Analyses of Transient Behaviors of No-Insulation REBCO Pancake Coils During Sudden Discharging and Overcurrent

Abstract

Stability margin of a high-temperature superconducting (HTS) coil is two or three orders of magnitude greater than that of a low-temperature superconducting coil. In recent years, many papers have reported test results of turn-to-turn no-insulation (NI) HTS coils having extremely enhanced thermal stability, such that burnout never occurs in an NI coil, even at an operating current exceeding 2.5 times the critical current. Thus, The main goal of this paper is to clarify transient electromagnetic and thermal behaviors and mechanism of the high thermal stability in an NI REBCO coil. A partial element equivalent circuit (PEEC) model is proposed for the numerical simulation of an NI REBCO coil, which considers a local electrical contact resistance between turns, an I–V characteristic of an REBCO tape, and local self and mutual inductances of the NI REBCO coil. Using the PEEC model, we investigate the influence of the turn-to-turn contact resistance on the transient behavior of the NI REBCO coil during sudden discharging. We also perform thermal conduction analyses with the PEEC model to clarify the transient behavior of an NI REBCO coil during an overcurrent operation.

I. Introduction

RECENT achievements in high-temperature superconducting (HTS) coil technologies have enabled high efficiency and compactness of next-generation magnetic resonance imaging and nuclear magnetic resonance, whose required magnetic fields are above 7 and 24 T, respectively. In these applications, the operation reliability of an HTS coil, particularly at a high overall current density under high magnetic field, is crucial. It is well known that a premature quench, often initiated by mechanical disturbances in an low temperature superconducting (LTS) coil, is extremely rare in an HTS coil, mainly owing to its large enthalpy margin, typically an order of J/cm³. In consequence, however, the protection of an HTS coil becomes challenging, which is particularly important to secure operation reliability of high field HTS magnets [1].

Hahn, et al. have reported the so-called no-insulation (NI) HTS winding technique that significantly enhances thermal stability of an HTS pancake coil and thus the coil can be operated at a high overall current density [2]. The key idea is elimination of turn-to-turn insulation in an HTS coil, which allows currents to bypass a local hotspot through the turn-to-turn contacts in an event of a quench as shown in Fig. 1 [2], [3]. As a consequence, an NI coil requires a minimal amount of stabilizer layers, which makes the NI coil highly compact and yet more reliable than its insulated counterpart. The compactness of an NI coil significantly reduces the amount of HTS conductor required for its insulated version. To date, the advantages of the NI HTS winding technique have been demonstrated mostly in experimental manners [4], [5]; a simple circuit model that consists of a coil inductance and a parallel resistance, representing turn-to-turn contacts, was used to roughly analyze electromagnetic behaviors of NI coils [6]. Therefore, a more elaborate numerical approach is needed to accurately simulate transient thermal and electromagnetic phenomena in an NI coil.

Fig. 1.

Features in terms of (a) high current density and (b) high thermal stability in an NI REBCO pancake coil.

We have developed a partial element equivalent circuit (PEEC) model [7], [8] that takes into account the local turn-to-turn contact resistance, the I–V characteristic, and the self and mutual inductances of the local coil elements within the NI winding to discuss the transient behavior in the NI coil in detail. Using the developed PEEC model, the electromagnetic and thermal behavior during sudden discharging and overcurrent are discussed in this paper. To confirm the high stability of an NI HTS coil, it is important to clarify the transient behaviors of sudden discharging and overcurrent; such situations occur when a power supply is suddenly cut off or a normal-state transition occurs in an NI HTS coil.

II. Simulation Model

A. PEEC

As known, a greater number of discrete elements results in a longer computation time, although a higher accuracy is realized. Owing to the analysis efficiency enhancement in the PEEC analysis for sudden discharging, an NI pancake coil is subdivided into 18 azimuthal divisions per turn. Further, an NI pancake coil is subdivided into 72 azimuthal divisions per turn for the PEEC-thermal coupled analysis for overcurrent. The common PEEC model is shown in Fig. 2, and it is constructed on the basis of Kirchhoff’s first and second laws, which are expressed as follows:

$$R_{\theta }^{(i)}{I}_{\theta }^{(i)}+\sum _{j=1}^N{M}_{ij}\frac{\text{d}{I}_{\theta }^{(j)}}{\text{d}t}+R_{\text{r}}^{(i+1)}{I}_{\text{r}}^{(i+1)}=R_{\theta }^{(i+Div\theta )}{I}_{\theta }^{(i+Div\theta )}+\sum _{j=1}^N{M}_{(i+Div\theta )}j\frac{\text{d}{I}_{\theta }^{(j)}}{\text{d}t}+R_{\text{r}}^{(i)}{I}_{\text{r}}^{(i)}$$ (1)

$$I_{\theta }^{(i)}+I_{\theta }^{(i)}=\{\begin{array}{ll}{I}_{\text{op}}\hfill & (i=1)\hfill \\ I_{\theta }^{(i-1)}\hfill & (1<i⩽Div\theta )\hfill \\ I_{\theta }^{(i-1)}+I_{\theta }^{(i-Div\theta )}\hfill & (N-Div\theta +1<i<N)\hfill \end{array}$$ (2)

$$I_{\theta }^{(i)}=I_{\theta }^{(i-1)}+I_{\text{r}}^{(i-Div\theta )}(N-Div\theta +1<i<N)$$ (3)

$$I_{\theta }^{(i)}+I_{\text{r}}^{(i-Div\theta +1)}=I_{\text{op}}(i=N)$$ (4)

where R_θ and R_r are the electrical resistances of the winding in the azimuthal direction based on the I – V characteristic and the turn-to-turn contact resistance, respectively. M_ij is the matrix that includes the self and mutual inductances of the partial elements. I_θ and I_r are the currents flowing in the azimuthal and radial directions, respectively. Divθ is the division of each turn of the windings in the azimuthal direction. i and j are the indices of the local winding from 1 to N (N = Divθ × Turns). The turn-to-turn contact resistivity is supposed to be 70 μΩ · cm², as obtained from the experimental result in the previous study [9].

Fig. 2.

Sample of the PEEC model of an NI REBCO pancake coil, which has 8 azimuthal divisions per turn.

B. PEEC-Thermal Coupled Analysis

To investigate the transient behavior in detail during an over-current, it is necessary to add a heat conduction analysis to the PEEC model to accurately evaluate the thermal dependence of the physical parameters including the electrical resistance, heat capacity, thermal conductivity, and I–V characteristic in the local winding. Here, the I–V characteristic of a REBCO tape has a thermal as well as a magnetic field dependence, we calculated the magnetic field based on Biot-Savart law and import it to simulation model. The thermal conduction equation is given as follows:

$$C(T)\frac{\partial T}{\partial t}=\nabla \cdot (\lambda (T)\nabla T)+Q_{\text{C}}+Q_{\text{J}}$$ (5)

where T, C(T), λ(T), Q_C, and Q_J, are the temperature, heat capacity, thermal conductivity, cooling condition, and Joule heat generation, respectively. The heat capacity and thermal conductivity are considered as composed of REBCO, copper, and Hastelloy, i.e., the thermal conductivities in the azimuthal [λ_θ(T)] and radial [λ_r(T)] directions are different because of the anisotropy of heat conduction. In the simulation, λ_r(T) is assumed to be one-twentieth of λ_θ(T). In addition, an adiabatic condition (Q_C = 0) is adopted as the cooling condition for simplicity.

C. Calculations of the Self and Mutual Inductances

The self and mutual inductances of the partial elements shown in Fig. 2 are calculated on the basis of the Neumann formula as follows:

$$M_{ij}=\frac{{\mu }_0{r}_i{r}_j}{4\pi w^2}\underset{0}{\overset{\lambda }{{\displaystyle \int }}}\underset{{\Theta }_j}{\overset{{\Theta }_j+\Delta \Theta }{{\displaystyle \int }}}\underset{0}{\overset{\lambda }{{\displaystyle \int }}}\underset{{\theta }_i}{\overset{{\theta }_i+\Delta \theta }{{\displaystyle \int }}}\frac{\cos \left(\theta -\Theta \right)}{R_{ij}}d\theta dzd\Theta dZ$$ (6)

$$R_{ij}=\sqrt{r_i^2+r_j^2-2r_i{r}_j\text{cos}(\theta -\Theta )+{(z-Z)}^2}$$ (7)

where M_ij is the self or mutual inductance between the partial elements. θ, Θ, z, and Z are the positional variables related to the intersection angle and the width between the partial elements. R_ij is the distance between partial elements i and j. r_i and r_j are the distances from the central axis of the pancake coil to the partial elements i and j, respectively. w is the width of the coil windings. Δθ and ΔΘ are the angle intervals of partial elements i and j, respectively. In this calculation, we assumed that the thickness of the HTS tape is almost zero.

D. Calculation of the I–V Characteristic of a REBCO Tape

In this simulation, a percolation model [10] is used to express the I–V characteristic of a REBCO tape; its formulae are as follows:

$$E=\frac{{\rho }_{\text{ff}}J}{m+1}{\left(\frac{J}{J_0(T,B)}\right)}^m{\left(1-\frac{J_{\text{cm}}(T,B)}{J}\right)}^{m+1}\text{\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}}T<T_{GL}$$ (8)

$$E=\frac{{\rho }_{\text{f}f}J}{m+1}{\left(\frac{J}{J_0(T,B)}\right)}^m\text{\hspace{1em} for }T=T_{GL}$$ (9)

$$E=\frac{{\rho }_{\text{ff}}\left|J_{\text{cm}}(T,B)\right|}{m+1}{\left(\frac{\left|J_{\text{cm}}(T,B)\right|}{J_0(T,B)}\right)}^m\times \left\{{\left(1+\frac{J}{\left|J_{\text{cm}}(T,B)\right|}\right)}^{m+1}-1\right\}\text{\hspace{1em}\hspace{1em} for }T>T_{GL}$$ (10)

where ρ_ff is the differential resistivity during the flux flow. m is the shape parameter of the critical current density. J_cm(T, B) and J₀(T, B) are the minimum value of the variability in the local critical current and the range of the variability in the local critical current, respectively. T_GL is the glass-liquid transition temperature in the HTS. E, J, B, and T are the voltage of the unit-length tape, the current density, the magnetic field, and the temperature, respectively.

III. Analysis on Transient Behavior of NI REBCO Pancake Coil During Sudden Discharging

The sudden discharging is required because of the sudden power breakdown due to electrical failure or other unforeseeable accidents in practical application. The NI REBCO pancake coil, unlike turn-to-turn insulated HTS coil, has turn-to-turn conduction; therefore, the radial bypass current was inferred as a risk to result in a temperature increase of the coil owing to Joule heat generation during sudden discharging. However, the transient distributions of current and temperature have not been clarified yet. In this section, we introduce a numerical simulation which can predict the transient electromagnetic and thermal behaviors, and also can evaluate the influence of the turn-to-turn contact resistance on the transient behavior of the NI REBCO coil, based on the proposed PEEC model.

A. Comparison of the Experimental and Analysis Results

The experimental results of a 60-turn test coil obtained in a previous study [2] were adopted to verify the validity of the PEEC model, and a simulation was conducted to investigate the local electromagnetic behavior inside the NI REBCO coil. In the experiment, an operating current of 30 A was applied for 2 s, and the current breaker then tripped. On the basis of the experiment in [2], the operation current I_op of the power supply was assumed to decrease from 30 to 0 A within 0.01 s, and the time step was 10⁻⁴ s in the simulation.

Fig. 3 shows a comparison of the experimental and analysis results during sudden discharging. The analysis obviously reproduced the central magnetic field response in the sudden discharging test carried out in the previous study using the specifications of the NI pancake coil listed in Table I. The validity of the PEEC model is confirmed as shown in Fig. 3.

Fig. 3.

Comparison of the experimental and analysis results during the sudden discharging of NI REBCO pancake coil.

TABLE I.

Specifications and Simulation Results of 60-Turn NI REBCO Pancake Coil in Sudden Discharging Test

	60-turn NI pancake coil
Contact surface resistivity, Rct (μΩ·cm2)	70	350	700
Time constant, τ (ms)	810	170	80
Inductance, Lcoil (μH)		432
Operation current, Iop (A)		30
Stored magnetic energy (mJ)		194.5
Energy consumed by turn-to-turn contact resistance, QJ (mJ)		194.5
Coil temperature increase, ΔT (K)		0.017

For the PEEC analysis results, Fig. 4 shows the distributions of the azimuthal current and radial bypass current during the sudden discharging test at times t = 0.01 s and t = τ(= 0.81 s) after discharging, where τ is the current decay time constant of the NI coil. Generally, the azimuthal and radial bypass currents exhibit an azimuthally symmetrical but radially asymmetrical distribution as time progresses after the discharge. During the sudden discharging test, the azimuthal current in the interior turns decays ahead of that in the exterior turns owing to the larger turn-to-turn contact resistance and a smaller inductance of the interior turns than that of the exterior turns. The current gradient between the interior and exterior turns during sudden discharging results in the bypass current flows from the exterior to interior turns.

Fig. 4.

Current distribution in NI pancake coil during the sudden discharging:(a) 0.01 s and (b) τ(= 0.81 s) after sudden discharging at turn-to-turn contact surface resistivity of 70 μ Ω · cm². The bypass current flow from exterior to interior turns is set as negative.

When the NI REBCO coil suddenly discharges, the inductance maintains the azimuthal current for a while. Nonetheless, the azimuthal current does not exceed the critical current of the coil during the sudden discharging process. According to the results, we conclude that the turn-to-turn contact resistance can consume all the energy stored in the NI coil during sudden discharging, and the NI coil is very stable that a hotspot hardly appears on it even though the power supply is suddenly cut off owing to any accident.

B. Influence of Contact Surface Resistivity on Thermal and Electromagnetic Behavior

To clarify the influence of the turn-to-turn contact resistance on the transient behavior during sudden discharging, a numerical simulation using the PEEC model was carried out using various turn-to-turn surface resistivities. Fig. 3 shows the field response analyzed using turn-to-turn contact surface resistivities of 350 and 700 μΩ·cm² for a 60-turn NI REBCO pancake coil. Table I summarizes the influence of the turn-to-turn contact resistance on the current decay time constant, the energy consumed by the turn-to-turn contact resistance, and the increase in the coil temperature during sudden discharging; Fig. 5(a) shows their trend. We verify that the time constant of the NI coil during sudden discharging can be controlled by changing the turn-to-turn contact resistivity, but the increase in the turn-to-turn contact resistance never results in a risk of temperature increase in the NI coil. The results also verify that the turn-to-turn contact resistance consumes all the magnetic energy stored in the NI REBCO pancake coil during sudden discharging.

Fig. 5.

Dependence of (a) contact surface resistivity and (b) the number of turns on the decay time constant, the energy consumed by the turn-to-turn contact resistance, and the increase in the coil temperature during the sudden discharging test.

C. Influence of the Number of Turns on Thermal and Electromagnetic Behavior

To clarify the influence of the coil scale on the transient behavior during sudden discharging, the number of turns of the NI REBCO pancake coil was adopted as a parameter. Table II summarizes the influence of the number of turns on the current decay time constant, the energy consumed by the turn-to-turn contact resistance, and the coil temperature increase during the sudden discharging of the NI coil; Fig. 5(b) shows their trend. As the number of turns increased, both the current decay time constant and the energy consumed by the turn-to-turn contact resistance exponentially increase. However, the increase in the coil temperature was insignificant during sudden discharging although the number of turns increased. We can infer that the Joule heat generation of NI REBCO pancake during sudden discharging will never result in the temperature increase of the coil in the adiabatic condition. That also means that the NI REBCO pancake coil has a high thermal stability during the sudden discharging operation.

TABLE II.

Specifications and Simulation Results of 60-Turn NI REBCO Pancake Coils With Various Numbers of Turns in Sudden Discharging Test

	Number of Turns
	120	240	360
Contact surface resistivity, Rct (μΩ·cm2)		70
Time constant, τ (ms)	1620	3350	5320
Inductance, Lcoil (μH)	1623	6157	13706
Operation current, Iop (A)		30
Stored magnetic energy (mJ)	730.4	2770.7	6167.7
Energy consumed by turn-to-turn contact resistance, QJ (mJ)	730.4	2770.7	6167.7
Coil temperature increase, ΔT (K)	0.031	0.052	0.070

D. Influence of Partial Turn-to-Turn Contactless Area on Electromagnetic Behaviors

In practical manufacturing of NI REBCO coils, winding them in accordance with their design configuration is impossible because of the nonuniform thickness of the REBCO wire [11]. Therefore, an analysis based on a worst-case scenario was performed to investigate whether or not the contactless area in a non-impregnated 60-turn NI REBCO pancake coil affects the current distribution during sudden discharging.

Fig. 6(a) shows the current distribution in a local centralized contactless area that consists of 5.6% of the total area inside the NI REBCO coil. We confirmed that the decay in the radial bypass current in the turns in the local centralized contactless area was slower, but that of the azimuthal current was faster than those in the other turns; however the distribution of the bypass current was similar to that of the contactless area owing to the non-electrical conduction at the contactless area.

Fig. 6.

Current distribution in an NI pancake coil when (a) a local contactless area exists and (b) azimuthally inhomogeneous contactless areas exist because of the winding accuracy during sudden discharging (0.01 s immediately after sudden discharging).

Further, Fig. 6(b) shows the current distribution in the contactless areas that consists of 11.3% of the total area. The contactless areas were randomly distributed inside the coil, assuming a manufacturing error. The distribution of the radial bypass current in Fig. 6(b) is also identical to that of the contactless areas.

Comparing both results, the maximum azimuthal current in Fig. 6(b) is almost 5% higher than that in Fig. 6(a) during the sudden discharge. According to these results, we conclude that the partial turn-to-turn contactless areas results in a rapid decay in the azimuthal current in that turn, thereby impeding the radial bypass current. The current decay constant times are shown in Fig. 6(a) and (b) are 770 and 720 ms, respectively. From the resistance perspective, the increase in the contactless area can be interpreted as an increase in the overall turn-to-turn contact resistance of the NI coil.

In conclusion, this simulation demonstrated that the partial turn-to-turn contactless areas in an NI coil do not result in a risk to a local hotspot in the coil.

E. Discussion on Sudden Discharging

We have developed the PEEC model of NI REBCO pancake coil to investigate the transient behavior during sudden discharging. The PEEC model takes into account the turn-to-turn contact resistance, the I–V characteristic of HTS tape, and the self and mutual inductances of the partial elements. Because the other method proposed in [12], similar to our proposed PEEC model, has ignored the self and mutual inductances of partial elements, the central magnetic field instantly decreased to zero in the sudden discharging test. In our proposed PEEC model, the self and mutual inductances maintained the azimuthal current for a short time. Therefore, the central magnetic field is attenuated with an arbitrary time constant. Such attenuation was also observed in the experiments [2], [9].

Further, we confirm that when the power supply is suddenly disconnected for some reasons, the NI REBCO pancake coil has a high thermal stability during sudden discharging even if it has partial contactless areas. To employ NI REBCO coils in practical applications, we need to investigate the sudden discharging characteristics of NI REBCO coils under various turn-to-turn contact resistivities and large number of turns. High stability during sudden discharging is an important requirement to ensure safety in practical applications although an HTS coil is sufficiently stable.

IV. Analysis of Transient Behavior of NI REBCO Pancake Coil During Overcurrent

The overcurrent test was also performed to confirm the high thermal stability of the NI REBCO pancake coil in the study [2]. In this section, to investigate the transient electromagnetic and thermal behaviors of the NI coil during overcurrent and clarify the mechanism of high thermal stability, we present a numerical simulation using the PEEC-thermal coupled analysis. Table III lists the specifications of the NI coil summarized in [2]. The time step in the PEEC model and thermal conduction analysis was 0.01 s.

TABLE III.

Specifications of the NI REBCO Pancake Coil Used in the Overcurrent Test

Parameters	30-turn NI pancake coil
HTS Conductor	Super Power® SCS4050 REBCO wire
Conductor width; thickness (mm)	4.0; 0.1
Copper stabilizer thickness (μm)	40
Ic @ 77K, self-field (A)	85
Ic @ 77K, coil-field (A)	54
Insulation	bare (no insulation)
Number of turns	30
i.d.; o.d.; height (mm)	60.0; 66.0; 4.0
Bz per amp at center (mT/A)	0.60
Inductance (μH)	110.0

A. Comparison of the Experimental and Analytical Results

In the previous study [2], the central magnetic field response of the NI REBCO coil with liquid nitrogen (LN₂) cooling was measured as shown in Fig. 7(a), when the excitation current was increased to 125 A with a sweep rate of 0.5 A/s. The central magnetic field response in the overcurrent experiment is categorized into four phases: linear increase phase [to point “A” in Fig. 7(a)], saturation phase (from point “A” to point “B”), cliff-type falling phase (between points “B” and “C”), and stabilizing phase (after point “C”). At the stabilizing phase, the central magnetic field finally stabilized at approximately 2 mT [2].

Fig. 7.

(a) Experimental and (b) analysis results for the central field response during the overcurrent test.

Fig. 7(b) shows the simulation results using the PEEC-thermal coupled analysis. According to the simulation results, the central magnetic field begins to saturate at approximately 108.0 s (at point “a”), suddenly decreases at approximately 163.7 s (at point “b”), and finally stabilizes after 165.5 s (from point “c”). Points “a,” “b,” and “c” correspond to operating currents of 54.0, 81.85, and 82.75 A, respectively.

Compared with the experimental result shown in Fig. 7(a), the times at points “b” and “c” in the analysis are lightly smaller than those in the experiment, and the current differences are 5.65 and 17.25 A, respectively. In addition, the bottom of the cliff-type falling of the central field after point “c” is lower, and the time interval during the cliff-type falling is shorter than that in the experimental results. Because an adiabatic condition is adopted in this analysis, the propagation velocity of the local heat in the NI coil is larger than that under LN₂ cooling conditions. Therefore, the analysis result is considered to be a worst-case scenario. To improve the accuracy of this analysis, we will adapt the heat transfer condition for LN₂ cooling in a future study.

The study [12] mentioned that operating current flows only in the innermost and outermost turns (called “single-turn-coil mode”) after it exceeds the coil critical current. However, in our analysis results, the azimuthal current even in the middle turns was maintained, as shown in Fig. 8(c), because the self and mutual inductances in the partial elements were taken into account. Consequently, it took approximately 2 s to dissipate the central magnetic field, as shown in Fig. 7(b).

Fig. 8.

Transient electrical behavior during overcurrent: (a) distribution of the azimuthal current during central field saturation, (b) distribution of the azimuthal current during central field cliff-type falling, (c) distribution of the radial bypass current during central field saturation, and (d) distribution of the radial bypass current during central field cliff-type falling.

B. Transient Behavior on the Saturation Phase

Figs. 8(a) and (c) show the current distributions at point “a,” from “a” to “b,” and at point “b,” on the saturation phase. In Fig. 8, the anticlockwise direction of the azimuthal current and the outward direction of the radial bypass current are set to be positive. Fig. 8(a) shows that the azimuthal current in the interior and exterior turns of the winding starts decreasing, but that in the middle turns of the winding increases during the process from “a” to “b.” Further, the radial bypass current in the interior and exterior turns of the winding obviously increases during the process from “a” to “b,” as shown in Fig. 8(c). This occurs because the critical current in the interior and exterior turns of the winding decreases compared with that in the middle turns of the winding, due to the nonuniform magnetic field applied to the REBCO winding turns, as shown in Fig. 9.

Fig. 9.

Axial magnetic field applied to NI REBCO coil winding.

We can conclude that the current in the interior and exterior turns of the coil winding bypasses the electrical terminals or the middle turns with a relatively higher critical current than the other turns. The local decrease in the critical current causes the saturation of the central magnetic field. However, the smaller amount of current in the interior and exterior turns is maintained due to the self and mutual inductances. This result is obviously different from that in the study [12], whose proposed equivalent circuit neglected the self and mutual inductances.

Fig. 10(a) shows the transient thermal behavior of the NI coil during the saturation process. In this process, neither obvious quenching nor temperature increase in the coil is observed, whereas the operating current exceeds the coil critical current because the small amount of heat generated in the NI coil is dissipated by the heat capacity of the coil winding.

Fig. 10.

Transient thermal behaviors during overcurrent: (a) thermal distribution during the process of central magnetic field saturation, (b) magnified drawing of current vector with thermal distribution at an operation current of 81.9 A, (c) thermal distribution during the process of the cliff-type falling of central magnetic field, and (d) magnified drawing of current vector with thermal distribution at an operation current of 82.75 A.

C. Transient Behavior on the Cliff-Type Falling Phase

From point “b,” the central magnetic field exhibits a cliff-type falling. The current distributions at point “b,” from “b” to “c,” and at point “c” are shown in Fig. 8(a)–(d). By observing the current distribution during the cliff-type central field falling, wide-ranging quenching occurs from the exterior to the interior turns of the NI coil winding within 1.8 s from point “b” to “c,” as shown in Fig. 8(b). In addition, the azimuthal and radial currents concentrate towards the outer electrical terminal.

To clarify whether the quenching is caused by a local temperature increase in the NI coil and to locate the region of temperature increase, the temperature distributions from “b” to “c,” and at the point “c” are shown in Fig. 10(c). A local temperature increase occurs in the exterior turns near the outer electrical terminal. Fig. 10(b) and (d) respectively show the current vector with temperature distribution at 0.01 s immediately after point “b” and at point “c,” where the images are magnified. We can observe that the quenching propagation starts from the exterior turns, i.e., the temperature increases beyond point “b” at an operating current of 81.85 A, and the current concentrated at the quenching region near the outer electrical terminal causes an instantaneous temperature increase there. Because the current in the outermost turn cannot bypass the outer radial direction, a current concentration obviously occurs near the outer electrical terminal, i.e., the overloaded current flows in the outermost winding and results in a temperature increase there. Subsequently, as the operating current increases, the heat immediately propagates from the exterior to the interior turns, and quenching occurs similarly within a short time until point “c.”

In conclusion, the cliff-type falling of the central magnetic field is caused by the instantaneous wide-ranging quenching that occurs from the outmost turn to the interior turns of the winding due to the temperature increase at the outer electrical terminal within an extremely short time.

D. Transient Behavior on the Stabilizing Phase

Beyond point “c,” the central magnetic field decreases to approximately 0.60 mT and then stabilizes at that level. The stabilized magnetic field in the simulation is much smaller than that in the experiment [2], because of the adiabatic condition. At this time, an azimuthal current flows in a few interior turns, as shown in Fig. 10(d), because the superconducting state remains in these turns, i.e., the steep decrease in the magnetic field increases the critical current in these turns. However, the concentration of the current near the outer electrical terminal is more serious than that at point “b,” because of the higher operating current. After the point “c,” the azimuthal current in the few interior turns that has not been influenced by the temperature increase maintains the stability of the central magnetic field.

However, in the experiment in [2], the NI REBCO coil did not burn out when the overcurrent exceeded the critical current of the coil. The temperature in this simulation increased to almost 650 K in the adiabatic analysis between points “b” and “c,” which means that a risk is present that the exterior turns in the NI REBCO coil would burn out near the outer electrical terminal during overcurrent, even in practical conduction cooling environment. Hence, increasing the stabilizer thickness on the outermost turn or coating it with a metal deposition with a low resistance and high current capacity is desirable to protect the NI REBCO coil during an overcurrent. All of the above protection solutions will be discussed in a future study.

E. Discussion on Overcurrent

Using the proposed PEEC model taking into account the turn-to-turn contact resistance, the I–V characteristic of HTS tape, and the self and mutual inductances, it is possible to reproduce the four phases observed in the experiment: linear increase, saturation, cliff-type falling, and stabilizing phases. It is confirmed, from the analysis result, that even though the operating current exceeds the critical current, the NI HTS coil does not instantly burn out because of the current bypassing. Further, it is also clarified that the current in a small number of the interior turns generates the stabilizing magnetic field. From the productivity of these phenomena, the validity of the PEEC model is verified.

In a future work, we will evaluate the stability of NI HTS coil when a local hotspot appears, using the proposed PEEC model. Furthermore, an NI HTS coil of a design stage for large-scale practical application will also be investigated in terms of the stability.

V. Conclusion

We have proposed a PEEC model for an NI REBCO pancake coil. To enhance the analysis accuracy, the turn-to-turn contact resistance, I–V characteristic of the REBCO tape, and self and mutual inductances in the elements were taken into account. For the investigation of overcurrent excitation, thermal analysis was combined with the proposed PEEC model.

Using the proposed PEEC model, we confirm that the turn-to-turn contact resistance has the important function of coil protection by consuming all the energy stored in the NI coil during sudden discharging. Even if a power supply is stopped, the NI REBCO coil is safe and stable, and the stored energy dissipates by itself, in contrast to the turn-to-turn insulation REBCO coil. We also note that, though the contactless areas result in an increase in the current in the local areas of the coil winding, there is no risk of temperature increase in an NI REBCO coil. However, the longer the constant time is, the longer is the time needed to charge/discharge the NI coil. Therefore, a shorter current decay constant time is desirable because the large inductance and small turn-to-turn contact resistance result in an extremely long waiting time (e.g., a few days) during charging/discharging a large-scale NI REBCO pancake coil (e.g., of the meter-class) in a practical application.

For overcurrent excitation, the transient thermal and electromagnetic behavior is reproduced in the analysis. Similar to the previous studies, we confirm the high thermal stability of the NI coil by clarifying the transient behavior during overcurrent excitation. The saturation of the central magnetic field is due to the resistive area where the azimuthal current exceeds the local critical current. The cliff-type falling of the central magnetic field is caused by the large amount of generated heat near the outer electrical terminal, followed by quenching that occurs from the exterior to the interior turns in an extremely short time. Finally, the central magnetic field stabilizes owing to the current in the few interior turns.