Numerical analysis of the screening current-induced magnetic field in the HTS insert dipole magnet Feather-M2.1-2

The computational domain Ω representing the superconducting magnet is illustrated in figure 1. The domain is decomposed into the source and source-free regions ${\Omega}_{\mathrm{H}}$ and ${\Omega}_{\mathrm{A}}$ oriented with the unit vector $\mathbf{n}$, such that ${\overline{\Omega}}_{\mathrm{H}}\cup{\overline{\Omega}}_{\mathrm{A}} = {\overline{\Omega}}$. The source region ${\Omega}_{\mathrm{H}}$ corresponding to the coil is given by the union of the superconducting and normal-conducting parts ${\Omega}_{\mathrm{H,s}}$ and ${\Omega}_{\mathrm{H,c}}$. The source-free region ${\Omega}_{\mathrm{A}}$, containing the remainder of the magnet such as the iron yoke, the mechanical supports, and the air region, is given by the union of the normal-conducting and non-conducting parts ${\Omega}_{\mathrm{A,c}}$ and ${\Omega}_{\mathrm{A,i}}$. A constant magnetic permeability µ is assumed in ${\Omega}_{\mathrm{H}}$, whereas a non-linear dependency from the magnetic field $\mathbf{B}$ is considered for the iron yoke in ${\Omega}_{\mathrm{A}}$, as $\mu(\mathbf{B})$.

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The field problem is solved under magnetoquasistatic assumptions for the reduced magnetic vector potential $\mathbf{A}^{\ast}$ [40] in ${\Omega}_{\mathrm{A}}$, and for the magnetic field strength $\mathbf{H}$ [41, 42] in ${\Omega}_{\mathrm{H}}$, with suitable boundary conditions on the exterior boundary. The formulation reads

$$\begin{align} \nabla\times \mu^{-1}\nabla\times \mathbf{A}^{\ast} + {\sigma}\partial_{t}\mathbf{A}^{\ast} & = 0\ \text{in}\ {\Omega}_{\mathrm{A}}, \end{align} \tag{ 2 }$$

$$\begin{align} \nabla\times {\rho}\nabla\times \mathbf{H} + \partial_{t}{\mu}\mathbf{H} - \nabla\times \boldsymbol{\chi}\mathrm{u}_{\mathrm{s}} & = 0\ \mathrm{in}\ {\Omega}_{\mathrm{H}}, \end{align} \tag{ 3 }$$

$$\begin{align} \int\limits_{{\Omega}_{\mathrm{H}}} \boldsymbol{\chi}\cdot(\nabla\times \mathbf{H}) d{{\Omega}} & = \mathrm{i}_{\mathrm{s}}, \end{align} \tag{ 4 }$$

where $\boldsymbol{\chi}$ is a voltage distribution function [43, 44], $\mathrm{u}_{\mathrm{s}}$ is the source voltage treated as an algebraic unknown, and $\mathrm{i}_{\mathrm{s}}$ is the source current which is imposed via a constraint equation (i.e. a Lagrange multiplier). The sources are provided by means of the electrical ports ${\Gamma}_{\mathrm{J}}$ and ${\Gamma}_{\mathrm{E}}$. The fields $\mathbf{A}^{\ast}$ and $\mathbf{H}$ are linked via continuity conditions at the interface of the domains ${\Gamma}_{\mathrm{HA}}$, represented in figure 1 by a dashed line. In particular, the continuity of the normal component of the magnetic field $\mathbf{B}_{\mathrm{n}}$ and the current density $\mathbf{J}_{\mathrm{n}}$, and the tangential component of the magnetic field strength $\mathbf{H}_{\mathrm{t}}$ and electric field strength $\mathbf{E}_{\mathrm{t}}$ are imposed, ensuring the consistency of the overall field solution.

The magnetic field quality in accelerator magnets is defined as the set of Fourier coefficients, known also as field harmonics or multipole coefficients. The coefficients are derived from the solution of the field problem in the source-free magnet aperture, which is given by the Laplace equation $\nabla^{2}\mathbf{A} = 0$. In the two-dimensional approximation of accelerator magnets, the axial field variations are neglected along the z-direction (the longitudinal axis of the magnet). Thus, the field can be expressed as (e.g. [45])

$$\begin{equation} \mathrm{A}_{\mathrm{z}}(r,\varphi) = \sum_{k = 1}^{\infty}{r}^{\,k} ({\mathcal{A}}_{k} \sin{k}\varphi + {\mathcal{B}}_{k} \cos{k}\varphi). \end{equation} \tag{ 5 }$$

where $\mathrm{A}_{\mathrm{z}}$ is the longitudinal component of the magnetic vector potential, ${\mathcal{A}}_{k}$ and ${\mathcal{B}}_{k}$ are the multipole coefficients, and (r, ϕ, z) are spatial coordinates in a cylindrical reference system consistent with the magnet aperture. The field components are obtained from (5) as

$$\begin{align} \mathrm{B}_{r}(r,\varphi) & = \sum_{k = 1}^{\infty}{k}{r}^{\,k-1} ({\mathcal{A}}_{k} \cos{k}\varphi - {\mathcal{B}}_{k} \sin{k}\varphi), \end{align} \tag{ 6 }$$

$$\begin{align} \mathrm{B}_{\varphi}(r,\varphi) & = - \sum_{k = 1}^{\infty}{k}{r}^{\,k-1} ({\mathcal{A}}_{k} \sin{k}\varphi + {\mathcal{B}}_{k} \cos{k}\varphi). \end{align} \tag{ 7 }$$

The index k represents solutions of the Laplace equation which can be associated to field distributions generated by ideal magnet geometries. As an example, k = 1,2,3 correspond to the dipole, quadrupole, and sextupole field distributions. Once the radial field component (6) is known at a reference radius $r = \mathrm{r}_{0}$ (either by measurements or simulations), the skew and normal multipole coefficients $\mathrm{A}_{k}$ and $\mathrm{B}_{k}$ are obtained for k = 1, 2, 3, ..., as

$$\begin{align} \mathrm{A}_{k}(\mathrm{r}_{0}) & = \frac{1}{\pi} \int_{0}^{2\pi} \mathrm{B}_{r}(\mathrm{r}_{0},\varphi) \cos{k}\varphi\ {d}\varphi, \end{align} \tag{ 8 }$$

$$\begin{align} \mathrm{B}_{k}(\mathrm{r}_{0}) & = \frac{1}{\pi} \int_{0}^{2\pi} \mathrm{B}_{r}(\mathrm{r}_{0},\varphi) \sin{k}\varphi \ {d}\varphi, \end{align} \tag{ 9 }$$

where the radius $\mathrm{r}_{0}$ is usually chosen as 2/3 of the magnet aperture.

The coefficients are often combined in the complex notation $\mathrm{C}_{k}(\mathrm{r}_{0}) = \mathrm{B}_{k}(\mathrm{r}_{0})+i\mathrm{A}_{k}(\mathrm{r}_{0})$ as the skew and normal pairs are orthogonal to each other. Moreover, the coefficients are typically normalized with respect to the main field component $\mathrm{B}_{\mathrm{K}} (\mathrm{r}_{0})$, and denoted as $\mathrm{b}_{k}$ and $\mathrm{a}_{k}$. Thus, the field quality is quantified as a relative error $\mathrm{c}_{k}$, for k = 1, 2, 3, ..., as

$$\begin{equation} \mathrm{c}_{k}(\mathrm{r}_{0}) = \mathrm{b}_{k}(\mathrm{r}_{0}) + i \mathrm{a}_{k}(\mathrm{r}_{0}) = 10^4 \frac{\mathrm{C}_{k}(\mathrm{r}_{0})}{\mathrm{B}_{\mathrm{K}}}, \end{equation} \tag{ 10 }$$

and given in 1 × 10⁻⁴ units with respect to $\mathrm{B}_{\mathrm{K}}$ at the reference radius $\mathrm{r}_{0}$. It is worth mentioning that for reaching accelerator quality standards, the field multipoles shall be limited within a few units [45].

The magnetic field quality can also be conveniently given in terms of total harmonic distortion factor $\mathrm{F}_{\mathrm{THD}}(\mathrm{r}_{0})$, which is a scalar quantity defined as

$$\begin{equation} \mathrm{F}_{\mathrm{THD}}(\mathrm{r}_{0}) = \sqrt{ \sum_{{k} = 1;\ {k}\neq {K}}^{\infty} \mathrm{b}_{k}^{2}(\mathrm{r}_{0}) + \mathrm{a}_{k}^{2}(\mathrm{r}_{0})}, \end{equation} \tag{ 11 }$$

where K refers to the index of the main field component.

The 2D magnetoquasistatic model of the HTS tape is used for calculating the specific Joule losses per cycle, in the sinusoidal regime. The tape is composed only by one superconducting layer whose specifications are given in table 1. Two scenarios are considered, differing only in the source quantity applied to the tape: (1) an external magnetic field at zero current, (figure 2, left), and (2) a supply current, in self field (figure 2, right). The results are verified against analytical solutions and presented in sections 3.1.1 and 3.1.2.

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Table 1. Tape specifications.

Name	Unit	Value	Description
$\delta_{\mathrm{w}}$	[mm]	10	Tape width
$\delta_{\mathrm{t}}$	[mm]	1 × 10−6	Tape thickness
$\mathrm{J}_{\mathrm{c}}$	[kA mm−2]	100	Critical current density

The numerical model of the HTS tape is used over a frequency range of several orders of magnitude. For this reason, an adaptive mesh distribution is used in the tape. The mesh elements are denser at the tape edges, following a geometrical distribution of ratio 25. This allows for resolving the highly-non-linear current density distribution in the tape. About 500 elements are used for the simulations at low field and current, whereas about 20 elements are used for saturated tapes, in accordance with the relaxation of the magnetic field within the tape. The maximum time step size is given as ${\Delta{t}}_{\mathrm{\textrm{max}}} = (50{f})^{\,-1}$, where f is the frequency of the source quantity in the model.

3.1.1. Tape in perpendicular external field.

A single HTS tape with no supply current is exposed to a time-dependent, perpendicular external field. The layout of the simulated scenario is shown in the box of figure 3. The source term is given by a sinusoidal magnetic field ${B}(t) = \mathrm{B}_{\mathrm{p}}\mathrm{\textrm{sin}}(2\pi ft)$, applied perpendicularly to the tape. The specific loss per cycle $\mathrm{w}_{\mathrm{J}}$ is calculated for n = 5, 20 and 40. The case of ${n} = \infty$, which corresponds to the critical state model [37, 38], is calculated analytically. The theory of infinitely thin films with finite width and one dimensional current distribution in a perpendicular field [47–49] gives $\mathrm{w}_{\mathrm{J}}$ as

$$\begin{equation} \mathrm{w}_{\mathrm{J}} = {\delta}_{\mathrm{w}} \mathrm{J}_{\mathrm{c}}\mathrm{B}_{\mathrm{c}} \left(\frac{2}{\mathrm{b}_{\mathrm{p}}} \mathrm{\textrm{ln}}(\mathrm{\textrm{cosh}}\ \mathrm{b}_{\mathrm{p}}) -\mathrm{\textrm{tanh}}\ \mathrm{b}_{\mathrm{p}}\right), \end{equation} \tag{ 12 }$$

where $\mathrm{B}_{\mathrm{c}} = {{\mu}_{0}}({\delta}_{\mathrm{h}}\mathrm{J}_{\mathrm{c}})/\pi$ is the critical magnetic field, ${\delta}_{\mathrm{w}}$ and ${\delta}_{\mathrm{h}}$ are the width and the thickness of the tape, and $\mathrm{b}_{\mathrm{p}} = \mathrm{B}_{\mathrm{p}}/\mathrm{B}_{\mathrm{c}}$ is the normalized magnetic field.

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The losses $\mathrm{w}_{\mathrm{J}}$ are given in figures 3 and 4 as a function of $\mathrm{B}_{\mathrm{p}}$ and f. The losses converge to the theoretical solution in [47] for increasing n-values, which is to be expected given that the critical state model corresponds to the power-law equation (1) where n is set to infinite. For low field values, the losses follow a quartic scaling law with respect to the magnetic field amplitude. Once the magnetic field reaches $\mathrm{B}_{\mathrm{c}}$, it fully penetrates in the tape, and the screening current distribution is maintained. The losses grow proportionally with the amplitude of the applied magnetic field, as the model considers the critical current density to be constant and field-independent. For the sake of completeness, figure 3 reports also a trend line for a cubic scaling law, which is found in models accounting for finite n-values and two dimensional current density distributions in the tape [4, 50, 51].

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The losses presented in figure 4 are calculated for a peak field of 10 mT. The field is chosen below the penetration limit, such that the field-screening behavior of the tape is included in the simulation. For high n-values (see figure 4) the frequency dependency tends to vanish, in accordance with a hysteresis-like behavior, and $\mathrm{w}_{\mathrm{J}}$ converges to the theoretical solution in [47] for ${n} = \infty$.

3.1.2. Tape in self-field.

A source current is imposed to a single HTS tape, in self-field. The layout of the simulated scenario is shown in the box of figure 5. The source term is given by a sinusoidal current ${I}(t) = \mathrm{I}_{\mathrm{p}}\mathrm{\textrm{sin}}(2\pi ft)$, applied to the tape as source. The calculation of $\mathrm{w}_{\mathrm{J}}$ is done for n = 5, 20 and 40, whereas the case of ${n} = \infty$ is calculated analytically. The theory of infinitely thin films with finite width and one dimensional current distribution in self-field [49, 52] gives $\mathrm{w}_{\mathrm{J}}$ as

$$\begin{equation} \mathrm{w}_{\mathrm{J}} = \frac{{\mu}_{0}}{{\delta}_{\mathrm{w}}{\delta}_{\mathrm{h}}} \mathrm{I}_{\mathrm{c}}^{2} \frac{\mathrm{i}_{\mathrm{p}}^{4}}{{6}\pi}, \end{equation} \tag{ 13 }$$

where $\mathrm{I}_{\mathrm{c}}$ is the critical current of the tape and $\mathrm{i}_{\mathrm{p}} = \mathrm{I}_{\mathrm{p}}/\mathrm{I}_{\mathrm{c}}$ is the normalized supply current. The losses $\mathrm{w}_{\mathrm{J}}$ are given in figures 5 and 6 as a function of $\mathrm{I}_{\mathrm{p}}$ and f. Consistent with (13), for high n-values the numerical results show a quartic dependence for currents up to the critical current. Beyond this value, the current density distributes homogeneously in the tape, and the losses are proportional to $\mathrm{I}^{n+1}$, in accordance with the power-law behavior in (1).

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The losses presented in figure 6 are calculated for a sub-critical current $\mathrm{I}_{\mathrm{p}} = 0.5\ \mathrm{I}_{\mathrm{c}}$. From figure 5 it is clear that with increasing n-value, the simulation results converge to the analytical dependence given in literature [49, 52], whereas figure 6 shows that with increasing n-value, the frequency dependency vanishes as expected.

In numerical models, the multipole coefficients are obtained by applying the Fast Fourier Transform algorithm to the radial component of the magnetic field, calculated along the reference circumference in the magnet aperture (see section 2.2). Care has to be taken, as the finite resolution of the mesh in the spatial discretization introduces a numerical error which affects the calculation of the multipole coefficients [53]. For this reason, a mesh sensitivity analysis is carried out for a reference model where a known analytical field solution is simulated and calculated at the reference circumference. The relative error ${\epsilon}_{{\Delta{x}}}$ is defined as

$$\begin{equation} {\epsilon}_{{\Delta{x}}} = {\frac{|\mathrm{F}_{\mathrm{THD}}-\mathrm{F}_{\mathrm{THD}}^{{\Delta{x}}}|}{\mathrm{F}_{\mathrm{THD}}}}, \end{equation} \tag{ 14 }$$

where $\mathrm{F}_{\mathrm{THD}}$ and $\mathrm{F}_{\mathrm{THD}}^{{\Delta{x}}}$ are the total harmonic distortion factors in (11) for the analytical and calculated field solutions. The error is shown in figure 7 as a function of the reciprocal of the maximum element size $\Delta{\mathrm{x}}_{\mathrm{\textrm{max}}}$. Based on this investigation, triangular elements with ${\Delta\mathrm{x}}_{\mathrm{\textrm{max}}} = 1\ \mathrm{mm}$ are chosen for the mesh, yielding an estimated error of 3 × 10⁻⁵.

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The model is implemented for a 2D transverse field configuration, thus neglecting the magnetic effects of the end-coils. Due to the presence of the layer jumps connecting the lower and the upper windings in the coil, the magnetic symmetry in the cross-section of the magnet is not preserved. For this reason, the model accounts for a four-quadrants geometry, including the layer jumps in the first and third quadrant. The layer jump is visible in figure 9, as a cable slightly misaligned with respect to the coil decks.

HTS tapes feature a multi-material and multi-layer structure. At the same time, the tape used in the coil has a width-to-thickness ratio of about two orders of magnitude. This justifies approximating the geometry of the tape with a line [46]. In this way, the discretization of the thickness of the superconductor is avoided. At the same time, the physical properties of the materials composing the tape are homogenized. Such simplification is adopted to ensure an acceptable computational time, as the 2D model accounts for 648 tapes over four quadrants.

The cable used in the coil is made of 15 tapes, which are fully transposed using the Roebel technique [58, 59]. The cross-section of the cable used in the numerical model is sketched in the box of figure 8, where each line represents a tape. Each tape is electrically connected in a parallel configuration, allowing for the redistribution of the supply current. Moreover, the Roebel transposition enforces the same electrical impedance for each of the tapes composing the cable, providing an even current distribution. For this reason, the same fraction of the supply current is imposed in the numerical model to each of the tapes, excluding current redistribution phenomena. Coupling currents [3] are also excluded, since they represent a second-order effect with respect to persistent currents [4].

Within each tape, current sharing phenomena are modeled by means of an equivalent surface resistivity, which homogenizes the superconducting and normal-conducting layers, as detailed in [32]. The surface resistivity depends from the power law in (1), thus is affected by the n-value. From magnet measurements [14], an n-value of 5 was experimentally found, outside the expected range of 20–30 [60], and attributed to unbalanced tape joints. However, note that for persistent magnetization the local critical current density is the relevant quantity and so the joint resistance is not the relevant quantify for calculating the persistent magnetization. Unfortunately, the tape was not characterized individually and so the uncertainty of the superconducting properties of the tapes is significant and the n-value is not known. To overcome this issue, a parametric sweep is performed for 4 ≤ n≤30, quantifying the sensitivity of the model. The results are compared with measurements in section 5.

The critical current density $\mathrm{J}_{\mathrm{c}}$ in (1) affects the persistent currents dynamics and, ultimately, the field quality in the magnet. In ReBCO tapes, $\mathrm{J}_{\mathrm{c}}$ shows an anisotropic, field- and temperature-dependent behavior, as $\mathrm{J}_{\mathrm{c}}({B},{T},{\theta}_{\mathrm{B}})$, where ${\theta}_{\mathrm{B}}$ is the magnetic field angle with respect to the direction perpendicular to the tape wide surface.

The behavior of $\mathrm{J}_{\mathrm{c}}$ is included in the model by means of the numerical fit provided in [57]. The fit parameters, reported in table 3, are taken from [4], since no data was available for the used Sunam tape. For this reason, the fit is scaled in order to provide a critical current for the Feather-M2.1-2 coil which is consistent with measurements [14], as follows.

Table 3. Parameters used for the $\mathrm{J}_{\mathrm{c}}$ fit.

Name	Unit	Value	Name	Unit	Value
$\mathrm{g_0}$	–	0.03	ν	–	1.85
$\mathrm{g_1}$	–	0.25	$\mathrm{a}$	–	0.1
$\mathrm{g_2}$	–	0.06	$\mathrm{n_0}$	–	1
$\mathrm{g_3}$	–	0.06	$\mathrm{n_1}$	–	1.4
$\mathrm{T_{c0}}$	K	93	$\mathrm{n_2}$	–	4.45
$\mathrm{p_c}$	–	0.5	$\mathrm{p_{ab}}$	–	1
$\mathrm{q_c}$	–	2.5	$\mathrm{q_{ab}}$	–	5
$\mathrm{B_{i0c}}$	K	140	$\mathrm{B_{i0ab}}$	T	250
${\gamma_\mathrm{c}}$	–	2.44	${\gamma_\mathrm{ab}}$	-	1.63
${\alpha_\mathrm{c}}$	$\frac{\mathrm{MA\ T}}{\mathrm{mm^{2}}}$	1.86	${\alpha_\mathrm{ab}}$	$\frac{\mathrm{MA\ T}}{\mathrm{mm^{2}}}$	68.3

The magnetic characteristic of the magnet, known also as the load line, is calculated numerically by means of magnetostatic simulations. With respect to figure 10, the load line is given in terms of peak magnetic field $\mathrm{B}_{\mathrm{p,coil}}$ in the coil as a function of the supply current (dotted line). The critical current is then given for each temperature as the intersection of the load line (markers) with the critical current provided by the fit (solid lines). The magnetic field is assumed to be perpendicular to the cable, as ${\theta}_{\mathrm{B}} = 0^{\circ}$. In figure 11, the calculated critical current is compared with the measurements, and parametrized by the field angle. The assumption of field perpendicularity gives the best agreement with the measured data. The fitting factor is finally obtained as

$$\begin{equation} \mathrm{f}_{\mathrm{c}}({T}) = \left.\frac{\mathrm{I}_{\mathrm{c,meas}}({T})} {\mathrm{J}_{\mathrm{c}} (\mathrm{B}_{\mathrm{p,coil}}({T}),{T},{\theta}_{\mathrm{B}}) \mathrm{S}_{\mathrm{HTS}}} \right\rvert_{{\theta}_{\mathrm{B}} = 0^{\circ}}, \end{equation} \tag{ 15 }$$

where $\mathrm{I}_{\mathrm{c,meas}}$ is the critical current obtained from measurements, and $\mathrm{S}_{\mathrm{HTS}}$ is the superconducting cross-section of the cable. The fitting factor is shown as a function of temperature in figure 12, and parametrized by the field angle. The factor obtained for ${\theta}_{\mathrm{B}} = 0^{\circ}$ is used in the model for scaling the critical current density fit.

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The magnetic material used in the yoke of the Feather-M2 magnet is chosen to minimize the detrimental influence of the iron hysteresis on the magnetic field quality. No material characterization data was available, however the iron is similar to the one of the LHC main dipoles. For this reason, the numerical model considers the same non-linear $\mathrm{B(H)}$ curve used for the LHC [61]. The curve is shown as a dashed line in figure 13. In stand-alone operations, most of the outer iron yoke of the Feather-M2.1-2 magnet remains unsaturated up to the maximum current of $5\ \mathrm{kA}$. For this reason, the contribution of the iron hysteresis on the field quality cannot be neglected a priori, and is estimated as follows.

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A coercive field $\mathrm{H}_{\mathrm{c}}$ of ${40\ \mathrm{A}\ \mathrm{m}^{-1}}$ is assumed for the material used in the yoke, in accordance with the LHC specifications ($\mathrm{H}_{\mathrm{c}}\leq {60\ \mathrm{A}\ \mathrm{m}^{-1}}$ [45]). The iron hysteresis is included by using the Jiles-Atherton model [62], whose loop is determined by using the B(H) curve as reference, and shown in figure 13. The relevant parameters for the hysteresis model are obtained using the open-source algorithm from [63] and are reported in table 4.

Table 4. Parameters for the Jiles-Atherton hysteresis model.

Name	Unit	Value	Description
$\mathrm{M_{s}}$	[${\mathrm{A}\ \mathrm{m}^{-1}}$]	1.35 × 106	Saturation magnetization
$\mathrm{a}$	[${\mathrm{A}\ \mathrm{m}^{-1}}$]	90	Domain wall density
$\mathrm{k}$	[${\mathrm{A}\ \mathrm{m}^{-1}}$]	40	Pinning loss
$\mathrm{c}$	[–]	1 × 10−6	Magnetization reversibility
α	[–]	50 × 10−6	Inter-domain coupling

The hysteresis contribution is calculated on a simplified Feather-M2.1-2 model, including only the iron as non-linear effect. The multipole coefficients are obtained as function of the current for both the upper and the lower hysteresis curve, then the two data sets are compared. Their difference Δb provides the amplitude of the magnetization loop for each coefficient, giving an estimation for the effect of the iron hysteresis on the field quality.

With respect to figure 14, the hysteresis of the iron has a minor influence at low current on the main field component $\mathrm{b}_{1}$. A peak value of about 20 units is found and it rapidly decreases once the current is increased, since the with of the hysteresis loop narrows. Concerning the higher order multipoles $\mathrm{b}_{3}$, $\mathrm{b}_{5}$ and $\mathrm{b}_{7}$, there is almost no influence since the contribution is always less than one unit. As a consequence, the contribution to the field from the interaction between the iron hysteresis and the screening currents in the coil can be reasonably assumed a second order effect, thus negligible.

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The analysis shows a limited influence from the iron hysteresis on the magnetic field quality, at the price of an increased computational cost. For this reason, the iron hysteresis is excluded from the numerical model of the Feather-M2.1-2 magnet.

Rotating-coil magnetometers, also known as harmonic coils, are electromagnetic transducers for measuring the $\mathrm{B}_{k}$ and $\mathrm{A}_{k}$ field multipoles. The coil shaft is positioned parallel to the magnetic axis of the magnet, and it is rotated in the magnet aperture. The change of flux linkage Φ induces, by integral Faraday's law $\mathrm{U}_{\mathrm{m}} = -\mathrm{d}_{t}{\Phi}$, a voltage signal $\mathrm{U}_{\mathrm{m}}$ which is measured at the terminals of the coil. By integrating in time the voltage signal, the flux linkage is obtained and given as a function of the series expansion of the radial field [45]. Assuming a coil of negligible thickness, perfectly centered in the aperture of a magnet, and rotating with angular velocity ω, then for an arbitrary angle ϕ(t) = ωt + ϕ₀ the flux linkage is given at time t as

$$\begin{align} \Phi(t) & = \sum_{k = 1}^{\infty} \mathrm{f}_{\mathrm{s}} \left[\mathrm{A}_{k}(\mathrm{r}_{\mathrm{c0}}) \cos k\varphi -\mathrm{B}_{k}(\mathrm{r}_{\mathrm{c0}}) \sin k\varphi \right], \end{align} \tag{ 16 }$$

$$\begin{align} \mathrm{f}_{\mathrm{s}}(k) & = \frac{{2} \mathrm{N}_{\mathrm{c}} \mathrm{l}_{\mathrm{c}} \mathrm{r}_{\mathrm{c0}}}{k}, \end{align} \tag{ 17 }$$

where the coil sensitivity factor $\mathrm{f}_{\mathrm{s}}(k)$ embeds the coil geometric parameters, namely the number of turns $\mathrm{N}_{\mathrm{c}}$, longitudinal length $\mathrm{l}_{\mathrm{c}}$ and the mean radius $\mathrm{r}_{\mathrm{c0}}$. Such parameters are calibrated in a dipole and quadrupole reference magnet.

Encouraged by the results obtained from the flux sensors presented in [15], a dedicated rotating-coil magnetometer was developed and employed to test the Feather-M2.1-2 magnet in the variable temperature cryostat at CERN The constructed coil shaft is composed of a chain of five Printed-Circuit Boards (PCBs), (200 mm in length and 35 mm in width), that span the entire magnet length including the fringe-field areas. Every PCB board contains three coils mounted radially, with an active surface of 0.1817 m².

For the magnetic-field harmonics, the measurement sensitivity is improved by connecting two coils in anti series; for the dipole magnet measurement the external coil minus the central coil. CERN proprietary digital cards [65] integrate the induced voltages in the coils rotating at a frequency of 2 Hz. In this paper, the measurement results are taken from the longitudinal center of the magnet (the central element of the rotating shaft of 200 mm in length), delivering a measurement precision of a magnetic-field harmonic of ±0.05 units.

To match the experimental procedure, a current excitation is applied as a source for the numerical model. With respect to the example provided in figure 15, the current follows firstly a trapezoidal pre-cycle, then a staircase profile spanning from a minimum value of $0.25 \mathrm{kA}$ up to the peak current, and back. The aim of the pre-cyle is to remove the dependency of the superconducting coil on the first magnetization cycle. The staircase signal is composed of steps of steepness $10\ \mathrm{A\ s}^{-1}$, which increase the current by $\Delta{I} = 250\ \mathrm{A}$, and then keep it constant for $\Delta\mathrm{t}_{\mathrm{flat}} = 120\ \mathrm{s}$. For each midpoint in the staircase plateaus, sowed in figure 15 with a marker, the magnetic field quality is calculated and compared with measurements. The number of steps is adapted for each scenario, in order to reach the prescribed peak current. The shape of the current excitation and the evaluation points for the field quality are consistent with the ones used in the measurements.

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Following [15], the staircase current profile is used to quantify the influence of hysteresis phenomena occurring within both the superconducting coil and the iron yoke of the magnet, as follows. With respect to figure 15, for each current step the field quality is measured and simulated twice, once during the ramp-up and then during the ramp-down, and the results are grouped in pairs. Subsequently, the difference of the field multipoles is calculated for each pair of field quality evaluations. Thanks to this operation, the contributions of the non-ideal geometry of the coil and the iron saturation are canceled out, being the same for both evaluations in each pair, and the residual is attributed to the hysteresis phenomena. Since the iron hysteresis was previously found to produce only a second-order effect on the field quality (see section 4.4), the hysteresis contribution is fully attributed to the persistent magnetization of the superconducting coil.

The measured and simulated field multipole coefficients are given in figure 16. The markers represent the measurements which are split in the up and down datasets, accordingly to the upward and downward part of the current staircase (see figure 15). The shaded area gives the envelope of the numerical solutions obtained by a parametric sweep of the n-value between 4 and 30. As an example, the simulation results for n = 20 are highlighted with a solid line. The dashed line represents the ideal case in which the the screening currents do not have any influence on the field quality. This is obtained by artificially increasing the resistivity of the superconducting coil until a homogeneous current density distribution is achieved in the time domain simulation. The rows show, from top to bottom, the normal dipole field $\mathrm{B}_{\mathrm{1}}$ and the multipoles $\mathrm{b}_{\mathrm{3}}$, $\mathrm{b}_{\mathrm{5}}$ and $\mathrm{b}_{\mathrm{7}}$, as a function of the source current. The columns separate the results by the operational temperature of the magnet, namely 4.5, 9, 25, and $68\ \mathrm{K}$ or, in other words, the simulated scenario.

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The field multipoles keep qualitatively the same behavior through the different scenarios (see figure 16, row by row). Moreover, the $\mathrm{b}_{\mathrm{3}}$ and $\mathrm{b}_{\mathrm{5}}$ multipoles are reduced as the the current is increased. The $\mathrm{b}_{\mathrm{7}}$ coefficient is negligible with respect to the others. The scenario at $4.5\ \mathrm{K}$ shows the highest variation in the magnitude of the multipoles. At low current, the $\mathrm{b}_{\mathrm{3}}$ contribution increases of about a factor 2, from 200 to 400 units, and the $\mathrm{b}_{\mathrm{5}}$ multipole shows an increase of a factor 8, from 10 to 80 units. This might be explained as screening currents are higher at low temperature, due to the higher critical current density of the tape.

Hysteresis phenomena in the Feather-M2.1-2 magnet create the magnetization loops which are present in the measured and simulated data sets. The loops are found to be at least one order of magnitude smaller than the absolute value of the multipole coefficients. For this reason, the width of the loops is shown separately in figure 17. The layout and the meaning of symbols is the same as before for figure 16. The rows show from top to bottom the variation in units for the multipoles $\mathrm{b}_{\mathrm{1}}$, $\mathrm{b}_{\mathrm{3}}$, $\mathrm{b}_{\mathrm{5}}$ and $\mathrm{b}_{\mathrm{7}}$, as a function of the supply current. The columns separate the results by the operational temperature of the magnet, namely 4.5, 9, 25, and $68\ \mathrm{K}$.

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The width of the magnetization loops due to persistent currents does not exceed twenty units for $\mathrm{b}_{\mathrm{1}}$ and $\mathrm{b}_{\mathrm{3}}$, two units for $\mathrm{b}_{\mathrm{5}}$ and one unit for $\mathrm{b}_{\mathrm{7}}$. The trend is generally monotone, showing the multipoles decreasing as the current increases, and vanishing as the current reaches its peak value. The $\mathrm{b}_{\mathrm{1}}$ coefficient is an exception, as it has a peak around $3.5\ \mathrm{kA}$, when the pole of the iron yoke saturates. In the ideal case where the screening currents are neglected, the width of the magnetization loops is always zero.