Critical Current Density Analysis
Theory: Bean's critical-state model
Overview
Superconductors have zero resistance, so why do they have losses in AC use?
In practical (type II) superconductors, field enters as thin flux bundles pinned by material defects. Changing the field drives flux past the pinning, dissipating energy each time: AC (hysteresis) loss. Bean modelled this simply: inside the superconductor, current either flows at the critical density Jc or not at all. An applied field creates a surface layer carrying Jc that screens it; increasing the field spreads that layer inward. This simple model lets penetration, magnetisation and loss all be calculated by hand.
Penetration field, magnetisation and trapped field
a is slab half-width and R cylinder radius. B_p is the field at which penetration reaches the slab centre.
AC loss (slab, parallel field)
Bean's 1962 paper
In 1962 Bean published the critical-state model to explain magnetisation of hard (strongly pinned) superconductors. Its bold simplification — current density either critical or zero — explained hysteretic magnetisation curves and AC loss well, and it remains the starting point for superconducting wire and magnet design. It was later extended by Kim's model with field-dependent Jc and by power-law relations between electric field and current density, used in numerical analysis.
Worked examples
Example 1: NbTi AC loss (Jc = 3×10⁹ A/m², 50 Hz, density 6,500 kg/m³)
| Half-width a | B_p | 0.01 T amplitude | 0.1 T amplitude | 0.5 T amplitude |
|---|---|---|---|---|
| 5 μm | 0.019 T | 0.22 W/kg | 20.2 W/kg | 112 W/kg |
| 50 μm | 0.19 T | 0.022 W/kg | 21.6 W/kg | 864 W/kg |
| 0.5 mm | 1.89 T | 0.0022 W/kg | 2.2 W/kg | 271 W/kg |
When amplitude exceeds B_p, loss is roughly proportional to a; below it, inversely proportional. Loss peaks when amplitude is comparable to B_p.
Example 2: trapped field in a YBCO bulk (radius 15 mm)
| Condition | Jc | Trapped-field limit (long cylinder) |
|---|---|---|
| 77 K | 1×10⁸ A/m² | 1.88 T |
| 30 K | 1×10⁹ A/m² | 18.8 T |
Real bulks have finite height, so values are lower. At around 30 K, electromagnetic stresses are large and bulks need metal-ring reinforcement against cracking; a reinforced bulk trapped 17.6 T at 26 K (2014).
I thought finer filaments were always better, but at small amplitude thicker ones lose less.
At small amplitudes the field enters only a thin surface layer. Thicker slabs have larger B_p, so the same amplitude penetrates relatively less deeply and loss is smaller. When the amplitude exceeds B_p and field sweeps through the whole slab, loss scales with magnetisation, Jc·a, so finer is better. For applications exposed to large AC fields, like power cables and transformers, filaments are made fine and twisted to break coupling currents between them. For DC wires such as MRI magnets AC loss matters little, and fine filaments mainly prevent flux jumping. Numerical analysis computes such losses for real geometries using finite elements with power-law E–J relations.
Analysis workflow
- Supply Jc at operating temperature and field from wire measurements.
- Estimate penetration field and loss magnitude with the Bean model.
- For real cross-sections, run FE analysis with power-law E–J behaviour.
- Include field dependence of Jc (e.g. Kim model).
- Convert loss into cryocooler load and check temperature margin against quench.
“The cryocooler couldn't keep up”
A superconducting coil ran fine in DC but kept heating when its current was cycled. The cryocooler had been sized for DC losses only. A Bean-model estimate of hysteresis loss showed current-change losses reached nearly half the cooling capacity. Slowing the ramp rate and switching to finer-filament wire stabilised the temperature.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Constant Jc | Wrong loss at high field | Include field dependence |
| Sizing cooling for DC only | Temperature rises | Add AC loss |
| Assuming finer is always better | Counterproductive at low amplitude | Compare amplitude with B_p |
| Ignoring coupling currents | Loss underestimated | Account for twist pitch |
| Using long-cylinder formula as is | Trapped field overestimated | Analyse finite height |
I'd like to learn related topics.
Related pages include flux pinning, AC loss in superconductors, quench simulation, HTS vs LTS and persistent current mode.
Related Topics
Experience the theory firsthand with the interactive simulator for this field
All Simulators