Supercond. Sci. Technol. 10 (1997) 187–194. Printed in the UK
PII: S0953-2048(97)79837-3
Calculations of hysteresis in a short
cylindrical type II superconductor
I Younas and J H P Watson
Institute of Cryogenics and Energy Research, University of Southampton,
Southampton SO17 1BJ, UK
Received 26 November 1996, in final form 5 February 1997
Abstract. The phenomenon of hysteresis in magnetic materials has been widely
studied for many decades and several models have been proposed to find good
approximations for this phenomenon. The simulation presented in this paper
describes hysteresis in a cylindrical type II superconductor of small aspect ratio. A
fundamental approach, using energy minimization, has been applied by which the
demagnetization effects are naturally included. A comparison has been made
between the critical current derived from the magnetization using Bean’s
formulation for the infinitely long cylinder and that for the short cylinder used in our
calculations, and it indicates how the error in using the Bean model for a short
cylinder can be minimized.
1. Introduction
The distribution of magnetic vortices in a type II
superconductor at finite temperature is governed mainly
by the balance of electromagnetic driving forces, pinning
forces of the material and flux creep. When an external
applied magnetic field is varied between Hc1 and Hc2 the
vortices begin to enter or leave the superconductor through
its boundary. Whenever the driving forces overcome
the pinning forces, the system of vortices arranges itself
into another metastable state such that an equilibrium is
achieved with the external magnetic field. Hence, a change
in the applied magnetic field or temperature forces the
unpinned vortices to move rapidly into another equilibrium
state or to leave the superconductor so that a new quasistationary equilibrium of energy is obtained.
Numerical calculations of magnetic field distribution in
superconductors have been carried out by many authors.
Frankel [1] calculated the current and field distribution in
a thin superconducting disk in transverse fields by use of
the critical-state model. Daümling and Larbalestier [2]
extended the calculations by including the short-cylinder
geometry, but they did not include the hysteresis effects.
Fabbricatore et al [3] carried out hysteresis simulations
on infinite slab and cylinder, but this is essentially a onedimensional case with no demagnetizing effect included.
Zhu et al [4] looked at trapped flux, but their hysteresis
calculations were limited to the thin-disk case.
The present work is an extension of the earlier work
by Watson and Younas [5] who looked at current and field
profiles in short cylindrical superconductors in the remnant
state using Kim’s relation [6] for the dependence of current
on field. In the present work the Bean model [7] was used
for the current flow in the superconductor, i.e. the current
c 1997 IOP Publishing Ltd
0953-2048/97/040187+08$19.50 density never exceeded some critical value Jc which is
independent of the magnetic field. This implies that for
a moderate applied magnetic field (Hc1 < Ha < Hc2 ) the
superconductor is divided into two regions:
(i) the field-penetrated region where the current density is
either +Jc or −Jc depending on the magnetic history
and
(ii) the field-free region where the current is identically
zero.
The boundary separating the above two regions is
called the free boundary, which is highly sensitive to the
magnitude and spatial dependence of the magnetic field, as
well as the topology of the superconducting material. The
free boundary is easily determined for a superconductor
with a simple geometry, e.g. infinitely long cylinder or strip.
For a geometry where the demagnetizing effect needs to
be included, a determination of the free boundary can be
non-trivial. Several front-tracking algorithms have been
developed to determine the free boundary, e.g. [8, 9], some
better suited to a particular problem than others, but there
may be a topological form of the superconductor where the
use of a particular methodology is not at all appropriate.
In [10] the authors calculate the levitation force on a
permanent magnet above a HTS material. Although they
use the critical-state model and the current is dependent
on the electric and magnetic fields it is not at all clear
how they have included the demagnetizing effects. In
[11] a vector potential notation for the current and a
virtual conductivity together with a certain non-linear J –
E constitutive relation are used to describe the critical
state. The most comprehensive paper describing the field
equations for the type II superconductor is given by [12],
but unfortunately they only considered the infinitely long
cylindrical conductor configuration.
187
I Younas and J H P Watson
In the present work the problem of finding the free
boundary has been avoided, although it can easily be found
once a solution has been obtained. The problem is tackled
by applying a fundamental physical approach using energy
minimization of the system consisting of the superconductor
and an external source giving a uniform magnetic field over
the volume of the conductor. When the applied field varies
with time the energy of the system is assumed to achieve
a minimum configuration at each time step.
2. Theoretical background
Consider a body of fixed currents situated in a magnetic
medium of permeability µ and positioned at a finite distance
from an arbitrary origin. The volume occupied by this
body is denoted as V1 , and the magnetic energy T1 due to
excitation currents in the body can be expressed as [13]
Z
1
B1 · B1 dV
(1)
T1 =
2µ
another formulation will be outlined which overcomes this
difficulty.
Assume that the body situated in V1 is a solenoid,
and the body occupying the volume V2 is a type II
superconductor. The field due to excitation currents
in the solenoid gives rise to screening currents in the
superconductor. The magnitude of the screening currents
depends on the magnitude of the applied field of the
solenoid. It is assumed that the field never exceeds the
upper critical field Hc2 of the superconductor. Hence
the conductor remains superconducting for the range of
applied field. It will also be assumed that the screening
currents are all volume currents, and that the Meissner
state can be neglected (Hc1 ∼
= 0). In the following
derivation the subindices 1 and 2 refer to the solenoid and
the superconductor respectively.
The first term of equation (3) can be rewritten as
Z
Z
1
1
B2 · H2 dV =
∇ × A2 · H2 dV
(4)
E1 =
2
2
V
V
V
where the integration volume V is extended over all space.
It is assumed that the relation between the magnetic field
B and the field strength H is linear and the medium is
isotropic.
Now reduce the intensity of the excitation currents
to zero, and introduce into a suitable cavity of the
medium another body which is assumed initially to be
unmagnetized, and the relation between B and H is linear.
The volume occupied by the second body is denoted V2 .
The excitation currents of the first body are subsequently
restored to their original configuration. Since the energy of
the system consisting of the two bodies is determined only
by the initial and final values of the system variables, the
total energy of the final state is expressed as:
Z
1
(B1 + B2 ) · (B1 + B2 ) dV
(2)
T2 =
2µ
V
where B1 and B2 are magnetic fields due to excitations in
the two bodies.
The energy required to build up the currents of the first
body to their initial state in the presence of the second
body is the energy difference T = T2 − T1 , where T can
be written as
Z
Z
1
B22 dV + 2 B1 · B2 dV .
(3)
T =
2µ
V
V
The first term in equation (3) represents the energy due to
the field of the second body alone, and the second term
denotes the interaction energy between the two bodies.
In the above discussion the magnetic response of the
two bodies has been included in the expression for the field
vectors. Hence, to calculate the energy at a given time the
magnetic field must be integrated over all space. When a
numerical solution is needed, and calculations are carried
out, this becomes a cumbersome approach to determine the
total energy of the system. In the following discussion
188
which when using the vector identity
∇ · (a × b) = b · (∇ × a) − a · (∇ × b)
(5)
(where a and b are two real arbitrary vectors) can be
expressed as
Z
1
[∇ · (A2 × H2 ) + A2 · (∇ × H2 )] dV (6)
E1 =
2
V
where the integration is over all space. The first term in
equation (6) can be transformed into a surface integral by
applying the divergence theorem. This surface needs to be
extended to infinity to contain the whole space. Since A
falls as r −1 and H as r −2 for increasing r, while the surface
area increases as r 2 , this integral does not contribute to E1 .
The second term in equation (6) will only contribute in the
region V2 because curl H2 is zero outside V2 .
The second term in equation (3) denotes the energy due
to magnetic interaction between the fields of the solenoid
and the superconductor. This term can be rewritten as
Z
Z
(7)
E2 = B1 · H2 dV = ∇ × A1 · H2 dV .
V
V
Again, by applying identity (5) this expression can be
reformulated as
Z
E2 = [∇ · (A1 × H2 ) + A1 · (∇ × H2 )] dV . (8)
V
By an argument similar to that for E1 the first term in
equation (8) vanishes, and the second term contributes only
in the region V2 . Hence the field formulation of equation
(3) for the energy of the system can be recast in a current
and vector formulation as
Z
1
(9)
T =
(A2 · J + A1 · J ) dV
2
V2
Hysteresis in cylindrical type II superconductor
field, and they screen as much of the conductor from the
magnetic field as possible. It is assumed that for a given
value of the applied field the screening currents adjust in
such a way that the total energy of the system is minimized.
Practically, this energy minimum was found by employing
an iteration procedure using quasi-Newton and bisection
methods in an intelligent way [14].
A conceptual algorithm for energy minimization is
outlined below.
Figure 1. A sketch of the geometry where the
superconductor is immersed into the bore of the solenoid,
which produces a uniform field over the volume of the
conductor. Dimensions not to scale.
where the current density J is given as: J = curl H2 .
In this work the geometry of the superconductor is
chosen as a short cylinder of known dimensions. The
superconductor is positioned in the bore of the solenoid,
which is configured such that it produces a uniform field
over the volume of the superconductor (see figure 1). The
simulation consists of two parts.
(i) Minimize expression (3) or (9) for various values of
the applied magnetic field. This will give different
screening current patterns for the different values of
applied magnetic field.
(ii) Generate a complete hysteresis loop (M versus H )
curve of the trapped flux as a function of a timedependent uniform applied field. This is achieved by
sweeping the applied magnetic field between −Bm and
Bm where Bm is less than or equal to the full penetration
field Bp . The relation between B and H is assumed
linear and given as B = µ0 H where µ0 denotes the
magnetic constant.
3. Implementation procedure
The geometry of the superconductor was chosen as a short
cylinder, with dimensions of radius R = 10 cm and
height D = 20 cm. The dimensions of the solenoid were
subsequently adjusted in such a way that it produces a
uniform magnetic field over the volume of the conductor,
parallel to the z-axis (see figure 1). Other ways of
producing the uniform field are by means of a pair of
Helmholtz coils, or by placing the conductor in the air gap
between the poles of an electromagnet. It is assumed that
the applied field is generated with a magnitude B0 by some
means described above.
The conductor is cooled to a temperature below Tc
and the cylinder becomes superconducting and the applied
field is subsequently switched on. Screening currents are
generated in the superconductor in response to the applied
(a) Choose the magnitude of the applied field B0 , and
decide on the value of the critical current density Jc
of the superconductor. This can be done by choosing
experimentally determined values for a short cylindrical
superconductor in the mixed state. B0 is chosen such
that the penetration depth is small.
(b) Add a current with density Jc in a very thin outer shell
of the conductor as a starting guess for the current
configuration.
(c) Calculate the total energy of the system using equation
(3) or (9).
(d) Create another current shell next to the previous one,
but where the current has only penetrated some finite
distance towards the mid-plane, i.e. decreased in z with
some step symmetrically from the outer surface for a
constant value of r.
(e) Determine the best configuration for z using bisection
and Newton techniques which minimizes the total
energy.
(f) Increase the penetration depth radially towards the
centre by a fixed step and repeat (d) and (e) again.
(g) Repeat (d)–(f) until a global energy minimum is found
for the chosen value of the applied field.
The geometry of the concentric conductor shells was
established by using a commercial software package called
OPERA from Vector Fields Ltd in Oxford, UK [15].
4. Results
The results consist of two parts. The first task was to
derive a useful minimization procedure by which the total
energy of the system as given by equation (3) or (9) could
be minimized. As an illustration the applied magnetic
field B0 was chosen to be 3.0 T, and the critical current
density Jc was chosen to be 108 A m−2 (which is close
to experimentally measured values). The problem is now
to arrange the screening currents of the superconductor
such that the applied field is completely cancelled in the
interior of the conductor, where B and hence J are zero.
After applying the minimization procedure described in the
previous section the following results were obtained.
Figure 2 shows the screening current configuration of
the superconductor to screen out the applied field of 3.0 T
applied along the z-axis. The number of conductors N
to model the screening currents was chosen to be 20, so
that the computation effort was kept at a ‘manageable
level’. An increase in N would increase the accuracy of
the results obtained, but computation effort needed for the
minimization procedure would also grow at least in the
same proportion.
189
I Younas and J H P Watson
Figure 2. A cross-section along the z -axis of the cylindrically shaped superconductor. The boundary between the screening
currents and the interior of the conductor marks the border between the field-penetrated and field-free regions.
Figure 3. (a ) Variation of the axial magnetic field (created by the screening currents alone) along the z -axis. (b ) Variation of
the axial magnetic field (created by the screening currents alone) along the mid-plane (z = 0).
Figure 3 displays field plots of the axial field component
Bz (produced by the screening currents to cancel the applied
field of 3.0 T) along the z-axis and along the mid-plane
(z = 0). It can be seen from the figures that the axial
component of the magnetic field produced by the screening
currents is constant and equal to 3.0 T through most of the
conductor. Hence it can be concluded that the screening
currents have adjusted in such a way that the fields of these
have cancelled the applied magnetic field (B0 = −3.0 T)
in the current-free region.
The radial field component of the screening currents is
also zero (not shown) in the current-free region.
The true field distribution for this configuration can be
calculated by subtracting the applied magnetic field B0 =
3.0 T (applied axially, i.e. in the z-direction) from the field
190
generated by the screening currents in the superconductor.
Figure 4 shows the variation of the resultant magnetic field
across the superconductor. Since the applied magnetic field
does not have a radial component, and the net field is zero in
the interior of the conductor, it is concluded that the radial
component of the field created by the screening currents is
also zero in the current-free region. Hence the screening
currents have adjusted in such a way that the demagnetizing
effect of the geometry has been correctly accounted for.
The error due to discretization of the screening currents and
inaccuracy of the minimization procedure was estimated to
be less than 3%.
A set of simulations were carried out where the applied
field was increased by 1 T in each simulation and the
resulting screening current pattern was determined by the
Hysteresis in cylindrical type II superconductor
Figure 4. Variation of the net magnetic field (i.e. magnetic field generated by the screening currents and the applied field)
across the superconductor. It can be seen that the magnetic field is very close to zero in the current-free region of the
superconductor.
energy minimization procedure described earlier. It was
found that the superconductor was fully penetrated with
field for a value of 10.8 T of the applied magnetic field.
Figure 5 displays the different regions for various values of
the applied field. Only one-fourth of the geometry needs
to be shown, because of the axial and mirror symmetry of
the magnetic field.
When the applied magnetic field was decreased after
increasing it to the full penetration value, some of the
screening currents switched from +Jc to −Jc in the
following way. The screening currents will try to prevent
any changes in the established field pattern, which is also
a lowest-energy configuration for the system. When the
field is decreased to some value B2 after increasing it to
a value B1 , the conductor senses a net field change of
1B = B1 − B2 (positive) which is opposite to the original
field direction. To oppose the field changes some of the
screening currents switch sign to −Jc starting from the
edges and outer surface of the conductor (as this region
of the conductor is most immediately affected by the field
changes) such that a new energy minimum is achieved for
the resulting field configuration.
The average magnetization of the superconductor can
be calculated by the following known formula:
Z
1
(Btot − µ0 Ha ) dV
(10)
µ0 hMi =
V2
V2
where V2 denotes the volume occupied by the superconductor. Btot is the total magnetic field in the superconductor
Figure 5. Sketches for some free boundaries for various
values of the applied magnetic field.
and Ba = µ0 Ha is the applied magnetic field. When for
a virgin superconductor the applied magnetic field is increased from zero to +Bm and then reduced to −Bm and
lastly increased to +Bm again (i.e. a full field cycle) a hysteresis curve can be generated as shown in figure 6. The
magnitude of the field sweep must be less than or equal to
the full penetration field. Figure 6 shows a hysteresis curve
191
I Younas and J H P Watson
Figure 7. Penetration depth as a function of the applied
magnetic field for the infinitely long cylindrical
superconductor in a parallel-field configuration.
Figure 6. Magnetization versus applied field for the short
cylindrical superconductor. The magnetization is calculated
as µ0 M and the applied field as Ba = µ0 Ha , both having the
dimension tesla. The maximum applied field is 10 T.
(magnetization versus applied field) for Bm equal to 10 T.
It is now interesting to explore the situation which
arises experimentally. Usually magnetization curves have
been measured on a short cylinder and the Bean model
is subsequently applied to calculate Jc . We require
therefore to calculate the errors produced by estimating
Jc from magnetization of a short cylinder when using
the Bean model for an infinitely long cylinder. Let us,
as an illustration, calculate the critical current density Jc
using five different magnetization curves. An example
of a magnetization curve is shown in figure 6 with a
maximum applied field of 10 T. In the calculations it will be
assumed that the results obtained were derived using Bean
formulae for an infinitely long cylindrical superconductor
in a parallel-field configuration.
The trapped field for the cylinder depends on the ratio of
the magnitude of the applied field Ha to the full penetration
field Hp (figure 7). The average field in the infinite cylinder
can be calculated by the following formula
µ0
hBi =
πR 2
ZR
Hs (r)2πr dr
(11)
0
where Hs denotes the field distribution in the superconductor and R is the radius of the cylinder.
When the applied field is increased to Hm ≤ Hp after
which it is reduced to Ha ≥ 0 an expression for the average
flux density in the superconductor can be derived:
1
1
1
2µ0
hBi = 2 − Hp Ha2 + Hp Ha Hm + Hp Hm2
Hp
4
2
4
1 3 1
1
1
− Ha − Ha Hm2 − Hm3 + Ha2 Hm .
(12)
24
8
8
8
Equation (12) in the form shown is very general, and we
shall consider two special cases as listed below.
Assume that the applied field is increased to Hm ≤ Hp
and kept at this value, i.e. Ha = Hm ; then the magnetization
192
of the superconductor is derived using equations (10) and
(12) as
1
M = 2 Hp Ha2 − 13 Ha3 − Ha .
(13)
Hp
This equation is a second-order polynomial equation in
Hp = Jc R, when M and Ha are known. Equation (13)
reduces to the well-known expression for M from the Bean
model [7], i.e.
1
1
M = − Hp = − Jc R
3
3
(13a)
when the applied field Ha is set to the full penetration field
Hp .
Now, if the applied field is reduced to zero after
increasing it to Hm ≤ Hp , then an expression for the
remnant magnetization is
M=
1
Hp2
1
H H2
2 p m
− 14 Hm3
(14)
where Hm denotes the maximum applied field. Equation (14) is also a second-order polynomial equation in
Hp = Jc R when M and Hm are known. It is interesting to
note that (14) reduces to
M=
1
1
Hp = Jc R
4
4
(14a)
when the maximum applied field Hm equals the full
penetration field Hp .
Table 1 was generated using the calculated values for
magnetization for the short cylinder using Jc = 108 A m−2 ,
and then applying the Bean equations (13) and (14) to find
Jc . This is done in order to estimate the errors in Jc which
arise when using the Bean formulae on a short cylinder
where the demagnetizing field is high.
The first column in table 1 contains values for the
maximum applied field in the hysteresis cycle. The
second column presents the calculated magnetization using
equation (10) when the applied field is at the maximum
value. In the third column the magnetization is calculated
at zero field after the applied field is first increased to some
maximum value and then subsequently reduced to zero.
The fourth and fifth columns in table 1 contain values of Jc
Hysteresis in cylindrical type II superconductor
Table 1. Magnetization using equation (10) and derived values for Jc using equations (13) and (14) for different values for
the sweep field.
Maximum
applied field
Hm (A m−1 )
M using (10) at
H = Hmax for
short cylinder (A m−1 )
M using (10) at
H = 0 for short
cylinder (A m−1 )
Jc from (13) at
H = Hmax using
Bean formula (A m−2 )
Jc from (14) at
H = 0 using
Bean formula (A m−2 )
1.5915×106 (2.0 T)
−1.1331×106
2.3515×105
0.4931×108
50.7% error
0.4415×108
55.9% error
3.1831×106 (4.0 T)
−1.7237×106
6.4768×105
0.5636×108
43.6% error
0.5598×108
44.0% error
4.7746×106 (6.0 T)
−1.9934×106
1.1180×106
0.6035×108
39.7% error
0.6381×108
36.2% error
6.3662×106 (8.0 T)
−2.1823×106
1.5580×106
0.6547×108
34.5% error
0.7448×108
25.5% error
7.9577×106 (10.0 T) −2.2150×106
1.8795×106
(0.6585×108 )
34.2% error
1.0403×108
4.0% error
which have been calculated using the values for M in the
second and third columns and applying the Bean equations
(13) and (14) respectively. The critical current density is
found by first solving equations (13), (14) for Hp , i.e. the
full penetration field, and then using Hp = Jc R for the
infinitely long cylindrical superconductor in a parallel field.
Since a second-order polynomial equation has always two
solutions, two values for Jc are obtained. The Jc values
that leads to values of Hp less than the maximum applied
field (first column) are discarded. The derived values for
Jc are similar in the two methods for small applied fields,
but, as the magnitude of the maximum applied field is
increased, the value of Jc obtained in the remnant state (i.e.
Ha reduced to zero after some maximum value) becomes
closer to the actual value used in the short cylinder, namely
Jc = 108 A m−2 .
The first column contains values for the maximum
applied field. the second and third columns contain the
magnetization of the superconductor at H = Hm and H = 0
in the short-cylinder configuration. The fourth and fifth
columns contain derived values for Jc using Bean formulae.
The fourth and fifth columns in table 1 contain values
for Jc together with an estimated deviation from the actual
value of Jc = 108 A m−2 . The last value for Jc in the fourth
column is in parentheses because the corresponding Hp
value is less than the maximum applied field (first column).
The best value for Jc differs only by 4% from the value
used in this work, i.e. Jc = 108 A m−2 .
in earlier work in this context. The error arising from the
discretization of the cylinder into finite concentric current
segments was not greater than 3% in all calculations. A
comparison has been made between the critical currents
derived from the magnetization curves, assuming they were
obtained for an infinitely long superconducting cylinder in a
parallel magnetic field configuration (Bean model), with the
value of Jc = 108 A m−2 used in this work. The results
indicate that the demagnetizing effect of the cylinder is
dominant for low values of the applied magnetic field. As
the magnitude of the maximum applied field is increased,
the discrepancy between the derived value for Jc and the
reference value becomes smaller. For an applied field of
10 T the error in Jc is about 4% when the magnetization
is measured in the remnant state, i.e. at H = 0. This
indicates that the demagnetizing effect has been reduced to
a large extent when the maximum applied field is 10 T in
the current cylinder configuration. The real full penetration
field in the current cylinder configuration was found to be
10.8 T. This value differs by 20% from the Bean value
for the infinite cylinder configuration (13.07 T) when the
maximum applied field is 10 T.
The obtained results indicate that the best estimates
for Jc , using the experimental values of magnetization for
a short cylinder, are obtained when Hm ∼
= Hp and the
remnant magnetization (i.e. H reduced to zero) is used to
calculate Jc from the Bean formula.
5. Conclusion
The authors acknowledge with gratitude support from the
EPSRC for this project.
We have shown that the demagnetizing field arising from
the finite dimension of the type II superconductor can be
accounted for and the flux penetration depth calculated for a
given value of the applied field by an energy minimization
approach. The hysteresis is then naturally derived by
switching appropriate currents during the magnetic field
sweep cycle.
An enhancement of the minimization
procedure together with a higher discretization will lead
to higher accuracy of the energy minimization, but this
has deliberately been avoided as the intention of this work
was to describe a natural and obvious approach not used
Acknowledgments
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