MAGNETIC FIELD SIMULATION OF GOLAY AND MAXWELL COILS
CHEW TEONG HAN
UNIVERSITI TEKNOLOGI MALAYSIA
MAGNETIC FIELD SIMULATION OF GOLAY AND MAXWELL COILS
CHEW TEONG HAN
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Science (Physics)
Faculty of Science
Universiti Teknologi Malaysia
MAY 2010
iii
To the late Associate Professor Dr Rashdi Shah Ahmad.
iv
ACKNOWLEDGEMENT
I would like to express my gratitude and appreciation towards both my
supervisors, the late Assoc Prof Dr Rashdi Shah and Dr Amiruddin for their guidance
throughout the research. I would like to thank both my internal examiner, Assoc Prof
Dr Ahmad Radzi Mat Isa and external examiner, Assoc Prof Dr Ahmad Nazlim Yusoff,
for their advices and useful disscussions in improving my research and especially
the report. Apart from that, I would like to thank Kuok Foundation for its financial
assistance. I would also wish to acknowledge Universiti Teknologi Malaysia (UTM)
for the Initial Research Grant Scheme (IRGS), vote number 78917, through Research
Management Center (RMC, UTM) as well as the facilities provided. Special thanks
go to family members and close friends for their continuous support. Other personnels
we would like to acknowledge include Mr Yap, Mr Teo and Mr Ng for making the
thesis writing process easier and Mr Eeu, for introducing Python to me, open-source
communities in general for providing great tools (Python, VisIt, Ubuntu) for free.
Thank you all.
v
ABSTRACT
Magnetic field gradient coils are essential in obtaining accurate magnetic
resonance imaging (MRI) or nuclear magnetic resonance (NMR) signals by generating
magnetic field gradient in each x, y and z direction. Two of the parameters to determine
the performance of such gradient coils are the magnetic field linearity and magnetic
field gradient uniformity. This research emphasizes on the analysis of the geometrical
effect of the conventional Golay-Maxwell pair gradient coils to these two parameters
through computer simulation. The results show that the geometrical parameters of
θ and d affect Golay coil’s magnetic field gradient. Usable volume is improved 50%
while gradient strength is increased 11% when θ is 1600 compared to the original 1200 .
The increase of d results in increase of usable volume, which is a maximum of 3374
cm3 at 0.8r but a loss of gradient strength of 36% compared to -0.34 mT/m at 0.2r.
The other geometrical parameters of Golay coil are found not to affect much on the
magnetic field gradient generated because of two reasons; the longitudinal sections
of Golay coil do not contribute to Bz generation and the outer arcs are just acting
as current return paths. For Maxwell coil, the usable volume can be improved until
19196.128 cm3 when d is 2.0r although the gradient value obtained is lower compared
to a maximum of -0.066 mT/m at 1.2r. Application wise, the higher the gradient value
and the bigger the usable volume, the better since the resolution can be improved,
not to mention, a bigger specimen accomodation. A computer simulation is written
fully in Open-source environment and feature variation of output as well as faster
vectorized algorithm. The simulation results will definitely provide useful information
for gradient coil designers without the need for physical development of prototype.
vi
ABSTRAK
Lingkaran kecerunan medan magnet adalah penting dalam proses untuk
mendapatkan isyarat yang jitu dalam pengimejan resonans magnet (PRM) atau
resonans magnetik nuklear (RMN) dengan penghasilan kecerunan medan magnet pada
paksi x, y dan z. Dua parameter untuk menentukan prestasi lingkaran kecerunan
adalah kecerunan medan magnet dan keseragaman kecerunan medan magnet. Kajian
in menekankan kepada analisis kesan geometri lingkaran kecerunan konvensional,
iaitu Golay-Maxwell, terhadap kedua-dua parameter tersebut menggunakan simulasi
komputer. Keputusan menunjukkan bahawa parameter geometri θ dan d menjejaskan
kecerunan medan magnet gegelung Golay. Isipadu boleh-guna meningkat 50%
manakala nilai kecerunan ditingkatkan 11% apabila θ adalah 1600 berbanding dengan
1200 pada asalnya. Peningkatan nilai d mengakibatkan peningkatan isipadu bolehguna, semaksimum 3374 cm3 pada 0.8r tetapi pengurangan nilai kecerunan sebanyak
36% daripada nilai - 0.34 mT/m pada 0.2r. Parameter geometri gegelung Golay
yang lain didapati tidak memberi kesan yang yang nyata kepada kecerunan medan
magnet kerana dua sebab; bahagian melintang gegelung Golay tidak menyumbang
kepada penghasilan Bz dan lengkok luar hanyalah berfungsi sebagai sambungan
untuk menyempurnakan litar. Untuk gegelung Maxwell, isipadu boleh-guna boleh
ditingkatkan kepada 19196.128 cm3 apabila d adalah 2.0r namun nilai kecerunan yang
didapati rendah berbanding dengan nilai maksimum -0.066 mT/m pada 1.2r. Dari
segi aplikasi, nilai kecerunan yang tinggi dan nilai isipadu boleh-guna yang tinggi
akan meningkatkan resolusi di samping membolehkan penggunaan sampel yang lebih
besar. Simulasi komputer ini dihasilkan menggunakan perisian sumber terbuka dan
mempunyai ciri-ciri seperti pelbagai pilihan output dan juga algoritma yang lebih
cepat. Keputusan daripada simulasi ini pasti dapat memberi maklumat berguna tanpa
pembangunan prototaip secara fizikal.
vii
TABLE OF CONTENTS
CHAPTER
TITLE
DECLARATION
DEDICATION
ACKNOWLEDGEMENT
ABSTRACT
ABSTRAK
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
LIST OF ABBREVIATIONS
LIST OF SYMBOLS
LIST OF APPENDICES
PAGE
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iii
iv
v
vi
vii
x
xi
xiv
xv
xvii
1
INTRODUCTION
1.1
Introduction to Modeling
1.2
Open Source
1.3
Research Profile
1.3.1
Background of Research
1.3.2
Statement of Problem
1.3.3
Purpose of Research
1.3.4
Objectives of Research
1.3.5
Significance of Research
1.3.6
Scope of Research
1.3.7
Methodology of Research
1.4
Summary
1
1
2
3
3
3
4
4
4
5
5
6
2
LITERATURE REVIEW
2.1
Theory of Nuclear Magnetic Resonance
2.2
Nuclear Magnetic Resonance Hardware
7
7
9
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2.2.1
2.2.2
2.3
2.4
2.5
2.6
Main Magnet
Gradient Coils
2.2.2.1 Longitudinal Gradient Coil
2.2.2.2 Transverse Gradient Coil
2.2.3
Radio Frequency System
Biot-Savart’s Law
Biot-Savart’s Law for Finite Length Current Segment
Related Research
Summary
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10
11
12
13
13
14
16
19
3
RESEARCH METHODOLOGY
3.1
Simulation Model
3.2
Problem Formulations
3.3
Python Coding
3.4
Data Collection and Analysis Procedures
3.5
Summary
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26
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4
RESULTS AND DISCUSSION
4.1
Three-Dimensional Visualization
4.2
Results on Golay Coil
4.2.1
General Results
4.2.2
Variation of θ
4.2.3
Variation of r
4.2.4
Variation of a
4.2.5
Variation of d
4.2.6
Variation of l
4.3
Results on Maxwell Coil
4.3.1
General Results
4.3.2
Variation of r
4.3.3
Variation of d
4.4
Discussion
4.5
Summary
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5
CONCLUSIONS
5.1
Conclusions
5.2
Suggestions for Further Works
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68
69
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REFERENCES
Appendices A – F
71
75 – 103
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
3.1
Calculation Time Comparison
27
4.1
4.2
4.3
4.4
4.5
4.6
Correlation and Gradient value for various θ
Usable Volume for various θ
Correlation, Gradient value and Usable volume for various a
Correlation, Gradient value and Usable volume for various d
Correlation, Gradient value and Usable volume for various l
Correlation, Gradient value and Usable volume for various d
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42
44
48
54
60
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LIST OF FIGURES
FIGURE NO.
2.1
2.2
2.3
2.4
2.5
2.6
2.7
TITLE
Precession of (a) Nucleus with External Field, B0 and (b)
Spinning Top with Gravity
Block Diagram of a Simple MRI/NMR Hardware
z axis Maxwell Coil
x axis Golay Coil
y axis Golay Coil
Biot-Savart’s Law
(a) A Finite Wire Carrying a Current I with the Magnetic Field
at M is Out of the Paper and (b) The Limiting Angles θ1 and θ2
PAGE
8
9
11
12
13
14
15
20
21
21
22
22
23
3.8
3.9
3.10
3.11
x axis Golay Coil Simulation Model
x axis Golay Coil Simulation Model on xy Plane
z axis Maxwell Coil Simulation Model
z axis Maxwell Coil Simulation Model on xy Plane
Flow Chart of Research Methodology
Parameters for Parametric Equation of Circle
−−→
Parameters Related to Finite Length Segment AB and
Measurement Point, M
Execution Flow of the Simulation
Example Plot to Determine the Linearity Range
Example Plot to Determine the Usable Region on xy Plane
Example Plot to Determine the Usable Region on xz Plane
4.1
4.2
4.3
4.4
4.5
4.6
x axis Golay Coil with a Three-dimensional Calculation Grid
Three-dimensional Contour Surface Plot
Three-dimensional Contour Plot
Projected Two-dimensional Contour Plot on xy Plane
Projected Two-dimensional Contour Plot on xz Plane
Projected Two-dimensional Contour Plot on yz Plane
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
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26
28
29
29
xii
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25
4.26
4.27
4.28
4.29
4.30
4.31
4.32
4.33
4.34
4.35
4.36
4.37
4.38
4.39
4.40
4.41
4.42
4.43
(a) x axis and (b) y axis Golay Coil
x axis Golay Coil with (a) xz Calculation Plane and (b) xy
Calculation Plane
Bz versus x at (y, z) = (0, 0)
Bz versus x at z = 0 for various y
Bz versus x at y = 0 for various z
4% Contour Plot on xy Plane at 5% interval at z = 0
4% Contour Plot on xz Plane at 5% interval at y = 0
Bz versus x for various θ at (y, z) = 0
4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 800
4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 1000
4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 1400
4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 1600
4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 800
4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 1000
4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 1400
4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 1600
Different Focal Point of Two Arcs of Golay Coil
Bz versus x for various a at (y, z) = 0
4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 2.5r
4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 3.0r
4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 3.5r
4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 4.0r
4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 4.5r
4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 2.5r
4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 3.0r
4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 3.5r
4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 4.0r
4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 4.5r
Bz versus x for various d at (y, z) = 0
4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.2r
4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.4r
4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.6r
4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.8r
4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 1.2r
4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 1.4r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.2r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.4r
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xiii
4.44
4.45
4.46
4.47
4.48
4.49
4.50
4.51
4.52
4.53
4.54
4.55
4.56
4.57
4.58
4.59
4.60
4.61
4.62
4.63
4.64
4.65
4.66
4.67
4.68
4.69
4.70
4.71
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.6r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.8r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.2r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.4r
Bz versus x for various l at (y, z) = 0
4% Contour Plot on xy Plane at 5% interval at z = 0 for l = r
4% Contour Plot on xy Plane at 5% interval at z = 0 for l = 2r
4% Contour Plot on xy Plane at 5% interval at z = 0 for l = 4r
4% Contour Plot on xy Plane at 5% interval at z = 0 for l = 5r
4% Contour Plot on xz Plane at 5% interval at y = 0 for l = r
4% Contour Plot on xz Plane at 5% interval at y = 0 for l = 2r
4% Contour Plot on xz Plane at 5% interval at y = 0 for l = 4r
4% Contour Plot on xz Plane at 5% interval at y = 0 for l = 5r
z axis Maxwell Coil with (a) xz Calculation Plane and (b) yz
Calculation Plane
Bz versus z at (x, y) = (0, 0)
4% Contour Plot on xz Plane at 5% interval at y = 0
4% Contour Plot on yz Plane at 5% interval at x = 0
Bz versus z at (x, y) = (0, 0) for various d
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.8r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.2r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.4r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.6r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.8r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 2.0r
4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 2.2r
Magnetic Field Generated by a Straight Segment
Schematic of Gradient Value and Slice Thickness
Schematic of Gradient Value and Resolution
52
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xiv
LIST OF ABBREVIATIONS
NMR
-
Nuclear Magnetic Resonance
MRI
-
Magnetic Resonance Imaging
RF
-
Radio Frequency
ROI
-
Region of Interest
VOI
-
Volume of Interest
DSV
-
Diameter Spherical Volume
VTK
-
Visualization Toolkit
xv
LIST OF SYMBOLS
A
-
Ampere
T
-
Tesla
m
-
meter
mT/m
-
mili-Tesla per meter
mT/m/A
-
mili-Tesla per meter per Ampere
GB
-
Gigabytes
ω
-
Larmor precession frequency
γ
-
Gyromagnetic ratio
µ0
-
Permeability of free space
I
-
Current
rc
−−→
AB
-
Correlation coefficient
-
ûAB
−−→ −−→
AB • CD
−−→ −−→
AB × CD
-
Vector from A to B
−−→
Unit vector of AB
-
−−→
−−→
Dot product between vector AB and CD
−−→
−−→
Cross product between vector AB and CD
~
B
-
Magnetic field (vector)
B
-
Magnetic field (scalar)
Bx
-
î component of magnetic field
By
-
ĵ component of magnetic field
Bz
-
k̂ component of magnetic field
B0
-
Main magnetic field
B1
-
Oscillating magnetic field/RF pulses
Gx
-
Magnetic field gradient in x direction
Gy
-
Magnetic field gradient in y direction
xvi
Gz
-
Magnetic field gradient in z direction
η
-
Magnetic field gradient efficiency
4%
-
Magnetic field gradient uniformity
xvii
LIST OF APPENDICES
APPENDIX
A
B
C
D
E
F
TITLE
MODULE
CODE FOR GOLAY COIL
CODE FOR MAXWELL COIL
VTK FILE FORMAT
PUBLICATION A
PUBLICATION B
PAGE
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92
94
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CHAPTER 1
INTRODUCTION
1.1
Introduction to Modeling
The concept of modeling has a long history of its own, even before the
computer exists. Modeling works before the era of computer, however, were limited to
the manual theoretical and mathematical formulations. The theories and formulations,
usually, are in ideal forms. By using approximations and assumptions, certain specific
cases of the theories can be derived. Of course, the possibility of each theory is
wide depending of the total parameters involved. While the mathematical formulation
still rule the physics world today, a certain depth of knowledge (and imagination) is
required for understanding. Fortunately enough, computer modeling or simulation
helps in both “visualizing” the mathematics behind the theories and the calculation of
each possible cases. Complex modeling may be a daunting tasks for human but that
would not be a problem to a computer. As long as there is enough processing power,
the computer will be able to execute any number of iterations. One main shortcoming
of computer simulation is the inability to model the continuous functions. The most
computer simulation can do is to approximate such functions by representating the
continuous functions as discrete or finite functions. Numerical techniques such as
numerical differentiation or integration, finite element method and finite difference
method provide such representation. Generally, the smaller the steps are, the closer the
simulation to the actual phenomenon, but with sacrifice in resources and computing
time. When such things happens, parallel computing or clustering is preferred in which
the job is distributed among few computers.
2
1.2
Open Source
Many scientific based software and toolkits are available to aid scientists
and engineers in their research. The availability of such software provides useful
functions and libraries to cater a wide area of applications. The famous scientific based
commercial software includes Matlab, Maple, Electronic Workbench and AutoCAD.
These software, without a doubt, come in a complete package as possible. Matlab,
for example, has found its way into virtually any area of research; physics, statistics,
engineering, to name a few. However, these commercial software share one thing
in common, the high price for licensing. A full package of Matlab can cost up
to few thousand ringgit. For a medium size of computer lab which consists of 10
to 20 computers, the Matlab installations alone is already pass the tenth thousand
mark, not to mention the operating software and others. One more issue involving
the commercial scientific based software is the licensing of the research product or
outcome. Since the development of the product is based on commercial software, the
distribution of the product might be troublesome. Although the research outcome is the
effort of the researchers, it is still limited by the commercial licensing. As far as the
concept of continuing and collaborative research is concerned, such issue no longer
encourage knowledge spreading and sharing, if one do not own a legal copy of the
commercial development platform.
Fortunately, there are alternatives to this solutions which is the open source
software. Commercial software are known as close source software in which the
source code of the software are not available to public. Without the source code, the
users are not allowed to modify and distribute the software. Open source software are
just direct opposite of this. The source codes of the software are available and can be
freely modified and distributed as long as they are compiled under the license of the
original software. Any programs developed using such software would not face any
modification or distribution issue. In such circumstances, open source software are
more communities oriented and the communities themselves repay the open source
world by providing patches, updates as well as third party extension to the software.
Most open source software are freely available. Examples of such software which are
scientific based include Octave and Scilab (Matlab alternatives), Python (open source
programming language), as well as VisIt and OpenDX (open source visualization
programs). With such powerful software minus the price, scientific researchs take a
great step forward, needless to always depending on the commercial software. This
research takes the opportunity to hightlight the capability and importance of such
software.
3
1.3
Research Profile
1.3.1
Background of Research
In the field of magnetic resonance imaging (MRI), besides the main magnetic
field, B0 , and the radio frequency (RF) subsystem, the magnetic field gradient coils
subsystem also play an important role in accurate signal acquisition. The operation
of MRI is based on a physical theory known as nuclear magnetic resonance (NMR).
In this theory, the spin of nucleus can be altered by manipulating the magnetic field.
With the presence of the main magnetic field, B0 , and the magnetic field gradient,
Gx = ∂Bz /∂x, Gy = ∂Bz /∂y and Gz = ∂Bz /∂z, the spin at each voxel (grid point
in three-dimensional space) can be uniquely characterized in terms of frequency and
phase, according to the Larmor equation and detectable by the RF subsystem. The
gradient coil subsystem, therefore, is critical in making sure the magnetic field gradient
generated is as linear over as large volume as possible.
1.3.2
Statement of Problem
Two of the important specifications of a gradient coil system are the field
linearity and gradient uniformity. As the name suggests, the main purpose of a
gradient coil is to generate a linearly varied magnetic field (linearity) along certain
axis, in this case, x, y and z axis. Besides being linear along certain axis, the
magnetic field gradient generated have to be uniform within an area as large as possible
(uniformity). The conventional coils used for x and y axis (also known as transverse
gradient coils) are the Golay coil configuration whereas for the z axis (also known as
longitudinal gradient coil), Maxwell coil configuration is used. Turner revolutioned the
gradient coil design by introducing what is called target-field method [1]. This method
generally involve solving a Fourier-Bassel expansion using Fourier Transforms [2].
Further researches import Turner’s technique by incorporating stream functions by
Sanchez et al. [3], using hybrid optimization method by Qi et al. [4], using finiteelement method by Shi et al. [5]. The most recent development in gradient coil
design is the three-dimensional toroidal design proposed by White et al. [6]. However,
the disadvantage of such methods includes a high level understanding of various
mathematical functions since the design method is an inverse problem. Besides,
in certain applications, conventional gradient coils are still prefered due to their
simplicity [7, 8, 9]. Biot-Savart’s Law offers a simpler and faster forward solution
4
to gradient coil design problem, making the whole gradient coil simulation and design
process cost and time effective.
1.3.3
Purpose of Research
To map the magnetic field generated by gradient coils with emphasize on field
linearity and gradient uniformity.
1.3.4
Objectives of Research
The objectives of the research include:
1.
To observe how the coil parameters affect the magnetic field gradient, magnetic
field linearity and magnetic field gradient uniformity
2.
To develop a user friendly, open source and freely distributable magnetic field
calculation program
3.
To provide valuable information to gradient coils designers without physical
prototype development
1.3.5
Significance of Research
The research has come out with a user-friendly program to calculate and map
the magnetic field generated by the conventional gradient coils used in MRI and NMR.
Therefore, this will provide valuable data to coil designers and researchers without
physically constructing the whole gradient coils themselves, reducing cost and time
in general. Furthermore, those who use this program do not have to go through
complex mathematical equations for them to calculate the generated magnetic field.
By utilizing open-source software, this program can be freely distributed to promote
knowledge sharing and modification to suit their needs. Since the magnetic field
calculation in the program is based on Biot-Savart’s Law for finite length current
segment, the same algorithm can be used to calculate magnetic field generated by
any current carrying conductor as long as the conductors can be divided into a series
5
of finite length segment. This will surely solve the unavailability of analytical BiotSavart’s formulations to calculate the magnetic field generated by current carrying coils
at points of interest.
1.3.6
Scope of Research
This research focuses on computer simulation of the magnetic field generated
by conventional unshielded gradient coils. Both the Golay coil and Maxwell coil
have been modeled and their generated magnetic field and gradient were simulated
and mapped accordingly. The field linearity and gradient uniformity of the generated
magnetic field were calculated and mapped using some of the definitions suggested by
Shi et al. [5], Di Luzio et al. [10] and Bowtell et al. [11]. The effect of coil parameters
to field linearity and gradient uniformity have been investigated. The research solved
problem in a forward manner rather than most of the gradient coil design methods
which are based on inverse problem solving. Besides assisting gradient coils designers,
the outcome of the research also provide a complete investigation of the geometrical
parameters of the gradient coil.
1.3.7
Methodology of Research
Biot-Savart’s Law will be used in the research to simulate the magnetic field
generated by the conventional gradient coils. Golay coil and Maxwell coil are two of
the well-known gradient coils used in conventional system [12]. By refering to the
design, the gradient coils are divided into a series of finite length current segment.
Using Biot-Savart’s Law for finite length current segment, the calculations in this
simulation are iterated and the generated magnetic field are summed up. The results are
visualized appropriately in two-dimensional or three-dimensional contour plot. Data
has been extracted from the output of the simulation for suitable statistical analysis and
tabulated.
6
1.4
Summary
From the information, the conventional gradient coils were modeled
accordingly. Suitable research methodologies were utilized to match the intentions
of the research and come out with relevant results. Conclusions are then drawn from
the results and discussions. All these will be discussed in the next few chapters.
CHAPTER 2
LITERATURE REVIEW
2.1
Theory of Nuclear Magnetic Resonance
In the medical field, MRI is widely used as one of the non-invasive technique
to investigate abnormality of human body. Before MRI was introduced, techniques
such as X-ray imaging which utilize ionizing sources, were used. These techniques
will affect human in long term basis. MRI, on the other hand, make use of the spin
properties of nuclei (hidrogen in human body) by applying magnetic field and RF
pulses, which is safer. Nuclei that have net spin possess a tiny magnetic moment,
similar to those found in bar magnet. When a magnetic field (B0 ) is applied to
such nuclei, the magnetic moment will undergo a process called precession. This is
analogous to the wobbling (which is caused by the presence of gravity) of a spinning
top. This phenomenon can be explained with the fact that the non-zero spin possessed
by the nuclei implies circulation of charge. The charge circulation can be represented
as a tiny current loop that produce its own magnetic field. When this magnetic field
is subjected to an external magnetic field, a “torque” will align the spin, or more
specifically the magnetic moment with the external field. This is shown in Figure
2.1.
Nuclear magnetic resonance (NMR) is a physical theory that lives behind MRI.
Nuclei in a sample (or human body) are randomly arranged. Therefore, a main
magnetic field, B0 , is used to align the nuclei inside a sample. As a result, the
nuclei align themselves either parallel or anti-parallel with B0 . The aligned nuclei
will also precess at certain frequency with B0 . Such frequency is known as the Larmor
frequency, shown in Equation 2.1.
ω = γB
(2.1)
8
Figure 2.1: Precession of (a) Nucleus with External Field, B0 and (b) Spinning Top
with Gravity
where ω is the Larmor frequency, γ is the gyromagnetic ratio and B is the magnetic
field strength. After the nuclei alignment, the B in Equation 2.1 is B0 value. This
means that all the nuclei will precess at one Larmor frequency only because they are
subjected to the same value of B. The value of γ for 1 H is around 42.58 MHz/T [12].
A typical MRI scanner has a main magnetic field strength of few tesla, resulting in the
Larmor frequencies falling in the MHz or the RF region.
The main concern now is the Larmor frequencies. If the nuclei in a sample
are only subjected to a magnetic field (B0 ), they will only precess at one Larmor
frequency. Since the detectable signals depend on Larmor frequencies, the sample
will need to be spatially encoded with different values of magnetic field, so that the
nuclei in that sample will precess at a frequency range rather than one frequency value
only. Magnetic field gradients are applied for this purpose. For example, if B0 is
in z direction, and a magnetic field gradient is applied in the same direction, then
each point along z direction will have different values of magnetic field (as a result
of the superposition of main magnetic field and the magnetic field gradient). The
different magnetic field values will result in different precession frequencies. Usually,
a magnetic field gradient is applied for each x, y and z direction to spatially encode the
sample in three dimensions.
MRI or NMR signal acquisition is based on the “tipping” off the nuclear
magnetization. Each nucleus has its own magnetic moment. Inside a sample, the
vector sum of these magnetic moments is known as nuclear magnetization. When
B0 is applied, the magnetic moments of all the nuclei will be aligned, resulting in a
net longitudinal magnetization (in the same direction as B0 ). The net longitudinal
magnetization will be tipped off of its longitudinal axis by applying a transverse
oscillating magnetic field (B1 ) at the Larmor frequency. The transverse component
9
magnetization (as a result of the tipping off) will now rotate at Larmor frequency
(hence, the term resonance in NMR). The rotation will produce another oscilating
magnetic field, detectable by RF receiver coil. As B1 is switched off, the transversed
magnetization will decay back to zero and the signal will disappear. Each point (or
voxel) of the sample will have a different value of magnetic field which means a
different Larmor frequency. The normal practice, therefore, is to sweep the sample
with B1 of a range of RF frequencies.
2.2
Nuclear Magnetic Resonance Hardware
Figure 2.2 shows the block diagram of the basic components for a NMR or MRI
hardware. It consists of main magnet, gradient coils and RF system. The machine will
be controlled by a computer system including the switching on and off the components
and the visualization of the output. The main magnet will be responsible for generating
the main magnetic field, B0 whereas the gradient system will generate magnetic field
gradient along x, y, z or a combination of the axes depending on factors like imaging
technique or imaging plane. The RF system will transmit RF pulses (B1 ) and at the
same time, detect the RF pulses during the relaxation process.
Figure 2.2: Block Diagram of a Simple MRI/NMR Hardware
10
2.2.1
Main Magnet
The main magnet in a NMR or MRI machine can be either a permanent
magnet or a more advanced superconducting magnet. Permanent magnet can be
bulky as the strength of its magnetic field increases besides offering a small region
of field uniformity. However, for micro imaging which usually small samples are
used, permanent magnet would be a better choice because of its simplicity and
pricing [13, 14, 15, 16]. Bigger applications such as in medical field or MRI,
superconducting magnet is used as the main magnet. Superconducting magnet is
basically specifically shaped coils which are cooled until the resistance of the coils
becomes zeros. This is normally done by liquid helium. The current flow inside
the coils will generate magnetic field along the coils axis. The advantage of using
superconducting magnet is its stable, homogeneous and high strength magnetic field.
The construction of superconducting magnet, however, is not simple and requires
higher budget. For a typical clinical usage, the main magnet field strength is around 2
T [12]. Other applications sometimes require B0 value to be higher. Some researches
reported to have use 7 T for magnetic resonance microscopy [17], a 14.1 T system for
NMR microscopy [18] or a lower end MRI system at 0.21 T [13].
2.2.2
Gradient Coils
The introduction of gradient coils into a NMR or MRI machine causes a change
in the value of field strength in the imaging region. Based on Equation 2.1, if the field
strength is known, then the precession frequency can be determined which means the
range of the frequency of the RF pulses (or B1 ) is known (either during excitation or
detection). Normally, the process of applying magnet gradient to the existing main
magnetic field is known as spatial encoding. There are many designs of gradient
coils including the “finger print” type coils [16, 19], planar type coils [20, 21] and
the conventional saddle type coils [22]. Two sets of gradient coils are incorporated in
a NMR or MRI machine, namely the longitudinal gradient coil and the transverse coil.
11
2.2.2.1 Longitudinal Gradient Coil
Longitudinal gradient coil, as its name suggests, produces a magnetic field
gradient along the axis of the main magnetic field. A conventional longitudinal coil
is shown in Figure 2.3. It is basically two identical coils carrying current in opposite
direction with each other separated by a distance. Such configuration is also known as
Maxwell coil.
Figure 2.3: z axis Maxwell Coil
The magnetic field generated along the z axis by a Maxwell coil with radius
= a, separated with a distance = d is given by Equation 2.2.
Bz =
µ0 Ia2
µ0 Ia2
−
2[(d/2 − z)2 + a2 ]3/2 2[(d/2 + z)2 + a2 ]3/2
(2.2)
Equation 2.2 is derived from the original Biot-Savart’s Law (which will be discussed
later in this chapter). Generally, the Biot-Savart’s Law can be derived for any shape of
the current carrying coil for any point of interest. This is, however, will be a difficult
task to be accomplished. If every point of interest requires different derivation of BiotSavart’s Law, then for a magnetic field simulation or calculation at a hundred of points
would definitely require one hundred set of equations especially at points which are
not along the axis of the coil. By using the finite-length current segment approach,
12
though, will solve this complexity, as used in this research.
2.2.2.2 Transverse Gradient Coil
Tranverse gradient coil will produce magnetic field gradient along the two
orthogonal axes to the main magnetic field. A general case is that normally the main
magnetic field will be along the z axis and the tranverse gradient coil will be placed in
such a way that a magnetic field gradient is generated along the x and y axis. Figure
2.4 and 2.5 show the conventional Golay coil used as the tranverse gradient coils for x
and y axis respectively with the red arrows indicate the direction of the current.
There is no an appropriate analytical formulation to calculate the magnetic field
generated by the Golay coil shown in Figure 2.4 and 2.5. However, the analysis of such
coils can be derived from Equation 2.3.
Bz (r, θ, φ) =
∞ X
n
X
rn Pnm (cos θ)(Anm cos mφ + Bnm sin mφ)
(2.3)
n=0 m=0
Equation 2.3 is written in cylindrical coordinate and the solution require the
knowledge of Lengendre polynomial (Pnm (cos θ) with Anm and Bnm as the expansion
coefficients). Again, solving Equation 2.3 would require a significant understanding
of mathematical functions.
Figure 2.4: x axis Golay Coil
13
Figure 2.5: y axis Golay Coil
2.2.3
Radio Frequency System
The role of RF system in a NMR or MRI machine is to excite the nuclei and to
detect the NMR signal during relaxation process. RF system consists of coils just like
gradient coils, only with different functions. Conventional saddle coil and “birdcage”
coil are two types of coils that are being used as RF coils in NMR or MRI machine.
2.3
Biot-Savart’s Law
~ generated at a
The Biot-Savart Law states that the differential magnetic field dB
field point by a steady current I flowing through a differential length d~s of a conductor
has the following properties:
1.
~ is perpendicular to both d~s (the direction of the current element)
The vector dB
and to the unit vector r̂ directed from the element to the field point.
2.
~ is inversely proportional to r2 , where r is the distance from
The magnitude of dB
the element to the field point.
3.
~ is proportional to the current and to the length d~s of the
The magnitude of dB
element.
4.
~ is proprotional to sin θ, where θ is the angle between the
The magnitude of dB
vector d~s and r̂.
14
Equation (2.4) summarizes Biot-Savart Law.
~ =
dB
µ0 Id~s × r̂
4π r2
(2.4)
~ is the differential magnetic field induced, µ0 is the permeability of free space
where dB
which is equal to 4π × 10−7 H/m, I is the current in Ampere, d~s is the differential
length vector of current element segment, r̂ is the unit displacement vector from the
current element to the field point and r is the distance from the current element to
the field point. Equation (2.4) shows only the differential magnetic field generated
according to the Biot-Savart’s Law. The total magnetic field generated by a coil is the
sum of the magnetic field generated by all of the differential length of the coil, which
means the integration of Equation 2.4, resulting in Equation 2.5.
~ = µ0 I
B
4π
Z
d~s × r̂
r2
(2.5)
Biot-Savart’s Law can be graphically represented as Figure 2.6.
Figure 2.6: Biot-Savart’s Law
2.4
Biot-Savart’s Law for Finite Length Current Segment
Biot-Savart’s Law can be further derived for conductor of any shape. In this
section, the Biot-Savart’s Law for a finite length segment is derived. Consider Figure
2.7, an element d~s is at distance r from M . The direction of the field at M due to this
element is out of the paper, since d~s × r̂ is out of the paper, according to the right-hand
rule. In this case, all the elements will give contribution directly out of the paper at M .
15
By taking the origin O and letting M be along the positive y-axis, with k̂ being a unit
vector pointing out of the paper, Equation 2.6 is obtained.
Figure 2.7: (a) A Finite Wire Carrying a Current I with the Magnetic Field at M is
Out of the Paper and (b) The Limiting Angles θ1 and θ2
d~s × r̂ = k̂|d~s × r̂| = k̂(dx sin θ)
(2.6)
Subtituting Equation (2.6) into Equation (2.4), Equation (2.7) is obtained.
~ =
dB
µ0 I dx sin θ
k̂
4π
r2
(2.7)
In order to integrate Equation (2.7), the variables θ, x and r must be related. An easy
way to do this is by expressing x and r in terms of θ. From Figure 2.7(b), with some
simple trigonometry manipulations, Equation (2.8) is obtained.
r=
a
= a csc θ
sin θ
(2.8)
16
Since tan θ = −a/x from the right triangle in Figure 2.7(a),
x = −a cot θ
(2.9)
dx = a csc2 θ dθ
(2.10)
Substitute Equation (2.8) and (2.10) into (2.7) gives,
~ =
dB
µ0 I a csc2 θ sin θ dθ
k̂
4π
a2 csc2 θ
(2.11)
Now, the expressions have been reduced to one involving only the variable θ. The
total field at M can be calculated by integrating Equation (2.11) over all elements
subtending angles ranging from θ1 to θ2 as in Figure 2.7(b) and Equation (2.12) is
obtained.
Z θ2
µ0 I
~
sin θ dθ
(2.12)
k̂
B=
4πa θ1
And finally the Biot-Savart’s Law for a finite length segment is obtained.
~ = µ0 I (cos θ1 − cos θ2 ) k̂
B
4πa
2.5
(2.13)
Related Research
Researches on NMR or MRI gradient coils involve variations of methods
to design gradient coils, optimization techniques to enhance current design method
as well as development of unconventional type of gradient coils. The importance
of gradient coils development, especially in medical area can be seen from von
Schulthess’s points of view [23], which stated that a disease process is normally 1
cm or less (in resolution), meaning for MRI to be able to detect the abnormalities, the
gradient coils have to generated a significant high gradient strength (which will in turn
result in higher resolved MRI signals).
General consideration of linearity, uniformity, efficiency, inductance and
resistance need to be taken care of during gradient coils design [12, 24]. These few
parameters are inter-related; the increase of one parameter may result in the decrease
of other parameters. Researchers, therefore, are trying to come up with designs or
formulations that will minimize parameters such as inductance and torque and at the
same time, maximize parameters such as gradient strength and usable region. One of
17
the famous design technique is the target-field method, introduced by Turner [1, 2].
Target-field method is an inverse solution in which at first the region of interest (ROI)
with certain gradient strength is determined. The problem is then solved by using
Fourier Transforms on a Fourier-Bassel expansion. One of the major disadvantage of
Turner’s technique is that it assume the resultant gradient coil to be infinitely long as
a result of using Fourier Transforms. This has led to several improvement on Turner’s
technique, including the use of finite-element method [3, 5] as well as the use of some
optimization methods [4, 20, 25, 26]. There are also some new techniques proposed
for designing gradient coils [6, 27, 28].
Turner’s technique originally resulted in “finger-print” gradient coil on cylinder
surface. However, there are a number of gradient coils which are not restricted on
cylinder surface anymore. There are planar type of gradient coil [13, 20, 29, 30],
butterfly coil [10], twin gradient coil [31], gradient coil etched on a printed circuit
board (PCB) [32] and even plate type of gradient coil [33].
Although newer geometrical shapes of gradient coils are used, conventional
Maxwell-Golay is still used in several cases [7, 8, 9, 15, 18, 22, 34] with slight
modifications of their own. The favourable of Maxwell-Golay over other types of
gradient coil is because of the simplicity of the coil design and relatively bigger space
for RF system accomodation [18]. It, however, offers relatively lower volume of
linear gradient compared to the newer types of gradient coil which is why conventional
Maxwell-Golay is used when smaller region of interest is concerned.
The gradient coil will produce a magnetic field gradient along an axis. Based
on the Biot-Savart’s Law mentioned in Section 2.3, a current carrying conductor will
generate magnetic field. In three dimensional space, the magnetic field generated will
~ = Bx .î + By .ĵ + Bz .k̂ or simply B = (Bx , By , Bz ). In NMR
be in vector form, B
or MRI, the main magnetic field, B0 is generated along the z axis and because of this,
only Bz generated by the gradient coils is important for frequency encoding, phase
encoding or signal acquisition in general [35]. Hence, the magnetic field produced by
gradient coil can be defined as in Equation 2.14 to 2.16 with the subscript denoting the
axis.
∂Bz
(2.14)
Gx =
∂x
∂Bz
Gy =
(2.15)
∂y
18
Gz =
∂Bz
∂z
(2.16)
Linearity and uniformity (or non-uniformity) are two parameters that are being
discussed when dealing with gradient coil related research. Literally, linearity means
how “straight” the gradient is within certain range or region and uniformity means how
much the difference the gradient value is with a reference point; normally at (x, y, z) =
(0, 0, 0). A simple representation of how good the magnetic field gradient linearity is
to plot Bz versus x, y or z coordinate. Visually, the “straighter” the plot of the Bz vs x,
y or z is, the better the linearity. Statistically, correlation calculation can be used [33].
The value of correlation coefficient, rc , is between −1 and 1. Thus, the closer the
value of |rc | to 1, the better the linearity. This is not to be confused with the gradient
or the slope of the line. A perfectly linear gradient within certain range would result in
constant gradient over the same range. This is where the concept of uniformity comes
in. A uniform gradient within certain range means the value of the gradient is the same
(or almost the same in actual case) over the same range. The uniformity, 4% , can
be defined as in Equation 2.17 [5], with Gx(x,y,z) as the gradient at point (x, y, z) and
Gx(0,0,0) is the gradient at reference point, in this case at (x, y, z) = (0, 0, 0).
4% =
Gx(x,y,z) − Gx(0,0,0)
Gx(0,0,0)
(2.17)
However, there are other definitions of uniformity. One such definition is that
uniformity can be represented as the deviation of the gradient with reference with the
mean value of gradient [36]. Another approach is to define the relative error as in
Equation 2.18 [37]. Equation 2.18 is claimed to reflect the difference between the true
and measured position of an object relative to the true distance from the object to the
center of the gradient coil, with ~r is the position vector at that point.
Bz (~r) − G~r
G~r
(2.18)
Bowtell et al, on the other hand, used Equation 2.19 to determine the uniformity in
their research [11].
Bz
−1
(2.19)
Gz
Graphically, normal two dimensional plot of Bz versus an axis for different
distances from the origin can be used [5, 29]. This approach may be simple and
19
straightforward but the information is limited. To obtain additional information, a
repetition of such plot is required. Most, however, favour the two dimensional contour
plot for the calculation of the previously defined deviation. A good tolerance of
uniformity is around 5% of deviation [10, 17, 22, 24, 34]. Most of the contour plots
of magnetic field gradient, therefore, show iso-value lines with 5% interval. Some
applications can actually tolerate a higher gradient deviation, such as reported by
Chronik et al. [38] for magnetic resonance microscopy of mice.
2.6
Summary
The gradient coils play an important role in NMR or MRI signal acquisition.
Therefore, a certain knowledge of the behaviour of the gradient coils is essential in
obtaining suitable gradient coils for specific purposes. Both statistical analysis and
normal contour plot will help to determine several important parameters of gradient
coils, such as the magnetic field gradient value, its linearity and uniformity and the
usable region for an acceptable non-uniformity.
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Simulation Model
The calculation model used in this research includes the x axis Golay coil and
z axis Maxwell coil. The y axis gradient coil is omitted since it is similar to the x axis
gradient coil, only with different orientation. Figure 3.1 and 3.2 show the simulation
model for x axis Golay coil with several geometrical parameters; namely a, d, l, θ
and r (radius). The red arrows denote the current direction. Figure 3.3 and 3.4 show
the simulation model for z axis Maxwell coil with its geometrical parameters, d and
r (radius). The effect of these parameters on field linearity and gradient uniformity
was observed using the simulation written in Python programming language, with the
visualization provided by VisIt (three-dimensional) and matplotlib (two-dimensional).
The statistical analysis was done in OpenOffice Calc, similar to Microsoft Office Excel.
The program development was done in Ubuntu operating system but it was tested in
Windows XP as well.
Figure 3.1: x axis Golay Coil Simulation Model
21
Figure 3.2: x axis Golay Coil Simulation Model on xy Plane
Figure 3.3: z axis Maxwell Coil Simulation Model
3.2
Problem Formulations
Figure 3.5 summarizes the methodology of this research. Literature review
activities were done before hand so as to find out the current related researches.
Different sources of literature were found which mainly were journal articles. It is
found out that although most of the researches now are more focused on the higher
22
end type of gradient coils, namely the “finger-print” coils as well as the planar coils,
Golay-Maxwell pair still preferred in certain cases of applications.
Figure 3.4: z axis Maxwell Coil Simulation Model on xy Plane
Literature
Review
Formula
Derivation
End
Vector
Analysis
Data Collection
& Discussion
yes
Python
Coding
Testing
OK?
no
Figure 3.5: Flow Chart of Research Methodology
Most of the research also deal the problems in an inverse manner, in this case, settting
a region with certain values of linearity and uniformity and then find out the coil or
wiring configuration. This research’s appoarch, however, is in the forward manner,
pre-defined the gradient coils and calculate the parameters of interest. With this,
the formula derivation part of the research activities involve a slightly easier Biot-
23
Savart’s Law (compared to the inverse appoarch which requires additional optimization
formulations). A general Biot-Savart’s Law is valid for virtually any current carrying
coil or conductor. For a specific shape of coil, such as the solenoid or the single circular
coil, relevant Biot-Savart’s Law have been derived. These derived equations, on the
other hand, only valid for certain points, which mean, if the magnetic field at other
points will require a different form of Biot-Savart’s Law. For complex type of coils,
the derivation of specific Biot-Savart’s Law would be difficult, or even more complex
if the coil’s configuration is not symmetry. This research uses a more straight forward
formulation, which is the Biot-Savart’s Law for finite-length current segment, shown
in Section 2.4. The idea is simple yet versatile. For Golay coils, the four straight
segments should be not problem since they are finite-length current segment. For the
four arcs, extra formulations were done. Consider a circle as shown in Figure 3.6.
Figure 3.6: Parameters for Parametric Equation of Circle
The parameters shown in Figure 3.6 can be related as,
x = j + r cos t
(3.1)
y = k + r sin t
(3.2)
Assuming the four arcs in Golay coil have a certain focal point and radius, the arcs can
be divided into a series of finite-length current segment by manipulating the parametric
equations of a circle. With this step, the magnetic field generated by the arcs can also
be calculated using Biot-Savart’s Law for finite-length current segment and by adding
up each contribution, magnetic field at any point can be obtained. The derivation in
Section 2.4 is mainly in scalar form. The actual magnetic field generated, however,
will be in full three-dimensional form. As a result, the formulation in Section 2.4 have
to be redo in vector form, shown as follows. Based on Figure 3.7, all the parameters
24
can be derived.
−−→
Figure 3.7: Parameters Related to Finite Length Segment AB and Measurement Point,
M
−−→
First vector of the current segment AB is determined.
−−→ −−→ −−→
AB = OB − OA
(3.3)
and its correspondent unit vector is given by,
ûAB
−−→
AB
= −−→
|AB|
(3.4)
−−→
−−→
with O is the origin (0, 0). By using similar method, AM and BM as well as their
unit vectors are given by,
−−→ −−→ −−→
AM = OM − OA
(3.5)
−−→ −−→ −−→
BM = OM − OB
(3.6)
−−→
AM
ûAM = −−→
(3.7)
|AM|
−−→
BM
ûBM = −−→
(3.8)
|BM|
The cosines of the angles shown in Figure 3.7, cos θ1 and cos θ2 are calculated from,
cos θ1 = ûAB • ûAM
(3.9)
cos θ2 = ûAB • ûBM
(3.10)
25
The distance between the measurement point, M to the current segment is given by the
−−→
magnitude of LM,
−−→ −−→ −→
LM = AM − AL
(3.11)
−→
−−→
AL is the projection of AM in the direction of ûAB ,
Therefore,
−→
−−→
AL = (AM • ûAB )ûAB
(3.12)
−−→ −−→
−−→
LM = AM − (AM • ûAB )ûAB
(3.13)
The direction of the magnetic field at the measurement point, M is obtained by taking
−−→
the cross porduct of the two unit vectors; unit vector of AB and the unit vector along
the line joining the current segment to the measurement point, M ,
ûB = ûAB × ûLM
(3.14)
Finally, the magnetic field at measurement point, M is obtained.
→
−
B=
µ0 I
−−→ (cos θ1 − cos θ2 ) ûB
4π|LM|
(3.15)
Although the formulations above are fully three-dimensional, only the k̂ component
of magnetic field, Bz , is analyzed since the other components of magnetic field do
not have significant effect on the magnetic field gradient. Once the magnetic field
values at each point can be calculated the next step in the simulation is to calculate the
magnetic field gradient value at each point. Magnetic field gradient can be calculated
based on the fact that gradient is actually the “steepness” of a slope (in this case the
“steepness” of the magnetic field). Such computations computer simulation require the
usage of numerical differentiation method. For example, the magnetic field gradient
along x axis between point x and point (x + h) can be approximate through numerical
differentiation as follows.
Gx =
Bz (x) − Bz (x − h)
h
(3.16)
Equation 3.16 can be used to calculate the magnetic field gradient at point x by
decreasing the value of h to as small as possible compared to x. For example, the
value of h in this research is fixed at 1 × 10−6 .
26
3.3
Python Coding
The formulation in Section 3.2 were translated into Python programming
language. Among the reasons Python programming language is chosen because it
is free and open source, platform independent as well as extensive third party modules
or libraries. Two important modules that were used in this research is Scipy (for
computation) and matplotlib (for plotting). The research also comes up with one selfwritten module which mainly consists of a collection of functions used to calculate
the magnetic field generated by current carrying conductor based on the Biot-Savart’s
Law for finite-length current segment. The general flowchart of the Python program is
shown in Figure 3.8.
User selection
& input
no
Input enough/
appropriate?
End
yes
Data storage &
visualization
Importing
libraries/modules
yes
Iteration
finished?
Start iteration
no
Figure 3.8: Execution Flow of the Simulation
The very first version of the simulation was done using the for loop. The for loop
approach is very easy to be implemented due to its easy and straight forward algorithm.
However, for each for cycle, the instruction and data will have to be reload into the
clock cycle, thus reducing the speed. The advantages of using such approach is the
simplicity and the reduction in memory usage to store variables. The improved version
of this simulation adopt what is called vectorized processing. The idea is to convert
the for loop to a large array. This improves the simulation speed significantly. The
27
setback of vectorized processing includes the large memory usage for storing all the
arrays and slightly complicated algorithm. A simple illustration is shown as follows.
# for loop
for i in range(0:100):
c[i] = a[i] + b[i]
# vectorized processing
c[0:99] = a[0:99] + b[0:99]
Table 3.1 shows the calculation time comparison between the two approaches. The
result in Table 3.1 is obtained from the simulation run for a certain two-dimensional
calculation grid.
Table 3.1: Calculation Time Comparison
Dimensions
11 × 11
21 × 21
51 × 51
Time (Seconds)
for loop vectorized
0.344
0.0
2.422
0.063
35.079
0.938
Due to significant improvement, the final version of the Python program
adopted the vectorized algorithm. Based on Figure 3.8, the program requires the
users to input some of the variables need for the simulation. Among the variables
available as users’ input are the current value, the radius (of the Maxwell coil or of
the arcs of the Golay coil), the separation between first coil and second coil and so
on. After the variables input, the users are then prompted to choose the calculation
plane (whether it will be xy plane, xz plane, yz plane or a full three-dimensional
volume). Based on the choosen calculation plane, the users will have to input the range
of the calculation. The limit of the calculation range actually depends on the amount
of physical memory (RAM) of the personal computer. The program will then verify
all the inputs and options before starting the simulation. Only appropriate libraries
or modules will be imported based on the options choosen by users. The simulation
will start by dividing the coil into a series of finite-length segment. At each point
of calculation, the main calculation algorithm will be repeated until the contribution
of each finite-length segment is taken account into. After that, the calculation will
28
move to the second calculation. The process will repeat until all calculation points are
covered. The program will display the total time elapsed for the simulation as well
as the magnetic field gradient generated at point (x, y, z) = (0, 0, 0). The results of
the simulation will be saved mainly in a Visualization Toolkit (VTK) file format (as
shown as in Appendix D). The output file can then be used for visualization or analysis
purposes. The resultant Python programs are shown in Appendix B for Golay coil and
Appendix C for Maxwell coil with a self-written module shown in Appendix A.
3.4
Data Collection and Analysis Procedures
A few data collection and analysis algorithm were incorporated in the Python
program whereas the rest were done using external program manually. The Python
program has an option to save a simple xy values pair in a simple text file. The text files
were saved and imported into OpenOffice Calc in which the magnetic field linearity
analysis was done. First a simple two-dimensional xy graph was plotted. Here, the
range at which the gradient is almost the same, was observed. Figure 3.9 shows an
example of such plot. The different colour coded plots show a constant gradient at
x range roughly between x = −0.05m and x = +0.05m. Hence, the magnetic field
linearity for each plot is determined through the calculation of correlation coefficient,
rc . Plot with |rc | value closer to 1 will have a good magnetic field linearity.
2e-05
y = 0.00
y = 0.05
y = 0.10
y = 0.15
1.5e-05
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-2e-05
-0.2
-0.15
-0.1
-0.05
0
x/meter
0.05
0.1
0.15
0.2
Figure 3.9: Example Plot to Determine the Linearity Range
29
The analysis of the magnetic gradient uniformity uses a more graphical
approach. Two two-dimensional contour plots were used in this case. The contours
represent the gradient values deviation according to Equation 3.17 for x axis Golay
coil and Equation 3.18 for z axis Maxwell coil.
4% =
Gx(x,y,z) − Gx(0,0,0)
Gx(0,0,0)
(3.17)
4% =
Gz(x,y,z) − Gz(0,0,0)
Gz(0,0,0)
(3.18)
Figure 3.10: Example Plot to Determine the Usable Region on xy Plane
Figure 3.11: Example Plot to Determine the Usable Region on xz Plane
30
The magnetic field gradient values at each point will be compared to the
maximum gradient value which is at point (x, y, z) = (0, 0, 0). With this, the area
or region (which expands from the origin) of acceptable gradient uniformity can
be determined. Both Equation 3.17 and 3.18 represent the gradient uniformity in
percentage or fraction. These calculation were incorporated into the Python program
during the analysis and the output will be the mentioned contour plots at 5% interval.
For x axis Golay coil, the contour plots of xy and xz planes were used and for z axis
Maxwell coil, the contour plots of xz and yz planes were used. The values shown in
the contours in Figure 3.10 and 3.11 are the deviation of gradient values in percentage
as explained previously. The analysis of gradient uniformity was based on the normal
practise of MRI which is ±5%. In Figure 3.10 and 3.11, the hatched areas are the
areas within ±5% of gradient deviation. In Figure 3.10, the hatched area appears to
be a circle with certain radius whereas in Figure 3.11, the hatched area looks like a
rectangle. From the figures, the volume of usable region is calculated by combining
the usable regions (which are the hatched areas). The calculations were based on the
values (radius, width, and so on) of the contour plots. In this example, if the usable
regions from Figure 3.10 and 3.11 are combined, the three-dimensional usable region
would be a cylinder. The gradient uniformity analysis was represented in two values;
one is the exact value of the usable volume (which is in cm3 ) and the other, the ratio
of usable volume with the total volume occupied by the respective Golay or Maxwell
coil. That concludes the data collection and analysis of this research.
3.5
Summary
Appropriate simulation model of Golay coil and Maxwell coil were determined
based on the literature review. All the geometrical parameters of the gradient coils were
pre-defined. The computer simulation program was written using Python programming
language, based on the Biot-Savart’s Law for finite-length current segment. Additional
equations were derived, mainly the vector version of all the Biot-Savart’s Law.
The Python program features some variables input from the user as well as several
simulation options. The output from the simulation were then collected and analyzed.
Both the analysis of magnetic field linearity and gradient uniformity use different
approach.
CHAPTER 4
RESULTS AND DISCUSSION
4.1
Three-Dimensional Visualization
Both the simulation codes written as in Appendix B and C are capable of
calculating a full three-dimensional grid in cartesian coordinates. Since the simulation
codes are written using vectorized processing, there is an disadvantage of total memory
usage (but significantly faster than normal for loop). In this simulation, grid is defined
as an array and the total memory of that array is the memory occupied by a single
element inside the array times the size of the array. In a 32-bit platform, typically, a
process would only have access to around 2 gigabytes (GB) of the system’s memory.
The maximum physical memory recognizable by a 32-bit platform is 4 GB which
means the total variables inside this simulation must not exceed 4 GB. The possible
solution to this is either to decrease the steps so that the grid can expanded or to opt
for 64-bit platform, provided the platform have enough memory to accomodate the
large arrays. An example of three-dimensional calculation grid is shown in Figure 4.1
with the calculation grid shown as a yellow coloured cube. A simulation is executed
based on Figure 4.1, covering x ranged from -0.1 m to +0.1 m, y ranged from -0.1 m
to +0.1 m and z ranged from -0.1 m to +0.1 m. Figure 4.2 shows a three-dimensional
contour surface plot generated from VisIt visualization program using VTK file output
generated from the simulation. In Figure 4.2, the usable region in three-dimensional is
plotted, bounded by the green (+5%) and red (-5%) coloured surface. Figure 4.2 can be
set to be viewed in a normal contour line plot rather than contour surface plot, shown
in Figure 4.3. By observing either Figure 4.2 or Figure 4.3, the size of usable region
can be determined. Relevant parameters can be set within VisIt, such as the value
and colour of the contours, different view angle of the plot as well as even projected
two-dimensional plot for a clearer view such as Figure 4.4 to 4.6.
32
Figure 4.1: x axis Golay Coil with a Three-dimensional Calculation Grid
Figure 4.2: Three-dimensional Contour Surface Plot
Figure 4.3: Three-dimensional Contour Plot
33
Figure 4.4: Projected Two-dimensional Contour Plot on xy Plane
Figure 4.5: Projected Two-dimensional Contour Plot on xz Plane
Figure 4.6: Projected Two-dimensional Contour Plot on yz Plane
34
4.2
Results on Golay Coil
The results in this section and the next one will be focusing on two-dimensional
plot (generated by matplotlib) to ensure better investigations and observations of the
coil’s geometrical parameters to the generated magnetic field gradient. For the Golay
coil simulation, only the x axis Golay coil is investigated since the y axis Golay coil
is only the 900 degree rotation version of the x axis Golay coil. This is illustrated by
Figure 4.7.
Figure 4.7: (a) x axis and (b) y axis Golay Coil
Figure 4.8: x axis Golay Coil with (a) xz Calculation Plane and (b) xy Calculation
Plane
35
Note that the current flow in Figure 4.7 is not the same as in Figure 2.5. This is
not a major concern as long as the current flow in the upper saddle pair is anti parallel
with the current flow in the lower saddle pair. The only difference is that either the
positive magnetic field gradient or negative magnetic field gradient is generated. The
results in the next few sections will be based on the two-dimensional calculation grid
based on Figure 4.8. The simulation on x axis Golay coil will be run for x ranged
from -0.1 m to +0.1 m, y ranged from -0.15 m to +0.15 m and z ranged from -0.2 m to
+0.2 m. The simulation is not run on yz plane since it would not provide any relevant
information about the x axis magnetic field gradient.
4.2.1
General Results
For the following sections, the geometrical parameters of the x axis Golay coil,
as shown in Figure 3.1 and 3.2, are related as follows, unless stated otherwise:
1.
θ = 1200
2.
r = 0.2 m
3.
a = 2r
4.
d=r
5.
l = 3r
with r = radius. Such configuration is chosen so as to simulate the accomodation of a
typical human body. Figure 4.9 shows the resultant Bz versus x. The linear region of
Bz exists roughly in between x = −0.1 m and x = +0.1 m. Alternatively speaking, a
magnetic field gradient along x axis, Gx , is generated. In Figure 4.9, the magnetic field
gradient generated is around -0.18 mT/m. Figure 4.9 is based on (y, z) = (0, 0). The
linear region generally decreases as (y, z) increases as shown in Figure 4.10 and 4.11.
Both Figure 4.10 and 4.11 are plotted for positive y and z respectively. Similar results
will be obtained if the Bz is plotted for negative y and z because of the symmetry
of Golay coil. Refering to Figure 4.10, the magnetic field gradient value decreases
along y axis and the usable region (the x range where magnetic field gradient is linear)
decreases too. Figure 4.11, however, shows some slight difference in z direction.
Based on Figure 4.11, the magnetic field gradient value increases but then decreases as
z increases. The linear region appear to be unchanged.
36
2e-05
1.5e-05
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-2e-05
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x/meter
Figure 4.9: Bz versus x at (y, z) = (0, 0)
2e-05
y = 0.00
y = 0.05
y = 0.10
y = 0.15
1.5e-05
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-2e-05
-0.2
-0.15
-0.1
-0.05
0
x/meter
0.05
0.1
0.15
0.2
Figure 4.10: Bz versus x at z = 0 for various y
6e-05
z = 0.0
z = 0.1
z = 0.2
z = 0.3
4e-05
Bz/Tesla
2e-05
0
-2e-05
-4e-05
-6e-05
-0.2
-0.15
-0.1
-0.05
0
x/meter
0.05
0.1
0.15
Figure 4.11: Bz versus x at y = 0 for various z
0.2
37
The two-dimensional xy-plot might look confusing. Figure 4.12 and 4.13
provide a better picture of this. Both Figure 4.12 and 4.13 are plotted on a 2dimensional plane stated, using contour plotting. The contour shown is no longer
the values of Bz but the iso value lines of the deviation of gradient value, 4% , which
is the more appropriate parameter to be plotted. By using such plot, the usable region
within certain percentage of tolerance can be determined.
Figure 4.12: 4% Contour Plot on xy Plane at 5% interval at z = 0
Figure 4.13: 4% Contour Plot on xz Plane at 5% interval at y = 0
Figure 4.12 and 4.13 are plotted at 5% interval of 4% . Based on Figure 4.12, if
the tolerance of the gradient deviation is 5% (which is generally the case), a elliptical
38
shape of usable region is obtained on the xy plane at z = 0. This region is roughly
±0.041 m in the x direction and ±0.065 m in the y direction. In Figure 4.13, such
region is in rectangular shape, ±0.042 m in the x direction and ±0.041 m in the z
direction. By combining the results from Figure 4.12 and 4.13, the usable region
in 3-dimension volume would be an elliptical cylinder, subjected to 5% of gradient
deviation. For MRI or NMR application that can tolerate a higher gradient deviation,
then the usable region would become bigger. The usable volume is around 0.207%
of the total volume. If the 10% gradient deviation tolerance is considered, the usable
volume can be increased to 0.464% of the total volume, which is 124% improvement.
4.2.2
Variation of θ
In this section, the simulation will be ran for various value of θ with the other
geometrical parameters fixed at the original value. Table 4.1 shows the correlation
calculation for each θ simulation mentioned, between x = −0.1 m and x = +0.1
m. It is found that as θ increases, the correlation coefficient, rc , decreases which also
means the linearity decreases. A high linearity is desired as this affects the gradient
uniformity. If a perfect linearity is obtained, the |rc | value would be 1, then the gradient
at any point would be the same, leading to 0% gradient deviation and perfect gradient
uniformity.
Table 4.1: Correlation and Gradient value for various θ
r between x = −0.1 m
θ (0 ) c
Gx (mT/m)
and x = +0.1 m
80
-0.999647
-0.13
100
-0.999329
-0.16
120
-0.998978
-0.18
140
-0.998667
-0.19
160
-0.998455
-0.20
Figure 4.14 shows the plot of Bz versus x for various θ at (y, z) = 0. From Figure 4.14
and Table 4.1, the gradient value increases as θ increases. Figure 4.15 to 4.18 show
the contour plots of 4% on xy plane whereas Figure 4.19 to 4.22 show the same plots
on xz plane. Table 4.2 shows the summary of the usable volume in each case. The
usable volume defined here is the region where the gradient deviation is within the 5%
tolerance.
39
2e-05
θ = 800
θ = 10000
θ = 1200
θ = 1400
θ = 160
1.5e-05
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-2e-05
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x/meter
Figure 4.14: Bz versus x for various θ at (y, z) = 0
Figure 4.15: 4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 800
Figure 4.16: 4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 1000
40
Figure 4.17: 4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 1400
Figure 4.18: 4% Contour Plot on xy Plane at 5% interval at z = 0 for θ = 1600
Figure 4.19: 4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 800
41
Figure 4.20: 4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 1000
Figure 4.21: 4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 1400
Figure 4.22: 4% Contour Plot on xz Plane at 5% interval at y = 0 for θ = 1600
42
θ (0 )
80
100
120
140
160
Table 4.2: Usable Volume for various θ
Percentage of usable volume
Actual usable volume (cm3 )
to total volume (%)
435.304
0.158
554.117
0.179
703.277
0.207
893.017
0.256
1061.434
0.304
As the θ increases, the percentage of usable volume and the actual usable
volume increases. In this simulation, generally, the best configuration would be the
Golay coil with θ = 1600 since it generates the highest magnetic field gradient besides
providing the highest usable volume. Higher θ value is expected to provide a bigger
usable volume. High gradient value will result in high contrast in the MRI or NMR
images since a small increment in distance will cause a high value in magnetic field
(Bz ) value difference.
4.2.3
Variation of r
Simulation on variation of r is not run although it is one of the geometrical
parameter of Golay coil. This is because other geometrical parameters are quoted as
the function of r. Generally, when r increases, d and l will increase. It would be more
appropriate to run the simulation for a certain ratio of d/r and l/r as this will represent
the results better.
4.2.4
Variation of a
Original configuration of Golay coil is shown in Figure 3.1 and 3.2 in which
the two arc section shown in Figure 3.2 have the same focal point, thus, the value of
a is 2r. Variation of a will mean that the two arc will not have the same focal point,
shown in Figure 4.23.
Figure 4.24 shows the plot of Bz versus x for various a mentioned. From Figure
4.24, as the a is increased, the gradient tend to switch from negative value to positive
value. Normally, the negative or positive value of gradient is not a big concern, as
43
long as the region with certain tolerance of gradient deviation is available. For positive
valued gradient in Figure 4.24, the gradient value increases and then drops as a is
further increased as a function or r.
Figure 4.23: Different Focal Point of Two Arcs of Golay Coil
4e-05
3e-05
a = 2.5r
a = 3.0r
a = 3.5r
a = 4.0r
a = 4.5r
2e-05
Bz/Tesla
1e-05
0
-1e-05
-2e-05
-3e-05
-4e-05
-0.2
-0.15
-0.1
-0.05
0
x/meter
0.05
0.1
0.15
0.2
Figure 4.24: Bz versus x for various a at (y, z) = 0
Table 4.3 shows the correlation coefficient, rc , between between x = −0.05
m and x = +0.05 m for each a, their respective gradient value, Gx , the actual usable
volume and the percentage of usable volume. When a = 3.5r, the gradient value and
44
rc is maximum. However, the percentage of usable volume is greatly affected from the
original 0.207% (for a = 2r) to less than 0.1% as a is further increased. The similar
observation occurs as with the actual usable volume. Although, it may seem that a high
gradient is obtainable when a = 3.5r which also resulted in high linearity, the usable
volume is smaller compared to the original configuration (703.277 cm3 ).
Table 4.3: Correlation, Gradient value and Usable volume for various a
a (m)
rc between
x = −0.05 m and
x = +0.05 m
Gx (mT/m)
2.0r
2.5r
3.0r
3.5r
4.0r
4.5r
-0.998978
-0.999364
-0.926008
0.999990
0.997483
0.999750
-0.18
-0.19
-0.08
0.35
0.72
0.30
Actual
usable
volume
(cm3 )
703.277
138.970
3.530
197.696
37.962
60.103
Percentage of usable volume to
total volume (%)
0.207
0.034
0.001
0.009
0.007
0.003
Figure 4.25 to 4.34 show the contour plots produced by varying a. The shape of
usable regions change too. From a general shape of elliptical cylinder into a rectangular
cubic. From Figure 4.29, although the usable region within 5% is a rectangular
shape, the region is expected to be split into two non-continuous regions (left and
right), making the usable region “useless”. As for the xz plane, the usable region
shrinks quite significantly besides the contour shape begins to change into something
unrecognizable (Figure 4.32 to 4.34).
Figure 4.25: 4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 2.5r
45
Figure 4.26: 4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 3.0r
Figure 4.27: 4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 3.5r
Figure 4.28: 4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 4.0r
46
Figure 4.29: 4% Contour Plot on xy Plane at 5% interval at z = 0 for a = 4.5r
Figure 4.30: 4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 2.5r
Figure 4.31: 4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 3.0r
47
Figure 4.32: 4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 3.5r
Figure 4.33: 4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 4.0r
Figure 4.34: 4% Contour Plot on xz Plane at 5% interval at y = 0 for a = 4.5r
48
4.2.5
Variation of d
In this section, the parameter d will be manipulated and its effect on gradient
linearity and uniformity are observed. Figure 4.35 summarized the Bz versus x plot
for the mentioned respective d. From Figure 4.35, the gradient value decreases as d
increases. Table 4.4 shows some interesting observations.
0.00015
d = 0.2r
d = 0.4r
d = 0.6r
d = 0.8r
d = 1.2r
d = 1.4r
0.0001
Bz/Tesla
5e-05
0
-5e-05
-0.0001
-0.00015
-0.2
-0.15
-0.1
-0.05
0
x/meter
0.05
0.1
0.15
0.2
Figure 4.35: Bz versus x for various d at (y, z) = 0
Table 4.4: Correlation, Gradient value and Usable volume for various d
d (m)
rc between
x = −0.1 m and x =
+0.1 m
Gx (mT/m)
0.2r
0.4r
0.6r
0.8r
1.0r
1.2r
1.4r
-0.998677
-0.999460
-0.999967
-0.999804
-0.998978
-0.997682
-0.996075
-0.34
-0.31
-0.27
-0.22
-0.18
-0.13
-0.10
Actual
usable
volume
(cm3 )
302.669
565.044
1212.572
3374.389
703.277
300.048
175.854
Percentage of usable volume to
total volume (%)
0.102
0.185
0.766
2.090
0.207
0.086
0.050
Although, high gradient value is obtained at d = 0.2r both the actual usable volume
and the percentage of usable volume are rather poor. The usable volume peaked at
around a = 0.8r which is 2.090% (3374.389 cm3 ) which is higher than the original
49
configuration (where a = r). High linearity is, however, obtained at a = 0.6r. Further
simulation is ran for d = 0.7r and the usable volume increases to 2.26%. From Figure
4.36 to 4.47, it can be seen that the change of d directly affects the usable region on
both xy and xz plane.
Figure 4.36: 4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.2r
Figure 4.37: 4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.4r
50
Figure 4.38: 4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.6r
Figure 4.39: 4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 0.8r
Figure 4.40: 4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 1.2r
51
Figure 4.41: 4% Contour Plot on xy Plane at 5% interval at z = 0 for d = 1.4r
Figure 4.42: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.2r
Figure 4.43: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.4r
52
Figure 4.44: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.6r
Figure 4.45: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.8r
Figure 4.46: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.2r
53
Figure 4.47: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.4r
4.2.6
Variation of l
The l parameter will affect the total length of the Golay coil. Since both
saddle coils are identical, the total length would be 2l + d. Figure 4.48 shows the
Bz versus x plot for various l as a function of r. There is hardly any difference shown
in Figure 4.48. Data in Table 4.5 confirm this observation. However, the usable volume
decreases as l increases. Figure 4.49 to 4.56 show the results obtained for various l.
Similar size of usable regions are observed in Figure 4.49 to 4.56.
2e-05
l=r
l = 2r
l = 4r
l = 5r
1.5e-05
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-2e-05
-0.2
-0.15
-0.1
-0.05
0
x/meter
0.05
0.1
0.15
Figure 4.48: Bz versus x for various l at (y, z) = 0
0.2
54
Table 4.5: Correlation, Gradient value and Usable volume for various l
l (m)
rc between
x = −0.1 m and x =
+0.1 m
Gx (mT/m)
r
2r
3r
4r
5r
-0.999185
-0.999012
-0.998978
-0.998964
-0.998957
-0.176
-0.179
-0.177
-0.175
-0.174
Actual
usable
volume
(cm3 )
855.723
721.722
703.277
699.444
698.814
Percentage of usable volume to
total volume (%)
0.604
0.311
0.207
0.166
0.135
Figure 4.49: 4% Contour Plot on xy Plane at 5% interval at z = 0 for l = r
Figure 4.50: 4% Contour Plot on xy Plane at 5% interval at z = 0 for l = 2r
55
Figure 4.51: 4% Contour Plot on xy Plane at 5% interval at z = 0 for l = 4r
Figure 4.52: 4% Contour Plot on xy Plane at 5% interval at z = 0 for l = 5r
Figure 4.53: 4% Contour Plot on xz Plane at 5% interval at y = 0 for l = r
56
Figure 4.54: 4% Contour Plot on xz Plane at 5% interval at y = 0 for l = 2r
Figure 4.55: 4% Contour Plot on xz Plane at 5% interval at y = 0 for l = 4r
Figure 4.56: 4% Contour Plot on xz Plane at 5% interval at y = 0 for l = 5r
57
Further simulation was run for l = 0.1r. The gradient value obtained is
−0.048 mT/m with a 0.74% of usable volume. The gradient value is poor although the
percentage of usable volume might be acceptable. Further increasing the l parameter
is almost useless, since this will only decrease the percentage of usable volume.
4.3
Results on Maxwell Coil
Maxwell coil is used to generate a magnetic field gradient along the z axis, Gz ,
in a conventional setup. Figure 4.57 shows the xz and yz plane in which the simulation
will be ran on. Both the planes are actually identical because of the symmetry of
Maxwell coil. Therefore, the simulation is only needed to be ran once on either plane.
The simulation covered x from −0.3 m to +0.3 m, y from −0.3 m to +0.3 m and z
from −0.3 m to +0.3 m.
Figure 4.57: z axis Maxwell Coil with (a) xz Calculation Plane and (b) yz Calculation
Plane
4.3.1
General Results
There are only two geometrical parameters involved in Maxwell coil; the
radius, r and the separation distance between the two circular coils, d. In this section,
r is fixed at 0.4 m whereas d = r. Such configuration is chosen to accomodate the
58
Golay coil’s width (which is fixed at 0.4 m) in Section 4.2. The original Maxwell coil
√
setup requires d to be 3r [7], however, in this research, d is the geometrical parameter
for observation, it is then fixed at d = r before it is varied from less than r to more
√
than r. Furthermore, some researches did not use the separation d = 3r based on
several optimization, such as based on image distortion [39] and based on application
[8]. Figure 4.58 to 4.60 show the typical results obtained for this setup.
Figure 4.58 confirms that a magnetic field gradient is generated along the z
direction. Figure 4.59 and 4.60 are identical, as expected, with usable volume in a
rectangular cubic shape. The gradient value is approximately -0.067 mT/m with a high
percentage of usable volume over total volume of 0.961% (1962.668 cm3 ).
1.5e-05
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-0.3
-0.2
-0.1
0
z/meter
0.1
0.2
0.3
Figure 4.58: Bz versus z at (x, y) = (0, 0)
4%
0.3
Contour Plot on xz plane at y = 0
0.2
0
.000
-20
-1-15.000.0000-5.000 05.0.00000
.00
0
1
1
20.50.000
00
0.0
152.0.000
000
z/meter
0.1
0.1
.000
0.2 -5
0.30.3
10
.000
00
50.0.000
00
.0
0
1
-
0.2
0
5.00
-1-20
.000
0.1
0.0
x/meter
0.1
0.2
0.3
Figure 4.59: 4% Contour Plot on xz Plane at 5% interval at y = 0
59
4%
0.3
Contour Plot on yz plane at x = 0
0.2
0
.000
-20
-1-15.000.0000-5.000 05.0.00000
.00
0
1
1
20.50.000
00
0.0
152.0.000
000
z/meter
0.1
0.1
.000
0.2 -5
0.30.3
00
10.0
00
50.0.000
00
-10.0
0.2
0
5.00
-1-20
.000
0.1
0.0
y/meter
0.1
0.2
0.3
Figure 4.60: 4% Contour Plot on yz Plane at 5% interval at x = 0
4.3.2
Variation of r
The z gradient coil simulation is not run directly for the variation of r since
the other geometrical parameters, d, is quoted as a function of r. The results presented
in the next section should reflect the effect of the ratio of d/r to the linearity and
uniformity.
4.3.3
Variation of d
The parameter d basically determines how far both the circular coils are being
separated in a Maxwell coil configuration. A general effect of d to the magnetic field
generated is shown in Figure 4.61. The correlation coefficient, gradient value and
usable volume are tabulated in Table 4.6. A perfect correlation is obtained when d =
1.8r (|rc | = 1, true to 6 decimal points) but the maximum percentage of usable volume
to total volume occurs at d = 1.6r. The percentage of usable volume increases when
d is increased to a certain value and then drops as it is further increased. The same
trend happens with the value of correlation coefficient. The gradient value, however,
continues to drop as d is increased. Figure 4.62 to 4.68 show the contour plot obtained
on xz plane. The results on xy plane are not shown since they are identical to their xz
plane counterparts.
60
1.5e-05
d = 0.8r
d = 1.2r
d = 1.4r
d = 1.6r
d = 1.8r
d = 2.0r
d = 2.2r
1e-05
Bz/Tesla
5e-06
0
-5e-06
-1e-05
-1.5e-05
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
z/meter
Figure 4.61: Bz versus z at (x, y) = (0, 0) for various d
Table 4.6: Correlation, Gradient value and Usable volume for various d
d (m)
rc between
x = −0.1 m and x =
+0.1 m
Gz (mT/m)
0.8r
1.0r
1.2r
1.4r
1.6r
1.8r
2.0r
2.2r
-0.999657
-0.999810
-0.999913
-0.999970
-0.999995
-1.000000
-0.999995
-0.999987
-0.065
-0.067
-0.066
-0.061
-0.055
-0.048
-0.042
-0.036
4%
0.3
Actual
usable
volume
(cm3 )
1201.734
1962.668
3777.962
10610.302
19174.800
19164.600
19196.128
11907.174
Percentage of usable volume to
total volume (%)
0.744
0.961
1.543
3.842
6.043
5.205
4.744
2.697
Contour Plot on xz plane at y = 0
0.2
0
00.00
00 0
-20.0
.00
0 .000 .0100
-15
-10.000-5.00 0 01.500
2
z/meter
0.1
0.0
0.1
-5.000
00
0.00 00
520.0050.0.000
1
0
0
10.
0
0.00
-1-15
.00000
-20.0
5.00
0
0.2
0.30.3
0.2
0.1
0.0
x/meter
0.1
0.2
0.3
Figure 4.62: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 0.8r
61
4%
0.3
Contour Plot on xz plane at y = 0
0.2
-2-10.500.0000 000 .000000
0 . 0 5.
-10.00 -5
10.
0
15.0000
0
20.000
0.0
15.00
0
20.000
z/meter
0.1
0.1
00 0
5.00.00
.0.00000
000 -1-015
10.
0.2
0.30.3
0.2
-20.000
-5.00
0
0.1
0.0
x/meter
0.1
0.2
0.3
Figure 4.63: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.2r
10.00
.0000
2105.00
0.2
0.000
5.000
20.00
15.000
0
05.10.000.000
00 0
0.2
00
2105.0
.000
0
10.00
0.1
0.000
5.000
z/meter
-20.05.0000000 000
-1 10.0 -5.
-
0
0.1
0.0
Contour Plot on xz plane at y = 0
0
5 .0
10.0.000000
15.000
20.000
0
4%
0.3
-20.000
.000
00-015
-10.
0.30.3
-5.0
00
0.2
0.1
0.0
x/meter
0.1
0.2
0.3
Figure 4.64: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.4r
4%
0.3
0.000
-10.000
-5.000
-10.000
-5.000
5.0
00
-20.000000
-15. 0.000 .000
-1
-5
0.000
z/meter
0.1
0.
5.00000
10.0000
15.000
20.000
10.00 00
15.000
20.00
0.2
0.0
Contour Plot on xz plane at y = 0
0.30.3
000
-15.-20
.000
00
.0
-10
0.2
20.000
15.000
10.0000
5.00
0.2
20.000
15.000
10.000
05.0.000
00
0.1
0.1
0.0
x/meter
0.1
0.2
-5.00
00.3
Figure 4.65: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.6r
62
4%
0.1
-20.000
-15.000
-10.000
-5.000
000
20.5.000
1 00
10.0
5.000
0.2
Contour Plot on xz plane at y = 0
152. 0.000
10.0000
00
5.00
0
0.3
00
0.0
-5.000
-20.000
-15.000
-10.000
-20.000
-15.000
-10.000
-5.000
z/meter
0.000
0.0
0.000
0.2
5.000
00
10.0 00
0
15.0.000
2
0.30.3
0.2
00
0.0
-5.000
-10.000
-15.000
-20.000
0.1
0.0
x/meter
5.000
10.
15. 000
20. 000
000
0.1
0.1
0.2
0.3
Figure 4.66: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 1.8r
4%
0.3
0.2
-5.000
-20.000
-15.000
-10.000
-20.000
-15.000
-10.000
1 20
10. 5.000.000
000
0.000
0.1
10.
20. 15.00 000
000 0
0.2
0.000
-5.000
-10.000
-15.000
-20.000
00
0.30.3
00
5.0
-5.000
z/meter
-20.000
-15.000
-10.000
-5.000
0.000
000 0
20. 15.00 .000
10
5.000
0.000
0.1
0.0
Contour Plot on xz plane at y = 0
5.0
0.2
0.1
0.0
x/meter
000
10. 5.000000
1 20.
5.0
00
0.1
0.2
0.3
Figure 4.67: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 2.0r
4%
0.3
0.2
-10.000
-20.000
-15.000
z/meter
-10.000
00
-5.0 0.000
5.000
20.00015.000
0.2
0.30.3
2
15.000 0.000
0.2
0.1
-5.0
10.000
-20.000
-15.000
0.1
5.000
10.000
20.000 15.000
10.000
5.00
0.00 0
0
0.1
0.0
Contour Plot on xz plane at y = 0
00
15.000 0.000
2
10.000
5.000
0.0
x/meter
0.1
0.2
0.3
Figure 4.68: 4% Contour Plot on xz Plane at 5% interval at y = 0 for d = 2.2r
63
4.4
Discussion
The usable volumes obtained in this research are either cylindrical (for Golay
coil) or cuboid (for Maxwell coil) in shape. These shape generally provide the
neccessary working area for MRI in which superconducting main magnets are in
cylinder shape. There have been reports on the similar shape of usable region obtained
such as a z gradient coil with usable volume of diameter = 6 cm and length = 12
cm [10], a z gradient coil with usable volume of diameter = 4.5 mm and length = 4.5
mm [11], a z gradient coil with usable volume of diameter = 16.4 cm and length =
21.8 cm [40] and others [24, 36]. Cuboid shaped usable region is not unusual, too. For
example, y and z gradient coils with usable region of 36.3 cm × 17.2 cm × 24.4 cm
cuboid [21] as well as a cuboid of 12 cm × 15 cm × 1.8 cm [20]. Beside cylinder and
cuboid shaped usable volume, spherical usable region is also a common one, reported
by Rinard et. al. [41], Dodd et al. [17] and Harvey [31]. The size and shape of the
usable region can be varied based on the gradient coils design itself (whether they are
the conventional one or the more advanced planar type and “finger print” type) and also
the application (whether it is for medical imaging or micro imaging). There are some
reports which prefer the observation on the region of interest (ROI), volume of interest
(VOI) or diameter spherical volume (DSV). With this definitions, the researchers first
fixed a region in which the magnetic gradient uniformity is to be optimized (the size
of ROI, VOI or DSV is most likely based on the application in which mentioned by
DeMesster et al. [42]), and then the gradient coil is designed to achieve best possible
condition in that predefined region. For example, White et al reported a 1.56% field
error inside their DSV [6], Du et al. with VOI of diameter = 34 cm, length = 40 cm
and maximum 5% of deviation [39], Lopez et al. with a DSV of diamter = 21 cm and
nonuniformity of 2.63% [25] among others [10, 16, 29]. For comparison purposes,
the maximum cylindrical usable volume observed in this research is approximately a
elliptical cylinder with major axis = 5.31 cm, minor axis = 3.15 cm and length = 5.71
cm (Golay coil) whereas the maximum cuboid usable region observed is 28.4 cm ×
28.4 cm × 23.8 cm (Maxwell coil).
The magnetic gradient values obtained only serve as comparison to show the
trend affected by the geometrical parameters. The magnetic gradient value can be very
different based on applications. A typical magnetic gradient value can range between
20 mT/m and 100 mT/m for medical imaging [6, 7, 13, 25, 26, 43, 40] and it is directly
affected by the current supply. Higher gradient values are reported but not in medical
imaging, such as 2000 mT/m for magnetic resonance microscopy [38] and 52 T/m for
high spatially resolved NMR [24]. Some also prefer to quote the gradient efficiency, η
64
= gradient/current, rather that the gradient value. In this simulation, the lower magnetic
field gradient values are mainly because of the current is only 10 Ampere. Again, here
the trend is highlighted rather than the absolute gradient value instead.
The analysis for the Maxwell coil is clearly easier compared to the analysis
of Golay coil. This is mainly due to the geometrical parameters involved. Maxwell
coil only consists of two identical circular coil whereas Golay coil consists of a
combination of arcs as well as straight segments. In the simulation of Maxwell coil, as
the d increases, the gradient value drops. This is because as the distance increases, the
magnetic field value generated by individual coil is reduced based on the Biot-Savart’s
Law. The usable volume, however, generally increases as d increases since when the
gradient value decreases, the range of linear gradient increase, resulting in a bigger
usable region in general as shown as in Figure 4.58. The behaviour of Golay coil is
slightly complicated. First the straight segments of the Golay coil do not contribute
to the magnetic field gradient generated. This can be seen very easily from the result
shown in Figure 4.48. This can be explained using Figure 4.69.
Figure 4.69: Magnetic Field Generated by a Straight Segment
By using the right hand rule in Figure 4.69, there would not be any magnetic field
generated in the z direction. Thus, only the arcs in the Golay coil setup play the role
of generating the magnetic field gradient. The arcs can be further divided into two
sets; one, the inner arcs (which are the arcs situated near to the origin (0, 0, 0)) and the
outer arcs (those situated further away from the origin). The inner arcs actually affect
the gradient generated as can be seen in Section 4.2.5. The difference in d affects the
gradient generated (linearity, gradient value and uniformity) significantly. The outer
arcs, however, do not actually affect the gradient to any considerable extent. This
is observed as in Section 4.2.6. As l increases, the distance between the inner arcs
65
and the outer arcs also increases. However, the results show no significant change
in the gradient value, linearity and usable region. The outer arcs can be said to be
merely as the current return paths as with the straight segments of Golay coil. The
behavior in which the inner arcs are able to generate a gradient can be seen from the
fact that a circular coil generating a uniform field in the axial direction. If the circular
coil is splited into two concentric arcs, each carrying current in the same direction,
then a gradient will be generated. This also explain why when the θ of Golay coil
increases, the usable region increases since as the θ increases, it will move closer to the
actual circular coil but then again, a complete circular coil carrying current in opposing
direction in one half of the circular is impossible (because it is a closed loop). This
is the same case with Maxwell coil which is exact opposite of Helmholtz coil. The
gradient generated by Golay coil generally lose its uniformity rather quickly in the z
direction as oppose to x or y direction. This most likely is due to the effect of the outer
arcs. The increases of z also mean the nearer to the outer arcs. The magnetic field
generated by the outer arcs might be superposed with the magnetic field generated by
the inner arcs and thus causing the lost in uniformity.
All the simulations ran are based on manipulating one parameter while others
being fixed at certain value. The combination of two or more geometrical parameters
might prove otherwise. For example, from Table 4.2, when θ = 1600 , the usable
volume is higher compared to the rest but if the simulation is ran for θ = 1600 and
d = 0.8r (from Table 4.4 which have high usable volume), the usable volume is not
optimum. This is mainly because the geometrical parameters involved are being related
to each other. Optimized coil should be made from simulation run on simultaneous
parameters, rather than one parameter at a time.
The general preference is a large region of gradient uniformity within certain
acceptable deviation. However, larger usable region is normally resulted from larger
gradient coil which will face problem concerning the torque and inductance. The
usable region produced over the total region covered by the gradient coil, therefore,
reflects the optimum of such ratio when constructing conventional gradient coils. The
importance of the gradient value can be explained as shown in Figure 4.70 and 4.71.
Refering to Figure 4.70, the higher the gradient value, the “thinner” the slice
thickness. This means that for a fixed number of voxels (or grid points), the thinner
slice will have a higher resolution than the thicker slice. A more detailed imaging
of the sample can be done since the thinner slice imaging might provide information
66
about the sample that the thicker slice imaging could not. Figure 4.71 shows a different
point of view. If there are two points in the sample that undergo the imaging process,
the difference of the magnetic field values between those two points will determine
how well resolved they are. As seen in Figure 4.71, the higher the gradient value, the
higher the resolution between two fixed points.
Figure 4.70: Schematic of Gradient Value and Slice Thickness
Figure 4.71: Schematic of Gradient Value and Resolution
67
4.5
Summary
Results on the Golay coil and Maxwell coil are presented. Four geometrical
parameters of Golay coil and one geometrical of Maxwell coil are being investigated
in term of their effects on the magnetic field gradient generated. This is found out that
the configuration of Golay coil and Maxwell coil can be improved for better gradient
values and usable regions. Some geometrical effects on the gradient are also being
discussed with comparison with related reference.
CHAPTER 5
CONCLUSIONS
5.1
Conclusions
The effects of a total of four geometrical parameter for Golay coil and one
geometrical parameters for Maxwell coil to the magnetic field gradient generated
respectively have been investigated through a computer simulation written fully in
Python programming language. The geometrical parameter, r (radius), for Golay
coil and Maxwell coil, is omitted from the simulation since the other parameters are
expressed as a function of r. The field linearity is illustrated by two-dimensional
normal xy-plot and also expressed in the form of correlation coefficient, rc . The
straighter the line in xy-plot or the closer the value of |rc | to 1, the higher the field
linearity is. The gradient uniformity is highlighted using contour plot of the gradient
deviation, 4% , on specified plane. A good gradient deviation, which is around 5%,
is used to determined the usable region for that configuration. The usable region is
expressed as percentage of usable volume to the total volume covered by the gradient
coil as well as the actual usable region in cm3 . Such definition of usable volume
can provide data on another form of usable volume optimization. Although bigger
usable volume is always desirable, bigger coil generally create bigger usable volume
and bigger coil means longer coil windings which might cause problem with the
inductance, heat dissipation, coil efficiency among other parameters. Because of the
similar reason, the other geometrical parameters involved are expressed as a function
of r.
As shown in the results, the parameters that affects the gradient significantly
are the θ, a and d for Golay coil and the parameter d for Maxwell coil. Comparing
to the original configuration, with the increase of θ, the improvement of 50% in the
usable region and 11% in gradient value, is observed. The increase of a in Golay
coil, however, can reduced the usable region up to 99% although it can improved the
69
gradient value by 200%, as compared to the original configuration. The interesting
parameter is d which shows that the usable region can be increase from 703.277 cm3
to a siginificant 3374.389 cm3 . The improvement of gradient value is around 50%. As
for Maxwell coil, the parameter d play an important role in usable region but not in
gradient value. The biggest improvement in usable region is from 1962.668 cm3 to
a staggering 19196.128 cm3 . The improvement of gradient value is not much, only
around 0.002 mT/m from original’s 0.065 mT/m.
The program was written in a vectorized processing way so as to improve the
calculation time. User defined functions are stored in a Python module. By using this
module, gradient coil with virtually any shape can undergo the simulation as long as
the position of coil is known. This is possible because the simulation adopted the BiotSavart’s Law for finite length segment and coil of any shape can be simplified into a
series of finite length segment, either manually or using parametric equations. The
module also provides calculations along x, y and z axis, xy, xz and yz plane as well
as 3-dimensional xyz volume. The main program itself provides user with interactive
features whereby user can input their desired geometrical parameters and selection of
calculation mode. The output of simulation can be exported to a widely used VTK file
format (supported by a large number of visualization program) in case user prefers to
visualize their outputs using other programs.
5.2
Suggestions for Further Works
A multiple geometrical parameters optimization is missing from this
simulation. Since the geometrical parameters are inter-related, it would be a more
complete investigation if such simulation is carried out. Other missing features in the
Python program would include the development of the graphical user interface (GUI)
and also more option and control to the simulation being run.
A computer simulation would not be complete with a physical experiment.
A prototype based on the simulation can be developed in order to justify the results
obtained in a computer simulation. However, without proper simulation, developing a
test prototype based on a trial-and-error concept could be disastrous. The developed
prototype might not be feasible which leads to another prototype being developed.
The prototype development based on computer simulation can help reducing the cycle
significantly. The original intention of the research is to map the magnetic field or
70
gradient generated by conventional gradient coil. The simulation can be modified so
that it is not a normal simulation but a optimizing simulation for conventional gradient
coil. A number of 6 geometrical parameters are able to undergo the optimization
process which can help in designing a gradient coil with optimum usable volume,
with optimum efficiency and etc. A modification of conventional gradient coil (such as
multiple layers of coils) can also be done using the same simulation to observe whether
the modification will improve the gradient or not.
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Dodd, S. and Chien, H. Short Planar Gradient Coils for MR Microscopy Using
Concentric Return Paths. Journal of Magnetic Resonance, 2002. 156: 1–9.
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Lee, S. C., Kim, K. S., Kim, J. H., Lee, S. C., Yi, J. H., Kim, S. W., Ha, K. S.
and Cheong, C. J. One Micrometer Resolution NMR Microscopy. Journal of
Magnetic Resonance, 2001. 150: 207–213.
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Barbara, T. M. and Bronnimann, C. E. Target Field Design for Magic Angle
Gradient Coils. Journal of Magnetic Resonance, 1999. 140: 285–288.
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Fisher, B. J., Dillon, N., Carpenter, T. A. and Hall, L. D. Design of a Biplanar
Gradient Coil using a Genetic Algorithm. Magnetic Resonance Imaging, 1997.
15(3): 369–376.
21.
Williams, G. B., Fisher, B. J., Huang, C. L.-H., Carpenter, T. A. and Hall,
L. D. Design of Biplanar Gradient Coils for Magnetic Resonance Imaging of
the Human Torso and Limbs. Magnetic Resonance Imaging, 1999. 17(17):
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von Schulthess, G. Clinical MR in Year 2010. Magnetic Resonance Materials
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Snaar, J. E. M., Robyr, P. and Bowtell, R. Strong Gradients for Spatially
Resolved Diffusion Measurements. Magnetic Resonance Imaging, 1998.
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Lopez, H., Liu, F., , Poole, M. and Crozier, S. Equivalent magnetization
current method applied to the design of gradient coils for magnetic resonance
imaging. IEEE Transactions on Magnetics, 2009. 45(2): 767–775.
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Mansfield, P., Bowley, R. and Haywood, B. Controlled E-field Gradient Coils.
Magnetic Resonance Materials in Physics, Biology and Medicine, 2003. 16:
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High Efficiency - Gradient Uniformity Tradeoff In Multilayer Asymmetric
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Permanent Magnets and Their Applications. 2008. 419–422.
APPENDIX A
MODULE
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#
#
#
#
#
#
file name
written by
date written
description
last modified
:
:
:
:
grad.py
chew teong han (teonghan@yahoo.com)
04/01/2009
collection of functions to calculate B using
biot-savart’s law
: 05/01/2009
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# import module(s)
import scipy as s
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# constant(s)
mu_0=4.0*s.pi*1.0e-7
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# module 1: to calculate vector cross product
def vecCro (vecA,vecB):
a=vecA[1]*vecB[2]-vecB[1]*vecA[2]
b=vecA[2]*vecB[0]-vecA[0]*vecB[2]
c=vecA[0]*vecB[1]-vecA[1]*vecB[0]
vecC=s.array([a,b,c])
return vecC
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# module 2: to calculate B for a general case
def calB(vOA,vOB,vOM,current):
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i
j
k
l
=
=
=
=
vOM.shape[1]
vOM.shape[2]
vOM.shape[3]
vOA.shape[1]
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vAM = s.zeros([3,i,j,k,l],dtype=float)
vBM = s.zeros([3,i,j,k,l],dtype=float)
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vAB = s.array([vOB[0]-vOA[0], vOB[1]-vOA[1], vOB[2]-vOA[2]])
magAB = s.sqrt(vAB[0]*vAB[0] + vAB[1]*vAB[1] + vAB[2]*vAB[2])
uAB = s.array([vAB[0]/magAB, vAB[1]/magAB, vAB[2]/magAB])
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for a in range(0,l):
vAM[0,::,::,::,a] = vOM[0,::,::,::]-vOA[0,a]
vAM[1,::,::,::,a] = vOM[1,::,::,::]-vOA[1,a]
vAM[2,::,::,::,a] = vOM[2,::,::,::]-vOA[2,a]
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vBM[0,::,::,::,a] = vOM[0,::,::,::]-vOB[0,a]
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vBM[1,::,::,::,a] = vOM[1,::,::,::]-vOB[1,a]
vBM[2,::,::,::,a] = vOM[2,::,::,::]-vOB[2,a]
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magAM = s.sqrt(vAM[0]*vAM[0] + vAM[1]*vAM[1] + vAM[2]*vAM[2])
uAM = s.array([vAM[0]/magAM, vAM[1]/magAM, vAM[2]/magAM])
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magBM = s.sqrt(vBM[0]*vBM[0] + vBM[1]*vBM[1] + vBM[2]*vBM[2])
uBM = s.array([vBM[0]/magBM, vBM[1]/magBM, vBM[2]/magBM])
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cos_theta1=s.array(uAM[0]*uAB[0] + uAM[1]*uAB[1] + uAM[2]*uAB[2])
cos_theta2=s.array(uBM[0]*uAB[0] + uBM[1]*uAB[1] + uBM[2]*uAB[2])
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vLM = s.array([vAM[0]-(vAM[0]*uAB[0] + vAM[1]*uAB[1] +
vAM[2]*uAB[2])*uAB[0],
vAM[1]-(vAM[0]*uAB[0] + vAM[1]*uAB[1] +
vAM[2]*uAB[2])*uAB[1],
vAM[2]-(vAM[0]*uAB[0] + vAM[1]*uAB[1] +
vAM[2]*uAB[2])*uAB[2]])
magLM = s.sqrt(vLM[0]*vLM[0] + vLM[1]*vLM[1] + vLM[2]*vLM[2])
uLM = s.array([vLM[0]/magLM, vLM[1]/magLM, vLM[2]/magLM])
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uB = vecCro(uAB,uLM)
magB = (mu_0*current*(cos_theta1-cos_theta2))/(4*s.pi*magLM)
B = s.array([uB[0]*magB,uB[1]*magB,uB[2]*magB])
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return B
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# module 3: to calculate B at one point
def calB_point(vOA,vOB,x,y,z,current):
vOM = s.mgrid[x:x+1:1,y:y+1:1,z:z+1:1]
B = calB(vOA,vOB,vOM,current)
return B
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# module 4: to calculate B along x-axis
def calB_x(vOA,vOB,xMin,xMax,stepX,y,z,current):
vOM = s.mgrid[xMin:xMax+stepX:stepX,y:y+1:1,z:z+1:1]
B = calB(vOA,vOB,vOM,current)
return B
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# module 5: to calculate B along y-axis
def calB_y(vOA,vOB,x,yMin,yMax,stepY,z,current):
vOM = s.mgrid[x:x+1:1,yMin:yMax+stepY:stepY,z:z+1:1]
B = calB(vOA,vOB,vOM,current)
return B
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# module 6: to calculate B along z-axis
def calB_z(vOA,vOB,x,y,zMin,zMax,stepZ,current):
vOM = s.mgrid[x:x+1:1,y:y+1:1,zMin:zMax+stepZ:stepZ]
B = calB(vOA,vOB,vOM,current)
return B
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# module 7: to calculate B on xy plane
def calB_xy(vOA,vOB,xMin,xMax,stepX,yMin,yMax,stepY,z,current):
vOM = s.mgrid[xMin:xMax+stepX:stepX,yMin:yMax+stepY:stepY,z:z+1:1]
B = calB(vOA,vOB,vOM,current)
return B
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# module 8: to calculate B on xz plane
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def calB_xz(vOA,vOB,xMin,xMax,stepX,y,zMin,zMax,stepZ,current):
vOM = s.mgrid[xMin:xMax+stepX:stepX,y:y+1:1,zMin:zMax+stepZ:stepZ]
B = calB(vOA,vOB,vOM,current)
return B
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# module 9: to calculate B on yz plane
def calB_yz(vOA,vOB,x,yMin,yMax,stepY,zMin,zMax,stepZ,current):
vOM = s.mgrid[x:x+1:1,yMin:yMax+stepY:stepY,zMin:zMax+stepZ:stepZ]
B = calB(vOA,vOB,vOM,current)
return B
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# module 10: to calculate B in xyz volume
def calB_xyz(vOA,vOB,xMin,xMax,stepX,yMin,yMax,stepY,zMin,zMax,stepZ,current):
vOM =
s.mgrid[xMin:xMax+stepX:stepX,yMin:yMax+stepY:stepY,zMin:zMax+stepZ:stepZ]
B = calB(vOA,vOB,vOM,current)
return B
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# module 11: to generate vOA & vOB for a saddle coil
def saddle(startTheta,endTheta,thetaStep,zU,zL,radius,j,k):
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if startTheta > endTheta:
thetaStep = -thetaStep
else:
thetaStep = thetaStep
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vOA = s.array([[],[],[]],dtype=float)
vOB = s.array([[],[],[]],dtype=float)
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for theta in s.arange(startTheta,endTheta,thetaStep):
VOA = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [zU]])
vOA = s.append(vOA,VOA,axis=1)
theta = theta + thetaStep
VOB = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [zU]])
vOB = s.append(vOB,VOB,axis=1)
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VOA=s.array([[radius*s.cos((endTheta)*s.pi/180.0)+j],
[radius*s.sin((endTheta)*s.pi/180.0)+k], [zU]])
vOA = s.append(vOA,VOA,axis=1)
VOB=s.array([[radius*s.cos((endTheta)*s.pi/180.0)+j],
[radius*s.sin((endTheta)*s.pi/180.0)+k], [zL]])
vOB = s.append(vOB,VOB,axis=1)
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for theta in s.arange(endTheta,startTheta,-thetaStep):
VOA = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [zL]])
vOA = s.append(vOA,VOA,axis=1)
theta = theta - thetaStep
VOB = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [zL]])
vOB = s.append(vOB,VOB,axis=1)
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VOA=s.array([[radius*s.cos((startTheta)*s.pi/180.0)+j],
[radius*s.sin((startTheta)*s.pi/180.0)+k], [zL]])
vOA = s.append(vOA,VOA,axis=1)
VOB=s.array([[radius*s.cos((startTheta)*s.pi/180.0)+j],
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[radius*s.sin((startTheta)*s.pi/180.0)+k], [zU]])
vOB = s.append(vOB,VOB,axis=1)
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return vOA,vOB
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#module 12: to generate vOA & vOB for a circular coil
def circle(thetaStep,z,radius,j,k):
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vOA = s.array([[],[],[]],dtype=float)
vOB = s.array([[],[],[]],dtype=float)
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if thetaStep > 0:
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for theta in s.arange(0,360,thetaStep):
VOA = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [z]])
vOA = s.append(vOA,VOA,axis=1)
theta = theta + thetaStep
VOB = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [z]])
vOB = s.append(vOB,VOB,axis=1)
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else:
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for theta in s.arange(360,0,thetaStep):
VOA = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [z]])
vOA = s.append(vOA,VOA,axis=1)
theta = theta + thetaStep
VOB = s.array([[radius*s.cos(theta*s.pi/180.0)+j],
[radius*s.sin(theta*s.pi/180.0)+k], [z]])
vOB = s.append(vOB,VOB,axis=1)
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return vOA, vOB
APPENDIX B
CODE FOR GOLAY COIL
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#
#
#
#
#
file name
written by
date written
description
:
:
:
:
x-golay.py
chew teong han (teonghan@yahoo.com)
05/01/2009
to calculate Bz of x-gradient coil + visualization
+ vtk output
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import time
import scipy as s
from grad import saddle
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# universal constant
thetaStep = 1.0
gridStep = 0.01
h = 1e-6
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print "=============================================================="
print "Welcome to Magnetic Field Calculation Program for Golay Coil"
print "=============================================================="
print "Please input the required parameters"
print "=============================================================="
coilTheta = raw_input("Please input the theta of the arc: ")
current = raw_input("Please input the current value (in Ampere): ")
radius = raw_input("Please input the radius value (in meter): ")
d = raw_input("Please input the d value (in meter): ")
l = raw_input("Please input the l value (in meter): ")
j1 = raw_input("Please input the x coordinate of the 1st arc: ")
k1 = raw_input("Please input the y coordinate of the 1st arc: ")
j2 = raw_input("Please input the x coordinate of the 2nd arc: ")
k2 = raw_input("Please input the y coordinate of the 2nd arc: ")
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# user-defined variables conversion
thetaOffset = 90.0-(float(coilTheta)/2.0)
current = float(current)
radius = float(radius)
d = float(d)
l = float(l)
zU2 = (d/2)+l
zL2 = (d/2)
zU1 = -zL2
zL1 = -zU2
j1 = float(j1)
k1 = float(k1)
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j2 = float(j2)
k2 = float(k2)
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# theta calculation
# saddle 1
startS1 = 90.0+thetaOffset
endS1 = 270.0-thetaOffset
# saddle 2
startS2 = 90.0-thetaOffset
endS2 = -90.0+thetaOffset
# saddle 3
startS3 = 270.0-thetaOffset
endS3 = 90.0+thetaOffset
# saddle 4
startS4 = -90.0+thetaOffset
endS4 = 90.0-thetaOffset
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print "=============================================================="
print "Please select the calculation plane/volume"
print "=============================================================="
plane = input("Enter 1 for xyz volume, 2 for xy plane or 3 for xz plane: ")
print "=============================================================="
if plane == 1:
from grad import calB_xyz
print "Please input the calculation grid parameters"
print "=============================================================="
xMin = raw_input("Please input the xMin value (in meter): ")
xMax = raw_input("Please input the xMax value (in meter): ")
yMin = raw_input("Please input the yMin value (in meter): ")
yMax = raw_input("Please input the yMax value (in meter): ")
zMin = raw_input("Please input the zMin value (in meter): ")
zMax = raw_input("Please input the zMax value (in meter): ")
print "=============================================================="
print "Processing..."
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# grid
xMin =
xMax =
yMin =
yMax =
zMin =
zMax =
dimX =
dimY =
dimZ =
float(xMin)
float(xMax)
float(yMin)
float(yMax)
float(zMin)
float(zMax)
int((xMax-xMin)/gridStep)+1
int((yMax-yMin)/gridStep)+1
int((zMax-zMin)/gridStep)+1
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then = time.time()
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# saddle 1
vOA,vOB = saddle(startS1,endS1,thetaStep,zU1,zL1,radius,j1,k1)
B1 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,zMax,gridStep,
current)
Bh1 = calB_xyz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,zMax,
gridStep,current)
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# saddle 2
vOA,vOB = saddle(startS2,endS2,thetaStep,zU1,zL1,radius,j2,k2)
B2 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,zMax,gridStep,
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current)
Bh2 = calB_xyz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,zMax,
gridStep,current)
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# saddle 3
vOA,vOB = saddle(startS3,endS3,thetaStep,zU2,zL2,radius,j1,k1)
B3 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,zMax,gridStep,
current)
Bh3 = calB_xyz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,zMax,
gridStep,current)
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# saddle 4
vOA,vOB = saddle(startS4,endS4,thetaStep,zU2,zL2,radius,j2,k2)
B4 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,zMax,gridStep,
current)
Bh4 = calB_xyz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,zMax,
gridStep,current)
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# adding up each B component
B = B1 + B2 + B3 + B4
Bh = Bh1 + Bh2 + Bh3 + Bh4
i = B.shape[0]
j = B.shape[1]
k = B.shape[2]
l = B.shape[3]
m = B.shape[4]
BT = s.zeros([i,j,k,l],dtype=float)
BTh = s.zeros([i,j,k,l],dtype=float)
for a in range(0,m):
BT = BT + B[::,::,::,::,a]
BTh = BTh + Bh[::,::,::,::,a]
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BZ = s.transpose(BT[2])
BZh = s.transpose(BTh[2])
G = ((BZ-BZh)/h)
G0 = G[(dimZ-1)/2,(dimY-1)/2,(dimX-1)/2]
delG = (G-G0)*100/G0
delG = s.ravel(delG)
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# printing time and G0
now = time.time()
print ’time =’,now-then
print ’G0 =’,G0*1e3
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elif plane == 2:
from grad import calB_xy
print "Please input the calculation grid parameters"
print "=============================================================="
xMin = raw_input("Please input the xMin value (in meter): ")
xMax = raw_input("Please input the xMax value (in meter): ")
yMin = raw_input("Please input the yMin value (in meter): ")
yMax = raw_input("Please input the yMax value (in meter): ")
z = raw_input("Please input the z value (in meter): ")
print "=============================================================="
print "Processing..."
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# grid
xMin = float(xMin)
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xMax
yMin
yMax
zMin
dimX
dimY
dimZ
=
=
=
=
=
=
=
float(xMax)
float(yMin)
float(yMax)
float(z)
int((xMax-xMin)/gridStep)+1
int((yMax-yMin)/gridStep)+1
1
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then = time.time()
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# saddle 1
vOA,vOB = saddle(startS1,endS1,thetaStep,zU1,zL1,radius,j1,k1)
B1 = calB_xy(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,current)
Bh1 = calB_xy(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,current)
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# saddle 2
vOA,vOB = saddle(startS2,endS2,thetaStep,zU1,zL1,radius,j2,k2)
B2 = calB_xy(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,current)
Bh2 = calB_xy(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,current)
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# saddle 3
vOA,vOB = saddle(startS3,endS3,thetaStep,zU2,zL2,radius,j1,k1)
B3 = calB_xy(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,current)
Bh3 = calB_xy(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,current)
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# saddle 4
vOA,vOB = saddle(startS4,endS4,thetaStep,zU2,zL2,radius,j2,k2)
B4 = calB_xy(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,current)
Bh4 = calB_xy(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,yMax,gridStep,zMin,current)
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# adding up each B component (each coil)
B = B1 + B2
Bh = Bh1 + Bh2
# adding up each B component (each vAB)
i = B.shape[0]
j = B.shape[1]
k = B.shape[2]
l = B.shape[3]
m = B.shape[4]
BT = s.zeros([i,j,k,l],dtype=float)
BTh = s.zeros([i,j,k,l],dtype=float)
for a in range(0,m):
BT = BT + B[::,::,::,::,a]
BTh = BTh + Bh[::,::,::,::,a]
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BZ = s.transpose(BT[2])
BZh = s.transpose(BTh[2])
G = ((BZ-BZh)/h)
G0 = G[(dimZ-1)/2,0,(dimX-1)/2]
delG = (G-G0)*100/G0
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# printing time and G0
now = time.time()
print ’time = ’,now-then
print ’G0 =’,G0*1e3
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# 2D plot using matplotlib
import matplotlib.pyplot as plt
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plt.xlabel(’x (meter)’)
plt.ylabel(’z (meter)’)
x = s.arange(xMin,xMax+gridStep,gridStep)
y = s.arange(yMin,yMax+gridStep,gridStep)
delG = s.reshape(delG,[len(y),len(x)])
cs = plt.contour(x,y,delG,[-20,-15,-10,-5,0,5,10,15,20])
plt.clabel(cs, inline=1, fontsize=10)
plt.title(r’$\bigtriangleup_{\%}$ on $xy$ plane at z = ’+str(zMin))
plt.grid("true")
plt.show()
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delG = s.ravel(delG)
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elif plane == 3:
from grad import calB_xz
print "Please input the calculation grid parameters"
print "=============================================================="
xMin = raw_input("Please input the xMin value (in meter): ")
xMax = raw_input("Please input the xMax value (in meter): ")
y = raw_input("Please input the y value (in meter): ")
zMin = raw_input("Please input the zMin value (in meter): ")
zMax = raw_input("Please input the zMax value (in meter): ")
print "=============================================================="
print "Processing..."
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# grid
xMin =
xMax =
yMin =
zMin =
zMax =
dimX =
dimY =
dimZ =
float(xMin)
float(xMax)
float(y)
float(zMin)
float(zMax)
int((xMax-xMin)/gridStep)+1
1
int((zMax-zMin)/gridStep)+1
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then = time.time()
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# saddle 1
vOA,vOB = saddle(startS1,endS1,thetaStep,zU1,zL1,radius,j1,k1)
B1 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin,zMax,gridStep,current)
Bh1 = calB_xz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,zMin,zMax,gridStep,current)
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# saddle 2
vOA,vOB = saddle(startS2,endS2,thetaStep,zU1,zL1,radius,j2,k2)
B2 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin,zMax,gridStep,current)
Bh2 = calB_xz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,zMin,zMax,gridStep,current)
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# saddle 3
vOA,vOB = saddle(startS3,endS3,thetaStep,zU2,zL2,radius,j1,k1)
B3 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin,zMax,gridStep,current)
Bh3 = calB_xz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,zMin,zMax,gridStep,current)
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# saddle 4
vOA,vOB = saddle(startS4,endS4,thetaStep,zU2,zL2,radius,j2,k2)
B4 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin,zMax,gridStep,current)
Bh4 = calB_xz(vOA,vOB,xMin-h,xMax-h,gridStep,yMin,zMin,zMax,gridStep,current)
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# adding up each B component (each coil)
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B = B1 + B2
Bh = Bh1 + Bh2
# adding up each B component (each vAB)
i = B.shape[0]
j = B.shape[1]
k = B.shape[2]
l = B.shape[3]
m = B.shape[4]
BT = s.zeros([i,j,k,l],dtype=float)
BTh = s.zeros([i,j,k,l],dtype=float)
for a in range(0,m):
BT = BT + B[::,::,::,::,a]
BTh = BTh + Bh[::,::,::,::,a]
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BZ = s.transpose(BT[2])
BZh = s.transpose(BTh[2])
G = ((BZ-BZh)/h)
G0 = G[(dimZ-1)/2,0,(dimX-1)/2]
delG = (G-G0)*100/G0
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# printing time and G0
now = time.time()
print ’time = ’,now-then
print ’G0 =’,G0*1e3
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# 2D plot using matplotlib
import matplotlib.pyplot as plt
plt.xlabel(’x (meter)’)
plt.ylabel(’z (meter)’)
x = s.arange(xMin,xMax+gridStep,gridStep)
z = s.arange(zMin,zMax+gridStep,gridStep)
delG = s.reshape(delG,[len(z),len(x)])
cs = plt.contour(x,z,delG,[-20,-15,-10,-5,0,5,10,15,20])
plt.clabel(cs, inline=1, fontsize=10)
plt.title(r’$\bigtriangleup_{\%}$ on $xz$ plane at y = ’+str(yMin))
plt.grid("true")
plt.show()
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delG = s.ravel(delG)
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else:
print "Wrong input. Please re-run the program."
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# writing output file
f=open("z-maxwell.vtk","w")
f.write("# vtk DataFile Version 3.0\n")
f.write("Magnetic Field of Maxwell Coil\n")
f.write("ASCII\n")
f.write("DATASET STRUCTURED_POINTS\n")
f.write("DIMENSIONS " + str(dimZ) + " " + str(dimY) + " " + str(dimX) + "\n")
f.write("ORIGIN " + str(zMin) + " " + str(yMin) + " " + str(xMin) + "\n")
f.write("SPACING " + str(gridStep) + " " + str(gridStep) + " " + str(gridStep)
+ "\n")
f.write("POINT_DATA " + str(dimZ*dimY*dimX) + "\n")
f.write("SCALARS BZ float\n")
f.write("LOOKUP_TABLE default\n")
for i in range(0,delG.shape[0]):
f.write(str(delG[i]) + "\n")
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f.close()
APPENDIX C
CODE FOR MAXWELL COIL
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#
#
#
#
#
#
file name
written by
date written
description
last modified
:
:
:
:
z-maxwell.py
chew teong han (teonghan@yahoo.com)
15/08/2008
to calculate Bz of z-gradient coil + visualization
+ vtk output
: 21/01/2009
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import time
import scipy as s
from grad import circle
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# universal constant(s)
thetaStep = 1.0
gridStep = 0.01
j = 0.0
k = 0.0
h = 1e-6
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print "=============================================================="
print "Welcome to Magnetic Field Calculation Program for Maxwell Coil"
print "=============================================================="
print "Please input the required parameters"
print "=============================================================="
current = raw_input("Please input the current value (in Ampere): ")
radius = raw_input("Please input the radius value (in meter): ")
d = raw_input("Please input the d value (in meter): ")
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# user-defined variables conversion
current = float(current)
radius = float(radius)
d = float(d)
zU = +d/2
zL = -zU
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print "=============================================================="
print "Please select the calculation plane/volume"
print "=============================================================="
plane = input("Enter 1 for xyz volume, 2 for xz plane or 3 for yz plane: ")
print "=============================================================="
if plane == 1:
from grad import calB_xyz
print "Please input the calculation grid parameters"
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print "=============================================================="
xMin = raw_input("Please input the xMin value (in meter): ")
xMax = raw_input("Please input the xMax value (in meter): ")
yMin = raw_input("Please input the yMin value (in meter): ")
yMax = raw_input("Please input the yMax value (in meter): ")
zMin = raw_input("Please input the zMin value (in meter): ")
zMax = raw_input("Please input the zMax value (in meter): ")
print "=============================================================="
print "Processing..."
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# grid
xMin =
xMax =
yMin =
yMax =
zMin =
zMax =
dimX =
dimY =
dimZ =
float(xMin)
float(xMax)
float(yMin)
float(yMax)
float(zMin)
float(zMax)
int((xMax-xMin)/gridStep)+1
int((yMax-yMin)/gridStep)+1
int((zMax-zMin)/gridStep)+1
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then = time.time()
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# maxwell 1
vOA,vOB = circle(-thetaStep,zU,radius,j,k)
B1 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,zMax,gridStep,
current)
Bh1 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin-h,zMax-h,
gridStep,current)
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# maxwell 2
vOA,vOB = circle(thetaStep,zL,radius,j,k)
B2 =calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin,zMax,gridStep,
current)
Bh2 = calB_xyz(vOA,vOB,xMin,xMax,gridStep,yMin,yMax,gridStep,zMin-h,zMax-h,
gridStep,current)
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# adding up each B component (each coil)
B = B1 + B2
Bh = Bh1 + Bh2
# adding up each B component (each vAB)
i = B.shape[0]
j = B.shape[1]
k = B.shape[2]
l = B.shape[3]
m = B.shape[4]
BT = s.zeros([i,j,k,l],dtype=float)
BTh = s.zeros([i,j,k,l],dtype=float)
for a in range(0,m):
BT = BT + B[::,::,::,::,a]
BTh = BTh + Bh[::,::,::,::,a]
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BZ = s.transpose(BT[2])
BZh = s.transpose(BTh[2])
G = ((BZ-BZh)/h)
G0 = G[(dimZ-1)/2,(dimY-1)/2,(dimX-1)/2]
delG = (G-G0)*100/G0
delG = s.ravel(delG)
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# printing time and G0
now = time.time()
print ’time = ’,now-then
print ’G0 =’,G0*1e3
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elif plane == 2:
from grad import calB_xz
print "Please input the calculation grid parameters"
print "=============================================================="
xMin = raw_input("Please input the xMin value (in meter): ")
xMax = raw_input("Please input the xMax value (in meter): ")
y = raw_input("Please input the y value (in meter): ")
zMin = raw_input("Please input the zMin value (in meter): ")
zMax = raw_input("Please input the zMax value (in meter): ")
print "=============================================================="
print "Processing..."
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# grid
xMin =
xMax =
yMin =
zMin =
zMax =
dimX =
dimY =
dimZ =
float(xMin)
float(xMax)
float(y)
float(zMin)
float(zMax)
int((xMax-xMin)/gridStep)+1
1
int((zMax-zMin)/gridStep)+1
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then = time.time()
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# maxwell 1
vOA,vOB = circle(-thetaStep,zU,radius,j,k)
B1 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin,zMax,gridStep,current)
Bh1 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin-h,zMax-h,gridStep,current)
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# maxwell 2
vOA,vOB = circle(thetaStep,zL,radius,j,k)
B2 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin,zMax,gridStep,current)
Bh2 = calB_xz(vOA,vOB,xMin,xMax,gridStep,yMin,zMin-h,zMax-h,gridStep,current)
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# adding up each B component (each coil)
B = B1 + B2
Bh = Bh1 + Bh2
# adding up each B component (each vAB)
i = B.shape[0]
j = B.shape[1]
k = B.shape[2]
l = B.shape[3]
m = B.shape[4]
BT = s.zeros([i,j,k,l],dtype=float)
BTh = s.zeros([i,j,k,l],dtype=float)
for a in range(0,m):
BT = BT + B[::,::,::,::,a]
BTh = BTh + Bh[::,::,::,::,a]
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BZ = s.transpose(BT[2])
BZh = s.transpose(BTh[2])
G = ((BZ-BZh)/h)
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G0 = G[(dimZ-1)/2,0,(dimX-1)/2]
delG = (G-G0)*100/G0
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# printing time and G0
now = time.time()
print ’time = ’,now-then
print ’G0 =’,G0*1e3
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# 2D plot using matplotlib
import matplotlib.pyplot as plt
plt.xlabel(’x (meter)’)
plt.ylabel(’z (meter)’)
x = s.arange(xMin,xMax+gridStep,gridStep)
z = s.arange(zMin,zMax+gridStep,gridStep)
delG = s.reshape(delG,[len(z),len(x)])
cs = plt.contour(x,z,delG,[-20,-15,-10,-5,0,5,10,15,20])
plt.clabel(cs, inline=1, fontsize=10)
plt.title(r’$\bigtriangleup_{\%}$ on $xz$ plane at y = ’+str(yMin))
plt.grid("true")
plt.show()
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delG = s.ravel(delG)
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elif plane == 3:
from grad import calB_yz
print "Please input the calculation grid parameters"
print "=============================================================="
x = raw_input("Please input the x value (in meter): ")
yMin = raw_input("Please input the yMin value (in meter): ")
yMax = raw_input("Please input the yMax value (in meter): ")
zMin = raw_input("Please input the zMin value (in meter): ")
zMax = raw_input("Please input the zMax value (in meter): ")
print "=============================================================="
print "Processing..."
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# grid
xMin =
yMin =
yMax =
zMin =
zMax =
dimX =
dimY =
dimZ =
float(x)
float(yMin)
float(yMax)
float(zMin)
float(zMax)
1
int((yMax-yMin)/gridStep)+1
int((zMax-zMin)/gridStep)+1
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then = time.time()
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# maxwell 1
vOA,vOB = circle(-thetaStep,zU,radius,j,k)
B1 = calB_yz(vOA,vOB,xMin,yMin,yMax,gridStep,zMin,zMax,gridStep,current)
Bh1 = calB_yz(vOA,vOB,xMin,yMin,yMax,gridStep,zMin-h,zMax-h,gridStep,current)
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# maxwell 2
vOA,vOB = circle(thetaStep,zL,radius,j,k)
B2 = calB_yz(vOA,vOB,xMin,yMin,yMax,gridStep,zMin,zMax,gridStep,current)
Bh2 = calB_yz(vOA,vOB,xMin,yMin,yMax,gridStep,zMin-h,zMax-h,gridStep,current)
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# adding up each B component (each coil)
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B = B1 + B2
Bh = Bh1 + Bh2
# adding up each B component (each vAB)
i = B.shape[0]
j = B.shape[1]
k = B.shape[2]
l = B.shape[3]
m = B.shape[4]
BT = s.zeros([i,j,k,l],dtype=float)
BTh = s.zeros([i,j,k,l],dtype=float)
for a in range(0,m):
BT = BT + B[::,::,::,::,a]
BTh = BTh + Bh[::,::,::,::,a]
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BZ = s.transpose(BT[2])
BZh = s.transpose(BTh[2])
G = ((BZ-BZh)/h)
G0 = G[(dimZ-1)/2,(dimY-1)/2,0]
delG = (G-G0)*100/G0
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# printing time and G0
now = time.time()
print ’time = ’,now-then
print ’G0 =’,G0*1e3
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# 2D plot using matplotlib
import matplotlib.pyplot as plt
plt.xlabel(’y (meter)’)
plt.ylabel(’z (meter)’)
y = s.arange(yMin,yMax+gridStep,gridStep)
z = s.arange(zMin,zMax+gridStep,gridStep)
delG = s.reshape(delG,[len(z),len(y)])
cs = plt.contour(y,z,delG,[-20,-15,-10,-5,0,5,10,15,20])
plt.clabel(cs, inline=1, fontsize=10)
plt.title(r’$\bigtriangleup_{\%}$ on $yz$ plane at x = 0’)
plt.grid("true")
plt.show()
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delG = s.ravel(delG)
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else:
print "Wrong input. Please re-run the program."
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# writing output file
f=open("z-maxwell.vtk","w")
f.write("# vtk DataFile Version 3.0\n")
f.write("Magnetic Field of Maxwell Coil\n")
f.write("ASCII\n")
f.write("DATASET STRUCTURED_POINTS\n")
f.write("DIMENSIONS " + str(dimZ) + " " + str(dimY) + " " + str(dimX) + "\n")
f.write("ORIGIN " + str(zMin) + " " + str(yMin) + " " + str(xMin) + "\n")
f.write("SPACING " + str(gridStep) + " " + str(gridStep) + " " + str(gridStep)
+ "\n")
f.write("POINT_DATA " + str(dimZ*dimY*dimX) + "\n")
f.write("SCALARS BZ float\n")
f.write("LOOKUP_TABLE default\n")
for i in range(0,delG.shape[0]):
f.write(str(delG[i]) + "\n")
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f.close()
APPENDIX D
VTK FILE FORMAT
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#vtk DataFile Version 2.0
Bz in xyz volume
ASCII
DATASET STRUCTURED_POINTS
DIMENSIONS nx ny nz
ORIGIN x y z
SPACING sx sy sz
SCALARS Bz
LOOKUP_TABLE default
data(0)
data(1)
.
.
.
data(n-1)
Explanations:
1.
Line 1 shows the version of the VTK data file (header)
2.
Line 2 shows the user-defined name (title) for the VTK data file
3.
Line 3 shows the type of the VTK data file (ASCII or BINARY)
4.
Line
4
shows
the
dataset
format
(STRUCTURED POINTS,
STRUCTURED GRID, RECTILINEAR GRID, POLYGONAL DATA
or
UNSTRUCTURED GRID)
5.
Line 5 to 7 shows the required field based on the dataset format
93
6.
Line 8 shows dataset attribute format (SCALARS, VECTORS, NORMALS,
TEXTURE COORDINATES, TENSORS or FIELD DATA), followed by userdefined dataset name
7.
Line 9 shows the table that will be used to color-code the dataset
8.
Line 10 and above shows the dataset components based on the type of DATASET
Note that there are other options available for different form of data but for this
research, the above format is sufficient.
APPENDIX E
PUBLICATION A
Journal of
Fundamental
Sciences
Magnetic Field Simulation of Golay Coil
*
Rashdi Shah Ahmad, Amiruddin Bin Shaari, Chew Teong Han
Department of Physics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia.
* Author to whom correspondence should be addressed; E-mail: rashdishah@yahoo.com, amiryeng@yahoo.com and
teonghan@yahoo.com
Received: 30 October 2008
ABSTRACT
In the field of magnetic resonance imaging (MRI), besides the main magnetic field and the radio frequency (RF) system,
the gradient coils system also plays an important role in signal acquisition. Generally, there are 2 transverse gradient
coils and 1 longitudinal gradient coil. In this research, the magnetic field (or gradient) generated by the conventional
transverse coil, namely the Golay coil was mapped. The calculation algorithm of the magnetic field generated was
written in C-programming language, compiled by the GNU-compiler collection (GCC) and was based on a forward
analytical approach by using the Biot-Savart Law. The data was then visualized by Gnuplot. The magnetic field was
successfully calculated and plotted using open-source softwares running under Ubuntu operating system and will
provide valuable data especially for gradient coil designer.
| Magnetic resonance imaging (MRI) | Golay coil | simulation | GCC | Gnuplot |
1. Introduction
Gradient coils play an important role in magnetic resonance imaging (MRI) since they directly affect the
resolution of the signal. There are usually three sets of gradient coils in a MRI system, each generating a
magnetic gradient along each orthogonal axis, namely x, y and z. The longitudinal gradient coil will generate a
magnetic gradient along z-direction which is derived from the shape of Maxwell coil. The two other transverse
gradient coils however consist of double saddle configuration, known as the Golay coil [1], made up from two
saddle coils. The main purpose of gradient coils in a MRI system is to produce a magnetic gradient along each
axis, helping in slice selection, frequency encoding and phase encoding. Technically speaking, the gradient
would be preferred to be as linear as possible over a large volume. However, due to several constraints such as
the inductance and the size of the coil, these few characteristics will have to be traded off with each other.
2. Methodology
The magnetic field generated by a finite current segment can be calculated using an analytical approach by
using the Biot-Savart Law [2]. Any coil shape can be divided into a series of finite current segment and by using
the Biot-Savart Law for finite current segment, the magnetic field generated by the coil at any point can be
Article
Available online at
http://www.ibnusina.utm.my/jfs
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calculated. Equation 1 shows the Biot-Savart Law for a finite current segment [3] with Figure 1 showing the
related parameters.
B=
µ0 I
(cos θ 1 − cos θ 2 )
4πa
(1)
From Figure 1, the magnetic field induced by the finite segment AB at point M can be calculated using
Equation 1. However, before doing any calculation, further vector analysis is required to obtain all the necessary
parameters.
Figure 1: Parameters Related to Finite Segment with Calculation Point, M
B is obtained by taking the cross product of the unit vector of AB and the unit vector
M ( LM ). The vector of the current segment,
AB and its unit vector, û AB , are given by Equation 2 and 3:
The direction of
along a line joining the current segment to the calculation point,
AB = OB − OA
uˆ AB =
Similarly,
AB
(2)
(3)
AB
AM and BM as well as their respective unit vectors are given by Equation 4 to 7:
AM = OM − OA
(4)
BM = OM − OB
(5)
uˆ AM =
uˆ BM =
AM
(6)
AM
BM
BM
(7)
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The cosine of the angles
θ1 and θ 2
are calculated from Equation 8 and 9:
cosθ1 = uˆ AB • uˆ AM
(8)
cos θ 2 = uˆ AB • uˆ BM
(9)
The distance between the calculation point,
Equation 10:
M to the segment is given by the magnitude of LM in
LM = AM − AL
(10)
AL is the projection of AM in the direction of û AB , given by Equation 12:
AL = ( AM • uˆ AB )uˆ AB
(11)
LM = AM − ( AM • uˆ AB )uˆ AB
(12)
Therefore,
The direction of the magnetic field at the calculation point,
M is shown in Equation 13:
uˆ B = uˆ AB × uˆ LM
Finally, the magnetic field at point
(13)
M is obtained as:
B=
µ0 I
4π LM
(cos θ1 − cos θ 2 )uˆ B
(14)
Figure 2 shows the calculation model used in this simulation. The arrows denote the current flow in the
Golay coil. The calculations are done on the xy-plane, divided in grids, shown in Figure 2.
The calculation grid for both xy-plane and xz-plane will be the size of (0.4 m X 0.4 m) with a step of 0.01 m.
A total of five coil parameters will be varied and simulated accordingly, namely the separation between two
saddle coils, a, the length of Golay coil, b (as shown in Figure 2), the arc angle of the coil, θ, the radius of the
coil, r and the separation between each half of saddle coil, c (as shown in Figure 3). For each simulation, only
one parameter will be varied while the rest will be fixed at certain value. The C-code of the program would
iterate the calculation and export the data into a text file which was then visualized by Gnuplot.
3. Results and Discussions
Figure 4 to 9 show the some of the results obtained by the simulation. Figure 4 to 6 show the magnetic
contour plot of the results to show how uniform is the gradient generated by altering one of the coil geometry
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mentioned above. Figure 7 to 9 show the normal xy-plot of the same results to show how linear is the gradient,
the gradient value itself and also the gradient uniformity. Figure 4 to 9 were plotted for an x-range with different
values of y. The graphs were then plotted on the same axis, overlapping each other. The idea was that for
different values of y, if the xy-plots were perfectly overlapped, and then one can consider the gradient was very
uniform along the y-direction.
Figure 2: Calculation Model
Figure 3: Top View of Calculation Model with z-axis pointing out
From Figure 4 and 7, as a increases, the gradient uniformity decreases, the gradient linearity increases
whereas the gradient value decreases. From Figure 5 and 8, as b increases, the gradient uniformity slightly
decreases almost no change in gradient linearity whereas the gradient value slightly changes. From Figure 6 and
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357
9, as c increases, the gradient uniformity decreases, almost no change in gradient linearity whereas the gradient
value decreases.
Figure 4: Magnetic Contour Plot for Various a
Figure 5: Magnetic Contour Plot for Various b
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Rashdi Shah Ahmad et al. / Journal of Fundamental Sciences 4 (2008) 353-361
Figure 6: Magnetic Contour Plot for Various c
Figure 7: xy-plot for various a
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Rashdi Shah Ahmad et al. / Journal of Fundamental Sciences 4 (2008) 353-361
Figure 8: xy-plot for various b
Figure 9: xy-plot for various c
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Other results obtained by varying the r parameter and the θ parameter. As r increases, almost no change in
gradient uniformity, gradient linearity increases whereas the gradient value decreases. As expected, θ = 1200
offered the best gradient uniformity among the values of θ. Figure 10 and 11 show the ideal data plot generated.
Analytically speaking, if the magnetic field mapping obtained, both experimentally or theoretically (through
computer simulation), is as similar to these plots, the better the gradient coil is, in terms of both field linearity
and gradient uniformity.
Figure 10: Ideal Contour Plot for x-Golay Coil
Figure 11: Ideal xy-plot for x-Golay Coil
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361
4. Conclusion
The simulation is considered success with few rooms for improvement including numerical analysis for
more accurate gradient calculations, mapping for magnetic gradient values rather than the magnetic field and etc.
The simulation also proves to be a great tool and to provide assistance especially to gradient coil designers.
5. References and Citations
[1] Adamiak, K., Rutt, B. K., and Dabrowki, W. J., Design of Gradient Coils for Magnetic Resonance Imaging,
IEEE Transactions on Magnetics, Vol 28, pg 2403-2405, 1992
[2] Rashdi Shah Ahmad and Chew, T. H., Algorithm for Magnetic Field Visualization of a Flat Plane Induced
by Finite Dipole Segment Using GCC and Gnuplot, Regional Annual Fundamental Science Seminar, 2007
[3] Sadiku, M., 1995. Elements of Electromagnets. Oxford University Press.
APPENDIX F
PUBLICATION B
Magnetic Field Simulation of Maxwell Coil
1
1
1,*
R. S. Ahmad , A. Shaari and T. H. Chew
1
Department of Physics, Faculty of Science,
Universiti Teknologi Malaysia,
81310 UTM Skudai, Johor,
Malaysia.
*email: teonghan@yahoo.com
Abstract
A configuration of two identical circular coils, carrying
current with anti-parallel direction, separated by a distance of 1.73 times its radius, is known as Maxwell coil.
Such configuration is able to generate a magnetic field
gradient along the axis of the coil. Because of this characteristic, Maxwell coil is widely used in magnetic resonance
imaging (MRI) or nuclear magnetic resonance (NMR) instruments; to generate longitudinal magnetic field gradient.
This research investigates the effect of the geometrical parameters of Maxwell coil to the magnetic field gradient
generated. For this purpose, a computer simulation is written in Python programming language. It is found that with
certain configuration, the magnetic field gradient generated can be improved within certain range compared to the
original configuration of Maxwell coil. Besides that, the
simulation also visualizes a full 3-dimensional magnetic
field gradient generated. In short, the research provides
useful information without actually constructing the physical prototype of gradient coil which maybe costly and time
consuming.
rents with anti parallel direction, separated at a distance
equal to 3 of the coil’s radius.
METHODOLOGY
This simulation will emphasize the mapping of the generated magnetic field gradient in 2 dimensions with the capability of extending to 3 dimensions. Besides contour plotting, data for simulation is extracted for correlation calculation to determine the linearity of the magnetic field gradient
generated.
Current carrying conductor is able to generate magnetic
field according to Biot-Savart’s Law. The Biot-Savart’s
Law can be further derived for finite length current segment
shown in Equation 1[1].
µ0 I
(cos θ 1 − cos θ 2 )
(1)
4πa
The parameters in Equation 1 are related as shown in Figure 1.
B=
Keywords
Gradient Coil, Maxwell Coil, magnetic resonance imaging
(MRI), Python
INTRODUCTION
Magnetic resonance imaging (MRI) is a well known noninvasive imaging technique widely used in medical area
which is based on nuclear magnetic resonance (NMR). In
NMR phenomenon, atomic spin is manipulated by external
magnetic field and excited by the matching radio frequency
pulses based on the Larmor equation. In actual practice, a
main magnetic field is applied to align the spins in the sample with a magnetic field gradient applied in certain direction to “encode” the spins in terms of phases and frequencies. RF system will then transmit RF pulses to excite the
spins. RF pulses released after the spins undergo relaxation
processes are picked up for mapping. Gradient coils are
responsible for generating the magnetic field gradient; one
coil for each x, y and z axis. Conventionally, the Maxwell
coil configuration is used as the z axis gradient coil, known
also as longitudinal gradient coil. A Maxwell coil configuration consists of two identical circular coils carrying cur-
Figure 1: Finite Current Segment AB and Measurement
Point, M
Vector analysis is used in order to obtain all necessary parameters needed to execute the calculations. The idea of
this simulation is to divide the Maxwell coil into a series of
finite current segments. The calculation planes (or volume)
are then split into grids of points. For each point in the grid,
the magnetic field contribution of each current segment is
calculated and summed up. The Biot-Savart’s Law after
undergoing vector analysis is shown in Equation 2 [2].
µ0 I
B=
4π LM
(cosθ 1 − cos θ 2 )uˆ B
(2)
With,
uˆ B = uˆ AB × uˆ LM
(3)
104
Figure 2 and 3 show the simulation model with the arrows
denoting the current direction.
Original Maxwell coil configuration
The result for original Maxwell coil configuration is shown
in Figure 4.
Figure 2: Maxwell Coil in 3D view
Figure 4: Gradient Deviation on xz Plane at 5% interval
for d = 1.73r
Figure 3: Maxwell Coil viewed on xy plane
For data analysis, the simulation is ran for xz plane and yz
plane. No simulation is ran on xy plane since z axis is not
involved (since Maxwell coil in Figure 2 and 3 will only
generate magnetic field gradient along z direction). Simulation will be ran for variation of d as a function of r (radius)
to observe the effect on the gradient generated.
RESULTS AND DISCUSSIONS
Magnetic field gradient along z direction can be defined as
in Equation 4.
The result on yz plane is not shown since the contour plot is
the same as on xz plane because of the symmetry of Maxwell coil. The gradient generated with this configuration is
approximately -0.050 mT/m (subjected to current = 10 A).
The correlation coefficient, r, between z = -0.1 m and z =
+0.1m is -1 (true to 6 decimal points), meaning the magnetic field is perfectly linear within the range. The usable
region within 5% is defined in this research as the ratio of
usable volume within 5% to the total volume covered by the
entire Maxwell coil (which is a cylinder). The usable region
for this configuration is around 5.83%.
Variation of d
The same simulation is ran for various d as a function of r.
Table 1 summarized the results for each variation.
Table 1: Correlation, gradient and usable volume for
various d
d (m)
r
G z (mT/m)
Usable volume (%)
0.8r
-0.999657
-0.065
0.744
(4)
1.2r
-0.999913
-0.066
1.543
The magnetic field linearity in this research is determined
by the correlation coefficient, r, for a certain range. The
closer the value to 1 or -1, the more linear the magnetic
field is. The gradient uniformity is calculated based on
Equation 5 [3].
1.4r
-0.999970
-0.061
3.842
1.6r
-0.999995
-0.055
6.043
1.8r
-1.000000
-0.048
5.205
2.0r
-0.999995
-0.042
4.744
2.2r
-0.999987
-0.036
2.697
∂B
Gz = z
∂z
∆% =
G z ( x , y , z ) − G z (0,0,0 )
G z ( 0 , 0, 0 )
(5)
A good gradient uniformity is within 5% tolerance [4, 5].
Therefore, the contour plot of magnetic field gradient is
usually done in 5% interval to determine the usable region
within acceptable tolerance.
From Table 1, as d increases, the gradient value decreases.
Both r and usable volume, however, increase and the decrease as d is further increased. The usable volume for d =
1.6r is 6.043% which is better than the original Maxwell
coil configuration. If the application is not usable-regioncritical, but gradient-value-critical, the d of Maxwell coil
can be decrease to achieve higher gradient value. This,
however, will decrease the usable volume and also the linearity of the magnetic field. Practically, higher gradient
105
value will result in higher resolution during the imaging
process. Higher usable volume basically means larger usable area for samples whereas higher linearity literally will
affect the accuracy of the image representation.
3 dimensional visualization
The simulation written has the capability of visualizing the
magnetic field or gradient generated in 3 dimensional volume. The main problem, however, is due to the limitation of
32 bit platform. The simulation is written using the vectorized processing method in which resulting in faster execution of code significantly compared to the normal “for”
loop. The disadvantage of using vectorized processing is
the high usage of system memory in which a 32 bit platform
limits it to be around 2 Gigabytes. This is almost equivalent
to calculating around 10 thousand grid points. Anything
beyond this will result in “memory error”. The practical
solution would be switching to 64 bit platform or to reduce
the grid size. Figure 5 show the sample 3 dimensional visualization.
simulation can be ran for a full 3 dimensional analysis in
one shot.
CONCLUSIONS
A simulation has been written in Python programming language to map the magnetic field and gradient generated by
Maxwell coil. The geometrical parameter, d, is found to
affect the magnetic field gradient generated in terms of
linearity and also the usable region. With d = 1.6r compared to original d = 1.73r, the usable region within 5%
gradient deviation can be improved as long as the ratio between the region and overall volume is concerned.
The simulation is written based on the Biot-Savart’s Law
for finite length current segment. This generally mean as
long as the shape of coil is known and can be divided in
series of finite length current segment, the same simulation
code can be used to simulate the magnetic field generated
regardless of the application. Since the whole simulation is
written using open-source software, it can be freely modified and distributed and the most important is that the simulation will provide valuable data without physically construct the actual Maxwell coil.
ACKNOWLEDGMENTS
The authors would like to acknowledge Kuok Foundation
for its financial assistance and Faculty of Science, UTM for
its facilities.
REFERENCES
[1] Sadiku, M., Elements of Electromagnets. Oxford University Press. 1995
Figure 5: 3 Dimensional Contour Plot of Gradient Deviation
Figure 5 is generated by VisIt visualization program developed as open-source software by the Lawrence Livermore
Laboratory. The simulation exported the data calculated
into VTK file format, supported by VisIt (and many more
visualization programs). With 3 dimensional visualization
program and appropriate working platform (64 bit), the
[2] Ahmad, R. S., Shaari, A., and Chew, T. H., Magnetic
Field Simulation of Golay Coil, Journal of Fundamental Sciences, 2008. 4: 353-361
[3] Shi, F. and Ludwig, R., Magnetic Resonance Imaging
Gradient Coil Design by Combining Optimization
Techniques with the Finite Element Method. IEEE
Transactions on Magnetics, 1998. 34(3): 671-683
[4] Di Luzio, S. e. a., A Novel, Cylindrical, Transverse
Gradient Coil Design for Magnetic Resonance Imaging
of Large Samples. Measurement Science Technology,
1998. 9: 1663-1671.
[5] Dodd, S. and Chien, H. Short Planar Gradient Coils for
MR Microscopy Using Concentric Return Paths. Journal of Magnetic Resonance, 2002. 156: 1-9.