3D Quench Simulation in No-Insulation HTS Coils: FiQuS/Pancake3D

Here is the question this lecture is built around. A coil wound from high temperature superconducting tape is carrying current with no measurable resistance. Then one short stretch of that tape stops superconducting. What happens next, and where? Start with the object. A pancake coil is one long flat tape, four millimetres wide and about a tenth of a millimetre thick, wound as a flat spiral in a plane, with one terminal at the inner end and one at the outer end. This drawing has five turns so that you can count them, and the coil in the paper has forty. Current enters one terminal, runs around and around the spiral, and leaves by the other. In the case study it is one hundred and thirty amperes, with the terminal faces held at four kelvin. The superconductor carries that current with no measurable resistance, but only inside limits: below a critical temperature, below a critical field, and below a critical current. Cross any one of those limits and a short stretch of tape turns resistive within milliseconds. Current keeps flowing through it, so it heats itself, and heat lowers the critical current further. That runaway is what we call a quench. Now the design choice this paper is about. Wind the turns with no insulation between them, so neighbouring turns touch through a thin resistive contact layer. The current then has an alternative. It can cross sideways into the turn beside it, run along that turn past the trouble, and cross back. That bypass is what makes a no insulation coil partly self protecting. It is also what makes it hard to model. Those currents flow radially, across contact layers only tens of micrometres thick, and every one of them dissipates heat where it crosses. They also press turn against turn, which is a mechanical problem sitting on top of the thermal one. So the questions are quantitative. How hot does that spot get? How do the currents redistribute through the winding? What does the terminal voltage do while it happens? Answering that needs a three dimensional, time dependent model in which the magnetic problem and the thermal problem are solved together. The paper delivers that as open source software: Pancake3D, a module of the finite element quench simulator FiQuS, built inside CERN's STEAM framework. You describe the coil in one text file, and the tool generates the geometry, meshes it, solves the transient, and plots the results. On the left is what that file says: two pancakes, a five millimetre inner radius, a hundred and twenty micrometre tape, twenty turns. On the right is the geometry it produced, cut open so you can see the grey windings between the two copper terminals. The rest of this lecture follows that path. First, why this simulation is hard. Then how the tool is put together. Then one full transient, with a defect switched on at sixty seconds, and a sweep over the size of it.

Obstacle one. A quench does not begin everywhere at once, it begins somewhere. In the case study, the damaged stretch is four tenths of a single turn, and here it is in red, on one winding of the spiral. Four tenths of a turn at that radius is about sixteen millimetres of tape, in a spiral almost two metres long. A model that averages around the winding cannot see it, and neither can a flat slice, because the whole point is that current leaves that stretch sideways and comes back. Every turn has to exist separately in the mesh. Obstacle two is that contact layer. The blue blocks are two neighbouring turns of tape, a hundred and twenty micrometres thick each. The red band between them is the contact layer, about ten micrometres, and this drawing is to scale across the thickness. Along the tape, that same layer runs the whole spiral. So one model holds a length of two metres and a thickness of ten thousandths of a millimetre, a ratio of about two hundred thousand to one. Fill that gap with ordinary volume elements and you get either an unusable number of unknowns, or elements so distorted the answer is worthless. Obstacle three. The tape is a laminate, and only a sliver of it superconducts. Seventy five micrometres of the thickness is a nickel alloy substrate, forty two is copper, one and a half is silver, and the superconducting film itself is one and a half micrometres of rare earth barium copper oxide. The drawing keeps the relative thicknesses of the two thick layers, and exaggerates the two thin ones so that they can be seen at all. Resolving all four in a three dimensional mesh is out of the question, so the tool replaces the stack with one equivalent material. That is homogenisation, and it has a consequence. Here is why. The superconductor does not switch from zero resistance to resistive at a threshold. Its electric field grows as a power of the current density it carries, with an exponent of about thirty for this material. The critical current is simply where that field reaches a chosen criterion, a hundred microvolts per metre. Below it the tape looks perfectly superconducting. A few percent above it the field is already many times larger, so the current shares itself between the film and the metal around it, and how it shares depends on the answer you are solving for. And obstacle four is a practical one. Getting all of that right means a coupled, non-linear, three dimensional finite element model, which is not what a magnet engineer wants to spend a year writing. So, four obstacles. A quench is local, so every turn has to be in the mesh. The contact layers are ten micrometres thick against a coil of two metres. The tape is a laminate that has to be homogenised. And the superconductor is non-linear, so every time step is an iteration. Everything in the next section is an answer to one of those four.

The input file has four sections, and they are exactly the four things the tool does: geometry, mesh, solve, and postprocess. Here is what the first two produce: one winding, seen from above, cut into elements around it and across its thickness. Underneath, three open source pieces do the work. Gmsh builds the geometry and the mesh, through the OpenCASCADE kernel. GetDP assembles and solves the finite element problem. And the STEAM material library supplies properties of copper, steel and the superconductor as functions of temperature and field. Now the physics. In the conducting parts, the magnetic field obeys one equation: the rate of change of the flux density, plus the curl of resistivity times the curl of the field, is zero. That is Faraday and Ampere, written for the field alone. The current density is the curl of that same field, so nothing extra is solved for it. And the heat it generates, resistivity times current density dotted with itself, is the source term of the heat equation, which is solved on the windings and the terminals at the same time. Outside the conductors no current flows, so the curl of the field is zero there and it can be written as the gradient of one scalar potential. Three unknowns per node become one across most of the volume, and that is why this is called an H phi formulation. The air still has to stop somewhere: either you mesh a large enough box, or a shell transformation maps the rest of space onto a thin outer layer and lets the air mesh stay small. And the coupling runs both ways. The electrical solution heats the material, and the properties, resistivity, thermal conductivity, heat capacity, all follow the temperature, while the resistivity also follows the local field and the current density itself. So how do you mesh a gap ten micrometres wide? On the left, the honest way, with the gap exaggerated so we can see it at all: real volume elements inside the layer. Each of them is thin and wide, exactly the shape a finite element hates, and there is a layer like it between every pair of turns. On the right, the approximation the tool uses. The layer is collapsed onto a single surface, and its physics is put back as conditions across that surface: a voltage drop that sets the radial current, and a temperature drop that sets the heat crossing it. The surface carries no thickness of its own, so the mesh stays sane, and the radial currents that make these coils interesting are still there. The papers behind this one check the approximation against fully meshed layers, thermally, magnetically, and coupled, and it holds. Back to the laminate. Instead of meshing four layers, the tool computes one equivalent material, and that material is not the same in every direction. Along the tape, current has a superconducting path. Through the thickness, it has to cross substrate and copper. So resistivity and thermal conductivity differ along the three directions of the tape. That is what this matrix says: one resistivity along the length, another across the width, another through the thickness. The tool builds it from the layer list and rotates it into place element by element, because in a spiral, along the tape means a different direction at every point. And because resistivity depends on the current density, every time step is a solve inside a solve. Most properties are updated by simple fixed point, or Picard, iteration. The conductor resistivity gets Newton-Raphson, linearised with respect to current density, which is what makes that power law converge in a sensible number of steps.

Here is the case the paper runs. Two pancake coils stacked half a millimetre apart, forty turns each, wound from four millimetre wide tape with a ten micrometre stainless steel layer between the turns. At four kelvin in zero field, the critical current of that tape is seven hundred and seventy amperes. The coil is driven in current mode. The supply ramps from zero to one hundred and thirty amperes over fifty seconds, holds there for ten, and then ramps back down, reaching zero at eighty seconds. The terminal faces are held at four kelvin throughout, which is one of three cooling options; the tool also offers a fully adiabatic model, or a cryocooler load map from a manufacturer. Now watch the operating point walk up the ramp. Fifty seconds, one hundred and thirty amperes, and the ramp is finished. Ten seconds of plateau follow, with the coil fully superconducting at about seventeen percent of its critical current. Then, at sixty seconds, the defect. Across four tenths of one turn in the first pancake, the critical current is set to zero, as though that stretch of tape had been damaged. Thirty milliseconds later the supply starts down, and it is at zero by eighty seconds. The run then continues to one hundred seconds, because the coil has not finished. Here is the first result, on one pair of axes. The grey curve is the supply current, scaled to fit. The green curve is the axial magnetic field at the centre of the coil, about zero point seven tesla at the top of the ramp. Look at where the green curve sits against the grey one. The field lags. Part of that is ordinary inductance. Part of it belongs to this design: current entering the winding does not immediately follow the spiral, some of it crosses the contact layers instead, and the field only builds as that sharing settles. Now the end of the run. The supply current is at zero from eighty seconds, and the central field is not. Something is still circulating in there, and it is still circulating when the run ends. On the left, the current density inside the windings ten seconds after the supply was switched off. There is no source current at all, and the coil is still full of current: loops of it, trapped, running around the turns. These are screening currents. A superconductor opposes a change in the field through it by setting up currents that cancel the change, and once they are established, with no resistance to speak of, they have no reason to stop. On the right is what they do to the field. Thirty millitesla at the coil centre, and after a steeper first second the computed decay follows this exponential, whose time constant is near twenty seven seconds. For a magnet that is supposed to be off, that is a number worth knowing before the magnet exists. Here is the temperature at the middle of the defect, across the whole run. During the ramp it drifts up from four kelvin to about five, before any defect exists, because radial currents and the terminal joints are already dissipating. At sixty seconds it jumps, to about five point four kelvin. That is the defect: the resistivity along that stretch rises, current redistributes across the contact layers, and heat appears both in the resistive stretch and in every layer the current crosses on its way round it. On the right is where the heat actually is, thirty milliseconds after the defect appears. Notice that the hottest ring is not the defect at all. It is the inner terminal, because that joint has the smaller area, so its contact resistance dissipates more than the outer one. And notice the size of the rise: a fraction of a kelvin, on a coil at four. This defect is survivable, which is what the no insulation design is for. A model like this is how you find out whether it is still survivable at higher current, or with a longer defect. The last quantity is the one you can actually measure: the voltage between the two terminals. It rises during the ramp, transiently at first and then linearly, to about three and a half millivolts on the plateau. That plateau reading is not the superconductor. The winding is still fully superconducting there; what you are measuring is the copper terminals and their contact layers. Any detection threshold has to sit on top of that offset. At sixty seconds the voltage steps up, to about three point nine millivolts. That step is the whole basis of quench detection, and it is four tenths of a millivolt riding on three and a half. Then it goes negative on the way down, because the coil's inductance opposes the falling current and the ramp down is faster than the ramp up. So the signal you would trigger on is a small step on a large, moving background, which is exactly why you want it simulated.

One simulation is one data point. So the last thing the paper does is turn the length of the defect into a parameter, and run the same case ten times, with the damage growing from a tenth of a turn to a full turn. On the left, the terminal voltage, with three of those ten runs drawn. Every curve steps up at sixty seconds, and the step grows with the defect, from about three point eight millivolts for the shortest to about four point two for the longest. On the right, the temperature at the middle of the defect. The ordering is the same: a longer normal zone dissipates more, and the peak climbs from about five point two five kelvin to about five point six. And look at where the curves come back together. By eighty seconds the voltages have converged, because by then the reading is dominated by the ramp down and the defect barely enters it. The size of the damage shows up in the peak, not in the tail. So what does this give an engineer. Every turn resolved in three dimensions, with the contact layers carried as surfaces rather than volumes, which is what makes a forty turn double pancake tractable at all. The magnetic and thermal problems solved together, with material properties that follow the temperature and the field instead of being frozen at their nominal values. And the entire model in one text file, which is what turned that ten run study into a loop rather than a project. This picture, incidentally, is a four pancake coil with sixty turns each, from the same tool. It is open source, and the input files for every run in the paper are published, so these results are reproducible rather than merely reported. For anyone designing a no insulation HTS magnet, that is the point: put a defect anywhere, give it any length, and get the coil's answer before the coil is wound.