How is superconducting magnet design carried out? From FEM analysis to thermal and structural verification
Superconducting magnets are not merely devices that generate strong magnetic fields. In particle accelerators, medical beamlines, MRI and NMR systems, and high-field research equipment, superconducting magnets are critical systems that must stably form target magnetic fields at precise locations and maintain magnetic field quality even during prolonged operation. Therefore, the design of superconducting magnets is not merely a process of calculating magnetic field strength, but an integrated engineering process that extends from electromagnetic field analysis, magnetic field uniformity optimization, structural stress review, thermal analysis, cooling system design, to post-fabrication magnetic field measurement (Xu et al. 2025; Takagi et al. 2025).
In particular, because superconducting magnets operate in cryogenic environments, their design is more challenging than that of conventional electromagnets. Strong Lorentz forces act on current-carrying coils, generating mechanical stress on the coils and supporting structures. Simultaneously, external heat inflow, Joule heat of current leads, radiant heat, contact thermal resistance, and the efficiency of cooling paths all affect operational stability. If any of these factors are not sufficiently verified, it can lead to magnetic field performance degradation, insulation damage, thermal instability, and the risk of quenching.
Why Designing Superconducting Magnets Is Difficult
The difficulty in designing superconducting magnets lies in the fact that multiple physical phenomena interact simultaneously. First, in electromagnetic field design, the target center magnetic field, effective magnetic field length, good field region, field uniformity, and harmonic error must be satisfied. For quadrupole or dipole magnets used in accelerators, the magnetic field distribution in the region through which the particle beam passes directly determines the beam’s trajectory and focusing performance; therefore, spatial distribution and higher-order harmonic components become more important than a simple average magnetic field.
Next, structural stability is critical. Superconducting coils are subjected to Lorentz forces under large currents and high magnetic fields, and these forces can push the coils outward or twist them. The coil windings, support structures, epoxy impregnations, and cooling structures must be designed to withstand these loads. Von Mises stress, hoop stress, tensile stress, and bending stress are factors that must be reviewed under actual operating conditions.
Thermal management is also a major factor. The superconducting state is maintained only below a specific critical temperature. If the heat load generated during operation exceeds the cooling capacity, the coil temperature rises, and localized quenching may occur, causing the superconducting state to break. Therefore, thermal analysis must encompass the selection of the cooling method, heat conduction paths, thermal shielding structures, chiller capacity, and current lead design.
Step 1: Electromagnetic Design
The first step in designing a superconducting magnet is defining the field requirements. These include the target center magnetic field, magnetic field gradient, effective magnetic field length, tolerance, operating current, reference radius, and good field region. For medical beamlines or particle accelerator magnets, these requirements are determined in conjunction with beam optics design, while for MRI and NMR magnets, magnetic field uniformity in the imaging or sample region serves as a key indicator.
Once the field requirements are determined, the coil geometry is designed. The coil radius, length, number of turns, conductor cross-section, spacing, and end structure are fundamental elements that form the target magnetic field. In this stage, the magnetic field distribution is calculated through 2D or 3D FEM analysis, and the coil layout is iteratively reviewed to ensure it satisfies the requirements (Diehl et al. 2024; Xu et al. 2025). For quadrupole magnets, pole geometry, shim geometry, yoke saturation, and current density all affect magnetic field quality. Recent research also utilizes genetic algorithms and multi-objective optimization techniques to simultaneously consider both magnetic field quality and fabrication efficiency.
The items primarily reviewed during the electromagnetic field design phase are as follows:
- Verify whether the center magnetic field or magnetic field gradient satisfies target values.
- Calculate magnetic field uniformity and multipole harmonics within the Good Field Region.
- Iteratively optimize coil geometry, number of windings, current density, and core or yoke saturation conditions.
- Rapidly review the basic geometry using 2D FEM and reflect the end field and actual fabrication structure using 3D FEM.
The key to this phase is to design a “magnet that creates the required shape of the magnetic field at the required location,” rather than a “strong magnet.” Even with identical 3T or 5T magnets, actual system performance can vary significantly depending on differences in coil placement and magnetic field distribution.
Step 2: Optimization of Magnetic Field Uniformity
Once the basic electromagnetic field design is complete, Field Uniformity and Harmonic Optimization are performed. Magnetic field uniformity refers to how consistently the magnetic field is maintained within a specific region. In accelerator magnets, magnetic field errors in the region through which the particle beam passes can affect beam trajectory, focus, emittance, and energy stability. In high-field research equipment or MRI/NMR magnets, uniformity dictates image quality or analytical resolution.
To improve magnetic field quality, higher-order harmonic components must be analyzed. Ideally, dipole, quadrupole, and hexpole magnets should have specific multipole components dominant; however, in actual fabrication, unwanted harmonic components arise due to coil position errors, conductor deformation, core saturation, end structure, and assembly tolerances. These components must be predicted in advance during the FEM analysis phase and reduced by adjusting coil placement or pole shape.
Additional considerations are required for HTS magnets. Tape-type high-temperature superconducting conductors, such as REBCO, can induce screening currents, which can lead to low-order magnetic field errors and temporal drift. Therefore, for HTS-based magnets, not only simple static magnetic field analysis but also time stability, screening-current-induced field, and shimming strategies must be considered. Methods such as Passive Shimming, Ferroshim, and Current Sweep Reversal can be utilized as practical strategies to improve magnetic field uniformity (Kim et al. 2026; Zhang et al. 2016; Diehl et al. 2024).
Step 3: Mechanical Analysis
If the electromagnetic field design of a superconducting magnet is the starting point for performance, structural analysis is the step that verifies whether that performance can be maintained under actual operating conditions. When current flows through a coil, a Lorentz force is generated through the interaction between the magnetic field and the current. This force pushes the coil radially or pulls it axially, creating complex stresses in the windings and supporting structures.
Structural analysis typically examines Von Mises stress, hoop stress, tensile stress, bending stress, strain, and the safety factor of the supporting structures. In particular, since superconducting conductors such as Nb3Sn can be sensitive to deformation, they must be designed to avoid exceeding allowable strain. In the case of epoxy-impregnated coils, coil expansion caused by the Lorentz force can be suppressed, but the risk of epoxy cracking or delamination must also be considered. Therefore, the coil windings, impregnation material, skeleton, suspension structure, and cooling links must be reviewed as a single structural system (Xu et al. 2025; Liang et al. 2025).
For magnets used in accelerators, structural analysis is also linked to magnetic field quality. If the coil is slightly deformed, the center of the magnetic field may shift or higher-order harmonic components may increase. In other words, structural stability is not merely a condition to prevent failure, but an essential condition for maintaining magnetic field quality.
Step 4: Thermal Analysis
Thermal analysis is a critical step in determining the operational stability of superconducting magnets. Since superconducting magnets operate stably only below their critical temperature, the cooling system must be able to maintain a constant coil temperature. To achieve this, designers must calculate the heat load, heat conduction paths, thermal shielding structures, contact thermal resistance, heat inflow through current leads, and chiller cooling capacity.
In Cryogen-Free Systems, coils are cooled using GM cryocoolers, GM/JT cryocoolers, heat conduction links, heat exchangers, and cooling plates instead of liquid helium. While conduction cooling can simplify the system and enhance operational convenience, localized temperature rises may occur if the heat conduction paths are not adequately designed. Conversely, for large HTS systems or cases requiring long heat transfer distances, a helium gas circulation cooling model may be considered.
In thermal analysis, both steady-state and transient states are important. Cooldown times, AC losses during current ramping, temperature changes during charging and discharging, and energy dissipation paths during quenching must be reviewed. The Quench Protection design consists of voltage taps, temperature sensors, protection circuits, heaters, diodes, energy discharge resistors, etc. The goal of this stage is not to prevent heat generation, but to create a structure in which the generated heat is rapidly and stably dissipated so as not to break the superconducting state (Takagi et al. 2025; Xu et al. 2025).
Step 5: Magnetic field verification after fabrication
Superconducting magnets are not considered complete once the design is finished. After actual fabrication, it is essential to verify the difference between the design values and the actual values through magnetic field measurements. Factors such as winding errors, assembly tolerances, conductor position deviations, epoxy impregnation status, shrinkage after cooling, and differences in core saturation can all affect the actual magnetic field. Therefore, magnetic measurement is the final stage of design verification and a core process of quality control.
For accelerator magnets, Rotating Coil, Field Mapping, and Harmonic Measurement are widely utilized. Rotating Coil Measurement is a method that calculates multipole components, such as dipole, quadrupole, and sextupole, by analyzing the voltage induced in a coil rotating inside the magnet bore. In particular, for systems like accelerator magnets where higher-order harmonic components directly affect performance, Rotating Coil Measurement serves as an essential verification tool.
During the magnetic field verification stage after fabrication, the following items are checked:
- Verify that the central magnetic field, magnetic field gradient, and effective magnetic field length match the design values.
- Measure the field error and harmonic components within the Good Field Region. • Compare the design FEM results with the actual Field Mapping data.
- If necessary, improve magnetic field quality through shimming, current correction, assembly position correction, etc.
It is important in this process not to view measurement results merely as simple pass or fail criteria. Measurement data serves as feedback for subsequent design improvements and as a standard for ensuring reproducibility between magnets during iterative production (Arpaia et al. 2019; Rogacki et al. 2020; Sorti et al. 2020).
KR TECH's Integrated Design Capabilities
Superconducting magnet design cannot be completed with a single analysis alone. No matter how precise the electromagnetic field analysis may be, excessive structural stress makes actual operation difficult, and insufficient thermal analysis increases the risk of quenching. Furthermore, without the capability to measure magnetic fields after fabrication, it is difficult to verify whether the design values have been realized in the actual device. Therefore, in the development of superconducting magnets, Electromagnetic Analysis, Thermal Analysis, Mechanical Analysis, Manufacturing, and Magnetic Measurement must be integrated into a single process.
In line with the integrated design perspective required in the fields of electromagnets for particle accelerators and superconducting magnets, KR TECH performs magnet design, fabrication, and verification based on the customer’s operating conditions and system requirements. This integrated approach, extending from defining magnetic field requirements to FEM-based electromagnetic field analysis, thermal and structural stability reviews, manufacturing process management, and magnetic field measurement, is a key differentiating factor in the development of high-performance superconducting magnets.
The performance of a superconducting magnet is not determined simply by “how many Tesla it can generate.” Actual performance is determined by how uniformly the required magnetic field is generated, how stably it is maintained in cryogenic environments, and how accurately its performance can be verified after fabrication. Design capabilities that connect FEM analysis, thermal and structural verification, and magnetic field measurement are the key to developing high-performance superconducting magnets.
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