Numerical evaluation of the deformation of REBCO pancake coil, considering winding tension, thermal stress, and screening-current-induced stress

Bending tolerance is one of the most important mechanical properties of REBCO tape for application to superconducting magnets. As such, many groups have evaluated the bending properties and uniaxial strain dependence of a REBCO tape [23–26]. As shown in figure 1, the bending strain, ${\varepsilon _\textrm{b,i}}$, of the $i$th-component layer in a REBCO tape is expressed as

$$\begin{equation}{\varepsilon _\textrm{b,i}} = \frac{{{\rho _i} - {\rho _0}}}{{{\rho _0}}},\end{equation} \tag{ 1 }$$

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where ${\rho _{\textit{i}}}$ and ${\rho _0}$ are the radii of the ith component layer and neutral axis, respectively. Therefore, ${{{\rho }}_{\textit{i}}} - {{{\rho }}_0}$ is the distance between the $i$th-component layer and neutral axis. The REBCO layer off the neutral axis is subjected to either compressive or tensile strain. To estimate the bending strain, the position of the neutral axis must be determined; the neutral axis is not located at the centre of the REBCO tape because of the tape's asymmetric layered structure. When all the components of the composite conductor deform only elastically, the neutral axis position does not move and is determined based on the stress balance over the cross-section. However, plastic deformation of a component results in a shift of the neutral axis to maintain stress balance. The neutral axis of the composite material of n layers with different elastic properties and thicknesses is generally expressed as

$$\begin{equation}{\rho _0} = \frac{{\mathop \sum \nolimits_{i = 1}^n \left( {{\rho _{2i}} + {\rho _{1i}}} \right){E_i}{h_i}}}{{2\mathop \sum \nolimits_{i = 1}^n {E_i}{h_i}}},\end{equation} \tag{ 2 }$$

where ${E_i}$, ${h_i}$, ${\rho _{1i}}$, and ${\rho _{2i}}$ are the Young's modulus, thickness, inner radius, and outer radius of the $i$th layer, respectively. The bending axis, at which the resultant force, due to bending is null, is located at the weighted average of the layers' mid-points position, $\left( {{\rho _{2i}} + {\rho _{1i}}} \right)/2$ [27]. In this study, by assuming elastic deformation, the neutral axis determined by (2) shifts by −0.1 μm from the geometric center of the tape, therefore the effective bending strain in the REBCO could easily be calculated by using geometric center of the tape.

If the winding loosens during the cooling and excitation stages, the wire moves, causing a temperature rise owing to frictional heat. In LTS magnets, a quench may occur owing to wire movements, whereas in HTS magnets, a quench rarely occurs due to wire movements. However, stresses and strains may exceed acceptable values owing to wire deformation [28]. Therefore, in this study, by applying tension in the winding that exceeds the thermal shrinkage at cooling and the electromagnetic force at excitation, we could prevent gaps between the windings when thermal shrinkage or electromagnetic force is applied, thus suppressing wire movement. To set the winding tension, we must consider factors such as the thermal stress or electromagnetic force as load as well as the mechanical stiffness of wire, looseness of wire after winding, and internal stress after winding [29]. Co-winding of the metal tape is also an effective method. When the tape is wound, as shown in figure 2, the following one-dimensional governing equation is formulated in the radial direction of stress [30]:

$$\begin{equation}{\rho ^{\,2}}\frac{{{\textrm{d}^2}\left( {{{\Delta }}{\sigma _\rho }} \right)}}{{\textrm{d}{\rho ^2}}} + 3\rho \frac{{\textrm{d}\left( {{{\Delta }}{\sigma _\rho }} \right)}}{{\textrm{d}\rho }} - \left( {\frac{{{E_\theta }}}{{{E_\rho }}} - 1} \right)\left( {{{\Delta }}{\sigma _\rho }} \right) = 0,\end{equation} \tag{ 3 }$$

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where ${{\Delta }}{\sigma _\rho }$ represents the incremental radial stress with increasing winding, $\rho$ is the radius of a winding turn, and ${E_\theta }$ and ${E_\rho }$ are the azimuthal and radial Young's modulus, respectively. The REBCO tape has a laminated architecture and can therefore be approximated to an equivalent homogeneous material with transverse isotropy. Therefore, ${E_\theta }$ and ${E_\rho }$ are calculated based on the rule of mixtures as

$$\begin{equation}{E_\theta } = \mathop \sum \limits_i^{} \frac{{{E_i}{h_i}}}{h},\end{equation} \tag{ 4a }$$

$$\begin{equation}\frac{1}{{{E_\rho }}} = \mathop \sum \limits_i^{} \frac{h}{{{E_i}{h_i}}},\end{equation} \tag{ 4b }$$

where ${E_i}$ and ${h_i}$ are Young's modulus and thickness of the $i$th layer, which is the substrate layer, REBCO layer, and so on in REBCO tape, respectively. $h$ is the thickness of the REBCO tape. A rule of mixtures is a weighted mean used to Young's modulus of a composite material. Equation (4 a) represents the overall property in the direction parallel to the REBCO tape's wide face, corresponds to a axial loading. ${h_i}/h\,$is volume fraction of a component layer in the REBCO tape. Equation(4 b) is the inverse rule of mixtures in the direction perpendicular to the REBCO tape's wide face, corresponds to a transverse loading. The boundary conditions are as follows:

$$\begin{equation}{\left. {\left( {{{\Delta }}{\sigma _\rho }} \right)} \right|_{\rho = {\rho _s}}} = - \frac{T}{{{\rho _s}}}h,\end{equation} \tag{ 5a }$$

$$\begin{equation}{\left. {\frac{{\textrm{d}\left( {{{\Delta }}{\sigma _\rho }} \right)}}{{\textrm{d}\rho }}} \right|_{\rho = {\rho _0}}} = \left( {\frac{{{E_\theta }}}{{{E_c}}} - 1 + \nu } \right)\frac{{\left( {{{\Delta }}{\sigma _\rho }} \right)}}{{{\rho _0}}},\end{equation} \tag{ 5b }$$

where ${\rho _0}{{\,}}$ and ${\rho _s}$ are the inner radius and radius of the winding turn, respectively; $T$, $h$, and $\nu$ are the winding tension, thickness, and Poisson's ratio of REBCO tape, respectively; and ${E_\textrm{c}}$ is the Young's modulus of the bobbin. The radial and azimuthal stresses, i.e. ${\sigma _{\rho ,i}}$ and ${\sigma _{\theta ,i}}$ in $i$th turn due to winding tension are calculated as follows:

$$\begin{equation}{\sigma _{\rho ,i}} = \mathop \sum \limits_{j = i + 1}^n \left( {{{\Delta }}{\sigma _{\rho ,ij}}} \right),\end{equation} \tag{ 6a }$$

$$\begin{equation}{\sigma _{\theta ,i}} = {\rho _i}\frac{{d{\sigma _{\rho ,i}}}}{{d\rho }} + {\sigma _{\rho ,i}},\end{equation} \tag{ 6b }$$

In this study, equation (3) is solved using the differential method, and the stress distribution is expressed using equations (6 a) and (6 b).

We evaluated the thermal and electromagnetic stress by considering the (a) mechanical stress due to the winding tension and the bending strain during winding process, (b) thermal stress during cooling down, and (c) electromagnetic stress including screening current-induced stress. The stress was analyzed using two-dimensional axisymmetric FEM, considering the winding configuration, as shown in figure 7. The bobbin, upper and lower frames, and spacer between pancake coils are made of glass fibre reinforced plastic. The mechanical properties of the materials used in these simulations are listed in table 3 [42]. In this simulation, the Young moduli of metal and metal alloy are used at 10 K. The mean linear thermal expansion is coefficient between room temperature (300 K) and operating temperature (10 K). In this study, the transient behaviours during cooling process and the non-linear elastoplastic property were not considered.

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Generally, in a numerical simulation, the coil winding is considered as a single elastic body by the rule of mixture. Table 4 listed the equivalent Young's modulus calculated the rule of mixture for four cases. As show in table 4, the equivalent Young's modulus is lower with entering soft material in between turns. Especially, the radial equivalent Young's modulus dramatically decreases. The numerical simulations on REBCO coil as an anisotropy single elastic body were reported in [43]. However, the winding turns separate or contact in a non-impregnated coil and NI coil. In this study, the contact and separation of turns in a pancake is considered.

The behaviours of deformation contact and separation of turns in a pancake is simulated. One numerical contact algorithm is the penalty method, which is well-known in FEM. However, a simple algorithm using a gap element as shown in figures 8 and 9 was adopted in this study. Contact pairs are defined between adjacent turns in a pancake. The contact problems are generally non-linear owing to changing boundary conditions. However, the contact states before and after the load application can be assumed, and if limited to a specific contact state, the state can be calculated using a linear FEM by introducing a contact element (gap element). Even if there is a contact state, at which the behavior is completely integrated before and after the load application and the analysis can be performed by approximately replacing the contact conditions with force or displacement, conventional stress analysis may be performed. A thin contact element (gap element) was inserted between corresponding nodes and elements. The thickness of the contact element was set to about 0.01–0.001 times the thickness of the nearest element so as not to affect the overall structure. In the case of an initial gap, the value of that gap is taken as the thickness.

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The contact modelling is illustrated in figure 8. The $x$- and $y$-axes show the anistropy that are perpendicular and parallel to the contact surface, respectively. The Young's modulus and Poisson's ratio are represented as $E$ and $\nu$, which in this case are represented as ${E_x}$, ${E_y}$, ${\nu _{xy}}$, and ${\nu _{yx}}$ for the contact element, and the shear modulus is represented as ${G_{xy}}$. In the following, ${E_x} = 0$ can be ignored compared to that in the structural part, such as ${E_x} = {10^{ - 4}}\,E$ in the numerical calculation. The same applies to ${E_y} = 0$ and ${G_x} = 0$.

• (a)  
  Bonding contact: We assumed that the contact surfaces behave as one:

  $$\begin{equation}{E_x} = {E_y} = E,\,{G_{xy}} = \frac{E}{{2\left( {1 - \nu } \right)}},{{\,}}{\nu _{xy}} = {\nu _{yx}} = v.\end{equation} \tag{ 9a }$$
• (b)  
  Separation: We assumed that the contacted part opens after deformation and achieves the free-boundary condition:

  $$\begin{equation}{E_x} = {E_y} = {G_{xy}} = 0,{{\,}}{\nu _{xy}} = {\nu _{yx}} = 0.\end{equation} \tag{ 9b }$$
• (c)  
  Sliding contact with friction coefficient $\mu = 0$: The contact portion does not open after deformation and slides freely in the direction of the contact surface:

  $$\begin{equation}{E_y} = 0,\,{E_x} = {G_{xy}} = 0,{{\,}}{\nu _{xy}} = {\nu _{yx}} = 0.\end{equation} \tag{ 9c }$$

Although the sliding contact with friction coefficient $\mu \ne 0$ could be considered, the friction force between adjacent turns was ignored in this study.

Figure 9 shows an algorithm of calculation of contact between turns in winding. First, the stress, as assumed in (a), was calculated; if it is a compressive stress, no separation occurs, so that the assumed contact state was appropriate. If tensile stress occurs, mechanical properties are changed to those in (b) and the stress is reanalysed. Next, when the element assumes (b) and the calculated stress is tensile, the assumed contact state is appropriate. If it is a compressive stress, as the separation between adjacent turns changes the contact state, the stress in (a) or (c) is recalculated. If a compressive stress is applied to the element assumed in (c), the separation between adjacent turns does not occur, and thus the assumed contact conditions are appropriate. If the stress is tensile, as it represents the separation between adjacent turns, we assume (b) and recalculate the stress. By repeating the above-mentioned procedure, we obtain a result indicating the correct contact state. The thermal stress and Lorentz force are gradually added at the final value by several steps in order to prevent divergence.

Next, we performed simulation on the deformation of the epoxy impregnated REBCO coil. The impregnation filling space is 10 μm. A model coil for the simulation is wound with a copper-plated REBCO tape with a winding load of 1 kg, and impregnated epoxy resin. We assumed that between turns is adhered, and the winding is adhered to the bobbin and frame. In this case, the compositions of the coil are glued, resulting in deforming coil integrally. Figure 13 shows the deformation of REBCO coil after the cooling down, and excitation. The coil shrunk during the cool down from 300 K to an operating temperature of 10 K. As shown in figure 13, the coil hardly deforms after the excitation because the epoxy impregnation reduces the deformation. Figures 14(a) and (b) shows the circumferential and radial stresses of REBCO layer in the winding on the corresponding line in figure 7(b). The radial stress is negative (i.e. compressive) after winding process and changes from compression to tension after cooling down. If the radial stress exceeds the acceptable transverse stress for the REBCO tape, i.e. a typical value of ∼10 MPa, degradation occurs. This is because the thermal contraction of impregnation is larger than that of the components of REBCO tape in shown in table 3. And the bonding strength of the impregnation is stronger than the delamination strength of REBCO tape. From the numerical results, it is confirmed that the degradation of the critical current of impregnated REBCO coils occurs because the radial stresses during the cooling process exceeds the acceptable transverse stress for the REBCO tape. The REBCO coil is carefully impregnated by considering stress or not impregnated in the dry wound [31, 32]. After an excitation, the circumferential stress increases because the winding expands outward due to Lorentz force.

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Figure 14(c) shows the circumferential stress due to electromagnetic force including screening-current effect of lower and upper sides of winding in uppermost single pancake coil. The stress is not affected with the electromagnetic force due to a screening current. This is assumed to be the reason that deforming the winding integrally defuse nonuniform effect of screening current.

The next case is non-impregnated coil wound by polyamide insulated tape. The thickness of insulator is 35 μm. A model coil for the simulation is wound with an insulated coated copper-plated REBCO tape with a winding load of 1 kg, and winding is not impregnated. Each winding in non-impregnated coils can deform separately and move freely. The contact between turns is modelled as mentioned in section 3.2.2. Figure 15 shows the deformation of coil after cooling and excitation. As seen in figure 15, after cooling down, the coil is not only shrinking but also distorted because of low rigidity due to non-impregnation. The outer diameter side deforms in the axial direction. Some winding is away from the adjacent turn after excitation. This reason is that screening current leads to generate the opposite direction of electromagnetic force in the tape as shown in figure 11. The circumferential and radial stresses of REBCO layer at 0.5 mm from upper side of tape in uppermost single pancake is shown in figures 16(a) and (b). The winding turns separate or lightly contact due to the thermal contraction because the stress and deformation are absorbed in the polyimide tape, therefore the radial stress almost unchanged after winding process.

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Figure 16(c) shows the circumferential stress due to electromagnetic force including screening-current effect of lower and upper sides of winding in uppermost single pancake coil. Effect of the electromagnetic force due to a screening current are remarkably appeared. The insulation coating REBCO tape shrinks during a cooling process and easily deforms against the force received by the REBCO tape during an excitation process. Therefore, the REBCO tapes in winding move and deform according to the nonuniform electromagnetic force distribution due to a screening current as shown in figure 11. Then, the stress distributions are different in the lower and upper sides of winding as shown in figure 16(c).

The NI winding technique provides a high thermal stability to the REBCO pancake coils [19]. A radial bypass current prevents the NI pancake coils from burning-out after a local normal zone appears or the coils quench. Here, the model coil is wound with a copper-plated REBCO tape with a winding load of 1 kg, and winding is not impregnated. Because the NI coil is not impregnated, each winding can deform separately and move freely. Figure 17 shows the coil deformation. As shown in figure 17, the deformation of this NI coil is reduced more than that of the non-impregnated coil wound with an insulated tape. While the NI coil is wound with a bare tape, in the non-impregnated coil wound by insulated tape, the polyamide on the tape shrinks in the cooling process, resulting in gaps between the coil turns owing to insufficient winding tension. Figures 18(a) and (b) shows the circumferential and radial stresses of the REBCO layer. The radial stress is negative (i.e. compressive) after winding process and increases to almost zero after cooling process. This is because the additional stress due to thermal contraction.

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Figure 18(c) shows the circumferential stress due to electromagnetic force including screening-current effect of lower and upper sides of winding in uppermost single pancake coil. Effect of the electromagnetic force due to a screening current are slightly appeared.

In an NI winding, the elimination of low modulus, high thermal contraction materials, such as polyamide tape or impregnation, significantly increases the mechanical stiffness of the winding. Forces are directly transmitted between conductor turns resulting in near integral coil deformation. Therefore, NI winding technique was also to significantly increase the mechanical stiffness of the winding. This is significant for magnets designed for use in high electromagnetic stress environment.

Although the NI REBCO pancake coils show high thermal stability, the charging delay is an issue. Hence, to improve the charging delay, we co-wound an SS tape between the turns to increase the turn-to-turn contact resistivity [44–46]; this is termed as 'metal-as-insulation' or 'metal-insulation (MI).' A thickness of SS tape is 70 mm. Here, the model coil is co-wound with a copper-plated REBCO tape and SS tape with a winding load of 1 kg, and winding is not impregnated. The results of SS-tape co-winding with NI are shown in figure 19. The addition of stainless steel tends to decrease the overall current density as shown in table 2. However, the overall current densities of SS-tape co-winding coil is the same as that of non-impregnated coil wound by insulated tape. Comparing between figures 15 and 19, the strength of stainless steel has a larger effect on increasing effective circumferential strength than the decreasing the overall current density.

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The stress distributions of REBCO layer are shown in figure 20. As shown in figure 20(a), the circumferential stress dramatically reduced after cooling process compared with figure 18(a). In this case, the thermal contraction of co-wound SS tape canceled the stress of winding tension. After an excitation, an additional circumferential stress is smaller than that of NI coil. And as shown in figure 20(c), effect of the electromagnetic force due to a screening current is slightly appeared as the case of NI coil.

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