⬇ Download Markdown · ⬇ Download LaTeX · ⬇ original · view original
1
Siena Hurwitz 1 , Matt Landreman 1 , Thomas M. Antonsen, Jr 1 ,
1 Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 21043 USA The design of electromagnetic coils may require evaluation of several quantities that are challenging to compute numerically. These quantities include Lorentz forces, which may be a limiting factor due to stresses; the internal magnetic field, which is relevant for determining stress as well as a superconducting coil’s proximity to its quench limit; and the inductance, which determines stored magnetic energy and dynamics. When computing the effect on one coil due to the current in another, these quantities can often be approximated quickly by treating the coils as infinitesimally thin. When computing the effect on a coil due to its own current (e.g., self-force or self-inductance), evaluation is difficult due to the presence of a singularity; coils cannot be treated as infinitesimally thin as each quantity diverges at zero conductor width. Here, we present novel and well-behaved methods for evaluating these quantities using non-singular integral formulae of reduced dimensions. These formulae are determined rigorously by dividing the domain of integration of the magnetic vector potential into two regions, exploiting appropriate approximations in each region, and expanding in high aspect ratio. Our formulae show good agreement to full finite-thickness calculations even at low aspect ratio, both analytically for a torus and numerically for a non-planar coil of a stellarator fusion device, the Helically Symmetric eXperiment (HSX). Because the integrands of these formulae develop fine structure as the minor radius becomes infinitely thin, we also develop a method of evaluating the self-force and self-inductance with even greater efficiency by integrating this sharp feature analytically. We demonstrate with this method that the self-force can be accurately computed for the HSX coil with as few as 12 grid points.
Index Terms—Electromagnetic coil, self-inductance, self-force, Lorentz force, stellarator, nuclear fusion, critical current
T HE self-field, self-force, and self-inductance are char- acteristic quantities of conducting coils that have both theoretical and practical importance. One approach to solve for these quantities is to use finite element techniques. For coils with complicated geometries, this approach can lead to high-dimensional numerical grids and a computationally intensive system of equations. Alternatively, one can use equations such as the Biot-Savart Law with a finite-thickness coil representation to solve for these quantities, though numerical evaluation is generally difficult due to the high dimensionality of the integrals and an integrable singularity. Notably, these finite-thickness equations can be reduced to one- and two-dimensional integrals when considering effects between separate coils (such as the mutual inductance) by treating each coil as an infinitely thin filament. These cases also do not suffer from singularities. However, one cannot perform a similar filamentary treatment for the self-force, internal magnetic field, and self-inductance as these quantities diverge when the conductor approaches zero thickness. Instead, one must consider the three-dimensional structure of the conductor and integrate over the singularity to obtain a correct result. To expedite these calculations, in the present paper we present novel well-behaved methods for evaluating these quantities using non-singular integral formulae of reduced dimensions.
Some prior research has been done to simplify calculations of the self-force, self-inductance, and internal magnetic field. In [1], [2], [3], the internal magnetic field of conductors was computed in a limited geometry: rectangular cross-section and a center-line made of straight segments and circular arcs. In [4], an energy-based derivation of the self-force was presented to obtain a one-dimensional integral approximation. A simpler integral form of the self-inductance was found in [5] using a methodology not dissimilar to our own. In these last two works, a section of the domain was removed from the final integral, whereas in our approach the integral will be performed over the full length of the filamentary coil. Integrating over the full periodic filament is advantageous as spectral convergence can be achieved with a uniform quadrature grid, and data on fixed quadrature points can be reused to evaluate the self-field at other points along the coil. Ref [6] examined a current sheet rather than discrete coils, and the self-force of a current sheet was computed in terms of several surface integrals.
In this paper, we rigorously derive filamentary integral models for the self-inductance (10), magnetic field inside and near the coil (16)-(19), and self-force (21), for thin coils with circular cross-section and uniformly-distributed current density. These integrals are of reduced dimensionality and generally resemble the standard expressions in the limit in which the minor radius is zero, though they have an added regularization term in the denominator that depends on the square of the minor radius. This regularization term therefore prevents the integrand from diverging when the source and evaluation points are the same, and the scale of the minor radius determines the domain in which the regularization term is non-negligible. A circular cross-section is not representative of many real-world coils so our formulae may therefore be less accurate in these cases. We present the circular crosssection case here as the algebraic complexity is less than for other cross-section shapes, allowing a clearer emphasis on the asymptotic methods. The more complicated case of a rectangular cross-section will be addressed in a subsequent accompanying paper [7].
Our proofs, discussed in Section III and completed in the Appendices, rely on partitioning the domain of the integral for 2 the magnetic vector potential across the length of the coil into a “near region” and a “far region.” The integral in the near region may be evaluated analytically, while the integral in the far region can be simplified to a 1D integral with no singularity. These results show remarkable numerical agreement with the standard finite thickness calculations, referred to throughout this paper as “exact” calculations. We show this correspondence analytically for a circular loop and numerically for a non-planar coil of a stellarator plasma confinement device, the Helically Symmetric eXperiment (HSX) at the University of Wisconsin [8]. Figure 1 illustrates the self-force on one of the HSX coils. Because the integrands of our novel formulae become sharp at the evaluation point when the minor radius approaches zero, we also discuss methods of making the numerical evaluations even more efficient in Section IV by eliminating this fine-scale structure in the integrals. We demonstrate that quadrature for the self-force can be performed accurately with this method for the HSX coil with a grid size of as small as 12 points. The code used for generating data in this project
3 where primes indicate differentiation with respect to ϕ. The vectors satisfy the Frenet-Serret formulas,
where the curvature and torsion, κ(ϕ) and τ (ϕ), are given by
These vectors are demonstrated in Figure 2 for the HSX coil. All results of this paper behave smoothly as κ → 0, and τ does not appear in the final expressions, so results are applicable even if the coil center-line has straight points or intervals.
In this Section, we discuss the simplified integral formulae for the self-inductance, self magnetic field, and self-force density of thin electromagnetic coils. These coils are chosen to carry a uniformly distributed current I, where the current density can be written as
It can be shown that this expression satisfies ∇ · J = 0. We expand in the small parameter a/L, which behaves as an inverse aspect ratio, where L represents length scales associated with r c such as |r ′ c |, κ −1 , and τ −1 . To obtain a concrete measure of the scale length L, we introduce the quantity R, which is defined to be the coil length divided by 2π, and can be conceptualized as an approximate major radius.
4
integrals. Approximating the right-hand side for α ≪ 1,
Applying (13) to (11), the self-inductance reduces as expected to the known analytic formula (given on page 234 of [19]),
The formulae in our work are determined by rigorously simplifying the three-dimensional integral form of the magnetic vector potential (29) under the assumption a/L ≪ 1. The key idea in this method is to partition the domain of the integral over ϕ̃ in (29) into a “near region” in which
∆r ≡ r − r̃. This partition occurs at some angle ϕ 0 such that a/L ≪ ϕ 0 ≪ 1. A detailed derivation for the vector potential is given in Appendix A. It is then possible to derive the selfinductance, self magnetic field, and self-force from the vector potential. These derivations are given in Appendices B-D.
It is shown in Appendix B that the self-inductance of thin coils is given by
For a circular conductor of major radius R, this becomes
We may apply the integral identity
where L reg,f is given in (62). A similar logarithmic dependence on a can be shown to hold for the self-force.
3.0 1e 6 Exact Method
2D Integral Method 2.5 Inductance [H] Fig. 3. Demonstration that dropping the singular grid point at r̃ c = r c for a numerical integration of (7)-(8) is an invalid method of determining selfforce. This is shown by comparing the results of this method at various grid resolutions to the analytic solution (9), a finite-thickness numerical evaluation (95), and our method (21) and (26) for a circular coil. All methods agree with
The regularized self-inductance (10) also shows close numerical agreement with the exact form of the self-inductance (54) for the HSX coil in Figure 4. Interestingly, the agreement holds true even for values of a/R that are only slightly below 1, even though (10) was was determined under the assumption a/L ≪ 1. When the minor radius is varied, the self-inductance depends on a purely through a term proportional to ln(l/a), where l is the length of the coil: from (57) and (70), it can be shown that
2.0 1.5 1.0 10 3 10 1 10 2
Inverse aspect ratio (a/R) [unitless] Fig. 4. The self-inductance of the HSX coil is compared between the exact method (54) and the regularized 2D method (10) for various values of the inverse aspect ratio a/R. These two methods agree extraordinarily well, even for coils where the inverse aspect ratio is slightly below 1. It is also demonstrated that, in agreement with 15, the self-inductance is logarithmic in a/R.
It is shown in Appendix C that the magnetic field for thin coils in the region s ≲ a is given by
5 where
5 0 4 1 3 2 2
3 1 Exact Method 1D Integral Method 3 2 1 0
x [m] 0
Fig. 5. These contour plots compare the magnitude of the magnetic field in the cross-section of the HSX coil between the exact Biot-Savart law (20) and the one-dimensional integral method for B (16). The axes are x = s cos θ and y = s sin θ, with the black circle indicating the conductor. The plots were evaluated at ϕ = 0 for I = 10 6 A and for inverse aspect ratios of 10 −1 and 10 −2 . These plots demonstrate the convergence of the 1D integral method for B to the exact result with respect to small a/R.
B reg , a fact that can be demonstrated for a circular coil. Therefore, it is reasonable to approximate the location of max s,θ |B| as the point that instead maximizes the quantity B (0) ≡ B reg +B cyl . This point is found to be given by s max = a
Substituting this solution into (16) and keeping terms up to zeroth order in a/L,
6 2 Exact Method
1D Integral Method 2.5 1 dF x /dl [N/m] Regularized Biot-Savart Law Integrand 1e6 2.0
0 1.5
1 1.0 2 3 0.5 0 1 2 3 4 phi [unitless] 5 6 Fig. 6. The self-force in the x̂-direction for the HSX coil displays excellent agreement between the exact method (98) and the one-dimensional integral formula (21). The current was taken to be 10 6 A and the inverse aspect ratio
The accuracy of (22) is demonstrated in Figure 7 by comparing it to numerical maximizations of the exact Biot-Savart law (20) and one-dimensional magnetic field calculation (16) for the HSX coil. The analytic maximization (22) agrees closely with the numerical maximization of (16), which indicates that the decision to find the maximum of |B (0) | instead of |B| was reasonable. The agreement between the exact Biot Savart law (20) and the one-dimensional methods is weaker, though it may be sufficiently close for (22) to be useful for estimates and within optimization.
Exact Result Semi-Analytic Fully Analytic 65.5
65.0 0.0
3 2 0 phi [radians] 1 2 3 Fig. 8. The integrand of the regularized Biot-Savart law (17) is shown for a circle with a/R = 1/10, where R is the major radius. This demonstrates the fine-scale structure of the integrand near r̃ c ≈ r c .
singularity at r̃ c = r c . Even so, quadrature is still challenging as the integrands have fine-scale structure near r̃ c ≈ r c for a/L ≪ 1. Figure 8 demonstrates the fine-scale structure of the integrand of the regularized Biot-Savart law (17).
One method to circumvent this issue relies upon subtracting and adding back some analytically integrable function, Λ(ϕ, ϕ̃), that fits the integrand well near the reference point. For the regularized Biot-Savart law, we choose Λ based on the integrand of (17) as follows. First, we expand the numerator and denominator about ϕ̃ = ϕ using the result of (37) and keep terms to second order in ∆ϕ. Comparing this expression to (3a), we find
max | B| [T] s, 1 64.0
(23)
63.5 In order to recapture the 2π-periodic behavior of the integrand
Comparing to the formulae for self-force of a circular coil (9, 21), we see
The advantage of (24) is that it allows us to subtract Λ from the integrand of (17) and add it back to obtain a modified regularized Biot-Savart law, 7
where we applied (25). Although algebraically identical to (17), (26) is easier to evaluate as the new integral has a significantly smoother integrand at r̃ c ≈ r c .
The efficiency of this modified method in comparison to the unmodified regularized Biot-Savart law (17) is demonstrated in Figure 9, where the results of both methods are shown for several quadrature grid resolutions. For each value of ϕ in the figure, B reg is computed using Gauss-Legendre quadrature on the interval ϕ̃ ∈ [ϕ, ϕ + 2π]. Compared to highly resolved calculations, the modified method achieves accuracy within 1% for a grid of only 12 points, while the unmodified method requires 40 points to do the same. For uniformly spaced quadrature points, similar results are obtained with a slightly larger number of points.
4.0 1e6 Modified, 12 Grid Points
Unmodified, 12 Grid Points Unmodified, 40 Grid Points 3.5 B reg [T] 3.0 Figure 10 demonstrates the efficiency of this method in comparison to the unmodified 2D integral form of the selfinductance (10) by comparing their predictions for the selfinductance of the HSX coil at various choices of quadrature grid resolution.
Unmodified Method Modified Method 10 1 Relative Error in L 10 3 10 5 10 7 10 9 10 0 10 1
2.5 2.0 1.5 1.0
0 1 2 3 4 phi [unitless] 5 6 Fig. 9. This plot compares evaluations of the modified (26) and unmodified (17) regularized Biot-Savart laws across the length of the HSX coil for various choices of quadrature grid resolution in order to demonstrate relative rate of convergence. The modified method performs significantly better than the unmodified method, a 12-point grid for the modified method yielded comparable results to a 40-point grid for the unmodified method. The current is taken to be I = 10 6 A and the inverse aspect ratio a/R is 10 −2 .
Using the same methodology, a modified expression can be obtained for the self-inductance,
sThe self-inductance, self magnetic field, and self-force of electromagnetic coils are important properties yet they are difficult to evaluate numerically due to the high dimensionality of the integrals and the presence of a singularity in the integrands. In this paper, we rigorously derived novel integral formulae for these quantities using an expansion in large coil aspect ratio. The main results are summarized in Section III, with (10) for the self-inductance, (16)-(19) for the field, (21) for the self-force, and (22) for the maximum field strength in the conductor. These formulae are described by integrals of significantly reduced dimensionality that no longer contain a singularity. We demonstrated that these formulae show remarkable agreement to the “exact” finite-thickness integrals even at low aspect ratios, both analytically for a torus and numerically for the HSX coil. In Section IV, we subsequently derived expressions to compute the self-inductance (27) and regularized Biot-Savart law (26) with even greater efficiency, and demonstrated that the self-force can be accurately computed with a grid of as few as 12 points.
A key direction of future research would be to incorporate measures of stress and critical current, both of which may be computed from our simplified integral formulae, into coil shape optimization. For application to fusion energy, the design of stellarators in particular may benefit from this work as there is significant flexibility in coil shape. Prior to this work, it would have been infeasible to optimize for stress or critical current as the calculations would have been too costly. The modified forms of the self-inductance and regularized Biot-Savart law now allow for optimization as they can be 8 evaluated rapidly. An additional direction for research would be to generalize these results for electromagnetic coils with different geometries. For example, a similar methodology can be applied to coils in which the cross-section is rectangular, as will be discussed in a separate paper [7].
A PPENDIX A R EDUCED I NTEGRAL F ORMULA FOR THE M AGNETIC
V ECTOR P OTENTIAL
The magnetic vector potential is given by a threedimensional integral,
We consider a closed conductor with a circular cross-section of minor radius a, carrying a uniformly distributed current I in the e 1 direction, (6). In our coordinates, the magnetic vector potential for this coil is given by,
where ∆r ≡ r − r̃. In this Appendix, we show that when s ≲ a and the minor radius is significantly smaller than the length scales L associated with r c (which includes |r ′ c |, κ −1 , and τ −1 ), there exists a reduced form of the magnetic vector potential,
where ϕ 0 is defined ahead. This formula will be important in Appendices B, C, and D for finding simplified integral forms of the self-inductance, magnetic field, and self-force, respectively.
To reduce (29) for a/L ≪ 1, different simplifying assumptions can be applied in different regions of the domain. In the
dimensional structure of the conductor is negligible and the coil can be approximated by a one-dimensional filament. By
the three-dimensional structure is not negligible and must be considered. In order to take both domains into account, we introduce an intermediate angle ϕ 0 such that a/L ≪ ϕ 0 ≪ 1. Then, we split the integral over ϕ̃ in (29) in two: the near
words, we write
where
A similar idea of identifying an intermediate scale for the self-inductance was used on page 123 of [20] and in [5]. We proceed to simplify these integrals in each region.
From the position vector (4),
where ∆r c ≡ r c − r̃ c . Performing a Taylor expansion in a/L and keeping terms up to first order,
Next, we substitute (34) into (32b), perform the integral over θ̃, and keep terms up to first order in a/L. Performing the next integral over s̃,
Comparing this expression to (34), we note that (35) may more simply be written as
In the near region, we use the assumption that |∆ϕ| ≪ 1,
First, we perform a Taylor expansion of r c to second order in
where we determined r̃ ′′ c by evaluating the e ′ 1 term of (2) with (1). Substituting (37) into (33), performing Taylor expansions of e 2 and e 3 , and keeping terms up to second order in ∆ϕ,
9 where the derivatives of the basis vectors can be found from the Frenet-Serret formulas (2). We assume all length scales
where e i is any one of the Frenet-Serret basis vectors. As a result, we can neglect the terms in (38) containing second derivatives of the basis vectors; these terms are second order in ∆ϕ and are additionally one order higher in the small quantity a/L than the remaining (∆ϕ) 2 terms. Neglecting these second derivative terms and using (2) to evaluate the first derivative terms, we find
and keeping all terms up to second order in ∆ϕ and first order
where ∆θ ≡ θ − θ̃. It is helpful to abbreviate this expression as
where
(42a)
By expanding r̃ ′ c and κ̃ about ϕ̃ = ϕ and applying (40), (32a) can be expressed in terms of integrals of the form
where m is an integer ≥ 0. Let us seek to simplify the first few cases of (43). For m = 0, [21]
simplifies under the assumption a/L ≪ ϕ 0 ≪ 1 to
√ Substituting in (42) and performing a Taylor expansion on γ,
For the m = 1 and m = 2 cases, [21] gives
(47a)
(47b)
which, under the assumptions that a/L ≪ ϕ 0 ≪ 1, yield
Notably, the m = 1 and m = 2 integrals (48) are smaller than (45) by factors of (a/L) 2 and ϕ 20 . We therefore use this observation to argue that only the m = 0 integral (45) contributes meaningfully to an evaluation of (32a). As a result, in (32a) we can expand r̃ ′ c and κ̃ about ϕ̃ = ϕ and only keep the zeroth order term of ∆ϕ,
Next, we apply to (49) our m = 0 integral formula (46), perform a Taylor expansion in a/L and keep terms up to first order,
Here, we make use of the identities
the integral of (50) with respect to θ̃ then s̃, we find that
where
Adding the near and far terms gives the desired result, (30).
10 A PPENDIX B
(58a) Applying the far form of the vector potential (35) and noting
Keeping only leading-order terms in a/L and performing the when a/L ≪ 1, for appropriate choice of k.
integrals over θ and s,
(68)
In the far region we may apply the assumption a/L ≪ 1 to the denominator of (61b), giving
In the near region, we apply (40) and write the denominator of (61a) as
Applying the near form of the vector potential (53) and keeping terms up to first order in a/L, the self-inductance in the near region (58a) becomes
Integrating over θ and s,
11 We apply[21] D. The Total Self-Inductance
where we Taylor expanded κ̃ and |r̃ ′ c | and kept terms up to second order in ∆ϕ. Applying this result (76) and |∆r c | 2 =
as in the first line of (83) and [∇ × A f ] dom refers to the differentiation of the domain as in the last two lines of (83).
Let us start with finding [∇ × A f ] int , which is given by
12 Applying the product rule for a curl and taking the gradient in Cartesian coordinates, we find
Using the definition of the position vector (4) and the far region assumption, a ≪ |∆r|, this simplifies to
Next, let us consider [∇ × A f ] dom . Applying the formulas for the position vector (4) and the Jacobian (5), we can write with some algebra that
In order to simplify this expression, let us expand r̃ ′ c and |r − r̃ c | −1 about ϕ̃ = ϕ and keep terms up to ∆ϕ. Noting that r̃ ′ c = |r̃ ′ c |ẽ 1 and applying the Frenet-Serret formula (2),
We next perform a Taylor series expansion of |∆r c | −1 ,
where |∆r c | can be found from (37). Combining (89) and (90),
(91)
Substituting this result into (88) and keeping terms up to first order in a/L, we find that
At last, in accordance with (84), we sum the two contributions to the curl, (87) and (92), to find the total curl in the far region,
Here we wish to evaluate ∇ × A n , where the vector potential in the near region is specified in (53). Applying (82),
keeping terms up to first order in a/L, we find
As stated in (31), the total magnetic vector potential is given by the sum of its near and far parts. As a result, the total magnetic field, taken as B = ∇ × A, is given as the sum of curls of the far part (93) and the near part (94),
Comparing this expression to the expanded form of the regluarized Biot-Savart law, which is given as the sum (73) of its far (75) and near (80) parts, we write the magnetic field as (16).
A PPENDIX D R EDUCED I NTEGRAL F ORMULA FOR THE S ELF -F ORCE The self-force per unit length (abbreviated here as selfforce) can be calculated in at least two ways. An elegant approach is to use the principle of virtual work, examined in a similar context in section 3 of [4], in which the shape of the coil is perturbed and the change to the configuration’s energy is examined. Expressing the latter in terms of the inductance, and applying (10), one obtains the self-force (21) with (17). Alternatively, we can directly use the formula for total force as a volume integral of J × B,
13 where we have applied our choice of uniform current density in the e 1 direction. Comparing this expression to the total force as a line integral of the force per unit length,
the self-force must be given by
Next, we substitute in the one-dimensional integral formula for the magnetic field (16)-(19) and keep terms up to first order
where the regularized Biot-Savart law is given in (17). Performing the integrals over θ and s, we find the force to be (21).
A CKNOWLEDGMENT We thank Robert Granetz, Stuart Hudson, Thomas Kruger, Jorrit Lion, Nicolo Riva, Felix Warmer, Yuhu Zhai, and Caoxiang Zhu for useful discussions. This work was supported by a grant from the Simons Foundation (No. 560651, T. A.). This work was also supported by the U.S. Department of Energy under Contract DE-FG02-93ER54197.
R EFERENCES
[1] S. J. Sackett, “Calculation of electromagnetic fields and forces in coil
systems of arbitrary geometry,” California Univ., Tech. Rep., 1975. [2] L. Urankar, “Vector potential and magnetic field of current-carrying
finite arc segment in analytical form, Part III: Exact computation for rectangular cross section,” IEEE Transactions on Magnetics, vol. 18, no. 6, pp. 1860–1867, 1982. [3] J. Lion, F. Warmer, H. Wang, C. Beidler, S. Muldrew, and R. Wolf, “A
general stellarator version of the systems code process,” Nuclear Fusion, vol. 61, no. 12, p. 126021, 2021. [4] D. A. Garren and J. Chen, “Lorentz self-forces on curved current loops,”
Physics of plasmas, vol. 1, no. 10, pp. 3425–3436, 1994. [5] R. Dengler, “Self inductance of a wire loop as a curve integral,”
Advanced Electromagnetics, vol. 5, no. 1, pp. 1–8, 2016. [6] R. Robin and F. A. Volpe, “Minimization of magnetic forces on
stellarator coils,” Nuclear Fusion, vol. 62, no. 8, p. 086041, 2022. [7] M. Landreman, S. Hurwitz, and T. M. Antonsen, Jr., “Efficient calcu-
lation of the internal magnetic field, self-force, and self-inductance for electromagnetic coils with rectangular cross-section,” arXiv, 2023. [8] F. S. B. Anderson, A. F. Almagri, D. T. Anderson, P. G. Matthews, J. N.
Talmadge, and J. L. Shohet, “The helically symmetric experiment, (hsx) goals, design and status,” Fusion Tech., vol. 27, p. 273, 1995. [9] National Academy of Engineering and National Academies of Sciences,
Engineering, and Medicine, Bringing Fusion to the U.S. Grid. Washington, DC: The National Academies Press, 2021. [10] N. Riva, R. Granetz, R. Vieira, A. Hubbard, A. Pfeiffer, P. Harris,
velopment of the first non-planar REBCO stellarator coil using VIPER cable,” Superconductor Science and Technology, 2023. [11] A. Creely, M. Greenwald, S. Ballinger, D. Brunner, J. Canik, J. Doody,
T. Fülöp, D. Garnier, R. Granetz, T. Gray et al., “Overview of the SPARC tokamak,” Journal of Plasma Physics, vol. 86, no. 5, 2020.
[12] A. Panin, W. Biel, P. Mertens, F. Nunio, K. Sedlak, L. Zani, and M. Cole-
man, “Approaches to analyze structural issues of the european demo toroidal field coil system at an early design stage,” IEEE transactions on applied superconductivity, vol. 26, no. 4, pp. 1–5, 2016. [13] T. S. Pedersen, M. Otte, S. Lazerson, P. Helander, S. Bozhenkov,
of the topology of the wendelstein 7-x magnetic field to better than 1: 100,000,” Nature communications, vol. 7, no. 1, p. 13493, 2016. [14] S. S. Hidalgo-Tobon, “Theory of gradient coil design methods for
magnetic resonance imaging,” Concepts in Magnetic Resonance Part A, vol. 36, no. 4, pp. 223–242, 2010. [15] X. Chen and M. Steckner, “Electromagnetic computation and modeling
in MRI,” Medical Physics, vol. 44, no. 3, pp. 1186–1203, 2017. [16] S. Russenschuck, Field computation for accelerator magnets: analytical
and numerical methods for electromagnetic design and optimization. John Wiley & Sons, 2011. [17] E. Kreyszig, Differential Geometry, ser. Dover Books on Mathematics.
Dover Publications, 1991, p. 35–43. [18] V. Shafranov, “Plasma equilibrium in a magnetic field,” Reviews of
plasma physics, vol. 2, p. 103, 1966. [19] J. D. Jackson, Classical Electrodynamics, 3rd ed. American Association
of Physics Teachers, 1999, p. 215. [20] L. D. Landau, J. S. Bell, M. Kearsley, L. Pitaevskii, E. Lifshitz, and
J. Sykes, Electrodynamics of continuous media. elsevier, 2013, vol. 8. [21] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and
products. Academic press, 2014. [22] P. Khare, “Divergence and curl in nonorthogonal curvilinear coordi-
nates,” American Journal of Physics, vol. 38, no. 7, pp. 915–916, 1970.