---
title: "Why stellarator coils bend in three dimensions"
description: "Wendelstein 7-X shapes a twisted magnetic field with external coils. Repeated crossings of one fixed plane reveal the field-line structure and its winding."
canonical_url: "https://sandboxphysics.com/stories/instruments/w7x-stellarator/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/instruments/w7x-stellarator/"
last_modified: "2026-11-13"
---

[How physics instruments measure](https://sandboxphysics.com/stories/instruments/)

# Why stellarator coils bend in three dimensions

Imagine winding thread around a tube, then bending the tube into a ring. The thread travels around the large ring while also turning around the tube. A stellarator creates a magnetic version of that route to guide a hot charged gas. The curved coils around Wendelstein 7-X supply the twist.

Sandbox Physics · 2026-11-13

## Closing the route into a ring is only the beginning

A hot gas with mobile electrons and ions is a plasma. Its charged particles gyrate while moving along the magnetic field. Joining a straight magnetic channel into a ring removes the two open ends through which particles could escape.

A purely toroidal field is stronger on the inside of the ring. Its curvature and changing strength cause slow drifts of the particles’ gyration centres. A suitable confinement field also winds around the tube’s cross-section. In a regular region, field lines cover nested toroidal layers, called magnetic surfaces, providing routes that remain inside the device.

Sources: [1]

## External coils supply the twist

Each current-carrying coil contributes to the field at every position. A stellarator combines those contributions through three-dimensional coil shapes and chosen currents. It supplies the twist without relying on a large toroidal current in the plasma.

Wendelstein 7-X has 70 main coils in a pattern that repeats five times around the ring. The calculation geometry contains five non-planar coil types and two planar types. The planar coils help adjust the configuration; they remain part of the apparatus when their current is zero in the standard reference.

Changing the current ratios changes how the combined field winds. Following that field reveals the interior structure produced by the surrounding hardware.

Sources: [2] [3] [4]

## Mark each return to the same plane

Long field lines soon crowd a three-dimensional picture. Instead, fix a vertical half-plane extending outwards from the device’s central vertical axis. Keep that plane fixed in space throughout the calculation. Record each crossing in the same direction as a line returns around the ring. The resulting return plot is a Poincaré section.

Crossings on a regular magnetic surface trace a closed curve. Several starting positions reveal nested curves; small loops around other centres form magnetic islands. A field line can cover a surface without ever closing exactly on itself. The closed curve shows the slice of that surface through the chosen plane.

Sources: [5] [6]

## Count winding about an axis that moves

At the centre of nested surfaces is a closed field line called the magnetic axis. In a stellarator it moves vertically and radially as it circles the device. The angle of a neighbouring line must be measured about the axis at each position.

Choose counterclockwise travel around the ring, viewed from above, as positive. In each cross-section, take the outward horizontal direction from the magnetic axis as zero and rotation towards the top as positive. Follow this angle continuously, retaining every completed turn. Dividing these signed turns by full circuits of the ring gives a finite-trajectory estimate of rotational transform.

One repeating section of Wendelstein 7-X occupies a fifth of the ring. A return to the next repeated section and a return to the original global plane count different distances. Winding estimates need the same full-ring scale for comparison.

In the Lab’s default reference, the line makes 64 full circuits and crosses five physical half-planes spaced a fifth of a turn apart, giving 320 crossings. The plot overlays these sections using rotational symmetry; 64 crossings return to the same base half-plane. The winding estimate already uses the accumulated global toroidal angle, so it needs no further factor of five.

$$
\iota_N=\frac{\Delta\theta_{\mathrm{unwrapped}}}{2\pi N}
$$

N is the signed number of full circuits, obtained by dividing the accumulated toroidal angle by 2π, with the positive direction chosen above. Δθ is the continuous signed angle change, in radians, about the moving magnetic axis, with whole turns retained. The ratio ι\_N describes this finite path. On a regular magnetic surface, a value that settles as the path grows longer estimates the surface’s rotational transform.

Sources: [5] [6]

## What a finite return plot can tell us

A few crossings may reveal only an arc, and a short path can hide slow changes. Numerical integration adds error. Longer paths, tighter integration and a checked axis help establish whether a pattern and winding estimate persist. A cloud of points on an unfinished plot needs these checks before it can be classified as chaotic.

This calculation describes the field of specified coils and currents. Plasma pressure and current modify the real field, while particles also gyrate, collide and move across it. Measuring how long heat remains and where it reaches material needs particle and thermal diagnostics. The return plot supplies a geometric map for that work.

Sources: [5] [6] [3]

## [Follow vacuum field lines in Wendelstein 7-X](https://sandboxphysics.com/labs/w7x-stellarator/)

Compare two reference current sets, follow lines in the resulting field, and inspect fixed-plane crossings and finite winding estimates.

Traceable single-filament CAD coils with prescribed currents provide a vacuum field, magnetic axis and finite trajectories. Numerical results retain convergence checks; wall geometry and particle transport are outside this model.

## Sources and further reading

1. [IPP · Der magnetische Einschluss](https://www.ipp.mpg.de/11728/mageinschluss)

   Checked 13 November 2026: gyration and motion along the field, the limitation of a purely toroidal field, twisted lines and nested magnetic surfaces. Used for the geometric picture, without treating it as a guarantee for real particle confinement.

2. [IPP · Der Stellarator](https://www.ipp.mpg.de/9792/stellarator)

   Checked 13 November 2026: external coils provide field-line twist without relying on a toroidal plasma current, and three-dimensional shaping adds design freedom. Principle only; no performance comparison or power-plant forecast is repeated.

3. [IPP · Reference equilibria for Wendelstein 7-X](https://www.ipp.mpg.de/4631651/Equilibria)

   Checked 13 November 2026: the short official account of 70 main coils, design geometry and reference equilibria at different plasma pressures. Supports the distinction between a prescribed vacuum field and a field including plasma effects; it is not presented as the download source of the coil file.

4. [Simsopt 1.10.2 · W7-X configuration data](https://simsopt.readthedocs.io/v1.10.2/simsopt.configs.html)

   Checked 13 November 2026: five non-planar and two planar base-coil types, zero planar current in the standard configuration, and the stated provenance of the single-filament CAD geometry coils.w7x\_v001 supplied by Geiger on 27 September 2022. Version, rights and current-sign details remain in the data documentation.

5. [Smiet, Rais, Loizu and Davies · Turnstile area as a measure for chaotic transport in magnetic confinement fusion devices (arXiv v3)](https://arxiv.org/html/2503.15792v3)

   Checked 13 November 2026: introduction definitions of the axis, Poincaré section and transform per full toroidal turn; section VI.4, figure 1/10 captions and the conclusions on W7-X structure and particle transport. Uses v3 submitted 15 April 2025. Published figures are not presented as runs of this interactive model; none is reproduced.

6. [Smiet et al. · W7-X figure-generation code](https://github.com/smiet/fusion_turnstile_figures/blob/414a89d5ea8ec0c5150a58ade389bd70d0fc109d/W7-X_plot.py)

   Checked 13 November 2026: pinned W7-X\_plot.py, including the common coil source, direct field, axis solve, section iterations and winding-angle treatment. Read to check the provenance and conventions behind a numerical plot; no paper reproduction was executed and no auxiliary wall file was adopted.

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## Keep reading

How physics instruments measure · Reading path

### How physics instruments measure

1. [Article **How a vernier reads between millimetres**](https://sandboxphysics.com/stories/instruments/length-metrology/)
2. [Article **Air helps hold up the object on a balance**](https://sandboxphysics.com/stories/instruments/precision-balance/)
3. [Article **How Cavendish Measured the Attraction Between Spheres**](https://sandboxphysics.com/stories/instruments/cavendish-balance/)
4. [Article **Which pressure does a liquid column measure?**](https://sandboxphysics.com/stories/instruments/manometer/)
5. [Article **The cup absorbs heat too**](https://sandboxphysics.com/stories/instruments/calorimeter/)
6. [Article **Rotational rheometers need both torque and speed**](https://sandboxphysics.com/stories/instruments/rotational-rheometer/)
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