1. Introduction
Owing to their high current-carrying density and high operating temperature, HTS magnets wound by the second-generation rare-earth barium copper oxide (REBCO) coated conductors have been attractive for many DC applications, e.g., nuclear magnetic resonance (NMR) [1], [2], electrical aircraft propulsion [3], and electro-dynamic suspension (EDS) system [4], [5]. Among these, closed-loop coils can operate in the PCM [6], [7], [8], thus eliminating considerable heat leakage from current leads [9] and allowing a power-off operation [10], in comparison to open-loop coils [11], [12]. Additionally, considering the good self-protection capability [13], [14] of the NI winding technique [15], NI closed-loop coils are promising in easily-quenching DC applications, which were employed in our EDS system to generate magnetomotive forces of 320-360 kA [10]. In such applications, the performance of coils is determined by their magnetomotive forces, total number of turns, and dimensions. Because total number of turns and dimensions of coils are immutable in practice, the operational current in the PCM is almost the primary interface parameter to the application system. Hence, it is crucial to obtain a reliable accuracy of measuring the operational current for NI closed-loop coils.
The operational current of an HTS coil is calculated by dividing the measured magnetic field by its coil constant, denoted as $C_c$ [16], [17]. For insulated coils, a simple method for $C_c$ is to transport a small current (e.g., 1 A) when the coils are in the normal state and measure the magnetic field [17], [18], thereby eliminating the impact of the SCIF [19]. But for NI coils, this method cannot be adapted because the turn-to-turn contacts [20], [21]. Table 1 compares the measurement methods for $C_c$ of the NI coils, where all methods would be influenced from the SCIF, owing to its reduction on the total magnetic field [22], [23], [19]. Even ignoring the impact of the SCIF, the simulation method [17] may also be disturbed by the resolution, relative position, operating temperature, and calibration of the Hall sensors. For instance, there was a difference of approximately 10% between two types of sensors (HT90150 produced by QD Honor Top Magnetic Technology Co. Ltd. and HGCA-3020 produced by Lake Shore Cryotronics Inc.). This may cause a 10% difference in the measured results, which is unacceptable for applications. The experimentally conventional method can mitigate the impact of Hall sensors. But in practice, it is indispensable to ensure the NI coils operate in steady state and eliminate the radial currents, owing to the nonlinear $C_c$ caused by the turn-to-turn contacts. Accordingly, the conventional method is to constantly transport current when the coils are in the superconducting state [24], [25], [26]. Until the magnetic field stabilizes, $C_c$ can be obtained by dividing the real-time value of the magnetic field by the current value of the power supply. However, it is very time-consuming for this conventional method, especially for NI closed-loop coils with a long charging-delay time; thus, this method is burdensome on refrigeration systems owing to the heat leakage of current leads. Therefore, a relatively accurate and fast method for measuring the operational current is needed for NI closed-loop coils.
Table 1. Measurement methods for coil constants of HTS NI coils. |
| Methods | Coil types | Impactors on measurement accuracy | Measurement Speed |
|---|---|---|---|
| Simulation [17] | All NI coils | Resolution, relative position, operating temperature, and calibration of the Hall sensors, as well as SCIF | Fast |
| Conventional method [24], [25], [26] | All NI coils | SCIF | Slow, decided by the charging delay |
| Proposed method | NI closed-loop coils | SCIF (but can be ignored when directly measuring the operational current) | Moderate |
Based on adjusting the output current of the adjustable power supply and monitoring the coil voltage as the indicator, a measurement method for the operational current of HTS NI closed-loop coils was proposed in this study, where the coils operate in the steady PCM, i.e., the operational current only flows in the azimuthal approach without discrepancy. Detailed operational procedures were proposed, as well as the criteria for guiding those procedures. The proposed method was verified simultaneously and experimentally, whose current accuracy is equivalent to the resolution of the power supply. In comparison to the simulation method, the proposed method has significantly reduced requirements for Hall sensor's performance and location, and has a more reliable accuracy. Additionally, this method presents a significantly faster measuring speed, compared to the conventional method. That is, the larger the inductance and the lower the turn-to-turn resistivity of NI coils, the greater the speed advantage of utilizing the proposed method. The influence of the SCIF was proven to be negligible for measuing $C_c$ of the tested pancake coils. Even for coils whose $C_c$ vibrates significantly owing to the SCIF, this method will also be adoptable to directly measure various operational currents and ignore the impact of the SCIF. Furthermore, it was discussed that the measurement error can be influenced by the current discrepancy among turns when the coil is not in the steady PCM [27], and a procedure with a criterion was proposed to reduce the measurement error.
2. Methods
2.1. Proposed measurement method
The circuit topology of the proposed method is shown in Fig. 1, which consists of an HTS NI coil, Hall sensor, adjustable power supply, and persistent current switch (PCS) [28]. Before measuring the operational current ($I_{op}$), the coil has been charged to operate in the steady PCM and the PCS is in the superconducting state. The key principle of this method is that when the PCS is in the non-superconducting state if the output current of the power supply ($I_{power}$) equals $I_{op}$, the coil would keep a same operating state as that before the measurement. Thus, in this condition, $I_{op}$ can be measured and equal the real-time value of $I_{power}$. For judging the operating state of the HTS coil, the coil voltage ($U_{coil}$) can be employed and monitored as an indicator. According to Kirchhoff's voltage law, the coil voltage is equal to the PCS voltage, which can be derived using Eq. (1):
$ U_{\text {coil }}=R_{\text {PCS }} I_{\text {PCS }}=R_{\text {PCS }}\left(I_{\text {power }}-I_{\mathrm{op}}\right),$
Fig. 1. Circuit topology of the proposed method. |
where $R_{PCS}$,$I_{PCS}$,$I_{power}$, and $I_{op}$ denote PCS resistance, PCS current, current of the power supply, and operational current of the NI coil, respectively. Assuming that the PCS is heated to the non-superconducting state, $U_{coil}$ will be influenced by $I_{power}$. Based on Eq. (1), $U_{coil}$ presents a positive or negative value when $I_{power}$ is larger or smaller than $I_{op}$, respectively. Moreover, when $I_{power}$ is closer to $I_{op}$, the absolute value of $U_{coil}$ is smaller. Hence, the operational current can be measured indirectly by adjusting the output current of the power supply and monitoring coil voltage as an indicator.
The flowchart of the proposed method is presented in Fig. 2, including the detailed procedures and criteria for guiding the procedures. Besides, the definitions and determining principles of the proposed method are summarized in Table 2. The details of Judgement 1 and 2&3 would be discussed in Section 4.4 Measurement error if the coil is not in the steady PCM, 2.2 Numerical description, respectively. First, an HTS NI closed-loop coil assembled with a PCS should be already charged by the power supply. Judgement 1 in Fig. 2 ensures the NI coil is in the steady PCM and no current discrepancy among turns of the coil. Then, the power supply can be initialized to output current and the PCS can be heated to the non-superconducting state. In Judgement 2, the absolute value of $U_{coil}$ is compared to a criterion ($U_c$) for judging whether it is approximately zero or not. If not, this means that $I_{power}$ is different to $I_{op}$; so the value of $I_{power}$ should be decreased or increased after Judgement 3. On the contrary, when the absolute value of $U_{coil}$ is smaller than $U_c$, the real-time value of $I_{power}$ is considered to be the result of $I_{op}$. Then $C_c$ is calculated as:
$ C_{\mathrm{c}}=\frac{B_{0}}{I_{\mathrm{op}}},$
Fig. 2. Flowchart of proposed method for operational current of HTS NI closed-loop coil. |
Table 2. Definitions and determining principles of proposed method. |
| Variables | Definitions | Determining principles | Values for tested coils |
|---|---|---|---|
| B | Real-time magnetic field | / | / |
| $I_{power}$ | Output current of power supply | / | / |
| $U_{coil}$ | Real-time coil voltage | / | / |
| $I_{op}$ | Operational current | / | / |
| $B_0$ | Initial value of magnetic field | Coil is in steady PCM | / |
| $I_{power}$,0 | Initial output current of power supply | Estimating by dividing B0 with ideal Cc | / |
| τ | Time constant of coil in steady PCM | L/Rj,L is coil inductance and Rj is joints resistance | ∼9.42×106 s for Coil B |
| $U_c$ | Criterion of $U_{coil}$ | k×AmRtRPCS/(Rt+RPCS),Rt is total turn-to-turn resistance and RPCS is PCS resistance | 0.005 mV for Coil A, 0.021 mV for Coil B |
| $A_m$ | Current accuracy of measurement | Equalling output resolution of power supply | 0.1 A |
| k | Safety coefficient | / | 0.3 |
where $B_0$ represents the initial value of the magnetic field measured by the Hall sensor. And the $C_c$ value can be employed in other measurements of various operational currents.
2.2. Numerical description
An equivalent circuit model coupled with a 3D FEM model was utilized for describing the electro-magnetic behaviors of the NI coil in this research. Fig. 3 illustrates the topology of the equivalent circuit model, where the NI coil is composed of m repeatable circuit elements connected in series. And each circuit element has a radial branch (i.e., a turn-to-turn branch [15]) and an azimuthal (spiral) branch connected in parallel. The voltage of the $i_{th}$ circuit element is derived as follows:
$ \begin{aligned} U_{\mathrm{e}, i} & =\sum_{k=1, k \neq i}^{m}\left(M_{k, i} \frac{\mathrm{dI}_{\theta, k}}{\mathrm{dt}}\right)+L_{i} \frac{\mathrm{d}_{\hat{\theta}}}{\mathrm{dt}}+R_{\theta, i} I_{\theta, i} \\ & =R_{\mathrm{t}, i} \boldsymbol{I}_{\mathrm{r}, i}, \end{aligned}$
$ R_{\theta, i}=\frac{R_{\mathrm{sc}, i} R_{\mathrm{m}, i}}{R_{\mathrm{sc}, i}+R_{\mathrm{m}, i}}+R_{\mathrm{j}, i},$
Fig. 3. Topology of equivalent circuit model of HTS NI closed-loop coil. |
Where $ U_{\mathrm{o}, i}, I_{\theta, i}, I_{\mathrm{r}, i}, L_{i}, M_{k, i}, R_{\theta, i}, R_{i, i}, R_{\mathrm{sc}, i}, R_{\mathrm{m}, i}$, and $ R_{\mathrm{j}, \mathrm{i}}$ represent the element voltage, azimuthal current, radial current, self-inductance, mutual inductance with the $k_{th}$ circuit element, equivalent azimuthal resistance, turn-to-turn resistance, superconductor resistance, matrix-layer resistance, and joint resistance, respectively. The turn-to-turn resistances of the coil were calculated using Eq. (5) as follows [20]:
$ R_{\mathrm{t}, i}=\frac{\rho_{i}}{S_{i}}, $
where $ \rho_{i}$ denotes the turn-to-turn resistivity and $ S_{i}$ denotes the contact area of the $ i_{\mathrm{th}}$ circuit element. Additionally, the superconductor resistance was calculated by FEM model with considering the E–J power law [29] and the angle dependence of the critical current [30].
Notably, the operational current mainly flows into the azimuthal path in the steady PCM, which implies that $ I_{\theta, i}$ equals $I_{op}$ and $I_{r,i}$ is approximately zero. If the measurement process is initiated and $I_{op}$ does not equal $I_{power}$, the excess current will flow into the turn-to-turn path, and the current change in the azimuthal path is gradual; this is owing to the impedance effect of the azimuthal inductance. So the peak value of $U_{coil}$ could be approximately derived as:
$ U_{\text {coil,peak }} \approx\left(I_{\text {power }}-I_{\mathrm{op}}\right) \frac{R_{\mathrm{t}} R_{\mathrm{PCS}}}{R_{\mathrm{t}}+R_{\mathrm{PCS}}},$
where $R_t$ represents the total turn-to-turn contact resistance. Therefore, the criterion of $U_{coil}$ can be set as:
$ U_{\mathrm{c}}=k \times A_{\mathrm{m}} \frac{R_{\mathrm{t}} R_{\mathrm{PCS}}}{R_{\mathrm{t}}+R_{\mathrm{PCS}}},$
here, $A_m$ is the current accuracy of measurement (unit: Ampere) and k is the safety coefficient ranging from 0 to 1 (set as 0.3 in this study).
3. Demonstration test
To establish a basic understanding for the readership and validate the feasibility of the proposed method, the typical demonstration tests were conducted to measure the operational current of HTS NI closed-loop coils. Two HTS NI closed-loop coils, named Coil A and Coil B, were fabricated for test. Coil A is a single-pancake circular coil, and Coil B is a practical double-pancake racetrack coil employed in the EDS project [27]. Both coils were obtained to operate in the PCM before the demonstration tests. The difference is, the operational current of Coil A was measured previously using the conventional method, whereas that of Coil B was unknown. This is owing to the relatively small charging-delay time of Coil A, the operational current of which is easily charged to be consistent with the output current of the power supply. However, it is much more time-consuming to charge Coil B if utilizing the conventional method.
3.1. Experimental settings
The experimental specifications are listed in Table 3. The schematic of experimental setup in the demonstration test is exhibited in Fig. 4, where the two coils were placed in an LN2 bath, and low-temperature Hall sensors were located inside the coils to measure the magnetic field. The tested coil was connected via current leads to an adjustable power supply (Model G10-340, TDK-Lambda Corporation) with a 0.1-A output resolution. The voltage leads were soldered across the tested coil; they and the Hall sensor were connected to a data acquisition (DAQ) device (Keithley 2700 Multimeter, Tektronix Co Ltd). Using the NI LabView platform, the coil voltage and magnetic field can be monitored in real-time. The thermal-controlled PCSs were employed to charge both closed-loop coils. The PCS of Coil A contained a segment of REBCO conductor, whereas that of Coil B incorporates a solenoid coil. The PCS resistances of Coil A and B were measured to be approximately 0.4 mΩ and 11 mΩ, respectively, after they were heated and their temperatures stabilized above 100 K in an LN2 bath. Because it is rather difficult to accurately measure the PCS resistance when its temperature changes quickly, this study simulated the variation in the PCS resistance of Coil B, as illustrated in Fig. 5, and the simulation results were in good agreement with the experimental data. Additionally, the total turn-to-turn resistances of Coil A and B were previously measured to be approximately 336 and 744 μΩ, respectively; hence, the $U_c$s of both coils were determined as 0.005 and 0.021 mV, respectively.
Table 3. Experimental specifications. |
| Items of conductor | Parameters or details |
|---|---|
| Type | BHO-doped REBCO tape, Shanghai Superconductor Technology Co. Ltd |
| Width and thickness | 6 and ∼0.19 mm |
| Items of Coil A (single-pancake) | Parameters or details |
| Number of turns | 360 turns |
| Inner and outer radius | 35 and 102 mm |
| Inductance | 17.5 mH |
| Conductor length in PCS | ∼10 cm |
| PCS resistance when heated, $R_{PCS}$ | 0.4 mΩ |
| Total turn-to-turn resistance, $R_t$ | 336 μΩ |
| Hall sensor | HT90150 (QD Honor Top Magnetic Technology Co. Ltd.) |
| Items of Coil B (double-pancake) | Parameters or details |
| Number of turns | 460 turns per pancake coil |
| Length of linear edge | 140 mm |
| Inner diameter of arc edge | 180 mm |
| Inductance | 393.5 mH |
| Joints resistance | ∼41.4 nΩ |
| Number of repeatable circuit elements | 18 |
| Inner diameter of PCS | 50 mm |
| Turns of PCS | 80 |
| PCS resistance when heated, $R_{PCS}$S | 11 mΩ |
| Total turn-to-turn resistance, $R_t$ | 744 μΩ |
| Hall sensors | HGCA-3020 (Lake Shore Cryotronics Inc.) and HT90150 |
Fig. 4. Schematic of experimental setup: (a) apparatus; (b) dimensional drawing of Coil A; dimensional drawings of Coil B in (c) front view and (d) lateral view. |
Fig. 5. Variation of PCS resistance of Coil B for simulations. |
3.2. Demonstration test for coil A
Before the test, Coil A has been charged to be 20.0 A and operated in the PCM. Fig. 6 presents the output current of the power supply and the coil voltage during the demonstration test for Coil A; here, the 0 s of the time axis is denoted as the moment when there is a noticeable change in coil voltage, indicating the transition of the PCS from a superconducting state to a non-superconducting state. It was depicted that $I_{power}$ initially outputted 21.0 A, and $U_{coil}$ increased to exceed $U_c$ before 20 s, owing to the increase of resistance of PCS when heated. This implied that $I_{power}$ is larger than $I_{op}$ and should be decreased, according to Judgement 3 in Fig. 2. By adjusting $I_{power}$,$U_{coil}$ changed accordingly. Even when $I_{power}$ was 20.1 A or 19.9 A at 85–110 s, close to the preset value of 20.0 A, the absolute value of $U_{coil}$ was obviously larger than $U_c$. Until when $I_{power}$ was 20.0 A after 110 s, the absolute value of $U_{coil}$ reached below $U_c$, which concluded that the operational current of Coil A was 20.0 A, according to Judgement 2 in Fig. 2. Furthermore, because the magnetic field was measured as 80.905 mT by sensor HT90150, the $C_c$ of Coil A was calculated as 4.0453 mT/A. Hence, the measured result was in agreement with the preset value, and the proposed method was experimentally validated.
Fig. 6. Output current of power supply and coil voltage during demonstration test for Coil A, with boundary of voltage criterion denoted by $U_c$ (dashed lines). |
3.3. Demonstration test for coil B
Coil B was charged and operated in the stable PCM before the test, and its operational current was unknown. The results of simulations and experiments of the demonstration test are exhibited in Fig. 7. $U_{coil}$ increased until 100 s, owing to the increased resistance of PCS when heated. By adjusting $I_{power}$,$U_{coil}$ changed accordingly. Until $I_{power}$ reached 22.1 A after 330 s, the absolute value of $U_{coil}$ reached below $U_c$. Hence, the operational current was measured as 22.1 A according to the Judgement 2 in Fig. 2. By inputting this result of 22.1 A and the measurement settings into the simulation, the simulated coil voltage was in good agreement with the experimental measurement.
Fig. 7. Output current of power supply and coil voltage during demonstration test for Coil B, with boundary of voltage criterion denoted by $U_c$ (dashed lines). |
4. Discussion
4.1. Accuracy of proposed method
The accuracy of the proposed method was discussed in this section through simulations. Assuming that there were two additional cases of original values for the operational current ($I_{op,0}$) of Coil B in the demonstration test of Section 3.3, referring to 22.00 and 22.20 A. By applying the measurement process described in Section 3.3, the simulated results of coil voltages are presented in Fig. 8. It was depicted that when $I_{op,0}$ was set to 22.00 or 22.20 A, the simulated profiles nearly bounded the experimental coil voltage. Although there is a mismatch between the simulations and the experiment before 150 s in Fig. 8, it is mainly owing to the difficulty in accurately simulating the PCS resistance. Therefore, the actual operational current may fall between 22.00 and 22.20 A. And the measurement error in the demonstration test is within 0.1 A according to the measured result of 22.1 A in Section 3.3; thus, the current accuracy of the demonstration test was considered as 0.1 A.
Fig. 8. Simulated results of coil voltages in cases of setting $I_{op,0}$ as 22.00 and 22.20 A, compared to the experimental data (denoted as “Exp”). |
Further virtual measurement tests on Coil B were conducted to validate the current accuracy of the proposed method; here, the virtual measurement refers to simulating the behaviors of the coil by the equivalent circuit model coupled with the FEM model, and then manually adjusting $I_{power}$ and monitoring $U_{coil}$ to measure $I_{op}$. By setting up several cases of $I_{op,0}$, the results and errors of these virtual measurements are detailed in Table 4. Notably, all presented measurement errors were within 0.1 A, which is consistent with the resolution of the adjustable power supply. This is attributed to the fact that the proposed method concernes the relative value of $U_{coil}$ in each output step of $I_{power}$. Hence, the current accuracy of this method is verified to be equivalent to the resolution of the power supply.
Table 4. Results and errors of operational current in five virtual measurement cases. |
| $I_{op,0}$ | Measured $I_{op}$ | Error |
|---|---|---|
| 22.00 A | 22.1 A | 0.1 A |
| 22.05 A | 22.1 A | 0.05 A |
| 22.10 A | 22.1 A | 0 A |
| 22.15 A | 22.2 A | 0.05 A |
| 22.20 A | 22.2 A | 0 A |
Moreover, the initial value of the magnetic field ($B_0$) was measured by two types of Hall sensors; consequently, the $C_c$ was obtained according to the $I_{op}$ result of 22.1 A, as listed in Table 5. It was illustrated that the $C_c$ value calculated by the simulation method was different from the measured values by Hall sensors, with discrepancies ranging from approximately 1.6% to 11.8%. These errors may arise from the inherent accuracies and calibration inconsistencies of the Hall sensors, as well as the deviation between actual and simulated positions. The deviation between the experimental and simulated values of $C_c$ can adversely affect the performance of HTS coils in engineering applications, where the magnetomotive force represents a pivotal parameter, notably in systems such as EDS and NMR. For instance, when utilizing the simulated $C_c$ and HT90150, the magnetomotive force of HTS coils could be measured as approximately 10% below the realistic value. In an EDS system, this can diminish the vehicle's levitation force, whereas in NMR, it might degrade the instrument's resolution. Hence, if utilizing the simulation method, the performances of Hall sensors must be good enough and the positions must be exact enough as that in simulation, then a reliable measurement accuracy can be obtained. In comparison, the proposed method significantly reduces the demands on the performances and locations of Hall sensors. As long as the sensors exhibit good linearity, it is possible to obtain the corresponding $C_c$ values for different sensors, and the measurement accuracy of operational current is reliable.
Table 5. Comparison of measured results of Coil B by simulation and proposed method. |
| Method | Measured $B_0$ | Measured $C_c$ |
|---|---|---|
| Simulation | 89.741 mT | 4.061 mT/A |
| Proposed method | 100.311 mT (using sensor HT90150) | 4.539 mT/A |
| Proposed method | 91.239 mT (using sensor HGCA-3020) | 4.128 mT/A |
4.2. Speed of proposed method
The speed of the proposed method was discussed and compared with that of the conventional method in this section using virtual measurements. Using the proposed method and the conventional method, the virtual measurements were conducted for the EDS magnet employed in our project [10], which was composed of three double-pancake coils comparable to Coil B in size. Here, the EDS magnet had an inductance of 3.4 H and a turn-to-turn resistivity of 10 μΩ·cm2. In the virtual measurement with using the conventional method, the power supply constantly transported a current of 20.0 A into the EDS magnet. The variations of the magnetic field and its deviation from the ideal value (denoted as $B_{ideal}$) are presented in Fig. 9, noting that the unit of time axis is hour; here, $B_{ideal}$ of EDS magnet was calculated as 231.225 mT when $I_{op}$ was 20.0 A. It was depicted that B kept increasing and its deviation to $B_{ideal}$ decreased with transporting period, demonstrating the magnetic field of EDS magnet was becoming stable. Accordingly, the measured value of $C_c$ will be more precise when the transporting period is longer. Besides, for achieving a 0.1-A current accuracy as the demonstration test in Section 3.3, it is necessary to constantly transport current for at least 13 h. This is because the deviation between B and $B_{ideal}$is smaller than approximately 1.156 mT after 13 h, corresponding to an $I_{op}$ of approximately 0.1 A. The long transporting period is owing to the turn-to-turn contacts of NI coils [20], so the coils have a significant charging delay. Besides, this long measuring period of the conventional method is unacceptable for industrial applications and burdensome for refrigeration systems.
Fig. 9. Magnetic field and its difference to $B_{ideal}$ of EDS magnet during virtual measurement when using the conventional method. |
However, because the proposed method monitors the value of $U_{coil}$ at the initial moment after switching $I_{power}$, as derived in Eq. (6), the measuring period is relatively short. For instance, if utilizing the proposed method for EDS magnet in the virtual measurement, the variations of $I_{power}$ and $U_{coil}$ are presented in Fig. 10. It was illustrated that only when $I_{power}$ reached 20.0 A after 309 s, the absolute value of $U_{coil}$ approximately reached below $U_c$ (0.012 mV for the EDS magnet). Hence, the operational current was measured as 20.0 A, consistent with the initial setup. The total measuring period was demonstrated to be several hundred of seconds, much shorter than that of using the conventional method. Therefore, the proposed method presents a significantly faster measuring speed compared to the conventional method. And it is notable that the charging-delay time of the NI coil is determined by its inductance and turn-to-turn resistivity. Therefore, the larger the inductance and lower the turn-to-turn resistivity of the NI coils, the greater the speed advantage of the proposed method.
Fig. 10. Output current of power supply and coil voltage of EDS magnet during virtual measurement when using proposed method. |
4.3. Impact form SCIF
$ B_{\text {actual }}=B_{\text {ideal }}+B_{\mathrm{SCIF}}=C_{\mathrm{c}, \text { iedal }} \times I_{\mathrm{op}}+B_{\mathrm{SCIF}} $
Where $B_{\text {actual }}, B_{\mathrm{SCIF}}$, and $C_{c, i e d a l}$ represent the actual magnetic field, SCIF, and ideal coil constant, respectively. Hence, the impact of the SCIF should be considered for measurement on the actual $C_c$, and additional experiments and simulations were conducted for discussion.
Owing to the simple circular geometry of Coil A, a 2D-axisymmetric homogeneous H-formulation FEM model [32] was built to calculate the SCIF of Coil A. The homogeneous model artificially expands the superconducting layer's thickness to the conductor's thickness, hence the critical current density is derived as:
$ J_{\mathrm{c}}=\frac{J_{\mathrm{c} 0}}{\left(1+\frac{\sqrt{k_{\mathrm{s}}^{2} B_{\perp}^{2}+B_{\|}^{2}}}{B_{\mathrm{a}}}\right)} \times \frac{T_{\mathrm{HTS}}}{T_{\text {conductor }}}, $
where $J_c$ is the critical current density, $B_⊥$ and $B_{\|}$ are the parallel and perpendicular components of magnetic flux density with respect to the conductor's surface, respectively, $T_{HTS}$ and $T_{conductor}$ are the thickness of superconducting layer and conductor, respectively. And for the conductor employed in this study, $T_{HTS}$ equals 1 μm, $T_{conductor}$ equals 186 μm, $B_a$ equals 106 mT, $J_c$ equals 3.5158×1010A/m2,$k_a$ equals 0.518, and α equals 0.74. Hence, the nonlinear resistivity of the superconductor follows the equation as:
$ \rho=\frac{E_{0}}{J_{\mathrm{c}}}\left|\frac{J}{J_{\mathrm{c}}}\right|^{n-1},$
where J is the transporting current density and n equals 24 for the conductor in this study. For the NI coil, the equivalent circuit model was coupled to calculate the transporting current density. Besides, a rectangular mesh with 8×50 elements for the coil was set in the model.
The value of $C_c$ was experimentally measured in several cases of operational current, where the charging processes were the same as that in Section 3.2. The simulated and experimental results of $C_c$ of Coil A are depicted in Fig. 11. It was illustrated that simulated values of $C_c$ were smaller than ideal Cc, owing to the field reduction from the SCIF. As the operational current increased, the simulated $C_c$ would also increase, owing to the hysteresis of the SCIF [33], [34]. And the variation was approximately 0.1% for values of $C_c$ with considering the SCIF, which was small enough to be ignored. This is because the SCIF occupies only a minor fraction of $B_{actual}$ for the pancake-shaped coil [19]; thus, its hysteresis change in different cases of operational current is negligible. For instance, the SCIF constituted only 3.1% of $B_{actual}$ for a pancake coil in the reference [19] with an inner diameter of 18 mm, outer diameter of 109 mm, and height of 8.3 mm. Whereas, the experimental values of $C_c$ were all larger than ideal Cc, which may arise from the inherent accuracy, relative position, or calibration of the Hall sensor. And the experimental values did not increase with the operational current, which may be attributed to the measurement error and SCIF. The variation of experimental values was approximately 0.2%, which is also negligibly small to be ignored for engineering applications. Hence, it can demonstrate that the SCIF has a minor impact on $C_c$ for the pancake-shaped Coil A. For the pancake-shaped Coil B, the SCIF was also calculated to have a negligible impact, so the proposed method was employed to calculate the $C_c$ of EDS magnet in our project. However, for coils whose $C_c$ vibrates significantly owing to the SCIF, it is suggested to utilize the proposed method to measure the operational current directly, rather than employing Cc. This is because the proposed method only monitors $U_{coil}$ as the indicator, which cannot be influenced by the SCIF.
Fig. 11. Simulated and experimental results of coil constant of Coil A, as well as ideal coil constant (solid line). |
4.4. Measurement error if the coil is not in the steady PCM
The above results and discussions are established on the cases of steady PCM of the NI closed-loop coil; however, a condition has not been considered that when the coil is not in the steady PCM. According to our previous study [27], when the NI coil has been charged and the PCS has returned to the superconducting state, the coil would undergo a transient process before reaching the steady PCM, i.e., there is a discrepant distribution of azimuthal currents. For instance, Fig. 12 exhibits the azimuthal currents, output current of the power supply, and magnetic field in the experimental charging and subsequent transient processes of Coil B; these processes were before the demonstration test of Section 3.3. In addition, the maximum output current of the power supply ($I_{power,max }$) was 30 A, and the PCS began to turn into the superconducting state at 1250 s, when the transient process was initiated. It was illustrated that the discrepant azimuthal currents redistributed and converged during the transient process after charging; thus, the magnetic field underwent a rapid decrease and then keep a natural decay.
Fig. 12. Variations of currents and magnetic field of Coil B in charging and subsequent transient processes. |
For discussing the measurement error when the coil is not in the steady PCM, a typical virtual measurement was simulated and initiated on the moment of 1400 s of the time axis in Fig. 12, when PCS had returned to the superconducting state. The results of this virtual measurement are exhibited in Fig. 13. According to Fig. 13a, the operational current was easily mistaken as 22.9 A according to the variation in $U_{coil}$ at 1510–1530 s, which increased from a negative value to be in the range of ±$U_c$. Whereas, $U_{coil}$ continuously increased and exceeded $U_c$ after 1530 s, illustrating that there was an error between the measurement result and $I_{op}$. Similarly, in the next step of $I_{power}$ switching to 22.7 A, $U_{coil}$ also exhibited an increase from a negative value to a positive value, and was in the range of ±$U_c$ at 1570–1600 s. This would result in a measured value of 22.7 A. However, $I_{op}$ was eventually measured as 22.1 A according to discussions in Section 3.3 Demonstration test for coil B, 4.1 Accuracy of proposed method. This inconsistency was attributed to that the discrepant azimuthal currents made the coil voltage vibrate keenly, as shown in Fig. 13b. To characterize the discrepancy of azimuthal currents, their maximum difference (denoted as $I_{diff}$) was employed. It was illustrated that $I_{diff}$ ranged from approximately 3 to 10 A during the virtual measurement, as also presented in Fig. 13b.
Fig. 13. Results of virtual measurement of Coil B, when coil was not in steady PCM. |
Additional virtual measurements for Coil B were conducted and initiated at various moments during the transient process, with results listed in Table 6. And the variations of normalized magnetic field and $I_{diff}$ in the transient process are presented in Fig. 14. The magnetic field of the coil underwent a rapid decrease in the transient process and then stabilized, along with $I_{diff}$. And it was illustrated that the measurement accuracy was improved when $I_{diff}$ was smaller. When $I_{diff}$ was negligible after 1800 s, the measurement error kept within 0.1 A, thus achieving the current accuracy of the proposed method. Hence, it can be concluded that the current discrepancy during the measurement process can enlarge the measurement error. Besides, when the current discrepancy is relatively large, the non-uniform operational current is not suitable for calculating the magnetomotive force of the coil, hence, the measurement is meaningless when the NI coil is not in the steady PCM.
Table 6. Measured results of operational current of Coil B during transient process. |
| Initiated moment | Measured $I_{op}$ | Error compared to result in demonstration test |
|---|---|---|
| 1400 s | 22.9 A | 0.8 A |
| 1500 s | 22.5 A | 0.4 A |
| 1600 s | 22.4 A | 0.3 A |
| 1700 s | 22.3 A | 0.2 A |
| 1800 s | 22.2 A | 0.1 A |
| 2000 s | 22.2 A | 0.1 A |
| 2300 s | 22.2 A | 0.1 A |
Fig. 14. Variations of normalized magnetic field and maximum difference of azimuthal currents during transient process. |
To guide the moment of initializing the measurement process and eliminate the impact of the current discrepancy, the decay of the magnetic field was employed. Three cases of charging Coil B with $I_{power,max }$ of 10, 30, and 50 A were simulated. The corresponding variations of normalized magnetic fields in the transient process and steady PCM are presented in Fig. 15. It was illustrated that all the normalized fields underwent a rapid decrease and then maintained an identical stable decay with a same rate of approximately 1.052×10-7 s-1. This is because the current discrepancy becomes negligible, and then the coil steps into the steady PCM; here, the magnetic field of coil undergoes a natural decay and can be derived as follows:
$ B(t)=B_{0} e^{-\frac{t}{t}},$
where $B_0$ and τ are the initial value of the magnetic field and time constant of the NI coil in the steady PCM, respectively. Besides, the time constant is mainly decided by the joints resistance and inductance of the coil, which is approximately 9.42×106 s for Coil B. The time constant of NI coil is usually large, hence according to the Taylor Formula, the Eq. (11) can be simplified as follows:
$ B(t) \approx B_{0}\left(1-\frac{t}{\tau}\right).$
In conclusion, if the decay rate of the normalized magnetic field equals 1/τ (Judgement 1 in Fig. 2), the current discrepancy can be negligible and the coil is in the steady PCM, thus the measurement process can be initiated with a reliable accuracy. And the impact of the charging/discharging level should be small enough in the measurement process, otherwise, a larger measurement error is still inevitable.
Fig. 15. Variations of normalized magnetic fields in the transient process and steady PCM in simulated cases of charging coil with $I_{power,max }$ equaling 10, 30, and 50 A. |
5. Conclusion
In this study, a method for measuring the operational current of HTS NI closed-loop coils in the steady PCM with reliable accuracy and moderate speed was proposed, whose procedures and criteria were detailed in Fig. 2. By adjusting the output current of the adjustable power supply, the operational current can be obtained when the absolute value of the monitored coil voltage is below $U_c$. Two HTS NI closed-loop coils were fabricated and tested to validate the proposed method, and the experimental data presented good agreements with the simulated results, which were obtained using the equivalent circuit model coupled with the FEM model. It is demonstrated that the current accuracy of the proposed method is equivalent to the resolution of the power supply. Compared to the simulation method, the proposed method significantly reduces requirements for Hall sensor's performance and location. That is, as long as the linearity of the sensor is good enough, the proposed method would present a reliable accuracy. Compared with the conventional method, the proposed method exhibits a faster measuring speed, especially for NI coils with large inductance and low turn-to-turn resistivity. Moreover, the influence of the SCIF was proven to be negligible for the $C_c$ of the tested pancake coils; whereas, for coil whose $C_c$ changes significantly because of the SCIF, the proposed method would also be adapted for direct measurement of various operational currents. Lastly, a procedure with a criterion for decay rate of normalized B was proposed to reduce the measurement error from current discrepancy and ensure that the coil is within the steady PCM.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work is supported by National Natural Science Foundation of China (NSFC) under project 51977130.
