A. F. Lietzke, Lawrence Berkeley National Laboratory

**Abstract:** Virgin superconducting magnet coils frequently require
several "training" quenches in order to achieve their conductor-limited
field level. In the complicated force-fields of non-solenoidal magnets,
"stick-slip" wire motions are often postulated to explain the temporary,
spontaneous loss of superconducting margin.

A simple mechanical model is presented to infer the parametric
dependence of "stick-slip" induced conductor temperature rises. A
reduction of this temperature rise is expected to reduce the probability
of "stick-slip" induced quenching, and thereby reduce the number of
training quenches that are required for a virgin magnet to achieve its
conductor-limited field.

Several theoretical implications are enumerated for purposes of reducing
"stick-slip temperature spiking. These implications are then applied to
un-potted cosine-theta , and potted race-track coil magnet designs, as a
series of specific engineering design guidelines for building
superconducting magnets that "train" faster.

Introduction

Virgin superconducting accelerator magnets are frequently unable to
reach their conductor limit until after several "training" quenches have
occurred. For some magnets, this may require hundreds of training
quenches (1). Accelerator magnet geometries usually have complex,
non-uniform 3-dimensional (radial, azimuthal, and axial) stress
distributions at the time of their initial excitation. These stress
distributions are often significantly modified in a complex fashion by
the Lorentz stresses on the conductor during magnet excitation. Lossy
straining of the magnet system elements can convert magnetic and
mechanical stored energy into heat. Localized, "stick-slip" heating is a
primary suspect for temperature spiking that is believed to cause
"spontaneous", temporary, localized margin-loss in the high-quality
superconductor that is available today.

Parametric Dependence of Stick-Slip Heating

The mean conductor temperature rise during a frictional, one
dimensional, adiabatic motion can be estimated:

*T~slip~ = F~fric~ \* ~Dslip\ ~*/ *C~heat~ / V~heat\ ~*,

in terms of the mean frictional force (*F~fric~*) in the direction of
the slippage (*D~slip~*), in the presence of the effective local heat
capacitance (*C~heat~*) and the effective volume that is heated
(*V~heat~*).

One can express the mean frictional force:

*F~fric~ = P~load~* \* *A~slip~ \* K~fric~* ,

in terms of the perpendicular loading pressure (*P~load~*), the
effective slippage area (*A~slip~*), and the mean kinetic friction
coefficient (*K~fric~*).

The force balance:

*F~mech~ + F~fric~ + F~Lorentz~ = 0*

between the mechanical restoring force (*F~mech~*), the friction force
(*F~fric~*), and the Lorentz force (*F~Lorentz~*) can be applied
twice: 1) just prior to slippage, and 2) just prior to re-sticking.
Linearization and subtraction, allows an estimate of the slippage
strain:

*D~slip~ = P~load~ \*(K~stick~ - K~slip~)*

*/ \[(dF~mech~/dx + dF~Lorentz~/dx) / A~slip~\]*,

in terms of the loading pressure (*P~load~*), the difference in friction
coefficients (*K~stick~ - K~slip~*), and the stiffness (*dF/dx/A~slip~*)
of the magneto-mechanical system in the slippage direction under
consideration.

The heated volume can be estimated:

*V~heat~ = A~slip~ \* D~heat~* ,

in terms of the effective slippage area (*A~slip~*) and the distance
that the heat must penetrate before reaching the conductor (*D~heat~*).

Inserting these relations into the temperature relation, yields the
parametric dependence of the "stick-slip" temperature rise:

*T~slip~ = (P~load~)\^2*

*\* K~slip~ \* (K~stick~ - K~slip~)*

*/ \[(dF~mech~/dx + dF~Lorentz~/dx) / A~slip~\]*

/ *C~heat~* / *D~heat~*,

in terms of some fundamental mechanical properties.

Implications and Limitations

One can expect sizable variations in this temperature rise, due to
sizable local variations in any of the parameters. It is postulated that
a reduction in the maximum "stick-slip" temperature rise will reduce
"stick-slip"-induced training quenches. Toward this aim, the following
parametric changes are suggested by this model:

1.  Reduce the loading pressure against those surfaces that might slip
    (i.e., those surfaces where the change in the Lorentz-induced force
    can exceed the maximum static force balance). The reduced loading
    helps in two ways: 1) not only will the temperature rise be smaller,
    but 2) much of the low-modulus-associated slippage will occur at a
    lower field, where the temperature margin will be more tolerant.

2.  Reduce the shearing friction coefficient on those surfaces that are
    likely to slip.

3.  Chose a shearing system whose static and kinetic friction
    coefficients are nearly equal.

4.  Increase the stiffness of the coil in the direction of slippage.

5.  Increase the heat capacitance between the slippage region and the
    conductor.

6.  Increase the distance from the slippage area to the conductor.

Note that two parameters have quadratic dependencies that make them
especially influential: 1) the loading pressure (*P~load~*) that is
perpendicular to the permitted slippage, and 2) the shearing friction
slippage coefficient, *K~slip~* (since the difference *\[K~stick~ -
K~slip~\]* tends to be somewhat proportional to *K~slip~*)*.*

Magnet Applications

The Lorentz-induced differential slippage is most likely where the
slippage-inducing force increases during magnet excitation, while the
slippage-restraining friction decreases. In cosine-theta magnets, there
are three distinct areas where this condition may exist:

1.  If a bore-tube is present, its radial loading pressure will reduce
    during magnet excitation, while the azimuthal (slippage-inducing)
    force will grow. The aggregate radial stiffness of the coil and
    radial support system relative to the bore, controls the rate of
    reduction of radial loading, while the azimuthal stiffness of the
    coil (relative to the bore tube controls the azimuthal strain during
    excitation. Removing the bore tube easily eliminates this source of
    "stick-slip" friction; and the associated improvement in training
    has been reported several times (Ref. 1). If a bore tube is
    required, one would be advised to design it so that its friction
    against the coil is minimized (e.g., allowing the coil to
    effortlessly lift off the bore-tube, and/or incorporating an easily
    sheared interface.

2.  If pole-spacers are present, the azimuthal loading on the pole
    reduces during magnet excitation, while the **radial**
    (slippage-inducing) force at the pole increases at a rate that is
    determined by the magnetic design details. Since the slippage is
    **radial**, the **radial** stiffness of the coil with respect to the
    **radial** stiffness of the pole-structure controls the relative
    strain. One is advised to either remove the pole, or lower the pole
    loading as much as practical (4), with easily sheared slip-planes as
    diffusively "far" from the conductor as possible (e.g., as with
    break-away shims that ride with the conductor), and high **radial**
    stiffness coils (relative to the pole).

3.  If a pole is present, the azimuthal loading on the pole reduces
    during magnet excitation, while the **axial** (slippage-inducing)
    tension in the coil will increase (most dramatically in the pole
    turn, and near the end). Since the slippage in this mode is
    **axial**, the relevant stiffness is the **axial** stiffness of the
    coil with respect to the axial stiffness of the pole-structure. One
    is advised to either remove the pole, or lower the pole loading as
    much as practical, and maximize the **axial** stiffness (as in
    stretching the coils before collaring D19A, Ref. 2). Slip-planes are
    not expected to be as effective for this mode, since axial straining
    of a un-potted, braided or Rutherford cable generally causes the
    conducting strands to rub on each other, not just the pole-piece.

In the case of potted, over/under dual-bore racetrack-coil magnets,
Lorentz-induced differential slippage is most likely in similar
locations:

1.  The analog of the cosine-theta bore-tube in this geometry is the
    bore plate. Magnet excitation reduces the horizontal loading
    pressure against the bore-plate, while increasing the vertical
    (slippage-inducing) force. The horizontal stiffness of the coil
    relative to any horizontal stiffening plates, controls the rate of
    reduction of horizontal loading, while the vertical stiffness of the
    coil with respect to the vertical stiffening plates controls the
    vertical strain. Removing the bore plate is the best solution, since
    it eliminates this source of friction. Otherwise, one is advised to
    reduce the horizontal load as much as practical, maximize the
    vertical coil modulus, and provide easily sheared slip-planes
    against a low modulus bore plate (as thermally insulated from the
    conductor as practical).

2.  If a pole is present, magnet excitation reduces the vertical loading
    on the pole, while increasing the **horizontal** (slippage-inducing)
    force. Since the slippage is **horizontal**, the relevant stiffness
    is the **horizontal** stiffness of the coil with respect to the
    **horizontal** stiffness of the pole-structure. One is advised to
    either remove the pole, or lower the pole loading as much as
    practical, provide easy pole slip-planes as diffusively "far" from
    the conductor as practical (e.g., as with break-away shims that ride
    with the conductor), and maintain high **horizontal** coil stiffness
    (relative to the pole).

3.  If a pole is present, magnet excitation reduces the vertical loading
    on the pole, while increasing the **axial** (slippage-inducing)
    tension in the coil (most dramatic in the pole turn, and near the
    end). For this mode, the relevant stiffness is the **axial**
    stiffness of the coil with respect to the **axial** stiffness of the
    pole-structure. One is advised to either remove the pole, or lower
    the pole loading as much as practical, and maximize **axial**
    stiffness. Pole slip-planes may help this mode in a potted coil,
    since the potting compound keeps the strained cable strands from
    rubbing on each other.

The potting operation is expected to produce opposite effects. While the
potting compound increases the coil modulus, it is expected
simultaneously to reduce the effective conductor heat capacitance (since
the heat capacitance of the potting compound at 4 K is considerably
smaller than LHe).

Coil stiffness is best increased by increasing the conductor fraction in
the composite. This can be done by 1) reducing the net insulation
thickness (e.g., thicker, higher-current cables), and 2) increasing the
cable's compaction. Higher cable compaction requires increasing the
strand deformations. This can be done to some degree during the cabling
process. However, Nb~3~Sn internal-tin coils are more safely compacted
by judiciously applying pressure during the soft bronze phase of the
coil-reaction cycle, where a relatively lower pressure is expected to
result in less tin-barrier rupture. Furthermore, pressure during this
phase also safely compacts the insulation.

Conclusions

The parametric dependence of a "stick-slip" conductor temperature rise
is determined by means of a simple model. Several implications are
enumerated that promise to reduce "stick-slip"-triggered training
quenches. These include 1) reducing of the friction forces in those
areas that are most likely to slip during Lorentz excitation, 2)
increasing the coil modulus (to further reduce the "stick-slip" strain
increments), and 3) keeping the slip-planes as thermally "far" from the
conductor as practical. These theoretical implications were applied to
both cosine-theta , and potted race-track coil magnet designs as a
series of engineering design guidelines for building magnets that train
faster.

References

1.  Some magnets "train" very slowly.

2.  Improved training when the bore tube is removed.

3.  Improved training in D19A.

4.  A practical lower loading limit may be imposed by conditions that
    the pole remain in place, or the magnet not detrain while warm.