Inverse analysis of critical current density in a bulk high-temperature superconducting undulator

In order to optimize the design of undulators using high-temperature superconductor (HTS) bulks we have developed a method to estimate the critical current density ($J_c$) of each bulk from the overall measured magnetic field of an undulator. The vertical magnetic field was measured along the electron-beam axis in a HTS bulk-based undulator consisting of twenty Gd-Ba-Cu-O (GdBCO) bulks inserted in a 12-T solenoid. The $J_c$ values of the bulks were estimated by an inverse analysis approach in which the magnetic field was calculated by the forward simulation of the shielding currents in each HTS bulk with a given $J_c$. Subsequently the $J_c$ values were iteratively updated using the precalculated response matrix of the undulator magnetic field to $J_c$. We demonstrate that it is possible to determine the $J_c$ of each HTS bulk with sufficient accuracy for practical application within around 10 iterations. The precalculated response matrix, created in advance, enables the inverse analysis to be performed within a practically short time, on the order of several hours. The measurement error, which destroys the uniqueness of the solution, was investigated and the points to be noted for future magnetic field measurements were clarified. The results show that this inverse-analysis method allows the estimation of the $J_c$ of each bulk comprising an HTS bulk undulator.

Figure 1

Principle of the HTS-bulk staggered-array undulator, HSAU (side view). The solenoid coil induces the magnetization of the HTS bulks which then generate the $y$-direction field.

Figure 2

Experimental setup. The 5-mm-thick copper holders, in which the 4-mm-thick HTS bulks are placed, are stacked on the end of the measurement system with a temperature sensor. This is installed into the cryostat with the 12-T solenoid.

Figure 3

Simulation geometry for the inverse analysis. Twenty bulks having different parameters, $p_i$, are aligned in a large air domain. The $z$-direction solenoid field is applied to the top and bottom boundaries and a perfect magnetic field constraint (such that $\mathbf{n}\times \mathbf{H}=0$) is applied to the left and right boundaries.

Figure 4

Flowchart of the inverse analysis.

Figure 5

Response of $B_y$ to $J_c$ change. This example is calculated from the change in $B_y$ when each lift factor $p$ is changed from 0.5 to 0.75 ($\alpha =0.25$) for $B_{s,\text{start}}=8\mathrm{T}$ and $B_{s,\mathrm{end}}=3\mathrm{T}$. The blue lines are for the upper (odd $i$) bulks and the red lines are for the lower (even $i$) bulks.

Figure 6

RMS difference of (a) given and estimated $\mathbf{p}$ and (b) input and output $B_y$, for ${\sigma }_p=0.05$, 0.2 and 0.4. (c) Comparison of the conversion of the matrix method with the general-purpose optimization algorithms.

Figure 7

Measured magnetic field $B_y$ when the solenoid field is changed to $B_s$ after field cooling with 8 T for operating temperatures of (a) 10 K and (b) 20 K; the relationship between (c) $\mathrm{\Delta }{B}_s$ and $B_0$, and (d) $\mathrm{\Delta }{B}_s$ and the relative standard deviation of $B_0$.

Figure 8

Results of inverse analysis; (a) the comparison of measured and simulated $B_y$, (b) the average and the standard deviation of the estimated $\mathbf{p}$ for each $T$ and $B_s$; each value of estimated (c) $\mathbf{p}$ and (e) ${\mathbf{p}}^{*}$ and (d, f) the local weighted average of these to cancel the zigzag error.

Figure 9

Current density distribution ($J_x$) after the 10th iteration of the inverse analysis for $T=10\mathrm{K}$ and $B_s=3\mathrm{T}$.

Figure 10

The zigzag effect of the local offset to the inverse analysis; (a) five input data sets for the inverse analysis, and the estimated $\mathbf{p}$ for (b) the single positive or negative offset at the left end and (c) the single or double offset in the middle.

Figure 11

Error sources for the measurement. Error due to (a) the tilt of the measurement line, (b) the probe offset in the $y$-direction, and (c) the contamination of $B_z$ due to the probe pitching. The simulation was carried out with $B_s=3\mathrm{T}$ and the estimated $\mathbf{p}$ in Sec. 3b.

Figure 12

Comparison of the results of the inverse analysis with the ANSYS backward computation method for 20 K and $B_s=2\mathrm{T}$.