Physica C 306 Ž1998. 154–162
Analytical approach to AC loss calculation in high-Tc
superconductors
V. Sokolovsky ) , V. Meerovich, S. Goren, G. Jung
Physics Department, Ben-Gurion UniÕersity of the NegeÕ, P.O.B. 653, 84105 Beer SheÕa, Israel
Received 7 April 1998; revised 25 June 1998; accepted 29 June 1998
Abstract
Using a linear spline approximation for the E–J characteristic of a superconductor and representing the solution in form
of series, analytical expressions for AC losses have been obtained. The expression explains experimentally observed
frequency and magnetic field dependencies of AC losses. Cases of complete and incomplete magnetic field penetration have
been distinguished. AC losses per cycle decrease with increasing frequency in the case of incomplete penetration, the case
relevant to thick slabs and low amplitude magnetic fields, while in thin slabs and large magnetic fields they increase with
increasing frequency, the case of complete penetration. The analysis of the analytical solutions obtained has given a simple
criterion for the applicability of the critical state model ŽCSM. to the calculations. This criterion involves characteristics of
both superconductor and applied magnetic field. The physical meaning of the criterion in terms of ratios between the
characteristic decay times of magnetic field energy in a superconductor and the period of applied magnetic field has been
established. It has been shown that the analytical solutions can be applied for various forms of E–J characteristics by means
of defining effective critical current density and flux flow resistivity. q 1998 Elsevier Science B.V. All rights reserved.
Keywords: Critical state model; AC losses; Superconductor
1. Introduction
The classical critical state model ŽCSM. introduced by Bean w1x has been successfully used for the calculation
of magnetic characteristics of classical type-II superconductors. This simple CSM model provides analytical
solutions for most practical cases, even for those where the critical current density depends on magnetic field.
However, pronounced flux motion in high temperature superconductors ŽHTSC. causes significant deviations
from the CSM predictions w2–5x. In particular, one notes a pronounced frequency dependence of AC losses per
cycle in HTSC materials w2,5x. To take into account the non-linear electromagnetic properties of HTSC in
calculations of AC losses, Maxwell’s equations are completed by equations describing flux motion effected by
pinning and viscous forces w6x, or a non-linear dependence of electric field E on current density J w7,8x.
Several models have been developed to describe the behavior of HTSC in the flux creep and flux flow
regimes w4,8–10x. The simplest approximation, referred to as the extended critical state model ŽECSM.,
)
Corresponding author. Tel.: q972-7-647-2458; Fax: q972-7-647-2903; E-mail: victorm@bgumail.bgu.ac.il
0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 3 4 7 - 5
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
155
accounts for the viscous vortex motion in the flux flow regime at currents exceeding the critical current and
neglects any motion at currents smaller than the critical current w9x. More elaborated approach accounts also for
the flux creep at currents close to the critical current resulting in a non-linear shape of current–voltage
characteristics. Using such approach several numerical solutions for AC losses and for the magnetic field
penetration problem have been found. In general, these solutions are based on the approximation of E–J
characteristic by a defined functional form w5,8,10,12x. The most commonly used approximations are the power
and exponential laws w4,11x.
The analytical description of AC losses that accounts for the vortex motion has been presented in several
papers w6,7,13x. Each approach starts from imposing particular limitations andror conditions on the solutions
obtained.
It was shown that in many cases there are regions of frequencies and amplitudes of the external magnetic
field at which the CSM can be used w4,7,14,16,17x. For example, Dresner in Ref. w14x has shown that the CSM
can be used for materials demonstrating power-law E–J dependence since in this case AC losses per cycle are
relatively insensitive to the actual value of the power-law exponent. Rhyner in Ref. w7x obtained the formula
based on the CSM approach containing an additional factor which has to be calculated numerically separately.
However, there is no general approach to the determination of the limits of applicability of the CSM to
calculations of magnetic characteristics of HTSC.
In our previous papers we have analyzed the penetration of time-increasing magnetic field into a superconducting slab and have shown that the region of the CSM applicability is determined both by the properties of the
superconductor itself and by the rate of change of external magnetic field w3,15x. In this paper we discuss an
analytical approach to the calculation of the losses in HTSC samples in external sinusoidal magnetic field and
provide a simple criterion for the applicability of the CSM to such calculation. As a basic model, we use the
ECSM with field-independent resistivity and critical current, and then generalize our results to other non-linear
models.
2. Approximation based on extended critical state model
Consider a superconducting slab with thickness 2 D whose inward surface normal points in the positive
x-direction. The external sinusoidal magnetic field He s H0 sinŽ v t . is applied in the z-direction, which is
parallel to the superconductor surface, while electric field E and current J are induced along the y-axis.
Let us start by approaching the AC losses problem in the framework of the ECSM w9x. This model
approximates the current–voltage characteristic of a superconductor by:
J r f Ž < J < y Jc .
Es
, < J < ) Jc ;
<J<
E s 0, < J < - Jc ,
Ž 1.
where r f is the flux flow resistivity and Jc is the critical current density. This characteristic, marked by the
label ECSM, is shown in Fig. 1 along with the CSM approximation and real E–J characteristic.
We assume the linear relation, B s m 0 H, between the magnetic induction inside type-II superconductor, B,
and magnetic field intensity H, with m 0 being the vacuum permeability. This is a good approximation when
everywhere inside the superconductor the field intensity H is larger than the lower critical field Hc1 of the
superconductor w4x. Dimensionless Maxwell’s equations for magnetic field inside a superconductor are:
Eh
s yj,
Ž 2.
Ej
Ee
Eh
s ya
,
Ž 3.
Ej
Eq
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
156
Fig. 1. Approximations of a real E – J characteristic according to the CSM and the extended CSM.
where j s JrJc , q s v t, h s HrH0 , e s ErŽ Jc r f ., a s vm 0 H02rŽ r f Jc2 ., and j s xrx p , where x p s H0rJc is
the maximum depth of penetration of magnetic field into the superconductor according to the CSM. Dimensionless external sinusoidal magnetic field on the superconductor surface takes the value h e s sinŽq .. In the current
section we will assume that r f and Jc do not depend on magnetic and electric fields. The relation Ž1. in the
dimensionless units becomes
° jŽ < j < y I .
< j<
¢0
e s~
if
< j < ) 1;
if
< j < F 1.
Ž 4.
Dimensionless Maxwell’s equations contain only a single parameter a, and the solution of Eqs. Ž2. and Ž3.
can be sought in the form of expansion in power series of parameter a.
h s h c q ah1 q . . . ;
e s a Ž e c q ae1 q . . . . ;
j s jc q aj1 . . . ,
Ž 5.
where zero-order approximations, h c for the magnetic field and e c for the electric field, are given by the CSM
solutions obtained from Eqs. Ž2. and Ž3. with j s jc s "1. To analyze the conditions at which the deviations
from the CSM are small, we will consider only the first two terms in expansions Ž5..
The problem is symmetric and it is sufficient to consider hereafter only one-half of the slab thickness and
one-half of the period of applied AC magnetic field, e.g., between the minimum h e s y1 to the maximum
h e s 1. According to the CSM, for phase q s 0 current density j s y1 for all j reached by the penetrating
magnetic field. This initial profile of the magnetic field is marked in Fig. 2 as 1. As the magnetic field increases,
the profile changes and starts to contain a region with j s 1. Let us denote by j 1 the point at which current
density reverses its sign. We have j s 1 for j F j 1 and j s y1 for j ) j 1. In the case of incomplete
penetration of magnetic field, the region with j s y1 extends to j s 1. In the case of complete penetration
ŽBean’s penetration depth x p greater than the half-thickness of the slab D ., the region with jc s y1 extends to
d s Drx p . The zero-order solution of Maxwell’s Eqs. Ž2. and Ž3. is
°0;
¢ dq Ž j y j . ;
y1 q j ;
hc s
and e c s
he y j ;
½
~ dh
e
1
if j ) j 1
if j F j 1 .
Ž 6.
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
157
Fig. 2. Bean’s profiles of AC magnetic field in a superconducting slab. The external magnetic field h e increases from y1 Žline 1. to q1
Žline 4.. The field h p corresponds to the case of complete penetration Žline 3..
The next term in series of current density expansion can be determined by putting in Eq. Ž4. the electric field
from Eq. Ž6.,
°
¢y1
d he
1qa
Žj
j s~
dq
1yj
if j F j 1
.
Ž 7.
if j ) j 1 .
Note that the term proportional to the parameter a goes to zero whenever external magnetic field reaches its
maximum or minimum. Consequently, functions h1 and e1 in Eq. Ž5. are zero for j ) j 1. For j F j 1 magnetic
field can be determined from Eqs. Ž2. and Ž7. and reads:
h s he y j y a
d he
dt
ž
j2
j1 jy
/
2
.
Ž 8.
The coordinate j 1 can be obtained from the continuity condition for the magnetic field at this point as:
he q 1
j1s
ya
2
d he Ž he q 1.
dt
2
16
.
Ž 9.
The electric field inside a superconductor follows from Eqs. Ž3. and Ž8. and is given by:
° dh
a
e
dt
e s~
Ž j1yj . qa
d he
2
¢ a dt Ž dyj . qa
d 2 he
dt2
d 2 he
2
dt2
ž
ž
j1 j 2
j3
y
2
6
dj 2
j3
y
2
y
j 13
3
d3
y
6
3
/
d he
/ ž /
q
dt
2
Ž j 2 y j 12 .
4
if h e - h p ;
Ž 10 .
if h e G h p .
The value of the complete penetration field h p , i.e., the applied field at which the j ) 0 in the entire slab, is
determined from Eq. Ž9. under the condition j 1 s d .
The surface density of AC losses per cycle, normalized to P0 s r f Jc H0rv , can be expressed using the
Poynting vector
ps2
2p
H0
eh e dq ,
Ž 11 .
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
158
where the electric field in Eq. Ž11. is taken as the field on the surface of the slab at j s 0. From Eq. Ž11. we
obtain
ps2 a
h 3p
q
3
h2p
q d Ž1 y h p . y
2
3sin Ž 4q1 .
q a2
y
64
25q1
q
2 a2d 3
1
q
6
3
sin Ž 2q1 .
48
q
6
p y q1 y
11sin3q1
y
sin Ž 2q1 .
2
11sin5q1
18
90
.
Ž 12 .
The above expression was obtained in the framework of the ECSM, the approximation which takes into
account only the viscous motion of vortices and assumes zero electric field below Jc thus neglecting flux creep
processes.
As indicated by many experimental results w4,11x, the pronounced flux creep is observed only in the close
vicinity of Jc and the characteristic time of the magnetic relaxation is much longer than the period of applied
magnetic fields. In this conditions creep processes during a cycle do not cause marked deviations of the current
density from the critical one and consequently, do not influence the distribution of magnetic field. Within this
approximation the solutions Ž8. – Ž12. remain unchanged. In order to evaluate losses due to creep processes, we
assume that these processes contribute to AC losses only in the region where j ) j 1 , see Fig. 2. Within this
scenario contributions to AC losses by flux creep and flux flow processes became spatially separated.
To estimate the upper limit of creep contribution to AC losses, the electric field caused by flux creep is taken
to be constant Ecr s E0 and can be determined from the real E–J characteristic of the superconductor at J s Jc ,
see Fig. 1. The creep contribution to AC losses is
pcr s 2
q1
H0
dq
d
c cr d j s ye cr
Hj j e
Ž 2 dq1 y q1 q sin q1 . ,
Ž 13 .
1
where ecr s yE0rŽ Jc r f .. Upon integrating Eq. Ž13. we have taken into account that the critical current density
in the region j ) j 1 is negative.
The expressions describing AC losses have different forms for the cases of complete and partial penetration
of the magnetic field into a superconducting specimen. In the case of incomplete penetration h p s 1, q1 s p ,
d s 1 and we obtain for the surface density of losses per cycle, determined as the sum of Eqs. Ž12. and Ž13.,
25p a 2
4a
y e cr p .
3
48
In the case of complete penetration, for d < 1, the losses take the form:
pt s
y
pt s 4 a d 2 1 y
ž
2d
3
/
2p a 2d 3
q
y
3
8 e cr d 3r2
3
.
Ž 14 .
Ž 15 .
3. Extension to power-law and exponential E–J characteristics
In the CSM model approach the current density equals rigorously the critical one. The above solutions for
AC losses in the framework of the ECSM are valid when the current density differs only slightly from a
constant. Numerical calculations performed in Ref. w10x demonstrate that the current density is indeed close to a
constant within certain ranges of external field parameters. Rhyner in Ref. w7x noted the possibility of
introducing the effective critical current which was determined numerically. In general, whenever the current
density does not change significantly during AC period, the AC losses problem may be reduced to the solution
obtained above. For this purpose one has to determine the effective values of the flux flow resistivity, critical
current density, and coordinates of the point E0 in Fig. 1. Several procedures can be employed for reducing a
real E–J characteristic to the form used in our first-order approximation. Our approach is illustrated in Fig. 3.
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
159
Fig. 3. Procedure employed for determining effective critical current and flux flow resistivity. Linearized E – J dependence is tangent to the
real curve at coordinates Ž² J :,² E :.. The intercept with the J-axis defines the effective critical current density Jc . The effective flux flow
resistivity r f is determined as the slope of thus obtained linear approximation.
The electric field intensity varies in time and depends on the spatial coordinate. Let us define the space- and
time-average electric field ² E : in the slab. For sinusoidal external magnetic field with the amplitude H0 , the
time-average electric field can be estimated as:
² E :t s 2 m 0 v H0 Ž D y x . rp .
Finally, averaging over the slab thickness gives:
² E : s m 0 v H0 Drp .
Ž 16 .
Next, we build the linear approximation of the characteristics as the tangent to the real E–J curve at the
coordinates Ž² J :,² E :.. The intersection of the linear approximation with the J-axis gives an effective value of
the critical current density. The value of E0 follows as the electric field on the real E–J characteristic at this
effective critical current density.
Applying our procedure to the power-law characteristic E s Ec Ž JrJc0 . n , we obtain the analytical expressions
for the effective resistivity and effective critical current as
rf s
nEc
² E:
Jc0
Ec
Ž ny1 .rn
ž /
ž /ž /
Jc s Jc0 1 y
1
² E:
n
Ec
,
1rn
.
Ž 17 .
Describing the real E–J curve by an exponential characteristic E s Ec expwŽ J y Jc0 .rJ1 x a 4 yields
rf s a
² E:
Jc0
ln
² E:
ž /
ž /
Jc s Jc0 q J1 ln
Ž a y1 .r a
,
Ec
² E:
Ec
1r a
q
J1
a
ln
² E:
ž /
Ec
Ž1y a .r a
.
Ž 18 .
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
160
Note that the effective critical current density does not have the physical meaning of a critical current. This
quantity represents a constant current density around which the real current density varies. The effective critical
current is always smaller than the critical current density Jc0 . The effective critical current and resistivity depend
on magnetic field and frequency because ² E : is proportional to the frequency and amplitude of the external
magnetic field.
4. Discussion
To discuss our results, we will rewrite Eqs. Ž14. and Ž15. in the dimensional form. We obtain for incomplete
penetration:
Ps
4m 0 H03
3 Jc
ž
1y
75p vm 0 H02
r f Jc2
192
3p
q
E0 Jc
4 vm 0 H02
/
Ž 19 .
and for complete penetration, neglecting term 2 dr3 in comparison with 1 in Eq. Ž15.:
P s 4m 0 H0 Jc D 2 1 q
p vm 0 H02 Jc D
6
r f Jc2
H0
2
q
E0 Jc
3 vm 0 H02
H0
ž /
Jc D
1r2
.
Ž 20 .
Expressions Ž19. and Ž20. allow one to establish a simple criterion of applicability of the CSM as well as to
analyze the influence of vortex motion on AC losses. By neglecting all frequency dependent terms in Eqs. Ž19.
and Ž20. one gets the well-known expressions for hysteresis losses obtained in the framework of the CSM. The
frequency dependent terms in the brackets appear as a result of subsequent corrections to the CSM approximation. The first frequency dependent terms represent losses related to the flux flow, while the last terms describe
losses due to the flux creep. The criterion for the applicability of CSM in the dimensional form becomes:
E0 Jc < m 0 v H02 < r f Jc2 .
Ž 21 .
The criterion Ž21. is a general criterion providing the applicability of the CSM for all cases. For complete
penetration, the applicability range of the CSM is extended because the additional terms in expression Ž20.
contain factor d s D JcrH0 , the ratio of the slab thickness to the Bean penetration length. With decreasing the
slab thickness, the contribution of the flux flow decreases while the flux creep influences more.
In a marked difference from other approaches relating the applicability of the CSM to the type of the E–J
characteristic of a superconductor, see e.g., Refs. w4,9,12,14x, we find that the criterion includes both the
characteristics of a superconductor Žresistivity r f , critical current density Jc , and the characteristic electric field
in the flux creep regime E0 . as well as parameters of the external magnetic field Žfrequency v and amplitude
H0 .. The criterion relates the applicability of the CSM to values of the frequency and amplitude of applied
magnetic field. However large the frequency may be, there are amplitudes of the magnetic field at which the
CSM is valid. In the same time, even at very low frequencies there are amplitudes at which the conditions of the
CSM applicability break down. The additional terms have inverse dependence on frequency and magnetic field:
the influence of the flux flow increases with frequency orrand field, whereas the contribution of the flux creep
decreases. The term related to the flux creep is always positive while the sign of the term associated with flux
flow is negative for thick slabs and reverses its sign with decreasing the slab thickness. This unusual behavior is
explained by competition of two different phenomena. On the one hand, the increase of the current density ŽEq.
Ž7.. leads to the increase of AC loss volume density. On the other hand, the same current density increase
causes to narrowing the region inside the slab where losses are observed Žthe second member in expression Ž9.
for j 1 is negative.. These patterns of the penetration are well confirmed by numerical calculations w3x. In the
case of complete magnetic field penetration, after the field penetrates the whole of the slab, further process is
associated only with the increase of the current density.
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
161
The obtained results explain the experimentally observed dependencies of AC losses on frequency, magnetic
field and sample thickness. So, the observed decrease of losses with increasing frequency w5x can be related to
the influence of the flux flow in the case of incomplete penetration. This case corresponds to large thickness
slabs or sufficiently small amplitudes of magnetic field. In the case of complete penetration, expression Ž20.
predicts the increase of losses with frequency and it corresponds to thin slabs or high magnetic fields. The same
results were obtained by numerical calculations in Refs. w8,12x.
Note that the factor m 0 H02r2 appearing in the frequency dependent terms of Eqs. Ž19. and Ž20. is the density
of magnetic field energy, while the factors r f Jc2 and E0 Jc are the densities of Joule losses. The first frequency
dependent terms are proportional to the ratio between the characteristic flux flow decay time of the magnetic
field energy tc1 s m 0 H02rr f Jc2 and the period T s 2prv . The second terms are proportional to the ratio
between the period T and the characteristic time of the energy decay due to losses E0 Jc in the creep flux regime
tc2 s m 0 H02rE0 Jc . Physical meaning of our criterion is now clear. The CSM can be applied when the
characteristic magnetic energy flux flow decay time is small with respect to the AC field period, tc1 < T, and
simultaneously, the characteristic flux creep decay time is much larger than the AC period tc2 4 T.
The approach developed above assumes that the critical current density Jc and the flux flow resistivity r f do
not depend on a local magnetic field. Let us evaluate now how the results will be affected by allowing for the
dependence of Jc and r f on the magnetic field. The general form of these dependencies can be written as w15x:
Jc s
rf s
Jc0 Hjn
Ž < H < q Hj .
n
,
r 0 Ž < H < q Hr .
Ž 22 .
m
Hrm
,
Ž 23 .
where J0 and r 0 are the values of the critical current and flux flow resistivity at zero magnetic field, Hj , and Hr
are constants.
Note that Eqs. Ž2. and Ž3. remain unchanged when the dimensionless parameter a in Eq. Ž3. is replaced by
the parameter
aX s
m 0 v H02
r f Ž H0 . Jc2 Ž H0 .
s
m 0 v H02 Hrm Ž H0 q Hj .
m
2n
r 0 Ž H0 q Hr . Jc02 Hj2 n
Ž 24 .
that takes into account the magnetic field dependence of the critical current and flux flow resistivity.
The initial approximation given by the CSM with the dependencies Ž22. and Ž23. can be presented in an
analytic form but the analytic solution for the next approximations is very complicated. Nevertheless, we can
formulate the criterion for the CSM applicability as the condition aX < 1. The frequency dependence of the
additional terms remains as before. The dependence on magnetic field is changed crucially at small values of the
constants Hj and Hr .
5. Conclusion
Based on the segment-linear E–J characteristic of a superconductor, we have obtained an analytical solution
that explains the basic features of AC losses in HTSC and gives the simple criterion for the applicability of the
classical CSM for such calculations.
162
V. SokoloÕsky et al.r Physica C 306 (1998) 154–162
Acknowledgements
This research was supported by the Israeli Science Foundation founded by the Israeli Academy of Sciences
and Humanities, and the G.I.F., German–Israeli Foundation for Scientific Research and Development.
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