Carmelo_Barbagallo_IFCCs_modelling_in_superconducting_filamentary_Nb3Sn_strands_and_magnets_final_version.pptx

Inter-filament coupling currents
modelling in superconducting filamentary
Nb3Sn strands and magnets
Author: Carmelo Barbagallo
Supervisors: Matthias Mentink and Arjan Verweij
TE-MPE-PE, 30th August 2019
Short-Term Internship Programme – March-September 2019
1
Acknowledgments: Lorenzo Bortot, TE-MPE-PE

• Introduction on Inter-Filament Coupling Currents (IFCCs)
• Aim of the study and roadmap
• Proposed analyses:
• Single round strand modelling:
• Study 1: Equivalent Magnetization Formulation;
• Study 2: Equivalent Resistivity Formulation – 1st approach;
• Study 3: Equivalent Resistivity Formulation – 2nd approach.
• Stack of Rutherford cables:
• Homogenized vs Multi-strand model;
• Modified IFCCs equation for the homogenized model.
• Conclusions (1/2) – Single round strand and Rutherford cable modelling
Outline 1/2 – Single strand and Cable stack
2

• Introduction on magneto-thermal analysis of FRESCA2 magnet;
• Aim of the study and roadmap – FRESCA2 magnet;
• Proposed analyses (Energy Extraction and CLIQ simulations):
• Homogenized SIGMA model;
• Homogenized SIGMA restructured model;
• Multi-strand model;
• Conclusions (2/2) – FRESCA2 magnet;
• Overall conclusions.
Outline 2/2 - Magnet
3

Introduction on IFCCs
4
[1] E. Ravaioli, B. Auchmann, G. Chlachidze, M. Maciejewski, G. Sabbi, S. E. Stoynev, and A. Verweij, Modelling of Interfilament
Coupling Currents and Their Effect on Magnet Quench Protection, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY,
VOL. 27, NO. 4, JUNE 2017.
• Superconducting wires are composed by a large number of
filaments embedded in a normal-conducting matrix (usually copper).
• When they are exposed to a time-varying external and transverse
magnetic field
𝑑𝐵a
𝑑𝑡
, a magnetic field 𝐵i is induced, which opposes the
applied magnetic field change. The resulting total magnetic field is
[1]:
𝐵t = 𝐵a − 𝐵i
𝜏IFCC =
𝜇0
2𝜌et
𝑙p
2𝜋
2
where 𝑙p [m] is the filament twist pitch, 𝜌et [Ωm] is the effective transverse resistivity of the
strand matrix and 𝜇0 = 4π10−7
[TmA-1] the magnetic permeability of vacuum.
CERN
• The induced magnetic field 𝐵i is generated by local coupling currents between
superconducting filaments, i.e., Inter-Filaments Coupling Currents (IFCCs), which flow
through the resistive wire matrix generating ohmic losses.
• The Inter-Filaments Coupling Currents (IFCCs) develop with a characteristic time constant:

Introduction on IFCCs
5
[2] M. Wilson, Superconducting magnets. Oxford University Press, 1983.
[3] B. Auchmann, L. Bortot, I. C. Garcia, L. D’Angelo, H. De Gersem, M. Mentink, S. Schöps and A. Verweij, Ad-hoc Homogenisation
for Interfilament Coupling Currents, STEAM collaboration - BMBF project, Internal report, 2019.
𝑀IFCC = −𝜈𝜏IFCC
𝜕𝐵i
𝜕𝑡
• By means of the internal field 𝐵i, we can calculate the magnetization 𝑀IFCC (in Am-1) due
to the IFCCs from direct integration of the coupling current distribution [2]:
• The magnetization is added as an additional source term in the Magnetoquasistatic (MQS)
partial differential equation (PDE), i.e.,
𝛻 × 𝜈𝛻 × 𝐴 + 𝜎
𝜕𝐴
𝜕𝑡
= 𝐽s + 𝛻 × 𝑀IFCC
𝛻 × 𝜈𝛻 × 𝐴 + 𝛻 × 𝜈𝜏IFCC𝛻 ×
𝜕𝐴
𝜕𝑡
= 𝐽s
• This magnetization term is due to small closed current-carrying loops, and the
corresponding magnetizing current 𝐽m = 𝛻 × 𝑀IFCC is divergence-free [3].
where 𝐴 is the magnetic vector potential, 𝜎 the conductivity and 𝐽s the current density
applied to the strand.
where 𝜈 is the reluctivity and 𝜏IFCC the IFCC decay time constant.
• Assuming no further eddy-current effects and considering the relation 𝐵 = 𝛻 × 𝐴, the
following MQS equation is obtained and solved for individual strands by using Finite
Element Method (FEM):

Aim of the study and roadmap
6
• Aim of the study:
1. Understanding of different magnetization formulations;
2. Modelling of IFCCs physics in Rutherford cables and magnets;
3. Show the impact of the homogenization on IFCCs behavior.
MQS equation
FEM simulation
Rutherford cable FRESCA2 magnet
Single round
strand
Cable stacks
Homogenized
model
Multi-strand
model
Homogenized
model
Multi-strand
model

Overview on different formulations to model IFCCs
7
• Equivalent Magnetization 𝑀IFCC approach:
1. M. Wilson, A. Verweij (W-V) formulation [2]
For single strand, multi-strand cable stack, and multi-strand magnet models.
2. M. Wilson, A. Verweij formulation for homogenized models (W-V-homo)
For homogenized cable stack and homogenized magnet models.
3. Modified (W-V-homo) formulation:
For homogenized cable stack and homogenized magnet models.
• Equivalent Resistivity 𝜌eq approach:
1. 1st approach (M. Mentink) (M-M): 𝜎eq =
1
𝜌eq
=
4𝜏IFCC
𝑅𝑠
2𝜇0
2. 2nd approach (C. Barbagallo, L, Bortot) (B-B):
Both for single strand, multi-strand cable stack and multi-strand magnet models.
𝜕𝐵i
𝜕𝑡
𝑀IFCC = −𝑘ht𝜈𝜏IFCC
𝜕𝐵i
𝜕𝑡
𝑑𝑀IFCC
𝑑𝑡
= 𝑘ht
−𝑝
−
1
2𝜏
𝑀IFCC − 𝑘ht
𝑝 𝑑𝐻
𝑑𝑡
𝜌eq = 𝑅if𝑆annulus
(W-V-homo-mod)

Single round strand modelling
8

Single round strand modelling
9
• Strands submitted to an external varying magnetic field (ramping from 0 T at 1 T/s),
obtained by the application of a current density 𝐽ext 𝑡 to the rectangular plates;
• Superconducting filaments are not modelled within the strands (strand cross-section is
considered homogeneous);
• In the center: conductor with IFCC physics (on the right), compared with same conductor
without special physics (on the left), mirrored over the center-line of the magnet
(∆𝐵 = 𝐵right,avg − 𝐵left,avg).
no IFCC IFCC
Model: Matthias Mentink

Study 1: Strand without imposed current 𝐼s - Equivalent
Magnetization Formulation (W-V)
10
• Strand diameter: Dstrand={0.5, 0.7, 0.9} mm;
• Imposed time-constant: 𝜏𝐼𝐹𝐶𝐶=10 ms (for all simulations);
• Free triangular mesh + mesh refinement in strand border
(150k elements, 1 min 27 s of computation time)
• Magnetic dipole field lines (red arrows);
• Changing the strand diameter, ΔB amplitude and the time
constant 𝜏IFCC remain the same (𝝉𝑰𝑭𝑪𝑪=10 ms, ΔBmax=10 mT);
• Single round strand: everything is as expected for this
particular case.

Study 1: Strand without imposed current Is – Governing PDE
11
• Study 1: The governing PDE used in this case is [3]:
where 𝑀IFCC = (𝑀IFCCx
, 𝑀IFCCy
, 0)
𝐼m =
Γ
𝑀IFCCx
𝑡x + 𝑀IFCCy
𝑡y 𝑑Γ = 0
*thanks to Lorenzo Bortot for the idea and help in model implementation.
𝐼m =
Ω
𝛻 × 𝑀IFCC ∙ 𝑑Ω = 0
Stokes’ theorem
Ω
𝛻 × 𝑀IFCC ∙ 𝑑Ω =
Γ
𝑀IFCC ∙ 𝑡Γ𝑑Γ = 0
𝑡Γ = (𝑡x, 𝑡y, 0)
Equation implemented in COMSOL
𝛻 × (𝜈𝛻 × 𝐴) = 𝛻 × 𝑀IFCC
• According to Wilson’s textbook [1], we expect the total current 𝐼m
due to the IFCCs in the strand to be zero*.
Ω
Γ
𝑥
𝑦
𝑧
𝑡
𝐼𝑠 = 0

12
• The total current 𝐼m can be alternatively written as [3]:
𝐼m =
Γ
𝑀IFCC ∙ 𝑡Γ𝑑Γ = −𝜏IFCC
𝑑
𝑑𝑡 Γ
𝜈𝐵 ∙ 𝑡Γ𝑑Γ = 0
• This means that, if we assume that initial condition are such that:
Γ
𝜈𝐵 ∙ 𝑡Γ𝑑Γ = 0
and this condition will remain the same for the other analyzed cases.
𝐼IN 𝐼OUT
𝐵
𝑀
𝐼m = 𝐼IN − 𝐼OUT = 0
𝐼IN
𝐼OUT
𝐼m
• Our FEM analyses show that IFCCs are localized on the strand border and perfectly
balanced as expected.
Study 1: Strand without imposed current Is

Study 1.2: Self-field (strand with imposed current Is and not
subjected to external field)
13
• Study 2: If we consider a strand with an imposed current Is, the governing PDE is [3]:
𝛻 × 𝜈𝐵 = 𝐽s − 𝛻 × 𝜈𝜏IFCC
𝜕𝐵
𝜕𝑡
Ω
Γ
𝑥
𝑦
𝑧
𝑡
𝐼𝑠 ≠ 0
where 𝐽s = χs𝐼s is the impressed current density and χs is a winding function.
• Integrating this expression along the strand’s cross-section Ω and using
Stoke’s theorem, we obtain:
Γ
𝜈𝐵 ∙ 𝑡Γ𝑑Γ = 𝐼s −
Γ
𝜈𝜏IFCC
𝜕𝐵
𝜕𝑡
∙ 𝑡Γ𝑑Γ = 𝐼s − 𝐼m
• This equation shows a paradox [3]. If the magnetization current (third term) should be zero
and if 𝐼s changes in time, the lefthandside term has to change in time as well. But this
implies that the magnetization current cannot remain zero, as assumed in the previous
slide.
• This paradox comes from IFCCs formulation itself, because it is implemented only for the
case in which IFCCs are due to an external varying magnetic field, and not for the strand’s
own field.

14
• From FEM perspective, we investigated a case in which we consider a time-varying current
𝐼s 𝑡 = 𝐼0𝑡 [A], where 𝐼0 = 300 A, applied to a strand in z direction (no external varying magnetic field is
applied);
• The magnetic flux density 𝐵 can be expressed as sum of 𝐵s, generated by the transport current density
𝐽s, and 𝐵i, due to the magnetization current density 𝐽m:
𝐵 = 𝐵s + 𝐵i
• The magnetization current is not zero in this case (i.e., at t=100 ms, 𝐼𝑚 = 0.19𝐼𝑠 A). This contradicts
the divergence-free assumption of the magnetizing current. FE implementation confirms that IFCCs
formulation presented by Wilson [2] cannot be used in the case of strand’s own field, indeed IFCCs are
not balanced in this case.
Field 𝐵 in the strand Magnetization current density
[2] M. Wilson, Superconducting magnets. Oxford University Press, 1983.
𝐵s
𝐵i
𝐵
𝐼𝑠
𝐼𝑚

15
• 𝐵 is the total magnetic field, 𝐵𝑠 is the imposed-current-field contribution and 𝐵i the IFCC-field
contribution.
𝐵
= +
𝐵s 𝐵i
𝐽tot = 𝐽s + 𝐽m
= +
𝐽tot 𝐽s 𝐽m
• 𝑩𝒊 is in the order of mT, so we did not notice so far the inconsistency of the magnetization formulation
for the case 𝐼𝑠 ≠ 0 because in our magnets the produced field is in the order of T.

Study 2: Equivalent Resistivity Formulation (M-M) for a single round
strand without imposed current Is – 1st approach (M. Mentink)
16
• An alternative approach to model IFCCs is the use of an Equivalent Resistivity Formulation. We made
use of COMSOL coil feature, which imposes the following conditions within the strand:
• 𝐼𝑐𝑜𝑖𝑙 = 0 (coil excitation);
• An equivalent electrical conductivity (in S/m) is applied to the conductive layer of
thickness 𝑡ℎ =
𝑅𝑠
2
:
𝜎eq =
1
𝜌eq
=
4𝜏IFCC
𝑅𝑠
2𝜇0
where 𝜌eq is the electrical resistivity and Rs the strand radius.
𝑥
𝑦
𝑧
𝑡ℎ
Γ
• Field amplitude and time
constant are in accordance
with the expectations
( ∆𝐵 = 10 mT and 𝜏IFCC =
10 ms);
• Equivalent Resistivity
Formulation returns results
in accordance with the
Equivalent Magnetization
Formulation for IFCCs
modelling in single strand.

17
𝐵𝑡
𝐵t = 𝐵a + 𝐵if
Total
field
Applied
field
Induced
field
𝑡eq
Study 3: Equivalent Resistivity Formulation (B-B) for a single round strand
without imposed current Is – 2nd approach (C. Barbagallo, L. Bortot)
M-model K-model
• M-model (Equivalent Magnetization model):
𝑀y = −
1
𝜇0
𝜏if
𝑑𝐵a
𝑑𝑡
𝜏if = 𝜇0𝛽if
𝛽if =
𝑙p
2𝜋
2
1
𝜌et
[4] E. Ravaioli, B. Auchmann, M. Maciejewski, H.H.J. ten Kate, A.P. Verweij, “Lumped-Element Dynamic Electro-Thermal model of
a superconducting magnet”, Cryogenics, vol. 80, 346-356, 2016.
• K-model (Current density model - inter-filament dissipative loops [4])
𝜏kif
=
𝐿if
𝑅if
= 𝜇0𝛽if = 𝜏if 𝐿if =
𝜇0𝜋
8
=
𝜇0𝑎𝑠
2𝑑𝑠
2
𝑅if =
𝜋
8
1
𝛽if
𝜌eq = 𝑅if𝑆annulus 𝑆annulus = 𝑆strand − 𝜋
𝑑s
2
− 𝑡eq
2
𝑡eq =
𝑑s
2
1
2𝜋
Equivalent thickness of the strand annulus
• We are supposing that IFCCs flows in the strand annulus of
thickness teq and equivalent resistivity 𝜌eq.
COMSOL coil feature

18
M-model/K-model
Jm is divergence-free
M-model/K-model
without imposed current Is – 2st approach (C. Barbagallo, L. Bortot)
• Magnetic field map is identical for the two
models;
• Consistent variation field and time constant
between the two models;
• Divergence-free assumption of Jm is verified
for both models.

19
• Changing the value of time constant 𝜏if (1 ms, 1s, 1000 s), the two different approaches return
consistent field variation for the three investigated cases;
• Magnetizing current is balanced along the strand border and could be represented as a cosinusoidal
function:
𝐾 = 𝐾 𝜑 = 𝐾0sin(𝜑)
without imposed current Is – 2st approach (C. Barbagallo, L. Bortot)

20
• From FEM perspective, we investigated a case in which we consider a piecewise-current function
𝐼𝑠 𝑡 =
𝐼0𝑡, 0 ≤ 𝑡 ≤ 1
𝐼0, 𝑡 > 1
[A], where 𝐼0 = 300 A, applied to a strand in z direction (no external varying
magnetic field is applied);
• The magnetic flux density 𝐵 can be expressed as sum of 𝐵s (self-field), generated by the transport
current density 𝐽s, and 𝐵i (induced field), due to the magnetization current density 𝐽m:
• The total magnetizing current is zero along the strand border. The divergence-free assumption of the
magnetizing current is verified. This IFCCs formulation could be used to simulate IFCCs also in the case
of strand’s own field, indeed IFCCs result well balanced and, as expected, the decay of IFCCs starts
once current plateau is reached.
K-model
with imposed current Is – 2st approach (C. Barbagallo, L. Bortot)

K-model
21
• For 𝜏if = 10 ms, after 10 ms from time instant (t=1 s) in which current plateau is reached, IFCCs start
decaying, as expected.
• Magnetizing current is balanced along the strand border and is once again representable as
cosinusoidal function:
𝐾 = 𝐾 𝜑 = 𝐾0sin(𝜑)
with imposed current Is – 2st approach (C. Barbagallo, L. Bortot)

22
Summary on IFCCs formulation for single strand model – Is=0
• Equivalent Magnetization formulation (M. Wilson - A. Verweij): 𝑀IFCC = −𝜈𝜏IFCC
𝜕𝐵i
𝜕𝑡
• Equivalent Resistivity formulation (M. Mentink): 𝜎eq =
1
𝜌eq
=
4𝜏IFCC
𝑅𝑠
2
𝜇0
• Equivalent Resistivity formulation (C. Barbagallo - L. Bortot): 𝜌eq = 𝑅if𝑆annulus
(W-V)
Equivalent Resistivity
(M-M or B-B)
 
(W-V)
(B-B)
• For a single strand subjected to an external time-varying
magnetic field and without an imposed current, the three
analyzed formulations return consistent results in terms of
magnetic field variation and IFCC time constant.
(M-M)
Legend:
 Correct x Wrong …To be studied
~ Almost Correct

23
Summary on IFCCs formulation for single strand model – Is ≠ 0
𝜕𝐵i
𝜕𝑡
• Equivalent Resistivity formulation (M. Mentink): 𝜎eq =
1
𝜌eq
=
4𝜏IFCC
𝑅𝑠
2
𝜇0
• Equivalent Resistivity formulation (C. Barbagallo - L. Bortot): 𝜌eq = 𝑅if𝑆annulus
(W-V)
(M-M or B-B)
x 
(W-V)
(M-M)
(B-B)
• For a single strand subjected to an external time-varying
magnetic field and with an imposed current, only
Equivalent-Resistivity-based formulations seem to
reproduce consistent results to simulate IFCCs.
𝐵
Legend:
~ Almost Correct

Rutherford cable modelling
25
Rutherford cable (real image)
How could we model it?
Round strands Octagonal strands [5]
[5] C. Barbagallo, Quench Protection Heaters FE Analysis and Thermal Conductivity Measurements of Epoxy-Impregnated Nb3Sn
Cables, EDMS nr. 2066640 v.1.
CERN
How did we model it for IFCC physics simulations?
Homogenized cable

Study 4: Stack of four Rutherford cables – Homogenized vs Multi-strand
model – Equivalent Magnetization Formulation
26
• Stack of four Rutherford cables submitted to an external varying magnetic field;
• Dstrand=0.9 mm, 𝜏IFCC = 10 ms (imposed in the simulation);
• Homogenisation density factor 𝑘ht =
Ωst
Ωht
= 0.55 (for this case) to compensate
discretization error in the homogenized model;
• Same field variation amplitude (ΔB=5.47 mT) but lower time constant in the
homogenized model (𝜏IFCC = 5.7 ms) than the multi-strand one (𝜏IFCC = 9.6 ms). Multi-
strand model has almost the same time constant of single strand model.
Multi-strand model
Homogenized model
𝜕𝐵
𝜕𝑡
𝜕𝐵
𝜕𝑡
(W-V) formulation
(W-V-homo) formulation

Study 4.1: Stack of four Rutherford cable – Homogenized model - Modified
IFCC equation for the homogenized model – (W-V-homo-mod)
27
𝑀IFCC = −
2𝜏IFCC
𝜇0
𝜕𝐵
𝜕𝑡
• IFCC equation:
→
equivalent 𝑑𝑀IFCC
𝑑𝑡
= −
1
2𝜏IFCC
𝑀IFCC − (1)
𝑑𝐻
𝑑𝑡
because 𝐵 = 𝜇0 𝐻 + 𝑀
Decay term Driving
term
• Modified IFCC equation for homogenized
model:
• Equations were implemented in COMSOL with
Dode (Domain ODEs and DAEs) module;
• By using this approach, in which p=0.5 is a fitting
parameter for this case, we obtained same field
variation ( ∆𝐵 = 5.47 mT) and time constant
(𝜏IFCC=9.6 ms) for both models;
• This approach could represent a valid candidate
for modelling magnets by using homogenized
cables.
Case Dstrand=0.9 mm
Homogenization + modified IFCC equation
𝑑𝑀IFCC
𝑑𝑡
= 𝑘ht
−𝑝
−
1
2𝜏
𝑀IFCC − 𝑘ht
𝑝 𝑑𝐻
𝑑𝑡

28
Summary on IFCCs formulations for Rutherford cable stack – Is=0
𝜕𝐵i
𝜕𝑡
• Equivalent Magnetization formulation for homogenized
models (M-V-homo):
(W-V)
Equivalent
Magnetization-Homo
(W-V-homo)
Modified Equivalent
Magnetization-Homo (W-V-
homo-mod)
Formulation (M-M or B-B)
 x  …
(W-V)
Legend: • For a stack of Rutherford cables, the (W-V) formulation applied to the multi-
strand model returns consistent results with the single strand model. The (W-
V-homo) formulation applied to the homogenized model gives a correct field
variation amplitude, but incorrect time constant. However, this problem is
fixed applying the (W-V-homo-mod) formulation to the homogenized model.
 Correct
x Wrong
… To be studied
𝜕𝐵i
𝜕𝑡
(W-V-homo)
• Modified Equivalent Magnetization formulation for homogenized
models (M-V-homo-mod):
𝑑𝑀IFCC
𝑑𝑡
= 𝑘ht
−𝑝
−
1
2𝜏
𝑀IFCC − 𝑘ht
𝑝 𝑑𝐻
𝑑𝑡
(W-V-homo-mod)

29
Summary on IFCCs formulations for strand and Rutherford cable
Is=0, time varying Bext applied
Single strand
model
Equivalent
Magnetization
(W-V)
Equivalent Magnetization-
homo
(W-V-homo)
Modified Equivalent
Magnetization-homo
(W-V-homo-mod)
(M-M or B-B)
 N/A N/A 
Cable stack  x  …
Is≠0, no Bext applied
Single strand
model
Equivalent
Magnetization
(W-V)
homo
(W-V-homo)
Modified Equivalent
Magnetization-homo
(W-V-homo-mod)
(M-M or B-B)
x N/A N/A 
Cable stack … … … …
Legend:
~ Almost Correct N/A Not Available

Conclusion (1/2) – Single round strand and Rutherford cable modelling
30
Single round strand and Rutherford cable modelling:
1. The Equivalent Magnetization formulation (W-V) reproduces results in line with
expectations when it is applied to a single strand subjected to an external time
varying magnetic field and no current is imposed in the strand. The total
magnetizing current Im is zero as expected;
2. The Equivalent Magnetization formulation (W-V) is inconsistent when a time
varying current is imposed to a strand and no external magnetic field is considered.
In this case the magnetizing current Im is not zero anymore;
3. Equivalent Resistivity formulation (M-M or B-B) fixes the issue of the diverge-free
of magnetizing current density. It could represent a valid candidate for IFCCs
modelling in a single strand as well as in magnets. The study is still on going;
4. Using the Equivalent Magnetization formulation for homogenized cables (W-V-homo)
in cable stacks, we obtained correct magnetic amplitude, but an incorrect time
constant in the homogenized model. However, proposed modified IFCC equation
for homogenized cable (W-V-homo-mod) returns values of magnetic amplitude and
time constant in line with the multi-strand model.

Magneto-Thermal Analysis of FRESCA2 Magnet
– Energy Extraction and CLIQ simulations
31

Magnet simulation overview
32
• Developed within a collaboration
between CERN and CEA Saclay;
• Nb3Sn dipole magnet;
• Magnetic field: 13 T;
• 100 mm clear bore.
• Block coil design with 4 double
pancake coils:
‒ 42 turns outer layer;
‒ 36 turns inner layer;
‒ Cable: 40 strands of 1 mm;
‒ Bladder and key concept for
loading.
FRESCA2 Nb3Sn dipole magnet
Image: G. Willering

Magneto-thermal model
33
Magnetoquasistatic general PDE:
𝛻 × 𝜈𝛻 × 𝐴 + 𝜎
𝜕𝐴
𝜕𝑡
= 𝐽s + 𝛻 × 𝑀IFCC
Heat balance equation:
𝐶𝑝
𝜕𝑇
𝜕𝑡
+ 𝛻 ∙ 𝑞 = 𝑄
The magnetoquasistatic field solution determines the magnet’s electrodynamics and the
related thermal losses [6].
[6] L. Bortot et al, A 2-D Finite-Element Model for Electrothermal Transients in Accelerator Magnets, IEEE Transaction on Magnetics,
2018, 54.3: 1-4.
Investigated cases for EE (Energy Extraction) (@Imag=12’108 A and Tbath=1.9 K,
Imag=10’000 A and Tbath=4.5 K, Imag=6’000 A and Tbath=1.9 K):
• Quench all (all the turns are quenched at the same instant);
• IFCCs (Inter-Filament Coupling Currents physics).
Investigated cases for CLIQ (The Coupling-Loss-Induced Quench) @ Tbath = 1.9 K:
• Imag = 12’108 A, VCLIQ=1’250 V, C=50 mF;
• Imag = 5’000 A, VCLIQ=1’250 V, C=10 mF.

Homogenized model vs Multi-strand model
34
Homogenized coil* Multi-strand coil
*built by Lorenzo Bortot
Magnetic implementation (all domains):
• Magnetization 𝑀IFCC (in Am-1) is applied to superconducting cable domains;
• The external current density 𝐽𝑧 (in Am-2) is applied to cables;
• Magnetic insulation condition are applied to the external boundaries of the geometry;
• As main result, magnet current discharge is obtained.
Thermal implementation (only superconducting coil):
• 𝑇bath = 1.9 K (helium bath temperature), thermal insulation on external boundaries;
• A volumetric heat source 𝑄 = 𝑄ohm + 𝑄IFCC + 𝑄ISCC (in Wm-3) is applied to the
superconducting coil;
• As main result, IFCC losses are obtained.

Aim of the study and roadmap – FRESCA2 magnet
35
• Aim of the study:
1. Restructuring of the pre-existent homogenized model of FRESCA2 magnet;
2. Modelling of the strands in FRESCA2 superconducting coil in order to determine the
impact of homogenization on IFCC behavior;
3. Simulate EE (Energy Extraction) and CLIQ (The Coupling-Loss-Induced Quench).
SIGMA
SIGMA-
restructured
Magnetization formulation
Homogenized
model
Multi-strand
model
𝜕𝐵
𝜕𝑡
𝜕𝐵
𝜕𝑡
𝑑𝑀IFCC
𝑑𝑡
= 𝑘ht
−𝑝
−
1
2𝜏
𝑀IFCC − 𝑘ht
𝑝 𝑑𝐻
𝑑𝑡
Magnetization formulation for
homogenized cables (W-V-homo)
Modified IFCC equation for homogenized cables
(W-V-homo-mod)
Magnetization for multi-strand
cables (W-V)

FRESCA2 homogenized model – Magneto-thermal simulation
36
In the homogenized model, cables are modelled considering homogeneous thermal
properties. The model was generated by using SIGMA (STEAM Integrated Generator of
Magnets for Accelerator) [7].
[7] cern.ch/STEAM
SIGMA-homogenized model SIGMA-homogenized model

FRESCA2 – SIGMA restructured model
37
[8] M. Mentink, “Development of the STEAM COMSOL Magnet Simulation Tool”, CERN Technical note, EDMS no. 2054126, 2018.
[9] B. Bokharaie , “Enhanced automation of magnet model generation”, CERN Training Report, Aarhus University School of
Engineering, 2019.
• Starting from the existing model in SIGMA, we performed the minimum amount of changes to the
model in order to make a restructuring with the aim to reduce computation time;
• Group identical properties together in common variables groups to reduce the computation time of
the model [8,9].
• Steps to follow:
‒ Disablement and deletion of variable groups and global parameters related to single turns;
‒ Adjustments to physics engine: deletion of ISCC physics and persistent magnetization;
‒ Definition of global material properties (i.e., heat capacity of the entire turn);
‒ Definition of Jz, i.e., the direction of the electrical current through the magnet coils;
‒ Definition of new junction boxes for magnet current;
‒ Definition of new variable group Voltages.
Before restructuring After restructuring

FRESCA2 Magneto-thermal simulation: comparison between
SIGMA and SIGMA restructured homogenized model
38
• Homogenized SIGMA restructured model returns the same results of the homogenized
SIGMA model for the two investigated cases (Quench all, IFCCs);
• Number of mesh elements: 5’563.
Quench all IFCCs
Homogenized SIGMA model 18 min 3 s 2 h 1 min 20 s
Homogenized SIGMA restructured 1 min 12 s 3 min 49 s
• Computation time was
significantly reduced after
restructuring operations.

Magneto-thermal analysis results –
Energy Extraction simulations
39

Homogenized vs. Multi-strand model – (W-V-homo) vs. (W-V)
formulations – Energy Extraction simulations
40
• Homogenized model: IFCC magnetization and
losses are considered at a cable level, using
homogenization approximation;
• Multi-strand model: IFCC magnetization and losses
are considered at a strand level;
• How well does the homogenized approach
approximate the multi-strand approach?
Equivalent Magnetization equation
𝜕𝐵
𝜕𝑡
𝜕𝐵
𝜕𝑡
Homogenized model
(W-V-homo)
Multi-strand model
(W-V)
Homogenized model Multi-strand model

Comparison between homogenized and multi-strand model –
Magneto-thermal analysis – Energy Extraction
41
IFCCs losses [W]
• For considered discharge (see table), multi-strand model gives
larger IFCCs losses than the homogenized model. In
particular, the peak value is 10% higher than the homogenized
model;
• Multi-strand model: more IFCCs losses → large coil fraction
quenches → higher quench resistance → faster
𝑑𝐼
𝑑𝑡
(see next
few slides);
• These results represent a quantitative estimation of the impact
of homogenization formulation on IFCC behavior.
10%
Circuit parameters
Circuit parameter Symbol Value U.M.
Initial magnet current I0 12,108 A
Circuit inductance Lcir 1E-06 H
Energy extraction resistance REE 81E-03 Ω

42
Temperature [K]
5%
• Quench all: temperature is the
same in both homogenized
and multi-strand models.
• IFCCs: temperature is higher
of 5% in the multi-strand
model than in the
homogenized model.
Quench resistance [Ω]
7%
• Quench all: quench resistance
is the same in both
homogenized and multi-strand
models.
• IFCCs: quench resistance is
higher of 7% in the multi-
strand model than in the
homogenized model.

43
Magnet current discharge [A]
6.3%
Magnetic flux density [T]
5.3%
ZOOM

44
• Quench all: electrical
resistivity of copper is the
same in both homogenized
and multi-strand models.
• IFCCs: electrical resistivity of
copper is higher of 7% in the
multi-strand model than in the
homogenized model.
Electrical resistivity of copper [Ω·m]
7%
Voltage at magnet terminal [V]
4%
ZOOM

45
MIITs [MA2s]
1%
Magneto-thermal simulation results – Energy Extraction
Quench all
(Homo vs Multi-
strand)
QIFCC Tpeak Rquench Imag Bave ρCu Vpeak MIITs
NA = = = = = = =
IFCCs
(Homo vs Multi-
strand)
↑ of 10 %
in Multi-
strand
↑ of 5 %
in Multi-
strand
↑ of 7 % in
Multi-
strand
↓ of 6.3 %
in Multi-
strand
↓ of 5.3 %
in Multi-
strand
↑ of 7 %
in Multi-
strand
↑ of 4 % in
Multi-
strand
↓ of 1 % in
Multi-strand
Legend:
N/A Not available ↑ Higher ↑ Lower
ZOOM

Comparison with experimental data (EE simulations)
46
• Both multi-strand and homogenized models have a reasonable match with experimental data, despite the
differences in IFCCs physics implementation.
• Simulation results follow the SIGMA input parameters (i.e. no parameter optimization to match the
experimental results)

Final overview on magneto-thermal simulation – Homogenized
model vs. Multi-strand model – Computation time
47
• No. of mesh elements: Homogenized model (9’357 elements), Multi-strand model (1’500’707 elements);
Computation time for the investigated cases
• Computation time increases
after modelling strands in the
cables.

Modified Equivalent Magnetization formulation
for FRESCA2 Magnet -Energy Extraction
48

Modified Equivalent Magnetization formulations – (W-V-homo-
mod) formulation – Energy Extraction
49
• Homogenized model current
discharge curve, obtained by
using modified IFCC equation for
homogenized model, better
approaches the curve of multi-
strand model.
• Modified IFCC equation for
homogenized model (p=0.5):
• Equations were implemented in COMSOL with Dode (Domain ODEs and
DAEs) module;
𝑑𝑀IFCC
𝑑𝑡
= 𝑘ht
−𝑝
−
1
2𝜏
𝑀IFCC − 𝑘ht
𝑝 𝑑𝐻
𝑑𝑡
IFCCs
Homogenized SIGMA 2 h 1 min 20 s
Homogenized SIGMA restructured 3 min 49 s
Multi-strand 22 h 28 min 47 s
Homogenized model
(modified IFCC equation - Dode)
10 min 25 s
• We reduced a lot the computation time
in case of EE (Energy Extraction) with
respect to multi-strand model.

50
Summary on IFCCs formulations – from strand to magnet level
Single strand
model
Equivalent
Magnetization
(W-V)
homo
(W-V-homo)
Modified Equivalent
Magnetization-homo
(W-V-homo-mod)
(M-M or B-B)
 N/A N/A 
Single strand
model
Equivalent
Magnetization
(W-V)
homo
(W-V-homo)
Modified Equivalent
Magnetization-homo
(W-V-homo-mod)
(M-M or B-B)
x N/A N/A 
FRESCA2
(EE)  ~ ~ …
Legend:
~ Almost Correct
• FRESCA2 (EE): equivalent magnetization based formulations could be considered almost correct
considering the reasonable match with experimental data.
N/A Not Available

Magneto-thermal analysis on FRESCA2
Magnet - CLIQ protection system
51

Magneto-thermal analysis with CLIQ protection system for
FRESCA2 magnet – Homogenized vs. Multi-strand model
52
Homogenised model
(W-V-homo)
Multi-strand model
(W-V)
• All simulations were performed
including IFCCs physics;
• No. of mesh elements: 13’876
elements (Homogenised model),
619’346 elements (Multi-strand
model).
• Computation time: 14 min 29 s
(Homogenised model), 20 h 29
min 10 s (Multi-strand model).

Magneto-thermal analysis with CLIQ: Homogenized model vs Multi-strand
model – Imag=12’108 A, VCLIQ=1’250 V, C=50 mF
53
• The slope
𝑑𝐼𝑚𝑎𝑔
𝑑𝑡 t=0
is 38% lower in the multi-strand model. Furthermore, after the initial
fluctuation, the discharge in the two models is similar. This outcome explains why we have
noticed so far a higher
𝑑𝑡
than expected in all homogenized models of magnets;
• Faster current decay in the multi-strand model.
• Losses due to IFCCs are, as consequence, lower of a factor two in the multi-strand model
than in the homogenized one.
𝑑𝑡 𝑡=0 𝐻𝑜𝑚𝑜.
𝑑𝑡 𝑡=0 𝑆𝑡𝑟𝑎𝑛𝑑𝑠

54
• The slope
𝑑𝑡 t=0
is 38% lower in the multi-strand model than the homogenized
model → faster current decay in the multi-strand model.
• Faster CLIQ current decay in the homogenized model.
• Losses due to IFCCs are, as consequence, lower of a factor two in the multi-strand
model than in the homogenized one.
𝑑𝑡 𝑡=0 𝐻𝑜𝑚𝑜.
𝑑𝑡 𝑡=0 𝑆𝑡𝑟𝑎𝑛𝑑𝑠
Magneto-thermal analysis with CLIQ: Homogenized model vs Multi-strand
model – Imag=5’000 A, VCLIQ=1’250 V, C=10 mF
Faster dampening in
homogenized model

Modified Equivalent Magnetization formulation
for FRESCA2 Magnet –CLIQ simulations
55

Modified Equivalent Magnetization formulations – (W-V-homo-
mod) formulation – CLIQ simulation
56
• Modified IFCC equation for
homogenized model (p=0.5):
• Equations were implemented in COMSOL with Dode (Domain ODEs and DAEs) module;
𝑑𝑀IFCC
𝑑𝑡
= 𝑘ht
−𝑝
−
1
2𝜏
𝑀IFCC − 𝑘ht
𝑝 𝑑𝐻
𝑑𝑡
• Homogenised-dode-model current discharge
curve approaches better the multi-strand curve
only in the first part of the simulation (until 25
ms), then the standard homogenised model curve
better fits the curve of multi-strand model.
• Computation time was reduced by using modified
IFCC equation – Dode with respect to multi-strand
model.

57
Summary on IFCCs formulations – from strand to magnet level
Single strand
model
Equivalent
Magnetization
(W-V)
homo
(W-V-homo)
Modified Equivalent
Magnetization-homo
(W-V-homo)
(M-M or B-B)
 N/A N/A 
Single strand
model
Equivalent
Magnetization
(W-V)
homo
(W-V-homo)
Modified Equivalent
Magnetization-homo
(W-V-homo)
(M-M or B-B)
x N/A N/A 
FRESCA2
(EE)  ~ ~ …
FRESCA2
(CLIQ)
To be compared
to Exp. data x x …
Legend:
~ Almost Correct N/A Not Available

Conclusions (2/2) – FRESCA2 magnet
58
• Restructuring operations performed on FRESCA2 magnet homogenized model generated by
SIGMA have permitted to reduce significantly the computation time of the magneto-thermal
simulation;
• Multi-strand model was implemented in FRESCA2 magnet. In this model, the computation time
increases for the high number of mesh elements required.
• For EE simulations, we noticed several differences in IFCCs simulations between homogenized
and multi-strand model, due the different used formulations (i.e., faster current discharge in the
multi-strand model).
• CLIQ discharge simulations implemented in the multi-strand model revealed a different slope
in the magnet current curve, with respect to the homogenized model. As a consequence, IFCCs
losses are quantitatively different. The current decay is faster in the multi-strand model. The CLIQ
current decay is slower in the multi-strand model, especially at lower currents.
• Modified IFCC equation for homogenized cable (Dode-formulation) reduces the computation
time with respect to the multi-strand and reasonably approaches the magnet current discharge
when EE (Energy Extraction) is simulated. In case of CLIQ discharge, this formulation permits to
approach better the magnet curve discharge than the standard magnetization formulation for
homogenized cables, but only in the first part of the discharge.

Overall conclusions
59
• Extensive study was done to investigate IFCC physics at the strand, cable stack, and magnet
level, with emphasis on the implications of homogenization.
• Strand level, comparison of equivalent magnetization to equivalent resistivity formulations :
• Without self-field: either formulation is consistent;
• With self-field, equivalent resistivity is correct.
• Cable stack, comparison between homogenized conductor, modified homogenized conductor and
multi-strand formulations
• Homogenization results in an incorrect time-constant and a correct final field amplitude;
• Modified homogenization approach with fitting factor p produces consistent results with multi-
strand model, with a reduced computational cost;
• Magnet (FRESCA2), comparison between homogenized conductor, modified homogenized
conductor and multi-strand formulations:
• EE:
• Quantitative difference between homogenized and multi-strand approach;
• Reasonable consistency for either approach with experimental observations;
• Modified homogenized approach gives greater consistency with multi-strand model;
• CLIQ:
• Homogenized approach gives higher initial dI/dt, and faster ICliq dampening at lower
currents, somewhat slower overall current decay;
• Modified homogenized approach gives better initial consistency, but worse after 20 ms.

Carmelo_Barbagallo_IFCCs_modelling_in_superconducting_filamentary_Nb3Sn_strands_and_magnets_final_version.pptx

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