Planar Coil Stellarator Architecture

• Planar Coil Stellarator Architecture is a design using only coils with planar windings to generate 3D magnetic fields for effective plasma confinement.
• The system leverages two coil families—encircling and shaping coils—optimized via sparse regression and HTS technology to minimize field errors and mechanical forces.
• Advances in control, cryogenics, and reactor integration have demonstrated promising confinement metrics and scalability for reactor-scale fusion applications.

A planar coil stellarator is a toroidal magnetic confinement device in which the requisite three-dimensional (3D) magnetic field shaping is achieved using only coils whose winding curves lie entirely in planes, in contrast to traditional modular coil stellarators that require nonplanar, spatially contorted windings. The planar coil approach exploits recent advances in sparse coil optimization, high-temperature superconductor (HTS) technology, and robust engineering analysis to realize fusion-relevant configurations with minimized field error, mechanical force, and optimized maintenance and flexibility. This architecture dramatically simplifies coil manufacturing, reduces assembly complexity, and enables new strategies for plasma control and reactor integration.

1. Coil System Topologies and Parameterization

A defining property of planar coil stellarator architectures is their decomposition of the coil set into two main families:

• Encircling ("TF-like") planar coils: Large, toroidally extended planar coils provide the principal toroidal field and bulk rotational transform. In designs such as Helios (Swanson et al., 8 Dec 2025) and Eos (Wu et al., 11 Feb 2025), 12 such coils are arranged evenly in toroidal angle; each lies in a concentric plane at a constant toroidal angle, with inboard and outboard contours parameterized by low-order Fourier modes (typically 1–4).
• Shaping ("correction" or "field-shaping") planar coils: A large number (dozens to hundreds) of smaller planar coils are positioned on planes offset from the plasma boundary (e.g., at constant normal distance of 0.5–1.2 m), densely arrayed in poloidal and toroidal columns. Shaping coils may adopt simple circular or rounded-rectangle geometry (defined via Fourier or superellipse parameterizations), with center positions and normal vectors as free placement parameters.

This arrangement enables full coverage of the toroidal volume by planar elements, avoiding three-dimensional manifold winding. Reactor-scale systems for quasi-axisymmetric (QA) or quasi-helically symmetric (QH) targets typically employ 2–3 planar TF coils per field period, and O(100) planar shaping coils (Kaptanoglu et al., 2024).

Table: Representative coil system parameters (Helios power plant) (Swanson et al., 8 Dec 2025)

Coil Family	Count	Geometry	Typical Radius (m)	Max. Field (T)
Encircling planar	12	Convex Fourier	5–8	≤ 20
Field-shaping planar	324	Planar circles	0.4–0.8	≤ 10

Reactor-scale planar dipole arrays (e.g., in (Kaptanoglu et al., 2024)) generalize this: each planar coil is parameterized by its radius (usually circular for optimization), spatial center, orientation (unit quaternion), and independently adjustable current.

2. Magnetic Field Generation, Optimization, and Confinement Metrics

Magnetic field computation uses the Biot–Savart law. The superposition principle applies: the total field is the sum of each coil's contribution, where the path of each winding is explicitly known and lies in a plane. This modular structure lends itself to efficient calculation and Jacobian construction for optimization.

The essential constraint for effective plasma confinement is minimization of the normal component of the vacuum field at the target plasma surface:

$$\| \mathbf{B} \cdot \hat{n} \|_{\text{rms}} = \sqrt{\frac{1}{A}\iint (\mathbf{B}\cdot\hat{n})^2 dA} \leq 10^{-3} B_0.$$

Optimization problems for finding the shaping coil currents (or placement) are typically framed as sparse regression or joint geometric/current minimization. For sparse shaping coil optimization (Wu et al., 11 Feb 2025), the linear map $A$ relates shaping coil currents to $\mathbf{B}\cdot \hat n$ at surface quadrature points, enabling a wide range of compressed-sensing approaches:

• Convex least-squares: Direct minimization of $\|A i - b\|_2^2$ subject to engineering current limits.
• LASSO ($\ell_1$) regularization: Promotes sparsity and minimizes total current.
• $\ell_0$ relax-and-split and MIQP: Directly minimize coil count, yielding checkerboard current distributions for robust performance.

This decomposition routinely produces $B_n$ values within 1–2% of the theoretical minimum for a given coil number, and solutions robust to misalignment (Wu et al., 11 Feb 2025). For fully 3D field shaping or multiple magnetic configurations, planar coils can be optimized jointly with modular (nonplanar) coils and multi-target objectives, sustaining both quasisymmetry and large nested-surface volume (Lee et al., 2022).

3. Engineering Constraints: Superconducting Materials, Mechanical Forces, and Shielding

All practical large-scale planar coil stellarators rely on high-field HTS materials—primarily REBCO tape (YBCO, GdBCO)—operated at 20 K or below for maximal current density and high mechanical strength. Key constraints include:

• Peak on-coil field: For Helios, $B_\text{max} \leq 20$ T (empirically achieved in large-bore REBCO coils) (Swanson et al., 8 Dec 2025).
• Engineering current density: $J_\text{eng} \approx 150$–200 A/mm$^2$ (Swanson et al., 8 Dec 2025), 180 A/mm$^2$ for Canis prototypes (Nash et al., 19 Mar 2025).
• Thermal margins: $\Delta T_\text{margin} \sim 5$–10 K.
• Mechanical limits: Von Mises stress $\sigma_\text{VM} \leq 800$ MPa, strain $\epsilon \leq 0.4$\% for cryostat and support structures.
• Coil lifetime: Neutron-induced damage threshold at a minimum 40-year operational life with sufficient blanket and shielding (1.2–1.5 m minimum plasma–coil gap).

Mechanical optimization now formally includes not only coil–coil distance ($d_{cc,0}$) and coil–surface distance ($d_{cs,0}$) but also pointwise and net Lorentz force ($|d \mathbf F/dl|$) and torque ($|d \boldsymbol \tau/dl|$) constraints (Kaptanoglu et al., 2024). Jointly optimizing coil position and orientation under these objectives eliminates designs with unmanageable mechanical loads, ensuring feasibility for reactor-scale operation.

4. Prototyping, Experimental Validation, and Control

The Canis 3×3 planar HTS coil array (Nash et al., 19 Mar 2025) provides critical experimental validation:

• Geometry: Nine planar REBCO coils in a rectangular array, with 1,500 turns each, operated at 20 K.
• Field performance: Closed-loop real-time control (FSCS, matrix-based feedback) yields RMS field errors $E_\text{RMS}$ < 1%, sufficient for integrated stellarator operation.
• Fabrication: Planar tape-wound coils, mass-manufactured on demountable stainless steel frames, with negligible mutual inductance beyond nearest neighbors.
• Cryogenic engineering: Parallel He cooling, FEA-validated structural stiffness, and robust SMI insulation for quench safety.
• Implication: Demonstrates scalability of planar-shaping coil arrays to the hundreds-of-coils regime, underpinning full devices like Eos and Helios.

Individual control of each shaping coil (as in Helios: 324 individually powered coils) allows for real-time compensation of error fields, flexible divertor strike-point shaping, and magnetic tuning for relaxed tolerance to manufacturing and assembly errors (Swanson et al., 8 Dec 2025).

5. Confinement Properties and Transport

High-performance confinement is achievable with planar coil arrays, provided that optimization metrics targeting $B_n$ minimization or effective helical ripple $\epsilon_{eff}^{3/2}$ are employed:

• QA/QH optimization: Helios (QA), Eos (QA), and dipole array (QA, QH) architectures, when fully optimized, can achieve $\epsilon_B \sim 10^{-3}$ and quasisymmetry error $f_{QS} \sim 10^{-4} - 10^{-3}$ (Kaptanoglu et al., 2024, Swanson et al., 8 Dec 2025).
• Neoclassical confinement: The four-planar-coil ZCS compact stellarator (Yu et al., 2021) achieved $\epsilon^\frac{3}{2}_{eff}$ in core $<$ $10^{-4}$, an order-of-magnitude improvement over the Columbia Non-neutral Torus (CNT) and resulting in factor of 10 reduction in neoclassical particle flux.
• Robustness: Sparse LASSO solutions for the shaping array exhibit resilience under $1$ cm displacement and $1^\circ$ tilt of coils; performance decline is minimal for moderate sparsity (Wu et al., 11 Feb 2025).

Quasi-isodynamic and quasi-symmetric fields can now be synthesized using only planar coils, relying on the latest near-axis expansions and 3D equilibrium solutions.

6. Reactor-Scale Applications and Design Trade-Offs

Planar coil architectures have matured to enable practical fusion power plant design, as exemplified by Helios (Swanson et al., 8 Dec 2025):

• Performance metrics: $P_{fus}\approx958$ MW, $P_{therm}\approx1.1$ GW, net electric power $P_{net}\approx390$ MW, $Q_{\text{startup}}\approx96$, $Q_{\text{ignite}}\approx960$, recirculating power fraction $\sim$13%.
• Maintenance: Sector-based toroidal maintenance, with sector mass <1000 t, 2-year cycles, and 88% capacity factor.
• Shielding and blanket integration: 1.2–1.5 m radial clearance for tritium breeding, neutron attenuation, and biological shielding.
• Scalability: Planar coil topologies (IL–PF, TF-dipole arrays) are extendable to both compact and reactor-sized plasmas, sustaining favorable quasisymmetry and transport with mass-producible components.
• Limitations: Dipole (planar) field magnitude decreases as $r^{-3}$, so increasing plasma–coil gap requires higher coil current or more coils; this must be balanced against conductor limits and support complexity.

A major trade-off is between structural simplicity and increased number of coils, each needing independent power and precise alignment; so, advanced optimization and control are essential for practical realization.

7. Outlook and Future Directions

Future research directions identified in (Kaptanoglu et al., 2024, Wu et al., 11 Feb 2025), and (Swanson et al., 8 Dec 2025) include:

• Hybrid supports and winding surfaces: Allow limited spatial freedom for planar array placement to balance optimization flexibility and mechanical simplicity.
• Explicit optimization for reactor integration: Including penetrations for blanket, heating, and diagnostics as design constraints.
• Extension to multiple magnetic configurations: Planar coils integrated into flexible coil sets supporting wide transform range and quasisymmetry (Lee et al., 2022).
• Material advances: Exploration of passive superconducting, hybrid, or ferromagnetic elements to further reduce active current requirements.
• System engineering: Increasing number of shaping coils imposes demands on power electronics, quench detection, and active control; co-design across all systems is essential.

The planar coil stellarator paradigm now encompasses reactor-scale devices with validated field quality, transport, and maintainability. Where modular coils imposed order-of-magnitude greater manufacturing complexity and cost, planar coil arrays—enabled by modern optimization, HTS, and robust engineering analysis—provide a compelling alternative path for practical stellarator-based fusion power (Kaptanoglu et al., 2024, Wu et al., 11 Feb 2025, Nash et al., 19 Mar 2025, Swanson et al., 8 Dec 2025).