**3. Results**

Figure 2 displays the normalized quasi-static levitation force obtained by calculation and experiment. In the experiment, four rectangular three-seeded YBaCuO bulks, fabricated by ATZ GmbH in Germany, were mounted in a sample holder fixed above the circular PMG of the SCML-03. The bulk size is 64 × 32 × 13 mm3. The cross-section of the PMG is the same as the geometry in the simulation, which is shown in Figure 1. The experiment process was the same as the quasi-static process of Figure 1. The YBaCuO bulks were first field cooled with a height of 30 mm, which is the distance between the top surface of the PMG and the bottom of the YBaCuO bulks. Afterwards, the bulks were brought down at 1 mm/s from the FCH to the WH (15 mm), and then relaxed for 10 min.

**Figure 2.** Normalized quasi-static levitation force versus time under different *J*c–*T* relationships of: *J*c = *J*c1(*1*-(*T*/*T*c)2)<sup>α</sup>, α = 1, 3/2, 2; *J*c = *J*c0((*T*c-*T*)/(*T*c-*T*0)); *J*c = *J*c0. The PMG in the simulation (or bulks in the experiment) moved from the FCH (30 mm) to the WH (15 mm) at the vertical 1 mm/s speed, and then relaxed for over 100 s. The original values of the levitation force at 15 s were 126.40 N (measured), 224.40 N (α = 1), 216.58 N (α = 3/2), 203.62 N (α = 2), 216.52 N (*J*c0((*T*c-*T*)/(*T*c-*T*0)), and 216.51 N (*J*c = *J*c0), respectively.

The numerical results are computed based on the *J*c–*T* functions of Equations (6) and (7) with α = 1, 3/2, and 2, as well as the case without considering thermal effect. The levitation forces obtained by calculation and measurement are normalized by dividing its maximum force at 15 s. The original values of the levitation force at 15 s were 126.40 N (measured), 224.40 N (α = 1), 216.58 N (α = 3/2), 203.62 N (α = 2), 216.52 N (*J*c0(*(T*c*-T*)/(*T*c*-T0*)), and 216.51 N (*J*c = *J*c0), respectively. So, all the normalized levitation forces equaled 1 only at 15 s, to better compare the difference between the force trends.

It is noted that the levitation force increases gradually during the PMG moving from the field-cooling height to the working height. Then, it shows clear attenuation at the first few seconds during the relaxation process. This is due to the grea<sup>t</sup> change of the external magnetic field caused by the large-range movement of the PMG or the HTS bulk, which leads to the redistribution of the flux inside the bulk. The inset in Figure 2 is the partial enlargement of the maximum force region. From the inset, we can see that the calculated levitation force with α equals 2 by Equation (7), which agrees best with the measurements in the quasi-static condition.

Figure 3 further displays the normalized dynamic levitation forces from 115 to 155 s at 2 and 60 Hz by five different calculations of different *Jc–T* formulas. Figure 3a,c show the general view of the dynamic process, while Figure 3b,d zoom in the vibration end to better compare the different levitation force changes. The corresponding movement of the PMG is depicted in Figure 1. It is found that the continuous vibration of the PMG can lead to the levitation force attenuation. Table 2 collects the attenuation values of each case. The attenuation is the difference between the results obtained at the beginning (120 s) and the end (140 s) of the dynamic condition process, which can be seen in Figure 3b,d. We can find in Table 2 that with the increase of the frequency, the attenuation gets more obvious. This is because the flux inside the HTS bulk under higher vibration frequency is more intense. Other studies in the literature [15,16] have concluded that the levitation force is a little higher when not considering the thermal effect than when accounting for the thermal effect, because the attenuation caused by thermal loss is not considered. From this point of view, under the experimental vibration condition, the force attenuation without the thermal effect is a little smaller than the case considering the thermal effect, in which *Jc* retains some mathematic relationships with temperature *T* during the calculations. As shown in Table 2, the calculated force attenuation without considering the thermal

effect is 0.009 at 2 Hz, and 0.065 at 60 Hz. However, the calculation results considering the thermal effect, such as 0.006 under α = 1 at 2 Hz and 0.062 under α = 3/2 at 60 Hz, are smaller. In addition, 0.010 under the linear function and 0.009 under α = 3/2 at 2 Hz are almost the same, with 0.009 at 2 Hz without any thermal effect. Therefore, only the results calculated by Equation (7) with α = 2 are satisfied and reasonable, which is the same conclusion by the quasi-static comparison in Figure 2.

**Figure 3.** Normalized levitation force profiles by different *J*c–*T* relationships of *J*c = *J*c0*; J*c = *J*c1(*1-*(*T*/*T*c)2)<sup>α</sup>, α = 1, 3/2, 2; *J*c = *J*c0((*T*c*-T*)/(*T*c*-T*0)) under (**a**) 2 Hz and (**c**) 60 Hz; (**b**) and (**d**) emphasize displaying the force attenuation caused by the vibration process under 2 Hz and 60 Hz, respectively. The dotted lines represent the force value at the end of Relaxation I (120 s). FCH = 30 mm, WH = 15 mm. 1, 2, 3, 4, and 5 represent five *Jc–T* calculation conditions.

**Table 2.** The normalized levitation force attenuation under different *J***c**–*T* relationships.


The current density and the temperature distribution of the HTS bulk under 60 Hz at the working height (15 s), the end of the first relaxation process (120 s), and the time when the PMG first reached the vibration peak (120.005 s) are shown in Figure 4a. It is seen that all the currents and the temperature rise appear from the bottom of the bulk, and the temperature gradually spreads to the entire bulk. All the modeling results indicate that the maximum temperature occurs at 15 s, when the PMG arrives at the working height after the quasi-static process. While the thermal effect by that tiny-amplitude vibration is small, it is concluded that compared to the small-amplitude high-frequency vibration of the

PMG, a long-distance movement between the FCH and the WH is more likely to lead to temperature rise, because it causes a distinct magnetic field variation despite the speed being only 1 mm/s. Figure 4b displays the maximum temperature inside the HTS bulk calculated by different *J*c–*T* relationships under 60 Hz. It is seen that temperature rise occurs mainly around the time when the PMG first reached the working height, and the first upward moved 1 mm during the dynamic process. The inset in Figure 4b further indicates the temperature variation difference after the beginning of the vibration. Although the vibration process excites the second temperature rise, the temperature rise is much smaller than that in the first movement from the 30-mm FCH to the 15-mm WH, because the vibration amplitude is only 1 mm. Furthermore, the temperature curves kept decreasing gradually after the PMG first reached the peak position during dynamic conditions.

**Figure 4.** Comparison of different *J*c–*T* relationships of *J*c = *J*c1(*1*-(*T*/*T*c)2)α with α = 1, 3/2, 2 and *J*c = *J*c0((*T*c*-T*)/(*T*c*-T*0)) under 60 Hz. (**a**) Current density and temperature profile inside the HTS bulk at 15 s, 120 s, and 120.005 s; (**b**) maximum temperature *T*max inside the HTS bulk versus time.
