The document presents techniques for designing multiple resonant multiconductor transmission line resonators. It discusses analyzing port admittance matrices using block matrix algebra methods like the 2n-port method and reduced dimension method. These methods decompose large port matrices into smaller matrices to efficiently determine resonance conditions. The document reviews existing multiconductor transmission line resonator designs and motivates developing analytical models for their resonance conditions.

1. Multiple resonant multiconductor transmission line
resonator design using circulant block matrix algebra
Multiple Resonant Multiconductor Transmission line
Resonator Design using Circulant Block Matrix Algebra
Sasidhar Tadanki

2. Outline for this presentation
• Introduction
• Basics of MRI Scanner and RF Coils
• Review
• State-of-the-art techniques to design
RF coils
• Review of existing MTL resonators
• Problem statement
• Theory
• Port admittance matrix
• Block matrix algebra
• 2n-port method
• Reduced Dimension method
• Analysis
• 2n-port method for equal and unequal
admittance terminations
• Reduced dimension method for dual
CMTL
• Results
• Conclusion

3. 3
• MRI is a noninvasive imaging
technique based on the NMR
principle
• MRI uses static magnetic fields,
time varying EM fields, and a
digital computer to produce
images of soft tissues
• MRI scanners are usually
expensive.
• MRI scanners can be
claustrophobic and can be loud.
• Not ideal for people with
implants and pacemakers
MRI is particularly useful
 In imaging soft tissue structures, such as ligaments and cartilage
 organs such as the brain, musculoskeletal, heart, and eyes
 Functional imaging of the brain (fmri)
Introduction: Magnetic Resonance Imaging(MRI)

4. 4
Introduction: MRI Instrumentation
Magnet
Produces uniform static magnetic field
with a strong Z component
Shim coils
Used to correct any inhomogeneities in
the main magnet’s field
Gradient coils
Gradient coils create a controlled non-
uniform magnetic fields in space which
are used to encode the spatial spin
density information.
RF coils
RF coils produce time-varying EM
fields with predominant x and y
components to perturb the nuclear spins
or receive signals from the relaxing
spins
Z axis
X axis
Y axis

5. 5
Introduction: RF coils
• RF coils are key components of a MRI clinical
scanner
• RF coils excite the spin magnetization, or
• Receive signals from the relaxing spins.
• RF coils are classified
• Based on their resonance conditions
• Single resonant or Multiresonant
• Based on their mechanical configurations
• Volume Coil or Surface Coil
• Based on their operations
• Transmit only
• Receive only
• Transmit/Receive

6. 6
Introduction: Why RF coil design is a non trivial task?
• RF coils needs to be efficient
• During Transmit cycle they need to produce a strong B1 fields with a little input power
• For commercial scanners: Input power will be in KW’s and output will be in microtesla (uT)
• During receive cycle they need to detect tiny fluctuations of spin magnetizations
• For commercial scanners: Induced signal voltages will be on the order of nanovolts to microvolts
• RF coils are limited by mechanical constraints
• RF coils need to fit anatomical profiles
• At higher fields, the wavelength becomes comparable to coil dimensions, requiring a shielded coil
design or a transmission line resonator design
• RF coils need to operate under limited field modes or configurations
• RF coils should have minimum E-fields
• RF coils need to produce uniform or characterizable fields

7. 7
Introduction: Motivation & Objective
To develop an analytical
model to determine
resonance conditions for
RF coils
Motivation • To present a simple design approach for
single and multiple resonant MTL coils
• Validate the design approach by constructing
a multiple resonant volume MTL coil and a
multiple resonant surface MTL coil
Objective of this research
• Review of existing techniques to determine the
resonant conditions
• Review of existing MTL resonators
Literature survey

8. 8
Techniques Explanation Disadvantages
1
Input
impedance/admittance
method
At the input port
canceling the reactive part of input
impedance/admittance Utilize iterative and
numerical techniques to
arrive at resonant
conditions
2 Ladder mesh network
Writing ladder mesh network
equations (like KVL, KCL) and
solving them for resonant
conditions
State-of-the-art techniques to determine the resonant conditions
Review: Techniques to determine the resonant conditions

9. 9
Configuration MTL structure Termination Example
1 Single resonant MTL
Multiconductors
surrounded by a
shield
Single-valued
2 Double resonant MTL Multi-valued
3 Multichannel MTL
decoupling and
terminating networks
Review of existing MTL resonators
Review: MTL resonators designs

10. Problem statement
10
How to efficiently determine the
terminating conditions for different
resonant modes of multiconductor
transmission line structures

11. Theory: Port admittance matrix
11
• An MTL structure with terminations can be represented with a port admittance matrix
• The eigenvalues of the port admittance matrix will give the conditions for resonance
• A simple mathematical proof can be shown with a 2-conductor transmission line terminated with shunt
capacitors.

12. Theory: Port admittance matrix
12
• Why port admittance matrix?
• To reduce the complexity of computations by utilizing matrix algebra
• To allow the formulation of analytical equations for conditions of resonance
• Open port admittance matrix can be generated by employing
• 3D full wave EM simulations
• Requires expensive commercial software like HFSS, CST studio
• Using MTL theory and 2D EM simulations
• More economical but requires homegrown code
• Experimentally determining for a given structure the scattering (S) parameters
• Optimization will become a problem

13. Theory: Port admittance matrices
13
Port admittance matrix and corresponding method of determining efficiently its eigenvalues
Port admittance matrix
Method to simplify the eigenvalue
determination
1 Circulant matrix Fourier matrix
2 Block circulant circulant block matrix(BCCB) 2n-port method
3 Circulant block matrix (CB) Reduced dimensional method

14. Theory: Block matrix algebra
14
• Circulant Matrix
Matrix in which each row vector is shifted one
element to the right relative to the preceding
row vector
• Block Matrix
Each element is a matrix
• Fourier matrix
Fourier matrix is an expression of the discrete
Fourier transform (DFT) as a transformation
matrix
𝐀3𝑥3 =
1 2 3
3 1 2
2 3 1
𝐁 𝟐𝒙3 =
𝐀 𝐁
𝐂 𝐃
=
1 2 3
4 5 6
7 8 9
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑘 𝑙 𝑚
𝑜 𝑝 𝑞
𝑟 𝑠 𝑡
11 12 13
14 15 16
17 18 19
𝑭 𝒏 =
1
2
𝑛
1 1 ⋯ 1 1
1 𝜔1
⋱ 𝜔 𝑛−2
𝜔 𝑛−1
⋮ ⋱ ⋱ ⋱ ⋮
1 𝜔 𝑛−2
⋱ 𝜔 𝑛−2 𝑛−2
𝜔 𝑛−2 𝑛−1
1 𝜔 𝑛−1
⋯ 𝜔 𝑛−2 𝑛−1
𝜔 𝑛−1 𝑛−1

15. 15
• Block circulant (BC) matrix
Block matrix in which blocks are arranged in
circulant fashion
𝑩𝑪 𝟐𝒙𝟑 =
𝐀 𝐁
𝐁 𝐀
=
1 2 3
4 5 6
7 8 9
𝑎 𝑏 𝑐
𝑐 𝑎 𝑏
𝑏 𝑐 𝑎
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
1 2 3
4 5 6
7 8 9
• Block circulant with circulant block (BCCB)
Block matrix in which blocks are arranged in
circulant fashion and each block is a circulant
matrix
• Circulant block (CB) matrix
Block matrix in which each block is a circulant
matrix
𝑩𝑪𝑪𝑩 𝟐𝒙𝟑 =
𝐀 𝐁
𝐁 𝐀
=
1 2 3
3 1 2
2 3 1
𝑎 𝑏 𝑐
𝑐 𝑎 𝑏
𝑏 𝑐 𝑎
𝑎 𝑏 𝑐
𝑐 𝑎 𝑏
𝑏 𝑐 𝑎
1 2 3
3 1 2
2 3 1
𝑪𝑩 𝟐𝒙𝟑 =
𝐀 𝐁
𝐂 𝐃
=
1 2 3
3 1 2
2 3 1
𝑎 𝑏 𝑐
𝑐 𝑎 𝑏
𝑏 𝑐 𝑎
𝑘 𝑙 𝑚
𝑚 𝑘 𝑙
𝑙 𝑚 𝑘
11 12 13
13 11 12
12 13 11
Theory: Block matrix algebra
Theory

16. 16
• Matrix Kronecker product
Kronecker product, denoted by ⊗, is a tensor product of matrices A and B
Theory: Block matrix algebra
𝐀 ⊗ 𝐁 =
𝑎 𝑏
𝑐 𝑑
⊗
1 2 3
3 1 2
2 3 1
=
𝑎𝐁 𝑏𝐁
𝑐𝐁 𝑑𝐁
=
𝑎1 𝑎2 𝑎3
𝑎3 𝑎1 𝑎2
𝑎2 𝑎3 𝑎1
𝑏1 𝑏2 𝑏3
𝑏3 𝑏1 𝑏2
𝑏2 𝑏3 𝑏1
𝑐1 𝑐2 𝑐3
𝑐3 𝑐1 𝑐2
𝑐2 𝑐3 𝑐1
𝑑1 𝑑2 𝑑3
𝑑3 𝑑1 𝑑2
𝑑2 𝑑3 𝑑1
• Polynomial representor of a circulant matrix
Polynomial representor of a circulant is 𝑝 𝛾 𝜔 = c1 + c2ω + c3ω2
+ ⋯ + cnωn−1
• Circulant block matrix
𝐀 𝐉 =
𝐚 𝟏𝟏 𝐉 ⋯ 𝐚 𝟏𝐦 𝐉
⋮ ⋱ ⋮
𝐚 𝐦𝟏 𝐉 ⋯ 𝐚 𝐦𝐦 𝐉
where 𝐚 𝐤𝐥 𝐉 is the polynomial representor of a circulant block located at kth row and lth column of A and J is
the permutation matrix of length n by n

17. 17
• A circulant matrix A can be converted into a diagonal matrix
𝒅𝒊𝒂𝒈 𝑴 = 𝑭 𝒎 𝐀(𝑭 𝒎 )∗
• A BCCBm.n matrix A can be converted into a diagonal matrix
𝒅𝒊𝒂𝒈 𝑴 = 𝑭 𝒎 ⊗ 𝑭 𝒏 𝐀(𝑭 𝒎 ⊗ 𝑭 𝒏 )∗
𝐀 =
1 5
5 1
3 9
9 3
3 9
9 3
1 5
5 1
𝐌 = 𝐅𝟐 ⊗ 𝐅𝟐 𝐀(𝐅𝟐 ⊗ 𝐅𝟐 )∗
↔
18 0
0 −10
0 0
0 0
0 0
0 0
−6 0
0 2
Theory: 2n-Port method

18. 18
• A CBm,n matrix A can be converted into block diagonal matrix M (where each block is a diagonal matrix)
𝐌 = 𝐅 𝐦 ⊗ 𝐅𝐧 𝐀(𝐅 𝐦 ⊗ 𝐅𝐧 )∗
𝐀 =
1 5
5 1
3 9
9 3
11 13
13 11
23 29
29 23
𝐌 = 𝑭 𝟐 ⊗ 𝑭 𝟐 𝑨(𝑭 𝟐 ⊗ 𝑭 𝟐 )∗
↔
47 0
0 −9
−17 0
0 3
−29 0
0 −1
11 0
0 −1
Theory: 2n-Port method
• The determinant of a block matrix X of the form
𝐀 𝐁
𝐂 𝐃
can be given as
det(X) = det(D)det(A-BD-1
C)
If CD = DC, then det(X) = det(AD-BC)
If A, B, C and D are diagonal matrices, then det(X) = 𝑖=1
𝑛
𝑎𝑖𝑖 𝑑𝑖𝑖 − 𝑏𝑖𝑖 𝑐𝑖𝑖

19. To compute eigenvalues and eigenvectors of a circulant block matrix CBm,n A(J),
• The matrix A(J) is decomposed into matrices of smaller dimensions,
• The elements of the smaller dimensional matrices are generated from polynomial representors of
circulant blocks, which are located at the corresponding locations in A(J)
• Eigenvalues of A(J) are the union of eigenvalues of smaller dimensional matrices
• Eigenvectors would be the Kronecker product of eigenvectors of smaller dimensional matrices with
eigenvectors of permutation matrix J
• A detailed mathematical explanation is given in the dissertation
19
Theory: Reduced dimension method

20. Reduced dimension method
𝑪𝑩 𝟐𝒙𝟒=
𝐀 𝐁
𝐂 𝐃
=
1 2 3 5
5 1 2 3
3 5 1 2
2 3 5 1
5 9 11 17
17 5 9 11
11 17 5 9
9 11 17 5
5 9 11 17
17 5 9 11
11 17 5 9
9 11 17 5
1 2 3 5
5 1 2 3
3 5 1 2
2 3 5 1
𝑪𝑩 𝟐𝒙𝟒 is reduced into 4 matrices of size 2 by 2 by the following operations
1 . 𝜔0
+ 2 . 𝜔1
+ 3 . 𝜔2
+ 5 . 𝜔3
5 . 𝜔0
+ 9 . 𝜔1
+ 11 . 𝜔2
+ 17 . 𝜔3
5 . 𝜔0
+ 9 . 𝜔1
+ 11 . 𝜔2
+ 17 . 𝜔3
1 . 𝜔0
+ 2 . 𝜔1
+ 3 . 𝜔2
+ 5 . 𝜔3
∀ 𝜔 = 1, 1𝑖 , −1, −1𝑖 T
20
Theory: Reduced dimension method example

21. Reduced dimension matrix for a corresponding eigenvalue
𝜔 = 1 is
11 42
42 11
its eigenvalues are
−31
53
& Eigenvectors:
− 2 2
2 2
𝜔 = 1𝑖 is
−2 + 3𝑖 −6 + 8𝑖
−6 + 8𝑖 −2 + 3𝑖
its eigenvalues are
4 − 5𝑖
−8 + 11𝑖
& Eigenvectors:
− 2 2
2 2
𝜔 = −1 is
−3 −10
−10 −3
its eigenvalues are
−13
7
& Eigenvectors:
− 2 2
2 2
𝜔 = -1𝑖 is
−2 − 3𝑖 −6 − 8𝑖
−6 − 8𝑖 −2 − 3𝑖
its eigenvalues are
4 + 5𝑖
−8 − 11𝑖
& Eigenvectors:
− 2 2
2 2
21
Theory: Reduced dimension method example

22. • The overall eigenvalues are
−31, 53 , 4 − 5𝑖 , −8 + 11𝑖 , −13 , 7 , 4 + 5𝑖 , −8 − 11𝑖 T
• The corresponding eigenvectors are found according to
− 2 2
2 2
⊗
0.5 −0.5 −0.5 0.5
−0.5 −0.5𝑖 0.5𝑖 0.5
0.5 0.5 0.5 0.5
−0.5 0.5𝑖 −0.5𝑖 0.5
22
Theory: Reduced dimension method example

23. Analysis: Introduction
23
• To demonstrate the utility of the 2n-port method, 3 special cases of MTL structures are considered
Open port admittance
matrix
Termination Configuration Validation
1
Block circulant circulant
block (BCCB) matrix
Equal value admittance Single resonance Standard MTL
2 Unequal value admittance Single resonance Standard MTL
3 Unequal value admittance Multiple resonance New DTDE coil
Open Port
admittance matrix
Termination Configuration Validation
1
Circulant block
(CB) matrix
Unequal value admittance Multiple resonance
New Dual CMTL
coil
• To demonstrate the utility of the reduced dimension method, the following MTL structure is considered

24. Analysis: 2n-Port method for equal admittance terminations
24
• The port admittance matrix is a BCCBm.n matrix with equal terminations of all elements
𝐘 =
Y11 Y12
Y12 Y11
Y13 Y14
Y14 Y13
Y13 Y14
Y14 Y13
Y11 Y12
Y12 Y11
+
YT 0
0 YT
0 0
0 0
0 0
0 0
YT 0
0 YT
𝐌 + 𝐘𝐬𝐋 =
M11 0
0 M22
0 0
0 0
0 0
0 0
M33 0
0 M44
+
YT 0
0 YT
0 0
0 0
0 0
0 0
YT 0
0 YT
𝐷𝑒𝑡 𝐌 + 𝐘𝐬𝐋 =
𝑖=1
2𝑛
Mii + YT
• The kth eigenvalue is the kth term in 𝐷𝑒𝑡 𝐌 + 𝐘𝐬𝐋
• The kth eigenvector is the kth row vector of its diagonalizing matrix Fm ⊗ Fn
• The required terminating value can be determined by solving the kth eigenvalue of 𝐌 + 𝐘𝐬𝐋
• After diagonalization, 𝐌 + 𝐘𝐬𝐋 becomes a diagonal matrix

25. Analysis: 2n-Port method for unequal admittance terminations
25
• The port admittance matrix is a CBm.n matrix for different terminating admittance on either end
𝐘 =
Y11 Y12
Y12 Y11
Y13 Y14
Y14 Y13
Y13 Y14
Y14 Y13
Y11 Y12
Y12 Y11
+
YT1 0
0 YT1
0 0
0 0
0 0
0 0
YT2 0
0 YT2
𝐌 + 𝐘𝐬𝐋 =
M11 0
0 M22
0 0
0 0
0 0
0 0
M33 0
0 M44
+
a 0 b 0
0 a 0 b
b 0 a 0
0 b 0 a
𝐷𝑒𝑡 𝐌 + 𝐘𝐬𝐋 =
𝑖=1
𝑛
Mi,i + 𝑎 Mn+i,n+i + 𝑎 − 𝑏 2
• The required terminating values can be found by solving the kth eigenvalue of 𝐌 + 𝐘𝐬𝐋 , assuming a linear
relationship between YT1 and YT2
• After diagnolazition, 𝐌 + 𝐘𝐬𝐋 becomes a diagonal matrix
• The kth eigenvalue is the kth term in 𝐷𝑒𝑡 𝐌 + 𝐘𝐬𝐋
where a =
(YT1+YT2)
2
and b =
(YT1−YT2)
2

26. Analysis: 2n-Port method for unequal admittance terminations
26
• The port admittance matrix is a CBm.n matrix of different terminating admittance on each end
𝐘 =
Y11 Y12
Y12 Y11
Y13 Y14
Y14 Y13
Y13 Y14
Y14 Y13
Y11 Y12
Y12 Y11
+
YT1 0
0 YT1
0 0
0 0
0 0
0 0
YT2 0
0 YT2
• The terminating values are obtained by solving the kth eigenvalue of 𝐌 + 𝐘𝐬𝐋 at both frequencies
• In mode space, 𝐌 + 𝐘𝐬𝐋 becomes a diagonal matrix
• The kth eigenvalue is the kth term in 𝐷𝑒𝑡 𝐌 + 𝐘𝐬𝐋
𝐌 + 𝐘𝐬𝐋 =
M11(𝜔) 0
0 M22(𝜔)
0 0
0 0
0 0
0 0
M33(𝜔) 0
0 M44(𝜔)
+
a(𝜔) 0 b(𝜔) 0
0 a(𝜔) 0 b(𝜔)
b(𝜔) 0 a(𝜔) 0
0 b(𝜔) 0 a(𝜔)
𝐷𝑒𝑡 𝐌 + 𝐘𝐬𝐋 =
𝑖=1
𝑛
Mi,i(𝜔) + 𝑎(𝜔) Mn+i,n+i(𝜔) + 𝑎(𝜔) − 𝑏(𝜔) 2
where a =
(YT1+YT2)
2
and b =
(YT1−YT2)
2

27. 27
• Open port admittance 𝐘 and termination matrix 𝐘𝐋 are given by
𝐘 =
𝐘 𝐀𝐀 𝐘 𝐀𝐁
𝐘 𝐀𝐁 𝐘 𝐀𝐀
𝐘 𝐀𝐂 𝐘 𝐀𝐃
𝐘 𝐀𝐃 𝐘 𝐀𝐂
𝐘𝐂𝐀 𝐘𝐂𝐁
𝐘𝐂𝐁 𝐘𝐂𝐀
𝐘𝐂𝐂 𝐘𝐂𝐃
𝐘𝐂𝐃 𝐘𝐂𝐂
& 𝐘𝐋 =
𝐘 𝐓𝟏 + 𝐘 𝐓 𝟎 −𝐘 𝐓 𝟎
𝟎 𝐘 𝐓𝟏 + 𝐘 𝐓 𝟎 −𝐘 𝐓
−𝐘 𝐓 𝟎 𝐘 𝐓𝟐 + 𝐘 𝐓 𝟎
𝟎 −𝐘 𝐓 𝟎 𝐘 𝐓𝟐 + 𝐘 𝐓
• YT1 is termination on wide strips
• YT2 is termination on narrow strips
• YT is admittance between the strips
Analysis: Port admittance matrix for multiresonant CMTL
• 𝐘 cannot be diagonalized by Kronecker products because of the above observation
• There are 3 unknowns ( YT1 ,YT2 & YT) and two equations (with two frequencies) can be obtained from a
smaller dimensional matrix
𝑌 =
𝑦11 𝜔 + 𝑦 𝑇1 𝜔 y12 𝜔
y21 𝜔 𝑦22 𝜔 + 𝑦 𝑇2 𝜔

28. 28
In order to demonstrate the usefulness of the proposed methods, the following approaches were applied
Results: Implementation
2n-Port method Reduced dimension method
• A standard single resonant published coil with
equal and unequal terminations are studied
• Results were compared with the published
values and verified by HFSS EM simulations
• A standard single resonant published coil with equal
terminations are analyzed
• Two new dual tuned head coils are simulated and
constructed
• Phosphorus-31 (165.09 MHz) and Sodium-23 (105.9
MHz)
• Sodium-23 (105.9 MHz) and Oxygen-17 (54.3MHz)
• A new dual tuned dual element (DTDE) surface
coil design is proposed and developed
• Role of inter-strip lumped elements for
optimization of mode separation and field
profiles are analyzed

29. 29
Small Coil Big Coil
Length (strips and former) 152.4 mm 152.4 mm
Outer diameter 105 mm 177.8 mm
Inner Diameter 72.5 mm 133.3 mm
Number of elements 12 12
Strip Width 6.4 mm 21.6 mm
Copper Thickness 1.5 mil 1.5 mil
Former Wall Thickness 2.8mm 2.8 mm
[1] Bogdanov G, Ludwig R. "Coupled microstrip line transverse electromagnetic resonator model for high- field magnetic resonance imaging." Magn Reson Med. 2002 Mar;47(3):579-93.
Single resonant MTL volume coil parameters [1]
Results: Validation of 2n port model using MTL volume coils

30. 30
Results: Frequency vs termination for different modes
Variation of terminating capacitance as a function of terminating capacitance ratio
𝑪 𝑻𝟏
𝑪 𝑻𝟐

31. 31
Results: Port Current distribution
equal admittance mode1
anti-rotational mode
equal admittance mode13
co-rotational mode
unequal admittance mode1
anti-rotational mode
unequal admittance mode13
co-rotational mode
Port Current distribution
Axial field plots in XY plane

32. 32
Results: Current distribution along the length of coil
Coronal field plots in XZ plane
Sagittal plots in YZ plane
equal admittance mode1
anti-rotational
equal admittance mode13
co-rotational mode
unequal admittance mode1
anti-rotational mode
unequal admittance mode13
co-rotational mode

33. 33
𝐘 =
Y11 Y12
Y12 Y11
Y13 Y14
Y14 Y13
Y13 Y14
Y14 Y13
Y11 Y12
Y12 Y11
+
YT1 0
0 YT1
0 0
0 0
0 0
0 0
YT2 0
0 YT2
Results: Dual tuned dual-element (DTDE) coil
Schematic of the proposed dual tuned dual element coil. Both elements are
terminated with capacitors
Front view of the dual element coil loaded with a phantom
Parameter Value
Length of Coil 152.4 mm
Inner diameter of coil 140 mm
Shield diameter 152.4 mm
Number of elements 2
Width of elements 12.7 mm
1H Proton Frequency 298 MHz
23Na Sodium Frequency 78.6MHz
DTDE coil parameters
• Port admittance matrix 𝐘 is

34. 34
Results: Dual tuned dual-element coil analysis
23Na image of a loading phantom on axial plane
1H image of an oil phantom 1H image of a saline loading phantom
Network analyzer measurements for the proposed DTDE coil
• The following capacitance values are predicted
• 166.2 pF for element 1, and
• 10.32 pF for element 2
• The obtained resonant frequencies using the
eigenmode solver, as well as the constructed coil,
are within 1 % of error tolerance

35. Dimensions of proposed coils
35
Parameter P31- Na23 Na23 – O17
Length of Coil 7 in 7 in
Inner Diameter of coil 9.5 in 9.5 in
Middle Strip diameter 10 in 10 in
Outer Diameter of coil 12.5 in 12.5 in
Number of Inner Strips 12 12
Number of Outer Strips 12 12
Width of inner Strips 1.67 in 0.567 in
Width of middle strips 1.67 in 0.567 in
Resonant High frequency (P31) 162.09 MHz ( Na23) 105.9 MHz
Resonant Low frequency ( Na23) 105.9 MHz (O17) 54.3MHz
Results: Proposed designs for multiresonant CMTL
3D CAD representation of dual tuned coil.
Photographs of dual tuned head coil (left).
Back side and front sight (right)

36. 36
Wide strip
termination
Narrow strip
termination
capacitance between
wide strip and narrow
strip
12.5 pF 31.24 pF 30 pF
6.95 pF 34.71 pF 35 pF
3.16 pF 36.38 pF 40 pF
-0.18 pF 37.15 pF 46 pF
-3.52 pF 25.89 pF 79 pF
3.46 pF 15.15 pF 87 pF
Termination values for the P31- Na23 coil as a function of
lumped capacitor values connecting narrow and wide strips
Results: Effect of inter-strip admittance
Wide strips are terminated with high
impedance and narrow strips with low
impedance capacitors
Narrow strips are terminated with high
impedance and wide strips with low
impedance capacitors

37. 37
Phosphorus channel of the P31- Na23 coil
Results: HFSS simulations for multiresonant CMTL
Oxygen channel of the Na23 – O17 coil Sodium channel of the Na23 – O17 coil
Sodium channel of the P31- Na23 coil

38. Manually shimmed Bo homogeneity values in terms of FWHM
38
Na23 Channel P31 Channel O17 Channel
P31 – Na23 Coil 8.26 Hz / 0.08 ppm 4.45 Hz/0.03 ppm
Na23 – O17 Coil 9.21 Hz/0.09 ppm 56.90 Hz/1.05 ppm
SNR Comparison
Na23 Channel P31 Channel O17 Channel
P31 – Na23 Coil 31.966 5.045 --
Na23 – O17 Coil 34.384 -- 10.75
CMRR Na23 Coil 36.2595 -- --
CMRR – O17 Coil -- -- 13.6
Results: Multiresonant CMTL scanner measurements

39. 39
Results: Na23 SNR maps for different designs
SNR map of sodium channel of the Na23 – O17
coil
SNR map of sodium channel of the P31- Na23
coil
SNR map of CMRR single tuned sodium coil

40. 40
Results: O17 SNR maps for different designs
SNR map of oxygen channel of the Na23 – O17 coil SNR map of CMRR single tuned oxygen coil

41. 41
Results: P31 SNR maps for the P31 –Na23 coil
SNR map of phosphorus channel of the P31- Na23 coil

42. 42
Major
contributions
in thesis
1. Introduction of port
admittance method for finding
resonant conditions
2. Application of Kronecker
product of Fourier matrices
3. Reduced dimension method
1. Dual Tuned Dual Element Coil
2. Multiresonant CMTL design
1. Predicts all port resonant
modes
2. Effect of unequal terminations
on resonant conditions
3. Role of inter-strip lumped
elements for optimization of
mode separation and field
profiles
Conclusion

43. In my research, I proposed an efficient way to determine resonant conditions.
Another alternative technique is to run full-wave 3D EM simulations using solvers like HFSS.
For example, Simulation of a Dual CMTL coil in HFSS required meshing up to 1 million mesh
elements, and 58 minutes of CPU time on a CPU running on an I7 processor with an 8GB RAM
memory.
For my model, generation of per-unit values, computation of port admittances and determination of
component values can be done in few minutes using Matlab on a CPU running on an I7 processor.
43
Conclusion

44. 44
Future
applications
and unsolved
problems
1. Partial volume Coils
2. Planer coils
3. Volume coils with unequal
conductor lengths
1. Circulant block matrix applications
2. Transmission line resonators applications
1. Interleaved multi-resonant coils
with unequal elements
2. Interleaved multi-resonant coils
with uneven number of
elements
3. Mode Mixing
Conclusion

45. 45