## 2 Machine Structure Operating Principle and Design of the Flux-Controllable NI HTS FSM
### Design of the Flux-Controllable FSM  
In this study, a three-phase FSM with a rated output power of 20 kW, and a rotation speed of 1000 rpm, was considered. The design was carried out through the following process [32,33]. 1. The design specifications of the NI HTS FSM, in this case the output power, flux regulation ratio and phase, are determined. 2. The maximum radius \(r_{r,max}\) and length \(l_{max}\) of the rotor are calculated considering rotor material's mechanical properties, using Equations (1) and (2), respectively. \[r_{r,max}=\sqrt{4\sigma_{yield}/((3+v)\rho\Omega^{2})}\] (1) \[l_{max}=\sqrt{\pi^{2}/k\Omega\sqrt{EI/\rho S}}\] (2)
3. The slot pitch is calculated based on the pole pitch and the numbers of slots. The number of conductors also are calculated using Equation (3). \[S_{slot}=S_{Cu}\cdot n_{0}/\beta=I_{0}\cdot n_{0}/(J_{Cu}\cdot\beta)=d_{slot} \cdot w_{slot}\] (3)
4. The dimensions of the armature conductor and the slot are determined by Equation (4). \[d_{Cu}=2\cdot d_{slot}\cdot\sqrt{\beta}/n_{0},w_{Cu}=w_{slot}\cdot\sqrt{\beta}/2\] (4)
5. The tangential stress in the rotor is calculated using the rotor dimension and the torque via Equation (5). \[\sigma_{F\tan}=T/(2\pi r^{2}l)\] (5)  
Figure 5: Flux regulation principle: (**a**) main magnetic flux by the NI HTS field coil, (**b**) magnetic flux by the additional field coil, and (**c**) regulated magnetic flux. The required air-gap field density, \(B_{0}\), is then calculated by Equation (6). \[B_{0}=\sqrt{2}\pi\sigma_{F\tan}(2r)/(I_{S}n_{0}q\cos\varphi) \tag{6}\]
7. Based on the required flux regulation ratio and \(B_{0}\), the NI HTS field coil and the additional field winding are designed by means of FEM under a no-load condition. 8. The cryostat for the NI HTS field coil is designed based on the HTS field coil and other thermal loads. The thermal loads of the cryostat are calculated by the equations below [39; 40; 41]. \[Q_{CL}=(\sqrt{P_{CL}l_{CL}l^{2}/A_{CL}}-\sqrt{k_{CL}A_{CL}(T_{H}-T_{L})/L_{CL}}^{ 2}+2I\sqrt{\rho_{CL}k_{CL}(T_{H}-T_{L})}\] (7) \[Q_{K}=N_{L}A_{L}k_{L}(T_{H}-T_{L})/L_{L}\] (8) \[Q_{R}=\sigma(T_{H}^{4}-T_{L}^{4})/((1-\varepsilon_{H})/ \varepsilon_{H}A_{H}+(1/\varepsilon_{L}+2N/\varepsilon_{N}-N)/A_{L})\] (9) Here, \(Q_{CL}\), \(Q_{K}\), and \(Q_{R}\) are the thermal loads from the conduction heat transfers by a pair of current leads, the leading-in tubes for the signal lines, and the radiation heat transfer. 9. Characteristic analyses of the deigned machine are carried out in the form of a transient motion analysis, using a commercialized FEM software package. In this design, an NI HTS field coil was considered for field excitation. Therefore, the characteristic resistance, \(R_{C}\), of the NI HTS field coil is required as a design parameter. For this reason, an NI HTS field coil was fabricated to obtain an \(R_{C}\) value. The \(R_{C}\) value can be calculated from a charging time constant (\(\tau\)) by a charging test. Figure 6 shows the NI HTS field coil fabricated in this study. The HTS coil was fabricated using 2G HTS wire with a 12 mm width. It has a double-layered structure with 40 winding turns. Both the effective length and width of the G10 bobbin for the NI HTS coil are 600 mm and 150 mm, respectively. Table 1 shows the specifications of the NI HTS field coil. To determine the characteristic resistance of the fabricated NI HTS field coil, the current charging test was carried out in liquid nitrogen. Figure 7 shows the normalized central magnetic field after current charging at 100 A, with a cryogenic Hall sensor used to measure the magnetic field. After the power supply current reached the target current of 100 A, the magnetic field relaxes over time while holding the current. The field increases logarithmically to approximately 25 mT at 600 s. This outcome indicates that the fabricated NI HTS field coil is affected by the charging delay phenomenon. The typical method for calculating \(R_{C}\) in NI HTS coils is to use a decay time constant after a sudden discharge of the power supply current.