Superconducting Magnet Design for MRI
Theory: how to make a uniform field
Overview
I thought an MRI magnet was just a long solenoid. Wouldn't making it longer make it uniform?
Longer does approach uniformity, but MRI demands homogeneity of a different order: within a 40–50 cm sphere, variation must be held to a few to a dozen or so parts per million. At 1.5 T, 1 ppm is 1.5 µT. Meeting that with a plain solenoid would make it too long for a patient. So real magnets place several coils within a limited length, adjusting each one's position and turns to cancel, order by order, the components by which the field changes away from centre.
On-axis field
On-axis field of a solenoid of radius $a$, length $L$, $n$ turns per unit length and current $I$.
Expanding the field near centre as a power series in $z$, a symmetric layout leaves only even terms. Choosing coil positions and currents to zero $c_2$, $c_4$, $c_6$ … in turn is the basis of homogeneous magnet design. Fixing the on-axis coefficients fixes the field inside the sphere as well.
The day superconductivity came to hospitals
Nuclear magnetic resonance, the principle behind MRI, was discovered in 1946, and research applying it to imaging the body advanced in the 1970s. The earliest scanners used resistive magnets with weak fields, but in the 1980s magnets of niobium–titanium superconducting wire cooled by liquid helium became practical, and 1.5 T scanners spread through hospitals. With zero resistance, a superconducting coil keeps its current flowing for years after the supply is disconnected, holding an extremely stable field. MRI became one of the largest-scale practical uses of superconductivity.
Worked example: homogeneity from coil layout
Example 1: a plain solenoid
For a 0.5 m-radius solenoid, field variation over ±0.2 m along the axis from centre:
| Length | Variation over ±0.2 m |
|---|---|
| 1.0 m | 60,800 ppm |
| 1.5 m | 23,582 ppm |
| 2.0 m | 9,939 ppm |
| 3.0 m | 2,454 ppm |
Example 2: splitting into coil pairs
Thin 0.5 m-radius coils were placed symmetrically; with the first pair's current set to 1, the others were chosen to cancel from $c_2$ upward:
| Layout | Terms cancelled | Relative currents | Variation over ±0.2 m |
|---|---|---|---|
| 3 pairs (±0.15, 0.45, 0.75 m) | z², z⁴ | 1 : 0.40 : 2.80 | 1,591 ppm |
| 4 pairs (±0.1, 0.3, 0.55, 0.8 m) | z², z⁴, z⁶ | 1 : 0.96 : 2.02 : 0.97 | 77 ppm |
Four pairs get it to 77 ppm. Why are the end currents larger?
The field weakens toward the coil ends, so the outer coils are strengthened to compensate — the same idea as adding turns at the ends of a long solenoid. Each additional cancelled term cuts variation by roughly an order of magnitude. Real magnets are designed to a few ppm with six or more pairs, and deviations from manufacturing tolerance are trimmed after installation by measuring the field and attaching iron pieces (passive shims) or driving correction coils (active shims).
Example 3: ampere-turns and fringe field
| Item | Value |
|---|---|
| Ampere-turns for 1.5 T (0.5 m radius, 1.5 m long) | 2.15 MA-turns (1.79 by infinite-solenoid estimate) |
| Field energy density at 1.5 T | 0.895 MJ/m³ |
| Distance to 0.5 mT unshielded (1.5 T, on axis) | 8.8 m |
| Same (3 T) | 11.1 m |
Design workflow
- Set centre field, homogeneity (sphere diameter and ppm), fringe-field extent, and magnet length and bore.
- Optimise coil positions and turns to cancel the on-axis expansion coefficients in turn.
- Add outer reverse-current coils to suppress the fringe field (active shielding).
- Check electromagnetic forces on coils and margins against peak field and critical current of the conductor.
- Analyse temperature and voltage during a quench (loss of superconductivity) and design the protection circuit.
“Designed for 5 ppm, installed at 300 ppm”
It's not unusual for an MRI magnet whose homogeneity was verified at the factory to show much greater variation once installed in the hospital. The cause is surrounding magnetic material — floor rebar, nearby steel columns, equipment next door — distorting the field. So after installation the field is measured at hundreds of points on a sphere and the arrangement of iron pieces is calculated (shimming). Including the room's steelwork in the analysis at the design stage avoids deviations too large for shims to correct.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Relying on a long solenoid | Magnet too long | Cancel terms with multiple coils |
| Ignoring surrounding iron | Homogeneity worsens on site | Model room steelwork |
| Choosing conductor by centre field | Peak field on coil exceeds critical | Check peak field on the coil |
| Ignoring electromagnetic forces | Coil motion triggers quench | Analyse support structure |
| Fringe field on axis only | Radial extent misjudged | Map full contours |
I'd like to try related calculations.
Use the solenoid field calculator and Biot–Savart law simulator for coil fields and the MRI SNR and coil tool for image SNR. On the superconducting side, see quench simulation, persistent current mode, critical current density and HTS versus LTS.
Related Topics
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