Modeling of airgap influence on DC voltage generation in a dynamo-type flux pump

In this paper, the MEMEP method in two dimensions has been employed for modeling. This variational method is able to take materials with nonlinear E(J) relation like superconductors into account. MEMEP is based on the calculation of current density J by minimizing a functional containing all the variables of the problem, such as the magnetic vector potential A, current density J, and scalar potential φ. It has been proved that the minimum of this functional in the quasimagnetostatic limit is the unique solution of the Maxwell differential equations [25]. Since the current density is only inside the superconducting part, discretization of mesh in the domain around the superconducting region is not needed in this method. The general equation for the current density and the scalar potential are as follows:

$$\begin{eqnarray}&&{\bf{E}}({\bf{J}})=-\dot{{\bf{A}}}-{\rm{\nabla }}\varphi \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\bf{J}}=0\end{eqnarray} \tag{ 2 }$$

The vector potential A in equation (1) has two contributions including A_a and A_J, which are the vector potential due to applied field and vector potential due to the current density in the superconductor, respectively. Here, 2D geometry has been assumed so that J = Je_z and A = Ae_z. Furthermore, the Coulomb's gauge (∇ · A = 0) has been assumed, and hence for infinitely long problems A_J follows [23]:

$$\begin{eqnarray}&&{A}_{J}({\bf{r}})=-\displaystyle \frac{{\mu }_{0}}{2\pi }{\int }_{S}{dS}^{\prime} J({\bf{r}}^{\prime} )\ \mathrm{ln}| {\bf{r}}-{\bf{r}}^{\prime} | .\end{eqnarray} \tag{ 3 }$$

For cross-sectional 2D problems with current constraints, equation (2) is always satisfied, and hence only equation (1) needs to be solved. To solve this equation, the following functional needs to be minimized [25, 26]:

$$\begin{eqnarray}&&L={\int }_{S}{ds}\left[\displaystyle \frac{1}{2}\displaystyle \frac{{\rm{\Delta }}{{\bf{A}}}_{J}}{{\rm{\Delta }}t}\cdot {\rm{\Delta }}{\bf{J}}+\displaystyle \frac{{\rm{\Delta }}{{\bf{A}}}_{a}}{{\rm{\Delta }}t}\cdot {\rm{\Delta }}{\bf{J}}+U({{\bf{J}}}_{0}+{\rm{\Delta }}{\bf{J}})\right],\end{eqnarray} \tag{ 4 }$$

where U is the dissipation factor defined as [25]

$$\begin{eqnarray}&&U({\bf{J}})={\int }_{0}^{J}{\bf{E}}({\bf{J}}^{\prime} )\cdot d{\bf{J}}^{\prime} .\end{eqnarray} \tag{ 5 }$$

This dissipation factor can include any E − J relations in superconductors including the critical state model [25].

The functional is minimized in discrete time steps. If the functional variables are J₀, A_J0 and A_a0 in a particular time step like t₀, then on its next time step t = t₀ + Δt, the variables will become J = J₀ + ΔJ, A_J = ${{\bf{A}}}_{{J}_{0}}$ + ΔA_J and A_a = A_a0 + ΔA_a, respectively, where ΔJ, ΔA_J and ΔA_a are the change of the variables between two considered time steps and Δt is the time period between the two time steps. In this modeling, the non-uniform applied magnetic field B_a caused by the rotating magnet appears in the functional in the form of A_a [25].

As previously explained, meshing was performed only inside the superconductor region in rectangular shape elements and uniformly across the object. For this modeling, only one element along the thickness of the superconducting tape suffices and the optimum number of elements along the width was chosen to be around 200 elements for small gaps. This number has been increased up to more than 3000 elements, when higher precision is needed, especially in larger airgaps where the magnetic field density is weak. Therefore, the whole number of elements in the modeling was between 200 and 3200 elements, which reduces the computation time significantly compared to conventional FEM. Besides, the optimum number of time steps that were chosen were 360 per cycle, being the lowest number of time steps that the computations are independent on the number of time steps.. For this number of mesh and time step, the computation time was less than 20 min per cycle.

The vector potential and magnetic flux density generated by the magnet can be calculated by the magnetization sheet current density K = M × e_n, where M is the magnet magnetization and e_n is the unit normal vector to the surface. For uniform magnetization, the vector potential A_M and magnetic flux density B_M generated by the magnet are

$$\begin{eqnarray}&&{\bf{A}}({\bf{r}})=-\displaystyle \frac{{\mu }_{0}}{2\pi }\,M{\int }_{\partial S}{dl}^{\prime} \,{{\bf{e}}}_{m}\times {{\bf{e}}}_{n}({\bf{r}}^{\prime} )\mathrm{ln}| {\bf{r}}-{\bf{r}}^{\prime} | ,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\bf{B}}({\bf{r}})=\displaystyle \frac{{\mu }_{0}}{2\pi }\,M{\int }_{\partial S}{dl}^{\prime} \,\displaystyle \frac{[{{\bf{e}}}_{m}\times {{\bf{e}}}_{n}({\bf{r}}^{\prime} )]\times ({\bf{r}}-{\bf{r}}^{\prime} )}{| {\bf{r}}-{\bf{r}}^{\prime} {| }^{2}},\end{eqnarray} \tag{ 7 }$$

where e_m is the unit vector in the direction of magnetization, ∂S represents the edges of the magnet cross-section, and dl' is the length differential on the edge. Note that since both e_m and e_n are in the xy plane, the cross-product e_m × e_n is always in the z direction, and hence A follows the z direction. In this article, the intergrals for A_M and B_M were evaluated numerically.

The magnetic field generated by the magnet acts as an applied field. Then, the program only needs to calculate this vector potential and magnetic field, once for each time step within the first cycle. The impact on the total computing time is negligible, since minimization takes most of the computing time.

To define the nonlinear superconductor characteristic, the model uses the isotropic E − J power law

$$\begin{eqnarray}&&{\bf{E}}({\bf{J}})={E}_{c}{\left(\displaystyle \frac{| {\bf{J}}| }{{J}_{c}}\right)}^{n}\displaystyle \frac{{\bf{J}}}{| {\bf{J}}| },\end{eqnarray} \tag{ 8 }$$

where E_c = 10⁻⁴ V m⁻¹ is the critical electric field , J_c is the critical current density and n is the power law exponent or n-value. With this E(J) relation, the nonlinear resistivity defined from E(J) = ρ(J)J is

$$\begin{eqnarray}&&\rho ({\bf{J}})=\displaystyle \frac{{E}_{c}}{{J}_{c}}{\left(\displaystyle \frac{| {\bf{J}}| }{{J}_{c}}\right)}^{n-1}.\end{eqnarray} \tag{ 9 }$$

For the E − J power law, the disspiation factor of the functional in the equation (5) becomes

$$\begin{eqnarray}&&U({\bf{J}})=\displaystyle \frac{{E}_{c}{J}_{c}}{n+1}{\left(\displaystyle \frac{| {\bf{J}}| }{{J}_{c}}\right)}^{n+1}.\end{eqnarray} \tag{ 10 }$$

The J_c(B) dependence of the critical current density of the HTS tape has been obtained from measurements in the temperature of 77.5 K (liquid nitrogen) performed in [22] and has been utilized in our model (figure 5). The tape critical current was measured in the range between 0° to 180° and the critical current dependence was assumed symmetrical from −180° to 0° .

The current density J flows in the z direction, and hence the vector potential A and electric field E are parallel to the z axis. The magnetic field B is perpendicular to the current density and its direction belongs to the xy-plane. Due to infinitely long symmetry, the following relations can be derived

$$\begin{eqnarray}\begin{array}{rcl}{\bf{E}}({\bf{r}}) & = & E(x,y){{\bf{e}}}_{z}\\ {\bf{J}}({\bf{r}}) & = & E(x,y){{\bf{e}}}_{z}\\ {\bf{A}}({\bf{r}}) & = & A(x,y){{\bf{e}}}_{z}.\end{array}\end{eqnarray} \tag{ 11 }$$

Taking these relations into account and equation (1), it is found that

$$\begin{eqnarray}&&{\rm{\nabla }}\varphi ({\bf{r}})={\partial }_{z}\varphi {{\bf{e}}}_{z},\end{eqnarray} \tag{ 12 }$$

where ∂_zφ is uniform within the superconductor. Since Coulomb's gauge is assumed, the scalar potential becomes the electrostatic potential. Figure 1, shows the sketch of the modeling configuration.

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To obtain open-circuit voltage, the gradient of the electrostatic potential needs to be calculated from equation (12).

In 2D modeling, the fact that the gradient of the electrostatic vector potential is uniform along the z axis, allows us to easily calculate the gradient per unit length. Afterwards, the voltage (or open-circuit voltage in our case) can be simply calculated by

$$\begin{eqnarray}&&V=-\left(\displaystyle \frac{\partial \varphi }{\partial z}\right)\cdot l,\end{eqnarray} \tag{ 13 }$$

where l is the length of the tape in between the voltage taps, which in the presented 2D modeling is related to the dimension of the used magnet.

The permanent magnet used for this study is a square magnet with the dimension of 10 × 10 mm. Note that although in [9] a cylindrical shape of magnet has been used, the conversion of a cylindrical shape in 2D would be a rectangular shape, which of course is not a perfect transformation, but accurate enough for our estimations. Furthermore, as shown in [22], the flux pumping mechanism can be well explained by a 2D model. The magnet type is a N42 magnet (Nd–Fe–B magnet), which offers the remanence magnetic field of around 1.3 T.

The parameters that have been selected to model the YBCO superconducting tape are 12 mm width, 1 μm thickness, n-value 30, and critical current at self field as 281 A. The Jc(B, θ) data used for HTS tape were obtained from the experimental data in [22], which belongs to the same type of the HTS tape (not identical tape) as in [9].

The rotating radius of the magnet rotor is 40 mm and the magnet outer surface has been placed on the circumference of this rotor. The rotating frequency is 12.3 Hz. The set-up configuration has been shown in figure 1.

The results presented in this section were calculated for airgap of 3.3 mm. The airgap was defined as the distance between the magnet surface and the HTS tape. Using the procedure explained in section 2.5 the voltage was calculated considering the J_c(B, θ) dependence. Besides, all the results belong to the second cycle to skip the transient state in the first cycle. The calculations start with the zero-field cooling condition, being J = 0 at t = 0. We tested and verified that the results at the following cycles are the same as those at the second cycle.

The calculated output V is comprised of two terms

$$\begin{eqnarray}&&V=-l\ {\partial }_{z}\varphi =l\ {\partial }_{t}A+l\ E(J),\end{eqnarray} \tag{ 14 }$$

where A is the total magnetic vector potential and E is the electric field, which is obtained by the E − J relation considered for the HTS tape and is dependent on the local current density of the tape. For the modeled flux pump, this voltage V has been calculated in figure 2 during the second cycle.

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However, an important parameter for flux pumping is the dc voltage component, V_dc. This dc component is defined as follows

$$\begin{eqnarray}\begin{array}{rcl}{V}_{{\rm{dc}}} & = & {fl}{\displaystyle \int }_{0}^{1/f}[{\partial }_{t}A+E(J)]{dt}={fl}{\displaystyle \int }_{0}^{1/f}E(J){dt}\\ & = & \ {fl}{\displaystyle \int }_{0}^{1/f}\rho (J){Jdt},\end{array}\end{eqnarray} \tag{ 15 }$$

where l is the tape length and ρ is the nonlinear resistivity of HTS tape. In the equation above, the vector potential in steady state is used, which is periodic and hence ${\int }_{0}^{1/f}{\partial }_{t}A\,=\,A[1/f]-A[0]=0$. The cause of the periodicity of vector potential is that both A from the magnet and the currents in the superconducting tape are periodic after the first cycle.

Based on equation (15), the output V_dc only depends on the electric field generated by the resistivity of the HTS tape, which itself is a function of the tape current density. An interesting feature is that V_dc can be calculated from the time integral of ρ(J)J on any point, the integrals for all points yielding the same result. Then, we can also use the cross-section average of the electric field to calculate V_dc, being

$$\begin{eqnarray}&&{E}_{{av}}(t)=1/S{\int }_{S}{ds}\,\rho \left[J({\bf{r}})\right]J({\bf{r}}),\end{eqnarray} \tag{ 16 }$$

where S is the cross-section surface and ds its differential. For the modeled flux pump, this contribution of voltage in the HTS tape is shown in figure 3.

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The instantaneous output voltage of a flux pump can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}V(t) & = & l[{\partial }_{t}A+E(J)]=l\displaystyle \frac{1}{S}{\displaystyle \int }_{S}{dS}[{\partial }_{t}A+E(J)]\\ & = & \ l[{\partial }_{t}{A}_{{av}}+{E}_{{av}}(J)],\end{array}\end{eqnarray} \tag{ 17 }$$

where A_av is the average total vector potential in the tape and A = A_a + A_J where A_a is the vector potential due to external magnetic field and A_J is the vector potential due to local current density in the tape. Using the same arguments as equation (15), we find that the dc voltage follows

$$\begin{eqnarray}&&{V}_{{\rm{dc}}}={fl}{\int }_{0}^{1/f}{E}_{{av}}(J).\end{eqnarray} \tag{ 18 }$$

For a linear material we have

$$\begin{eqnarray}&&{E}_{{av}}(J)=\rho {J}_{{av}}=\rho \displaystyle \frac{I}{S},\end{eqnarray} \tag{ 19 }$$

and hence V_dc vanishes for the open-circuit case because I = 0. However, V_dc does not vanish for the superconductor nonlinear E(J) relation.

The output voltage difference between the tape at 77 K (superconducting mode) and at 300 K (normal mode), being very relevant for measurments, is

$$\begin{eqnarray}&&{\rm{\Delta }}V={V}_{77{\rm{K}}}-{V}_{300{\rm{K}}}\end{eqnarray} \tag{ 20 }$$

which is equal to

$$\begin{eqnarray}&&{\rm{\Delta }}V=l\left[{\partial }_{t}({A}_{{av},J,77{\rm{K}}}-{A}_{{av},J,300{\rm{K}}})+{E}_{{av}}(J)-{\rho }_{300{\rm{K}}}\frac{I}{S}\right].\end{eqnarray} \tag{ 21 }$$

For open-circuit mode, the last term vanishes. In addition, since the metal resistivities are large at 300 K, the vector potential generated by currents at 300 K (A_{av,J,300 K}) will be negligible compared to those from superconductor at 77 K (A_{av,J,77 K}). Therefore we will have

$$\begin{eqnarray}&&{\rm{\Delta }}{V}_{{oc}}\approx l[{\partial }_{t}{A}_{{av},J}+{E}_{{av}}(J)],\end{eqnarray} \tag{ 22 }$$

where the sub-index oc denotes the open-circuit mode.

If V_dc of the graph in figure 3 is calculated using equation (18), it can be observed that the flux pump has a DC value equal to 33.6 μV. Over many cycles, this value of DC voltage will be accumulated to inject the magnetic flux into a superconducting circuit connected in series to the pump [9]. In figure 4, this trend can be noticed over 10 cycles for the modeled flux pump. The accumulated voltage is calculated by

$$\begin{eqnarray}&&{V}_{{accumulated}}(t)={\int }_{0}^{t}{V}_{{\rm{dc}}}(t^{\prime} )\ {dt}^{\prime} .\end{eqnarray} \tag{ 23 }$$

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The performance of the flux pump is also studied with the assumption of constant J_c at self-field. For this case, the generated V_dc in each cycle is equal to 16.2 μV, which is almost half of the case for J_c(B, θ) dependence. In this section, the reason behind this difference is investigated.

The flux pumping phenomena occurs because of overcritical eddy current generated in HTS tape while the magnet traverses over the tape [22]. According to the I_c(B, θ) data showed in figure 5, which was obtained from [22] and measured at 77.5 K in magnetic fields up to 7 T, the magnetic field significantly redueces the local J_c across the tape. For instance, in the case of airgap equal to 3.3 mm, the maximum perpendicular magnetic field density in the tape is around 260 mT (marked with blue dashed line in figure 5).

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Figure 6 depicts the comparison of the current density profiles, J_z, in the case of constant J_c (red solid line) and dependent J_c(B, θ) (blue dashed line) when the magnet is just on the top and middle of the tape with the airgap of 3.3 mm. Although the magnet is concentric with the tape, the current density is not symmetrical because of the hysteretic nature of the screening currents. In the case of constant J_c, the local J_c across the tape remains the same (green dashed line in figure 6). Therefore, the overcritical eddy currents that results in generation of electric field, and thus voltage, belong to the regions where the red solid line exceeds the green dashed line. However, in the case of considering J_c(B, θ) dependence, the value of J_c will diminish up to around 0.8 of J_c in the middle part (yellow dashed line in figure 6), due to perpendicular magnetic field density up to 260 mT. Therefore, the overcritical current occurs in regions where the blue dashed line exceeds the yellow dashed lines. Experiments and modeling results confirm that the latter leads to higher amount of electric field and voltage generated by the flux pump.

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In this section, we investigate the impact of the critical current density of the HTS tape on the generated open-circuit voltage of the flux pump. We consider several different cases with constant J_c with critical currents ranging from 70 to 1120 A, including the experimental critical current in self-field of 280 A. The rest of the parameters are the same as in the previous sections.

Figure 7 shows the signal of E_avl for three different values of J_c corresponding to critical currents of 140, 280 and 560 A. An important result is that this signal is almost the same, taking into account the high variation of J_c. The cause is that, as long as the tape is saturated with magnetization currents, only the value and time dependence of the magnetic field from the magnet is important. When looking at the DC component, which is responsible for flux pumping, the difference is even smaller (table 1), being almost no difference for I_c between 70 and 560 A. The decrease in the DC voltage at 1120 A is expected to be due to significant shielding of the magnetic field from the magnet, since the self-field is of the order of 100 mT and the maximum magnetic field from the magnet is around 250 mT (see figue 9).

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To confirm the modeling results, the DC value of the open-circuit voltage for different airgaps have been compared to experimental studies obtained from [9], which can be observed in figure 12.

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From figure 12, it is apparent that there is good agreement between DC values for airgaps equal to or larger than 3.3 mm. However, there is a big discrepancy in airgap of 2.4 mm. The reason for this discrepancy can be explained by several factors. The first one, as explained before in section 3.1 comes from the shape of the magnet. The magnet used in experiments was cylindrical while in 2D it is not possible to model this shape and, instead, the square bar shape has been used. This difference becomes bolder when the magnet gets very close to the tape because the shape of flux lines in very close distances are different in cylindrical and infinite rectangular magnets (the modeled one). The other error originates from measurements. This mostly happens due to error in measuring airgaps because of thermal contraction and also changing airgap value during movement of the magnet. In the end, also the modeling error cannot be neglected. This partly originates from the limitations of 2D modeling and assuming infinite tape and magnet in z direction (modeling was performed in xy-plane), and hence the flux lines are not identical to the real magnet. The variations of J_c(B, θ) within the measured tape could also cause discrepancy.