# Development of a cryogenic induction motor for use with a superconducting magnetic bearing

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3 748 T. Matsumura et al. / Physica C (2005) Motor model Richards and Tinkham [5] analyzed the torque due to eddy currents that is applied to an infinitely large conductive plane when a sinusoidal magnetic field wave with a group velocity m 0 travels above it. We adapt their formalism to describe our motor. The equation of motion of the rotor can be written as s 2p ¼ I _ m f ¼ Aðf 0 f Þ Bf C 2p ; ð1þ where s is the torque on the rotor, I m is its moment of inertia, and f 0 = m 0 /2pr, where r is the radial location on the rotor where the motor applies torque. The first term on the right hand side is called the torque due to ÔslipÕ, that is the difference between the frequency of the traveling wave f 0 and the instantaneous rotation frequency f. There is no torque when the two friction terms on the right hand side balance the driving term coming from slip. The coefficient A can be expressed as A ¼ Aðr; dþi 2 c ð2þ where I c is the RMS of the current applied to the coils, r is the electrical conductivity of the conductive plane, and d is its skin depth. The friction terms describe torques that originate from the SMB system and arise from eddy current losses, which are proportional to Bf and hysteresis losses, which are quantified by C. When using this model a correspondence needs to be made between the frequency of the alternating current in the coils of the motor and the frequency of the traveling wave of the magnetic field. For our physical geometry and assuming coils that are equally spaced circumferentially around the entire disk this correspondence gives that when the frequency of the alternating current is F the equivalent traveling wave frequency is f 0 = 0.16F. Because there are only three coils around a small part of the circumference of the disk we expect this correspondence to be only approximate. A solution of Eq. (1) gives f ðtþ ¼ðf i f f Þe at þ f f ; ð3þ where f i is the initial frequency, and f f is the final frequency of rotation, a ¼ A þ B I m ; f f ¼ f 0A C 2p A þ B ¼ f 0AI 2 c C 2p AI 2 c þ B. ð4þ According to Eqs. (3) and (4) a steady state frequency of rotation f f is attained exponentially. The exponential time constant is determined by the RMS of the current, by the skin depth, by the electrical conductivity, and by friction. For sufficiently large currents in the coils Eq. (4) gives f f = f 0, which for our geometry is f f = 0.16F. 4. Measurements and results 4.1. Validity of motor model We measured the temporal evolution of the frequency of rotation by applying periodic changes in the RMS of the current supplied to the coils and then letting the rotor stabilize to a new final rotation speed. The current was changed in steps between 15 ma and 77 ma RMS and the experiment was repeated for alternating current frequencies of F = 12.5, 25, and 50 Hz. Data from one such experiment is shown in Fig. 2. Following Eq. (3) we fit the data with exponentials and extracted a from the exponent of the fit. The agreement between the fit and the data shown in Fig. 2 is representative of the quality of the fit in all cases Source of friction To measure which of hysteresis or eddy currents is the major source of friction in our motor we let the rotor spin-down freely with the coils not energized. In Fig. 3 we plot the one sample of the data of deceleration as a function of rotation frequency f. According to Eq. (1) we expect df dt ¼ a 0 þ a 1 f. ð5þ A fit to this data gives the fit parameters a 0 and a 1 as s 2 and s 1, respectively.

4 T. Matsumura et al. / Physica C (2005) Fig. 2. The rotation frequency as a function of time. The RMS current changed from 32 ma to 53 ma at t = 470 s. The rotation frequency increased from its initial state to the final state exponentially. The solid line is a fit to the data after t = 470 s using Eq. (3). Fig. 3. The deceleration as a function of rotation frequency during a free spin-down of the rotor Long term stability of rotation It is important to characterize the stability of the rotation over an extended period of time. The top panel of Fig. 4 shows the rotation frequency with a fixed frequency F of 25 Hz and an RMS current of 39 ma. The rotation frequency stays approximately constant for over 8 h of measurement. The bottom panel shows the power spectral density of the rotation frequency after subtraction of a mean over 8 h. The RMS variation is Hz for time scales between 2 s and 8 h.

5 750 T. Matsumura et al. / Physica C (2005) Fig. 4. Top panel: stability of rotation frequency as a function of time. The inset shows a zoom on the data between 4 and 5 h with an expanded scale for the vertical axis. Bottom panel: power spectral density of the data in the top panel after subtraction of a mean Variation of rotation frequency within a single period Using our data we extracted the variation of the rotation frequency within a single period of rotation. A description of the technique is given by Matsumura et al. [4]. The fractional speed variation, d, is defined as d ¼ Df pp ; ð6þ f where Df pp is the peak to peak rotation frequency variation from its mean f within one cycle of rotation. Over the 8 h of data shown in Fig. 4 we find d ¼ 3.4 1% Heat dissipation in the coils In Fig. 5 we plot the temperature of coil as a function of the frequency F. We have also found that the temperature of the coils remains below 5 K when the RMS current to the coil is below 60 ma and the frequency is below 25 Hz. 5. Discussion and conclusions There appears to be agreement between our model for the motor and the data. The temporal Temperature of coil [K] ma rms 5 15 ma rms 20 ma rms 60 ma rms Frequency [Hz] Fig. 5. The temperature of the coil as a function of frequency of the applied current for RMS currents of 7.5 ma, 15 ma, 20 ma, and 60 ma. evolution of the rotation speed follows an exponential, as expected from Eq. (3). Fig. 6 shows the exponent a as a function of the RMS of the AC current for the three different frequencies f 0. The data shows a quadratic dependence on the current in agreement with Eqs. (2) and (4). When we apply sufficiently large currents the frequency f f does approach the expected value f 0 as predicted by Eq. (4). The interpretation of the data in terms of the model is subject to some uncertainty because of

6 T. Matsumura et al. / Physica C (2005) exponent α [sec -1 ] Hz drive 25 Hz drive 50 Hz drive current to coil [ma rms] Fig. 6. The exponent a as a function of the RMS current for frequencies of 12.5, 25, and 50 Hz. The continuous lines are quadratic fits to the data of each of the frequencies. uncertainty in the temperature of the rotor. For a fixed frequency f 0 the torque of the motor depends on the electrical conductivity of the disk, which is a function of its temperature. The change in the RMS of the driving current results in a change in the RMS of eddy currents in the disk and therefore a change in heat dissipation. Because the disk is levitated and has no conductive or convective thermal path to the environment its heating or cooling time scale is long compared to the duration of accelerations or decelerations, and therefore the temperature of the disk may not stabilized during our measurements. Our experiments show that the coefficient of friction in our bearing system is dominated by losses due to eddy currents and not due to hysteresis. This indicates that the presence of the motor increases the friction due to eddy currents because the same experimental setup without the motor gave friction which was dominated by losses from hysteresis [2]. It is interesting to measure the fractional speed variation d because it informs the decision about the angular encoding of the SMB [4]. Previous measurements by our group [4] gave an upper limit of 1% for d at rotation frequency above 1 Hz when the rotor was not driven. The 1% limit was due to noise that originated from the translational vibration of the rotor. We find an increase of d to a level of 3.4% in the presence of drive torque due to the motor. If we assume that the entire contribution to d comes purely from an actual change in rotation frequency, and is not induced by e.g. a translational motion of the rotor, we calculate that an angular encoding of the position of the rotor only once during a full period would give an upper limit on the uncertainty in the determination of angular position of only 1. Such accuracy may be adequate for many polarimeters. The temperature rise of the coil depends on both the RMS and frequency of the applied AC current. Possible heat inputs are Joule heating in the coils, hysteresis loss in the core and in the coils, and eddy currents in the core, in the copper cladding of the coils, and in metallic elements near the coils. The data of Fig. 5 shows a quadratic nature of the increase in temperature with respect to frequency F, which suggests that the source of heat are eddy currents. This conclusion relies on the assumption that the thermal conductance between the source of heat and the thermometer stays approximately constant as a function of temperature. We have identified a range of operation parameters for the motor where the temperature of the coil remains stable and far below the critical temperature of the superconducting NbTi wire of the coils. Acknowledgement The author would like to thank Dr. Ki Ma for the useful discussion. References [1] J.R. Hull, Supercond. Sci. Technol. 13 (2000) R1. [2] S. Hanany, T. Matsumura, B. Johnson, T. Jones, J.R. Hull, K.B. Ma, IEEE Trans. Appl. Superconduct. 13 (June) (2003) [3] Trust Automation Inc., TA 310 linear amplifier. [4] T. Matsumura, S. Hanany, J.R. Hull, B. Johnson, T. Jones, presented in Applied Superconductivity Conference 2004, Jacksonville Florida, IEEE Trans. Appl. Superconduct, in press. [5] P.L. Richards, M. Tinkham, J. Appl. Phys. 43 (1972) 2680.

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