RF coil design using circulant and block circulant matrix algebra

Sasidhar Tadanki^{1}

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.

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.

[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.

Fig. 1:Modes vs required
termination value. Mode 13 which is required mode for MRI values resonates for
a capacitance value of 10.52 pf which agrees with published value of 10.46 pf.

Fig. 2: Normalized
current and field plots. Left column is for mode13 (ʎ/2
open resonance) and right column is for mode 1 (ʎ/2 short resonance). A) Port currents B) Current
distribution along axis C) Axial field plot in XY D) Coronal
field plot in XZ E) Sagittal field plot in YZ

Fig. 3: Left)
Schematic diagram of dual resonant dual strip coil Right) photograph of dual resonant dual strip coi

Fig. 4: Bench
measuremnt of dual
resonant dual strip coil tuned and matched for 7T sodium ( 78.6MHz) and
Proton(298 MHz) frequencies

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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