potential on strips conductor and potential Published online at http://journal.sapub.org/ijea in two slot region. Then, Galerkin method is applied to find Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved the unknown coefficient of chare density basis function. The 175 International Journal of Electromagnetics and Applications 2012, 2(6): 174-181 total charge on the coplanar waveguide is sum of charge at ~ ~ where Vd is potential on dielectric slot region and Vc is upper and lower strip of centre conductor. Therefore, the potential on conductor strip region. The potential at lower quasi-static SDA obtains the total charge on the central strip. ~ ~ and upper surface of the conductor are 1 β + 1 β By using the total charge on the central strip the per unit Vc ( n ) Vd ( n ) ~ 2 ~ 2 length capacitance is computed. The effective dielectric and Vc (β n )+ Vd (β n ) respectively. constant and characteristic impedance of the CPW ~ ~ ~ ~ V1 (β )+ V1 (β ) transmission line can be determined using capacitance per β + β = c n d n Vc ( n ) Vd ( n ) ~ 2 ~ 2 (3) unit length. Vc (β n )+ Vd (β n ) The Green’s function for multilayer CPW in Fourier domain is obtained by transverse transmission line (TT L) technique. The Green’s function for the multilayer CPW shown in Figure 1 is given by ~ ~ ~ G11 (βn ) G12 (βn ) G(βn ) = ~ ~ (4) G 21 (βn ) G 22 (βn ) where ~ 1 G11 (β n ) = (5a ) β n (GF1 + GF3 ) ~ 1 G 22 (β n ) = (5b) β n (GF2 + GF4 ) ~ ~ ε r3 sinh(βn h 3 ) G12 (βn ) = G 22 (βn ) (5c) ε r3 coth(βn h 3 ) + GF1 ~ ~ ε r3 sinh(βn h 3 ) G 21 (βn ) = G11 (βn ) (5d) ε r3 coth(βn h 3 ) + GF2 ε + ε coth(β h ) coth(β h ) GF = ε r2 r1 n 1 n 2 (5e) Fi gure 1. Mult ilayer Shielded Coplanar Waveguide 1 r2 ε r1 coth(βn h1 ) + ε r2 coth(βn h 2 ) The charge density basis functions are confined only on ε r4 + ε r5 coth(βn h 4 ) coth(βn h 5 ) GF = ε (5f) conductors strip and are zero outside the conductor strips. 2 r4 ε coth(β h ) + ε coth(β h ) The basis function for the charge for centre strip W, left r4 n 4 r5 n 5 ground G and right ground G strip is ρ (x), ρ ( x), ρ (x) ε + coth(β h )GF 1 2 s1i s2i s3i GF = ε r3 n 3 2 (5g) respectively, where i = 1, 2, 3.... the number of basis terms 3 r3 ε r3 coth(βn h 3 ) + GF2 of the corresponding strips. The Fourier transform of these ε r3 + coth(βn h 3 )GF1 basis functions of charge density on central conductor of GF = ε (5h) 4 r3 ε β + width W , left ground G1 and right ground G2 are r3 coth( n h 3 ) GF1 ~ ~ ~ ρs1i (βn ), ρs2i (βn ), ρs3i (βn ) respectively. In the spectral On substituting Eq.(1), Eq.(3) and Eq.(4) in Eq.(2), ~ ~ domain, the charge density basis function for lower and G11 (β ) G12 (β ) ~ n ~ n × upper surface of conductor plane is given by- G β G β (6) 21 ( n ) 22 ( n ) ρ~ (β )= si n N N N 1 1 1 N N N ∑ a1ρ~ (β )+ ∑ b1ρ~ (β )+ ∑ c1ρ~ (β ) 1 1~ 1 1~ 1 1~ i s1i n i s2i n i s3i n ∑ a ρ (β )+ ∑ b ρ (β )+ ∑ c ρ (β ) i = 1 i = 1 i = 1 = i s1i n i s2i n i s3i n (1) N N N i = 1 i = 1 i = 1 1 1 1 N N N ∑ a2ρ~ β + ∑ b2ρ~ β + ∑ c2ρ~ β 1 1 1 i s1i ( n ) i s2i ( n ) i s3i ( n ) ∑ a2ρ~ β + ∑ b2ρ~ β + ∑ c2ρ~ β i = 1 i = 1 i = 1 i s1i ( n ) i s2i ( n ) i s3i ( n ) i = 1 i = 1 i = 1 ~ ~ V 1 β )+V 1 β ) 1 1 1 2 2 2 c ( n d ( n where, a i , bi , ci and a i , bi , ci are unknown constant ~ ~ V 2 (β )+V 2 (β ) coefficients associated with central conductor, left ground c n d n Now take the inner product of Eq. (6) with testing function and right ground respectively for lower and upper surface of ~ ~ ~ ρt1j (βn ), ρt2 j (βn ) , ρt3j (βn ) and applying the Parseval’s the conductor, N1 is number of basis function. The Green’s function and the basis function in Fourier domain are related identity. By Galerkin’s method solve for the unknown to potential on strip and slot region by following expression, constant of the charge basis function The line capacitance ~ ~ ~ per unit length of finite conductor thickness CPW is G(β )~ρ (β ) = V (β ) + V (β ) (2) n si n c n d n computed by substituting the constants in Eq. (7). A. K. Verma et al.: Loss Computation of Multilayer Coplanar Waveguide using Single Layer Reduction Method 176 N1 where q is filling factor for coplanar waveguide. The C = (a1Q + a 2Q ) dielectric loss[15] of the coplanar waveguide is given by, ε ∑ i i i i (7) o 8.686π ε′req (ε′ −1) i=1 α = reff tan(δ ) (14) d λ ε′ − eq 0 ε′reff ( req 1) G1 +S1 +W The conductor loss of CPW is computed by perturbation Q = ρ (x)dx where i ∫ s1i method (Pertb. M.),[16-18]. G1 +S1 R s α c = × 2 ( ) − 2 The following charge distribution basis function on central 16ZK k (1 k ) and ground conductor strip is used[14]. 1 2a b − a + ∆ ln −1 + (15) − − (x S1 G1 ) a ∆ b + a + ∆ cos (i −1) π W ρ = 1 2b b − a + ∆ s1i (x) ln −1 2 (8a) ∆ + − ∆ 2 (x −S1 − G1 )− W b b a 1− W where K(k) is co mplete elliptic integral o f first kind, ∆ is (G1 + S1 ≤ x ≤ G1 + S1 + W) stopping distance, k = a b , a = W 2 , b = S1 + W 2 .
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