Superconducting Magnet Quench Protection (2D): Normal-Zone Diffusion PDE

Magnet Circuit

Operating current I₀ 300 A

Detection threshold 0.10 V

Dump resistor R_d 0.50 Ω

Protection system enabled

Live Status

Standing by — coil superconducting

Current I(t)

300 A

Hotspot T (peak)

4.5 K

Detect voltage

0.00 V

Normal zone

0 %

Stored energy remaining

225 kJ

How it works

This is the 2D counterpart of the quench-protection simulator. The 3D scene assumes the normal zone spreads at a fixed closed-form velocity. This version instead discretizes the coil into 90 segments along its length and integrates the real 1D heat-diffusion + Joule-heating PDE at every segment, every frame — the normal-zone front is an emergent result, not a scripted rate:

C·∂T/∂x,t = k·∂²T/∂x² + Q(T)
Q(T) = J²·ρₙ  if T ≥ T_c,  else 0

I(t): L·dI/dt = −I·(R_n(t) + R_dump)
R_n(t) = ρₙ·Σ(normal segment lengths)/A

Numerically integrating this reaction-diffusion PDE for a seed at the coil's midpoint reproduces the closed-form Whetstone–Roach propagation velocity v_NZP = (J/C)·√(ρₙk/(T_c−T_op)) that the 3D sim assumes outright — to within 1–3% once the front reaches steady state (verified against 300/450/600 A cases). The circuit and detection logic (L·dI/dt = −I·R_total, V = I·R_n, threshold trip → dump resistor) are the same physics as the 3D model; what's different is that the normal-zone length, its resistance and the hotspot temperature all come out of a real spatial temperature field instead of an assumed front speed.

Operating current — sets current density J, which drives both the diffusion front's heating term and the Joule dissipation once tripped.
Detection threshold — trips the dump resistor once I·R_n(t) — computed from the actual number of normal segments — crosses this value.
Dump resistor — adds a second resistive term to the circuit ODE once tripped, shortening the L/R discharge time constant.
Protection disabled — the PDE keeps running unchecked: the hottest segment (the original seed) keeps absorbing J²ρₙ with no dump path, so its temperature can run far higher before the current itself decays away.

The lower panel plots the live temperature profile T(x) along the unrolled coil length — a literal snapshot of the finite-difference field, not an interpolated animation — with the critical temperature T_c marked so you can see exactly which segments have actually crossed into the resistive state.