Analysis of Microstrip Lines on Substrates Composed of Several Dielectric Layers under the Application of the Discrete Mode Matching Subtitle

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Master Program in Electronics/Telecommunications Examiner: Olof Bengtsson

Supervisor: Marcos V. T. Heckler

DEPARTMENT OF TECHNOLOGY AND BUILT ENVIRONMENT

Analysis of Microstrip Lines on Substrates Composed of Several Dielectric Layers under the Application of the

Discrete Mode Matching Subtitle

Manuel Gustavo Sotomayor Polar September 2008

Master’s Thesis in Electronics/Telecommunications

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Abstract

Microstrip structures became very attractive with the development of cost-effective dielectric materials. Among several techniques suitable to the analysis of such structures, the discrete mode matching method (DMM) is a full-wave approach that allows a fast solution to Helmholz equation. Combined with a full-wave equivalent circuit, the DMM allows fast and accurate analysis of microstrips lines on multilayered substrates.

The knowledge of properties like dispersion and electromagnetic fields is essential in the implementation of such transmission lines. For this objective a MATLAB computer code was developed based on the discrete mode matching method (DMM) to perform this analysis.

The principal parameter for the analysis is the utilization of different dielectric profiles with the aim of a reduction in the dispersion in comparison with one-layer cylindrical microstrip line, showing a reduction of almost 50%. The analysis also includes current density distribution and electromagnetic fields representation. Finally, the data is compared with Ansoft HFSS to validate the results.

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ii

Acknowledgement

It would be unfair to start this thesis without the appropriate acknowledgement to all the people who helped in one way or another in the accomplishment of this thesis, but it is equally unfair not to acknowledge expressly many people who were keystones in this work.

I would like to express my gratitude to my supervisor Marcos Heckler, his help, wise advices and encouragements make possible the culmination of this thesis.

I would also like expressly to thank my dear friends Efrain, Juan Felipe, Juan Jose and Olof for sharing wise advices and good times. To Enrique and Nikola, who have supported me during my stay in Germany, making workdays nice and funny, especially at lunch time. To The Orates group, for sharing a lifetime plenty of true friendship.

This thesis is dedicated to my beloved family, which always supported me in its realization. I am very grateful for their love, understanding, help, support and to the examples of my father F. Carlos Sotomayor Campana, my mother Ana Maria Polar de Sotomayor, my brother Carlos and my sisters Victoria and Ana, their unflinching courage and conviction will always inspire me. It is to them that I dedicate this work.

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List of Figures

Figure 1: (a) Outside cylindrical microstrip line. (b) Inside cylindrical microstrip line...5

Figure 2: Discretization of a cylindrical microstrip line ...7

Figure 3: Stratified dielectric layer ...7

Figure 4: Full-wave Equivalent Circuit model ... 10

Figure 5: Full wave equivalent circuit, with only one current source. ... 10

Figure 6: Dielectric profile of a multilayered microstrip line... 14

Figure 7: a) Discretization considering N=2, b) Discretization considering N=10, N: number of e-lines on the microstrip... 16

Figure 8: Extrapolation of the propagation constant when infinite number of e-lines on the microstrip ... 20

Figure 9: Sequence to calculate the fields in the substrate. ... 22

Figure 10: Sequence to calculate the fields in the superstrate. ... 23

Figure 11: Interpolation of both the electric and magnetic field depending on the discrete angles ... 24

Figure 12: Determination of the improvement in the dispersion. ... 25

Figure 13: Deviation from the perfect circular cylinder ... 25

Figure 14: Substrate and superstrate configuration of the microstrip line used for the validation ... 26

Figure 15: a) Electric permittivity profiles of one-layer microstrip lines, b) Dispersion of the transmission lines. ... 27

Figure 16: Dispersion changes because of different profiles of the electric permittivity in the substrate... 28

Figure 17: Dispersion changes due to different profiles of the electric permittivity in the superstrate.. ... 28

Figure 18: Dispersion improvements because of different profiles of the electric permittivity in both the substrate and the superstrate. ...29

Figure 21: Dispersion improvements because of considering one continuous profile. ...31

Figure 22: Validation of the results, comparative between DMM and HFSS results in three different cases... 31

Figure 23: Electric Field Distribution of a microstrip line with air as both substrate and superstrate... 32

Figure 24: Magnetic Field Distribution... 33

Figure 25: Real power distribution, a) Results obtained from DMM, b) Results obtained from HFSS. ... 33

Figure 26: a) Electric field distribution, b) Electric permittivity profile, c) Dispersion improvement. ... 34

Figure 27: Magnetic field distribution ... 34

Figure 28: Real power distribution, a) Results obtained from DMM, b) Results obtained from HFSS. ... 35

Figure 29: Invested time per simulation (average) using DMM and HFSS to calculate the propagation constant. ... 35

Figure 30: Second Propagation Mode, a) Electric Field. b) Dielectric Profile. c) Magnetic Field. d) Real Power Flow. ... 38

Figure 31: Electric permittivity profile, Dispersion and real power. ... 41

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Chapter 1 Introduction

1.1. The Problem

Microstrip lines are also utilized in applications where a cylindrical body is present as vehicles, aircrafts, missiles or sounding rockets, because of their light weight and conformability. The knowledge of properties like dispersion, electromagnetic fields and current distributions, is essential in their implementation.

1.2. Goal

The goal of this work is to investigate of the dispersion properties of conformal microstrip lines printed on cylindrical structures, which consist of multiple dielectric layers. For this purpose a MATLAB computer code will be developed based on the discrete mode matching method (DMM) to compute the propagation constants.

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Chapter 1: Introduction 2

_______________________________________________________________________

The main parameter for the analysis will be the utilization of different dielectric profiles. Finally a validation of DMM’s results will be carried out comparing them with results given by HFSS.

1.3. Previous Research

As showed in [1], an extension of the Discrete Mode Matching for the analysis of structures in cylindrical coordinates has been presented, giving very accurate results when determining both the propagation constant and the effective dielectric constant in cylindrical microstrip lines. It also suggests a way to be implemented, so the analysis of microstrip lines composed of several dielectric layers could be done.

Although there are several methods and approaches to analyze transmission lines with cylindrical structure, some of them treated rectangular microstrip lines and some others cylindrical microstrip lines and even cylindrical striplines; every one of them has advantages and drawbacks.

The method of lines is suitable for the analysis of asymmetric cylindrical homogeneous and inhomogeneous guided wave structures using rectangular or cylindrical structures [2]. Another approach makes the microstrip with multi- layers dielectric equal to the common one simplifying the boundary conditions using optical theory [3]. In [4] an analysis of frequency dependent propagation characteristic of microstrip lines anisotropic substrate and overlay that uses the Galerkin’s procedure given a good agreement with the results, but it only considers just one layer in the substrate and one layer in the superstrate. A full- wave two dimensional Finite-Difference-Time-Domain method in cylindrical coordinate system is presented in [5], this method proved to be efficient and economical in both CPU time, temporary storage requirement and it can also be used to study the cylindrical optical fiber.

There is also a patent of a method to analyze the properties of cylindrical transmission lines based on the use of green’s functions.

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1.4. Justification of the Project

Cylindrical microstrip lines have been gaining more attention lately because of the need for new kinds of antennas and/or devices which can be mounted in curved surfaces. That is why a fast and accurate analysis is required not only to obtain the intrinsic characteristics, but also to determine how the utilization of several dielectric layers affects them.

This kind of analysis can be achieved under the application of the Discrete Mode Matching (DMM) method, which is well suited to the analysis of multilayered structures, since the fields must be only sampled at the interfaces between the dielectric layers. The analytical solution is obtained with the Green's functions using a full-wave equivalent circuit.

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Chapter 2 Theory

2.1. Discrete Mode Matching

2.1.1. Cylindrical Microstrip Lines

Basically there are two kinds of cylindrical microstrip lines; the outside cylindrical microstrip line and the inside cylindrical microstrip line [6], as it is shown in Figure 1. These transmission lines have different response; because of its special configuration. On the other hand, since the outside cylindrical microstrip line has a more familiar configuration; it was chosen to be the basic model for the analysis in this thesis work; although the method of analysis can be applied in both cases indistinctly.

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Figure 1: (a) Outside cylindrical microstrip line. (b) Inside cylindrical microstrip line

2.1.2. Wave Equation in Cylindrical Coordinates

Considering an infinite long transmission line in z-direction, the propagation with e^^{jk z}^z^ and assuming time harmonic variations, a full wave solution is expected as a result of the wave equation within every source-free layer normalized by k₀[1, 7].

 

2

2 _d , ,z 0

     

  

      

  

      

 

(2.1) Where  represents each of the components E_z or H_z

2

d r r kz

   

r: is the relative permeability

r: is the relative permittivity

k : is the propagation constant along the z-direction z

The solution of the wave equation can be written as the modal expansion in the

-direction.



^,



¹ ^^{ }^{( )}

2

i ji

i

e ^

    













 ^(2.2)

Where the tilde indicates that, the field components are in the spectral domain.

In order to apply the Discrete Mode Matching formulation, N terms have to be

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Chapter 2: Theory 6

_______________________________________________________________________

taken in (2.2), which correspond to the number of e-lines used to discretize the whole cross section [1], see Figure 2.

For a multilayered structure, the electromagnetic fields within every layer “k”

in the spectral domain are given by:

^{ }^{( )}



k

 

k



i

k A J kk i _ B Y kk i _

      (2.3)

Where J x and _i( ) Y x are the Bessel functions of first and second kind _i( ) respectively [1].

The modal expansion, in j -direction, is carried out using the next equation [8].

( ) 2

0 ( , ) ^{j k z}^z ^jⁱ

i ^ ^  z e ^ e^{ }dzd

^ 

 

  ^(2.4)

2.1.3. Tangential Field Components

The other field components which are tangential to the dielectric interfaces are given in spatial domain by [8]:

2

2 2

0 0

1

r

z r r

z r

z j

z



   

   

   

   

 

    

      

       

     

  

 

E E

H H (2.5)

2.1.4. Discretization

There are three basic rules to perform the discretization in order to use the Discrete Mode Matching method.

The discretization starts at 0.25() from the edges of the microstrip line, where “” is the angular distance between discretization lines of the same kind.

The first discretization point from any of the edges of the microstrip line is designated to e-lines.

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The e-lines and h-lines are related to the transformation matrices which will be explained later on.

The next figure shows, a discretization example following the guidelines described before.

-10 -8 -6 -4 -2 0 2 4

4 6 8 10 12 14 16 18

X-axis (mm)

Y-axis (mm)

Discretization

e-lines h-lines metallization

0.2 5(j)0.5(j)

-15 -10 -5 0 5 10 15

X-axis (mm)

Y-axis (mm)

Discretization

0.5 0.25

Figure 2: Discretization of a cylindrical microstrip line

In order to perform the discretization, some designations have to be made in the limits of a given layer. The next figure depicts the designations of the distance in- direction for each layer.[8]

Figure 3: Stratified dielectric layer

2.1.5. Matrix Formulations

The E_z and H_z matrices containing the information of the fields (z-direction) in the spatial domain are given by the pair of equation in (2.6), the transformation matrices T and _e T are found by the set of equations in (2.7) _h [1].

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Chapter 2: Theory 8

_______________________________________________________________________



z e z

z h z



E T E

H T H

(2.6)

 

, ,

h

e

h ji

h h

e ji

e e

i e i e



 

 



 

T T

(2.7)

Where ^e and ^h are the angles of the discretization lines (e-lines or h-lines) according to Figure 2.

These transformation matrices will be used to transform the fields and the Green’s functions in the spectral domain back to the spatial domain in which it is needed to apply the boundary conditions later on.

In this case, when transforming the dyadic Green matrix to the spatial domain, the transformation process is given by the next equation.

z z

z zz z zz

   

 



   

    

   

 

 

    

1

h h

1

e e

G G T 0 G G T 0

G G 0 T G G 0 T

 

  (2.8)

2.1.6. Hybrid Matrix

The Hybrid Matrix K , using different intrinsic characteristics of the layers _k (_r, _r, physical dimensions), gives information about the relationship between the fields in both sides of every single layer; so, it is possible to track down the evolution of the fields through the layers. There is one K hybrid _k matrix per layer which is composed by submatrices [8],

  

 

k k k

k k

 

  

 

 

V Z

K

Y B

(2.9)

where the submatrices:V_k,Z_k, Y_k and B_k are 2x2 matrices given by the next set of equations.

 

2

z k

k

 k k 



 ^^ ^ ^ ^ ^ ^ ^

  

 

r ν rν q

V 0 q

 (2.10)

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

² ²



²

2 2

r r z z

k

r z k

k k k

k k

   

  

  



       

  

  

 

 

s ν p νp

Z

νp p

 (2.11)

 

2

2 2 2

2

z k

r z r r z

k

k k

k k k

  

   



   ^

   

  

    

 

 

p νp

Y νp s ν p

 (2.12)

 

2 2 k

z k

k k



 





 

    ^ν ^ν  

q 0

B ν r q r

 (2.13)

Where the recurrence relations for cross-product of Bessel functions can be found in [9].

Now that hybrid matrices are known in each layer, the equivalent hybrid matrix is obtained by a multiplication of each of them as shown in equation (2.14).

1 n

eq k

k

eq eq

eq

eq eq



 



 

  



K K

V Z

K Y B

 

  

(2.14)

Taking into account both the influence of the external medium and the inner ground cylinder, the correspondent admittance matrices are found using the following equations [10].

1 2

1

n z

vn n

n n n z n

k k



  



 

   

 

I ν

Y u

ν y

 (2.15)

2

2 2 2

2 1 z

n rn rn n n

n

k

k_    _ ^

 

y ν u (2.16)

     

¹

'( 2) (2 )

1 1

n k n n k n n

  _ν   _  _ν   _ ^

u H H (2.17)

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Chapter 2: Theory 10

_______________________________________________________________________

2.2. Full-Wave Equivalent Circuit

The Full-Wave equivalent circuit is a special representation of layered structure including the metallizations and dielectric layers of the microstrip line.

Because of DMM requires an evaluation of the fields just next to the interfaces of each given layer, the model of the equivalent circuit relates current densities and fields at those interfaces. The next figures show the full-wave equivalent circuit of one-layer microstrip line.

Keq

Em

Hm^

Hm^

Jm Y⁽⁰⁾ H0

-5

0

5 x 10^-3

0.006 0.008 0.01 0.012 0.014 0.016

X (m)

Y (m) Microstrip Line

1



r



_r₂

Figure 4: Full-wave Equivalent Circuit model

The grounded metallization is represented with a short circuit on the left hand side of the diagram, the substrate is represented by its equivalent hybrid matrix, the microstrip metallization is represented by a current source and the surrounding medium is represented by an admittance element. Figure 5 shows a detailed diagram of a multilayered microstrip line.

E0

H0 H₁

1

Em



• • •

1

Hm

K1

Km E_m

1

Km

Hm^

Hm^

Jm

1

Em 1

Hm

• • •

1

Kn 2

Hn

1

En

En 1

Hn

Hn

Yn

Ku



K0





Em

 Hm^

Hm^

Jm Y^{(0 )}

( )u

Y

( )a

( )b

Figure 5: Full wave equivalent circuit, with only one current source.

(a) Expanded Circuit, (b) Simplified Circuit.

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The admittance matrices Y^{( )}^u and Y⁽⁰⁾ are determined using the next equations. Regarding that there is a metallic cylinder in the interior of the structure [11], then:

( )u 1

u u

Y Z V  ^ (2.18)

where Z_u_and

Vu represent the components of the hybrid matrix K_u_{which is} the equivalent hybrid matrix of the substrate layers.

And considering that there is no an exterior metallic cladding surrounding the structure, Y can be represented as [11]: ⁽⁰⁾

  

¹

(0)

0 0 n 0 0 n

Y  Y B Y  V Z Y  ^ (2.19)

where the elements with sub-index “0” are the components of the hybrid matrixK , which is the equivalent hybrid matrix of the superstrate layers; and ₀ Y is the admittance matrix of the last layer that extends to infinity, generally it n

is considered the layer of air (surrounding medium).

According to Figure 5b, the magnetic field in spectral domain can be found using:

( )

(0) u

m m

H Y E

 

 

 



 

  (2.20)

2.2.1. Propagation Constant

Using a simple circuit analysis, the Green’s functions can be derived to obtain the relationship between the electric field and current density at the microstrip interface.

 





0

z

z z

z zz

j ^ ^ ^ ^



 



    

   

 

    

   

G G J E

J

G G E

(2.21)

Where G is the dyadic Green’s function in the spectral domain, and J and E represent the electric surface current densities and the electric field at the interfaces respectively [1].

Using equation(2.8), the last equation in the spatial domain is given by:

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_______________________________________________________________________

0

z

z zz z z

j ^ ^ ^ ^



 



     

    

   

 

G G J E

G G J E (2.22)

Regarding that tangential component of the electric field must vanish in the metallization, two systems equations are obtained since the boundary conditions are complementary [1]:

and/or

red red  red red 

G J 0 Y E 0 (2.23)

The only value of k_z, which fulfills either of these equations, is the propagation constant of the line; the dispersion can be obtained when calculating it for different frequencies.

2.2.2. Fields Analysis

From the definition of a point source considered in [11], the relationship of the electric field and the current density in the microstrip interface is given by:

( ' ' )

' '

'

( ) ( ) 1 ˆ ^j ^{k z}^z

n z nk z k

k

k k e ^



 

   ^ν ^I

E G J (2.24)

Expanding the previous equation:

( ' ' )

0

( ' ' )

' '

ˆ ˆ

z

j k z

k n z

j k z

z zz

k

zn k z

e J

j

e J



  



 

 

 

 

  

  

   

 

    

     

ν I

G G

E

G G

E

 



 

 (2.25)

Considering a single microstrip line composed of only one layer in the substrate, the magnetic field in spectral domain is related to the electric field also in the spectral domain by the next equation [11].

1 0

1

n n n

zn n n

n

n n zn

j ^



 





 

    

      

   

 

H Y E

H E

H Y E

  

 

  

(2.26)

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Where: Y is the admittance matrix of the most external layer (surrounding _n medium), and generally considered as air (cf. Equation (2.15)).

The relationship between the fields in one given interface and the fields in the next interface within either the substrate or the superstrate is given by the equivalent hybrid matrix between those two interfaces as the next equations present [11]:

1

k k k k



     

    

     

E V Z E

H Y B H

   

    (2.27)

0

^k ^k

k

zk

zk k

k k

j ^





 

 

  

 

  

   

 

 

E E

E H H

H

 



 



(2.28)

Regarding that  and z components are known parameters, the components of both electric field and magnetic field in - direction can be found from the z- components of both fields as indicated in the next equations [11].

2

0 0

1 1

1

z rk

k k zk

k zk

rk z

jk

k jk



 

  

 

  

  

  

 

    

 

   



 

   

      

I ν

E E

H H

ν I

 

  (2.29)

In order to perform the derivative of the fields in the spectral domain, the general solution for the wave equation also in spectral domain is applied as follows.

1 1

1 k k k

k

k k k

k

 

      

    

   

 

ν ν

J Y A

Ψ

J Y B

Ψ



 (2.30)

1 1

1 k k k

k

k k k

k

k_



 

  

     

        

      

ν ν

J Y A

Ψ

J Y B

Ψ



 (2.31)

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_______________________________________________________________________

 

1 1

k k k

k k

  



  

 

C C

(2.32)

Where, Ψ represents any of the electromagnetic field in z-direction within _k any given interface; and C__k represents the Bessel function of first or second kind of order .

Equations (2.30) and (2.31) form a complete set of linear equations, which guaranties a unique solution.

2.2.3. Concept of Profile

Given a multilayered microstrip line, the profile is obtained when the electric permittivity is plotted for every layer. Considering that the layers are in a certain distance from center of the cylindrical structure, the profile can also be obtained when plotting the electric permittivity vs. the distance of each layer.

Taking into account that each layer is formed by a homogeneous medium, when a great number of layers are considered, an approximation to a continuous profile can be achieved.

In all the analyzed cases, the grounded cylinder was considered to be on the left of the chart and the microstrip metallization is always in the center.

The next figure shows the concept of profile.

8 9 10 11 12 13 14 15 16

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

10

8

6

4

2

1 2 3 4 5 1 2 3 4 5

Layer

ElectricPermittivity

Dielectric Profile

Figure 6: Dielectric profile of a multilayered microstrip line

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Chapter 3 Software Implementation

3.1. Method

The method of analysis is composed mainly of three steps where each of them depends on the previous ones.

Dispersion Analysis Currents Analysis Fields Analysis

3.1.1. Basic Model

The basic microstrip line model taken for all the analysis has the next physical dimensions.

 Separation between the metallization: 0.762 mm.

 Ratio between the grounded cylinder radius and the microstrip metallization radius: 0.935.

 Ratio between width of the microstrip interface and the separation of the metallization: 5.37.

 The frequency range of the analysis is 20 GHz.

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Chapter 3: Software Implementation 16

_______________________________________________________________________

3.1.2. Dispersion Analysis

a) Normalization

The first action to take is the normalization of the physical dimensions of the microstrip in order to avoid handling big numbers in every single analysis. So, the wave number is chosen to be the normalization factor.

k0

 (3.1)

k_ k_ k0 (3.2)

0

z z

k k k (3.3)

Where  refers to cylindrical coordinates, k_z is the normalized propagation constant and k_ is the wave number in -direction and it is given by [8, 12]:

2

r r z

k_     k (3.4)

The sign of k_ in the most external layer



^   has to be chosen



considering ^Im

 

^k ⁰ according to the Sommerfeld’s radiation condition [8].

b) Discretization and Transformation Matrices

The discretization process was carried out following the concepts explained in the theory section (cf. Figure 2), the number of discretization lines on the strip was considered from two (quicker analysis), up to ten for a detailed analysis and electromagnetic fields analysis.

-10 -5 0 5 10

X-axis (mm)

Y-axis (mm)

Discretization

(a)

-10 -5 0 5 10

Discretization

X-axis (mm)

Y-axis (mm)

(b)

Figure 7: a) Discretization considering N=2, b) Discretization considering N=10, N:

number of e-lines on the microstrip.

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The next pair of equations shows a small example of the transformation matrices building process.

1 max 1

min 1

2 max 2

min 2

max max

( )

(2 1) (2 1)

e e

e

i

e e

e

i

j j

j

j j

j e

e e e

T e e e

   

 

   

 

 

 



 



  

 

 



 

 

 

    

(3.5)

1 max 1

min 1

2 max 2

min 2

max max

( )

(2 1) (2 1)

h h

h

i

h h

h

i

j j

j

j j

j h

e e e

T e e e

   

 

   

 

 

 



 



  

 

 



 

 

 

    

(3.6)

c) Dyadic Green’s Matrix

To obtain the dyadic Green’s matrix, first the intrinsic properties of each layer like_r, _r, k_, _k and _k_₁ are calculated in both the substrate and the superstrate. The hybrid matrices for each layer are calculated using equations (2.10) - (2.13), after that the equivalent hybrid matrix is determined using equation (2.14). Since, the transmission line is a cylindrical microstrip line composed of a grounded cylinder in the interior, the equivalent circuit for the first layer next to such cylinder is consider short-circuited as in Figure 5, the equivalent admittance matrix for the substrate Y^{( )}^u is found using equation (2.18), and the equivalent admittance matrix for the superstrate Y⁽⁰⁾ is found with equations (2.15) and (2.19).

Finally the Green function matrix is obtained taking the inverse of the equivalent admittance matrix of the model. Regarding those calculations are carried out just for one point in the spectrum, the dyadic Green’s matrix is assembled joining every point of the spectrum in one bigger matrix. The next equation shows the distribution when assembling the dyadic Green matrix according with [11].

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_______________________________________________________________________

( ) ( )

0 0 0 0

z

i i

z

z zz

i i

z zz

 



 

 

 

 











G G

G

G G

     

         

 

   

         

 

   



     

         

 

   

         

 

   







(3.7)

d) Spatial Domain Transformation and Reduced Dyadic Green Matrix

Once the dyadic Green’s matrix is obtained, it has to be transformed into de spatial domain using equations (2.7); so, in conjunction with the currents density distribution and the boundary conditions, a new reduced G matrix can be obtained. The boundary conditions state that the tangential electric field should vanish on the metallization. Considering the currents are normalized to one and using equation (2.22), the reduced form of the dyadic Green matrix can be acquired.

0

red red

z

red red

z z zz z

 j   



 



 

     

   

     

   

    

E G G J 0

E G G J 0 (3.8)

Expanding the last equation, a small example of the application of the previous concept can be exposed in equation (3.9).

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,11 ,12 ,13 ,11 ,12 ,13

,21 ,22 ,23 ,21 ,22 ,23

,31 ,32 ,33 ,31 ,32 ,33

,11 ,12 ,13 ,11 ,12 ,13

,21 ,22 ,23 ,21 ,22 ,23

,31 ,32 ,33

z z z

z z z zz zz zz

z z z z

G G G G G G

G G G G

     

  

1

2

1

,31 ,32 ,33

0 0

1 0

0

1 0

z

z zz zz

E E

E

G G





 

 

   

 

   

 

   

 

   

 

  

 

   

 

   

 

   

 

   

   

 

 

(3.9)

Taking the elements of the dyadic Green matrix related to positions of the currents on the microstrip, the reduced form is obtained.

,33 ,32 ,33

,23 ,22 ,23

,33 ,32 ,33

z z

z zz zz

G G G

  



 

 

  

 

 

Gred (3.10)

e) Determining the Propagation Constant

In order to find a solution to equation (3.8), the determinant of G^red must be zero as described in theory section (equation (2.23) ).

The Green’s function depends, among some other variables, on the propagation constant; so, evaluating the determinant of G^red while varying the propagation constant, a solution to equation (3.8) can be found when that determinant is zero, cf. equation (3.11). The propagation constant should vary from a value of one until a solution is found. When a solution is obtained, the propagation constant represents the first mode of propagation.

 

det 0

kz 

Gred (3.11)

Considering that the discretization points are a finite number, a good way to simulate an infinitesimal discretization is giving a picture of the propagation constant versus the inverse of the number of discretization points in the metallization. Then, using a curve fitting method is easy to extrapolate the value of the propagation constant when an infinite discretization is assumed.

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_______________________________________________________________________

0 0.1 0.2 0.3 0.4 0.5

1.6 1.605 1.61 1.615 1.62 1.625 1.63 1.635

Extrapolation of the Propagation Constant

1/(Num. e-Lines on the microstrip) Normalized Propagation Constant: k z

Figure 8: Extrapolation of the propagation constant when infinite number of e-lines on the microstrip

3.1.3. Currents Analysis

Although the reduced form of dyadic Green’s matrix and the value of the electric field are known in the microstrip interface, the current density in equation (3.8) can not be determined because of the determinant of G^redis zero and there is no inverse for such matrix. To solve equation (3.8), the eigenvalues and eigenvectors of G^red are needed. According to the next equation, some properties of the eigenvalues and eigenvectors are shown.

Given a matrix A:

Avv (3.12)

Where, v is the eigenvector and  is the eigenvalues.

Taking the smallest of the eigenvalues ( 0) and its correspondent eigenvector, the multiplication of the second term in equation (3.12) will tend to a null vector, so that same concept can be applied to equation (3.8) to find a solution. The next equations explain this case.

0 min min

0 0

red red

z

red red

z

z zz

J

G G

j v

J

G G



 



 



     

 

     

    

 

(3.13)

0

min min 0

red red

z

red red

z

z zz

j J

G G

j J v

G G



 



 

 

   

   

   

 

(3.14)

Analysis of Microstrip Lines on Substrates Composed of Several Dielectric Layers under the Application of the Discrete Mode Matching Subtitle

Master Program in Electronics/Telecommunications Examiner: Olof Bengtsson

Supervisor: Marcos V. T. Heckler

DEPARTMENT OF TECHNOLOGY AND BUILT ENVIRONMENT

Analysis of Microstrip Lines on Substrates Composed of Several Dielectric Layers under the Application of the

Discrete Mode Matching Subtitle

Manuel Gustavo Sotomayor Polar September 2008

Master’s Thesis in Electronics/Telecommunications

Abstract

Acknowledgement

Table of Contents

List of Figures

Chapter 1 Introduction

1.1. The Problem

1.2. Goal

1.3. Previous Research

1.4. Justification of the Project

Chapter 2 Theory

2.1. Discrete Mode Matching

2.1.1. Cylindrical Microstrip Lines

2.1.2. Wave Equation in Cylindrical Coordinates

 









 



 

2.1.3. Tangential Field Components

2.1.4. Discretization

2.1.5. Matrix Formulations

 

 

2.1.6. Hybrid Matrix

 





 

 



     

2.2. Full-Wave Equivalent Circuit







  

2.2.1. Propagation Constant

2.2.2. Fields Analysis

 

 

2.2.3. Concept of Profile

Chapter 3

Software Implementation

3.1. Method

3.1.1. Basic Model

3.1.2. Dispersion Analysis





 

 

3.1.3. Currents Analysis