Kobayashi and Komori (Cryogenics 47, 101, 2007) demonstrated a liquid-nitrogen pump driven by a stepping motor whose rotor, three stacked cylindrical YBCO bulk superconductors, is magnetized in situ by a capacitor-discharge current pulse delivered through the motor's own stator electromagnets. Performance is nonmonotonic in the applied magnetization voltage: the maximum stepping speed, the rotor's trapped flux, and the pump head and flow all peak near 350 V and fall at 400 V, and a larger stator driving current leaves less retained flux in the pulsed-field-magnetized rotor. The paper offers no quantitative theory for these observations. The open problem is to establish a validated, coupled electromagnetic-thermal theory of the trapped-flux dynamics of this motor-pump class (pulse magnetization of the multi-bulk rotor through the stator circuit, its modification by the four-phase drive currents during rotation, the resulting torque and speed limits, and the role of axial cracks); once validated, the theory can be applied to predict and optimize operation beyond the tested ranges of magnetization voltage, driving current, and rotational speed.

Background

Melt-grown single-grain Y-Ba-Cu-O (YBCO) bulks are ceramic high-temperature superconductors (HTS) with a superconducting transition near 92 K, so they can be cooled by liquid nitrogen at 77 K. They are type-II superconductors: magnetic flux lines penetrate as quantized vortices and are immobilized by crystal defects (flux pinning), so persistent currents up to the critical current density JcJ_c can circulate without dissipation. After a strong enough applied field is removed, such a bulk keeps a persistent current pattern and acts as a quasi-permanent 'trapped-field' magnet whose field is set by JcJ_c and the sample dimensions; the trapped field slowly relaxes by thermally activated flux creep.

Two generic behaviors of trapped-field bulks matter here. First, pulsed-field magnetization (PFM) magnetizes a bulk in situ by discharging a capacitor bank through a coil rather than by a slowly ramped superconducting magnet; during the pulse, fast flux motion dissipates heat in the low-thermal-conductivity ceramic, lowering JcJ_c so that freshly intruded flux escapes. Consequently the trapped field first grows with applied pulse amplitude, peaks at an optimum amplitude, and then falls, a behavior established by coupled electromagnetic-thermal experiments and simulations (Fujishiro and Naito 2010; Ainslie and Fujishiro 2015) and still driving refinements such as multi-pulse sequences and crossed-field schemes (Yang and Deng 2024; Jiang et al. 2024). Second, cross-field demagnetization: transverse or time-varying fields applied on top of a trapped field redistribute the shielding currents and erode the trapped field, as measured and modeled in 3D for bulk samples (Kapolka et al. 2018) and assessed for tape-stack rotor poles operating in liquid nitrogen (Smara et al. 2019), with mitigation strategies still under development (Jiang et al. 2026).

Against this backdrop, motors and pumps immersed in cryogenic liquid have been explored since the early 2000s because a drive submerged in its own working fluid avoids shaft-feedthrough heat leaks and lubrication problems (Jung et al. 2003; Kovalev et al. 2004; Komori and Uchino 2004). Komori's group at Kyushu Institute of Technology built a stepping motor (a motor that advances in discrete angular steps as its stator electromagnets are energized in sequence) with an eight-electromagnet stator driven in a four-phase sequence and a rotor containing three cylindrical YBCO bulks, each 9 mm in diameter and 11 mm long, stacked in a 10 mm diameter, 46 mm long, 18 g rotor, with an impeller (the pump's rotating bladed wheel that imparts momentum to the fluid) running in liquid nitrogen. Their 2007 Cryogenics paper asked what changes when, instead of zero-field cooling (ZFC, cooling below the transition with no applied field so the rotor starts unmagnetized) or field cooling (cooling in an applied field so that flux is trapped as the superconducting transition is crossed), the rotor is magnetized in place by a capacitor-discharge pulse through the motor's own pulsed-field magnetization circuit.

They measured: the pulse waveform (surface flux density of the electromagnet peaking at about 10 ms, with peak amplitude proportional to the applied voltage over 175-400 V); the maximum stepping speed versus magnetization voltage (100-400 V) and stator driving current (1.0-4.0 A), which rises above the ZFC limit and peaks at 1350 rpm at 350 V, falling to 1050 rpm at 400 V; the rotor's trapped flux (largest at 350 V among 200, 350, and 400 V; a rotor run under ZFC also becomes magnetized by the stator field alone, more strongly at 4.0 A than 1.0 A, whereas after PFM a 4.0 A drive leaves slightly less flux than 1.0 A); the axial flux profile (roughly constant over 10-20 mm but non-smooth because the bulks contain many cracks along the axial direction); and the pump output (the head, meaning the pressure rise expressed as the height of a column of the pumped liquid, and the flow rate, both proportional to speed and independent of driving current; head about 42 mm at 1000 rpm under ZFC versus about 55 mm at 1350 rpm after PFM at 350 V, and maximum flow 0.75 L/min at 1050 rpm under ZFC versus 1.3 L/min at 1350 rpm after PFM).

The paper reports these results without a quantitative model: the origin of the optimum voltage, the interplay between pulsed-field-prepared trapped flux and the stator currents during rotation, the effect of the cracks, and the behavior outside the tested envelope are all left unexplained. Kamijo and Fujimoto (2001) had already shown for isolated bulks that pulsed-field magnetization is heating-limited and can be improved by temperature-controlled repetition, but that result was never connected to motor dynamics.

Problem Statement

Establish a quantitative, experimentally validated theory of the operating mechanism of a pulsed-field-magnetized bulk-YBCO stepping motor driving a liquid-nitrogen pump, of the type demonstrated by Kobayashi and Komori (2007). Use of the validated theory to optimize the system (item 4 below) is an application of the theory rather than a required completion condition. The theory must explain, within one coupled account of the trapped-flux dynamics, the following linked observations: (1) Optimum magnetization voltage: why the maximum stepping speed, the rotor's trapped flux, and the pump output are nonmonotonic in the applied magnetization voltage, peaking near 350 V and falling at 400 V, given the measured pulse-circuit behavior (surface flux density peaking at about 10 ms with peak amplitude proportional to the applied voltage over 175-400 V) and the three-bulk cylindrical rotor geometry. (2) Interaction with the stator drive current: how the four-phase stepping currents, which simultaneously provide torque and act as time-varying transverse fields on the rotor, modify the pulsed-field-prepared trapped flux during rotation, including why a zero-field-cooled rotor becomes progressively magnetized by the stator field alone while a pulsed-field-magnetized rotor retains less flux at a driving current of 4.0 A than at 1.0 A, and how the resulting rotor magnetization sets the maximum stepping speed and the generated torque. (3) Effect of bulk inhomogeneity: to what extent axial cracks and other imperfections in the bulk superconductors limit the amplitude and uniformity of the trapped magnetization and hence the motor and pump performance. (4) Regime beyond the tested envelope (application): once validated against the demonstrated data, the theory can be used to predict operation outside the measured ranges of magnetization voltage, driving current, and rotational speed, including pulse-waveform and pulse-sequence choices, and to explore operating conditions and design changes that improve pump head and flow rate for wider cryogenic pumping duty. Items 1-3 are the explanatory components of the single objective, a predictive model of the trapped-flux dynamics from pulse magnetization through rotating operation to hydraulic output; item 4 is an application of the validated theory rather than a completion boundary, and neither extrapolation beyond the tested envelope nor any new experiment is required for acceptance.

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