Dongyean Koh^{1}, Jörg Felder^{1}, Chang-Hoon Choi^{1}, and N. Jon Shah^{1,2,3}

A Dual Series (DS) Composite Right/Left Handed (CRLH) Microstrip Transmission Line (MTL) coil for MRI was designed and verified in a commercial 3T Scanner. According to metamaterial theory, we demonstrate the existence of the right-handed leaky mode to be in the UHF region. For design proposes we propose an equivalent circuit parameter extraction method and demonstrate that the phase velocity of the DS-CRLH MTL coil’s guided mode is located in a fast wave region.

The 1-D periodic structure
shown in Fig. 1-(a) can be described in the form of a complex immittance transfer
function with the relations given in Fig. 1-(b) to (d). As can be seen from
Fig. 1-(e) the TL propagation constant has two solutions allowing propagation
of two modes which makes the circuit a Dual-series (DS) CRL^{3-5}. The
dispersion relation gives information about the group/phase velocities as
function of operating frequency. In order to analyze this, it is necessary to
extract the CRLH parameters (L_{s}, C_{s}, L_{p}, and C_{p})
for the equivalent circuit shown in Fig. 1-(a). The serial inductance (L_{s})
and capacitance (C_{s}) can be obtained by extracting the Y_{21}
value from the full EM simulation of the unit cell structure, e.g. using CST
MWS (CST AG, Darmstadt Germany), and calculating the series inductance/capacitance
value by using the reactance definition: L_{s} = Im(1 / Y_{21})
/ 2πf and C_{s} =1/(2πf * Im(1 / Y_{21}) ). The parallel
capacitance (C_{p}) and inductance (L_{p}) can be freely chosen
to suit the desired TL inductance^{6}. Finally it has to be verified
that the chosen dual series reactance, Im(Z_{DS}), is a monotonically
increasing function that satisfies the Foster theorem^{7}.

On the basis of the theory
presented above, a λ/40
length CRLH
unit-cell was designed and used to construct a TL coil element shown in Fig. 2-(b).
Dimensions are 4.25cm for the unit cell and the entire coil spans 6 unit cells
with 5mm gap between shield and coil. 10pF and 3.9pF
capacitors were used in parallel to tune
the coil to 123.2MHz. A
5.6pF capacitor was used to match the element to 50Ω. The coil response
was characterised on the bench by measuring the return loss (S_{11}) using a
vector network analyzer. MR experiments were carried out on a 3T clinical MRI scanner
using a uniform cylinder phantom (outer diameter = 11cm, length = 25cm). A
standard 2D FLASH sequence (TR/TE = 65ms/2.81ms, matrix size = 256x256, FOV
= 250x250mm^{2} and slice thickness = 3mm) was used to acquire central
transverse, coronal and sagittal images.

^{ }

1. Adriany, G., et al., Transmit and receive transmission line arrays for 7 Tesla parallel imaging. Magnetic resonance in medicine, 2005. 53(2): p. 434-445.

2. Shajan, G., et al., Design and evaluation of an RF front‐end for 9.4 T human MRI. Magnetic resonance in medicine, 2011. 66(2): p. 594-602.

3. Damm, C., Artificial Transmission Line Structures for Tunable Microwave Components and Microwave Sensors. 2010: Shaker.

4. Lai, A., T. Itoh, and C. Caloz, Composite right/left-handed transmission line metamaterials. IEEE microwave magazine, 2004. 5(3): p. 34-50.

5. Caloz, C., Dual composite right/left-handed (D-CRLH) transmission line metamaterial. IEEE microwave and wireless components letters, 2006. 16(11): p. 585-587.

6. Grover, F.W., Inductance calculations: working formulas and tables. 1946: New York : D. Van Nostrand Co.

7. Foster, R.M., A reactance theorem. Bell Labs Technical Journal, 1924. 3(2): p. 259-267.

Circuit diagram of the CRLH
structure and its corresponding analytic solution: (a) Equivalent circuit
diagram, design equations expressing (b) impedance, (c) propagation constant, (d)
immittance and (e) dispersion relation. Z_{ω}, Z_{CRLH}, γ_{ω}, γ_{CRLH}, μ, and ε are plane wave impedance,
CRLH transmission line (TL) impedance, plane wave propagation constant, CRLH TL
propagation constant, permeability and permittivity, respectively.

(a) Picture of the coil on the
phantom, (b) picture of λ/40, 6 unit cell CRLH TL coil, (c) top and (d) bottom-view of the
physical structure.

Dispersion diagrams of the
CRLH parameters (a) reactance (b) propagarion
constant (c) Dispersion Diagram (d) Ratio between Series and Pararrel resonant frequency

Measured return loss

Acquired transverse, coronal
and sagittal MR Images and the corresponding
profiles.