1. Introduction
The contactless rotation of a permanent magnet (PM) above a superconducting bulk in a vacuum seems to be lossless, however, AC (rotational) losses primarily generated by magnetic field inhomogeneity, gradually decelerate the rotor of SMBs. The loss mechanism in SMBs is rather complicated and has been investigated in many studies [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. It has been said that rotational loss has two main components; hysteresis loss and eddy current loss. The former, which arises from flux pinning in the superconducting bulk, is said to be independent of rotational velocity [13]. However, some researchers claimed that there would be a small velocity dependence in hysteresis loss [10]. The latter can be created in all electrically conducting parts which are subjected to an AC magnetic field from either a rotating PM or an inhomogeneous HTS bulk magnetization. The inhomogeneous magnetic field of the rotating PM may induce eddy currents in the HTS bulk and all surrounding conductors. The magnetic field trapped in the HTS bulk may also be inhomogeneous, due to either the discrete structure of the HTS bulks arrangement or the inhomogeneous shielding current of a single HTS bulk, which can induce eddy currents in rotating conductors such as PM rotor [14], [15].
Wilson [16] formulated the hysteresis loss generated in a superconducting slab and a superconducting disc subjected to an AC magnetic field as:
$Q=\frac{B_{m}^{2}}{2 \mu_{0}}\left(\frac{2 \beta}{3}-\frac{\beta^{2}}{3}\right) \text { for } \quad \beta<1$
$Q=\frac{B_{m}^{2}}{2 \mu_{0}}\left(\frac{2}{3 \beta}-\frac{1}{3 \beta^{2}}\right) \text { for } \beta>1$
where $\beta=\frac{B_{m}}{B_{p}}$, with $B_m$ being the peak-to-peak amplitude of the AC component of the magnetic field and $B_{p}=2 \mu_{0} J_{c} r$ being the penetration field. Afterward, many researchers utilized this formula alongside different methods to simulate and calculate the hysteresis loss [1], [3], [17], [18]. Eqs. (1), (2), however, do not completely capture the physics of hysteresis loss in SMBs. Other contributing factors such as the periodicity of the magnetic field in one complete rotation and the distribution of the magnetic field over the bearing surface are missed out.
The coefficient of friction (CoF) was presented as a parameter to characterize rotational loss, and as a figure of merit to evaluate SMB performance [1]:
$C o F=\frac{F_{D}}{F_{L}}$
where $F_D$ is the drag force, $F_L$ is the levitation force. $F_D$ can be calculated from the drag torque:
$\tau_{D}=F_{D} R_{D}=-I \alpha$
where
$R_D$ is the mean drag force radius, I is the moment of inertia of the rotating magnet, and α is the angular acceleration:
$\alpha=d \omega / d t=2 \pi d f / f t$
where f is the rotational frequency of the levitated object. The moment of inertia of the disc-shaped object can be described as:
$I=M R_{\gamma}^{2}$
where M is the mass of the rotating object and $R_γ$ is the radius of gyration. $F_L$ in Eq. (3) can be described as the weight to be lifted:
$F_{L}=M g$
where g is the acceleration of gravity. Substituting Eqs. (4), (5), (6), (7) yields:
$C o F=\frac{-2 \pi R_{T}^{2}}{g R_{D}} \frac{d f}{d t}$
$R_D$ has been usually taken to be the outer radius of the rotating part of the SMB because the AC loss mechanism and its distribution over the bearing radius has not been fully explored [10], [1]. Since then, the CoF, which is obtained from spin-down tests, has been used by other researchers as a reliable tool when studying rotational loss in SMBs [4], [19], [20], [21], [22], [5], [2], [23]. The main challenge in performing spin-down tests in SMBs is to engineer a driving system to accelerate and decelerate the levitating PM. Gas jets [1], [24], electrical motors [1], [6], [25], [13], a 3-phase handmade brushless motor [7], [26], eddy current clutches [4], [6], [27], [28], [29], [30], and magnetic gears [31] have been employed to accelerate the rotating part of superconducting bearings. These methods are associated with complex coupling structures, loss creation, limited maximum speed, and a lack of precision when it comes to speed control.
In this paper, a new expression is proposed for hysteresis loss in SMBs, which takes magnetic field periodicity and loss surface distribution into account. This new expression is tested by performing spin-down measurements in a novel test setup with PMs of the same size but different levels of inhomogeneity.
2. Methodology
The SMB in this experiment is a thrust bearing that consists of a disc-shaped PM and a disc-shaped HTS bulk. In this configuration, the axis of rotation is parallel to the axis of levitation force. Two experimental rigs were designed and built for this study, namely the spin-down rig and the magnetic field mapping rig.
2.1. Spin-down rig
The main challenge in this rig was to spin up the PM, release it, then capture it again. Fig. 1, Fig. 1 (c) show the schematic and a photo of the driving system in this rig. A high-speed brushless DC motor is used to spin the PM. Because the motor operates in a vacuum where heat dissipation through convection is not possible, the motor is encased in an aluminium heat sink to cool it. This heat sink with the motor inside is mounted on a vertical linear displacement stage.
Fig. 1. Schematic demonstration of the spin-down rig (a), the driving system (b), and a photo of the driving system (c). |
A two-part power transmission mechanism was designed and built to transfer power from the motor shaft which is 4 mm in diameter to the PM which is 17 mm in diameter. The first part includes the male side of the PM coupling. This part is a hollow cylinder inside which the PM is bonded. The male part is made of acetal (polyoxymethylene) because it is a high-strength, low-friction, and easy-to-machine engineering plastic. Furthermore, it prevents eddy currents from being induced in the male part by the magnetic field trapped in the HTS bulk. The top surface of the cylinder is machined to form a P4C-like polygon based on DIN 32712. The upper edge of the polygon was chamfered with an aspect ratio of 1:1 and dimensions of 3 mm ×45∘ to help self-centering during the coupling and decoupling processes. The second part of the power transmission mechanism consists of a 65 mm aluminium shaft. The bottom surface of the shaft is a cut-through P4C-like polygon with the same chamfer characteristics as that of the male part and serves as the female side of the PM coupling. A jaw-type coupling connects the aluminium shaft to the motor shaft. The aluminium shaft is supported by two ceramic ball bearings embedded at two ends of an aluminium box. This box improves the bearings axial alignment and enhances their heat dissipation.
A thin aluminium spacer mounted on a vertical linear displacement stage is provided to adjust the cooling height of the PM above the HTS bulk surface. This spacer together with the linear stage is mounted on another lateral linear stage, which can move the spacer aside after reaching the desired temperature.
The HTS bulk is a melt-textured $Y B a_{2} C u_{3} O_{7-\delta}$ disc [32] and is 28 mm in diameter and 10 mm thick. The HTS disc which forms the stator part of the SMB is partially recessed into a copper plate mounted on the Coolstar 0/40 cold head of a GM cryocooler supplied by Oxford Cryosystems [33].
The whole test setup is housed inside a cylindrical stainless steel vacuum chamber. Two windows are machined on the side wall of the chamber, allowing non-vacuum-rated laser displacement sensors to be installed outside the vacuum environment. The base plate of the chamber allows the cold head of the cryocooler and electronic connections to be fed through (Fig. 1 (a)). A two-stage Edwards T-Station 85 [34] is used to evacuate the chamber. This system is capable of providing vacuum to below 1×10-6 mbar, thereby ensuring the stability of the cryogenic environment.
An S950-SM silicon diode sensor [35] mounted on the copper plate is used as the temperature reference. The temperature of the HTS bulk is controlled by a Cryocon-26 controller [36]. A Monarch Instrument remote optical sensor model ROS-W [37] mounted on the setup frame measures the rotational frequency of the PM by detecting a reflected pulse from a reflective tape attached to the side surface of the male part of the PM coupling. A more detailed description of this setup can be found in [38].
2.2. Magnetic field mapping rig
Studying AC loss mechanisms in SMBs requires comprehensive measurement of the applied magnetic field profiles. Considering the rotation of the levitated magnet around its axis of symmetry in this study, magnetic field profiles in the azimuthal direction need to be characterized. In order to achieve this, a benchtop setup (Fig. 2) was designed and built. The magnetic field was measured by a P15A Hall sensor supplied by AHS [39]. The Hall sensor was driven by a Keithley 2400-c sourcemeter [40] at 1 mA, and was read by a NI USB-6009 card supplied by National Instruments [41]. The sensor is mounted on the tip of a cylinder made of G10. 3D motion (along x, y, and z axes) with a resolution of 0.05 mm is realized by linear stages. Two linear stages work in parallel along the x-axis. A rotary table for rotating an object beneath the sensor with the resolution of 1.8∘ gives a fourth axis around which the table rotates. PMs of different sizes can be installed on the center of the rotary table. The sensor mount and PM are aligned axially with an accuracy of 10μm. The rotary table and all linear stages are controlled by an MC 405 controller supplied by TrioMotion [42]. There are some limitations in this setup. Mounting the Hall sensor parallel to the G10 cylinder surface and PM surface was challenging. In order to keep the sensor level on the tip of the G10 cylinder, a thin PCB circuit was used on top of the sensor, which increased the minimum distance between the sensor and the top surface of PM to 2.6 mm.
Fig. 2. Photo of the magnetic field mapping setup. This setup was used to measure the magnetic field of PMs around their circumference. |
3. Results and discussion
When an inhomogeneous magnet rotates above the HTS surface, the superconductor experiences a steady (DC) magnetic field with an alternating (AC) component [43]. This AC component can create hysteresis loss, which according to the Bean model for a superconducting disc in the critical state, can be analytically described (per cycle per unit volume) by Eqs. (1), (2). According to Eqs. (1), (2), the hysteresis loss in an SMB is not dependent on the rotational frequency of the bearing, however, the periodicity of the magnetic field in one complete (2π) rotation has not been taken into account in these equations. In other words, the frequency of the AC component of the magnetic field ($B_m$) has been assumed to be $f_m=1$ Hz. In order to take the magnetic field periodicity into account, the magnetic field of the PMs was measured circumferentially by the magnetic field mapper along the z-axis for $r=r_i$ to $r=r_f$ in 1.8∘ and 1 mm intervals along the azimuthal and radial directions, respectively. Discrete Fourier transform analysis was performed on $B-B_{mean}$ measured along each radius to derive the amplitude ($A_i$) and frequency ($f_i$) of its components. Using the Fourier transform results, Eq. (1) can be modified as:
$Q=\frac{1}{n_{r}} \sum_{r_{i}}^{r_{f}} Q_{r} \quad \text { with } \quad Q_{r}=\sum_{i} f_{i} \times\left(\frac{B_{m i}^{2}}{6 \mu_{0}}\left(2 \beta_{i}-\beta_{i}^{2}\right)\right)$
where $Q_r$ is the hysteresis loss along each radius, $n_r$ the number of radii along which the magnetic field has been measured, $B_{m i}=2 A_{i}$, and $\beta_{i}=\frac{B_{m i}}{B_{p}}$. The total inhomogeneity of the PM, ϕ, is given by:
$\phi=\frac{\sum_{n r} 2 A_{\text {max }, r}}{n_{r}}$
where, $A_{\max x, r}$ is the maximum component of the magnetic field measured along each radius.
As a first-order approximation, the frequency of the magnetic field along each radius can be assumed as $f_{m}=1$ Hz and the peak-to-peak ΔB along each radius can be taken as Bm. Applying these assumptions, Eqs. (9), (10) can be written as:
$Q=\frac{1}{n_{r}} \sum_{r_{i}}^{r_{f}} Q_{r} \quad \text { with } \quad Q_{r}=\frac{\Delta B_{r(p-p)}^{2}}{6 \mu_{0}}\left(2 \beta_{r}-\beta_{r}^{2}\right)$
$\phi=\frac{\sum_{n r} \Delta B_{r(p-p)}}{n_{r}}$
where, $\Delta B_{r(p-p)}$ is the peak-to-peak amplitude of the AC component of the magnetic field measured along each radius, and $\beta_{r}=\frac{\Delta B_{r(p-p)}}{B_{p}}$.
The best PMs have been reported to have an AC component of almost 1% of the average field at a fixed radius above the rotating surface [10]. This magnetic field inhomogeneity acts as a drag force on the rotating body in a superconducting bearing, causing a gradual deceleration. The stored energy in the rotating PM is given by:
$E=\frac{1}{2} I \omega^{2}$
where $I=m R_{\gamma}$ is the moment of inertia, $\omega=2 \pi r f$ is the angular velocity. Here, m is the mass of the rotor, $R_γ$ is the radius of gyration, and f is the rotational frequency. The energy loss per cycle is given by:
$P_{\text {cycle }}=-\frac{1}{f} \frac{d E}{d t}=-2 m R^{2} \pi^{2} \frac{d f}{d t}$
Here, R is the outer radius of the rotor.
Eqs. (1), (9), (11) can be used to calculate the hysteresis loss per cycle per unit volume. According to Eq. (14), the AC loss (hysteresis loss + eddy current loss) per cycle in a superconducting bearing can be determined by measuring the deceleration rate ($d f / d t$) of the freely spinning PM.
The magnetic field of six PMs was measured from r=1 mm to r=12 mm in 1.8∘ and 1 mm intervals along azimuthal and radial directions, respectively. These PMs are referred to as PM1 to PM6 in this paper. All PMs were N45-grade NdFeB, 17 mm in diameter, and 10 mm thick [44]. The vertical separation between the PM and the Hall sensor was 2.7 mm, equal to the vertical separation between the PM and the HTS bulk during spin-down measurements. Fig. 3 shows the magnetic field of PM number 6 measured along 12 discrete radii, and the associated Fourier analysis. Fig. 4 provides visual information on magnetic field irregularities of magnet number 6. A threshold of 1×10-4 T was defined to determine the data points that contribute to Eq. (9). Because of the 3rd and the 4th power of $B_{mi}$ in Eq. (9), the contribution of components with amplitudes smaller than the threshold is negligible. The same technique was applied to all magnets. Fig. 5 shows the magnetic field of all PMs studied in this paper measured azimuthally at r=7 mm.
Fig. 3. Left: magnetic field (in the Z-direction at z=2.7 mm) of PM6 measured azimuthally over its surface with r=1 mm being the innermost radius. The radius of the magnet is 8.5 mm. Right: the discrete Fourier transform of $B-B_{mean}$ over three radii. The red dashed line indicates the 1×10-4 T threshold, above which the data points are included in Eq. (9). |
Fig. 4. Magnetic field (Bz) of PM6 measured at 2.7 mm distance from its surface. Measurement was performed in a 30×30 mm2 surface with 0.5 mm intervals in both x and y directions. |
Fig. 5. Magnetic field (Bz) of all PMs measured azimuthally at r = 7 mm and 2.7 mm distance from surface with r=1 mm being the innermost radius. The radius of the magnets is 8.5 mm. |
In order to compare AC loss measurements with analytical values, Eqs. (1), (9), (11) were used to calculate hysteresis loss from magnetic field inhomogeneities, and Eq. (14) was used to calculate AC loss from deceleration rates below resonance, because AC losses in SMBs are mainly dominated by hysteresis loss below the resonance frequency of the rigid body. AC losses measured below resonance were used because AC losses above resonance are believed to be influenced by additional factors excited by external vibrations [10]. In this comparison, the value of Jc = 6.46×107 A/m2, estimated from levitation force decay measurements [45], was used to calculate $B_p$. Fig. 6 shows the AC loss measured in spin-down tests below resonance frequency in comparison with hysteresis loss calculated with Eqs. (1), (9), (11). In Fig. 6 (a), PM inhomogeneities on the x-axis were taken as the maximum peak-to-peak ΔBi. In all graphs, experimental values are in good agreement with those calculated analytically, however, the order of magnets on the graphs varies because the computed values of inhomogeneity are not the same. In Fig. 6 (a) the theoretical values are larger than the experimental data points for PM6, PM5, and PM3. This is inconsistent with the nature of spin-down experiments. Other factors of magnetic field inhomogeneity (radial ($ΔB_r$) and vertical ($ΔB_z$)) are involved by the experimental procedure and contribute to the total loss, whereas in analytical calculations these factors are ignored and only the azimuthal magnetic field inhomogeneity has been considered. As a result, the experimental data points are expected to be larger than theoretical values. Furthermore, Eq. (1) does not completely capture the physics of hysteresis loss because it ignores the periodicity of the magnetic field and the distribution of the magnetic field inhomogeneity over the surface of the bearing. In Fig. 6, Fig. 6 (c), however, all experimental data points are larger than theoretical values, which is consistent with the physics and nature of the experiments. In Fig. 6 (b), although PM6 has the lowest inhomogeneity level, it generates a larger AC loss than PM4 and PM1. This can be explained by Eq. (9) which captures the physics of hysteresis loss by considering the impact of magnetic field periodicity and the distribution of AC loss over the surface. PM6 shows that a magnet with a smaller inhomogeneity level but a more periodic magnetic field can generate a larger AC Loss. Fig. 6 (c) shows that Eq. (11) which is the first-order approximation of Eq. (9), can still provide an acceptable prediction of AC loss by only considering the distribution of magnetic field inhomogeneity.
Fig. 6. AC loss generated in an HTS bulk 28 mm in diameter and 10 mm thick versus magnetic field inhomogeneity calculated analytically and measured in spin-down experiments. |
The largest discrepancy is seen for PM3 and PM2 in Fig. 6 (a), however, this discrepancy happens for PM2 and PM5 in Fig. 6, Fig. 6 (c). Considering the more comprehensive definition of hysteresis loss by Eqs. (9), (11), the discrepancy for PM3 in Fig. 6 (a) can be caused by Eq. (1) itself. The discrepancy for PM2 and PM5 in Fig. 6, Fig. 6 (c), however, might be caused by the challenging experimental procedure and the PM coupling system.
Fig. 7 shows the CoF as a function of rotational frequency for the six PMs of the same size but with different levels of inhomogeneity. PM5 could not be released at frequencies well above the resonance frequency, due to coupling imbalance and vibrations at higher frequencies. Comparing Fig. 6 and Fig. 7, it can be seen that although PM3 generates a larger loss by all definitions (Eqs. (1), (9), (11)), it has a smaller CoF than PM5. This can be explained by the definition of CoF. In CoF calculations, $R_D$ is usually taken as the outer radius of the rotating PM [10], [1]. This assumption simply ignores the distribution of inhomogeneity and AC loss over the surface of the bearing. Fig. 8 shows the hysteresis loss distribution over PM3 and PM5 surface calculated by Eqs. (9), (11). This figure shows that the distribution of hysteresis loss is more towards the center of PM3, whereas it is more towards the circumference of PM5.
Fig. 7. The CoF versus rotational frequency for six PMs of the same size but with different levels of inhomogeneity. The separation between PM and HTS was 2.7 mm. |
Fig. 8. Hysteresis loss distribution over the surface of PM3 (a) and PM5 (b) calculated by Eqs. (9), (11). |
The radius of drag force, $R_D$, is given by [46]:
$R_{D}=\frac{\sum r_{i} \Delta B_{i}^{3}}{\sum \Delta B_{i}^{3}}$
where, $r_i$ is the $i^{th}$ radius over which the magnetic field is measured circumferentially, and $ΔB_i$ is the peak-to-peak AC component of the magnetic field over radius $r_i$. This equation is also missing the periodicity of the magnetic field over radius $r_i$. In order to take the periodicity of the magnetic field into account, Eq. (15) can be updated by using Eq. (9) as:
$R_{D}=\frac{\sum r_{i} Q_{i}}{\sum Q_{i}}$
where $Q_i$ is the hysteresis loss over $r_i$ calculated by Eq. (9). Fig. 9 shows the CoF values calculated by using the updated values of $R_D$. As can be seen, by taking the AC loss distribution and magnetic field periodicity into account, the CoF curves are updated in an order matching the AC loss order shown in Fig. 6.
Fig. 9. The CoF versus rotational frequency updated by using Eq. (16) to calculate $R_D$ values. In this figure, the CoF of PM3 is larger than that of PM5. |
The dynamic behaviour shown in Fig. 9 has been widely reported for disc-shaped SMBs [10]. The CoF shows a major resonance behaviour in all cases and is almost independent of rotational frequency below and above the resonance region. This main resonance mode has been said to be the radial resonance frequency for disc-shaped bearings [10]. The large CoF value in the resonance region is due to radial vibrations of large amplitude. The CoF is larger at frequencies above resonance than below resonance. This can be explained by the transition of the center of rotation. Below resonance, the PM rotates around its center of magnetism (or center of geometry), and above resonance, it rotates around its center of mass. This can add a radial factor to magnetic field inhomogeneity [10]. Other vibration modes can be induced by the coupling mechanism which consequently adds to the hysteresis loss. As can be seen in PM2, PM4, and PM6 graphs, the CoF is mostly independent of frequency below and above the resonance region. In all graphs, a slight increase in the CoF was observed below the resonance region at very low frequencies which might be caused by cogging torque created by magnetic field inhomogeneity [46].
As shown in Fig. 9, the resonance region widens and the peak rotational frequency decreases with AC loss generated by the magnets. This means that the bearing dynamic stiffness, which is related to the resonance frequency by $K=m \omega_{n}^{2}$, decreases with magnetic field inhomogeneity. Here, m is the mass of the rotor, $\omega_{n}=2 \pi f_{n}$ is the resonance angular velocity, and $f_n$ is the resonance frequency. The larger the inhomogeneity, the deeper the AC component penetrates into the HTS bulk and the more flux lines are influenced. This AC component helps flux lines escape from pinning centers which in turn decreases bearing stiffness. This is consistent with the work reported in [47], [48], [49] where the resonance frequency of a levitated PM was found to be inversely proportional to the amplitude of excited vibrations. Lower levels of inhomogeneity can result in higher stiffness, however, passing through higher resonance frequencies in practical applications requires extra safety measures. Other local peaks can be observed (mainly 15 Hz – 20 Hz, and 70 Hz – 80 Hz regions) in the CoF graphs which might be attributed to vertical, whirling, and other modes of motion excited externally. PM6 shows more local resonance modes than the other magnets. These modes might have been excited by external vibrations during an imbalanced release process.
4. Conclusion
AC losses in superconducting magnetic bearings are said to be generated by magnetic field inhomogeneity. Analytical models based on the Bean model for the critical state have been developed to calculate AC losses at low frequencies in superconducting bulks. However, these models do not take the magnetic field periodicity, nor the distribution of AC loss into account. Furthermore, no data has been reported about the relationship between AC losses and magnetic field inhomogeneity in an actual superconducting bearing environment. In this paper, the analytical hysteresis loss model in HTS bulks was updated which, now, takes the distribution of AC loss and the frequency of the magnetic field during one complete rotation into account. The relationship between magnetic field inhomogeneity and AC loss was studied experimentally in spin-down measurements by using permanent magnets of the same size but with different levels of inhomogeneity. The results show that the relationship measured experimentally was consistent with the updated analytical calculations. The data reported in this paper also reveals a relationship between magnetic field inhomogeneity and the dynamic behaviour of the superconducting bearing. The resonance region widens and the resonance frequency of the rigid body, and hence the bearing stiffness, decreases with inhomogeneity. This is of further importance when designing and building superconducting bearings for practical applications.
Declaration of Competing Interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: [All authors reports financial support was provided by Ministry of Business Innovation and Employment.]
Acknowledgements
This work was financially supported by the New Zealand Ministry of Business, Innovation, and Employment. The authors wish to acknowledge the invaluable technical advice and assistance provided by Grant Kellett and Mike Davies.
