WEBVTT

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Here is the question
this lecture is built around.

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A coil wound from high temperature
superconducting tape is carrying current

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with no measurable resistance.

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Then one short stretch
of that tape stops superconducting.

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What happens next, and where?

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Start with the object.

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A pancake coil is one long flat tape,

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four millimetres wide and about a tenth of
a millimetre thick, wound as a flat spiral

00:00:26.777 --> 00:00:31.004
in a plane, with one terminal
at the inner end and one at the outer end.

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This drawing has five turns
so that you can count them,

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and the coil in the paper has forty.

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Current enters one terminal,
runs around and around the spiral,

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and leaves by the other.

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In the case study it is one
hundred and thirty amperes,

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with the terminal faces
held at four kelvin.

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The superconductor carries that current
with no measurable resistance, but only

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inside limits: below a critical
temperature, below a critical field,

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and below a critical current.

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Cross any one

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of those limits and a short stretch of
tape turns resistive within milliseconds.

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Current keeps flowing through it,

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so it heats itself, and heat
lowers the critical current further.

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That runaway is what we call a quench.

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Now the design choice this paper is about.

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Wind the turns with no
insulation between them,

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so neighbouring turns touch
through a thin resistive contact layer.

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The current then has an alternative.

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It can cross sideways into the turn beside
it, run along that turn past the trouble,

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and cross back.

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That bypass is what makes a no
insulation coil partly self protecting.

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It is also what makes it hard to model.

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Those currents flow radially,
across contact layers only tens

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of micrometres thick, and every one
of them dissipates heat where it crosses.

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They also press turn against turn,

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which is a mechanical problem
sitting on top of the thermal one.

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So the questions are quantitative.

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How hot does that spot get? How do the
currents redistribute through the winding?

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What does the terminal
voltage do while it happens?

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Answering that needs a three
dimensional, time dependent model

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in which the magnetic problem and
the thermal problem are solved together.

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The paper delivers
that as open source software:

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Pancake3D, a module of the finite
element quench simulator FiQuS,

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built inside CERN's STEAM framework.

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You describe the coil in one text file,
and the tool generates the geometry,

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meshes it, solves the transient,
and plots the results.

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On the left is what that file says:

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two pancakes, a five
millimetre inner radius,

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a hundred and twenty
micrometre tape, twenty turns.

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On the right is the geometry it produced,

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cut open so you can see the grey
windings between the two copper terminals.

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The rest of this lecture
follows that path.

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First, why this simulation is hard.

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Then how the tool is put together.

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Then one full transient,

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with a defect switched on at sixty
seconds, and a sweep over the size of it.

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Obstacle one.

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A quench does not begin everywhere
at once, it begins somewhere.

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In the case study, the damaged
stretch is four tenths of a single turn,

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and here it is in red,
on one winding of the spiral.

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Four tenths of a turn at that radius
is about sixteen millimetres of tape,

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in a spiral almost two metres long.

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A model that averages around
the winding cannot see it,

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and neither can a flat slice, because
the whole point is that current leaves

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that stretch sideways and comes back.

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Every turn has to exist
separately in the mesh.

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Obstacle two is that contact layer.

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The blue blocks are two
neighbouring turns of tape,

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a hundred and twenty
micrometres thick each.

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The red band between them is the
contact layer, about ten micrometres,

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and this drawing
is to scale across the thickness.

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Along the tape, that same
layer runs the whole spiral.

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So one model holds a length of two
metres and a thickness of ten thousandths

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of a millimetre, a ratio
of about two hundred thousand to one.

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Fill that gap with ordinary volume
elements and you get either an unusable

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number of unknowns, or elements
so distorted the answer is worthless.

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Obstacle three.

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The tape is a laminate,
and only a sliver of it superconducts.

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Seventy five micrometres of the
thickness is a nickel alloy substrate,

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forty two is copper, one and a half
is silver, and the superconducting film

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itself is one and a half micrometres
of rare earth barium copper oxide.

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The drawing keeps the relative
thicknesses of the two thick layers,

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and exaggerates the two thin ones
so that they can be seen at all.

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Resolving all four in a three
dimensional mesh is out of the question,

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so the tool replaces the stack
with one equivalent material.

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That is homogenisation,
and it has a consequence.

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Here is why. The superconductor does not
switch from zero resistance to resistive

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at a threshold.

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Its electric field grows as a power
of the current density it carries,

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with an exponent of about
thirty for this material.

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The critical current is simply where
that field reaches a chosen criterion,

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a hundred microvolts per metre.

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Below it the tape looks
perfectly superconducting.

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A few percent above it the field
is already many times larger,

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so the current shares itself between
the film and the metal around it,

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and how it shares depends
on the answer you are solving for.

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And obstacle four is a practical one.

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Getting all of that right means a coupled,
non-linear, three dimensional finite

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element model, which is not what a magnet
engineer wants to spend a year writing.

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So, four obstacles.

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A quench is local, so every
turn has to be in the mesh.

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The contact layers are ten micrometres
thick against a coil of two metres.

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The tape is a laminate
that has to be homogenised.

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And the superconductor is non-linear,
so every time step is an iteration.

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Everything in the next section
is an answer to one of those four.

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The input file has four sections, and they
are exactly the four things the tool does:

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geometry, mesh, solve, and postprocess.

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Here is what the first two produce:

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one winding, seen from above,

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cut into elements around it
and across its thickness.

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Underneath, three open
source pieces do the work.

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Gmsh builds the geometry and the mesh,
through the OpenCASCADE kernel.

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GetDP assembles and solves
the finite element problem.

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And the STEAM material library
supplies properties of copper,

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steel and the superconductor
as functions of temperature and field.

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Now the physics.

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In the conducting parts,

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the magnetic field obeys one equation:
the rate of change of the flux density,

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plus the curl of resistivity times
the curl of the field, is zero.

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That is Faraday and Ampere,
written for the field alone.

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The current density
is the curl of that same field,

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so nothing extra is solved for it.

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And the heat it generates, resistivity
times current density dotted with itself,

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is the source term of the heat equation,

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which is solved on the windings
and the terminals at the same time.

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Outside the conductors no current flows,
so the curl of the field is zero there

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and it can be written
as the gradient of one scalar potential.

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Three unknowns per node become
one across most of the volume,

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and that is why
this is called an H phi formulation.

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The air still has to stop somewhere:

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either you mesh a large enough box,
or a shell transformation maps the rest

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of space onto a thin outer layer
and lets the air mesh stay small.

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And the coupling runs both ways.

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The electrical solution heats the
material, and the properties, resistivity,

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thermal conductivity, heat capacity,

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all follow the temperature, while the
resistivity also follows the local field

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and the current density itself.

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So how do you mesh a gap
ten micrometres wide?

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On the left, the honest way, with the gap
exaggerated so we can see it at all:

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real volume elements inside the layer.

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Each of them is thin and wide,
exactly the shape a finite element hates,

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and there is a layer like it
between every pair of turns.

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On the right,
the approximation the tool uses.

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The layer is collapsed
onto a single surface,

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and its physics is put back
as conditions across that surface:

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a voltage drop
that sets the radial current,

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and a temperature drop
that sets the heat crossing it.

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The surface carries no thickness
of its own, so the mesh stays sane,

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and the radial currents that make
these coils interesting are still there.

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The papers behind this one check the
approximation against fully meshed layers,

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thermally, magnetically,
and coupled, and it holds.

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Back to the laminate.

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Instead of meshing four layers,
the tool computes one equivalent material,

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and that material is not
the same in every direction.

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Along the tape, current
has a superconducting path.

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Through the thickness,
it has to cross substrate and copper.

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So resistivity and thermal
conductivity differ along

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the three directions of the tape.

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That is what this matrix says:

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one resistivity along the length,

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another across the width,
another through the thickness.

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The tool builds it from the layer list and
rotates it into place element by element,

00:09:56.927 --> 00:10:01.931
because in a spiral, along the tape means
a different direction at every point.

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And because resistivity
depends on the current density,

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every time step is a solve inside a solve.

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Most properties are updated by simple
fixed point, or Picard, iteration.

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The conductor resistivity
gets Newton-Raphson, linearised

00:10:18.715 --> 00:10:20.350
with respect to current density,

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which is what makes that power law
converge in a sensible number of steps.

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Here is the case the paper runs.

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Two pancake coils stacked half
a millimetre apart, forty turns each,

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wound from four millimetre wide tape

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with a ten micrometre stainless
steel layer between the turns.

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At four kelvin in zero field,

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the critical current of that tape
is seven hundred and seventy amperes.

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The coil is driven in current mode.

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The supply ramps from zero to one hundred
and thirty amperes over fifty seconds,

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holds there for ten, and then ramps back
down, reaching zero at eighty seconds.

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The terminal faces are held
at four kelvin throughout,

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which is one of three cooling options; the
tool also offers a fully adiabatic model,

00:11:05.828 --> 00:11:09.125
or a cryocooler load
map from a manufacturer.

00:11:09.740 --> 00:11:12.782
Now watch the operating
point walk up the ramp.

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Fifty seconds, one hundred and thirty
amperes, and the ramp is finished.

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Ten seconds of plateau follow,
with the coil fully superconducting

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at about seventeen percent
of its critical current.

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Then, at sixty seconds, the defect.

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Across four tenths of one turn
in the first pancake, the critical

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current is set to zero, as though
that stretch of tape had been damaged.

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Thirty milliseconds later
the supply starts down,

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and it is at zero by eighty seconds.

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The run then continues
to one hundred seconds,

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because the coil has not finished.

00:11:48.107 --> 00:11:51.393
Here is the first result,
on one pair of axes.

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The grey curve is the supply
current, scaled to fit.

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The green curve is the axial
magnetic field at the centre

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of the coil, about zero point
seven tesla at the top of the ramp.

00:12:03.579 --> 00:12:07.945
Look at where the green curve sits
against the grey one. The field lags.

00:12:08.142 --> 00:12:10.929
Part of that is ordinary inductance.

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Part of it belongs to this design:

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current entering the winding
does not immediately follow the spiral,

00:12:17.337 --> 00:12:20.064
some of it crosses
the contact layers instead,

00:12:20.147 --> 00:12:23.247
and the field only builds
as that sharing settles.

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Now the end of the run.

00:12:26.099 --> 00:12:31.243
The supply current is at zero from eighty
seconds, and the central field is not.

00:12:31.439 --> 00:12:33.388
Something is still circulating in there,

00:12:33.471 --> 00:12:36.281
and it is still
circulating when the run ends.

00:12:37.678 --> 00:12:41.834
On the left, the current density inside
the windings ten seconds after the supply

00:12:41.834 --> 00:12:43.228
was switched off.

00:12:43.425 --> 00:12:47.208
There is no source current at all,
and the coil is still full of current:

00:12:47.291 --> 00:12:50.566
loops of it, trapped,
running around the turns.

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These are screening currents.

00:12:53.336 --> 00:12:57.400
A superconductor opposes a change in the
field through it by setting up currents

00:12:57.400 --> 00:13:01.962
that cancel the change, and once they are
established, with no resistance to speak

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of, they have no reason to stop.

00:13:05.627 --> 00:13:08.031
On the right is what they do to the field.

00:13:08.228 --> 00:13:13.406
Thirty millitesla at the coil centre, and
after a steeper first second the computed

00:13:13.406 --> 00:13:18.619
decay follows this exponential, whose time
constant is near twenty seven seconds.

00:13:18.816 --> 00:13:20.082
For a magnet that is supposed

00:13:20.082 --> 00:13:24.355
to be off, that is a number worth
knowing before the magnet exists.

00:13:25.151 --> 00:13:29.436
Here is the temperature at the middle
of the defect, across the whole run.

00:13:29.633 --> 00:13:34.184
During the ramp it drifts up from four
kelvin to about five, before any defect

00:13:34.184 --> 00:13:39.084
exists, because radial currents and the
terminal joints are already dissipating.

00:13:39.996 --> 00:13:44.014
At sixty seconds it jumps,
to about five point four kelvin.

00:13:44.211 --> 00:13:49.389
That is the defect: the resistivity along
that stretch rises, current redistributes

00:13:49.389 --> 00:13:53.741
across the contact layers, and heat
appears both in the resistive stretch

00:13:53.824 --> 00:13:57.435
and in every layer the current
crosses on its way round it.

00:13:58.139 --> 00:14:00.715
On the right is where
the heat actually is,

00:14:00.798 --> 00:14:03.747
thirty milliseconds
after the defect appears.

00:14:03.944 --> 00:14:07.242
Notice that the hottest
ring is not the defect at all.

00:14:07.346 --> 00:14:10.978
It is the inner terminal,
because that joint has the smaller area,

00:14:11.061 --> 00:14:14.649
so its contact resistance
dissipates more than the outer one.

00:14:15.260 --> 00:14:20.648
And notice the size of the rise:
a fraction of a kelvin, on a coil at four.

00:14:20.844 --> 00:14:25.837
This defect is survivable, which is
what the no insulation design is for.

00:14:26.034 --> 00:14:27.706
A model like this is how you find

00:14:27.706 --> 00:14:32.513
out whether it is still survivable at
higher current, or with a longer defect.

00:14:33.910 --> 00:14:36.659
The last quantity is the one
you can actually measure:

00:14:36.742 --> 00:14:39.237
the voltage between the two terminals.

00:14:39.320 --> 00:14:43.405
It rises during the ramp,
transiently at first and then linearly,

00:14:43.488 --> 00:14:46.553
to about three and a half
millivolts on the plateau.

00:14:47.350 --> 00:14:49.937
That plateau reading
is not the superconductor.

00:14:50.020 --> 00:14:53.156
The winding is still
fully superconducting there;

00:14:53.608 --> 00:14:57.614
what you are measuring is the copper
terminals and their contact layers.

00:14:57.718 --> 00:15:01.225
Any detection threshold
has to sit on top of that offset.

00:15:02.022 --> 00:15:06.991
At sixty seconds the voltage steps up,
to about three point nine millivolts.

00:15:07.188 --> 00:15:08.639
That step is the whole basis

00:15:08.639 --> 00:15:13.504
of quench detection, and it is four tenths
of a millivolt riding on three and a half.

00:15:14.417 --> 00:15:18.330
Then it goes negative on the way down,
because the coil's inductance opposes

00:15:18.330 --> 00:15:22.231
the falling current and the ramp
down is faster than the ramp up.

00:15:22.428 --> 00:15:26.629
So the signal you would trigger on is a
small step on a large, moving background,

00:15:26.712 --> 00:15:29.383
which is exactly why
you want it simulated.

00:15:30.442 --> 00:15:32.773
One simulation is one data point.

00:15:32.857 --> 00:15:36.813
So the last thing the paper does is turn
the length of the defect into a parameter,

00:15:36.897 --> 00:15:38.856
and run the same case ten times,

00:15:38.940 --> 00:15:42.494
with the damage growing
from a tenth of a turn to a full turn.

00:15:43.197 --> 00:15:47.645
On the left, the terminal voltage,
with three of those ten runs drawn.

00:15:47.842 --> 00:15:51.659
Every curve steps up at sixty seconds,
and the step grows with the defect,

00:15:51.743 --> 00:15:55.284
from about three point eight millivolts
for the shortest to about four point two

00:15:55.284 --> 00:15:56.561
for the longest.

00:15:57.358 --> 00:16:00.668
On the right, the temperature
at the middle of the defect.

00:16:00.864 --> 00:16:04.496
The ordering is the same:
a longer normal zone dissipates more,

00:16:04.580 --> 00:16:09.178
and the peak climbs from about five point
two five kelvin to about five point six.

00:16:09.603 --> 00:16:12.146
And look at where the curves
come back together.

00:16:12.598 --> 00:16:15.907
By eighty seconds the voltages
have converged, because by then

00:16:15.907 --> 00:16:20.215
the reading is dominated by the ramp
down and the defect barely enters it.

00:16:20.412 --> 00:16:24.035
The size of the damage shows up
in the peak, not in the tail.

00:16:25.339 --> 00:16:27.348
So what does this give an engineer.

00:16:27.661 --> 00:16:29.875
Every turn resolved in three dimensions,

00:16:29.959 --> 00:16:33.393
with the contact layers carried
as surfaces rather than volumes,

00:16:33.477 --> 00:16:37.321
which is what makes a forty turn
double pancake tractable at all.

00:16:38.117 --> 00:16:40.576
The magnetic and thermal
problems solved together,

00:16:40.660 --> 00:16:44.004
with material properties that follow
the temperature and the field instead

00:16:44.004 --> 00:16:46.478
of being frozen at their nominal values.

00:16:47.274 --> 00:16:49.629
And the entire model in one text file,

00:16:49.713 --> 00:16:53.731
which is what turned that ten run study
into a loop rather than a project.

00:16:53.927 --> 00:16:55.515
This picture, incidentally,

00:16:55.599 --> 00:17:00.058
is a four pancake coil with sixty
turns each, from the same tool.

00:17:00.855 --> 00:17:05.043
It is open source, and the input files
for every run in the paper are published,

00:17:05.127 --> 00:17:08.866
so these results are reproducible
rather than merely reported.

00:17:09.179 --> 00:17:14.485
For anyone designing a no insulation HTS
magnet, that is the point: put a defect

00:17:14.485 --> 00:17:19.501
anywhere, give it any length, and get the
coil's answer before the coil is wound.