GENERALIZED ANALYTIC METHOD FOR NEAR-FIELD RECONSTRUCTION OF
ANTENNA CURRENTS IN MICROWAVE HYPERTHERMIA CANCER
TREATMENT SYSTEMS
Pawan Kavikondala
B.E, Jawaharlal Nehru Technological University, India, 2007
Parimala Ritika Sen Pesaramalli
B.E, Jawaharlal Nehru Technological University, India, 2006
PROJECT
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
ELECTRICAL AND ELECTRONIC ENGINEERING
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
SPRING
2010
GENERALIZED ANALYTIC METHOD FOR NEAR-FIELD RECONSTRUCTION OF
ANTENNA CURRENTS IN MICROWAVE HYPERTHERMIA CANCER
TREATMENT SYSTEMS
A Project
by
Pawan Kavikondala
Parimala Ritika Sen Pesaramalli
Approved by:
__________________________________, Committee Chair
Preetham B. Kumar, Ph.D.
__________________________________, Second Reader
Milica Markovic, Ph.D.
___________________________
Date
ii
Students: Pawan Kavikondala
Parimala Ritika Sen Pesaramalli
I certify that these students have met the requirements for format contained in the
University format manual, and that this project is suitable for shelving in the Library and
credit is to be awarded for the Project.
___________________, Graduate Coordinator
Preetham B. Kumar, Ph.D.
Department of Electrical and Electronic Engineering
iii
____________
Date
Abstract
of
GENERALIZED ANALYTIC METHOD FOR NEAR-FIELD RECONSTRUCTION OF
ANTENNA CURRENTS IN MICROWAVE HYPERTHERMIA CANCER
TREATMENT SYSTEMS
by
Pawan Kavikondala
Parimala Ritika Sen Pesaramalli
Currently chemotherapy radiation and surgery are the standard treatments for cancer that
are widely accepted and used. However, a fourth modality is emerging as an adjuvant
tool against cancer. Hyperthermia treatment is currently used as an added form of
treatment during radiation or chemotherapy, and works as a booster, sometimes even
doubling the effects of these standard treatments.
Hyperthermia treatment currently explored in this reference is done by exposing the
cancer region to precisely controlled RF microwave radiation which treats the area. Our
aim in this project is to write algorithms in MATLAB software to generate appropriate
antenna currents for different geometries like planar, cylindrical and spherical, with the
aim to focus at the desired tumor area. Several simulations were carried out to test the
validity of the algorithms and results verified the accuracy.
, Committee Chair
Preetham B. Kumar, Ph.D.
______________________
Date
iv
ACKNOWLEDGEMENT
We would like to take this opportunity to thank the people who had aided us and have
been instrumental in the successful completion of our Master’s project.
Firstly, we owe our deepest gratitude to our advisor, committee chair and graduate
coordinator, Dr. Preetham B. Kumar. Thank you Dr. Preetham B. Kumar for your
constant guidance, patience and understanding. Thank you for making this experience
knowledgeable and a smooth curve to tread. We are truly fortunate to have you as our
mentor.
We would like to thank our second reader Dr. Milica Markovic, for taking the time in
reviewing, proving us valuable feedback on our report; consequentially helping us
succeed in our goals.
Finally, we would like to thank our parents. They are the sole reason for where we stand
in our lives today. Thank you, for accepting our failures and providing us with
infinite opportunities. Although, we can never thank you enough for the unlimited
amount of time spent, caring for us and the unconditional love shown, we would
like to dedicate our success to you as small token of our gratitude.
v
TABLE OF CONTENTS
Page
Acknowledgement……………………………………………………………………….v
List of Figures……………………………………………………………………...…....vii
Chapter
1. INTRODUCTION ....................................................................................................... ..1
2.SYNTHESIS OF EXCITATION CURRENTS OF PLANAR ARRAYS ..................... 3
2.1 Mathematical Analysis............................................................................................. 3
3. SYNTHESIS OF EXCITATION CURRENTS OF CIRCULAR CYLINDRICAL
PATCH ARRAYS...........................................................................................................8
3.1 Analysis of Cylindrical Array ................................................................................. 8
4. SYNTHESIS OF EXCITATION CURRENTS OF SPHERICAL ARRAYS............. 14
4.1 Anaysis of Spherical Array .................................................................................... 14
5. NUMERICAL RESULTS AND DISCUSSIONS…………………………………..19
5.1 Simulation Results for a Planar Array....................................................................19
5.2 Simulation Results for Cylindrical Array...............................................................21
5.3 Simulation Results for Spherical Array..................................................................24
6. CONCLUSION.............................................................................................................27
Appendix MATLAB Programs………………………………………………………….29
A.1. MATLAB program for Planar Array…................................................................29
A.2. MATLAB program for Cylindrical Array............................................................35
A.3. MATLAB program for Spherical Array...............................................................40
References ........................................................................................................................ 44
vi
LIST OF FIGURES
1. Figure 2.1 Planar Array……………………………………………….………………..4
2. Figure 3.1 Cylindrical array ………………………………….……………….……....9
3. Figure 4.1 Spherical Array……………………………………………………….……15
4.Figure 5.1 Exact (a) and reconstructed (b) currents of planar array……….....………20
5. Figure 5.2 Exact (a) and reconstructed (b) currents of cylindrical array …..….…….23
6. Figure 5.3 Exact (a) and reconstructed (b) currents of spherical array …….…..….…25
vii
1
Chapter 1
INTRODUCTION
Treatment of cancer has always been targeted by using radiation, chemotherapy
and surgery, but alternative treatments for cancer are in the pipeline too, amongst them
Hyperthermia treatment is the most promising and in current times it is transitioning from
the experimental to clinical stage [1-3] This heating of the cancer tissue subsequently
sensitizes the cells to the following radiation or chemotherapy treatment, sometimes even
doubling the response , as compared to standalone radiation or chemotherapy.
Hyperthermia treatment for cancer currently serves as a supplemental or adjuvant
treatment to chemotherapy and radiation; this treatment uses antenna arrays to project
microwaves to heat the area of the cancer tissue up to
F.
For efficient hyperthermia application, it is necessary for the projected
microwaves to be focused on the tumor area without having significant energy distributed
over normal (non-cancerous) tissue and also to have control over the motion of the beam
around the tumor area for successful therapeutic effect. In order to achieve this, the
appropriate distribution of currents to the antenna arrays needs to be supplied; to generate
accurately postured radiation beams.
2
We need to remember that we are treating humans or maybe even animals therefore the
areas of the surface required to be treated can be of a various shapes. Therefore the array
geometry being considered in this aspect has to be preferably conformal to planar,
cylindrical and spherical surfaces. Now by processing the algorithms for every shape
individually by using MATLAB software we can obtain the simulation models of the
reconstruction currents from the source near zone electric fields. The algorithms are
designed for every pattern and every algorithm takes the desired fields and then applies
the reconstruction algorithms to calculate them. Once calculated, these currents can be
electronically changed i.e. using the hardware (in the arrays) and the array will focus on
the desired tumor area.
3
Chapter 2
SYNTHESIS OF EXCITATION CURRENTS OF PLANAR ARRAYS
In the synthesis of excitation currents of planar arrays, we take into account the numerical
implementation of the technique proposed to reconstruct the excitation currents of planar
arrays or apertures with near-field data. This technique is based on a two-dimensional
FFT (Fast Fourier Transform) algorithm [4], and can be implemented with high accuracy
and speed.
2.1 Mathematical Analysis
2.1.1 Array with currents along x-axis:
Let us consider a uniformly spaced planar array with an inter-element spacing ‘d’ and
with x-directed electric point dipoles as shown in figure 2.1. Let there be M=
number
of elements in the array with N elements arranged along the x-axis (with y held constant)
and N elements along the y-axis ( for any fixed value of x) . The exact expression for the
near-field electric field components of this array at an observation point
(x,y,z)
obtained from the potential integral solution are given by [5]
=
+
…(2.1)
4
Y
Sm
…. …...
………….
. P(x,y.zo) ...
…… …..
……………
….
……
.P(x,y.zo)
X
Sm
Xn
Z
Z=0
Z=zo
Source Plane
Position vector of point dipole is
Observation Plane
=
Figure 2.1 Planar Array
=
... (2.2)
=
where AX represents the magnetic vector potential.
The vector potential is [5]
=
...(2.3)
5
Where
=
...(2.4)
Sm= field distance between the source and observant plane, k=2π/λ where λ is the
wavelength and
is the amplitude and phase of the excitation current of the
element located at P (
).Hence substituting 2.3 in 2.1 we get,
=
+
-
+[
+
…(2.5)
]
The near field of the array reference element located to
=
+
[
+
;y-
; )
=0;
-
= 0 is given by [4],
-
]
...(2.6)
From (2.4) and (2.6) we obtain:
(
)=
(x -
...(2.7)
By Fourier transformation on both sides of (2.7) with respect to the spatial frequency
variable ‘u’ and ‘v’ we obtain:
(u,v) =
Where
(u,v)
(u, v) =
(x,y)
...(2.8)
dx dy
...(2.9)
We rewrite (2.8) as
(u,v) =
(u,v) [ FT
where FT [g( , )] =
and
( , )]=
( , )
(u,v)
d
(u,v)
...(2.10)
d
( , )= am denotes the x-directed surface current density of the radiating aperture.
6
2.1.2 Array with currents along arbitrary axis:
In this case, the radiating aperture comprises the M array sources. The analytical
formulation described above could be generalized to a planar array in which the
excitation current of the
element is oriented in an arbitrary direction
in the x-y
plane. The expression for the aperture excitation current of such an array is given by
J=
] δ (x-
+
where
and
=
) and
) δ (y=
)
(
...(2.11)
)
is the magnitude and phase of the excitation current of the
element of the
array. Carrying out an analysis similar to the one employed previously for the array with
x-directed excitation currents, we obtain:
(x,y,z) =
(x-
; y-
;
)+
(x-
; y-
;
)]
...(2.12)
(x,y,z) =
(x-
; y-
;
)+
(x-
; y-
;
)]
...(2.13)
Where we define
(i)
(x,y,z) as the x-component of the near-field generated at (x,y,z) by a -
directed electric point dipole with unit current amplitude located
(ii)
as the x component of the near-field generated at (x,y,z) by a
point dipole with unit current amplitude located at
(iii)
=0;
= 0 and z=o
–directed electric
= 0 and z=0.
(x,y,z) as the y-component of the near-field generated at (x,y,z) by a
directed point electric dipole with unit current amplitude located at
z=0.
=0;
= 0;
-
= 0 and
7
(iv)
(x,y,z) as the y-component of the near-field generated at (x,y,z) by a
directed electric point dipole with unit current amplitude located at
=0;
-
= 0 and
z=0.
In (2.12) and (2.13)
is given by (2.6) with
(x,y,z) =
xy ( 3+3jk
(x,y,z) =
(x,y,z)
=
-
+
= 1 and
)
...(2.14)
...(2.15)
[
+
-
-
]
...(2.16)
Fourier Transforming on either sides of (2.12) and (2.13) yields
(u,v, ) =
(u, v,
)
+
(u,v, )
...(2.17)
(u,v, ) =
(u, v,
)
+
(u,v, )
...(2.18)
We can rewrite (2.17) and (2.18) as
(u,v, ) =
(u, v, ) [ FT
( , )]+
(u, v, ) [ FT
( , )]
...(2.19)
(u,v, ) =
(u, v, ) [ FT
( , )]+
(u, v, ) [ FT
( , )]
...(2.20)
8
Chapter 3
SYNTHESIS OF EXCITATION CURRENTS OF CIRCULAR CYLINDRICAL
PATCH ARRAYS
In the previous chapter, the excitation currents of planar arrays were synthesized
using the near field data. In the present chapter we synthesize the excitation currents of a
circular cylindrical patch array [6]. Here, the geometry is different but the process for
synthesis remains the same as the synthesis of planar arrays. We use Fourier transform
methods and therefore avoid solution of a system of linear equations. A near field is
generated for array geometry of circular cylindrical patches by using the excitation
currents. These excitation currents are reconstructed back from the near field by using
inversion method. The reconstructed currents may or may not be same as the original
excitation currents. If the currents are not the same it means the antenna is faulty.
3.1 Analysis of Cylindrical Array:
The circular cylindrical surface of a two-dimensional curved array is built up of discrete
current elements distributed along the surface as depicted in Figure 3.1. These distributed
current elements are formed due to point dipoles. The spacing between the elements of
the array is assumed to be uniform and the current flowing in the array elements is in the
–direction
as shown in figure 3.1.a or in the
- direction
as shown in figure 3.1.b.
9
X
P(ρo,φ,z)
Y
Z
Source cylinder
(a)
X
P(ρo,φ,z)
Y
Z
Y
(b)
Figure 3.1 Cylindrical Array
10
3.1.1 Direction of current flow of the array element along
- direction:
The curved two-dimensional array has an inter element spacing of dz in the ẑ
direction and a spacing of d  along the ˆ direction as depicted in Figure 3.1.a.The
current element is considered to be located at a position Pi (a, i , zi ) where a is the
radius
of
the
 ( zi  z ')
Js  ˆ
cylinder,
whose
surface
current
density
 ( i   ') Ii
is
given
by
...(3.1)
a
Where Ii is the excitation current of the ith array element. At a position P (, , z) in
near field we obtain from equation 3.1:
Using
we get,
 jk    i
 Ii sin( i   ) e
A i 
4    i
...(3.2a)
Using
we get,
 jk    i
 Ii cos( i   ) e
A i 
4    i
...(3.2b)
1
where
   i  [( x  xi ) 2  ( y  yi ) 2  ( z  zi ) 2 ] 2
...(3.3)
by using transformation of variables to cylindrical coordinate system,
1
  i  [  2  a 2  2  a cos(  i )  ( z  zi ) 2 ] 2
...(3.3a)
11
The electric field at P (, , z) is given by
ˆ zi
Ei  ˆEi  zE
...(3.4)
Where
E i  E1i  E2i
...(3.5)
Ezi  E1zi  E2 zi
...(3.6)
E1 i   j A i
...(3.7)
E2i 

j 
1
[
1

 2 Ai

( Ai ) 
]

 2
E1zi  0
E2 zi
...(3.8)
...(3.9)

1

[
j z 
1
 2 Ai

( Ai ) 
]

z
...(3.10)
we define
Li (  ,  , z; a, i , zi ) 
1


( Ai )

...(3.11)
12

( Ai )

M i (  ,  , z; a, i , zi ) 
...(3.12)
2 
  A (  ,  , z; a,  , z ) e
Ai (  , u, v; a, i , zi ) 
i
i
 j (u vz )
i
d dz
…(3.13)
d dz
…(3.14)
0 
Li (  , u, v; a, i , zi ) 
2 
  L (  ,  , z ; a,  , z ) e
i
i
 j (u vz )
i
0 
M i (  , u, v; a, i , zi ) 
2 
  L (  ,  , z ; a,  , z ) e
i
i
 j (u vz )
i
d dz
...(3.15)
0 
Using the Fourier Transform relations (The Fourier relations are provided in the
Appendix) from 3.5 to 3.15, we get
2
Ezi 

 E
0
zi
( , , z) e
 j (u vz )
d dz 

and Ei  Ii [ j Ao 
where
Mo 
A o 
Ii  0
1
j
juLo 
M i (  , u, v; a,0,0)
Ii
Ai (  , u, v; a, 0, 0)
Ii
1
j
u2

[ jvM o  uvA o ] I i e
Ao ] e
 jui  jvzi
 jui  jvzi
...(3.16)
...(3.17)
...(3.18)
...(3.19)
13
Finally we have
M
E (  , u, v)   Ei  [ j Ao 
i 1
1
j
{ juLo 
u2

Ao }]  I i e
 jui  jvzi
i
...(3.20)
Ez (  , u , v ) 
1
j
[ jvM o  uvAo ]
I e
 jui  jvzi
i
...(3.21)
i
The equations 3.20 and 3.21 can be inverted to solve for the array currents Ii(φ,z)
3.1.2. Discrete point dipoles with direction of current flow along ẑ direction:
Analyzing current flow using a similar analysis employed in section 3.1 previously gives
us the following:
Js  zˆi
 ( z  zi )
 (  i ) Ii
a
...(3.22)
 jk   i
Ii e
Azi 
4   i
...(3.23)
1 uv
v2
 jui  jvzi
Ei (  , u, v; a, i , zi )  [ˆ(
)(
) Azo  zˆ( j  ) Azo ]Ii e
j 
j
...(3.24)
M
E (  , u, v)   Ei (  , u, v; a, i , zi ) 
i 1
1 uv
v2
 jui  jvzi
Azo [ˆ( )(
)  zˆ ( j 
)]I i e
j 
j i
which can be inverted to field the array current Ii(φ,z)
...(3.25)
14
Chapter 4
SYNTHESIS OF EXCITATION CURRENTS OF SPHERICAL ARRAYS
This chapter details the reconstruction algorithm to obtain the array currents from the
near-field data of a spherical array. The analysis of spherical arrays is more complex than
planar or cylindrical arrays, owing to the geometry. Hence, it does not involve a purely
two-dimensional Fourier transform inversion as in the case of the planar and cylindrical
arrays, but a Legendre-Fourier type of transformation [7-9]. The following section details
the analysis of the spherical array, and the inversion procedure to reconstruct the array
currents for the near-field data.
4.1 Analysis of Spherical Array:
Consider a spherical array as shown in Figure 4.1with array radius r o. The array elements
are point dipoles with currents along
direction. The near zone field of the array at a
point P(R, Ө, φ) is given by:
Electric field of array,
(θ, Φ) =
where f (
) f(
,
R=
and
[
]=
)=
[
,
)
]
...(4.1)
15
=
[
]
Finally f (
,
)=
[jKR + 1 ]
[
X
]
P(R, Ө, φ)
Y
Z
Figure 4.1 Spherical array
and the final equation for the electric field is :
(θ, Φ) =
where R =
) [jKR + 1 ]
[
…………...(4.2)
16
4.1 Synthesis of Excitation currents of the array:
From equation 4.1
(θ, Φ) =
[
Expanding the finite
using spherical wave expansion [10] we get
(θ, Φ) =
(
]
[
(
(
=
…(4.3)
(
(
Here
(
(
is the associated Legendre function and
,
are spherical Bessel
functions.
The synthesis procedure is shown below in a series of steps:
Step 1: Legendre Fourier Transform
F(n,m)=
(
=
(
sinθ d dΦ
(
m
(
…(4.4)
.
17
which finally yields
or Q(n,m)=
(x) and
=
(
…(4.5)
(
(x) are proportional ,since differential equations depends only on
and
m is an integer.It can be shown that
(x) =
(x)
The solution of Laplace’s equation was decomposed into a product of factors for the three
variables r,θ, and ∅. It is convenient to combine the angular factors and construct
orthonormal functions over the unit sphere. We will call these functions spherical
harmonics are sometimes called “tesseral harmonics”.
The functions
0
(∅) =
form a complete set of orthogonal functions in the interval
. The functions
form a similar set in the index l for each m value on
the interval
. Therefore their product
will form a complete
orthogonal set of the unit sphere in the two indices l, m. From the normalization
condition (3.52) it is clear that the suitably normalized functions, denoted by
are:
=
(
...(4.6)
18
The spherical harmonics obey the condition [10]
=
...(4.7)
The normalization and orthogonality conditions are
d
=
...(4.8)
The complete relation is;
= (∅ -
) (cos -cos
...(4.9)
Using the spherical harmonics in Equation (4.6) on equation (4.4), and utilizing the
orthogonality relation 4.9, we can reconstruct the currents of the array.
19
Chapter 5
NUMERICAL RESULTS AND DISCUSSIONS
5.1 Simulation Results for a Planar Array
In order to calculate numerically the array excitation currents from the near field
date, we use equations 2.8 to 2.10
(u,v) from
obtains
(x,y) and
(u,v) could be computed straight forwardly and one
from
using (2.10), Finally an inverse FT of
procedure is employed to obtain
(x,y) and
yields
(u,v). A similar
(x,y) from (2.19) and (2.20). In this case a
simultaneous equation is solved for each value of (u,v) to obtain FT [ ( ,
[ ( ,
) ] and FT
) ] from (2.19) and (2.20).
In this simulation, A uniformly spaced planar array consisting of point electric
dipoles with a current flow direction along the x-axis as shown in figure 2.1. The dipoles
were assumed to be excited with currents uniform in amplitude and phase .I this example
d = 0.25 is the distance between dipoles and
data was generated at
=
= 2 (figure 6.1) .The near field
= 2 . A 128*128 point FFT was used to obtain the array
excitation currents .The exact array excitation currents employed for generating the near
20
field data the synthesized array excitation currents are compared in figures 5.1.a and
5.1.b. A close agreement is noticed between the two.
(a)
(b)
Figure 5.1 Exact (a) and reconstructed (b) currents of planar array
21
5.2 Simulation Results for Cylindrical Array
A two dimensional cylindrical path array built of uniformly spaced current
elements is assumed with the following dimensions (Figure 3.1 b)
L = 15
a=2
0=225
=2/32 radians=11.25 degrees.
z=0.75
This array is considered with the current in each point dipole flowing along the
direction of ẑ . The near-field of the array Ez (ρ,φ,z) is utilized to reconstruct the array
currents by using the reconstruction algorithm given in equations 2.20-2.25.Near field
synthesis performed on the array gives numerical results, which are depicted in Figures
5.2a and 5.2b. The color bar comparison on the right shows that the array currents have
been reconstructed with good accuracy, thereby validating the rigorousness of the
algorithm.
The electric field components E (, u, v) or Ez (, u, v) are obtained from E (,
, z) = ˆ E (, , z) + ẑ Ez(, , z) to numerically calculate the excitation currents of
the array from the near field data. Azo and
Ao can be obtained directly by solving
equations (2.20), (2.21), (2.25).While the current flows in an array element in the
direction of , we apply a two dimensional inverse fast Fourier transforms procedure to
22
obtain Ii from equation (2.20).Synthesis of currents is dependent only on the near field
component E in this case as illustrated in equation (2.4).We have  Ez(, , z) <<  E
(, , z)and so it cannot consider other near field components but only E .Also,
practically the near field data is prone to measurement error and considering Ez would
result in a relatively higher error therefore these various reasons strengthen the
dependency of the currents on the component E
23
(a)
(b)
Figure 5.2 Exact (a) and reconstructed (b) currents of cylindrical array
The results in the graph are obtained by considering the excitation currents
distributed over one quarter of the patch (0  z  L; 0    0; =a) because of the
symmetry.
24
5.3 Simulation Results for Spherical Array
Finally the synthesis example was completed for a spherical array of ẑ directed
point dipoles situated around the surface of the array as shown in figure 4.3. The details
of the array are given below
5* 5 array = 25 elements of magnetic dipoles
Radius of Spherical array =
=2𝛌
Radius of Spherical array =
=4𝛌
Frequency of array = 1 GHz
Array Spacing
ΔΦ =
radians;
Δθ =
radians;
=0,
=0,
= ΔΦ,
= Δθ,
= 2ΔΦ,
= 2Δθ,
= 3ΔΦ,
= 3Δθ,
= 4ΔΦ
= 4Δθ
The array solution was done using equations 4.4 to 4.9 to yield the currents. The exact
and reconstructed currents are shown in figure 5.3a and 5.3b respectively and show good
agreement.
25
7
1
0.9
6
0.8
5
0.7
0.6
4
0.5
3
0.4
0.3
2
0.2
1
0.1
0
0
0.5
1
1.5
2
(a)
2.5
3
3.5
0
26
7
0.8
6
0.7
5
0.6
4
0.5
3
0.4
0.3
2
0.2
1
0
0.1
0
0.5
1
1.5
2
2.5
3
3.5
0
(b)
Figure 5.3 Exact (a) and reconstructed (b) currents of spherical array
27
Chapter 6
CONCLUSIONS
The main focus of this project is design of efficient algorithms for the near-field
reconstruction of array currents with planar, cylindrical and spherical arrays.
The
synthesis procedure was similar for all array geometries: The near-field of the array is
formed
by feeding proper current values to the antennas .Then the currents are
reconstructed from field to ensure that the reconstructed currents match the currents that
were initially fed to the antennas.
All the three arrays have been simulated and synthesized using MATLAB. The
code is simulated for various current values to obtain a graphical result. The graphical
representation of the simulated code for each array shows the extent to which
reconstructed currents match the initial currents. To a large extent, the reconstructed
currents and the initial currents have matched for each of the three arrays. Hence planar,
cylindrical and spherical arrays have been synthesized successfully.
FUTURE ENHANCEMENTS AND APPLICATIONS
The synthesis of arrays can be applied vastly in the medical field concerning the
treatment of tumors. It helps in focusing the antenna beam very precisely in the tumor
affected area. Otherwise the harmful rays can disperse over the unaffected area damaging
the good tissue and causing unwanted complications. This method of focusing the beam
by feeding the needed currents offers more controlled way of dealing with tumors. Its
future applications are extended towards treatment of carcinoma and hyperthermia.
28
Another application is pertaining to the space field, mostly satellites which have
/enormous number of antennas. In case anyone antenna or few antennas malfunction, it
becomes a tedious task to find the defective antennas amongst the large number of
antennas. To solve this problem all the antennas on the satellite are fed with currents and
the currents are reconstructed back. The non-defective antennas have the same
reconstructed and initial currents .Those antennas for which the initial and reconstructed
currents do not match are considered defective. Hence this method is very effective to
weed out the defective antennas when dealing with large number of antennas.
29
APPENDIX
MATLAB Programs
A.1.MATLAB program for Planar Array
clear all;
close all;
clc
M= 25;
f=1e9;
% frequency
l=3e8/f;
%lambda
k=2*pi/l;
w=2*pi*f;
% omega
a=[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];
% a1=zeros(128);
xb= [-0.5*l -0.250*l 0 0.250*l 0.5*l -0.5*l -0.25*l 0 0.250*l 0.5*l -0.5*l -0.250*l 0
0.250*l 0.5*l -0.5*l -0.250*l 0 0.250*l 0.5*l -0.5*l -0.250*l 0 0.250*l 0.5*l ];
yb =[0.50*l 0.50*l 0.50*l 0.50*l 0.50*l 0.25*l 0.25*l 0.25*l 0.25*l 0.25*l 0 0 0 0 0
-
30
0.25*l -0.25*l -0.25*l -0.25*l -0.25*l -0.50*l -0.50*l -0.50*l -0.50*l -0.50*l ];
z =5*l;
mu= 1.2566e-6;
% Meu-Knot
p=-j*w*mu/(4*pi);
%constant before first summation
E = 8.854e-12 ;
% Epsilon
q=1/(4*pi*j*w*E);
dx=0.25*l;
dy=0.25*l;
for ix=1:128;
x=(ix-64)*dx;
for jy=1:128;
y=(jy-64)*dy;
Ex(ix,jy)=0.0;
for m = 1:M;
s = ((x-xb(m))^2+(y-yb(m))^2+(z^2))^(1/2) ;
s1= p*((a(m)*exp(-j*k*s)/s));
sum1=(s1);
result1=p*sum1;
31
s2= (a(m)*exp((-j*k*s)));
sum2=(s2);
result2 =q*sum2;
t1 = (3*j*k*(x-xb(m))^2)/(s)^4 ;
t2 =(3*((x-xb(m))^2)/(s^5));
t3 =-(((k^2)*(x-xb(m))^2)+1)/((s^3));
t4 = (j*k)/((s)^2);
result3 = t1+t2+t3+t4;
Ex(ix,jy) = Ex(ix,jy)+(result1)*(result2+result3);
end
end
end
for ix=1:128;
x=(ix-64)*dx;
xi(ix)=x;
for jy=1:128;
y=(jy-64)*dy;
yi(jy)=y;
Ex0(ix,jy)=0.0;
s = ((x)^2+(y)^2+(z^2))^(1/2) ;
s1= p*((exp(-j*k*s)/s));
32
sum1=(s1);
result1=p*sum1;
s2= (exp((-j*k*s)));
sum2=(s2);
result2 =q*sum2;
t1 = (3*j*k*(x)^2)/(s)^4 ;
t2 =(3*((x)^2)/(s^5));
t3 =-(((k^2)*(x)^2)+1)/((s^3));
t4 = (j*k)/((s)^2);
result3 = t1+t2+t3+t4;
Ex0(ix,jy) = (result1)*(result2+result3);
end
end
Exf=fft2(Ex);
Ex0f=fft2(Ex0);
acf=(Exf./Ex0f);
% ac=ifft2(acf);
ac=ifftshift(ifft2(acf));
subplot(2,1,1);
a1=zeros(128);
for i=63:67;
33
for j=63:67;
a1(i,j)=1.0;
end;
end;
% a3=zeros(1,8179);
% a4=zeros(1,8180);
% a5=[a3 a a4];
% a1=reshape(a5,128,128);
% a1=a1;
% 2D plot use view for this plot
%figure(1)
%subplot(2,1,1);
%surf(real(Ex0))
% view(0,90);
% colorbar;
subplot(2,1,1);
surf(xi,yi,abs(a1));
axis([-1,1,-1,1]);
view(0,90);
34
colorbar;
% 3D Plot for reference
% figure(2)
subplot(2,1,2);
%surf(real(Ex0))
%colorbar;
%subplot(2,1,2);
surf(xi,yi,abs(ac));
axis([-1,1,-1,1]);
view(0,90);
colorbar;
35
A.2.MATLAB program for Cylindrical Array
clear
T=pi*2;
Mx=5;
Mxa=2^Mx;
Xmx=Mxa;
M=7;
Ma=2^M;
Xma=Ma;
Dz=pi/2;
Dphi=T/Xmx;
Xz=-Dz*(Xma/2);
Xph=-pi;
A=1.0*T;
Rh=3.0*T;
for i1=1:Ma
Z=Xz+(i1-1)*Dz;
for j1=1:Mxa
Phi=Xph+(j1-1)*Dphi;
S1=0.0;
36
for k1=1:5
Zk=-Dz*2+(k1-1)*Dz;
for k2=1:5
Phik=-Dphi*2+(k2-1)*Dphi;
Z1=Z-Zk;
Phi1=Phi-Phik;
S=Rh^2+A^2-2*A*Rh*cos(Phi1)+Z1*Z1;
S=sqrt(S);
F1=exp(-j*S);
F2=3*j*Z1*Z1/S^4;
F3=3*Z1*Z1/S^5;
F4=-(Z1^2+1)/S^3;
F5=-j/S/S;
F=F1/S+F1*(F2+F3+F4+F5);
S1=S1+F;
end;
end;
A1(i1,j1)=S1;
37
Zk=0;
Phik=0;
Z1=Z-Zk;
Phi1=Phi-Phik;
S=Rh^2+A^2-2*A*Rh*cos(Phi1)+Z1*Z1;
S=sqrt(S);
F1=exp(-j*S);
F2=3*j*Z1*Z1/S^4;
F3=3*Z1*Z1/S^5;
F4=-(Z1^2+1)/S^3;
F5=-j/S/S;
F=F1/S+F1*(F2+F3+F4+F5);
A0(i1,j1)=F;
end;
end;
x=fft2(A1);
y=fft2(A0);
A2=x./y;
p1=ifft2(A2);
p=ifftshift(p1);
38
%surf(p);
for i=1:Ma;
for j=1:Mxa;
a1(i,j)=0.0;
end;
end;
for i=1:Ma
xi(i)=Xz+(i-1)*Dz;
for j=1:Mxa
yi(j)=Xph+(j-1)*Dphi;
end;
end;
for i=1:Ma;
for j=1:Mxa;
a1(i,j)=0;
end;
end;
for i=63:67;
for j=14:18;
39
a1(i,j)=1.0;
end;
end;
subplot(2,1,1);
surf(xi,yi,abs(a1'));
axis([-5,5,-2,2]);
view(0,90);
colorbar;
% 3D Plot for reference
% figure(2)
subplot(2,1,2);
%surf(real(Ex0))
%colorbar;
%subplot(2,1,2);
surf(xi,yi,abs(p'));
axis([-5,5,-2,2]);
view(0,90);
colorbar;
40
A.3.MATLAB program for Spherical Array
clear;
f=900e6;
c=3e8;
M=25;
lambda=c/f;
ro=2.3*lambda;
r=3.1*lambda;
k=2*pi/lambda;
a=[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];
N=90;
dth=pi/N;
dph=2*pi/N;
dthn=pi/6;
dphn=2*pi/6;
thetan= [dthn 2*dthn 3*dthn 4*dthn 5*dthn dthn 2*dthn 3*dthn 4*dthn 5*dthn dthn
2*dthn 3*dthn 4*dthn 5*dthn dthn 2*dthn 3*dthn 4*dthn 5*dthn dthn 2*dthn 3*dthn
4*dthn 5*dthn];
phin =[0 0 0 0 0 dphn dphn dphn dphn dphn 2*dphn 2*dphn 2*dphn 2*dphn 2*dphn
3*dphn 3*dphn 3*dphn 3*dphn 3*dphn 4*dphn 4*dphn 4*dphn 4*dphn 4*dphn];
% Generation of two-dimensional electric field E(theta, phi)
for i1=1:N;
theta=(i1-1)*dth;
for j1=1:N;
phi=(j1-1)*dph;
s=0.0;
41
for m=1:M;
R=sqrt(r^2+ro^2-2*r*ro*(cos(theta)*cos(thetan(m))+sin(theta)*sin(thetan(m))*cos(phiphin(m))));
Er=(r)*ro*(j*k*R+1)*(exp(-j*k*R)/R^3)*(sin(theta)*sin(thetan(m))*sin(phi-phin(m)));
s=s+a(m)*Er;
end;
e(i1,j1)=s;
end;
end;
% figure(1);
% surf(abs(e));
% Take the two-dimensional Legendre-Fourier Transform
Nr=round(k*r);
for n=1:Nr;
n1=(n-1);
for m1=1:n;
mr=m1-1;
for i1=1:N;
theta=(i1-1)*dth;
for j1=1:N;
phi=(j1-1)*dph;
theta1(i1)=theta;
phi1(j1)=phi;
eph(j1)=e(i1,j1)*exp(j*mr*phi)*(-1)^mr;
ephneg(j1)=e(i1,j1)*exp(-j*mr*phi)*(-1)^mr;
end;
legen=legendre(n1,cos(theta),'norm');
eth(i1)=trapz(phi1,eph)*legen(m1)*sin(theta);
ethneg(i1)=trapz(phi1,ephneg)*sin(theta)*legen(m1);
end;
hankelj=besselh(n1+0.5,2,k*r);
if mr~= 0
f(n,m1)=trapz(theta1,eth)*2*sqrt(r*ro)/besselj(n1+0.5,k*ro)/hankelj/mr/sqrt(2)/2/pi^2;
42
fneg(n,m1)=trapz(theta1,ethneg)*2*sqrt(r*ro)/besselj(n1+0.5,k*ro)/hankelj/(mr)/sqrt(2)/2/pi^2;
elseif mr==0
%f(n,m1)=trapz(theta1,eth)*2*r*ro/besselj(n1+0.5,k*ro)/hankelj/sqrt(2)/2/pi^2;
%fneg(n,m1)=trapz(theta1,ethneg)*2*r*ro/besselj(n1+0.5,k*ro)/hankelj/sqrt(2)/2/pi^2;
f(n,m1)=0.0;
fneg(n,m1)=0.0;
end;
end;
end;
for i1=1:5;
thetam=(i1)*dthn;
for j1=1:5;
phim=(j1-1)*dphn;
s=0.0;
for n=1:Nr;
n1=(n-1);
legen=legendre(n1,cos(thetam),'norm');
for m1=1:n;
mr=m1-1;
s=s+ (f(n,m1))*sqrt(2)*legen(m1)*exp(-j*mr*phim)*(-1)^mr;
end;
for m2=2:n;
mr=m2-1;
s=s+ (fneg(n,m2))*sqrt(2)*legen(m2)*exp(j*mr*phim)*(-1)^mr;
end;
end;
a1(i1,j1)=s;
end;
end;
t=(a1)/4/pi/4/sqrt(pi)/pi;
for i1=1:N;
theta(i1)=(i1-1)*dth;
for j1=1:N;
phi(j1)=(j1-1)*dph;
a(i1,j1)=0.0;
a2(i1,j1)=0.0;
end;
43
end;
for i1=1:18:N;
i2=1;
theta(i1)=(i1-1)*dth;
for j1=1:18:N;
j2=1;
phi(j1)=(j1-1)*dph;
a(i1,j1)=1;
a2(i1,j1)=abs(t(i2,j2));
j2=j2+1;
end;
i2=i2+1;
end;
figure(1);
% surf(abs(t));
%plot(2,1,1);
surf(theta,phi,abs(a));
%axis([-1,1,-1,1]);
view(0,90);
colorbar;
% 3D Plot for reference
figure(2)
%plot(2,1,2);
%surf(real(Ex0))
%colorbar;
%subplot(2,1,2);
surf(theta,phi,abs(a2));
%axis([-1,1,-1,1]);
view(0,90);
colorbar;
\
44
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C.A. Balanis, ‘Antenna Theory: Analysis and Design’, John Wiley & Sons, New
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45
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B.P. Kumar & G.R. Branner, ‘Generalized Analytical Technique for the Synthesis
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