AC Loss in Superconductors
Theory: energy lost as flux moves in and out
Overview
Superconducting cables have zero resistance, yet I hear cooling is hard with AC. Why?
With DC, loss is indeed zero. But with AC, every change in current's direction and size drives flux into and out of the superconductor's surface. Flux inside is 'pinned' by defects and loses energy when it moves — this is AC loss, a form of hysteresis loss. Loss per cycle is frequency-independent, so at 50 Hz you lose it 50 times a second. And removing that heat from liquid-nitrogen temperature costs many times the loss itself in electricity.
Norris's formulas
Transport loss per cycle per unit length (J/m). $I_c$ is critical current and $F$ the ratio of peak current to it. Use the upper formula for thin tapes (such as REBCO high-temperature superconductors) and the lower for elliptical cross-sections. Both rest on the critical-state (Bean) model.
Loss power is proportional to frequency $f$. Rejecting heat from 77 K to 300 K needs 2.9 times the loss in electricity even for an ideal Carnot refrigerator.
Norris's formulas and high-temperature superconductivity
Norris derived his formulas in 1970 in the UK, using the critical-state model to obtain analytical transport losses for elliptical and thin-strip superconductors. That was the era of helium-cooled metallic superconductors, and AC use was limited. When high-temperature superconductors were discovered in 1986 and liquid-nitrogen cooling became possible, AC applications — power cables, transformers, fault-current limiters — suddenly looked realistic. With thin tapes now dominant, Norris's formulas remain the reference for estimating AC loss and comparing with measurements.
Worked example: a 100 A critical-current conductor
Loss (W/m) for a conductor with 100 A critical current (77 K, self field) carrying 50 Hz AC:
| Peak current (F) | Thin tape | Elliptical | Ellipse / tape |
|---|---|---|---|
| 10 A (0.1) | 3.4×10⁻⁶ | 3.5×10⁻⁵ | 10.5 |
| 30 A (0.3) | 2.8×10⁻⁴ | 1.1×10⁻³ | 3.8 |
| 50 A (0.5) | 2.3×10⁻³ | 5.7×10⁻³ | 2.4 |
| 70 A (0.7) | 1.0×10⁻² | 1.9×10⁻² | 1.8 |
| 90 A (0.9) | 3.6×10⁻² | 5.3×10⁻² | 1.5 |
At low current, an elliptical wire loses ten times more than a tape?
Yes. A thin tape's loss grows roughly with the fourth power of current, an ellipse's with the third, and at low current the fourth power is far smaller — flux enters at the tape edges differently than in an ellipse. Conversely, running a tape near its critical current makes loss climb steeply: raising current 40% from 50 to 70 A multiplies loss 4.4 times. In cable design, the fraction of critical current you operate at dominates the loss.
Example 2: cooling burden of a power cable
A 1 km three-phase cable with 20 tapes per phase (50 A peak each, F = 0.5) carries 707 A rms per phase:
| Item | Value |
|---|---|
| AC loss at 77 K | 139 W |
| Cooling power (Carnot) | 404 W |
| Cooling power (real refrigerators, 10–20×) | 1.4–2.8 kW |
| Loss at F = 0.7 | 611 W (4.4×) |
These values come from the isolated-tape formula; in real cables, mutual fields between tapes, magnetisation loss from external fields, eddy-current loss in metallic stabilisers and heat leak through the cryostat add to it.
Analysis workflow
- Prepare critical current as a function of operating temperature and field magnitude and angle.
- Estimate transport loss with Norris's formulas and choose the operating fraction of critical current.
- Compute total loss including magnetisation with finite element methods such as the H or T-A formulation, including tape arrangement.
- Add stabiliser eddy-current loss and cryostat heat leak to size the cooling load.
- Compare with measurement (electrical or calorimetric).
“Loss several times the calculation”
A prototype superconducting coil measured AC loss several times the Norris estimate. Norris's formula covers transport loss in a single tape, but inside the coil, fields from other turns were perpendicular to the tape face. Thin tapes are very sensitive to perpendicular fields, and this magnetisation loss far exceeded the transport loss. Recomputing with finite element analysis including the full coil field matched measurements well. In coils and cables, always account for fields produced by other currents.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Estimating transport loss only | Magnetisation loss missed | Include external and neighbouring fields |
| Constant critical current | Loss underestimated | Include field and angle dependence |
| Comparing raw loss power | Cooling burden misjudged | Convert via refrigerator efficiency |
| Operating near critical current | Loss climbs steeply | Keep margin on operating fraction |
| Ignoring stabiliser eddy currents | Loss grows at higher frequency | Model metallic layers |
I'd like to learn related topics.
The magnetic hysteresis simulator illustrates hysteresis. On superconductivity, see critical current density, flux pinning, superconducting cables, HTS versus LTS and quench simulation.
Related Topics
Experience the theory firsthand with the interactive simulator for this field
All Simulators