Comparison of Superconductors and Permanent Magnets for Small-scale Magnetic Resonance Imaging Devices

Abstract

In this feasibility study, we use finite element method simulations to compare superconductors (SC) coils and permanent magnets (PM) assemblies as main field source for a small-scale magnetic resonance imaging scanner. The motivation behind this investigation is that for certain tissues, e.g., knee ligament, diagnosis can be performed with equipment operating with relatively weak magnetic fields. It is then interesting to assess whether the relaxed field requirement enables alternative technologies, which present some advantages over the well-established helium-cooled low-temperature superconductors. As prototypical magnetic systems, we consider the Helmholtz coil and the Halbach cylinder for the SC and PM cases, respectively. Each possibility is simulated over a wide range of combinations of the fundamental design parameters. We estimate field strength, field homogeneity, and total cost over the service life of the device. From these figures of merit, we evaluate the optimal combinations of design parameters. Finally, we compare the SC and PM systems and we establish the range of field requirement values for which one alternative is more suitable than the other one. The result of the present investigation provides the starting point for more detailed studies where more specific design considerations are taken into account.

Appendix

1.1 Constitutive Relation Model Functions

1.1.1 Superconductor

The data sets for the superconductor $J_c(B,T)$ relation (Fig. 2a) and the permanent magnet M-H relation (Fig. 2b) have been extracted from Ref. [12] and Ref. [29], respectively.

We decided to fit the $J_c(B,T)$ data with the following model function:

$$\begin{aligned} {J_c(B,T) = J_{c0}\times f_B(B) \times f_T(T), } \end{aligned}$$

(26)

where the non-dimensional functions $f_B(B)$ and $f_T(T)$ are defined as

$$\begin{aligned} f_B(B)= \,& {} \frac{1}{1+(B/B_0)^{\alpha }} \xrightarrow {B\rightarrow 0 } 1 \end{aligned}$$

(27)

$$\begin{aligned} f_T(T)= \,& {} \frac{\varepsilon (1+(T/\gamma )^2)-1}{(\varepsilon -1)(1+(T/\gamma )^2)} \xrightarrow {T\rightarrow 0 } 1. \end{aligned}$$

(28)

The function $f_B$ is a rational function of the flux density norm B. As long as $B_0>0$, the limit of $f_B$ for $B\rightarrow 1$ is 1. The exponent $\alpha$ and the parameter $B_0$ determine how quickly $f_B$ decreases as B increases.

The function $f_T$ is a Lorentz distribution with center in $T=0$, full width at half-maximum $2\gamma$, vertically translated by $\epsilon$ and then re-normalized so that $f_T(T) \rightarrow 1$ as $T \rightarrow 0$.

We assume that the parameters $B_0$ and $\alpha$ appearing in the expression of $f_B$ are also temperature dependent. The temperature dependence can be modeled by a third-degree polynomial:

$$\begin{aligned} B_0(T)= & {} \sum _{k=0}^{3} \beta _k \,T^k \end{aligned}$$

(29)

$$\begin{aligned} \alpha (T)= & {} \sum _{k=0}^{3} \alpha _k\, T^k. \end{aligned}$$

(30)

The parameters’ values giving the best match with the data points of Ref. [12] are

$$\begin{aligned} \begin{array}{c c c} J_{c0}=2.6931 \;\text {MA}/\text {cm}^2 \;;&{}\beta _0=0.2049 \text { T}\;;&{}\alpha _0=0.6924\\ \gamma = 15.1217\; \text {K}\;;&{}\beta _1=-2.4833\,10^{-2}\text { T}/\text {K}\;;&{}\alpha _1=-2.6761 \,10^{-3}\text { K}^{-1}\\ \varepsilon =0.0251\;;&{} \beta _2=2.6294 \,10^{-3}\text { T}/\text {K}^2\;;&{}\alpha _2=5.6932\,10^{-4}\text { K}^{-2}\\ &{} \beta _3=-2.8058\,10^{-5}\text { T}/\text {K}^3\;;&{}\alpha _3=-4.6994\,10^{-6}\text { K}^{-3}\\ . \end{array} \end{aligned}$$

(31)

It should be stressed that Eq. 26 represents the engineering current density (assuming wire thickness $80\;\mu$m), while Ref. [12] gives the total current $I_c$ carried by a 4-mm-wide tape. The proportionality factor can be easily calculated.

1.1.2 Permanent Magnet

For the permanent magnet, we fitted the experimental data with the following model function:

$$\begin{aligned} {M_{\parallel }(H_{\parallel }) = \left( \frac{M_0}{ \pi /2}\right) \text {atan}\left( \frac{H_{\parallel }-H_{\text {ci}}}{\Delta _{H}} \right) }. \end{aligned}$$

(32)

Here $M_0$ is the saturation magnetization, (i.e., the maximum value of magnetization corresponding to the limit $H_{\parallel } \rightarrow +\infty$), $H_{\text {ci}}$ represents the intrinsic coercivity (i.e., the intersection of the $M_{\parallel }(H_{\parallel })$ curve with the horizontal axis $M=0$), and $\Delta _{H}$ can be interpreted as a horizontal scale factor. The values giving the best match with the data from Ref. [29] are

$$\begin{aligned} \mu _0 M_0=\, & {} 1.4407\,\text {T} \end{aligned}$$

(33)

$$\begin{aligned} \mu _0 H_{\text {ci}}= & {} -1.2957\,\text {T} \end{aligned}$$

(34)

$$\begin{aligned} \mu _0 \Delta _H= \,& {} 0.0181\,\text {T}. \end{aligned}$$

(35)

This set of parameters corresponds to the demagnetization branch of the $M_{\parallel }(H_{\parallel })$ hysteresis loop shown in Fig. 2b.