Synopsis
In this work a simple, efficient method to designing a
transmission line volume resonator coil for MR applications is presented. A multiconductor
transmission line is represented as a multiport network using its port
admittance matrix. Closed form solutions for port resonant mode frequencies are
calculated by solving the eigenfunctions of the port admittance matrix using
block matrix and circulant block matrix algebra. Detailed analysis and
simulated results are presented and compared with standard published results. A
dual-tuned surface coil is developed to demonstrate the efficacy of the
proposed method.PURPOSE
The purpose of this
study is to provide a model to design multiconductor
transmission line (MTL) RF coils for MR applications.
METHODS
In
this work, a MTL structure is represented as a multi-port network using its
port admittance matrix. Resonant conditions are obtained by equating the
determinant of port admittance matrix to zero. A mathematical proof to prove that
the solution of the characteristic equation of port admittance matrix is
equivalent to solving source side input impedance can be obtained by writing
transmission chain parameter matrix for twin conductor transmission line and
solving them. Port admittance matrix can be written to take one of the forms
depending on type of MTL structure: a circulant matrix or a circulant block matrix
(where blocks are circulant) or a block circulant circulant block matrix BCCB
(each block is circulant matrix and blocks are arranged as an elements of
circulant matrix). A method to reduce the computations is to diagonalize or do
a direct computation of eigenvalues of the port admittance matrix.
A
circulant matrix can be diagonalized by a simple Fourier matrix where as a BCCB
matrix is diagonalized by using matrices formed from Kronecker product
operation of Fourier matrices [1]. A powerful
technique called “reduced dimension method” can be used to compute the
eigenvalues of circulant block matrix[2]. In the reduced
dimension method, the eigenvalues
of a circulant block matrices are computed as set of eigenvalues of
matrices of reduced dimension. The required reduced dimension matrices are created
using permutation matrix J and polynomial representor of a circulant
matrix. A detailed mathematical proof of reduced dimension method is derived
from the ref [2]. The main
advantage of reduced dimension method is for a 2n+1 MTL structure, computation
of eigenvalues of a 4n by 4n matrix are reduced to computation of eigenvalues
of 2n matrices of size 2 by 2. In addition to reduced computations, the model
also facilitate simplified analytical formulations for resonant conditions.
RESULTS
In
order to demonstrate the effectiveness of the proposed model a twostep approach
was followed. In first step a standard published coil [3] was analyzed
using the proposed model. The resonant conditions obtained are then compared
with the published values and are verified by full wave HFSS simulations. In
second step a new dual tuned dual element coil was designed, constructed and
bench tested.
For convenience
the standard published coil is called “small coil” and it had an inner diameter
of 7.25 cm, an outer diameter of 10.5 cm, and was 15.25 cm long with 12
elements composed of 6.4 mm wide copper strips and was tuned for 200MHz [3]. The
port admittance matrix is computed and arranged as a BCCB as well as a
circulant block matrix. Fig. 1 shows the terminating component values
as computed with both Fourier and reduced dimension methods. Fig.
2 shows HFSS simulations validating
the results.
As
a demonstration of the proposed theory, a new dual element, dual-tuned (DEDT) surface
coil resonating for 7T sodium (78.6 MHz) and proton frequencies (298 MHz) was
developed. Fig.
3 shows the schematic representation
of the coil. The coil consist of two copper strips 12.7mm in width and 152.4mm
in length. Both elements were placed on an arc of radius 140mm describing an
angle of 45 degrees (center to center). Shield is separated from elements by
25.4 mm. The port admittance matrix is computed and arranged as a circulant
block matrix at both frequencies. The specified
reduced dimension matrices are computed and then solved to obtain the termination
values for dual resonance. Fig.
4
shows the bench measurements of the DEDT coil.
DISCUSSION
The new design framework of using port admittance model to predict
resonant conditions offer the advantage of computing the intrinsic port modes,
some of which were not considered in earlier investigations. The model allows
theoretical formulation for unequal terminations, mode mixing and
multi-resonant coil designs.
Acknowledgements
The author would like to thank Prof John Gore, VUIIS and Prof Reinhold
Ludwig, WPI for their encouragement and support.References
[1] P.
J. Davis, Circulant Matrices. Chelsea, 1994.
[2] S. Rjasanow, Linear
Algebra Appl., vol. 202, pp. 55–69, Apr. 1994.
[3] G. Bogdanov and R.
Ludwig, Magn. Reson. Med., vol. 47, no. 3, pp. 579–593, Mar. 2002.