Quasi-TEM_Analysis_of_Shielded_Suspended

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Quasi-TEM Analysis of Shielded Suspended 
Substrate Microstrip Line 
 
Bououden Ali
(1,2) 
(1)
Centre de développement des Technologies avancées, 
CDTA 
Baba Hassen, Alger, Algérie. 
Email : abououden@cdta.dz 
 
Mohamed Lahdi Riabi 
(2)
Faculté des sciences de la technologie, Université 
Constantine 1, Campus Ahmed Hamani, 
Constantine, Algérie 
Email : ml.riabi@yahoo.fr 
 
Abstract— This paper describes a modified Least square 
boundary method (LSBRM) to the quasi-TEM approach analysis 
of shielded suspended substrate microstrip line (SSSML).This 
method is implemented to calculate the electric-field distribution, 
characteristic impedance and effective dielectric constant, as well 
as validated against CST EM STUDIO® (CST EMS) [22]. 
The computed results are in very good agreement compare 
with those obtained by other methods and EM simulators, HFSS 
and COMSOL. 
Key words—shielded suspended substrate microstrip line, 
modified least square boundary method, characteristic impedance, 
CST EM STUDIO. 
 
I. INTRODUCTION 
The suspended substrate microstrip line is used very often 
in microwave integrated (MMIC’s) interconnects structures , 
high-speed VLSI circuits, and wireless communication 
systems ,and plays an important role in the design different 
types of filters ,antennas,etc ,due to its characteristics having 
lower attenuation, lower dispersion ,lower propagation loss. 
Due to the air gap between the substrate and ground plane of 
the suspended microstrip lines.[1] 
Recently, the Quasi-TEM mode analysis is essential in the 
study planar transmission lines, optical integrated circuits, 
microwave sensors and dispersion analysis. 
 A number of methods have been proposed for analysis of 
these circuits, Such as the point matching method 
[7,8,9],methods of lines [16] , variational method [11,12],The 
generalized bioconjugate gradient method [6],finite element 
method [15,17,13], spectral domain method [5], EM 
simulators and CAD models[71,18 ,21] 
Among these, the least-squares boundary residuals LSBR 
is the variant of the moment method .This method was 
applied for the first time by Davies [2] ,and has been applied 
by others with success to the solution of the junction of 
rectangular and cylindrical waveguides ,the analysis and 
design of impedance transformers, coupled-line filters 
,antennas and microstrip couplers, etc. [19]. 
The reference [4] presents an analysis of planar structures 
with the consideration of the Gibbs phenomenon caused by 
those singularities by a modified LSBRM . 
The aim of this work is to compute the values of 
capacitance and inductance per unit length of SSSML 
structure employing a modified LSBRM on the Quasi-TEM 
mode analysis ,this structure is also analysis using CST EM 
STUDIO, once the capacitance per unit length is obtained, 
the characteristic impedance and effective dielectric constant 
are determined, and then compare our results with available 
theoretical results obtained using other methods and EM 
simulators 
 
II. METHODS OF ANALYSIS 
A. LSBRM method 
 
Formulation of method 
Fig1 shows the cross-section to analyze the shielded 
suspended substrate microstrip line. 
We divide this structure into three subregions 
Where R1 and R3 are free space and R2 is the substrate 
isotropic and is assumed lossless dielectric ,it has a relative 
dielectric constant ( .The layers are different thickness and 
have dielectric constants respectively 
The microstrip line consists of perfectly conducting strip 
of zero thickness ( and finite width w residing on top of 
dielectric substrate is of height h, which, in turn, is enclosed 
onside perfectly conducting box and of width a and height b. 
 
 
 
 
 
 
 
 
 
Fig1.Cross-section of symmetric SSSML 
 
 
The potentials 
 
 within each region represented as a 
sum of basic functions in terms of Fourier-sine series a 
satisfying the Laplace equation and boundary condition can be 
expressed ,respectively, as 
 ∑ 
 
 
 
 
 ∑ 
 
 
 
 ] 
 ∑ 
 
 
 
 
Where 
 
 
 , N is harmonics number. 
And are coefficients to be determined. 
The electric field vector ⃗⃗⃗ in the structure which can be 
represented by the potential as 
 ⃗⃗ ⃗⃗ ⃗⃗ = ⃗ 
 
The electric displacement vector ⃗⃗ 
 ⃗⃗ ⃗ 
 
 
 
We expresses the coefficients as a function of 
the unknown by applying the continuity conditions of 
the tangential electric fields and the normal electric 
displacement components in different interface layers 
 and are given in the Appendix 
The coefficient can be determined by using modified 
LSBR method are detailed in the papers [3, 4, 10] 
The charge density is given by applying the 
Gauss theorem in the interface , 
 ⃗ - ⃗ 
The static capacitance per unit length of a SSSML is 
obtained from the varational expression 
 
 
 
 
Where is the amount of the electrostatic field energy 
per unit length . 
The energy can be computed from 
 
 
 
∫ ∫ 
 
 
 
 
 
 
 
The charge Q distribution in equation is calculated from 
 ∫ ∫ 
 
 
 
 
 
 
 
 Where is the electrostatic potential and is the surface 
charge density on the microstrip conductor. 
The effective (dielectric) permittivity is 
 
 
 
 
Where is the capacitance of a SSSML and is the free 
–space capacitance of the line. 
The phase velocity is 
 
 
√ 
 
Where is the velocity of the light in free space. The 
characteristic impedance is 
 
 
 
= 
 
 √ 
 
The inductance per unit length L is 
 
 
 
 
Where are permeability and permittivity of free 
space respectively. 
B. CST EMS simulator 
Next, we modify the structure shown in Fig.1 by the 3D 
SSSML structure depicted in Fig.3 has been modeled and 
analyzed using Finite Integration Technique implemented in 
commercial software CST EM STUDIO® (CST EMS). 
The physical parameters used in simulation are: 
 
These modeling and simulation allowed us to calculate the 
matrix capacitance, then to compute the characteristic 
impedance and potential distribution of the SSSML. 
 
(1) 
(2) 
(3) 
(4) 
(5) 
(7) 
(6) 
(8) 
(9) 
(10) 
(11) 
 
Fig.4 3D structure of SSSML 
III. RESULTS AND DISCUSSION 
In this section , we have validated and checked the 
modified LSBRM. we first compare the accuracy of 
computation of the characteristic impedance and effective 
dielectric constant for SSSML structure with the results 
obtained using CAD model and other methods . 
Table 1 lists calculated the capacitance and 
characteristic impedance values of the SSSML with various 
width dimensions of microstrip line . 
Comparaisons were made between our calaculated 
results using the modified LSBRM and TOMAR [27]. 
They have very small differences. 
Table 2 shows other comparisons results with different 
dielectric constant and new physical parameters of the 
2D SSSML. These comparaisons were made between our 
calaculated results using the modified LSBRM and results 
obtained by various methods and authors, the results are 
good agreement, which were foundfor all the calculations. 
Figure 3 shows the variation of , these 
components of static electric field distribution as a function 
of in plane for various values ratios (
 
 
 by using 
modified LSBRM. 
These figures are normalized by the maximum value of 
all distributions. 
Figure 4 depicts the 2D surface potential distributions of 
3D SSSML obtained by simulation with using CST 
EMS. 
Table 3 lists the CST EMS results for calculating the 
characteristic impedance values of the SSSML with 
different ratios 
 
 
 .The results in Table 3 are very close 
values compared with other electromagneic simulators . 
 
 
 
 
 
 
 
 
 
 
 
From these results it appears clearly that all geometrical 
parameters of SSSML have large effects on the 
characteristics impedance and effectives dielectric values . 
The deviation of the results by using modified LSBRM are 
about 3% ,these are due to the assumptions of perfectly 
conducting of shielding and micrstrip line and for the other 
we neglect the metals thickness of them. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig.3. Electric field distribution components for SSSML structure 𝑎 𝑏 
 𝑚𝑖𝑙 𝑚𝑖𝑙 
𝑏
 
 𝜀𝑟 
 
Fig. 4. 2D surface potential distributions 
Table 1. Calculated characteristic impedance of SSSML 𝑎 𝑏 𝑚𝑖𝑙 𝑚𝑖𝑙 
𝑏
 
 𝜀𝑟 
 
 Modified LSBRM (our results) Empirical Model[21] Variational Method [21] 
𝒘
𝒉
 
𝑪(pF/m) 𝑪𝟎(pF/m) 𝑳 𝝁𝑯/𝒎) 𝒁𝒄 𝜴 √𝝐𝒆𝒇𝒇 𝒁𝒄 𝜴 √𝝐𝒆𝒇𝒇 𝒁𝒄 𝜴 √𝝐𝒆𝒇𝒇 
0.1 9.32 8.59 1.294 372.688 1.0415 329.11 1.2188 350.98 1.544 
1 15.90 12.70 0.878 234.912 1,1214 228.69 1.1550 242.22 1.1083 
5 21.70 19.00 0.583 163.952 1,0675 151.16 1.1142 161.59 1.0646 
10 27.90 25.2 0.440 125.754 1,0511 120.72 1.0539 124.20 1.0504 
20 40.90 37.5 0.296 85.087 1,0434 80.24 1.0890 84.81 1.0422 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
. 
 
 
 
 
 
 
 
 
 
 
 
IV. CONCLUSION 
This article has presented a detailed description of the quasi 
TEM analysis of SSSML .The results obtained using 
modified LSBRM and Finite Integration Technique 
implemented in commercial software CST EMS for 
computing the characteristic impedance and effective 
dielectric constant. Some examples have been included to 
illustrate the strength and versatility of the modified 
LSBRM. The validity of this method was checked by 
various comparisons with existing results obtained by other 
methods and electromagnetic simulators. 
 
APPENDIX 
The coefficients as a function of the 
unknown are given by: 
 . 
 
 (
 
 
) 
 
Where 
 )) 
 
 
 ) 
 
 
 
 (
 
 
) 
 
 
 
 (
 
 
) 
Table 3. Calculated characteristic impedance of SSSML using CST EMS 𝑎 𝑏 𝑚𝑖𝑙 𝑑 𝑚𝑖𝑙 𝑚𝑖𝑙 
𝑏
 
 𝑡 𝑚𝑖𝑙 𝜀𝑟 
 
 CST EMS HFSS[21] COMSOL[17 ] 
𝒘
𝒉
 
𝐂(pF/m) 𝐂𝟎(pF/m) 𝐋 𝛍𝐇/𝐦) 𝐙𝐜 𝛀 √𝛜𝐞𝐟𝐟 𝐙𝐜 𝛀 √𝛜𝐞𝐟𝐟 𝐙𝐜 𝛀 √𝛜𝐞𝐟𝐟 
0.1 11.81 9,00 1,234 323.153 1.1458 366.82 1.1485 366.82 1.1539 
1 15.53 12.71 0.8740 237.244 1.1052 255.46 1.1072 255.46 1.1079 
5 22.20 19.61 0.5664 159.751 1.0637 165.19 1.0625 165.19 1.0642 
10 28.58 25.90 0.4299 122.494 1.0505 126.23 1.0507 –– ––– 
20 41.87 38.42 0.2900 83.0978 1.0439 85.84 1.0445 –– ––– 
 
 
Table 2. Comparison of characteristic impedance of SSSML between our results and the values calculated using other methods ( 4𝑏 
 𝑏 6𝑏 𝑏 𝑤 
 
 𝜺𝒓 𝟑 𝟕𝟖 𝜺𝒓 𝟐𝟓 
𝑎
𝑤
 
𝒁𝒄 𝜴 
Modified LSBRM 
𝒁𝒄 𝜴 𝟏𝟐] 𝒁𝒄 𝜴 𝟏𝟒] 𝒁𝒄 𝜴 𝟐𝟎] 𝒁𝒄 𝜴 
Modified LSBRM 
𝒁𝒄 𝜴 𝟏𝟐] 𝒁𝒄 𝜴 𝟏𝟒] 𝒁𝒄 𝜴 𝟐𝟎] 
2 52.05 52.60 52.59 51.78 31,80 31.50 32.30 32.01 
5 54,10 54.80 55.15 54.35 40.95 40.70 40.84 40.71 
9 55,37 55.00 56.57 54.34 41,10 41.00 41.91 41.18 
 
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