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Application of an RF Current Mirror for MRI Transmit Coils
Roland Müller1, Tobias Lenich1, Evgeniya Kirilina1, and Harald E. Möller1

1Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany

### Synopsis

Some types of MRI transmit coils (e.g. Helmholtz coils) require equal currents in different coil elements. We present a novel feeding concept based on a passive RF current mirror, which ensures equal currents even if the loading and tuning of individual elements differ. Analytical equations are given for the dimensioning. It is demonstrated by simulations and experiments that the concept is viable, especially for ultra-high field imaging.

### Introduction

It is well known that some radio-frequency (RF) coils require equal currents in different elements. This applies, for example, to Helmholtz coils, although at lower fields their loops can simply be connected in series. At higher field or for coils with more distant elements, these are usually fed separately via a power splitter. In this case, however, equal currents are not guaranteed if the loading and tuning of the elements differ. Therefore, a circuit that provides equal output currents would yield superior performance than a conventional power splitter. Subsequently, we refer to such an arrangement as an RF current mirror.

### Theory

Since a certain spatial distance has to be bridged, it is apparent that the arrangement will include transmission lines. Therefore, for an initial analysis as illustrated in Figure 1A, we investigated if equal load currents can be achieved despite different loads $(Z_1≠Z_2)$. Upon setting $V_g=0$ and feeding a current $I_1$ into $Z_1$ (Figure 1B), the entire circuit becomes a network of cascaded 2-ports, and the ABCD-matrix$^\textbf{1}\:$representation is advantageous. The corresponding matrices for the components $Z_1$,$\:C_t$,$\:C_m$,$\:Tline_n$,$\:Z_g\:$and$\:Z_2$ are given by $\begin{pmatrix}1&Z_1\cr\:0&1\end{pmatrix}$,$\begin{pmatrix}1&0\cr\:i\,C_t\,\omega&1\end{pmatrix}$,$\begin{pmatrix}1&-\frac{i}{C_m\,ω}\cr0&1\end{pmatrix}$,$\begin{pmatrix}\mathrm{cos}\left(\beta\:l\right)\:&\:i\,Z_0\,\mathrm{sin}\left(\beta\:l\right)\:\cr\:\frac{i\,\mathrm{sin}\left(\beta\:l\right)\:}{Z_0}\:&\:\mathrm{cos}\left(\beta\:l\right)\:\end{pmatrix}$,$\begin{pmatrix}1&0\cr\frac{1}{Z_g}&1\end{pmatrix}$ and $\begin{pmatrix}1&Z_2\cr\:0&1\end{pmatrix}$, which have to be multiplied according to their order in the circuit. The element $D$ of an ABCD-matrix is defined as: $$D={\rm{ABCD}}_{2,2}=\left.\frac{I_1}{I_2}\middle|\mathop{}\limits_{V_2=0}\right.$$ Hence, considering the arrow direction (Figure 1B), both load currents are equal if: $$D=-1$$ Solving this equation for $\beta\:l$ yields, inter alia, solutions which contain neither $Z_1$,$\:Z_2\:$nor$\:Z_g$: $$\beta\:l\:=\:\frac{\pi}{2}-\mathrm{atan}\left(\frac{C_m\,C_t\,\omega\,Z_0}{C_t+C_m}\right)+k\pi,\:k=0,1,2,...$$ The result is the electrical length of both transmission lines (in radians) at which the original arrangement (Figure 1A) acts as the intended current mirror. No active components are required. We note that equivalent results for other kinds of matching networks can be obtained in a similar fashion.

### Experiments

To validate the theoretical finding, a setup was designed (Figure 2) that could be easily simulated and realized experimentally. A simple Helmholtz coil (25mm loop radius and spacing; two opposite gaps per loop) was selected for this purpose. The conductor was assumed to be 4mm-wide copper tape. An oil-filled cylinder (43mm diameter, 66mm height) served as phantom (Figure 3A). A particular goal was to easily switch between the conventional operation mode ($R_w=200Ω$) and the implemented current mirror ($R_w$ removed) for a direct comparison. In contrast to the simplified theoretical analysis, the transmission lines were modeled, taking the skin effect and dielectric losses into account. The 3D electromagnetic field (EMF) simulation and the RF circuit co-simulation were performed with openEMS$^\textbf{2}\:$(v0.0.35) and with LTspice (Linear Technology, Milpitas, CA, USA), respectively. A similar setup (Figure 4A) was built for use in a human-scale MRI scanner (MAGNETOM 7T, Siemens, Erlangen, Germany). In this case, balanced lines$^\textbf{3}\:$were used, and the loops were made of 2mm silver-plated copper wire. Pure polydimethylsiloxane$^\textbf{4}$ $(\textit{µ}=200~{\rm{mPa·s}},\:T_1=1.3~{\rm{s}})\:$ was used as phantom fluid. The coil adjustment was accomplished using a pick-up loop to achieve equal loop currents initially.

### Results and Discussion

The theoretical analysis predicts that the line length depends on the matching network but not on the coil loading. We note that this is similar to the concept of preamplifier decoupling in receive arrays, which requires identical cable lengths. This similarity can be employed for line-length adjustment. For further verification, simulations and experiments were compared for three conditions:

1. Power splitter, loops tuned:$~~~~~~R_w=200Ω,~C_{x1}=1{\rm{pF}},~C_{x2}=1\rm{pF}$
2. Power splitter, loops detuned:$~~R_w=200Ω,~C_{x1}=0,~~~~~~C_{x2}=2\rm{pF}$
3. Current mirror, loops detuned:$~R_w=\:∞,~~~~~C_{x1}=0,~~~~~~C_{x2}=2\rm{pF}$

The corresponding $B_1^+$ maps are shown in Figures 3B and 4B from left to right. All maps were scaled by setting the mean value inside a region-of-interest (ROI) within the coil center (Figure 3A) to 100% with a color scale between 30% and 170%. As expected, detuning (condition #2) significantly deteriorated the homogeneity. After switching to current mirror mode (condition #3), homogeneity was restored. The experimental results are in good agreement with the simulations. Furthermore, the results show that the proposed concept also applies to coupled coil elements.

In the arrangement discussed above, the line lengths of the current mirror circuit are determined by the (fixed) matching network. This limits the practical applicability. A solution to overcome this limitation is to separate the feed from the current mirror (Figure 5), which achieves matching for a wide load range by adjusting $C_{m1},\:C_{t1}\:$and$\:C_{t2}$. In addition, there are new degrees of freedom for dimensioning, which require further investigation.

### Conclusion

We have demonstrated a novel feeding concept, which is based on an arrangement that acts as an RF current mirror. It provides almost equal currents to individual coil elements (e.g. of a Helmholtz coil) which are independent of differences in loading or tuning of the individual elements.

### Acknowledgements

No acknowledgement found.

### References

$^\textbf{1}\:$ Pozar, D. M. (2005), Microwave engineering. Hoboken, NJ: J. Wiley.

$^\textbf{2}\:$Liebig, T., Rennings, A., Held, S. and Erni, D. (2013), openEMS – a free and open source equivalent-circuit (EC) FDTD simulation platform supporting cylindrical coordinates suitable for the analysis of traveling wave MRI applications. Int. J. Numer. Model., 26: 680–696. doi:10.1002/jnm.1875

$^\textbf{3}\:$Müller, R., Kozlov, M., Möller H.E. (2015), Balanced Feed Lines with Bridged Shield Gaps for RF Coil Arrays. Proc Intl Soc Magn Reson Med 23:1803

$^\textbf{4}\:$Skloss T. (2004), Phantom ﬂuids for high ﬁeld MR imaging. Proc Intl Soc Magn Reson Med 11:1635

### Figures

Figure 1. (A) Two loads (Z1, Z2, e.g. loops), each connected to a voltage source (Vg, Zg) via a matching network (Ct, Cm) and a (lossless) transmission line, which is characterized by Z0 (characteristic impedance), the phase constant β and the physical length l. The matching networks and transmission lines are dimensioned identically. (B) Modified schematic used for the investigation. Z1 resp. Z2 contain all reactances and losses of a loop outside its feed port. The loops are initially regarded as not coupled. I1 is a residual current caused by differences in tuning and loading.

Figure 2. Setup for circuit simulation. By setting Rw to 200Ω, Tline1 and Tline2 work as a special type of a Wilkinson power splitter. Removal of Rw switches the arrangement to the current mirror mode if the cable lengths are suitable. The 4-port block 2_Loops contains the coil model obtained from EMF-simulation, which includes loop coupling. By adjusting C1 and C2 while Cx1=Cx2=1pF the coil was tuned to resonance at 297.2MHz and also to equal loop currents. The loops can be detuned by a given amount by changing the values of Cx1 and Cx2.

Figure 3. (A) Coil geometry used for the EMF-simulation (gray: oil phantom, red: copper loops, green: ports). The B1+ field is mapped in a region from the coil center to the radius of the phantom. In the center of the phantom, the ROI used as a scaling reference is highlighted by a box. (B) Relative B1+ maps obtained by simulation.

Figure 4. (A) The coil used for experiments is mainly based on 3D-printed parts, which carry the wire-loops and hold the phantom. The feed ports are on the left side. The trimmer capacitors opposite to the feed ports are equipped with plug-in sockets for easy capacitor replacement without soldering. (B) Relative B1+ maps were obtained by a double-angle method (2D FLASH, tip angles 30° and 60°, TR=7.5s, 1mm isotropic resolution) . Note that the image of the coil is not aligned with the maps.

Figure 5. A Helmholtz coil (loop ø65mm), developed for 1H-imaging of tissue specimens at 7T, schematic (A) and practical realization (B). The (balanced) RF current mirror is here connected opposite the feed port. The dimensioning of the current mirror is obtained according to: 1/(2/Cs+1/(Cp+Copen)≈Cd. Copen denotes the capacitance at the terminals of an open-ended line with the electrical length of βl=π/2-atan(CpZdiffω), which is half the total length required. (C) Image of a kiwi fruit (TSE; TE=23ms, TR=1s, 0.156×0.156×0.5mm3 voxel size).

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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