A temperature-dependent multilayer model for direct current carrying HTS coated-conductors under perpendicular AC magnetic fields

The governing equations for the electromagnetic module are derived from Faraday's law and Ampere's law:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\rm{\nabla }}\times E=-{\mu }_{0}{\mu }_{r}\displaystyle \frac{\partial H}{\partial t}\\ {\rm{\nabla }}\times H=J\end{array}\right.,\end{eqnarray} \tag{ 1 }$$

where μ₀ is the permeability of free space, μ_r is the relative permeability. For the superconducting layer and non-magnetic layers, the relative permeability is μ_r = 1. The equations are solved by using the H-formulation [36]. In 2D geometry, H = [H_x, H_y], which represents the components of magnetic field in the x and y directions. The induced electric field and the current density are perpendicular to the cross section and are defined as E_z and J_z. By substituting H =[H_x, H_y], E = E_z and J = J_z into equation (1), we obtain:

$$\begin{eqnarray}&&\left\{\begin{array}{l}\left[\begin{array}{c}\displaystyle \frac{\partial {E}_{z}}{\partial {y}}\\ -\displaystyle \frac{\partial {E}_{z}}{\partial {x}}\end{array}\right]=-{\mu }_{0}{\mu }_{r}\left[\begin{array}{l}\displaystyle \frac{\partial {H}_{{x}}}{\partial t}\\ \displaystyle \frac{\partial {H}_{{y}}}{\partial t}\end{array}\right]\\ {J}_{z}=\displaystyle \frac{\partial {H}_{{y}}}{\partial {x}}-\displaystyle \frac{\partial {H}_{{x}}}{\partial {y}}\end{array}\right..\end{eqnarray} \tag{ 2 }$$

The resistivity (ρ) of the superconducting element in YBCO layer follows the E–J power law [42, 43]:

$$\begin{eqnarray}&&E={E}_{0}{\left(\displaystyle \frac{J}{{J}_{c}(B,T)}\right)}^{n},\end{eqnarray} \tag{ 3 }$$

where E₀ = 10⁻⁴ V m⁻¹, n = 21 [36–38], and J_c(B, T) is the temperature and magnetic field dependent critical current density, which will be explained in section 2.2.2. The resistivities (ρ) for the non-superconducting elements are set as constant values. The resistivity (ρ) is 8 × 10⁻⁹ Ωm for Cu-stabilizer layer, 1 × 10⁻⁶ Ωm for substrate layer, and 100 Ωm for air region.

The critical current density of the 2G HTS coated conductor is sensitive to and influenced by the variation of both the temperature and the magnetic field density. The magnetic field and temperature dependent critical current density are given as:

$$\begin{eqnarray}&&{J}_{c}(B,T)={J}_{c0}\cdot {J}_{c}(B)\cdot {J}_{c}(T),\end{eqnarray} \tag{ 4 }$$

where J_c0 is the self-field critical current density at 77 K, J_c(B) represents the coefficient of magnetic field dependence, and J_c(T) represents the coefficient of temperature dependence. J_c(B) ∈ [0, 1], and J_c(T) ∈ [0, 1]. The self-field critical current is set as 140 A, according to the experimental measured self-field critical current from a 2G HTS CC wire manufactured by SuperPower [7]. J_c0 is set as 2.33 × 10¹⁰ A m⁻², which is calculated according to self-field critical current (140 A) and the cross-section area of the YBCO layer (6 mm wide × 1 μm thick).

J_c(B) is calculated through an anisotropic model [44–46], which is given as:

$$\begin{eqnarray}&&{J}_{c}(B)=\displaystyle \frac{1}{{\left(1+\tfrac{\sqrt{{(k{B}_{para})}^{2}+{{B}_{perp}}^{2}}}{{B}_{c}}\right)}^{b}},\end{eqnarray} \tag{ 5 }$$

where B_para and B_perp represent the flux density parallel and perpendicular to the wide tape surface, respectively. k, B_c and b are three parameters controlling the J_c(B) curve. These three parameters are obtained by curve fitting on experimental measured critical current data for a specific 2G HTS CC wire from the HTS CC manufacturers. In our model, we use the values for k, B_c and b from a SuperPower coated conductor wire in [45]: B_c = 0.3 T, k = 0.25 and b = 0.6. In the experimental verification (section 2.4), a SuperPower SCS4050 2G HTS coated conductor is used, which is the same 2G HTS coated conductor wire with the one used in [45].

J_c(T) is calculated according to equation (6), which has been adopted in [37, 38]

$$\begin{eqnarray}&&{J}_{c}(T)=\left\{\begin{array}{l}1,T\leqslant {T}_{0}\\ \displaystyle \frac{{T}_{c}-T}{{T}_{c}-{T}_{0}},{T}_{0}\lt T\lt {T}_{c}\\ 0,T\geqslant {T}_{c}\end{array}\right.,\end{eqnarray} \tag{ 6 }$$

where T is the temperature for each element in the superconducting (YBCO) layer, which is calculated through the thermal module. T₀ = 77 K is the starting temperature of the 2G HTS CC in the model, which is the same with the temperature of the surrounding liquid nitrogen, T_c = 92 K is the critical temperature, above which the HTS element totally losses its superconductivity.

For the real multilayer 2G HTS coated conductor wire, a portion of transport current will flow into protective metal layers such as Cu-stabilizer layers under certain conditions. These conditions include when the YBCO layer enters a resistive stage due to the 'dynamic resistance effect', due to a large overcurrent exceeding the YBCO critical current, due to a temperature rise of the YBCO layer, or due to high external DC or AC field. Our four-layer model can simulate the current sharing among different layers of the CC, which will help further our understanding about the interaction between different layers in the 2G HTS CC. The transport current (I_t) is set as the sum of the current in each layer:

$$\begin{eqnarray}&&\begin{array}{l}{I}_{{t}}={I}_{{YBCO}}+{I}_{{Cu}\_{upper}}+{I}_{{Cu}\_{lower}}+{I}_{{Sub}}=\displaystyle \int {J}_{{YBCO}}ds\\ \,\,+\,\displaystyle \int {J}_{{Cu}\_{upper}}ds+\displaystyle \int {J}_{{Cu}\_{lower}}ds+\displaystyle \int {J}_{{Sub}}ds,\end{array}\end{eqnarray} \tag{ 7 }$$

where, s refers to the cross-section area of each layer. I_YBCO, I_{Cu_upper}, I_{Cu_lower} and I_Sub is the DC transport current in the YBCO layer, the upper Cu-stabilizer layer, the lower Cu-stabilizer layer, and the substrate layers respectively. J_YBCO J_{Cu_upper}, J_{Cu_lower}, and J_Sub are the current density in each element of above four layers. The transport current I_t is applied to the CC through a 'Pointwise Constraint' in COMSOL. To avoid non-convergence in the model calculation, it takes 0.01 s for the transport current to ramp linearly from zero to the preset constant value.

The applied ac perpendicular magnetic field (B_app) is generated by a pair of electro-magnets with a narrow flux gap (≈1 mm). Each magnet consists of a pair of windings and an iron core, which is similar to that described in [35]. A series of sinusoidal magnetic waveform (B_app) with different peak magnitudes and frequencies are applied perpendicular to the surface of the HTS CC tape in the model. The boundary of the air region is far away from the superconductor, therefore the perpendicular component of magnetic field to the boundary is very small and can be set as zero in the model.

The loss (q) per length meter (W m⁻³) generated in each element of the CC is the heating power source for each element in the thermal module. It is also a key parameter linking electromagnetic module and thermal module. q is calculated by equation (8):

$$\begin{eqnarray}&&q=E\cdot J.\end{eqnarray} \tag{ 8 }$$

In the multilayer 2G HTS coated conductor wire, different layers produce different types of loss: the superconducting (YBCO) layer generates a transportation loss and a magnetization loss. The transportation loss origins from the dynamic resistance and flux flow resistance. The magnetization loss normally refers to hysteresis loss during the magnetization process of the superconductor (YBCO). Meanwhile, metal layers such as Cu-stabilizer will incur a transportation loss due to the metal's (copper's) resistivity if carrying a transport current, and an eddy current loss if subjected to a perpendicular AC magnetic field (B_app). The existence of the eddy current in the Cu-stabilizer has a significant impact on the behavior of the direct current carrying HTS CC when subjected to a perpendicular AC magnetic field. It will be analyzed in detail in the sections 3.3 and 3.4. To sum up, the loss of the HTS CC contains three types of loss, including transportation loss, magnetization loss, and eddy current loss.

Total loss of each layer per length meter (Q_layer) (W m⁻¹) can be calculated by a surface integral, as shown in equation (9), where s refers to the cross-section area of each layer. The total loss of the CC (Q_cc) can be obtained from the sum of loss in four layers. Since the resistivity of substrate layer is much larger than Cu-stabilizer layer, current in substrate (I_Sub) layer is negligible. Q_Sub is also very small and negligible. In order to distinguish the transport loss from magnetization loss or the eddy current loss, the voltage per length-meter (V_cc/L) (V m⁻¹) of the CC along the tape length direction should be calculated first. It is the average electric field density along the length direction of the CC wire, which can be calculated by equation (10). s_YBCO refers to the cross-section area of YBCO layer. The transportation loss (Q_trans) in a certain layer can be calculated by equation (11), where I_transport refers to the transport current in the same layer, for example I_YBCO in YBCO layer, I_{Cu_upper} and I_{Cu_lower} in upper and lower Cu-stabilizer layer in figure 3. Afterwards, as shown in equation (12), the magnetization loss (Q_mag) in YBCO layer can be calculated by removing out the transportation loss (Q_trans) from the total loss of YBCO layer (Q_YBCO) which can be calculated through equation (9). Similarly, as shown in equation (13), the eddy current loss (Q_eddy) in the Cu-stabilizer layer can be calculated by removing the transportation loss (Q_trans) from the total loss of Cu-stabilizer layer (Q_Cu) which can be calculated through equation (9)

$$\begin{eqnarray}&&{Q}_{{\rm{layer}}}=\displaystyle \int qds=\displaystyle \int EJds,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{V}_{{CC}}}{L}=\displaystyle \frac{\displaystyle \int {E}_{{YBCO}}ds}{{s}_{{YBCO}}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{Q}_{{trans}}={I}_{{transport}}\cdot \displaystyle \frac{{V}_{{cc}}}{L},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{Q}_{{mag}}={Q}_{{YBCO}}-{I}_{{YBCO}}\cdot \displaystyle \frac{{V}_{{cc}}}{L},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{Q}_{{eddy}}={Q}_{{Cu}}-{I}_{{Cu}}\cdot \displaystyle \frac{{V}_{{cc}}}{L},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{{YBCO}}}{L}=\displaystyle \frac{\tfrac{{V}_{{CC}}}{L}}{{I}_{{YBCO}}}.\end{eqnarray} \tag{ 14 }$$

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The resistance per length-meter (R_YBCO/L) (Ω m⁻¹) of YBCO layer along the tape length direction can be calculated by equation (14). The waveform of the R_YBCO/L is an important parameter to determine the status of the YBCO layer, such as a simple status only containing a dynamic resistance, or a mixed status containing both the dynamic resistance effect and flux flow effect. The total resistance of the HTS CC is mainly determined by the parallel resistances, mainly the YBCO layer and two Cu-stabilizer layers. The impact of the high resistivity substrate layer is very small and can be ignored. The equivalent circuit is presented in figure 3, from which we know that V_cc/L = V_ybco/L = V_cu/L.

The thermal module calculates the temperature for each element in the coated conductor. In each calculation step, it is responsible for updating the temperature of the CC wire, which is important for updating the coefficient of temperature dependence J_c(T) as shown in equations (4) and (6). The thermal module is based on the general heat transfer equation (15):

$$\begin{eqnarray}&&dC\displaystyle \frac{\partial T}{\partial t}={\rm{\nabla }}\cdot (k{\rm{\nabla }}T)+q,\end{eqnarray} \tag{ 15 }$$

where d is the mass density (kg m⁻³), C is the heat capacity (J kg⁻¹K⁻¹)), k is the thermal conductivity (W m⁻¹K⁻¹). The mass density for YBCO layer is 5900 kg m⁻³, it is 8000 kg m⁻³ for Cu-stabilizer, and 8280 kg m⁻³ for substrate. The heat capacity C and thermal conductivity k of the YBCO, substrate, and copper (Cu-stabilizer) in this model are temperature dependent, which are presented in figure 4. The data for C and k are obtained from reference [47–50]. q (W m⁻²) is the heating power source in each element, which is calculated by equation (8) in the electromagnetic module. This general heat transfer equation (15) means that the temperature variation is a final result from two impacting factors: the heating power (q) generated inside the element, and the heat transfer between the element itself and the surrounding elements.

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An accurate setting for heat transfer boundary condition is critical for the accuracy of the thermal-coupled model. It significantly influences the thermal-electromagnetic behavior of the 2G HTS CC in the simulation model. In our model, a convective heat flux type boundary condition is adopted, which is highlighted by red line in figure 2(b). The convective heat flux boundary condition can be expressed as:

$$\begin{eqnarray}&&Q=h({T}_{ext}-T),\end{eqnarray} \tag{ 16 }$$

where Q is the heat transfer (W m⁻²) between the liquid nitrogen and the surface of the HTS CC tape, T_ext is the temperature of liquid nitrogen surrounding the coated conductor, and h is the heat transfer coefficient (W m⁻² K⁻¹) representing the cooling (heat transfer) capability of the liquid nitrogen by liquid-to-solid surface contact. T_ext is set as a constant value at 77 K. h value is an important parameter in heat transfer boundary condition. In reference [37], the h is set as 10 000 W m⁻² K⁻¹. But in our verification experiment, the CC tape is placed in a narrow gap of the magnet, so the cooling condition is not as good as in an open space in the liquid nitrogen. When the heating power is large, bubbles will be generated in the liquid nitrogen, which is very detrimental to the heat dissipation of the CC tape in the narrow gap. So, h value should be lower in this model. We have tried different values for h between 500 to 10 000, but finally got reasonable simulation results when h = 800 W m⁻² K⁻¹. Table 1 gives a brief summary of the specifications of the HTS coated conductor model.

In either the previous dynamic resistance model or flux flow model, the temperature rise in the superconductor is not considered. This is not valid when the applied field is with high frequency, high magnitude, and long duration. In this case, quench behavior may happen in the CC. By using our temperature-dependent model, the thermal-electromagnetic behaviors of 2G HTS coated conductors can be successfully simulated. Figure 6 shows the electric field response of the CC to 20 cycles of 1 kHz applied field. Our experimental results show that the impact of the temperature rise is not obvious when the field frequency is low. So, we focus on the high frequency range around 1 kHz here. Figures 6(a1)–(a3) are the simulation results when the applied field (peak) is 50 mT, 200 mT and 400 mT. Figures 6(b1)–(b3) are the experimental data for comparison. The transport current is set as 80 A (around 0.6 I_c0) in both simulation and experiment. The simulation results are in well accordance with the experimental data, especially in terms of waveform shape and recovery time. The difference in magnitude between the simulation and experiments is acceptable.

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When the field magnitude is relatively low, the electric field response of the CC is stable. And it recovers to zero immediately when the applied field is removed. When the applied field magnitude is increased to 200 mT, the electrical field peak value increases gradually with time, which is different from the case when the applied field is low. At the same time, the valley value gradually increases and does not return to zero ('floating') after a time period. Both are different phenomena as well. But the electric field still can recover to zero quickly when the field is removed. When the applied field increases to 400 mT, the electrical field is both increasing and 'floating' at the same time. When the field is removed, an obvious 90 ms time-delay of electrical field returning to zero is observed, which is regarded as thermal quench and recovery behavior. Both our numerical and experimental results show that the CC becomes unstable and even quenches when a magnetic field with high frequency, high magnitude, and long duration is applied. By comparing the simulation and experimental waveforms, our thermal-coupled model has been successfully validated to be capable of correctly simulating the electro-thermal response of the DC carrying CC conductor to the external applied field with various magnitudes.

Monitoring the temperature variation of the 2G HTS CC wire is another important function of this temperature-dependent model. Figure 7(a) shows the temperature rise of the CC when 20 cycles of 1 kHz applied field are applied. When the field magnitude is relatively low (50 mT), the temperature of the CC rises slightly, from 77 to 78 K. When the field is 200 mT, the temperature rise is moderate (6 K, from 77 to 83 K). But when the field increases to 400 mT, the temperature rise is considerable (12 K, from 77 to 89 K), and it decreases slowly after the field is removed. The high temperature rise and very slow cooling speed lead to the fast increase of the electrical field (Vcc/L) and the long recovery time (90 ms) of the HTS CC as shown in figures 6(a3) and (b3).

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Figures 7(b1)–(b3) show the coefficient of temperature dependence J_c(T) distribution along the length direction in the YBCO layer at different instants of time when the field magnitude is 50, 200, and 400 mT. Results show that the degradation of J_c(T) is more severe at two edges of the CC wire. It is caused by the faster and higher temperature rise at two edges of the CC wire than the temperature rise in the center. This is because the current density on the two ends is higher, then the electrical field is higher, and therefore the loss in two ends is larger. Our results indicate that the most vulnerable part of the HTS CC wire is at the two edges when carrying a DC transport current and subjected to a perpendicular AC field. This finding is very indicative for the design of HTS current switches and switches-based flux pump. Our model is also very useful in analyzing the quench and recovery process of HTS CC switches.

This multilayer structured model is very powerful to help study the current sharing and different loss components in different layers, which could not be accurately measured. Figure 8 shows the current distribution of the CC in YBCO layer and one Cu-stabilizer layer. We have verified that the current in the upper and lower Cu-stabilizer layers are exactly the same, so here only the current in one Cu-stabilizer (upper or lower) layer is plotted. When the field magnitude is relatively low (50 mT), the transport current of the CC mainly flows in the YBCO layer and a very small proportion flows into Cu-stabilizer. When the field increases to 200 mT, a larger proportion of transport current enters Cu-stabilizer. When the field increases to 400 mT, a great proportion of the transport current flows into the Cu-stabilizer. Moreover, when the temperature of the CC increases, more and more current enters Cu-stabilizer due to the temperature dependency of the critical current density of the YBCO layer. When the 400 mT field is removed, a long recovery time is needed for the transport current in Cu-stabilizer to return to YBCO layer, which is a typical characteristic of the thermal quench and recovery behavior.

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Figure 9 shows the resistance of YBCO layer and Cu-stabilizer. When the field magnitude is relatively low (50 mT), the resistance of the YBCO layer is very small and is solely due to dynamic resistance. When the field increases to 200 mT, the resistance of the YBCO layer increases to a higher level. When the field magnitude rises to 400 mT, the resistance of YBCO layer is comparable to the constant resistance of Cu-stabilizer layer and the instantaneous value finally surpasses the resistance of Cu-stabilizer due to the temperature rise. The waveform shows that the resistance of YBCO layer consists of both dynamic resistance and flux flow resistance. When the 400 mT field is removed, the resistance of the YBCO layer does not recover to zero immediately because the YBCO layer is still in flux flow region. The resistance variation of the YBCO layer compared to the constant resistance of Cu-stabilizer in figure 9 can explain the current distribution between YBCO layer and Cu-stabilizer in figure 8. Since the total resistance of the HTS CC is determined by the parallel resistances, mainly the YBCO layer and two Cu-stabilizer layers, the variation of the resistance of YBCO layer will result in the change of the total resistance of HTS CC.

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Figure 10 shows the critical current response of the YBCO layer to the applied field. The critical current waveform can help determine the working condition of the 2G HTS CC wire. When the field magnitude is relatively low (50 mT), the degradation in critical current of the YBCO layer is very slight. When the field magnitude increases to 200 and 400 mT, the critical current decreases greatly to different extent, which results in the current sharing in figures 7(b) and (c). When the 400 mT field is removed, the critical current of the YBCO layer is still much lower than transport current, so consequently the current flows through the Cu layer. During the recovery process, the critical current gradually increases, surpasses the transport current, and finally recovers to its initial state.

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This section investigates the shielding effect of the Cu-stabilizer layer. In order to better understand the impact of the Cu-stabilizer layer, it's better to separate the impact of the current sharing ability and field shield effect. In order to achieve this goal, a new set of comparison models are designed and simulated: (i) An HTS tape which only contains the YBCO layer and substrate layer, whose upper and lower Cu-stabilizer layers have been removed. (ii) An HTS CC tape which contains both upper and lower Cu-stabilizer layers, but the transport current (I_t) has been constrained to flowing in the YBCO (Superconducting) layer. In this case, the current sharing ability of the Cu-stabilizer has been removed. (iii) An HTS CC tape which contains both up and down Cu-stabilizer layers and the transport current (I_t) constraint has been cancelled, so I_t can be naturally shared by the Cu-stabilizers. It is the same as the model setting in previous sections. Above experiments and simulations have already shown that the temperature rise influences the behaviors of the HTS CC tape significantly. In order to get rid of the influence of the temperature difference between comparison simulations, the heat transfer coefficient h is increased from 800 W m⁻² K⁻¹ to 10 000 W m⁻² K⁻¹ (a more than 10 times higher level). Our simulation results below will prove that this very high heat transfer coefficient (h) value can help guarantee a same temperature rise for comparison simulation models, which is quite an important precondition for studying the shielding effect of the Cu-stabilizer layer. The peak magnitude of B_app is 35 mT and the frequency is set as 2 kHz for a more obvious shielding effect. The low frequency range like around 50 Hz has been studied, but the shielding effect is not as significant as in the high frequency range (kHz level). The transport current is still 80 A.

Figure 13(a) shows the temperature curves of three HTS tapes. They are almost the same, which is an important precondition for further analysis and discussion of shielding effect. Figure 13 (b) is the transport current (I_ybco) flowing in the YBCO (superconducting) layer in three HTS tapes. Only in tape model (iii), I_ybco fluctuates due to the current sharing of Cu-stabilizer layer. Both figures 13(a) and (b) verifies our temperature control and current injection in three comparison models for different HTS tapes.

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Figure 13(c) shows the electrical field (or voltage per meter length) (V_cc/L) generated on three HTS tapes. Results show that the existence of Cu-stabilizer layer greatly reduces the electrical field (V_cc/L) generated on DC carrying HTS CC tapes subjecting to AC external applied field (B_app). It is easy to understand that the current sharing ability of Cu-stabilizer will decrease the electrical field (V_cc/L) because the electrical field or is a product of the resistance (R_ybco/L) and current (I_ybco) in YBCO (superconducting) layer. However, mechanism is not so simple for the decrease of electrical field (V_cc/L). Our results in figure 14 reveal the existence of the magnetic field shielding effect of the Cu-stabilizer and its impacts on the dynamic resistance of the superconducting (YBCO) layer.

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Figure 14(a) shows the perpendicular field magnitude (B_y) at the center point of YBCO layer's up surface, which can tell the intensity of the field shielding effect of Cu-stabilizers. Figure 14(b) shows the temperature and magnetic field dependent critical current (I_c(B,T)) of HTS tape. The temperature curves are the same for three HTS tapes, therefore I_c(B,T) is a good monitor for field shielding effect. Both figures 14(a) and (b) successfully show that the perpendicular field magnitude at the center point of YBCO layer's up surface is reduced due to the upper and lower Cu-stabilizers' shielding effect. Figure 14(c) shows the resistance (R_ybco/L) of superconducting (YBCO) layer. Their waveforms show that the resistances are totally dynamic resistance. From [5], we know that the dynamic resistance in a certain fixed width HTS tape is determined by the magnitude of perpendicular component (By) of AC applied external magnetic field, if the field frequency is fixed as well:

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{{d}\perp }}{L}=\displaystyle \frac{4{\rm{a}}f}{{I}_{{C0}}}({B}_{a,\perp }-{B}_{{th},\perp }).\end{eqnarray} \tag{ 17 }$$

So, for tape (ii) (the transport current (I_t) only flows in the YBCO (Superconducting) layer), the dynamic resistance (R_ybco/L) decreases a lot simply because the up and down Cu-stabilizers shield a portion of the perpendicular field (B_y). For tape (iii), the transport current (I_t) can be shared by the Cu-stabilizers naturally. This situation is more complicated. Both the current sharing phenomenon and shielding effect of the Cu-stabilizers lead to a further reduced dynamic resistance (R_ybco/L). On one hand, the shielding effect of the Cu-stabilizer can reduce the perpendicular field (B_y) to some extent, which has already been verified in above discussion. On the other hand, when the dynamic resistance appears in the superconducting (YBCO) layer due to the AC perpendicular field, a portion of transport current (I_t) flows into the upper and lower Cu-stabilizer layers. As a result, I_ybco decreases and so the YBCO layer can screen more applied field. The screening effect of the YBCO layer becomes stronger under the same AC field, compared to tape (ii). This also further reduces the AC perpendicular field (B_y). Figure 14(a) provides good evidence for this explanation, where AC perpendicular field (B_y) in tape (iii) is further reduced, compared with tape (ii). The further reduced AC perpendicular field (B_y), caused by both the Cu-stabilizers' shielding effect directly and by their current sharing ability indirectly, finally results in a higher I_c(B,T) and a lower dynamic resistance (R_ybco/L) of superconducting (YBCO) layer, which are clearly presented in figures 14(b) and (c) respectively.

Now, let's move back to figure 13(c): the electric field (V_cc/L) generated on three HTS tapes. After the analysis and discussion above, the mechanism behind the decrease of electrical field (V_cc/L) for tape (iii) is clear now: (a) Firstly, Cu-stabilizers' shielding effect directly reduces the dynamic resistance (R_ybco/L) of superconducting (YBCO) layer, and the Cu-stabilizers' current sharing ability also indirectly reduces dynamic resistance (R_ybco/L) of superconducting (YBCO) layer. (b) Secondly, the Cu-stabilizers share a portion of transport current (I_t), which reduces the transport current (I_ybco) in YBCO (superconducting) layer. As a result, compared to tape (i) and tape (ii), the electrical field (V_cc/L) generated on HTS tape (iii) is further reduced due to the drop in both the resistance (R_ybco/L) and transport current (I_ybco) in YBCO (superconducting) layer. (For the multilayer coated conductor, V_cc/L = V_ybco/L = V_{cu_stabilizer}/L because the multilayers are electrically connected in parallel).

To conclude, our simulation models and results have deepened our understanding on the impact of the Cu-stabilizer on the behavior of the DC carrying HTS CC tapes subjecting to AC external applied field. The simulation results have verified the shielding effect and current sharing ability of Cu-stabilizer. The existence of the Cu-stabilizers will decrease the dynamic resistance and the dynamic voltage of the HTS coated conductors due to both the shielding effect and current sharing ability, directly and indirectly. Our analysis and discussion here are useful for the future design of HTS persistence current switches, flux pumps, and cables.

This section presents the frequency dependence of the shielding effect. Figure 15(a1)–(a3) shows the perpendicular field magnitude (B_y) at the center point of YBCO layer's up surface under 1, 2, and 4 kHz field frequency, for tape (i) and tape (iii). Figures 15(b1)–(b3) shows the dynamic resistance (R_ybco/L) of superconducting (YBCO) layer accordingly. Tape (ii) is not the real situation for the HTS coated conductor, so it is not discussed here. The magnitude of the applied perpendicular AC field is 35 mT, but the measured perpendicular field magnitude (B_y) at the center point of YBCO layer's up surface is around or less than 20 mT. It is caused by the shielding effect of the whole coated conductor, including the superconducting layer (predominately) and Cu-stabilizer layers (slightly). But it would not influence our study on the frequency dependence of shielding effect below.

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As shown in figure 15(a1)–(a3), when the field frequency is 1 kHz, the B_y decreases slightly due to the existence of Cu-stabilizers. When the frequency increases, the B_y further decreases due to the existence of Cu-stabilizer. When the field frequency increases to 4 kHz, the field B_y decreases greatly, by around ¼, due to the existence of Cu-stabilizers. Figure 15(b1)–(b3) shows that the frequency dependency for dynamic resistance (R_ybco/L) of superconducting layer is more significant as a result of the existence of Cu-stabilizers. When the field frequency increases to 4 kHz, the dynamic resistance (R_ybco/L) of the superconducting layer decreases by around half, as a result of existence of Cu-stabilizers on both sides of the coated conductor.

Figure 16 shows the percentage of the total eddy current loss (two Cu-stabilizer layers) to the total loss of the HTS CC wire when carrying DC current and subjecting to perpendicular AC external applied field. As a comparison, the percentage of the transport current loss of the two Cu-stabilizer layers is also presented. Results show that: as the field frequency increases from 1 to 4 kHz, the percentage of the eddy current loss in two Cu-stabilizer layers increases significantly from 4% to 12.5%. It means that the shielding effect of the Cu-stabilizers has a strong field frequency dependence in the high frequency working zone like kHz level. In contrast, the percentage of the transport loss of the two Cu-stabilizers increases slightly as the field frequency goes up. It shows that the current sharing ability of the Cu-stabilizers does not have an obvious field frequency dependence. The reason for the slight increases in percentage of the transport loss of two Cu-stabilizers might be: when the field frequency increases from 1 to 4 kHz, the dynamic resistance (R_ybco/L) of superconducting layer increases from 2.8 mΩ m⁻¹ to around 7.0 mΩ m⁻¹. A higher dynamic resistance (R_ybco/L) of superconducting layer pushes more transport current into the two Cu-stabilizer layers, which results in a slightly higher percentage of the transport loss from Cu-stabilizers.

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To conclude, this section reveals the strong frequency dependence of the field shielding effect of the Cu-stabilizer layer under kHz level high frequency field. As the frequency of the applied perpendicular AC field rises, the shielding effect becomes stronger, and the proportion of the eddy current loss from Cu-Stabilizer becomes higher and higher. Moreover, the strong shielding effect also decreases the dynamic resistance of the superconducting layer, and finally reduces the dynamic voltage (V_cc/L) of the HTS coated conductor. From the above results, we found that both the shielding effect of the Cu-stabilizers and its directly triggered eddy current loss are detrimental to the performance of the DC carrying HTS CC tapes subjecting to AC external applied field, especially when it is used as the AC field-controlled switch and in the switches-based HTS flux pumps.