analysis of microstrip lines on substrates composed of ...173580/fulltext01.pdf · compared with...

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

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.

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.

iii

Table of Contents

CHAPTER 1 ....................................................................................................... 1

Introduction..........................................................................................................................................1 1.1. The Problem................................................................................................................................1 1.2. Goal ............................................................................................................................................1 1.3. Previous Research .......................................................................................................................2 1.4. Justification of the Project ...........................................................................................................3

CHAPTER 2 ....................................................................................................... 4

Theory...................................................................................................................................................4 2.1. Discrete Mode Matching .............................................................................................................4

2.1.1. Cylindrical Microstrip Lines ................................................................................................4 2.1.2. Wave Equation in Cylindrical Coordinates ...........................................................................5 2.1.3. Tangential Field Components...............................................................................................6 2.1.4. Discretization.......................................................................................................................6 2.1.5. Matrix Formulations ............................................................................................................7 2.1.6. Hybrid Matrix......................................................................................................................8

2.2. Full-Wave Equivalent Circuit ....................................................................................................10 2.2.1. Propagation Constant .........................................................................................................11 2.2.2. Fields Analysis ..................................................................................................................12 2.2.3. Concept of Profile..............................................................................................................14

CHAPTER 3 ..................................................................................................... 15

Software Implementation ...................................................................................................................15 3.1. Method .....................................................................................................................................15

3.1.1. Basic Model.......................................................................................................................15 3.1.2. Dispersion Analysis ...........................................................................................................16 3.1.3. Currents Analysis...............................................................................................................20 3.1.4. Field Analysis ....................................................................................................................21 3.1.5. Base of Comparison ...........................................................................................................25 3.1.6. Validation of the Results ....................................................................................................25

CHAPTER 4 ..................................................................................................... 27

Results.................................................................................................................................................27 4.1. Dispersion Analysis...................................................................................................................27 4.2. Field Analysis ...........................................................................................................................32

iv

CHAPTER 5 ..................................................................................................... 36

Discussion ...........................................................................................................................................36

Future Work.......................................................................................................................................37 5.1. Characteristic Impedance...........................................................................................................37 5.2. Analysis of higher propagation modes .......................................................................................38

Conclusions.........................................................................................................................................39

References...........................................................................................................................................40

Appendix.............................................................................................................................................41

v

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

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.

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.

Chapter 1: Introduction

3 _______________________________________________________________________

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.

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.

Chapter 2: Theory

5 _______________________________________________________________________

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 zjk ze 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 0k [1, 7].

2

2 , , 0d z

(2.1)

Where represents each of the components zE or zH

2d r r zk

r : is the relative permeability

r : is the relative permittivity

zk : is the propagation constant along the z-direction

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

-direction.

1, ( )2

i ji

ie

(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

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

ik k i k iA J k B Y k (2.3)

Where ( )iJ x and ( )iY x are the Bessel functions of first and second kind

respectively [1].

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

2( )

0( , ) iz jj k zi 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 20 0

1

1

rz

r rz

r

jz

zj

z

E EH 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.

Chapter 2: Theory

7 _______________________________________________________________________

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 44

6

8

10

12

14

16

18

X-axis (mm)

Y-a

xis

(mm

)

Discretization

e-linesh-linesmetallization

0.2 5( )j0.5( )j

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

X-axis (mm)

Y-a

xis

(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 zE and zH 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 eT and hT are found by the set of equations in (2.7)

[1].

Chapter 2: Theory

8 _______________________________________________________________________

zz e

zz h

E T E

H T H (2.6)

,

,

h

e

h jih h

e jie 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

1h h

1e e

G G T 0 G G T 0G G 0 T G G 0 T

(2.8)

2.1.6. Hybrid Matrix

The Hybrid Matrix kK , using different intrinsic characteristics of the layers

( 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 kK hybrid

matrix per layer which is composed by submatrices [8],

kk

kk k

V ZK

Y B (2.9)

where the submatrices: kV , kZ , kY and kB are 2x2 matrices given by the next

set of equations.

2

2z

kk

k k

νr ν r q

V0 q

(2.10)

Chapter 2: Theory

9 _______________________________________________________________________

2 2 2

22r r z z

kr z k

k k k

k k

s ν p νpZ

νp p (2.11)

2

2 2 22z

kr z r r z k

k k

k k k

p νpY

νp s ν p (2.12)

22kz k

k k

ν ν

q 0B

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 kk

eq eqeq

eq eq

K K

V ZK

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].

21

1

n zvnn

n n n z n

k kk k

I νuYν y

(2.15)

2

2 2 212

zn rn rn n n

n

kk

y ν u (2.16)

1'(2) (2)1 1n n n n nk k

ν νu H H (2.17)

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.

eqK mE

mHmH

mJ (0)Y

0H

-5

0

5

x 10-3

0.006 0.008 0.01 0.012 0.014 0.016

X (m

)

Y (m)

Microstrip Line

1r 2r

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.

0E

0H 1H

1mE

• • •

• • •

1mH

1K mK mE1mK

mHmH

mJ

1mE

1mH

• • •

• • •

1nK

2nH

1nE

nE

1nH

nH

nY

uK

0K

mE

mHmH

mJ (0)Y( )uY

( )a

( )b

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

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

Chapter 2: Theory

11 _______________________________________________________________________

The admittance matrices ( )uY and (0)Y are determined using the next

equations. Regarding that there is a metallic cylinder in the interior of the

structure [11], then:

( ) 1uu uY Z V (2.18)

where uZ and uV represent the components of the hybrid matrix uK which is

the equivalent hybrid matrix of the substrate layers.

And considering that there is no an exterior metallic cladding surrounding the

structure, (0)Y can be represented as [11]:

1(0)0 0 0 0n nY Y B Y V Z Y

(2.19)

where the elements with sub-index “0” are the components of the hybrid

matrix 0K , which is the equivalent hybrid matrix of the superstrate layers; and

nY is the admittance matrix of the last layer that extends to infinity, generally it

is considered the layer of air (surrounding medium).

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

using:

( )

(0)

um m

m m

H Y EH 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.

0z

z zz zz

j

G G J E

JG 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:

Chapter 2: Theory

12 _______________________________________________________________________

0z

z zz z z

j

G G J EG 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 zk , 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( ) ( ) zj k zn z nk z k

k

k k e

ν IE G J (2.24)

Expanding the previous equation:

( ' ' )

0( ' ' )

' '

ˆ

ˆ

z

z

j k zzk n

j k zz zzkzn k z

e Jj

e J

ν I

ν I

G GEG GE

(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

01

n n n

zn n nn

n n zn

j

H Y E

H EY

H E

(2.26)

Chapter 2: Theory

13 _______________________________________________________________________

Where: nY is the admittance matrix of the most external layer (surrounding

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

1

k k k k

k k k k

E V Z EH Y B H

(2.27)

0

k kk

zk

zkk

k k

j

EE

E

HH

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].

20 0

11

1

z rkk k zk

k zkrk z

jk

k jk

I νE EH Hν 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 11 k k kk

k k kk

ν ν

ν ν

J Y AΨJ Y BΨ

(2.30)

1 11 k k kkk

k k kk

k

ν ν

ν ν

J Y AΨJ Y BΨ

(2.31)

Chapter 2: Theory

14 _______________________________________________________________________

1 1

k k k

k k k

k

k

C C

C C (2.32)

Where, kΨ represents any of the electromagnetic field in z-direction within

any given interface; and kC 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 15 160

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

Elec

tric

Perm

ittiv

ity

Dielectric Profile

Figure 6: Dielectric profile of a multilayered microstrip line

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.

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.

0k (3.1)

0k k k (3.2)

0z zk k k (3.3)

Where refers to cylindrical coordinates, zk is the normalized propagation

constant and k is the wave number in -direction and it is given by [8, 12]:

2r r zk k (3.4)

The sign of k in the most external layer has to be chosen

considering Im 0k 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

-10

-5

0

5

10

X-axis (mm)

Y-a

xis

(mm

)

Discretization


(a)

-10 -5 0 5 10

-10

-5

0

5

10

Discretization

X-axis (mm)

Y-a

xis

(mm

)


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

number of e-lines on the microstrip.


17 _______________________________________________________________________

The next pair of equations shows a small example of the transformation

matrices building process.

1 max 1min 1

2 max 2min 2

max max

( )( )

( )( )

(2 1) (2 1)

e eei

e eei

j jj

j jje

e e e

T e e e

(3.5)

1 max 1min 1

2 max 2min 2

max max

( )( )

( )( )

(2 1) (2 1)

h hhi

h hhi

j jj

j jjh

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 1k 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 ( )uY is found using equation

(2.18), and the equivalent admittance matrix for the superstrate (0)Y 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].


18 _______________________________________________________________________

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

z

i iz

z

z zz

i iz zz

z zz

G G

G G

G GG

G 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 redz

red redz zz zz

j

E JG G 0E JG G 0

(3.8)

Expanding the last equation, a small example of the application of the previous

concept can be exposed in equation (3.9).


19 _______________________________________________________________________

,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

z z z

z z z zz zz zz

z z z zz zz zz

zz z z

G G G G G GG G G G G G

G G G G G G

G G G G G G

G G G G G G

GG G G

1

2

1

,31 ,32 ,33

001 0

01 01 0

z

z zz zz

EE

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

z zz zz

G G G

G G GG G G

redG (3.10)

e) Determining the Propagation Constant

In order to find a solution to equation (3.8), the determinant of redG 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 redG 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 0zkredG (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.


20 _______________________________________________________________________

0 0.1 0.2 0.3 0.4 0.51.6

1.605

1.61

1.615

1.62

1.625

1.63

1.635Extrapolation of the Propagation Constant

1/(Num. e-Lines on the microstrip)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: 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 redG is zero

and there is no inverse for such matrix. To solve equation (3.8), the eigenvalues

and eigenvectors of redG 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

00

red redz

red redzz zz

JG Gj v

JG G

(3.13)

0min min

0

red redz

red redzz zz

j JG Gv

j JG G

(3.14)


21 _______________________________________________________________________

The previous equation is possible if and only if :

0min

0 z

j Jv

j J

(3.15)

Finally, the currents can be determined from the eigenvector of the reduced

dyadic Green matrix as it is presented in the next equation.

min0

1

z

Jv

J j

(3.16)

3.1.4. Field Analysis

a) Fields in the Metallization Interface

Now that the current densities on the microstrip metallization are determined,

they have to be transformed to the spectral domain in order to use them and

then to calculate both the electric and magnetic field in the interface where the

metallization lies. Equation (2.4) provides the method to transform the current

densities to spectral domain.

Using equation (2.25), the electric field in spectral domain is obtained for each

set of current densities ( and )zJ J . The total electric field is the addition of

the fields generated by each current. To find the magnetic field in spectral

domain, equation (2.26) is used.

b) Fields in the Substrate

To calculate the fields in other interfaces within the substrate, equation (2.27)

is needed. Once the electric field is found, equation (2.20) is used to find the

magnetic field in that interface as well. One has to notice that in order to use

equation (2.20), the admittance matrix ( )uY has to be recalculated, computing

again a new equivalent hybrid matrix considering all the layers from that very

layer to the most inner layer. This recalculation of a new ( )uY has to be done


22 _______________________________________________________________________

every time the fields in the next interface are required to be found. To

determine both the electric and magnetic field in - direction, equation (2.29)

is used.

To transform the fields into the spatial domain, transformation matrices are

used which are described by the next set of equations and equations (2.28),

(2.7).

z z

e

h

e

E T EE T E

E T E

(3.17)

z z

h

e

h

H T HH T H

H T H

(3.18)

The next figure shows the sequence which calculates the fields for inner

interfaces within the substrate.

• • •

• • •

1mK

1mH

mE ( )oY0E

mHmH

mJ

(0)Y

mK

2mH

1mE

1K

1H

( 1)mY ( 2)mY (1)Y ( 1) ( )m oY Y

2mE

1E

( ) ( )m uY Y

1

2

3

45

6

0H

( )um mH Y E ( 1)

1 1m

m mH Y E

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

1. Find the electric field in the microstrip metallization.

2. Calculate the required equivalent admittance element.

3. Calculate the magnetic field in the microstrip metallization.

4. Use the hybrid equivalent matrix between the actual interface and the

next one to find the next electric field.

5. Recalculate the equivalent admittance element using the equivalent

circuit.

6. Use the recalculated equivalent admittance element and the electric

field to obtain the magnetic field.


23 _______________________________________________________________________

c) Fields in the Superstrate

To calculate the fields in other interfaces within the superstrate, equation (2.27)

has to change a bit. Taking the inverse of the equivalent hybrid matrix and

place it in the second term of the equation, it is possible to calculate the fields

for a certain layer using the fields in the previous interface:

1

1

1

k k k k

k k k k

E V Z EH Y B H

(3.19)

Once the electric field is found, equation (2.26) is needed to find the magnetic

field in that interface as well. In order to use equation (2.20), the admittance

matrix (0)Y has to be recalculated, a new equivalent hybrid matrix considering

all the layers from that very layer to the most outer layer ( nY ). This

recalculation of a new (0)Y has to be carried out every time the fields in the

next interface are required to be found. To find both the electric and magnetic

field in - direction, equation (2.29) is used.

To transform the fields into the spatial domain, transformation matrices are

used as shown previously in equations (3.17) and (3.18), besides equations

(2.7) and (2.28).

The next figure shows the sequence to calculate the fields for outer interfaces

in the superstrate.

• • •

• • •

2nK

2nH

1nE

nE

1nH

nH

nYmE

mH mH

mJ( )uY 1nK

1nH

2nE

1mK

1mH

( )nnY Y ( 1)nY ( 2)nY ( 2)mY ( 1) ( )m oY Y

1nE

1mE

( ) ( )m uY Y

( )om mH Y E ( 2)

1 1m

m mH Y E

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

1. Find the electric field in the microstrip metallization.

2. Calculate the required equivalent admittance element.

3. Calculate the magnetic field in the microstrip metallization.


24 _______________________________________________________________________

4. Use the hybrid equivalent matrix between the actual interface and the

next one to find the next electric field.

5. Recalculate the equivalent admittance element using the equivalent

circuit.

6. Use the recalculated equivalent admittance element and the electric

field to obtain the magnetic field.

Regarding equations (3.17) and (3.18), the e-lines discretization points (cf.

Figure 2); only zE , E and H are known values in these points and in view

of the h-lines discretization points (cf. Figure 2); only zH , H and E are the

known parameters. In order to have a complete set of components of both the

electric field and the magnetic field, an interpolation is performed considering

two adjacent components.

The next figure shows how the interpolation is carried out.

zE

E

E

zH

H

H

-10 -8 -6 -4 -2 0 2 4 6 8 100

5

10

15

X-axis (mm)

Y-a

xis

(mm

)

Interpolation


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

angles

Considering just the electric field represented by blue vectors; as explained

before, the interpolated values are given by red vectors, and in the case of the

magnetic field, the red vectors represent the calculated values and the blue

vectors the interpolated values.

After the interpolation, every single discretization point counts with a full set

of vectors for each field. The vectors in and direction are used to

determine the poynting vector in each discretization point respectively.


25 _______________________________________________________________________

3.1.5. Base of Comparison

The dispersion produced by a one-layer microstrip line is taken as the base of

comparison which is represented by the line with symbols. The improvement in

the reduction of the dispersion is considered as the percentage reduction after

18GHz.

The next figure shows how the improvement is determined.

5 10 15 20

2.08

2.1

2.12

2.14

2.16

2.18

2.2

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

Figure 12: Determination of the improvement in the dispersion.

3.1.6. Validation of the Results

The dispersion and field analysis from DMM will be compared with the

commercial software Ansoft HFSS to validate the results.

The usual method of HFSS to handle curved surface deliberates a deviation

between the surface and the approximation. The next figure shows this

deviation from the perfect circular cylinder due to FEM discretization (HFSS

approximation).

Figure 13: Deviation from the perfect circular cylinder


26 _______________________________________________________________________

a) Parameters of the validation

HFSS deviation of 0.1mm and 0.01mm.

Three different microstrip lines composed of 4 layers in the substrate

and no special superstrate.

The surrounding medium is air.

The next table shows the dielectric constants for each layer of the analyzed

microstrip lines and Figure 14 presents the configuration of such layers.

Table 1: Analyzed Microstrip lines for validation.

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

X-Axis

Y-A

xis

Cylindrical Microstrip Line

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

validation

Dielectric Constant

Layer1 Layer2 Layer3 Layer4

Case 1 4.50 2.94 2.70 2.55

Case 2 10.00 6.00 6.00 4.50

Case 3 10.00 6.15 6.00 6.00

Chapter 4

Results

4.1. Dispersion Analysis

The next figure represents one of the simplest cases; when the microstrip line is

composed of just one layer in the substrate and without any special profile in the

superstrate except the surrounding medium which is considered to be air.

As it can be observed in blue; the dispersion, of the microstrip lines with high values of

electric permittivity, increases in a higher rate than those microstrip lines with low

values of electric permittivity as seen in the red case.

11 11.2 11.4 11.6 11.8 12 12.2 12.4

2

4

6

8

10

12

14

16

18

Distance (Ro Direction in mm.)

Elec

tric

Per

mitt

ivity

Electric Permittivity Profile: Substrate

5 10 15 20

2.6

2.8

3

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

5 10 15 201.48

1.5

1.52

1.54

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Er: 10Er: 2.54

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

the transmission lines.

Chapter 4: Results

28 _______________________________________________________________________

The next step in complexity is the case when different kinds of profiles are used only in

the substrate. Figure 16 shows the dispersion of a microstrip line in such case and just

air as surrounding medium. Taking into account this configuration, the next figure

presents the substrate profiles which give the best results. The reduction of the

dispersion reaches almost 7.6% in 18GHz.

As it can be seen, the dispersion is very sensitive to any variation in the substrate

profile.

11 11.5 120

2

4

6

8

10

12


Elec

tric

Per

mitt

ivity

Electric Permittivity Profile: Substrate

0 0.5 1 1.5 2

x 1010

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Frequency (GHz)N

orm

aliz

ed P

ropa

gatio

n C

onst

ant:

Kz

Dispersion

7.6%

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

the substrate. a) Electric permittivity profiles, b) Dispersion of the transmission lines.

Figure 17 shows the dispersion for different profiles of the superstrate maintaining

constant the one-layer profile in the substrate ( 10r ) in all cases.

As it can be observed, there is a reduction of the dispersion for every profile, with

almost 37% at best when the superstrate has a constant profile of the same width as the

substrate. Although there are big variations among the profiles in the superstrate, the

variations among the dispersion curves are not extensive, which indicates that the

dispersion is not very sensitive to changes of the superstrate profiles.

11 11.2 11.4 11.6 11.8 12 12.2 12.4

2

4

6

8

10

12

14


Ele

ctric

Per

mitt

ivity

Electric Permittivity Profile

11 11.2 11.4 11.6 11.8 12 12.2 12.4

2

4

6

8

10

12

14


Ele

ctric

Per

mitt

ivity

Electric Permittivity Profile

0.5 1 1.5 2

x 1010

2.95

3

3.05

3.1

3.15

3.2

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

36.9%

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

superstrate. a) and b) Electric permittivity profiles, b) Dispersion of the transmission lines.

Chapter 4: Results

29 _______________________________________________________________________

Figure 18 presents a configuration of the profiles where the maximum values of the

electric permittivity are the same in both the substrate and the superstrate. It also shows

the improvements of the dispersion for different profiles in the superstrate regarding the

substrate profile which gives the best performance among others of this same

configuration.

11 11.5 121

1.5

2

2.5

3


Elec

tric

Perm

itivity

Electric Permitivity Prof ile

11 11.5 121

1.5

2

2.5

3


Elec

tric

Perm

itivity

Electric Permitivity Prof ile

0.5 1 1.5 2

x 1010

1.16

1.165

1.17

1.175

1.18

1.185

1.19

1.195

1.2

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

49.4%

Figure 18: Dispersion improvements because of different profiles of the electric

permittivity in both the substrate and the superstrate. a) and b) Electric permittivity profiles, b) Dispersion of the transmission lines.

Taking into account a maximum value of the electric permittivity in the substrate

greater than the maximum value in the superstrate as Figure 19 shows, there is an

improvement of the reduction of the dispersion about 29.5% after 18 GHz when the

superstrate width is the same as the substrate width (cf. Figure 19b).

5 10 15 202.2

2.22

2.24

2.26

2.28

2.3

2.32

2.34

2.36

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

29.487%

0 5 10 15 202.15

2.2

2.25

2.3

2.35

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

5.0438%

11 11.5 121

2

3

4

5

6

7

8

9

10


Ele

ctric

Per

miti

vity

Electric Permitivity Profile of the Line: Cladding Width Reduced

11 11.5 121

2

3

4

5

6

7

8

9

10


Ele

ctric

Per

miti

vity

Electric Permitivity Profile


permittivity in both the substrate and the superstrate. a) Electric permittivity profiles, b) and d) Dispersion of the transmission lines, c) Electric permittivity profiles with reduction

of the superstrate width.

Chapter 4: Results

30 _______________________________________________________________________

Reducing the width of the superstrate and maintaining the same profile in the substrate

(cf. Figure 19c), an improvement of 5% is obtained (cf. Figure 19d).

Considering the maximum value of the electric permittivity of the superstrate greater

than the maximum value of r in the substrate as Figure 20 shows, there is an

improvement of the dispersion about 46.7% at best; even if there is a reduction of the

superstrate width, it is possible to get an improvement of the dispersion around 17.1%.

0.5 1 1.5 2x 1010

2

2.02

2.04

2.06

2.08

2.1

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

14.5292%17.1207%

5 10 15

2.05

2.1

2.15

2.2

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

46.7026%42.3195%

11 11.5 12

2

4

6

8

10


Ele

ctric

Per

miti

vity


11 11.5 121

2

3

4

5

6

7

8

9

10

Number of Layers

Ele

ctric

Per

miti

vity

Electric Permitivity Profile of the line


permittivity in both the substrate and the superstrate. a) Electric permittivity profiles, b) and d) Dispersion of the transmission lines, c) Electric permittivity profiles with reduction

of the superstrate width.

The next figures consider that both substrate and superstrate model one continuous

profile. It can be observed that when r decreases in distance ( direction) following

the different profiles (see Figure 21a,c), the best improvement of the dispersion reaches

31.9% at best and only 5% at worst; but, in the opposite way, when r increases in

distance the dispersion improves in 24.8% at worst, obtaining even better results at best.

Chapter 4: Results

31 _______________________________________________________________________

0 5 10 15 202.4

2.5

2.6

2.7

2.8

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

0 5 10 15 201.17

1.175

1.18

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

11 11.5 12

2

4

6

8

10

Distance (Ro Direccion in mm.)

Elec

tric

Per

miti

vity


0 5 10 15 202.95

3

3.05

3.1

3.15

3.2

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

0 5 10 15 201.51

1.52

1.53

1.54

1.55

1.56

Nor

mal

ized

Pro

paga

tion

Con

stan

t: K

z

Dispersion

0 5 10 15 201.51

1.52

1.53

1.54

1.55

1.56

11 11.5 12

2

4

6

8

10

Distance (Ro Direccion in mm.)

Elec

tric

Per

miti

vity


Figure 21: Dispersion improvements because of considering one continuous profile of the

electric permittivity for both the substrate and the superstrate in different configurations. a) and c) Electric permittivity profiles, b) and d) Dispersion of the transmission lines.

For the validation of these results, Figure 22 shows the dispersion of three different

microstrips, described in the previous section, and the results obtained from HFSS

taking into account its deviation in the approximation for curved surfaces.

As it can be observed, DMM gives very accurate results and HFSS results converge into

DMM’s when its deviation is reduced.

0 5 10 15 201.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5Validation

Frequency

Nor

mal

ized

Pro

paga

tion

Con

stan

t

DMMHFSS (dev=0.01mm)HFSS (dev=0.1mm)

Case 1

Case 3

Case 2

Figure 22: Validation of the results, comparative between DMM and HFSS results in three

different cases.

Chapter 4: Results

32 _______________________________________________________________________

4.2. Field Analysis

Figure 23 shows the analysis of the electric field of a microstrip line which substrate

and superstrate are air, using DMM and HFSS. The physical dimensions are the same as

described as the base model of analysis and 6 e-lines were used to for this analysis.

The electric field is more concentrated between the two metallization and the highest

values of the Electric field are just on the edges of the microstrip. The results are very

similar to one another.

-0.2 -0.1 0 0.1 0.2

0.4

0.45

0.5

0.55

Electric Field

X (Normalized)

Y (N

orm

aliz

ed)

Metallizations

-0.2 -0.1 0 0.1 0.2

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

X (Normalized)

Y (N

orm

aliz

ed)

Electric Field (Magnitude)

0.5

1

1.5

2

x 104

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

superstrate, a) Results obtained from DMM, b) Results obtained from HFSS, c) Normalized Magnitude of the electric field obtained from DMM, d) Magnitude of the electric field

obtained from HFSS.

Figure 24 shows the analysis of the magnetic field and the current density distribution

on the microstrip metallization.

The magnetic field is more concentrated between the two metallization as well. It can be

seen from Figure 24b that the highest values, of the current density zJ , are on the edges

of the microstrip which matches the theory. Figure 24c shows the current density

distribution does not vanish on the edges of the microstrip and it is because of the

number of discretization points and the distribution of the discretization h-lines which

do not coincide with the edges of the microstrip and that gives mostly this impression.

Chapter 4: Results

33 _______________________________________________________________________

-0.2 -0.1 0 0.1 0.2

0.4

0.45

0.5

0.55

Magnetic Field

X (Normalized)

Y (N

orm

aliz

ed)

Metallizations

1 2 3 4 5 61.4

1.5

1.6

1.7

1.8

1.9

2

2.1Current Density: Jz (Z-Direction)

Discretization Points in the Metallization

Mag

nitu

de

1 2 3 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-6 Current Density: Jphi (Phi-Direction)


Mag

nitu

de

Figure 24: Magnetic Field Distribution, a) Results obtained from DMM, b) Current Density Distribution in z-direction, c) Current Density Distribution in -direction.

Regarding that both the electric and magnetic field are more concentrated between the

two metallization, there will be more power flow in that place. Figure 25 shows the real

power flow of this particular microstrip. It can be observed that the real power is mostly

concentrated on the edges and also between the metallizations but in lower values.

-0.2 -0.1 0 0.1 0.2

0.4

0.45

0.5

0.55

X (Normalized)

Y (N

orm

aliz

ed)

Real Power (Poynting Vector)

2

4

6

8

10

x 106

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

from HFSS.

The electric field distribution changes slightly when there is a profile in the substrate

and/or in the superstrate. The physical dimensions are kept from the basic model, but 10

e-lines were used for the discretization. Figure 26a illustrates this case, the used profile

is shown in Figure 26b and the improvement of the dispersion is shown in Figure 26c.

The electric field maintains the characteristic distribution of the first mode of

propagation.

Chapter 4: Results

34 _______________________________________________________________________

11 11.5 12 12.5

2

4

6

8

10


Elec

tric

Perm

itivi

ty


5 10 15 202.24

2.26

2.28

2.3

2.32

2.34

2.36

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: Kz

Dispersion

-1.5 -1 -0.5 0 0.5 1 1.5

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Electric Field

X (Normalized)

Y (N

orm

aliz

ed)

Metallizations

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

improvement.

The magnetic field also changes slightly when a profile is applied as it is presented in

Figure 27a. The current density ( zJ ) has higher values near the edges of the microstrip

and J can be seen clearly, due to the number of discretization lines, that it vanishes on

the edges of the microstrip line.

-1.5 -1 -0.5 0 0.5 1 1.5

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Magnetic Field

X (Normalized)

Y (N

orm

aliz

ed)

2 4 6 8 100.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

Current Density: Jz (Z-Direction)


Mag

nitu

de

2 4 6 8

-3

-2

-1

0

1

2

3

x 10-4 Current Density: J

phi (Phi-Direction)


Mag

nitu

de

Figure 27: a) Magnetic field distribution, b) Current Density Distribution in z-direction, c)

Current Density Distribution in -direction.

Figure 28 presents the power flow in the microstrip, demonstrating that the real power is

concentrated mainly between the metallizations, but still there are two peaks just close

to the edges because of the high values of both electric and magnetic field.

Chapter 4: Results

35 _______________________________________________________________________

-1.5 -1 -0.5 0 0.5 1 1.5

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

X (Normalized)

Y (N

orm

aliz

ed)


500

1000

1500

2000

2500

3000

3500

-10

13.5 4 4.5 5

-1000

0

1000

2000

3000

4000

Y (Normalized)


X (Normalized)

Pow

er (N

orm

aliz

ed)

500

1000

1500

2000

2500

3000

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

from HFSS.

A comparative chart about the invested time in average per simulation is given by

Figure 29. For these comparative results between HFSS and DMM, the three different

cases, explained in the previous section, were considered. Each case takes into account

different profiles and five points in frequency. (cf. Table 1, Figure 14 and Figure 22).

HFSS was set to use a deviation of 0.01mm, and it was simulated in a Quad Core

microprocessor with 12 gigabytes RAM computer. DMM was performed in single core

Centrino microprocessor 512 megabytes RAM.

In spite of the great and unfair difference between both simulation platforms, DMM is

considerably faster than HFSS.

Invested Time per Simulation

0

5

10

15

20

25

30

35

40

45

Ave

rage

Tim

e (m

inut

es)

HFSSDMM

HFSS 41.7800DMM 0.3667

1

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

propagation constant.

Chapter 5

Discussion

Although this thesis work got a very good agreement of the data and simulations, what

could be done differently is the method of interpolation when determining the

electromagnetic field components. The bilinear and bicubic interpolation method were

considered for this work and even they have not showed a great difference, there might

be a better method to determine those fields’ components.

The propagation constant when infinite number of e-lines is considered on the

microstrip, the extrapolation process was done in the least square sense. Another

extrapolation method can be considered in this part as well.

The analysis of the improvements in the reduction of the dispersion was done using a

reference curve which compares the percentage of the reduction after some frequency

range. Considering that the dispersion analysis gives curves as results, a kind of

normalization based on decibels might be done in order to achieve lines instead those

curves and then to obtain a better figure of merit.

The conclusions in [3] says that the dispersion of microstrip with multi-layers media is

much large than those with single layer media. Which is only true in some given cases

as proved in this thesis work when the dielectric constant of the one-layer microstrip

line has low values. When that kind of microstrip has high values of dielectric constant,

there are profiles which can reduce the dispersion of that transmission line. (cf. Figure

16 - Figure 20).

Chapter 5: Future Work

37 _______________________________________________________________________

Future Work

5.1. Characteristic Impedance

To develop a computer code able to calculate the characteristic impedance

based on DMM method and high integration possibilities with this thesis work.

The complete dyadic Green matrix is composed of nine elements as shown in

equation (5.1). zG should be determined in order to find the characteristic

impedance of cylindrical microstrip lines [6, 11].

z

z

z z zz

G G GG G G G

G G G

(5.1)

The expression for the characteristic impedance is given by [6]:

2

1 12 2

1

sin ( / 2)2

zk zmM M

j jz zk zm

k mc M

zzm

m

V D I e I eZ

b D I

(5.2)

Where:

( , , )b

z zaV G k d (5.3)

Chapter 5: Future Work

38 _______________________________________________________________________

5.2. Analysis of higher propagation modes

To extend the study and the investigation of the dispersion properties in

conformal microstrip lines considering the analysis for higher propagation

modes.

The next figures show some cases.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6Electric Field

X (Normalized)

Y (N

orm

aliz

ed)

11 11.2 11.4 11.6 11.8 12 12.2 12.41

2

3

4

5

6

7

8

9

10

11


Ele

ctric

Per

miti

vity


-2 -1.5 -1 -0.5 0 0.5 1 1.5 25.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

Magnetic Field

X (Normalized)

Y (N

orm

aliz

ed)

-2 -1 0 1 2

5.5

6

6.5

7

7.5

X (Normalized)

Y (N

orm

aliz

ed)

Power (Poynting Vector)

0

50

100

150

200

250

Figure 30: Second Propagation Mode, a) Electric Field. b) Dielectric Profile. c) Magnetic

Field. d) Real Power Flow.

Chapter 5: Conclusions

39 _______________________________________________________________________

Conclusions

The investigation of the dispersion properties cylindrical microstrip lines was presented

in this thesis work, giving very good agreements of the data and the literature.

A MATLAB computer code was developed based on the discrete mode matching

method (DMM) to compute the propagation constants. The main parameter for the

analysis was the utilization of different dielectric profiles.

The theoretical model and the program can be modified without much effort to analyze

some more complex structures such as inside cylindrical microstrip line and coupled

lines.

The propagation constant as well as the dispersion are very sensitive to changes in the

substrate, but not when changes occur in the superstrate.

Improvements of more than 49% in the reduction of the dispersion are achieved by

some dielectric profiles.

The higher the number of discretization points, the better the accuracy.

Both electric and magnetic fields can be obtained without too much computational

effort.

Because of discretization method (e-lines and h-lines) and the interpolation, the fields

will suffer a small deviation from the real values.

DMM is considerably faster than HFSS, besides getting very accurate results.

HFSS converges into DMM’s values when its deviation is reduced in the approximation

for curved surfaces.

References

[1] M. V. T. Heckler and A. Dreher, "Analysis of Cylindrical Microstrip Lines Using the Discrete Mode Matching Method," Microw. and Wireless Compo. Lett., vol. 16 no.7, pp. 392 - 394, 2006.

[2] S. Xiao, R. Vahldieck, and J. Hesselbarth, "Analysis of Cylindrical Transmission Lines with the Method of Lines," IEEE Trans. Microw. Theory Tech., vol. 44, pp. 993-999, 1996.

[3] T. Ling, "The Analysis of the Microstrip Dispersion with Multi-Layers Dielectric," International Journal of Infrared and Millimeter Waves, vol. 20, 1999.

[4] V. Mathew, "Computation of Dispersion in Microstrip Line with Anisotropic Substrate and Overlay," International Journal of Infrared and Millimeter Waves, vol. 26, pp. 563 - 573, 2005.

[5] G. Zheng and K. Chen, "The Studies of Cylindrical Microstrip Line with the FD-TD Method in Cylindrical Coordinate System," International Journal of Infrared and Millimeter Waves, vol. 13, pp. 1421- 1431, 1992.

[6] K.-L. Wong, Design of Nonplanar Microstrip Antennas and Transmission Lines, vol. 1, First ed. Taiwan: John Wiley & Sons, inc., 1999.

[7] R. K. Wangsness, Electromagnetic Fields, 2nd Edition ed. Arizona, 1986. [8] M. Thiel and A. Dreher, "Dyadic Green's Function of Multilayer Cylindrical

Closed and Sector Structures for Waveguide, Microstrip-Antenna, and Network Analysis," IEEE Trans. Microw. Theory Tech., vol. 50 no.11, pp. 2576 - 2579, 2002.

[9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Ninth ed. New York: Dover, 1964.

[10] M. V. T. Heckler and A. Dreher, "Analysis of Graded-Index Fibers Using a Full-Wave Equivalent Circuit," J. Lightw. Technol., vol. 25 no.1, pp. 392-394, 2007.

[11] M. Thiel, "Die Analyse von zylinderkonformen und quasi-zylinderkonformen Antennen in Streifenleitungstechnik," vol. Ph.D. Dissertation. Munich, Germany: Tech. Univ. Munich, 2002.

[12] D. M. Pozar, Microwave Engineering, vol. 1, 3rd. ed: John Wiley and Sons, Inc., 2005.

Appendix

Recurrence Relation of Cross products of Bessel functions.

k k kq k p q (5.4)

1k k kr k p r (5.5)

21k k k ks k p p s (5.6)

' '

' '

' ' ' '

p J a Y b J b Y a

q J a Y b J b Y a

r J a Y b J b Y a

s J a Y b J b Y a

(5.7)

1k

k

a kb k

(5.8)

The next figure shows the profile, the dispersion improvement and real power flow in a

microstrip line.

5 10 15 20

1.165

1.166

1.167

1.168

1.169

1.17

1.171

1.172

1.173

Frequency (GHz)

Nor

mal

ized

Pro

paga

tion

Con

stan

t: Kz

Dispersion

11 11.5 121

1.5

2

2.5

3


Ele

ctric

Per

mitiv

ity


-0.2 -0.1 0 0.1 0.2

0.4

0.45

0.5

0.55

X (Normalized)

Y (N

orm

aliz

ed)


0.5

1

1.5

2

2.5

3

3.5

4

4.5x 107

-0.20

0.2 0.4 0.45 0.5 0.55

-1

0

1

2

3

4

5x 107

Y (Normalized)


X (Normalized)

Rea

l Pow

er (N

orm

aliz

ed)

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 107

(b)(a)

(d)(c)

Figure 31: a) Electric permittivity profile, b) Dispersion improvement, c) Normalized magnitude of the real power, d) 3D representation of the real power (normalized values).

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