{"version":1,"lectureId":"01M1FZVJEE0EB53FR7BQHEAVZ2","attempt":0,"publication":{"slug":"3d-quench-simulation-in-no-insulation-hts-coils-fiqus-pancake3d","title":"3D Quench Simulation in No-Insulation HTS Coils: FiQuS/Pancake3D","subject":"engineering","summary":"A structured walk through the 2024 paper by Atalay and colleagues on Pancake3D, the no-insulation pancake coil module of the open-source finite element quench simulator FiQuS. The lecture starts from the hardware: what a high-temperature superconducting pancake coil is, why leaving the turns uninsulated lets current bypass a damaged stretch, and what a quench actually is. It then works through the four numerical obstacles the paper identifies, the thin-shell and homogenisation strategies used to get past them, the coupled field and heat formulation, and one complete transient: ramp, plateau, a local defect switched on at sixty seconds, ramp-down, residual screening currents, hot spot, terminal voltage, and a ten-run sweep over the defect length. Written for engineers who are not superconductivity specialists.","metaDescription":"How FiQuS/Pancake3D simulates quenches in no-insulation HTS pancake coils: thin-shell contact layers, homogenised tape, one full transient.","transcript":"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.","watch":{"version":1,"scenes":[{"title":"The Coil and the Quench","start":0,"end":189.41656250000005,"objects":{"bottom_wind":"a Polygon [gray] drawn in stack (vertices=((1.8, 1.6), (7.0, 1.6), (7.0, 3.8), (1.8, 3.8)), fill_opacity=0.5)","build_label":"a Tex [text] that says \"What it builds\"","bypass_in":"an Arrow [green] drawn in coil (start=(0.7442017240009579, -0.24180579809839556), end=(0.6015432465566846, -0.19545324894215457))","bypass_out":"an Arrow [green] drawn in coil (start=(-0.7150000000000001, 4.3781123069517883e-16), end=(-0.865, 7.415236368837223e-16))","coil":"a Figure (x_range=(-1.5, 1.5), y_range=(-1.5, 1.5), aspect=(1.0, 1.0))","current_way":"an Orientation [blue] drawn in coil (path=((0.34, 0.0, 0.0), (0.31018824469502304, 0.16579907103051886, 0…, arrows=4, of='tape')","defect":"a ParametricCurve [red] drawn in coil (function=<function>, t_range=(3.0, 3.4))","defect_dot":"a Point [red] labelled \"upright(\"defect\")\" drawn in coil (location=(0.2533939353874575, 0.7798663433620258))","detour":"a ParametricCurve [green] drawn in coil (function=<function>, t_range=(1.95, 2.5))","head_tool":"a Heading that says \"One Input File, One 3D Model\"","inner_term":"a Polygon [red] drawn in stack (vertices=((1.0, 0.6), (1.8, 0.6), (1.8, 7.4), (1.0, 7.4)), fill_opacity=0.35)","inputs":"a Block [text] that says \"two pancakes, $0.5 thin upright(\"mm\")$ apart inner radius $5 thin upright(\"mm\")$ tape $120 thin mu upright(\"m\")$ thick twenty turns each\"","m_current":"a Math [text] that says \"$I = 130 thin upright(\"A\"), quad T = 4 thin upright(\"K\")$\"","m_tape":"a Math [text] that says \"$upright(\"width\") = 4 thin upright(\"mm\")$\"","m_thick":"a Math [text] that says \"$upright(\"thickness\") = 120 thin mu upright(\"m\")$\"","m_turns":"a Math [text] that says \"$upright(\"turns\") = 40$\"","outer_pad":"a Point [gray] labelled \"upright(\"terminal\")\" drawn in coil (location=(1.09, -1.3348650110706152e-15))","outer_term":"a Polygon [red] drawn in stack (vertices=((7.0, 0.6), (7.8, 0.6), (7.8, 7.4), (7.0, 7.4)), fill_opacity=0.35)","point":"a Point [yellow] drawn in coil (location=(0.34, 0.0))","point_2":"a Point [yellow] drawn in coil (location=(0.2533939353874575, 0.7798663433620258))","question":"a Panel that says \"A no-insulation HTS pancake coil is carrying $130 thin upright(\"A\")$. 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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.","live":["question"],"does":[[16.99,"coil is shown on the screen, written out."],[16.99,"tape is shown on the screen, drawn."],[22.226,"coil moves to a new place on the board."],[22.226,"m_tape is shown on the screen, written out."],[24.698999999999998,"m_thick is shown on the screen, written out."],[28.797,"point is shown on the screen, grown."],[29.9,"outer_pad is shown on the screen, written out."],[31.072379235093145,"point is hidden from the screen."],[35.798,"m_turns is shown on the screen, written out."]]},{"start":37.35,"say":"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.","live":["question","m_tape","m_thick","m_turns","coil","tape","outer_pad"],"does":[[39.52100000000001,"current_way is shown on the screen, written out."],[44.92,"m_current is shown on the screen, written out."]]},{"start":49.734,"say":"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.","live":["question","m_tape","m_thick","m_turns","m_current","coil","tape","outer_pad","current_way"],"does":[[62.934,"defect is shown on the screen, drawn."],[64.084,"defect_dot is shown on the screen, written out."]]},{"start":75.0865,"say":"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.","live":["question","m_tape","m_thick","m_turns","m_current","coil","tape","outer_pad","current_way","defect","defect_dot"],"does":[[88.47300000000001,"bypass_in is shown on the screen, written out."],[90.57400000000001,"detour is shown on the screen, drawn."],[92.81500000000001,"bypass_out is shown on the screen, written out."]]},{"start":94.55250000000001,"say":"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. 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You describe the coil in one text file, and the tool generates the geometry, meshes it, solves the transient, and plots the results.","live":[],"does":[[135.6765,"head_tool is shown on the screen, written out."],[148.633,"inputs is shown on the screen, written out."],[150.35100000000003,"stack is shown on the screen, written out."],[150.35100000000003,"inner_term is shown on the screen, written out."],[150.35100000000003,"outer_term is shown on the screen, written out."],[151.93,"top_wind is shown on the screen, written out."],[152.1300968405762,"bottom_wind is shown on the screen, written out."]]},{"start":156.3495,"say":"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.","live":["inputs","stack","head_tool","inner_term","outer_term","top_wind","bottom_wind"],"does":[[157.05800000000002,"write_label is shown on the screen, written out."],[166.648,"build_label is shown on the screen, written out."]]},{"start":173.42450000000002,"say":"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.","live":["write_label","inputs","build_label","stack","head_tool","inner_term","outer_term","top_wind","bottom_wind"],"does":[[188.37489583333337,"build_label is hidden from the screen — left the board."],[188.37489583333337,"head_tool is hidden from the screen — left the board."],[188.37489583333337,"inputs is hidden from the screen — left the board."],[188.37489583333337,"stack is hidden from the screen — left the board."],[188.37489583333337,"inner_term is hidden from the screen — stack left the board."],[188.37489583333337,"outer_term is hidden from the screen — stack left the board."],[188.37489583333337,"top_wind is hidden from the screen — stack left the board."],[188.37489583333337,"bottom_wind is hidden from the screen — stack left the board."],[188.37489583333337,"write_label is hidden from the screen — left the board."]]}]},{"title":"Four Obstacles","start":189.41656250000005,"end":380.21843750000005,"objects":{"crit":"a PlotPoint [yellow] labelled \"E_c\" drawn in ej (target='ej_curve', x=1.0)","cross":"a Figure (x_range=(0.0, 60.0), y_range=(-8.0, 34.0), aspect=(60.0, 42.0))","ej":"an Axes (x_range=(0.0, 1.2), y_range=(0.0, 6.5), x_ticks_every=0.5)","ej_curve":"a FunctionPlot [blue] drawn in ej (function=<function>, x_range=(0.0, 1.06))","gap_layer":"a Polygon [red] drawn in cross (vertices=((3.0, 14.0), (57.0, 14.0), (57.0, 15.0), (3.0, 15.0)), fill_opacity=0.9)","guide_h":"a Line [gray] drawn in ej (start=(0.0, 1.0), end=(1.2, 1.0), dashed=True)","guide_v":"a Line [gray] drawn in ej (start=(1.0, 0.0), end=(1.0, 6.0), dashed=True)","head_four":"a Heading that says \"Four Obstacles\"","head_gap":"a Heading that says \"Ten Micrometres Against Two Metres\"","head_law":"a Heading that says \"How a Superconductor Resists\"","head_local":"a Heading that says \"A Quench Is a Local Event\"","head_stack":"a Heading that says \"A Tape That Is Not One Material\"","hurt":"a ParametricCurve [red] drawn in spiral (function=<function>, t_range=(3.0, 3.4))","hurt_mark":"a Point [red] labelled \"upright(\"defect\")\" drawn in spiral (location=(0.24659556151120865, 0.7589431000035324))","l_ag":"a Polygon [blue] drawn in layers (vertices=((2.0, 7.0), (38.0, 7.0), (38.0, 7.7), (2.0, 7.7)), fill_opacity=0.6)","l_cu":"a Polygon [yellow] drawn in layers (vertices=((2.0, 4.5), (38.0, 4.5), (38.0, 7.0), (2.0, 7.0)), fill_opacity=0.4)","l_hts":"a Polygon [red] drawn in layers (vertices=((2.0, 7.7), (38.0, 7.7), (38.0, 8.4), (2.0, 8.4)), fill_opacity=0.9)","l_sub":"a Polygon [gray] drawn in layers (vertices=((2.0, 0.0), (38.0, 0.0), (38.0, 4.5), (2.0, 4.5)), fill_opacity=0.4)","law":"a Math [text] that says \"$E = E_c (J / J_c)^n, quad n = 30$\"","law_note":"a Text [text] that says \"No sharp switch: the electric field, and with it the resistivity, rises smoothly with current density, so the sharing of current between the film and the metal has to be resolved by iteration.\"","layers":"a Figure (x_range=(0.0, 40.0), y_range=(-4.0, 12.0), aspect=(40.0, 16.0))","lower_turn":"a Polygon [blue] drawn in cross (vertices=((3.0, 2.0), (57.0, 2.0), (57.0, 14.0), (3.0, 14.0)), fill_opacity=0.25)","m_arc":"a Math [text] that says \"$0.4 upright(\"turn\") approx 16 thin upright(\"mm\")$\"","m_layer":"a Math [text] that says \"$upright(\"contact layer\") = 10 thin mu upright(\"m\")$\"","m_len":"a Math [text] that says \"$upright(\"tape length\") approx 1.9 thin upright(\"m\")$\"","m_ratio":"a Math [text] that says \"$frac(1.9 thin upright(\"m\"), 10 thin mu upright(\"m\")) approx 2 times 10^5$\"","m_tape2":"a Math [text] that says \"$upright(\"tape\") = 120 thin mu upright(\"m\")$\"","obstacles":"a Block [text] that says \"A quench is local, so the model is three dimensional and every turn is in it. Contact layers are ten micrometres thick in a coil nearly two metres long. The tape is a laminate, so its properties must be homogenised. The superconductor is n…\"","spiral":"a Figure (x_range=(-1.4, 1.4), y_range=(-1.4, 1.4), aspect=(1.0, 1.0))","stack_table":"a Table [text] that says \"Layer Thickness ($mu upright(\"m\")$) Hastelloy substrate 75 Copper 42 Silver 1.5 REBCO 1.5\" (rows=(('Layer', 'Thickness ($mu upright(\"m\")$)'), ('Hastelloy substr…, header=True)","upper_turn":"a Polygon [blue] drawn in cross (vertices=((3.0, 15.0), (57.0, 15.0), (57.0, 27.0), (3.0, 27.0)), fill_opacity=0.25)","winding":"a ParametricCurve [blue] drawn in spiral (function=<function>, t_range=(0.0, 6.0))"},"beats":[{"start":189.41656250000005,"say":"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.","live":[],"does":[[189.41656250000005,"head_local is shown on the screen, written out."],[189.41656250000005,"spiral is shown on the screen, written out."],[189.41656250000005,"winding is shown on the screen, drawn."],[198.10056250000005,"spiral moves to a new place on the board."],[198.10056250000005,"m_arc is shown on the screen, written out."],[200.75956250000004,"hurt is shown on the screen, drawn."],[201.35592244587286,"hurt_mark is shown on the screen, written out."]]},{"start":204.19206250000005,"say":"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.","live":["m_arc","spiral","head_local","winding","hurt","hurt_mark"],"does":[[207.23356250000006,"m_arc (the \"16 thin upright(\"mm\")\" part) is emphasized."],[209.13756250000006,"m_arc (the \"16 thin upright(\"mm\")\" part) is no longer emphasized."],[210.06656250000006,"m_len is shown on the screen, written out."],[224.69506250000006,"head_local is hidden from the screen — left the board."],[224.69506250000006,"m_arc is hidden from the screen — left the board."],[224.69506250000006,"m_len is hidden from the screen — left the board."],[224.69506250000006,"spiral is hidden from the screen — left the board."],[224.69506250000006,"winding is hidden from the screen — spiral left the board."],[224.69506250000006,"hurt is hidden from the screen — spiral left the board."],[224.69506250000006,"hurt_mark is hidden from the screen — spiral left the board."]]},{"start":225.29506250000006,"say":"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.","live":[],"does":[[225.29506250000006,"head_gap is shown on the screen, written out."],[225.29506250000006,"cross is shown on the screen, written out."],[225.29506250000006,"lower_turn is shown on the screen, written out."],[230.10156250000006,"upper_turn is shown on the screen, written out."],[232.01756250000005,"cross moves to a new place on the board."],[232.01756250000005,"m_tape2 is shown on the screen, written out."],[234.25756250000006,"gap_layer is shown on the screen, written out."],[237.05556250000006,"m_layer is shown on the screen, written out."]]},{"start":241.75356250000004,"say":"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.","live":["m_tape2","m_layer","cross","head_gap","lower_turn","upper_turn","gap_layer"],"does":[[252.09856250000007,"m_ratio is shown on the screen, written out."],[254.82656250000005,"gap_layer is indicated — a transient flash."],[262.90756250000004,"cross is hidden from the screen — left the board."],[262.90756250000004,"lower_turn is hidden from the screen — cross left the board."],[262.90756250000004,"upper_turn is hidden from the screen — cross left the board."],[262.90756250000004,"gap_layer is hidden from the screen — cross left the board."],[262.90756250000004,"head_gap is hidden from the screen — left the board."],[262.90756250000004,"m_layer is hidden from the screen — left the board."],[262.90756250000004,"m_ratio is hidden from the screen — left the board."],[262.90756250000004,"m_tape2 is hidden from the screen — left the board."]]},{"start":263.50756250000006,"say":"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.","live":[],"does":[[263.50756250000006,"head_stack is shown on the screen, written out."],[263.50756250000006,"layers is shown on the screen, written out."],[266.20056250000005,"layers moves to a new place on the board."],[266.20056250000005,"stack_table is shown on the screen, written out."],[269.6605625000001,"l_sub is shown on the screen, written out."],[269.6605625000001,"stack_table is shown on the screen, written out."],[274.38556250000005,"l_cu is shown on the screen, written out."],[274.38556250000005,"stack_table is shown on the screen, written out."],[276.5335625000001,"l_ag is shown on the screen, written out."],[276.5335625000001,"stack_table is shown on the screen, written out."],[278.46056250000004,"l_hts is shown on the screen, written out."],[278.46056250000004,"stack_table is shown on the screen, written out."]]},{"start":284.39006250000006,"say":"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.","live":["layers","head_stack","l_sub","l_cu","l_ag","l_hts"],"does":[[285.76056250000005,"stack_table (the \"column=2\" part) is emphasized."],[295.18756250000007,"stack_table (the \"column=2\" part) is no longer emphasized."],[304.3820625000001,"head_stack is hidden from the screen — left the board."],[304.3820625000001,"layers is hidden from the screen — left the board."],[304.3820625000001,"l_sub is hidden from the screen — layers left the board."],[304.3820625000001,"l_cu is hidden from the screen — layers left the board."],[304.3820625000001,"l_ag is hidden from the screen — layers left the board."],[304.3820625000001,"l_hts is hidden from the screen — layers left the board."],[304.3820625000001,"stack_table is hidden from the screen — left the board."]]},{"start":304.98206250000004,"say":"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.","live":[],"does":[[304.98206250000004,"head_law is shown on the screen, written out."],[304.98206250000004,"ej is shown on the screen, written out."],[304.98206250000004,"guide_v is shown on the screen, written out."],[304.98206250000004,"guide_h is shown on the screen, written out."],[313.37656250000003,"ej_curve is shown on the screen, drawn."],[316.19756250000006,"ej moves to a new place on the board."],[316.19756250000006,"law is shown on the screen, written out."]]},{"start":319.3170625,"say":"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.","live":["law","ej","head_law","guide_v","guide_h","ej_curve"],"does":[[322.70756250000005,"crit is shown on the screen, written out."],[333.31856250000004,"law_note is shown on the screen, written out."]]},{"start":340.6875625,"say":"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.","live":["law","law_note","ej","head_law","guide_v","guide_h","ej_curve","crit"],"does":[[346.6325625,"law (the \"(J / J_c)^n\" part) is emphasized."],[351.33456250000006,"law (the \"(J / J_c)^n\" part) is no longer emphasized."],[353.6440625,"ej is hidden from the screen — left the board."],[353.6440625,"guide_v is hidden from the screen — ej left the board."],[353.6440625,"guide_h is hidden from the screen — ej left the board."],[353.6440625,"ej_curve is hidden from the screen — ej left the board."],[353.6440625,"crit is hidden from the screen — ej left the board."],[353.6440625,"head_law is hidden from the screen — left the board."],[353.6440625,"law is hidden from the screen — left the board."],[353.6440625,"law_note is hidden from the screen — left the board."]]},{"start":354.24406250000004,"say":"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.","live":[],"does":[[354.24406250000004,"head_four is shown on the screen, written out."],[355.4865625,"obstacles is shown on the screen, written out."],[357.90156250000007,"obstacles (the \"local\" part) is emphasized."],[362.5915625,"obstacles (the \"local\" part) is no longer emphasized."],[362.5915625,"obstacles (the \"ten micrometres thick\" part) is emphasized."],[368.3505625,"obstacles (the \"homogenised\" part) is emphasized."],[368.3505625,"obstacles (the \"ten micrometres thick\" part) is no longer emphasized."],[371.08956250000006,"obstacles (the \"homogenised\" part) is no longer emphasized."],[371.08956250000006,"obstacles (the \"non-linear\" part) is emphasized."]]},{"start":375.06806250000005,"say":"Everything in the next section is an answer to one of those four.","live":["obstacles","head_four"],"does":[[377.1585625,"obstacles (the \"non-linear\" part) is no longer emphasized."],[379.1767708333334,"head_four is hidden from the screen — left the board."],[379.1767708333334,"obstacles is hidden from the screen — left the board."]]}]},{"title":"Inside the Tool","start":380.21843750000005,"end":625.5370416666667,"objects":{"air_claim":"a Math [text] that says \"$upright(\"air\") : nabla times arrow(H) = 0 arrow.r arrow(H) = - nabla phi$\"","dep":"a Math [text] that says \"$rho = rho(arrow(B), T, arrow(J)), quad kappa = kappa(T)$\"","fields":"a Derivation [text] that says \"$frac(partial, partial t) (mu arrow(H)) + nabla times (rho nabla times arrow(H)) &= 0 \\ arrow(J) &= nabla times arrow(H) \\ C_V frac(partial T, partial t) - nabla dot (kappa nabla T) &= rho arrow(J) dot arrow(J)$\"","head_fields":"a Heading that says \"Two Fields, Solved Together\"","head_homog":"a Heading that says \"One Material Where There Were Four\"","head_shell":"a Heading that says \"The Contact Layer Becomes a Surface\"","head_under":"a Heading that says \"What Runs Underneath\"","iter_note":"a Block [text] that says \"Picard, or fixed point, iteration for most properties. Newton-Raphson for the conductor resistivity, which depends on the current density it carries.\"","mesh_cells":"a PolarGrid [gray] drawn in mesh_view (r_range=(0.55, 1.0), r_cells=3, theta_cells=28)","mesh_inner":"a Circle [blue] drawn in mesh_view (radius=0.55)","mesh_outer":"a Circle [blue] drawn in mesh_view","mesh_view":"a Figure (x_range=(-1.2, 1.2), y_range=(-1.2, 1.2), aspect=(1.0, 1.0))","s_cross1":"an Arrow [green] drawn in shell (start=(20.0, 14.0), end=(20.0, 22.0))","s_cross2":"an Arrow [green] drawn in shell (start=(40.0, 14.0), end=(40.0, 22.0))","s_face":"a Line [red] drawn in shell (start=(3.0, 18.0), end=(57.0, 18.0))","s_lower":"a Polygon [blue] drawn in shell (vertices=((3.0, 2.0), (57.0, 2.0), (57.0, 17.0), (3.0, 17.0)), fill_opacity=0.25)","s_upper":"a Polygon [blue] drawn in shell (vertices=((3.0, 19.0), (57.0, 19.0), (57.0, 34.0), (3.0, 34.0)), fill_opacity=0.25)","sections":"a Block [text] that says \"geometry mesh solve postprocess\"","shell":"a Figure (x_range=(0.0, 60.0), y_range=(-6.0, 36.0), aspect=(60.0, 42.0))","shell_label":"a Tex [text] that says \"As a surface: jump conditions across it\"","slab":"a Solid [blue] drawn in tape_frame (upper=<function>, x_range=(0.0, 4.0), y_range=(0.0, 1.5))","tape_frame":"an Axes3D (x_range=(-0.4, 4.4), y_range=(-0.4, 1.9), z_range=(-0.2, 1.4))","tensor":"a Math [text] that says \"$rho = mat(rho_l, 0, 0; 0, rho_w, 0; 0, 0, rho_t)$\"","tools":"a Block [text] that says \"Gmsh generates the geometry and the mesh. GetDP assembles and solves the finite element problem. The STEAM material library supplies the material properties.\"","v_cut1":"a Line [red] drawn in vol (start=(17.0, 14.0), end=(17.0, 22.0))","v_cut2":"a Line [red] drawn in vol (start=(30.0, 14.0), end=(30.0, 22.0))","v_cut3":"a Line [red] drawn in vol (start=(43.0, 14.0), end=(43.0, 22.0))","v_gap":"a Polygon [red] drawn in vol (vertices=((3.0, 14.0), (57.0, 14.0), (57.0, 22.0), (3.0, 22.0)), fill_opacity=0.2)","v_len":"a Vector [green] labelled \"rho_l\" drawn in tape_frame (start=(1.0, 0.75, 0.12), end=(3.2, 0.75, 0.12))","v_lower":"a Polygon [blue] drawn in vol (vertices=((3.0, 2.0), (57.0, 2.0), (57.0, 14.0), (3.0, 14.0)), fill_opacity=0.25)","v_thk":"a Vector [red] labelled \"rho_t\" drawn in tape_frame (start=(3.6, 0.75, 0.12), end=(3.6, 0.75, 0.95))","v_upper":"a Polygon [blue] drawn in vol (vertices=((3.0, 22.0), (57.0, 22.0), (57.0, 34.0), (3.0, 34.0)), fill_opacity=0.25)","v_wid":"a Vector [yellow] labelled \"rho_w\" drawn in tape_frame (start=(2.0, 0.2, 0.12), end=(2.0, 1.35, 0.12))","vol":"a Figure (x_range=(0.0, 60.0), y_range=(-6.0, 36.0), aspect=(60.0, 42.0))","vol_label":"a Tex [text] that says \"As a volume: real elements inside the gap\""},"beats":[{"start":380.21843750000005,"say":"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.","live":[],"does":[[380.21843750000005,"head_under is shown on the screen, written out."],[380.21843750000005,"mesh_view is shown on the screen, written out."],[380.21843750000005,"mesh_outer is shown on the screen, written out."],[380.21843750000005,"mesh_inner is shown on the screen, written out."],[385.30343750000003,"mesh_view moves to a new place on the board."],[385.30343750000003,"sections is shown on the screen, written out."],[395.27643750000004,"mesh_cells is shown on the screen, written out."]]},{"start":398.70893750000005,"say":"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.","live":["sections","mesh_view","head_under","mesh_outer","mesh_inner","mesh_cells"],"does":[[398.70893750000005,"tools is shown on the screen, written out."],[402.8184375000001,"tools (the \"Gmsh\" part) is emphasized."],[408.61243750000006,"tools (the \"GetDP\" part) is emphasized."],[408.61243750000006,"tools (the \"Gmsh\" part) is no longer emphasized."],[412.8614375000001,"tools (the \"GetDP\" part) is no longer emphasized."],[412.8614375000001,"tools (the \"STEAM material library\" part) is emphasized."],[418.7944375000001,"tools (the \"STEAM material library\" part) is no longer emphasized."],[420.54743750000006,"head_under is hidden from the screen — left the board."],[420.54743750000006,"mesh_view is hidden from the screen — left the board."],[420.54743750000006,"mesh_outer is hidden from the screen — mesh_view left the board."],[420.54743750000006,"mesh_inner is hidden from the screen — mesh_view left the board."],[420.54743750000006,"mesh_cells is hidden from the screen — mesh_view left the board."],[420.54743750000006,"sections is hidden from the screen — left the board."],[420.54743750000006,"tools is hidden from the screen — left the board."]]},{"start":421.1474375,"say":"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.","live":[],"does":[[421.1474375,"head_fields is shown on the screen, written out."],[426.34843750000005,"fields is shown on the screen, written out."]]},{"start":439.3134375000001,"say":"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.","live":["head_fields"],"does":[[441.02043750000007,"fields is shown on the screen, written out."],[451.01643750000005,"fields is shown on the screen, written out."]]},{"start":456.44643750000006,"say":"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.","live":null,"does":[[463.65543750000006,"air_claim is shown on the screen, written out."]]},{"start":485.30493750000005,"say":"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.","live":["air_claim","head_fields"],"does":[[495.88143750000006,"dep is shown on the screen, written out."],[501.87193750000006,"air_claim is hidden from the screen — left the board."],[501.87193750000006,"dep is hidden from the screen — left the board."],[501.87193750000006,"fields is hidden from the screen — left the board."],[501.87193750000006,"head_fields is hidden from the screen — left the board."]]},{"start":502.4719375000001,"say":"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.","live":[],"does":[[502.4719375000001,"head_shell is shown on the screen, written out."],[502.4719375000001,"vol is shown on the screen, written out."],[502.4719375000001,"v_lower is shown on the screen, written out."],[502.96705754662014,"v_upper is shown on the screen, written out."],[506.5704375000001,"vol moves to a new place on the board."],[506.5704375000001,"vol_label is shown on the screen, written out."],[508.8804375000001,"v_gap is shown on the screen, written out."],[511.7604375000001,"v_cut1 is shown on the screen, written out."],[511.89350443306705,"v_cut2 is shown on the screen, written out."],[512.1567708333334,"v_cut3 is shown on the screen, written out."]]},{"start":523.0989375,"say":"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.","live":["vol_label","vol","head_shell","v_lower","v_upper","v_gap","v_cut1","v_cut2","v_cut3"],"does":[[523.0989375,"shell is shown on the screen, written out."],[523.0989375,"s_lower is shown on the screen, written out."],[523.5676171945194,"s_upper is shown on the screen, written out."],[523.7834375,"shell_label is shown on the screen, written out."],[528.3004375,"s_face is shown on the screen, drawn."],[534.7084375000001,"s_cross1 is shown on the screen, written out."],[534.8858730025029,"s_cross2 is shown on the screen, written out."]]},{"start":539.4074375,"say":"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.","live":["vol_label","vol","shell_label","shell","head_shell","v_lower","v_upper","v_gap","v_cut1","v_cut2","v_cut3","s_lower","s_upper","s_face","s_cross1","s_cross2"],"does":[[544.2954375,"s_cross1 is indicated — a transient flash."],[544.5127875000001,"s_cross2 is indicated — a transient flash."],[556.9499375,"head_shell is hidden from the screen — left the board."],[556.9499375,"shell is hidden from the screen — left the board."],[556.9499375,"s_lower is hidden from the screen — shell left the board."],[556.9499375,"s_upper is hidden from the screen — shell left the board."],[556.9499375,"s_face is hidden from the screen — shell left the board."],[556.9499375,"s_cross1 is hidden from the screen — shell left the board."],[556.9499375,"s_cross2 is hidden from the screen — shell left the board."],[556.9499375,"shell_label is hidden from the screen — left the board."],[556.9499375,"vol is hidden from the screen — left the board."],[556.9499375,"v_lower is hidden from the screen — vol left the board."],[556.9499375,"v_upper is hidden from the screen — vol left the board."],[556.9499375,"v_gap is hidden from the screen — vol left the board."],[556.9499375,"v_cut1 is hidden from the screen — vol left the board."],[556.9499375,"v_cut2 is hidden from the screen — vol left the board."],[556.9499375,"v_cut3 is hidden from the screen — vol left the board."],[556.9499375,"vol_label is hidden from the screen — left the board."]]},{"start":557.5499375,"say":"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.","live":[],"does":[[557.5499375,"head_homog is shown on the screen, written out."],[557.5499375,"tape_frame is shown on the screen, written out."],[557.5499375,"slab is shown on the screen, written out."],[568.3704375000001,"v_len is shown on the screen, written out."],[572.4334375000001,"v_thk is shown on the screen, written out."],[580.0154375000001,"tape_frame turns to a new orientation in its own slot."],[580.0154375000001,"v_wid is shown on the screen, written out."]]},{"start":582.4844375,"say":"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.","live":["tape_frame","head_homog","slab","v_len","v_thk","v_wid"],"does":[[583.8544375,"tape_frame moves to a new place on the board."],[583.8544375,"tensor is shown on the screen, written out."],[586.8494375,"tensor (the \"rho_l\" part) is emphasized."],[588.5214375,"tensor (the \"rho_l\" part) is no longer emphasized."],[588.5214375,"tensor (the \"rho_w\" part) is emphasized."],[590.1234375,"tensor (the \"rho_t\" part) is emphasized."],[590.1234375,"tensor (the \"rho_w\" part) is no longer emphasized."],[594.1754375,"tensor (the \"rho_t\" part) is no longer emphasized."]]},{"start":602.3799375000001,"say":"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.","live":["tensor","tape_frame","head_homog","slab","v_len","v_thk","v_wid"],"does":[[606.0254375000001,"iter_note is shown on the screen, written out."],[612.7124375000001,"iter_note (the \"Picard\" part) is emphasized."],[616.9044375000001,"iter_note (the \"Newton-Raphson\" part) is emphasized."],[616.9044375000001,"iter_note (the \"Picard\" part) is no longer emphasized."],[622.0354375000001,"iter_note (the \"Newton-Raphson\" part) is no longer emphasized."],[624.4953750000001,"head_homog is hidden from the screen — left the board."],[624.4953750000001,"iter_note is hidden from the screen — left the board."],[624.4953750000001,"tape_frame is hidden from the screen — left the board."],[624.4953750000001,"slab is hidden from the screen — tape_frame left the board."],[624.4953750000001,"v_len is hidden from the screen — tape_frame left the board."],[624.4953750000001,"v_thk is hidden from the screen — tape_frame left the board."],[624.4953750000001,"v_wid is hidden from the screen — tape_frame left the board."],[624.4953750000001,"tensor is hidden from the screen — left the board."]]}]},{"title":"One Full Transient","start":625.5370416666667,"end":930.4066666666668,"objects":{"amps":"a VariableNumber (format_spec='.0f')","clock":"a VariableNumber (format_spec='.0f')","coil_map":"a Figure (x_range=(-1.35, 1.35), y_range=(-1.35, 1.35), aspect=(1.0, 1.0))","current_label":"a Tex [gray] that says \"Grey: the supply current, scaled.\"","decay_axes":"an Axes (x_range=(80.0, 100.0), y_range=(0.0, 40.0), x_ticks_every=5.0)","decay_curve":"a FunctionPlot [green] drawn in decay_axes (function=<function>, x_range=(80.0, 100.0))","decay_label":"a Math [text] that says \"$B_(z,c) approx 30 thin upright(\"mT\"), quad tau approx 27 thin upright(\"s\")$\"","field_axes":"an Axes (x_range=(0.0, 100.0), y_range=(0.0, 0.8), x_ticks_every=20.0)","field_curve":"a FunctionPlot [green] drawn in field_axes (function=<function>, x_range=(0.0, 100.0))","field_label":"a Tex [green] that says \"Green: the axial field at the centre.\"","head_case":"a Heading that says \"The Test Case\"","head_field":"a Heading that says \"The Field Does Not Follow the Current\"","head_hot":"a Heading that says \"The Hot Spot\"","head_screen":"a Heading that says \"What Is Left When the Supply Is Off\"","head_volt":"a Heading that says \"What the Terminals Report\"","hot_axes":"an Axes (x_range=(0.0, 100.0), y_range=(3.8, 5.8), x_ticks_every=20.0)","hot_curve":"a FunctionPlot [red] drawn in hot_axes (function=<function>, x_range=(0.0, 100.0))","j_label":"a Tex [text] that says \"Trapped current, with the supply off\"","loop_bore":"a Circle [gray] drawn in loops (radius=0.45)","loop_edge":"a Circle [gray] drawn in loops","loop_inner":"a Circle [red] drawn in loops (radius=0.6)","loop_inner_way":"an Orientation [red] drawn in loops (path=((0.6, 0.0), (0.5971108360033182, 0.05881028419773636), (0.5884…, closed=True, arrows=4)","loop_outer":"a Circle [red] drawn in loops (radius=0.85)","loop_outer_way":"an Orientation [red] drawn in loops (path=((0.85, 0.0), (0.8459070176713673, 0.08331456928012651), (0.833…, closed=True, arrows=4)","loops":"a Figure (x_range=(-1.25, 1.25), y_range=(-1.25, 1.25), aspect=(1.0, 1.0))","m_lag":"a Text [text] that says \"The field lags: coil inductance, and current crossing the contact layers before it settles into the spiral.\"","m_left":"a Math [text] that says \"$I = 0, quad B_(z,c) eq.not 0$\"","map_bore":"a Circle [gray] drawn in coil_map (radius=0.45)","map_defect":"a ParametricCurve [yellow] drawn in coil_map (function=<function>, t_range=(0.4, 0.9))","map_defect_label":"a Point [yellow] labelled \"upright(\"defect\")\" drawn in coil_map (location=(0.55, 0.55), show_marker=False)","map_edge":"a Circle [gray] drawn in coil_map (radius=1.05)","map_joint":"a Circle [red] drawn in coil_map (radius=0.5)","map_joint_label":"a Point [red] labelled \"upright(\"inner joint\")\" drawn in coil_map (location=(0.0, -0.5), show_marker=False)","params":"a Table [text] that says \"Quantity Value Coils 2 pancakes, 40 turns each Tape $4 thin upright(\"mm\") times 120 thin mu upright(\"m\")$ Contact layer $10 thin mu upright(\"m\")$ stainless steel Critical current $770 thin upright(\"A\")$ at $4 thin upright(\"K\")$ Cooling ter…\" (rows=(('Quantity', 'Value'), ('Coils', '2 pancakes, 40 turns each'),…, header=True)","point":"a Point [yellow] drawn in supply (location=(60.0, 130.0))","point_2":"a Point [yellow] drawn in field_axes (location=(60.0, 0.7))","point_3":"a Point [yellow] drawn in field_axes (location=(30.0, 0.403))","point_4":"a Point [yellow] drawn in hot_axes (location=(60.6, 5.4))","point_5":"a Point [yellow] drawn in volt_axes (location=(60.2, 3.9))","point_6":"a Point [yellow] drawn in volt_axes (location=(80.0, -0.5))","ramp":"a FunctionPlot [blue] drawn in supply (function=<function>, x_range=(0.0, 100.0))","ref_curve":"a FunctionPlot [gray] drawn in field_axes (function=<function>, x_range=(0.0, 100.0))","rider":"a PlotPoint [yellow] labelled \"(0, 0)\" drawn in supply (target='ramp', x=<VariableNumber clock = 100.0>)","supply":"an Axes (x_range=(0.0, 100.0), y_range=(0.0, 150.0), x_ticks_every=20.0)","temp_label":"a Math [text] that says \"$T : 4 arrow.r 5.06 arrow.r 5.4 thin upright(\"K\")$\"","tmap_label":"a Tex [text] that says \"Where the heat is at 60.03 s\"","v_down":"a Math [text] that says \"$upright(\"ramp down\") : V < 0$\"","v_plateau":"a Math [text] that says \"$upright(\"plateau\") : V approx 3.5 thin upright(\"mV\")$\"","v_step":"a Math [text] that says \"$upright(\"defect\") : V approx 3.9 thin upright(\"mV\")$\"","volt_axes":"an Axes (x_range=(0.0, 100.0), y_range=(-1.0, 4.6), x_ticks_every=20.0)","volt_curve":"a FunctionPlot [blue] drawn in volt_axes (function=<function>, x_range=(0.0, 100.0))"},"beats":[{"start":625.5370416666667,"say":"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.","live":[],"does":[[625.5370416666667,"head_case is shown on the screen, written out."],[625.5370416666667,"supply is shown on the screen, written out."],[625.5370416666667,"params is shown on the screen, written out."],[631.0050416666667,"params is shown on the screen, written out."],[632.8980416666667,"params is shown on the screen, written out."],[634.5930416666667,"params is shown on the screen, written out."],[642.3020416666667,"params is shown on the screen, written out."]]},{"start":644.9445416666667,"say":"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.","live":["supply","head_case"],"does":[[648.1840416666668,"ramp is shown on the screen, drawn."],[650.8080416666667,"rider is shown on the screen, written out."]]},{"start":657.2245416666667,"say":"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.","live":["supply","head_case","ramp","rider"],"does":[[658.6290416666667,"params is shown on the screen, written out."],[672.9790416666667,"rider is redrawn as the numbers it depends on change."],[672.9790416666667,"clock ticks to 50.0."],[672.9790416666667,"amps ticks to 130.0."]]},{"start":677.6945416666667,"say":"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.","live":null,"does":[[678.7110416666667,"rider is redrawn as the numbers it depends on change."],[678.7110416666667,"clock ticks to 60.0."],[685.3280416666666,"point is shown on the screen, grown."],[688.4860416666667,"params is shown on the screen, written out."],[688.5126852714543,"point is hidden from the screen."]]},{"start":696.1570416666667,"say":"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.","live":null,"does":[[698.3620416666668,"rider is redrawn as the numbers it depends on change."],[698.3620416666668,"clock ticks to 80.0."],[698.3620416666668,"amps ticks to 0.0."],[703.4700416666667,"rider is redrawn as the numbers it depends on change."],[703.4700416666667,"clock ticks to 100.0."],[705.4440416666667,"rider is hidden from the screen."],[706.4310416666667,"head_case is hidden from the screen — left the board."],[706.4310416666667,"params is hidden from the screen — left the board."],[706.4310416666667,"supply is hidden from the screen — left the board."],[706.4310416666667,"ramp is hidden from the screen — supply left the board."]]},{"start":707.6310416666668,"say":"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.","live":[],"does":[[707.6310416666668,"head_field is shown on the screen, written out."],[707.6310416666668,"field_axes is shown on the screen, written out."],[711.7640416666667,"field_axes moves to a new place on the board."],[711.7640416666667,"ref_curve is shown on the screen, drawn."],[711.7640416666667,"current_label is shown on the screen, written out."],[715.4900416666667,"field_curve is shown on the screen, drawn."],[715.4900416666667,"field_label is shown on the screen, written out."],[719.6000416666667,"point_2 is shown on the screen, grown."],[721.6806497997536,"point_2 is hidden from the screen."]]},{"start":723.2305416666667,"say":"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.","live":["current_label","field_label","field_axes","head_field","ref_curve","field_curve"],"does":[[724.1590416666667,"point_3 is shown on the screen, grown."],[725.8440060187233,"point_3 is hidden from the screen."],[726.8760416666668,"m_lag is shown on the screen, written out."]]},{"start":743.6955416666667,"say":"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.","live":["current_label","field_label","m_lag","field_axes","head_field","ref_curve","field_curve"],"does":[[749.7320416666666,"m_left is shown on the screen, written out."],[751.9500416666667,"A box is drawn around m_left."],[756.1295416666667,"current_label is hidden from the screen — left the board."],[756.1295416666667,"field_axes is hidden from the screen — left the board."],[756.1295416666667,"ref_curve is hidden from the screen — field_axes left the board."],[756.1295416666667,"field_curve is hidden from the screen — field_axes left the board."],[756.1295416666667,"field_label is hidden from the screen — left the board."],[756.1295416666667,"head_field is hidden from the screen — left the board."],[756.1295416666667,"m_lag is hidden from the screen — left the board."],[756.1295416666667,"m_left is hidden from the screen — left the board."]]},{"start":757.3295416666667,"say":"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.","live":[],"does":[[757.3295416666667,"head_screen is shown on the screen, written out."],[757.3295416666667,"loops is shown on the screen, written out."],[757.3295416666667,"loop_edge is shown on the screen, written out."],[757.3295416666667,"loop_bore is shown on the screen, written out."],[760.3950416666667,"loops moves to a new place on the board."],[760.3950416666667,"j_label is shown on the screen, written out."],[767.2910416666666,"loop_outer is shown on the screen, drawn."],[767.551548349912,"loop_outer_way is shown on the screen, written out."],[768.0460416666667,"loop_inner is shown on the screen, drawn."],[768.5198992980003,"loop_inner_way is shown on the screen, written out."]]},{"start":771.0140416666667,"say":"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.","live":["j_label","loops","head_screen","loop_edge","loop_bore","loop_outer","loop_outer_way","loop_inner","loop_inner_way"],"does":[]},{"start":785.1510416666667,"say":"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.","live":null,"does":[[785.9640416666667,"decay_axes is shown on the screen, written out."],[788.2280416666667,"decay_curve is shown on the screen, drawn."],[789.9340416666666,"decay_label is shown on the screen, written out."],[796.9000416666668,"decay_label (the \"tau approx 27 thin upright(\"s\")\" part) is emphasized."],[799.1410416666668,"decay_label (the \"tau approx 27 thin upright(\"s\")\" part) is no longer emphasized."],[804.2030416666666,"decay_axes is hidden from the screen — left the board."],[804.2030416666666,"decay_curve is hidden from the screen — decay_axes left the board."],[804.2030416666666,"decay_label is hidden from the screen — left the board."],[804.2030416666666,"head_screen is hidden from the screen — left the board."],[804.2030416666666,"j_label is hidden from the screen — left the board."],[804.2030416666666,"loops is hidden from the screen — left the board."],[804.2030416666666,"loop_edge is hidden from the screen — loops left the board."],[804.2030416666666,"loop_bore is hidden from the screen — loops left the board."],[804.2030416666666,"loop_outer is hidden from the screen — loops left the board."],[804.2030416666666,"loop_outer_way is hidden from the screen — loops left the board."],[804.2030416666666,"loop_inner is hidden from the screen — loops left the board."],[804.2030416666666,"loop_inner_way is hidden from the screen — loops left the board."]]},{"start":804.8030416666667,"say":"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.","live":[],"does":[[804.8030416666667,"head_hot is shown on the screen, written out."],[804.8030416666667,"hot_axes is shown on the screen, written out."],[805.6040416666667,"hot_curve is shown on the screen, drawn."],[811.1420416666667,"hot_axes moves to a new place on the board."],[811.1420416666667,"temp_label is shown on the screen, written out."]]},{"start":819.5900416666667,"say":"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.","live":["temp_label","hot_axes","head_hot","hot_curve"],"does":[[821.1920416666667,"point_4 is shown on the screen, grown."],[823.3970232998263,"point_4 is hidden from the screen."]]},{"start":837.8370416666667,"say":"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.","live":null,"does":[[838.4410416666667,"coil_map is shown on the screen, written out."],[838.4410416666667,"map_edge is shown on the screen, written out."],[838.4410416666667,"map_bore is shown on the screen, written out."],[840.7980416666667,"tmap_label is shown on the screen, written out."],[842.2140416666666,"map_defect is shown on the screen, drawn."],[842.5230642230576,"map_defect_label is shown on the screen, written out."],[847.8450416666666,"map_joint is shown on the screen, drawn."],[848.0649799693612,"map_joint_label is shown on the screen, written out."]]},{"start":855.0045416666667,"say":"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.","live":["temp_label","hot_axes","tmap_label","coil_map","head_hot","hot_curve","map_edge","map_bore","map_defect","map_defect_label","map_joint","map_joint_label"],"does":[[872.3615416666667,"coil_map is hidden from the screen — left the board."],[872.3615416666667,"map_edge is hidden from the screen — coil_map left the board."],[872.3615416666667,"map_bore is hidden from the screen — coil_map left the board."],[872.3615416666667,"map_defect is hidden from the screen — coil_map left the board."],[872.3615416666667,"map_defect_label is hidden from the screen — coil_map left the board."],[872.3615416666667,"map_joint is hidden from the screen — coil_map left the board."],[872.3615416666667,"map_joint_label is hidden from the screen — coil_map left the board."],[872.3615416666667,"head_hot is hidden from the screen — left the board."],[872.3615416666667,"hot_axes is hidden from the screen — left the board."],[872.3615416666667,"hot_curve is hidden from the screen — hot_axes left the board."],[872.3615416666667,"temp_label is hidden from the screen — left the board."],[872.3615416666667,"tmap_label is hidden from the screen — left the board."]]},{"start":873.5615416666667,"say":"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.","live":[],"does":[[873.5615416666667,"head_volt is shown on the screen, written out."],[873.5615416666667,"volt_axes is shown on the screen, written out."],[876.9160416666666,"volt_curve is shown on the screen, drawn."],[883.9400416666666,"volt_axes moves to a new place on the board."],[883.9400416666666,"v_plateau is shown on the screen, written out."]]},{"start":887.0015416666666,"say":"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.","live":["v_plateau","volt_axes","head_volt","volt_curve"],"does":[[900.1680416666666,"v_plateau is indicated — a transient flash."]]},{"start":901.6735416666667,"say":"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.","live":null,"does":[[903.5310416666666,"point_5 is shown on the screen, grown."],[904.8200416666666,"v_step is shown on the screen, written out."],[905.663585160556,"point_5 is hidden from the screen."]]},{"start":914.0105416666667,"say":"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.","live":["v_plateau","v_step","volt_axes","head_volt","volt_curve"],"does":[[915.0320416666666,"v_down is shown on the screen, written out."],[915.2528445395665,"point_6 is shown on the screen, grown."],[917.1363387589549,"point_6 is hidden from the screen."],[929.365,"head_volt is hidden from the screen — left the board."],[929.365,"v_down is hidden from the screen — left the board."],[929.365,"v_plateau is hidden from the screen — left the board."],[929.365,"v_step is hidden from the screen — left the board."],[929.365,"volt_axes is hidden from the screen — left the board."],[929.365,"volt_curve is hidden from the screen — volt_axes left the board."]]}]},{"title":"Sweeping the Defect","start":930.4066666666668,"end":1040.5701666666669,"objects":{"head_sweep":"a Heading that says \"One Parameter, Ten Runs\"","head_take":"a Heading that says \"What the Tool Gives You\"","pancake_1":"a Cylinder [blue] drawn in stack_frame (start=(0.0, 0.0, 0.1), end=(0.0, 0.0, 0.6), opacity=0.25)","pancake_2":"a Cylinder [blue] drawn in stack_frame (start=(0.0, 0.0, 0.7), end=(0.0, 0.0, 1.2), opacity=0.25)","pancake_3":"a Cylinder [blue] drawn in stack_frame (start=(0.0, 0.0, 1.3), end=(0.0, 0.0, 1.8), opacity=0.25)","pancake_4":"a Cylinder [blue] drawn in stack_frame (start=(0.0, 0.0, 1.9), end=(0.0, 0.0, 2.4), opacity=0.25)","stack_frame":"an Axes3D (x_range=(-1.4, 1.4), y_range=(-1.4, 1.4), z_range=(-0.2, 2.6))","sweep_left":"a Math [text] that says \"$V(t), quad l = 0.1 arrow.r 1.0 upright(\"turn\")$\"","sweep_right":"a Math [text] that says \"$T(t), quad l = 0.1 arrow.r 1.0 upright(\"turn\")$\"","t_long":"a FunctionPlot [yellow] drawn in temp_axes (function=<function>, x_range=(58.0, 82.0))","t_mid":"a FunctionPlot [green] drawn in temp_axes (function=<function>, x_range=(58.0, 82.0))","t_short":"a FunctionPlot [blue] drawn in temp_axes (function=<function>, x_range=(58.0, 82.0))","takeaways":"a Block [text] that says \"Every turn resolved in 3D, with contact layers carried as surfaces. Magnetodynamics and heat solved together, with properties that follow temperature and field. The whole model in one text file, so a study is a loop over that file. Open so…\"","temp_axes":"an Axes (x_range=(58.0, 82.0), y_range=(3.9, 5.8), x_ticks_every=5.0)","v_long":"a FunctionPlot [yellow] drawn in volt_axes (function=<function>, x_range=(58.0, 82.0))","v_mid":"a FunctionPlot [green] drawn in volt_axes (function=<function>, x_range=(58.0, 82.0))","v_short":"a FunctionPlot [blue] drawn in volt_axes (function=<function>, x_range=(58.0, 82.0))","volt_axes":"an Axes (x_range=(58.0, 82.0), y_range=(-1.0, 4.6), x_ticks_every=5.0)"},"beats":[{"start":930.4066666666668,"say":"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.","live":[],"does":[[930.4066666666668,"head_sweep is shown on the screen, written out."],[930.4066666666668,"volt_axes is shown on the screen, written out."],[940.1356666666668,"volt_axes moves to a new place on the board."],[940.1356666666668,"sweep_left is shown on the screen, written out."]]},{"start":942.8951666666668,"say":"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.","live":["sweep_left","volt_axes","head_sweep"],"does":[[953.5656666666667,"v_short is shown on the screen, drawn."],[953.8922880844647,"v_mid is shown on the screen, drawn."],[955.5156666666668,"v_long is shown on the screen, drawn."]]},{"start":957.0091666666667,"say":"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.","live":["sweep_left","volt_axes","head_sweep","v_short","v_mid","v_long"],"does":[[957.6946666666668,"temp_axes is shown on the screen, written out."],[958.3566666666668,"sweep_right is shown on the screen, written out."],[961.0726666666668,"t_short is shown on the screen, drawn."],[961.2797018425462,"t_mid is shown on the screen, drawn."],[961.6629681742045,"t_long is shown on the screen, drawn."]]},{"start":969.4401666666668,"say":"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.","live":["sweep_left","volt_axes","sweep_right","temp_axes","head_sweep","v_short","v_mid","v_long","t_short","t_mid","t_long"],"does":[[983.8366666666667,"head_sweep is hidden from the screen — left the board."],[983.8366666666667,"sweep_left is hidden from the screen — left the board."],[983.8366666666667,"sweep_right is hidden from the screen — left the board."],[983.8366666666667,"temp_axes is hidden from the screen — left the board."],[983.8366666666667,"t_short is hidden from the screen — temp_axes left the board."],[983.8366666666667,"t_mid is hidden from the screen — temp_axes left the board."],[983.8366666666667,"t_long is hidden from the screen — temp_axes left the board."],[983.8366666666667,"volt_axes is hidden from the screen — left the board."],[983.8366666666667,"v_short is hidden from the screen — volt_axes left the board."],[983.8366666666667,"v_mid is hidden from the screen — volt_axes left the board."],[983.8366666666667,"v_long is hidden from the screen — volt_axes left the board."]]},{"start":985.0366666666667,"say":"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.","live":[],"does":[[985.0366666666667,"head_take is shown on the screen, written out."],[987.6606666666668,"takeaways is shown on the screen, written out."],[991.5956666666667,"takeaways (the \"resolved in 3D\" part) is emphasized."],[994.3246666666668,"stack_frame is shown on the screen, written out."],[994.3246666666668,"pancake_1 is shown on the screen, written out."],[994.5046710904668,"pancake_2 is shown on the screen, written out."],[994.8700470249946,"pancake_3 is shown on the screen, written out."],[995.3757721742978,"pancake_4 is shown on the screen, written out."]]},{"start":997.7686666666667,"say":"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.","live":["takeaways","stack_frame","head_take","pancake_1","pancake_2","pancake_3","pancake_4"],"does":[[999.9166666666667,"takeaways (the \"resolved in 3D\" part) is no longer emphasized."],[999.9166666666667,"takeaways (the \"solved together\" part) is emphasized."]]},{"start":1006.9256666666668,"say":"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.","live":null,"does":[[1008.5396666666668,"takeaways (the \"one text file\" part) is emphasized."],[1008.5396666666668,"takeaways (the \"solved together\" part) is no longer emphasized."],[1015.9126666666667,"stack_frame turns to a new orientation in its own slot."]]},{"start":1020.5061666666668,"say":"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.","live":null,"does":[[1021.1566666666668,"takeaways (the \"Open source\" part) is emphasized."],[1021.1566666666668,"takeaways (the \"one text file\" part) is no longer emphasized."],[1029.7126666666668,"takeaways (the \"Open source\" part) is no longer emphasized."],[1039.5285000000001,"head_take is hidden from the screen — left the board."],[1039.5285000000001,"stack_frame is hidden from the screen — left the board."],[1039.5285000000001,"pancake_1 is hidden from the screen — stack_frame left the board."],[1039.5285000000001,"pancake_2 is hidden from the screen — stack_frame left the board."],[1039.5285000000001,"pancake_3 is hidden from the screen — stack_frame left the board."],[1039.5285000000001,"pancake_4 is hidden from the screen — stack_frame left the board."],[1039.5285000000001,"takeaways is hidden from the screen — left the board."]]}]}]},"durationSeconds":1041,"chapters":[{"title":"The Coil and the Quench","startSeconds":0,"narration":"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."},{"title":"Four Obstacles","startSeconds":189.41656250000005,"narration":"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."},{"title":"Inside the Tool","startSeconds":380.21843750000005,"narration":"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."},{"title":"One Full Transient","startSeconds":625.5370416666667,"narration":"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."},{"title":"Sweeping the Defect","startSeconds":930.4066666666668,"narration":"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."}]}}