Persistent Current Mode Analysis
Theory: the superconducting magnet as an L/R circuit
Overview
I've heard MRI magnets keep their field for years with the supply off. Is resistance really zero?
The superconducting wire itself has essentially zero resistance, but the whole loop isn't exactly zero. Joints between wires and slow flux motion inside the superconductor (flux creep) act as tiny resistances. With large coil inductance, current decays slowly with time constant L/R. Very small resistance gives time constants of hundreds of thousands of years, but in MRI even a few ppm of field change affects images, so that slow decay matters. Design sets an upper limit on loop resistance from the allowed decay and achieves it through joint quality.
Current and field decay
Field is proportional to current, so its relative rate of change is about R/L. Conditions: L = 25 H, I = 400 A (stored energy 2 MJ).
Charging time and the persistent switch
V is the voltage across the coil during charging and R_sw the resistance of the heated, normal-state persistent switch. During charging some current shunts through the switch and heats it.
Current that doesn't fade for years
That current in a superconducting ring does not decay has been studied since the days of Kamerlingh Onnes's group, which discovered superconductivity in 1911. Later experiments observing current in closed superconducting rings over long periods found decay too small to detect. MRI turned this into practice, and from the 1980s MRI scanners with persistent-mode superconducting magnets spread widely through hospitals.
Worked examples
Example 1: loop resistance and field decay (L = 25 H)
| Loop resistance | Time constant | Decay (ppm/h) | Decay (ppm/year) |
|---|---|---|---|
| 10⁻¹³ Ω | about 7.9 million years | 1.4×10⁻⁵ | 0.13 |
| 10⁻¹² Ω | about 790,000 years | 1.4×10⁻⁴ | 1.3 |
| 10⁻¹¹ Ω | about 79,000 years | 1.4×10⁻³ | 12.6 |
| 10⁻¹⁰ Ω | about 7,900 years | 1.4×10⁻² | 126 |
| 10⁻⁹ Ω | about 790 years | 0.14 | 1,262 |
The limit for 0.1 ppm/h is R < 6.9×10⁻¹⁰ Ω. Twenty joints of 5×10⁻¹⁴ Ω each total 10⁻¹² Ω, giving 1.3 ppm per year.
Example 2: charging and resonance-frequency drift
| Item | Value |
|---|---|
| Time to 400 A at 2 V | 5,000 s (about 83 min) |
| Current through switch (10 Ω) | 0.2 A |
| Switch heating | 0.4 W |
| Drift of 1.5 T resonance (63.87 MHz), R = 10⁻¹¹ Ω | 2.2 Hz/day |
| Same, R = 10⁻⁹ Ω | 221 Hz/day |
10⁻⁹ Ω seems practically zero, yet it shifts 221 Hz a day.
The rate of field change is R/L, so what matters is resistance relative to inductance, not its absolute value. MRI images distort with even small resonance shifts; scanners adjust frequency to track, but large drift requires daily adjustment and degrades stability. Hence the importance of joining superconducting wires while staying superconducting: for NbTi, the copper matrix is removed and superconductor pressed together or bonded with superconducting solder. For high-temperature superconductor tapes, superconducting joints remain difficult, and reducing residual joint resistance is a key challenge for persistent mode.
Design and analysis workflow
- Set the loop-resistance limit from the allowed field decay.
- Allocate joint count and resistance per joint.
- Check joint critical current margin at local field and temperature.
- Balance ramp rate, switch heating and cooling capacity.
- Analyse energy release and protection circuits during quench.
“Decay ten times the spec”
A research superconducting magnet decayed about ten times faster than specified in persistent mode. Each joint's resistance had been verified on samples, but in the assembled magnet one joint sat where the coil's stray field was strong. In that field the joint's critical current fell and it developed slight resistance. Moving the joint to a low-field location and adding magnetic shielding brought decay within spec.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Looking only at absolute resistance | Misjudged decay | Evaluate R/L |
| Joints in high field | Resistance appears | Place in low field |
| Joint checked only on samples | Degrades once assembled | Evaluate in actual field |
| Ramping too fast | Switch overheating, AC loss | Match cooling capacity |
| Ignoring flux creep | Early decay misjudged | Measure early operation |
I'd like to learn related topics.
Related pages include MRI magnet design, quench simulation, critical current density, flux pinning and HTS vs LTS.
Related Topics
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