Magnetic field calculations including the impact of persistent currents in superconducting filaments

Abstract

The magnetic field in the coils of superconducting magnets induces
so-called persistent currents in the filaments. Persistent currents are
bipolar screening currents that do not decay due to the lack of
resistivity. The NbTi-filaments are type II superconductors and can be
described by the critical state model. This paper presents an analytical
hysteresis model of the filament magnetization due to persistent
currents which takes into account the changing magnetic induction inside
the filament. This model is combined with numerical field computation
methods, taking local saturation effects in the ferromagnetic yoke into
consideration

Full text

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
European Laboratory for Particle Physics
MAGNETIC FIELD CALCULATIONS INCLUDING THE IMPACT
OF PERSISTENT CURRENTS IN SUPERCONDUCTING FILAMENTS
M. Aleksa, S. Russenschuck, C. Völlinger
The magnetic field in the coils of superconducting magnets induces so-called persistent currents in the
filaments. Persistent currents are bipolar screening currents that do not decay due to the lack of
resistivity. The NbTi-filaments are type II superconductors and can be described by the critical state
model. This paper presents an analytical hysteresis model of the filament magnetization due to persistent
currents which takes into account the changing magnetic induction inside the filament. This model is
combined with numerical field computation methods, taking local saturation effects in the ferromagnetic
yoke into consideration.
LHC Division
Presented at the 13th Conference on the Computation of Electromagnetic Fields (COMPUMAG)
2-5 June 2001, Lyon-Evian, France
Geneva,
Large Hadron Collider Project
Administrative Secretariat
LHC Division
CERN
CH - 1211 Geneva 23
Switzerland
LHC Project Report 494
Abstract
27 August 2001

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Magnetic Field Calculations Including the Impact of
Persistent Currents in Superconducting Filaments
M. Aleksa, S. Russenschuck, C. V¨ ollinger
CERN, 1211 Geneva 23, Switzerland
Abstract—The magnetic field in the coils of supercon-
ducting magnets induces so-called persistent currents in
the filaments. Persistent currents are bipolar screening
currents that do not decay due to the lack of resistivity.
The NbTi-filaments are type II superconductors and
can be described by the critical state model [1]. This
paper presents an analytical hysteresis model of the fil-
ament magnetization due to persistent currents which
takes into account the changing magnetic induction in-
side the filament. This model is combined with numer-
ical field computation methods, taking local saturation
effects in the ferromagnetic yoke into consideration.
I. Introduction
The Large Hadron Collider (LHC) [2], a proton-proton
superconducting accelerator, will consist of about 8400 su-
perconducting magnet units of different types, all operating
in superfluid helium at a temperature of 1.9K. The ap-
plied magnetic field induces currents in the filaments that
screen the external field. The filaments are made of type
II hard superconducting material with the property that
the magnetic field inside the filament does not vanish, but
differs from the external field due to the screening effect
of the persistent currents. Macroscopically, these persis-
tent currents (that do not decay but persist due to the lack
of resistivity) are the source of a magnetization M(B) of
the superconducting strands. One way to account for this
magnetization is to mesh the coil in finite elements and
solve the non-linear problem numerically by making use
of a measured M(B)-curve. This approach has two main
flaws: The numerical field computation has to be combined
with a hysteresis model and the coil has to be discretized
with highest accuracy also accounting for the existing gra-
dient of the current density due to the trapezoidal shape of
the cables, and the conductor alignment. Hence, we aimed
for computational methods that avoid the meshing of the
coil completely. In this paper, a model to calculate the
magnetization of the superconducting strands is presented,
taking into account alternating external fields, while keep-
ing the magnetization vector parallel to the outside mag-
netic induction. Though the flux pinning is not described
microscopically, this model fully considers the hysteretic
behaviour of the superconducting filaments and predicts
the measured magnetization.
putation of persistent currents is based on the measured
The method for the com-
critical current density jcof the superconductors and cal-
culates the filament magnetization by means of the critical
state model [1]. It is combined with the coupled boundary
element / finite element method (BEM-FEM-method) [3]
which avoids the representation of the coil in the finite ele-
ment mesh since the coil is positioned in the iron-free BEM
domain. The fields arising from current sources in the coil
are calculated by means of the Biot-Savart Law, while the
surrounding non-linear iron yoke has to be meshed in finite
elements. Hence, the discretization errors due to the finite-
element part in the BEM-FEM formulation are limited to
the iron-magnetization arising from the yoke structure.
The input function jc(B,T) only depends on the mate-
rial properties of the superconductor but is independent
of geometrical parameters such as the filament diameter
or shape, and the ratio λ of the superconductor to total
strand volume. The method reproduces the hysteretic be-
havior for arbitrary turning points in the magnet excitation
cycle including minor loops. In order to account for the im-
pact of the filament magnetization on the magnetic field an
M-iteration as a function of B is performed.
II. The Magnetic Induction Inside the Filament
The critical state model [1] postulates that the screening
currents in the superconductor always flow with the critical
current density jc. Magnetization models for superconduct-
ing filaments can be found in literature (e.g. [4]). These
models usually neglect the field-dependence of the critical
current density inside the filaments and give expressions for
the screening fields of slices of finite thickness. The model
introduced in this paper includes changing current densities
inside the filament and calculates the continuous course of
the magnetic field by means of a differential approach. The
dependence of the critical current density on the modulus
of the magnetic induction B = |?B| is given by the following
fit [5], where Bc= Bc0(1 − (T/Tc0)1.7):
jc(B,T) =jref
cC0Bα−1
(Bc)α
?
1 −B
Bc
?β?
1 −
?T
Tc0
?1.7?γ
.
(1)
The fit parameters for the LHC main-magnet cables are
a critical current density at 4.2K and 5T of jref
3 · 109A/m2, an upper critical field Bc0 = 14.5T, a crit-
ical temperature of Tc0= 9.2K, a normalization constant
C0= 27.04T and the fit parameters α = 0.57, β = 0.9 and
c
=
1

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γ = 2.32.
For small magnetic inductions B, where the persistent
currents influence the field quality most, Eq. (1) strives for
infinity with Bα−1= B−0.43∼ 1/√B for B → 0. For the
computation of the induction inside the filament, Eq. (1)
was approximated around the actual value of the applied
field Bout = |?Bout| with the following function (T is con-
stant):
jc(B,T) ∼ jc(Bout,T)
√Bout
√B
≡F(Bout)
√B
.
(2)
Fig. 1. Cross-section of one filament showing the intersecting ellipses
of the critical current in the individual penetrations of one layer (here:
virgin curve, non-fully penetrated state). For demonstration the slices
are shown with finite thicknesses, while in the model dq → 0.
Be q the relative penetration parameter which is zero at
the surface of the filament and equals one in the center
(see Fig. 1). From the expression of the perfectly uniform
dipole field produced by two ellipses with opposite current
densities shifted by the distance r dq with respect to each
other, the local field change dB(q) along the radial slice of
thickness r dq (dq → 0) can be derived:
dB(q) = ξµ0H jc(B(q)) dq =ξµ0H F(Bout) dq
?B(q)
,
(3)
where r is the filament radius. The geometry factor H =
r/2 corresponds to the ideal screening field of two intersect-
ing ellipses [4]. In our model H = r(2 − 2ln2) = 0.614 r,
correcting for the little spaces that are left when a round
filament is filled with a series of intersecting ellipses (see
Fig. 1, (dq → 0)). The parameter ξ equals −1 in case of
ramping up and ξ = 1 for ramping down. In the first case,
the orientation of the magnetic moment of the screening
current is opposite to the orientation of the outside field
Boutand B decreases inside the filament.
Equation (3) is a differential equation for B(q), consider-
ing the dependence of jcon B(q), that can be solved with
the known boundary condition B(0) = Bout in a closed
analytical form, yielding:
B(q) =
?
B3/2
out+3
2ξ H F(Bout)µ0q
?2/3
(4)
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
00.20.40.60.81
Penetration depth q
B(q) [T]
0
50
100
150
200
250
Jc(q) [kA/mm2]
Outside Field
Fig. 2. Magnetic induction B(q) as a function of the penetration depth
q (continuous line). The dashed line denotes the current density jc(q).
Figure 2 shows B(q) according to Eq. (4) and the de-
pendence of the critical current density jc(B(q)) on the
penetration depth q.The magnetic induction at q = 0
equals the external field Bout. The shown field distribution
is a fully penetrated state, reached for the case of increasing
the external field from negative field values to Bout= 0.08T
(ξ = −1). As is shown in Figure 2, this results in a decreas-
ing field B(q) along the penetration depth which produces
an increase of jc(B(q)) along q. At B(q) = 0, the criti-
cal current density reaches its maximum value. There the
strong increase of jcproduces a sharp decline of B(q). The
course of jc(q) shows the importance of expressing jcas a
function of q rather than assuming a constant value.
III. The Strand Magnetization and Hysteresis
From the analytic expression for the magnetic induction
B(q) inside the filament, the magnetization due to the ra-
dial slice of current jc(q) between the penetrations qiand
qi+1is derived. Individual slice magnetizations are needed
to describe the hysteresis after changes of the ramp direc-
tion. Such a change (∂Bout/∂t changes sign) will produce
a new layer of screening currents with opposite polarity (ξ
switches sign). For small changes the new current layer will
penetrate the filament only from q1= 0 to q2? 1 while the
currents inside persist. Figure 1 shows one such layer in
the cross-section of a filament for the non-fully penetrated
state. The values for qi are calculated using Eq. (4). For
minor excitation loops, the magnetization is obtained as
the superposition of n different layers,
M =
n
?
i=1
Mi=
n
?
i=1
qi+1
?
qi
mi(q) dq .
(5)
In Eq. (5) Mi denotes the modulus of the magnetization
? Mi; it has negative values if the orientation is opposite to
the external field.
Mi=4rξ
π
qi+1
?
qi
jc(B(q))(1 −q)2dq =4rξ F
π
qi+1
?
qi
(1 − q)2
?B(q)
dq.
(6)
2

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Equation (6) can be solved analytically and – together with
Eq. (4) – yields a closed expression for the filament magne-
tization:
Mi=
4rB(q)
5πF2µ2
??
0H3
?
4q2
B3
out+ ξHFµ0 ·
?
·
5 − 4q +5
ξHFµ0− (q − 4)B3/2
out
??????
q=qi+1
q=qi
(7)
Figure 3 presents computations of the filament magneti-
zation according to Eq. (7) multiplied with the filling fac-
tor λ. The virgin curve and several hysteresis loops are
displayed. The comparison of the calculation and the mea-
sured magnetization of a superconducting strand (dashed
line) shows good agreement apart from the region of B close
to zero. There the difference between the magnetizations of
one filament and a whole strand becomes significant. Since
the outside field for each filament varies slightly due to the
position in the strand cross-section and is additionally in-
fluenced by the field arising from the screening currents in
the neighboring filaments, Bout will be different for each
filament according to its exact position. This results in a
spread of filament magnetizations and hence in a smoothen-
ing of the region of B close to zero. Since the Rutherford
cables used in LHC magnets consist of many individual
strands, this region will be smoothened out automatically
due to the differences of Boutat the individual strand po-
sitions within the coil.
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
-1.6-1.2-0.8-0.40.00.40.81.2 1.6
Β [Τ]
µ0 Μ [Τ]
calculation
measurement
virgin curve
Bp1
Fig. 3. Computed magnetization curve for one filament (r = 3.5µm,
λ = 1/2.95, T = 1.9K) compared with measurements from a super-
conducting strand [6].
IV. Virgin Curve and Fully Penetrated State
From the expressions for the magnetic induction B(q)
(Eq. (4)) and the magnetization M (Eqs. (6) and (5)) both
parameters can be computed as a function of the pene-
tration depth q. Figure 4 (upper plot), shows the values
for increasing external fields Bout(ξ = −1) for the virgin
curve creating one layer extending from q1= 0 to q2(Bout).
Depending on Bout, the field decreases until a certain pen-
etration depth, where complete screening of the external
field is obtained. The remaining part of the filament stays
field free. In the lower plot the contribution m(q) of a slice
dq to the total magnetization M is presented. The value of
M can be obtained by integrating the presented curves (see
indication on the plot). Figure 5 illustrates the same quan-
tities as in Fig. 4, but for a filament that has already ex-
perienced a negative outside field before (different history)
and hence is fully penetrated. Since the currents inside the
superconductor persist, there is a remaining negative field
B(q) inside, whereas in the case of the virgin curve the field
remains zero for q > q2(Bout). The lower plot in Fig. 5 also
explains why the maximum magnetization does not occur
at Bout= 0: The magnetization is given by the integrated
area M under the m(q) curve, which is biggest for small
values of Bout ?= 0. This characteristic behaviour has al-
ready been observed in measurements (see Fig. 3), and is
in good agreement with the calculations.
Fig. 4. Magnetic induction B(q) and the magnetization contribution
m(q) as a function of the penetration depth q for the virgin curve.
V. Calculation of Bp1
The magnetic induction Bp1 denotes the value of the
outside magnetic induction where the modulus of the fila-
ment magnetization passes through its first maximum when
ramping up on the virgin curve (see Fig. 3). Since the mag-
netization has been calculated in a closed analytical form
(Eq. (7)), we can now derive the maximum of the virgin
curve by solving ∂ M(Bout)/∂ Bout = 0. For the virgin
curve the magnetization consists of only one layer, n = 1,
that penetrates from q1= 0 to q2= 2B3/2
obtained by solving B(q2) = 0. In this region of Bout= Bp1
we find that F?(Bout)∼= 0 and hence we obtain
(H F(Bp1) µ0)2/3(15 − 5√5)1/3
out/(HF(Bout)µ0),
Bp1
∼=
2
,
(8)
3

Page 5

Fig. 5. Course of the magnetic induction B(q) and the magnetiza-
tion contribution m(q) as a function of the penetration depth q for a
filament already been exposed to a magnetic field before (non-virgin
curve).
q2
∼=
?
5
6−5√5
18
∼= 0.46 .
(9)
The recursive Eq. (8) yields a good estimate for Bp1after
few iterations. From Eq. (9) can be seen that the maximum
modulus of the magnetization does occur at a penetration
depth of q2∼= 0.46 rather than in the fully penetrated state.
This fact is illustrated by the lower plot of Fig. 5, where
the area will be maximal for q → 0.46 (solid line). Note,
that the value of q2 is independent of the critical current
fit, provided ∂F(Bout)/∂Bout∼= 0 (i.e. the critical current
diverges with 1/√Bout, for Bout→ 0, see Eq. (2)).
VI. Combining the Model with Numerical Field
Computation
In the BEM-FEM coupling method the field in the non-
meshed air domain is the sum of the source fields from
the superconducting coils1and the reduced magnetic induc-
tion from the non-linear iron yoke. A third contribution,
originating from the magnetization of the superconducting
strands depending on the magnetic induction at the strand
position has to be added to the source fields. Re-iterations
are performed until convergence is obtained. For a descrip-
tion in detail see [7]. The presented model has been imple-
mented into the ROXIE field computation program [3].
Figure 6 shows the results of a BEM-FEM calculation
(obtained with ROXIE) of a quadrant of the LHC main
quadrupole during ramping up (I = 730A). The magnetic
induction was calculated as described above including the
1Calculated analytically by means of Biot-Savart’s Law.
influence of persistent currents. From this result, the mul-
tipoles in the aperture have been calculated. The global
effect of the persistent currents in the strands on the field
quality in the aperture of the magnet is a ∆bpers
calculated at 17mm reference radius and compares well to
the measured values in the main quadrupole prototypes
((∆bpers
6
)meas = −3.5, [8]). For the definition of the field
component b6see e.g. [2].
6
= −3.8
020406080100120140160180200220240
0.0000.021-
0.0210.043-
0.0430.065-
0.0650.086-
0.0860.108-
0.1080.130-
0.1300.152-
0.1520.173-
0.1730.195-
0.1950.217-
0.2170.238-
0.2380.260-
0.2600.282-
0.2820.304-
0.3040.325-
0.3250.347-
0.3470.369-
0.3690.390-
0.3900.412-
|Btot| (T)
Fig. 6. Modulus of the magnetic induction |B| of the main quadrupole
including the stainless steel collar during ramping up (I = 730A).
The effect of persistent currents has been taken into account.
VII. Conclusions
A hysteresis model for the filament magnetization due to
persistent currents has been developed and combined with
the BEM-FEM [3] numerical field-computation method.
Persistent currents influence the field quality of supercon-
ducting accelerator magnets. The model has been imple-
mented into the ROXIE field computation program and is
used to calculate the distortion of the magnetic field in the
aperture of the superconducting magnets for LHC.
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4