Force-Optimised Banana Coils

• Force-Optimised Banana Coils are non-planar coils used in hybrid tokamak–stellarator configurations to achieve precise magnetic perturbations while managing complex electromagnetic forces.
• The design integrates multiple optimization objectives, including coil current, curvature, spacing, and Lorentz-force cancellation, through filament and quadratically constrained modeling.
• Distinct coil families (Train Track and Dalí Clock) reveal trade-offs between force reduction and field accuracy, impacting overall device performance and engineering feasibility.

Searching arXiv for the cited papers to ground the article in the current literature. Force-optimised banana coils are non-planar coils for compact tokamak–stellarator hybrid configurations that are optimized not only to reproduce a target three-dimensional magnetic perturbation, but also to satisfy electromagnetic force constraints that are not captured by simple geometric measures alone. In the 2026 study of the Hybrid concept, banana coils are the additional coil family that sits in the helical grooves of a quasi-axisymmetric boundary, does not link the plasma, and is intended to provide the external rotational transform associated with stellarator-like confinement while preserving a tokamak-like coil set. In that setting, force optimization is treated as a coupled field-and-engineering problem involving coil current, curvature, spacing, wall proximity, and Lorentz-force management rather than as a purely geometric refinement (Zettl et al., 16 Jun 2026).

1. Hybrid setting and definition of banana coils

The Hybrid concept is described as an optimized tokamak–stellarator hybrid that retains the conventional toroidal-field (TF) and poloidal-field (PF) coils while adding only one new coil type, the banana coil, to generate a compact quasi-axisymmetric perturbation on the inboard side of the plasma. Because the perturbation is localized on the inboard side, the device remains tokamak-like from the outboard side, but can be switched into a stellarator-like mode by energizing the banana coils. The stated motivation is to combine tokamak and stellarator advantages in a compact quasi-axisymmetric design with low aspect ratio, large plasma volume, good particle confinement, and relatively simple coils (Zettl et al., 16 Jun 2026).

The 2026 analysis considers three compact quasi-axisymmetric equilibria, all with aspect ratio $A=5$, major radius about $R_\text{major}\approx 2.5\ \mathrm{m}$, on-axis field $B_0\approx 3.5\ \mathrm{T}$, and plasma current $I_\text{plasma}=800\ \mathrm{kA}$. The cases are H1 with $n_\text{fp}=4$ and $\iota_\text{ext}=0.05$, H2 with $n_\text{fp}=4$ and $\iota_\text{ext}=0.15$, and H3 with $n_\text{fp}=6$ and $\iota_\text{ext}=0.05$. In each case, the banana coils must reproduce the target boundary fields of the equilibrium while remaining mechanically feasible (Zettl et al., 16 Jun 2026).

A central design premise is that force-optimised banana coils cannot be understood solely through length or curvature penalties. The engineering problem explicitly includes coil length, curvature, coil–coil spacing, coil–wall proximity, force magnitude, and whether coils are linked or escape the TF region. This reflects the specific difficulty that banana coils may require tight turns, strong curvature, and close proximity to the TF field, all of which can generate large Lorentz forces (Zettl et al., 16 Jun 2026).

2. Electromagnetic force formulation

The general stellarator-coil force problem is addressed in a rigorous form by Robin and collaborators, who define the Laplace force on a current-carrying coil winding surface despite the magnetic-field discontinuity across a current sheet. For a coil winding surface $R_\text{major}\approx 2.5\ \mathrm{m}$0 with outward normal $R_\text{major}\approx 2.5\ \mathrm{m}$1 and surface current density $R_\text{major}\approx 2.5\ \mathrm{m}$2, they define the force per unit surface at $R_\text{major}\approx 2.5\ \mathrm{m}$3 as

$R_\text{major}\approx 2.5\ \mathrm{m}$4

They further prove that this limit exists in $R_\text{major}\approx 2.5\ \mathrm{m}$5 for any finite $R_\text{major}\approx 2.5\ \mathrm{m}$6 provided the current density belongs to the Sobolev space $R_\text{major}\approx 2.5\ \mathrm{m}$7, and they show consistency with a 3D finite-thickness current-layer model (Robin et al., 2021).

For banana coils specifically, the 2026 work adopts a filamentary force model. The force per unit length is written as

$R_\text{major}\approx 2.5\ \mathrm{m}$8

where $R_\text{major}\approx 2.5\ \mathrm{m}$9 is the tangent vector along the coil centerline. The decomposition separates self-force, mutual force from neighboring banana coils, TF force from the dominant toroidal field, and weaker PF and plasma-current contributions (Zettl et al., 16 Jun 2026).

Four force-reduction mechanisms are stated explicitly: lowering the coil current $B_0\approx 3.5\ \mathrm{T}$0, aligning the coil tangent with the field to reduce $B_0\approx 3.5\ \mathrm{T}$1, moving the high-force sections outward to weaker TF regions, and exploiting cancellation between self and TF fields (Zettl et al., 16 Jun 2026). The same study gives a parallel-wire picture for self-force,

$B_0\approx 3.5\ \mathrm{T}$2

showing that closely spaced, nearly parallel conductors can have large self- and mutual forces. For the TF contribution,

$B_0\approx 3.5\ \mathrm{T}$3

so the force is minimized when the coil is locally aligned with the toroidal direction and is strongest where the coil turns sharply across the toroidal field (Zettl et al., 16 Jun 2026).

A rough comparability criterion between TF and self/mutual forces is also derived, leading to an estimated critical radius

$B_0\approx 3.5\ \mathrm{T}$4

for the studied Hybrid cases, using approximately $B_0\approx 3.5\ \mathrm{T}$5, $B_0\approx 3.5\ \mathrm{T}$6, $B_0\approx 3.5\ \mathrm{T}$7, and $B_0\approx 3.5\ \mathrm{T}$8. The interpretation given is that, for these Hybrids, banana coils are best placed inside the TF coils rather than outside them. The paper also sketches “escaping coils,” whose endpoints pass between TF coils and make their turns outside the TF region, but notes that this would be challenging (Zettl et al., 16 Jun 2026).

3. Optimization formulations and objectives

In the banana-coil study, optimization is carried out in SIMSOPT with many randomized runs in which objective weights and threshold values are varied. The coils are represented as filament curves with Fourier coefficients, initialized from a self-stellarator-symmetric ellipse leaned into the groove, and the banana current is chosen randomly between $B_0\approx 3.5\ \mathrm{T}$9 and $I_\text{plasma}=800\ \mathrm{kA}$0 of the enclosed current, corresponding to

$I_\text{plasma}=800\ \mathrm{kA}$1

The full objective is

$I_\text{plasma}=800\ \mathrm{kA}$2

combining flux error, coil length, curvature, coil–coil distance, force, wall avoidance, and a Gauss-linking-type penalty (Zettl et al., 16 Jun 2026).

The force term is

$I_\text{plasma}=800\ \mathrm{kA}$3

with $I_\text{plasma}=800\ \mathrm{kA}$4 in the main optimizations and one additional H1 scan using a finite threshold $I_\text{plasma}=800\ \mathrm{kA}$5. This makes force an explicit objective rather than a post-processing diagnostic. The paper states that reducing forces is coupled to field reproduction: moving coils farther from the plasma can reduce ripple and improve field matching, but can also alter current requirements and increase forces because of stronger TF exposure or larger currents (Zettl et al., 16 Jun 2026).

This emphasis on explicit force objectives is consistent with broader stellarator-coil optimization. Robin et al. augment REGCOIL-style optimization by adding force-related costs such as

$I_\text{plasma}=800\ \mathrm{kA}$6

and an engineering-threshold penalty

$I_\text{plasma}=800\ \mathrm{kA}$7

with test values $I_\text{plasma}=800\ \mathrm{kA}$8 and $I_\text{plasma}=800\ \mathrm{kA}$9. Their generalized objective,

$n_\text{fp}=4$0

treats force minimization as compatible with field fidelity, current smoothness, and buildability (Robin et al., 2021).

At the winding-surface level, QUADCOIL extends this logic by formulating a quadratically constrained quadratic program that can directly minimize or constrain quadratic functions of the current, including Lorentz force, magnetic energy, curvature, field-current alignment, and maximum dipole density. The paper emphasizes that NESCOIL and REGCOIL are limited to norms of linear functions of $n_\text{fp}=4$1, whereas QUADCOIL can target force-related quantities unavailable in those methods, requires no initial guess, and is nearly $n_\text{fp}=4$2 faster than filament optimization (Fu et al., 2024).

4. Symmetry, geometry, and magnetic-field structure

A notable structural feature of the 2026 work is self-stellarator symmetry, which allows a banana coil to satisfy stellarator symmetry by itself rather than through a mirrored partner. For a self-stellarator-symmetric coil centered on a symmetry axis, the centerline satisfies

$n_\text{fp}=4$3

which in Cartesian coordinates implies even parity in $n_\text{fp}=4$4 and odd parity in $n_\text{fp}=4$5 and $n_\text{fp}=4$6. The associated Fourier parameterization is

$n_\text{fp}=4$7

This permits one banana coil per field period, but also implies that such coils cannot be entirely flat against the boundary and must cross the symmetry axis at least twice (Zettl et al., 16 Jun 2026).

To interpret magnetic-field generation, the same work analyzes three asymptotic regimes. A long straight segment gives a field scaling like $n_\text{fp}=4$8. A banana coil approximated as two long parallel wires with opposite currents yields an intermediate regime with $n_\text{fp}=4$9 at distances large compared with the wire spacing. At distances much larger than the coil size, the far field approaches the magnetic-dipole limit with $\iota_\text{ext}=0.05$0 (Zettl et al., 16 Jun 2026). These scalings are then used to interpret optimized geometries.

Three geometric descriptors are introduced to classify optimized coils: the central coil width $\iota_\text{ext}=0.05$1, the net twist angle

$\iota_\text{ext}=0.05$2

and the central coil–surface distance. The authors explicitly note that these are only proxies and can miss important width variations along the coil, especially for the $\iota_\text{ext}=0.05$3 case (Zettl et al., 16 Jun 2026).

5. Characteristic families and empirical trends

After filtering out coils with maximum normal-field error above $\iota_\text{ext}=0.05$4, excessive projected self-intersections, S-shaped turns at the outer center point, wrong sign of the local slope relative to the groove, curvature $\iota_\text{ext}=0.05$5, coil–coil distance below $\iota_\text{ext}=0.05$6, or coil–plasma distance below $\iota_\text{ext}=0.05$7, the study retains 5093 acceptable banana coils across the three configurations (Zettl et al., 16 Jun 2026).

The optimized coils fall into two characteristic families. Train track coils are narrow, nearly parallel-filament coils with small or slightly negative twist angle, higher curvature at the endpoints, and stronger field cancellation because the two filaments are close together. They behave like the double-wire model and typically satisfy

$\iota_\text{ext}=0.05$8

Dalí clock coils are wider, strongly twisted at the center, have a wave-like outer filament, and bend more gently at the endpoints, with lower endpoint curvature than train-track coils. These coils often transition toward

$\iota_\text{ext}=0.05$9

when the inner filament gets closer to the plasma (Zettl et al., 16 Jun 2026).

A representative comparison illustrates the distinction. The cited Dalí clock coil has $n_\text{fp}=4$0, $n_\text{fp}=4$1, length $n_\text{fp}=4$2, curvature $n_\text{fp}=4$3, twist $n_\text{fp}=4$4, current $n_\text{fp}=4$5, integrated force $n_\text{fp}=4$6, and maximum force $n_\text{fp}=4$7. The representative train-track coil has $n_\text{fp}=4$8, curvature $n_\text{fp}=4$9, twist $\iota_\text{ext}=0.15$0, current $\iota_\text{ext}=0.15$1, integrated force $\iota_\text{ext}=0.15$2, and maximum force $\iota_\text{ext}=0.15$3. The intermediate coil lies between these cases, with $\iota_\text{ext}=0.15$4, curvature $\iota_\text{ext}=0.15$5, twist $\iota_\text{ext}=0.15$6, current $\iota_\text{ext}=0.15$7, integrated force $\iota_\text{ext}=0.15$8, and maximum force $\iota_\text{ext}=0.15$9 (Zettl et al., 16 Jun 2026).

The paper identifies several empirical trends. The central coil–plasma distance tends to increase with coil current and approximately follows

$n_\text{fp}=6$0

The field strength per unit current, $n_\text{fp}=6$1, decreases with increasing coil–plasma distance, with narrow train-track coils following the expected $n_\text{fp}=6$2 trend and wider Dalí clock coils transitioning toward $n_\text{fp}=6$3. Force and field accuracy are traded against one another: lower forces usually come with larger normal-field error, whereas better field reproduction tends to require more severe coil loading (Zettl et al., 16 Jun 2026).

6. Design implications, broader context, and limitations

The force-optimization results imply that banana-coil design is governed by multiple local minima with different physical mechanisms. Train-track coils concentrate loads near the endpoints and have low force elsewhere, whereas Dalí clock coils distribute force more along the outer, farther-from-plasma segment. The maximum force usually occurs near the turning point, where the coil is nearly perpendicular to the toroidal field, but finite-threshold force optimization can shift the high-force location outward and reduce the angle to the TF field. The scalar product $n_\text{fp}=6$4 is used diagnostically: positive values indicate unfavorable alignment and larger forces, while negative values indicate cancellation. The finite-threshold optimizations produce more solutions with self/TF counteraction (Zettl et al., 16 Jun 2026).

These conclusions align with the broader stellarator literature, which argues that coil force should be treated as a first-class design objective rather than as a post-processing engineering check. Robin et al. report peak force reductions by up to $n_\text{fp}=6$5 for NCSX-like winding-surface optimization, with about $n_\text{fp}=6$6 reduction in peak tangential force and about $n_\text{fp}=6$7 reduction in peak normal force in their best combined case. They also observe that the tangential and normal maxima occur at different locations, suggesting that separate engineering constraints may be useful (Robin et al., 2021). A plausible implication for banana coils is that separate handling of tangential and normal loads may become increasingly important once finite-build conductors and support structures are modeled in more detail.

At the winding-surface level, QUADCOIL suggests a complementary screening strategy. Because it can directly optimize or constrain Lorentz force, curvature proxy, topology, magnetic energy, and field-current alignment, it can in principle penalize force-related quantities before a filament coil cut is performed. The paper does not present a dedicated banana-coil force-optimization case study, but it explicitly frames the method as a way to exclude equilibria that are hard to build coils for and to control contour topology that would otherwise complicate coil cutting (Fu et al., 2024).

Several limitations are stated explicitly in the banana-coil study. The self-force regularization may be stretched beyond its validity when the curvature radius is comparable to the conductor thickness. A more complete study should include torque, net force, and shear, and should compare the regularized filament model with finite-element calculations. The reported trends are based on only three Hybrid equilibria, so they are not presented as universal. The authors therefore leave open the question of which equilibrium features determine whether banana coils can be made compatible, simple, and mechanically feasible (Zettl et al., 16 Jun 2026).