EM Programmer's Notebook - IEEE Xplore

EM Programmer's Notebook

John L. Volakis

ElectroScience Lab

David B. Davidson

Elecrical Engineering Dept. The Ohio State University 1320 Kinnear Rd. Columbus, OH43212 +1 (614) 292·5846 Tel. +1 (614) 292-7297 (Fax)

Dept. E&E E ng inee ri ng University of Stellenbosch Stellenbosch 7600, South Africa (+27) 21 8084458 (+27) 21 8084981 (Fax)

[email protected] (e·mail)

[email protected] (e-mail)

Foreword by the Editors The reflection

and transmission of plane

waves from layered

media is a standard problem in advanced undergraduate/introduc­

tory

p ostgraduate

courses. Controlling the reflection

quency band is an interesting

probl em in

over a

fre·

optimization. This

A more advanced application of this type of work is in the

design of radar absorbers. Some years back, the design

consists of resistive sheets

month's contribution addresses a package developed by the authors

to address this. The authors have made the code described avai lab l e, an d we thank them for this submission.

publicly

of J auman

absorbers was reviewed in the Magazine [I). This type of absorber

1. L. J.

Du Toit,

sandwiched between dielectric layers.

"The Design of Jauman

Ab sorbers," IEEE Anten­ 17·25.

nas and Propagation Magazine, 36, 6, December 1994, pp.

A Graphical User Interface for Calculation of the Reflection and Transmission Coefficients of a Layered Medium Veysel Demir and Atef Z. Eisherbeni Center of Applied El ectromagnetic Systems Research (CASER) Electrical Engineering Department, The University of Mississippi, University, MS E-mail:

[email protected], [email protected]

38677 USA

Abstract This paper presents a software package that calculates the reflection and transmission coefficients of a layered medium, and optimizes reflection and transmission in given ranges of parameters. The algorithms to calculate fields in a layered medium due to normally incident plane waves are described, and the use of a software package developed based on these algorithms is demonstrated. Keywords; Electromagnetic scattering; electromagnetic scattering by nonhomogeneous media; electromagnetic reflection; nonhomogeneous media; optimization methods; graphical user interfaces

1£££ Antennas and Propagation Magazine, Vol. 48, No.1, February 2006

113

1. Introduction

D eflec tion and transmission of electromagnet ic waves by a lay­ -I'-e red medium is in the scop e of undergraduate/introductory

graduate-level electro magnetics courses,

solutions [I, 2).

straightforward

-

and has well known

,

L

X

�

It is easy to develop some algo­

rithms based on these solutions. Based on these algorithms, a soft­

ware packag e oped, using

reflection and

and FORTRAN, in order to calculate the

transmiss ion coefficients of

a layered medium, and

to optimize reflection and transmission in given ranges of parame­

.

ters The main emphasis of this package is to l et students learn how

5"p.,

---

x=O

utilizing a graphical user interface (GUl) was devel­

MATLAB

[] M .. .

3

2

N

£:2:')1" 83:.1ll B,-:':'" . N�JlN BN+l,/JN+l � � !!J. � Ii. � !V, � 3- 1. � L � �l 50,A

Figure

1.

d,

d,

..-�--- -

d,

d...

The layout of the layered medium,

and backward-traveliug

waves.

dN aud the forward ­

the optimization of a problem - commonly taught at the under­ graduate level - can lead to

useful

applications, and that various

a pplications Can also be attempted and fostered using the knowl­ edge they gained through their

then necessary to

education. A

graphical i nterface is

allow ease of use for ap plications , without a need

for reprogramming or teaching optimization at this early stage of teaching.

The developed program provides an ea

sy-to-use

order to find the

reflection

and

transmissio n

coefficients, the

forward- and backward-traveling wave amplitudes will be calcu­

lated .

These waves should satisfy the continuity of the tangential

components of the ries. The

total electric and magnetic fields on the bounda­

electric-field continuity equation can be written as

interface to

define and visualize the geometry of the problem, and to display the solutions for reflection and transmission coefficients, as well as

the field distributions in the medium. Furiliennore,

In

the

program

-

whereas the magnetic fiel d continuity equ ati on can be written as

helps the user define parameters to sweep, and optimizes reflection

or transmission for a given target value within the given ranges of

swept p arameters.

The term

2. Reflection and Transmission by a

kj

Layered Medium Consider a

z-polarized

x-traveling plane wave,

nonnaIIy incident on a layered medium, as

Eo, which is

shown in Figure 1. Each layer, "i", is defined by its p enn itti v ity E:i' penneabili ty J'j, electric conductivity at , magnetic conductiv ity at', and thickness dj -di-l' Backward- and forward-traveling waves in any layer j are denoted by Ai

and

waves are g iven by

iij, respectively, where expres sion s

for

these

(Ia)

EI01al,

in layer

i

is the

If

the

A- B- '(Ajejk.x Bie-jk-x) i+

i;;;; Z

J

+

I

.af)

( , E:j = E:j l 1-; -;-

, [

J'j=j.lj

(5)

1", are g iven by

,

)

.ar . 1-;-;-

(6a)

(6b)

Equations (4) can be rearranged into the forms

(7a)

sum of the forward- and

•

=mFfP{,

where (() is the angular frequency of the waves. The complex per­

(7b)

backward-traveling waves, such that E-i =

is the wavenumber in l ayer i, and can be written as

mittivi ty, 10;, and the permeability,

(lb) The total field,

kj

where

(2)

­

medium consists of N layers that are surrounded by semi

infinite dielectric spaces on both sides, the reflection and transmis­ sion coefficients are expressed as

(3 a)

(3b) 114

IEEE Antennas and Propagation Magazine, Vol. 48, No.1, February 2006

This procedure can be repeated until the amplitudes in the semi­ infinite layer

a are solved.

Then, Equation (3) can be used to cal­

culate the reflection and transmission coefficients.

Czh;

=

..Jei+dl'i+1 e-jk;.A.

In addition to Equ ations (7), w e assume

3. Reflection by a Layered Medium with PEe Termination

that

(Sa) (8b)

Using Eqnations (7) and (8), a matrix equation can be constructed.

For the configuration in Figure I, it can be written as shown in Figure

2,

and will be denoted

Equation (9).

So lutio n of

Equa­

tion (9) will give the amplitudes of the forward- and backward­ traveling waves in all layers. However, solution of Equation (9) is

not computationally efficient, and one can construc t another algo­ rithm, denoted as the back-lo-tront a lgorithm, which is more effi­ cient.

Let's assume that that AN+1

=

0, because

Eo

is

not known, and BN+1 =1.

there is

no reflection within the

ampl itudes

N can be since examination of Equation (7) reveals that if the Ai+l and Bi+l in layer i + 1 are known, Equation (7)

can be rearranged and solved simultaneously for the and

Bi

in layer

i, such that

1

spaces on both sides. However, the previous section can also be us e d to calculate the reflection co effi cient and the fields in a layered medium terminated by a PEe (perfect electrical conductor) boundary, as well . In this case, the boundary condition for the tangential electric field components needs to b e applied,

the

equations

and algorithms demonstrated in

such that

(1la)

which leads to

(llb)

We know

right semi­

infinite medium. Therefore, the wave amplitudes in layer determined,

So far, it has been assumed that the layered medium is sur­

rounded by semi-infinite dielectric

amplitudes A,.

Using Equation (11) together with Equation (7), one can co ns truct a

matrix

equati o n similar to Equation (9): th i s is

the previous section still can be employed to solve this problem

more efficiently instead of solving the forward.traveling wave in layer boundary on

[CYhi

CxeiCyh; - Cxh,·Cyei -Cxhj

Equation (12),

given in Figure 3. The back-Io-front algorithm that is described in

a side,

a

Equation (12). Assuming that N, the l yer touching the PEe

has unit amplitude, i.e.,

(13)

2; ] [=�:�

-

i

=�:::J[�::] (10)

and by using Equation (13) in Equation (11), one as

backward-traveling wave ampli tu de

can find the

Bo

EO 0

Cxeo

Cyeo

Czeo

CweO

Ao

Cxho

Cyho

Czho

Cwho

Bl

0

Cxel

Cye!

Cze!

Cwel

Al

0

Cxhl

Cyhl

Czhl

Cwhl

B2

0

A2

B3

A3

0 0

0

o o

BN

Figure 2, The matrix Equation

o

o

CxeN

CyeN

CzeN

CweN

AN

CxhN

CyhN

CzhN

CwhN

BN+I

o

1

AN+l

o

(9) for the amplitudes of the forward- and backward-traveling waves in all layers.

IEEE Antennas and Propagation Magazine, Vol. 48, No. t, February 2006

115

BO

Eo

Cxeo

Cyeo

Czeo

Cweo

Ao

0

Cxho

C y ho

Czho

Cwho

BI

0

Cxel

C ye l

Cze1

ewe1

Al

0

Cxhl

Cyh1

Czh1

Cw hl

B2

0

A2

0 0 0

Figure 3. The matrix Equation (12) for a

(14) Once the amplitudes AN and

BN

are calculated, one can continue

with Equation (10) until all other amplitudes are detennined.

Furthermore, it is worth no ting that the transmission-line

analogy is another technique that is often employed to solve these kinds of problems.

B4

0

Cxe4

Cye4

Cze4

Cwe4

�

0

Cxh4

Cyh4

Czh4

Cwh4

BN

0

CxeN

CyeN

AN

0

layered medium with a PEe

In th e main parameters button

termiuation.

problem-definition window, the pops up the

optimization

optimization parameters

window ,

as shown in Figure 5. All parameters of the layers are listed in this

window. The user can choose som e of these parameters as sweep­

ing parameters by clicking on the associate d check boxes. When a parameter is chosen as a swe epi ng parameter, the user can define

the range of this parameter and th e step size for the values t hat this parameter can take.

The

user can choose optimization on the

reflection coefficient or the transmission coefficient, u sing the

respective radio button.

Furthermore,

the user

can define a

target

4. Description of the Program The algorithms described

in the

p r evious sections have been

programm ed, and a software package with a

graphica l user inter­ GUl that is define multip le numbers

face has been constructed. Figure 4 shows the main used to define the problem. Th e user can

of layers. Each layer is defined by its relative permittivity, relative

permeability,

thickness.

The

electric

user

conductivity,

defines the

magnetic

conductivity

and

S1l!lf!1o��"nl;:r 8

�I

r

IlllCkrI�3

End FreqUII:lcy 12. , rel.£erm�I�,. -2..I'lOOIro:

#fHlqlltfidu C"·....... 41 III �rmililllint

0,1,-09'99°

�l!�;r ul"t.ara:s, ')

a1.� Cllfiod\lC11IIfIl' r",JVl -(:(lrt�U�I!'t'Ill 0.00000

o,�oo

units offrequency and length on base d By default, the layered

which the problem parameters are

.

medium is surrounded by free space on both sides. However, the

use r can choose a PEe tennination on the right boundary. The reflection and transmission

coe fficients are calculated due to plane

� t==5

waves incident on the left boundary. The user defines the frequen­

window that displays the results pops up, as shown in Figure 6. The

reflection

and transmission coefficients can be displayed as

functions of frequency in this window. The magnitudes and phases

can be displayed separately, using the dropdown menu. The mag­

nitude can be displayed using either a linear scale or a dB scale by using the toggle button. Furthennore, the spatial distribution of the

fields at a chosen frequency can be d i sp layed .

Any d is playe d

plot

a separate figur e by clicking the plot in s eparate in any other documents . can be exported to

figure button. This helps the user to manipUlate the plot for use

116

f.iW21 1�10 • ·4

ll;$;;;2 1->:=1 0'00 .;"00

cies of the plane waves. On clicking the plot fields button, another

:';;1

0'.0 .;".0

S----i----:,:--7:1O--,"'"S--"'''' O1S1af'lCe (millunSterS')

!orn:l�P¥:.rWl�] �

Figure 4. The GUI for defining the parameters of mUltiple lay­ ers of slabs.

IEEE Antennas and Propagation Magazine. Vol. 48, No.1. February 2006

where NF is the number of frequencies in the frequency range, and rn and Tn are the re fl ection and transmission coefficients at the

frequency defined by the index n. rtarget and T;arget are the target values for the reflection and transmission coefficients. The combi­ nation of parameters that satisfies the c onditio n in Equation (15) is considered to be optimum. Since sweeping through all pos sible combinations and cal cu latin g the reflection or transmission coeffi· cient for each combination is a time-consuming process, this task is programmed in FORTRAN rather than MATLAB, in orde r to obtai n a fast response.

o.

0.1

Once a problem is constructed and solved, it can be saved as a project file. The user thus does not need to redefine a pr ob l em that has been worked on before, and hence a [e· optimization for a different set of parameters can be started e a sily at a later time. nn

5. Sample Results

Figure Sa. Sweeping parameters for minimum reflection: Elec­

trical

couductivities are euabled as sweeping parameters and

their rauges are defined.

Figure 4 sh ows a two-layer medium in free space backed by a PEC wall. The material parameters of the layers are displayed in the figure , Since the layers were backed by a PEC wall and the layer media were lossless, the incident fields were going to be reflec ted without any losses. If it is was desired to minimize the reflection from these layers, the materials should include losses. Figure 5 shows the configuration of an optimization problem that was used to find the optimum electrical conductivity values o f the layers for minimum reflection. As shown in Figure 50, the electri· cal cond uctivity parameters were enabled as s weeping parameters. After the optimization was perfonned, optimum values of the elec­ trical conductivities were obtained within the spec ifi ed ranges of values, as shown in Figure 5b. Figure 6 shows the amplitudes of the reflection coefficients from the l ayers using these new electrical conductivity values, over a frequency range from 8 to 12 GHz, on a dB scale.

Qjoptimizereilection

I Fliotiosepartrtefiglll'e llko:�=� '--___--"--' �

Reflection CoeffiCIent . 1 6 ,---,..---,--,---,.----, ·181'-..·..·""· ;·.. ••• .. +··"·····

Figure 5b. After sweepiug has beeu performed, optimum val­ ues of the parameters are determined.

for optimization. The start optimization button runs exe cuta­ that sweeps all combinations of parameters, and finds the c om bination that gives the closest reflection or tran smi ssio n to the target value. It returns this combination back to the optimization parameters window. The condition for the optimum value is given by value

ble code

(15a) or

.20

·22 .......

+".,......,.;. .. ........

. . r��f_:j�,· · ·v, ijj" �

,

, ......... i..........; .. ........,

8.5

. (NF� ITn

mm

-

2J '

lIarget I

(15b)

IEEE Antennas and Propagation Magazine, Vol. 48, No. 1, February 2006

Figure

6.

9

The reflection

defined in Figure 4.

9.5

10

1D�

Frequen