Trapped field characteristics of an HTS magnet with two holes using four magnetization methods

Abstract

High temperature superconducting (HTS) magnets have widespread applications in a strong and steady magnetic field at low temperature. However, they can not be operated in persistent current mode (PCM) due to their immature joint technique without resistance. In order to realize the PCM, an HTS magnet stacked by double-hole rectangular HTS plates was proposed and fabricated. The trapped field of the HTS magnet was measured and simulated under four kinds of magnetization methods, field cooling (FC), zero field cooling (ZFC) and inner magnetization (a solenoid is placed at right/left hole of double-hole rectangular HTS magnet to magnetize HTS magnet) as well as combination of both methods. Meanwhile, the H-formulation is applied to the 3D model to analyze the electromagnetic behaviour. It was found that the trapped field of double-hole rectangular HTS magnet magnetized by FC with inner magnetization or ZFC with inner magnetization is higher than pure FC or ZFC magnetization. In inner magnetization, the trapped field of one hole in HTS magnet has no effect on the other one. In addition, the experiment results are in good agreement with numerical analysis, which can provide significant references for the magnetization method.

Keywords

Double-hole FC rectangular HTS magnet trapped field ZFC

1. Introduction

Strong magnetic field plays an extremely important role in the development of science and technology [1], which breeds many major scientific discoveries and new technologies. Meanwhile, high temperature superconducting (HTS) magnet has attractive applications in producing strong and steady magnetic field because of its high transition temperatures and higher critical current density in high field and low temperature [2]. Recently, research of HTS magnets has achieved some important progress in many fields, such as medicine (magnetic resonance imaging) [3,4], pharmacy (nuclear magnetic resonance) [5,6], power applications (such as the Superconducting Fault Current Limiter and Superconducting Magnetic Energy Storage) [7,8]. Nowadays, the maximum magnetic field of 45.5-tesla had been achieved by an HTS coil that generates a magnetic field of 14.4-tesla inside a 31.1-tesla resistive background magnet to obtain a direct current (DC) magnetic field of 45.5-tesla [9]. Furthermore, China has successfully developed a maximum full superconducting magnet with central magnetic field up to 32.35-tesla so far, the FBML laboratory had also carried out programs with HTS magnet applying into MRI [10] and NMR [11]. However, the HTS magnet with PCM has not been realized since the jointing technique without resistance is immature.

The analysis of the electromagnetic behaviour of superconducting materials based on the finite element method (FME) can predict their performance for practical devices, and assist in verifying experiments results. There are four formulations for FEM calculations: the H-formulation [12–14], the A-V formulation [15], the T −𝛺 formulation [16,17] and T-A formulation [18–20]. Maxwell’s equations can be written in each of these formulations and these formulations are equivalent in principle, but the solutions of the corresponding partial differential equations (PDEs) can be very different [21].

Since previous references have studied a Bitter-like HTS magnet stacked by the REBCO annular plates for multiple times [22–25]. In this paper, a novel HTS magnet stacked of 60 REBCO rectangular plates with two holes is proposed and no one ever studied this kind of structure so far. Four magnetization methods, field cooling (FC), zero field cooling (ZFC), FC with inner magnetization and ZFC with inner magnetization, are carried out under experiments and simulations respectively, which aim to analyze the characteristics of trapped field of the rectangular HTS magnet with two holes and explore the magnetization efficiency on this structure of HTS magnet. Furthermore, the trapped field at top surface center position of double-hole rectangular HTS magnet is measured and discussed. A 3D model is also used to analyze the magnetic field distribution using H-formulation that utilizes the real superconducting layer thickness. In addition, this technology can analyse the electromagnetic behaviour of this kind of superconducting magnet more easily.

2. Magnet design

2.1. Configuration of the rectangular HTS magnet with two holes

The double-hole rectangular HTS magnet is prepared by alternately stacking of 60 REBCO double-hole rectangular plates and insulating double-hole rectangular plates which is made by polypropylene laminated paper (PPLP). The REBCO double-hole rectangular plates were manufactured by Shanghai Creative Superconductor Technologies Co., Ltd. The REBCO rectangular plates were fabricated by cutting commercial REBCO tapes with cross-section of 12 mm × 0.05 mm. The radius of each hole is 3.5 mm, the distance between two holes is 6 mm. Figure 1a illustrates the schematic geometrical structure of single double-hole rectangular HTS plate. Figure 1b shows the photo of single double-hole rectangular HTS plate. Figure 1c illustrates the schematic geometrical structure of double-hole rectangular HTS magnet and Fig. 1d shows the photo of the double-hole rectangular HTS magnet. Table 1 lists the main specifications of the double-hole rectangular HTS magnet.

Fig. 1.

(a) Geometrical structure of single plate. (b) Photo of single plate. (c) Geometrical structure of double-hole rectangular HTS magnet. (d) Photo of double-hole rectangular HTS magnet.

Table 1

Specifications of the double-hole rectangular HTS magnet

Items	Value
Width of REBCO plate (mm)	12
Length of REBCO plate (mm)	30
Diameter of one hole (mm)	7
Numbers of rectangular plates	60
Stacked height (mm)	10
Distance between two holes (mm)	6
J _c0 of the REBCO tape and in self 77 K (A/cm)	310

2.2. Magnetization methods

In this paper, the double-hole rectangular HTS magnet was magnetized by the following four methods: (A) FC magnetization; (B) ZFC magnetization; (C) FC magnetization with inner magnetization; (D) ZFC magnetization with inner magnetization. Meanwhile, a DC source and a periodic pulse current source with triangular waveform were also used during magnetization process. The periodic pulse current source outputs an alternating current (AC) with triangular waveform and a maximum current is 10 A, whose rising and falling duration of AC current are 1 ms and 10 s, respectively.

3. Experimental process

3.1. FC and ZFC magnetization

For FC and ZFC magnetization, the coil section is a racetrack-shaped copper coil as outer magnetization. The copper coil container, which is 250 mm in length, 150 mm in width and 170 mm in height, reserves two joints for connecting current leads. When switching on a DC source, the racetrack-shaped copper coil is powered and provides magnetic field in the central gap. The Bitter-like HTS magnet with two holes is placed at top surface center of the racetrack-shaped copper coil. Both of them were fixed by foam plastic bulks that are insulated and have no influence on the magnetic field. Figure 2 displays the photo of the FC/ZFC magnetization with double-hole rectangular HTS magnet.

Fig. 2.

Picture of FC/ZFC magnetization with double-hole rectangular HTS magnet.

During FC/ZFC magnetization, a DC source is used to power the racetrack-shapedd copper coils with setting the voltage as 5 V and increasing the current up to 10 A. The racetrack-shaped coil was all immersed in liquid nitrogen (LN2) during the process of FC/ZFC magnetization to make sure the double-hole rectangular HTS magnet in superconducting state.

3.2. Inner magnetization

For inner magnetization, the coil section is a solenoid, which is wound by two layers of copper wire. The copper wire, 0.5 mm in diameter, was wound into a solenoid coil with 200 mm long and a total of 800 turns. The solenoid is also inserted with iron core which is stacked by 5 silicon strips with 0.4 mm in width and 200 mm in length each, and the outer diameter of the solenoid is 4 mm. By applying a pulsed current with triangular waveform, the trapped field at top surface top surface center position of two holes is measured by Gaussmeter. The measurement range of the Gaussmeter is 0.01 mG–300 kG, and its refresh rate is 18 times/second, and its frequency range is DC or AC 10 Hz–400 Hz. Since there are two holes in the magnet, the left hole is inserted with a solenoid, the Gaussmeter is used to measure the magnetic field of two holes respectively. Figure 3 shows the photo of the solenoid and one silicon strip. Figure 4a displays schematic configuration of inner magnetization with double-hole rectangular HTS magnet and Fig. 4b shows the photo of inner magnetization with double-hole rectangular HTS magnet.

Fig. 3.

Photography of solenoid and one silicon strip.

Fig. 4.

(a) Schematic configuration of inner magnetization with double-hole rectangular HTS magnet. (b) Photo of inner magnetization with double-hole rectangular HTS magnet.

3.3. FC with inner magnetization

Based on the previous experiments, combining FC magnetization with inner magnetization to magnetize double-hole rectangular HTS magnet has been studied in this paper as well. Figure 5a illustrates temperature variation and current waveform of FC magnetization with inner magnetization. Since the rectangular HTS magnet has two holes, a solenoid cannot be placed into two holes simultaneously. The left hole is inserted the solenoid firstly as left magnetization, the Gaussmeter is used to measure the magnetic field for one minute before attenuating for five minutes. Then, the right hole is inserted the solenoid as right magnetization, the Gaussmeter is used to measure the top surface centre magnetic field in two holes for one minute. Since the maximum current range of the periodic pulse current is 10 A, in order to simply compare the impact of inner excitation on the trapped field of double-hole rectangular HTS magnet, the periodic current source can be turned off once lasting for 1 minute and we measured the trapped field of two holes, also the magnetic field attenuation lasts for 10 minute. The reason that a short time was used to magnetize the double-hole rectangular HTS magnet under inner excitation is because the trapped field of Bitter-like HTS magnet reached approximately 90% of the maximum magnet field according to reference [23], so there is no need to magnetize this HTS magnet for over 2 hours. Considering the magnetic field measurement process and the magnetization process must be separately carried out, the measured value is a little smaller than the actual value. But the period of measurement in the process is quite short, the attenuation almost does not affect the observation of the changing of the magnetic field during the magnetization process.

Fig. 5.

(a) Temperature variations and current waveform of FC with inner magnetization. (b) Temperature variations and current waveform of ZFC with inner magnetization.

3.4. ZFC with inner magnetization

In addition, combining ZFC magnetization with inner magnetization to magnetize the rectangular HTS magnet with two holes has also been obtained in this paper. Figure 5b shows temperature variation and current waveform of ZFC magnetization with inner magnetization. The process of magnetization is similar to FC with inner magnetization, which did not detail here.

4. Simulation analysis using H-formulation

Since the double-hole rectangular HTS magnet is inserted with a solenoid coil that cannot be placed in two holes simultaneously in the inner magnetization, which means that the simulation model cannot be simplified into an axial symmetric geometry in a 2D model. Thereby a 3D model was proposed to simulate the HTS magnet under four magnetization methods. H-formulation is applied to this model, analyzing the electromagnetic behaviour. It is worthwhile to mention that only the REBCO layer (1 μm) has been considered in the mode in order to increase the computation speed, without taking other layers, including a substrate (Hastelloy) layer (50 μm), a buffer stack layer (100 nm), two copper (Cu) stabilizer layers (5 μm), into consideration. In order to simplify the 3D model and reduce the calculation time, the double-hole rectangular HTS magnets stacked with 60 plates has been simplified into one integral magnet with the total height in 61 μm × 60.

The governing equation of H-formulation is derived from Faraday’s law and Ampere’s law and it directly solves the magnetic field components H = [H _x, H _y, H _z] in 3D geometry in x-y-z plane. The relevant equations are displayed there: $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}{\nabla}\times E=-\frac{\partial B}{\partial t}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}{\nabla}\cdot B=0\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}{\nabla}\times H=J\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}E={\rho}_{\mathit{HTS}}J\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}B={\mu}_{0}{\mu}_{r}H.\end{eqnarray}$ (5) The resistivity of the superconducting material, 𝜌_HTS, is given by the E-J power law: $\begin{eqnarray}\displaystyle {\rho}_{\mathit{HTS}}=\frac{E_{0}}{J_{C}(B)}\cdot \left(\frac{J}{J_{C}(B)}\right)^{n-1} & & \displaystyle\end{eqnarray}$ (6) Where E ₀ is the critical electrical field, which is 1e-4 μV/m and n is a parameter indicating the sharpness of the transition from superconductor to the normal state, which is 27 in this paper. J is the current density flowing through the REBCO layer. J _c is the critical current density, which is relative with the magnetic field, follows by equation. $\begin{eqnarray}\displaystyle J_{c}(B)=J_{c0}\left(1+\frac{\sqrt{k^{2}B_{//}^{2}+B_{\bot }^{2}}}{B_{0}}\right)^{-{\alpha}} & & \displaystyle\end{eqnarray}$ (7) Where J _c0 is the critical current density of superconducting tapes, which is 310A/cm, B _∕∕ and B _⊥ are the parallel and perpendicular magnetic field applied to the double-hole rectangular HTS magnet while k = 0.23, B ₀ = 30 mT, 𝛼 = 0.6 [26].

Combined the E − J and B − H constitutive relationships, hence the equation can be described as follows: $\begin{eqnarray}\displaystyle {\nabla}\times ({\rho}_{\mathit{HTS}}{\nabla}\times H)=-\frac{\partial ({\mu}_{0}{\mu}_{r}H)}{\partial t}. & & \displaystyle\end{eqnarray}$ (8) And $\begin{eqnarray}\displaystyle {\nabla}\cdot ({\mu}_{0}{\mu}_{r}H)=0. & & \displaystyle\end{eqnarray}$ (9)

5. Results and discussion

5.1. FC magnetization results

Figure 6 illustrates the magnetic field at top surface center of the double-hole rectangular HTS magnet during FC magnetization between experimental and numerical results. It turns out to have the almost same magnetic field of two holes according to experimental and numerical outcomes, with the solid circle representing the measured results and the solid rectangular in symbol of the simulated data. When the background current is 10 A, the applied filed of racetrack-shaped coil under experiments is around 27.15 mT (approximately 2.7 mT/A) while being around 30 mT under the simulation data (around 3 mT/A). The difference might result from the measurement errors in the experiments. The trapped field of double-hole rectangular HTS magnet is 25.15 mT for right hole and 25.23 mT for the left hole respectively according to the experiments, which clearly represented the great capacity of trapping magnetic field of double-hole rectangular HTS magnet, with 90%. In terms of simulation results, the final values of trapped field in right hole and left hole are 24.59 mT and 25.4 mT, which has a minor difference with the experimental results.

Fig. 6.

Trapped field at top surface center of double-hole HTS magnet during FC magnetization.

Fig. 7.

Trapped field at top surface center of two holes of double-hole HTS magnet during ZFC magnetization.

5.2. ZFC magnetization results

Figure 7 represents the trapped field at two holes top surface center of the double-hole rectangular HTS magnet during ZFC magnetization under experimental and numerical results. The experimental results show that the applied field of racetrack-shaped coil under 10 A is 4.67 mT (around 0.5 mT/A) while the applied field in simulation procedure is 5.35 mT (roughly 0.5 mT/A), the trapped fields at right hole top surface center of the double-hole rectangular HTS magnet are 0.52 mT and 0.53 mT at left hole after attenuating for 10 minute. According to the simulation results, the trapped fields of right hole and left hole are 0.50 mT and 0.46 mT respectively. The results difference between FC and ZFC magnetization on the one hand is due to the frozen magnetic flux under the FC magnetization and Meissner effect under ZFC magnetization, and both magnetization methods are strongly dependent on flux pinning and critical current density. There still exists the trapped field under ZFC magnetization, which probably results from the non-ideal features of second generation HTS tapes.

Fig. 8.

Trapped field of two holes in FC with inner magnetization.

5.3. FC with inner magnetization results

Figure 8 shows the trapped field at top surface center position of two holes in FC magnetization with inner magnetization as the background current is 10 A under experimental and numerical analysis. It clearly manifests that the trapped field of two holes has no link according to the Fig. 8. The trapped field of two holes are around 25.15 mT as FC magnetization ended. As left hole inserted a solenoid called left magnetization, the trapped field at top surface center position of left hole sharply increased up to 28.27 mT, while the trapped field in the right hole stayed at 24.63 mT, nearly as same as the FC magnetization ended. As the right hole was inserted a solenoid called right magnetization, the trapped field of right hole instantly achieved to 28.51 mT, while the trapped field at left hole top surface center kept at 28.14 mT. After switching off the power source, the trapped field of two holes nearly stayed unchanged, which indicates the good stability of double-hole rectangular HTS magnet. Comparing with FC magnetization (25.15 mT), the trapped field in FC with inner magnetization (28.51 mT) has a 13.3% increase. The simulated results also show a similar data compared with the experimental results, which have 24.6 mT in right hole and 25.41 mT in the left hole after the end of FC magnetization. Then the trapped field of left hole rises to 29.21 mT during the process of left magnetization (24.6 mT in the right hole), the trapped field of right hole instantly increases to 28.51 mT after right magnetization.

Fig. 9.

Trapped field of two holes in ZFC with inner magnetization.

5.4. ZFC with inner magnetization results

Figure 9 describes the trapped field of two holes in ZFC magnetization with inner magnetization between experiments and simulations. Since the background current is also 10 A, the trapped field of right hole is 0.52 mT and the trapped field of left hole is 0.53 mT as the ZFC magnetization ended. Then the double-hole rectangular HTS magnet was magnetized by left magnetization at first, the trapped field at top surface center of right hole stays unchanged during left magnetization while the trapped field on the left hole increases to 3.88 mT instantly. During the right magnetization, the trapped field of right hole added up to 3.73 mT during right magnetization while the trapped field in left hole kept 3.85 mT. Comparing with the pure ZFC magnet magnetization (0.5 mT), the trapped field of ZFC with inner magnetization (around 3.85 mT) is way too much higher, which indicates its priority of ZFC with inner magnetization. The final values of trapped field under simulation in right hole and left hole are 3.21 mT and 3.53 mT respectively. As for the increase of magnetic field under ZFC with inner magnetization, the main reason probably attributes to the solenoid with iron core which can increase and centralize the background magnetic field in double-hole rectangular HTS magnet comparing with racetrack-shaped coil.

6. Conclusion

In this paper, a double-hole rectangular HTS magnet stacked by rectangular HTS plates with two holes was proposed and designed to study the characteristics of trapped field. The trapped field was measured in LN2 temperature in four kinds of magnetization methods. The experimental and numerical results on the one hand reveal that the trapped field of one hole has nothing to do with the other, on the other hand, the trapped field magnetized by FC with inner magnetization is 13.3% higher than pure FC magnetization; Moreover, the trapped field in four magnetization methods is arranged from high to low, FC with inner magnetization, FC, ZFC with inner magnetization, ZFC with inner magnetization. The results can provide an analytical and reliable foundation for further magnetization methods.

Footnotes

Acknowledgements

I am highly thankful for excellent advice and comments by Pro. Wang in the process of experiments. In addition, with the help of Guangyi Zhang, I can fulfill my whole experiments. This work was supported in part by the National Natural Science Foundation of China under Grant 51977078.

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