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Influence of Jc(B) on the full penetration current of superconducting tube
Physica C 443 (2006) 23–28 www.elsevier.com/locate/physc
Influence of Jc(B) on the full penetration current of superconducting tube B. Douine *, K. Berger, D. Netter, J. Le´veˆque, A. Rezzoug GREEN, University of Nancy, BP 239, F-54506 Vandoeuvre-le`s-Nancy, France Received 1 December 2005; received in revised form 29 March 2006; accepted 13 April 2006 Available online 21 June 2006
Abstract It is well known that the critical current density Jc of a superconducting material depends on the magnetic field B. If magnetic independent Jc is chosen for analytical calculation of current distribution, the critical current Ic corresponds to full penetration current Ip. Ic is a measured current with 1 lV/cm criterion and Ip is a calculated current. The aim of this paper is to calculate the influence of the Jc(B) variation on Ip of a superconducting tube. To calculate Ip, which is depending on the material itself, a linear function Jc(B) is sufficient to obtain realistic values by analytic way. We need to have a linear Jc(B) law that is close to the measured Jc(B) characteristics presented in this paper. The linear Jc(B) law chosen was used for the calculation of the distribution of both magnetic field B(r, t) and the current density J(r, t). These distributions allow the analytical calculation of Ip. The calculated results of magnetic field distribution and full penetration current with Bean model and linear model are compared. We also present the variation of critical current with the characteristic parameters of the material. The present results, allow to understand the relationship between the full penetration current variation of a sample and the variation of the Jc(B) characteristics. Ó 2006 Elsevier B.V. All rights reserved. PACS: 74.60.Jg; 74.25.Ha Keywords: HTc superconductor; Magnetic field dependence; Critical current
1. Introduction To relate the irreversible magnetization to the field and current profiles inside a superconducting sample, Bean [3] introduced the critical state concept and assumed that the critical current density at any point of the sample can only take one constant unique value, the critical current density Jc, among three different states Jc, Jc and zero. Thanks to this model, the calculation of the current distribution is possible in case of a superconducting cylinder [1] or tube [2] fed with transport current. With this model, when the fed current i(t) increases into a superconducting sample, the current density penetrates up to the critical current Ic. *
Corresponding author. Tel.: +33 (0)3 83 68 41 25; fax: +33 (0)3 83 68 41 33. E-mail address: [email protected] (B. Douine). 0921-4534/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.04.090
For Ic the current density is equal to Jc everywhere in the superconducting material and the critical current corresponds to the full penetration current Ip. This is the complete penetration state. So the relation between Jc and Ic is very simple, Ic = Jc Æ S, where S is the section of the sample. Ic is used to calculate the AC losses [8]. Experimentally Ic is defined though the critical electric field Ec. For high temperature superconductors this value is generally fixed to 1 lV/cm. Unfortunately, for superconducting materials, the real critical current density varies with the magnetic flux density B [4–6]. In this case Ic does not correspond to Ip. Ic remains a measured current with the 1 lV/cm criterion. Ip is the value of the applied current i(t) at which the current density arrives to the centre of the sample. It’s a calculated current. The aim of this paper is to calculate the influence of the Jc(B) variation on the full penetration current of a superconducting tube. Considering Jc(B)
24
B. Douine et al. / Physica C 443 (2006) 23–28
for complete penetration the current density is not constant in the superconducting material because the magnetic flux density is not constant. As consequence, the relation between Ip and Jc(B) is not easy to calculate. For the calculation of Ip, the first step consists in setting a Jc(B) law that must be close to the measured Jc(B) characteristics. In a second step, the analytical calculation of current density, electric field and magnetic flux density distributions are presented. Finally the formula of Ip taking into account the variation of Jc(B) is derived. 2. Analytical calculation of the flux density distribution in a superconducting tube 2.1. Studied sample The aim of this paper is to calculate the full penetration current Ip of a tube based on the Jc(B) variation. To simplify the calculations, without reducing the generality of the problem, the applied current i(t) to the tube increases from 0 to Imax (Fig. 1), which is sufficient to determine Ip. The internal radius Rin, the external radius Re. and the length h of the tube are defined in Fig. 2. The current i(t) circulates along [Oz]. The edge effects are neglected in our calculations; E, J and B do not depend on z.
J ¼ J c
and
ð4Þ
E>0
or E ¼ 0 and
J c < J < þJ c
When the current rises, the current density penetrates from the external radius Re toward the internal radius Rin and its direction is [Oz]. Because of symmetry, the current density being oriented along the [Oz] axis, the flux density B(r, t) has only one component ~ Bðr; tÞ ¼ Bðr; tÞ ~ uh First we are going to prove that an analytical expression can be found for B(r, t) with a linear model for Jc(B). Considering the relation (2) and (4) 1 oðr Bðr; tÞÞ Bðr; tÞ ¼ l0 J c ðBÞ ¼ l0 J c0 1 ð5Þ r or Bj0 where jBj = B since B > 0. 2.3. Flux density distribution B(r, t) The analytical solution of expression (5) is exp l0BJj0c0 r B2j0 þK Bðr; tÞ ¼ Bj0 r l0 J c0 r
ð6Þ
K is a constant which can be calculated using the boundary condition at r = Re. The Ampere law enables to write
2.2. Modeling of the problem The electromagnetic behavior of superconductor is governed by the Maxwell equations
Bðr ¼ Re ; tÞ ¼ be ðtÞ ¼
l0 iðtÞ 2pRe
ð7Þ
Considering (6) and (7), we can deduce
Bðr; tÞ ¼
Bj0 ðl0 J c0 r Bj0 Þ þ exp
l0 J c0 ðRe rÞ Bj0
B2j0 þ l0 J c0 Re be ðtÞ Bj0
o~ B !~ rot E ¼ ot !~ rot B ¼ l0 ~ J
ð1Þ ð2Þ
For the superconducting material, we consider
Bj0 ðl0 J c0 dðtÞ Bj0 Þ þ exp
ð8Þ
l0 J c0 r
l0 J c0 ðRe dðtÞÞ Bj0
This expression is valid for B(r, t) > 0. So we have to determine the point where B(r, t) = 0. To do that, one can remark that it is equivalent to solve B(r = d(t), t) = 0 with d(t) = Re pd(t), where pd(t) is the penetration depth. Using the relation (8) and mathematical software, we obtain
B2j0 þ l0 J c0 Re ðbe ðtÞ Bj0 Þ
l0 J c0 dðtÞ
¼0
ð9Þ
Bj0 dðtÞ ¼ ð1 þ W ðX 1 ÞÞ l0 J c0 B ¼ l0 H
ð3Þ
E(J) is given by the critical state model defining the relation between electric field E and current density J
X 1 ðtÞ ¼
ðB2j0 þl0 J c0 Re ðbe ðtÞBj0 ÞÞ B2j0
exp
l0 :J c0 :Re Bj0
1 , where X1 >
0. W(X1) is Lambert’s W-function and is well-known in many mathematical libraries.
B. Douine et al. / Physica C 443 (2006) 23–28
25
i(t)
Solenoidal inductor
IMAX
Bext I Superconducting tube U z
t1
t
Fig. 1. Current supply.
z Fig. 3. Jc(B) measurement device. Re
Rin
100 90
h
Bext
U(µV)
80
0 x
y
θ
0,085 T
70
0,043 T
60
0,025 T
50
0,014 T 0,011 T
40
0,007 T
30
0,004 T 0
20
Fig. 2. Studied superconductor tube.
1µV/cm * h
10 0
3. Experimental Jc(B) of the sample
0
100
I(A)
Table 1 Jc(B) laws JcB, JK0, BK, Jc0, and Bj0 are constants and depend on the superconducting material Eq. no.
Fig. 4. Measure of sample voltage drop versus direct current for different external magnetic flux density.
Jmc(Bext) and Jc(B) are different because – Jc(B) is a local law and Jmc(Bext) is a macroscopic law – B is not equal to Bext because of the self magnetic field BSF : ~ B ¼ ~ BSF þ ~ Bext Region 1
Region 2
5 4 J(A/mm2)
Several variation laws Jc(B) are presented in Table 1. The experimental tests are made using a cylindrical current lead of BiSCCO. The dimensions of this sample are: Rin = 3.8 mm, Re = 5 mm, tube section S = 33 mm2 and h = 11.7 cm. Without external magnetic field, the measured critical current is Imc0 = 96 A, with a 1 lV/cm criterion. To obtain the experimental curve Jc(B) of this tube, it was fed with direct current I and submitted to an external magnetic flux density Bext parallel to the axis [Oz] (Fig. 3). The sample voltage drop U versus I is measured for different external magnetic flux densities Bext (Fig. 4). From the measured critical current Imc(Bext) corresponding to the voltage drop equal to 1 lV/cm h (Fig. 4). We can ext Þ deduce the J mc ðBext Þ ¼ I mc ðB . Fig. 5 shows the curve S related to this function.
3 2 1 0
11
J c ðBÞ ¼ J cB J K0 J c ðBÞ ¼ 1 þ BjBjK
12
J c ðBÞ ¼
10
50
J c0 ðBj0 jBjÞ Bj0
Bean model [3]
0
0.005
0.01
0.015
0.02
B(T)
Kim model [7] Jc(B) Kim
Jc(B) Lin.
Jmc(Bext)
Linear model [5] Fig. 5. Measured Jc(Bext), Jc(B) with the Kim model and the linear model.
26
B. Douine et al. / Physica C 443 (2006) 23–28
When the value of Bext is sufficiently large, BSF becomes negligible and the values of Jmc are very close to Jc(B). Among the laws represented in Table 1, the Kim model is the most suitable to extrapolate Jmc(B). With the appropriate values of JK0 and BK, we propose the following expression of the Kim law J c ðBÞ ¼
4:6 106 jBj 1 þ 0:004
A=m2
0.005
B (T)
0.004
0.002 0.001
ð13Þ
r (m) 0.0038
As shown in Fig. 5, the values of Jc(B) provided by the Kim model are close to measurements for high values of B. An important discussion has to be set before going on. We have to find Ic in the region where the magnetic field is low, so we have to eliminate the self field effect. To do that we allow that the Kim law continues to be true near B = 0. To develop an analytical study in the previous region, we approximate the function by a linear one, with Jc0 = 4.6 A/mm2 and Bj0 = 7 mT J c ðBÞ ¼
4:6 106 ð0:007 jBjÞ A=m2 0:007
ð14Þ
Fig. 5 represents the different results of the previous approaches. The following part deals with the study of the flux density in the region 1. 4. Calculation of the magnetic flux density and current density penetration In this part, we present the calculations of the distributions of B(r, t) and J(r, t). The results provided by both the Bean model and the linear model are compared. For B(r, t) distribution with linear model, relation (8) is used
Bðr; tÞ ¼
rS ¼
0.003
Bj0 ðl0 J c0 r Bj0 Þ þ exp
l0 J c0 ðRe rÞ BJ 0
0.0042
0.0046
Rin
0.005
Re i(t) = Ip = 100A
i(t) = 50 A Bean model
Linear model
Linear model
Fig. 6. B(r) distributions with the Bean model and the linear model.
To compare the linear model and the Bean model for the B (r, t) distributions, the critical current density JcB has to be equal to Jc0 and so IpB = Ic = Jc0. S = 156 A. Fig. 6 represents B(r) for two values of current. For i(t) = 50 A the penetration is incomplete for both, the Bean model and the linear model. For i(t) = 100 A with the linear model there is complete penetration, so i(t) = Ip = 100 A. With the Bean model there is complete penetration for i(t) = Ip = Ic Æ S = 156 A. It follows that the magnetic flux density penetrates faster into the material with the linear model than with the Bean model. From B(r, t), the maximum penetration depth (Re rs) and Bmax(r) are deduced, where B(r, t) for i(t) = Imax
ðB2j0 þ l0 J c0 Re ðbe ðtÞ Bj0 ÞÞ
l0 J c0 r
Bj0 ð1 þ W ðX 1 ðt1 ÞÞÞ l0 J c0
rS < r < Re : Bmax ðrÞ ¼
Bj0 ðl0 J c0 r Bj0 Þ þ ðB2j0 þ l0 J c0 Re ðBe max Bj0 ÞÞ exp
6
l0 J c0 ðRe rÞ Bj0
ð15Þ
l0 J c0 r
ð0:007BÞ with J c ðBÞ ¼ 4:6100:007 ; J c0 ¼ 46 A=mm2 ;Bj0 ¼ 7 mT. From now, the current corresponding to Bean model full penetration current is named IpB, while the current corresponding to the linear model is named Ip. For a B(r, t) distribution with the Bean model [9], where Jc = JcB and IpB = Ic = JcB Æ S, there is
l0 J cB R2e R2e iðtÞ r 1 2 Bðr; tÞ ¼ 2 r Rin I pB
where Bemax = Bmax(r = Re). From Jc(B) and B(r, t), J(r, t) can be deduced 4:6 106 ð0:007 Bðr; tÞÞ A=m2 0:007 We represent J(r) for the two models at the same instant for i(t) = 50 A (Fig. 7). We understand that J(r) is not the same for the models because for linear model J(r) is weaker than for Bean model except where B(r) = 0. It follows that the current penetrates deeper in the case of linear model.
J ðr; tÞ ¼ J c ðBðr; tÞÞ ¼
B. Douine et al. / Physica C 443 (2006) 23–28
5. Influence of flux density on the full penetration current Ip 5.1. Analytical calculation of Ip This part shows the most important difference between the linear model and the Bean model. With the Bean model, the current corresponding to complete penetration of current density is IpB with IpB = Ic = JcB Æ S. The full penetration current Ip can be calculated with the linear model with
Bðr ¼ Rin ; iðtÞ ¼ I p Þ ¼ 0 ¼
Bj0 ðl0 J c0 Rin Bj0 Þ þ exp
27
ated by the tube (around 5 mT) is close to Bj0. Now, as shown in part 3, one has to use only the linear model for B much smaller than Bj0. So there is no sense to use this Ip formula (16) for very small values of Bj0, where B > Bj0. 6. Conclusion In this article the influence of Jc(B) on the current, electric field and magnetic field distributions in a superconductor tube fed by a current i(t), was studied. These
l0 J c0 ðRe Rin Þ Bj0
B2j0 þ l0 J c0 Re
l0 I p 2pRe
Bc0
l0 J c0 Rin
So
Ip ¼
2p Bj0 l0 J c0 ðRin Re Þ l J R B þ ðB l J R Þ exp ; c0 e j0 j0 c0 in 0 0 Bj0 l20 J c0
Ip is smaller than Ic0. 5.2. Ip variation with BJ0 I
Fig. 8 represents the ratio I c0p according to Bj0 for different internal radii for the same section. It shows that Ip is closer to Ic0 for large internal radii because both the magnetic flux density and the influence of Jc(B) are weaker. It also shows that Ip is close to Ic0 for large values of Bj0. On the other hand, Ip is much smaller than Ic0 for small values of Bj0. For these small values of Bj0, the self field cre-
ð16Þ
distributions were calculated using the linear model. The influence of Jc(B) is important for distributions of B(r, t), J(r, t) and E(r, t). It was found that the current penetrates deeper in the case of the linear model than for the Bean model. An analytical calculation of the influence of Jc(B) on the full penetration current Ip is given. For the Bean model, IpB = Ic = JcB Æ S, while for the linear model Ip < Jc0 Æ S. It follows that with the linear model there is full current penetration for smaller values of i(t) than for the
1.2 1
Ip/Ic
0.8 J (A/mm2)
4
0.6
3
0.4
2
0.2
1
0
0.0038
0.0042
0.0046
0.005
r (m)
0
0.005
0.01
0.015
Bj0(T) Linear model
Bean model
Fig. 7. Distribution of J(r) with the Bean model and the linear model for i(t) = 50 A.
Ip/Ico
Ip/Ico Rin1 Fig. 8. Ratio Ip/Ico versus Bj0.
Ip/Ico Rin2>Rin1
0.02
28
B. Douine et al. / Physica C 443 (2006) 23–28
Bean model. Thanks to these results, one is able to understand the relationship of the Ip variation with the variation of the Jc(B) characteristics. References [1] M.N. Wilson, Superconducting Magnets, Oxford Science Publications, 1983, 335 p. [2] B. Douine, D. Netter, J. Leveque, A. Rezzoug, AC losses in a BSCCO current lead: comparison between calculation and measurement, IEEE Trans. Appl. Supercond. vol. 12 (l) (2002). [3] C.P. Bean, Rev. Mod. Phys. (January) (1964) 31. [4] G. Fournet, A. Mailfert, J. Phys. tome 31 (April) (1970) 357.
[5] J.H.P. Watson, Magnetization of synthetic filamentary superconductors, the dependence of the critical current density on temperature and magnetic field, J. Appl. Phys. vol. 39 (7) (1968) 3406. [6] P. Vanderbemden, Determination of critical current in bulk high temperature superconductors by magnetic flux profile measuring methods, Ph.D. Thesis, 1999, U. Lg., 1999, 193 p. [7] Y.B. Kim, C.F. Hempstead, A.R. Strnad, Critical persistent currents in hard superconductors, Phys. Rev. Lett. 9 (1962) 306. [8] B. Douine, J. Leveque, D. Netter, A. Rezzoug, Calculation of losses in a HTS current lead with the help of the dimensional analysis, Physica C 399 (2003) 138. [9] J. Leveque, B. Douine, D. Netter, AC losses under self-field in a superconducting, in: A.V. Narlikar (Ed.), High Temperature Superconductivity 1, Materials, Springer-Verlag, 2004, p. 431.
Influence of Jc(B) on the full penetration current of superconducting tube B. Douine *, K. Berger, D. Netter, J. Le´veˆque, A. Rezzoug GREEN, University of Nancy, BP 239, F-54506 Vandoeuvre-le`s-Nancy, France Received 1 December 2005; received in revised form 29 March 2006; accepted 13 April 2006 Available online 21 June 2006
Abstract It is well known that the critical current density Jc of a superconducting material depends on the magnetic field B. If magnetic independent Jc is chosen for analytical calculation of current distribution, the critical current Ic corresponds to full penetration current Ip. Ic is a measured current with 1 lV/cm criterion and Ip is a calculated current. The aim of this paper is to calculate the influence of the Jc(B) variation on Ip of a superconducting tube. To calculate Ip, which is depending on the material itself, a linear function Jc(B) is sufficient to obtain realistic values by analytic way. We need to have a linear Jc(B) law that is close to the measured Jc(B) characteristics presented in this paper. The linear Jc(B) law chosen was used for the calculation of the distribution of both magnetic field B(r, t) and the current density J(r, t). These distributions allow the analytical calculation of Ip. The calculated results of magnetic field distribution and full penetration current with Bean model and linear model are compared. We also present the variation of critical current with the characteristic parameters of the material. The present results, allow to understand the relationship between the full penetration current variation of a sample and the variation of the Jc(B) characteristics. Ó 2006 Elsevier B.V. All rights reserved. PACS: 74.60.Jg; 74.25.Ha Keywords: HTc superconductor; Magnetic field dependence; Critical current
1. Introduction To relate the irreversible magnetization to the field and current profiles inside a superconducting sample, Bean [3] introduced the critical state concept and assumed that the critical current density at any point of the sample can only take one constant unique value, the critical current density Jc, among three different states Jc, Jc and zero. Thanks to this model, the calculation of the current distribution is possible in case of a superconducting cylinder [1] or tube [2] fed with transport current. With this model, when the fed current i(t) increases into a superconducting sample, the current density penetrates up to the critical current Ic. *
Corresponding author. Tel.: +33 (0)3 83 68 41 25; fax: +33 (0)3 83 68 41 33. E-mail address: [email protected] (B. Douine). 0921-4534/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.04.090
For Ic the current density is equal to Jc everywhere in the superconducting material and the critical current corresponds to the full penetration current Ip. This is the complete penetration state. So the relation between Jc and Ic is very simple, Ic = Jc Æ S, where S is the section of the sample. Ic is used to calculate the AC losses [8]. Experimentally Ic is defined though the critical electric field Ec. For high temperature superconductors this value is generally fixed to 1 lV/cm. Unfortunately, for superconducting materials, the real critical current density varies with the magnetic flux density B [4–6]. In this case Ic does not correspond to Ip. Ic remains a measured current with the 1 lV/cm criterion. Ip is the value of the applied current i(t) at which the current density arrives to the centre of the sample. It’s a calculated current. The aim of this paper is to calculate the influence of the Jc(B) variation on the full penetration current of a superconducting tube. Considering Jc(B)
24
B. Douine et al. / Physica C 443 (2006) 23–28
for complete penetration the current density is not constant in the superconducting material because the magnetic flux density is not constant. As consequence, the relation between Ip and Jc(B) is not easy to calculate. For the calculation of Ip, the first step consists in setting a Jc(B) law that must be close to the measured Jc(B) characteristics. In a second step, the analytical calculation of current density, electric field and magnetic flux density distributions are presented. Finally the formula of Ip taking into account the variation of Jc(B) is derived. 2. Analytical calculation of the flux density distribution in a superconducting tube 2.1. Studied sample The aim of this paper is to calculate the full penetration current Ip of a tube based on the Jc(B) variation. To simplify the calculations, without reducing the generality of the problem, the applied current i(t) to the tube increases from 0 to Imax (Fig. 1), which is sufficient to determine Ip. The internal radius Rin, the external radius Re. and the length h of the tube are defined in Fig. 2. The current i(t) circulates along [Oz]. The edge effects are neglected in our calculations; E, J and B do not depend on z.
J ¼ J c
and
ð4Þ
E>0
or E ¼ 0 and
J c < J < þJ c
When the current rises, the current density penetrates from the external radius Re toward the internal radius Rin and its direction is [Oz]. Because of symmetry, the current density being oriented along the [Oz] axis, the flux density B(r, t) has only one component ~ Bðr; tÞ ¼ Bðr; tÞ ~ uh First we are going to prove that an analytical expression can be found for B(r, t) with a linear model for Jc(B). Considering the relation (2) and (4) 1 oðr Bðr; tÞÞ Bðr; tÞ ¼ l0 J c ðBÞ ¼ l0 J c0 1 ð5Þ r or Bj0 where jBj = B since B > 0. 2.3. Flux density distribution B(r, t) The analytical solution of expression (5) is exp l0BJj0c0 r B2j0 þK Bðr; tÞ ¼ Bj0 r l0 J c0 r
ð6Þ
K is a constant which can be calculated using the boundary condition at r = Re. The Ampere law enables to write
2.2. Modeling of the problem The electromagnetic behavior of superconductor is governed by the Maxwell equations
Bðr ¼ Re ; tÞ ¼ be ðtÞ ¼
l0 iðtÞ 2pRe
ð7Þ
Considering (6) and (7), we can deduce
Bðr; tÞ ¼
Bj0 ðl0 J c0 r Bj0 Þ þ exp
l0 J c0 ðRe rÞ Bj0
B2j0 þ l0 J c0 Re be ðtÞ Bj0
o~ B !~ rot E ¼ ot !~ rot B ¼ l0 ~ J
ð1Þ ð2Þ
For the superconducting material, we consider
Bj0 ðl0 J c0 dðtÞ Bj0 Þ þ exp
ð8Þ
l0 J c0 r
l0 J c0 ðRe dðtÞÞ Bj0
This expression is valid for B(r, t) > 0. So we have to determine the point where B(r, t) = 0. To do that, one can remark that it is equivalent to solve B(r = d(t), t) = 0 with d(t) = Re pd(t), where pd(t) is the penetration depth. Using the relation (8) and mathematical software, we obtain
B2j0 þ l0 J c0 Re ðbe ðtÞ Bj0 Þ
l0 J c0 dðtÞ
¼0
ð9Þ
Bj0 dðtÞ ¼ ð1 þ W ðX 1 ÞÞ l0 J c0 B ¼ l0 H
ð3Þ
E(J) is given by the critical state model defining the relation between electric field E and current density J
X 1 ðtÞ ¼
ðB2j0 þl0 J c0 Re ðbe ðtÞBj0 ÞÞ B2j0
exp
l0 :J c0 :Re Bj0
1 , where X1 >
0. W(X1) is Lambert’s W-function and is well-known in many mathematical libraries.
B. Douine et al. / Physica C 443 (2006) 23–28
25
i(t)
Solenoidal inductor
IMAX
Bext I Superconducting tube U z
t1
t
Fig. 1. Current supply.
z Fig. 3. Jc(B) measurement device. Re
Rin
100 90
h
Bext
U(µV)
80
0 x
y
θ
0,085 T
70
0,043 T
60
0,025 T
50
0,014 T 0,011 T
40
0,007 T
30
0,004 T 0
20
Fig. 2. Studied superconductor tube.
1µV/cm * h
10 0
3. Experimental Jc(B) of the sample
0
100
I(A)
Table 1 Jc(B) laws JcB, JK0, BK, Jc0, and Bj0 are constants and depend on the superconducting material Eq. no.
Fig. 4. Measure of sample voltage drop versus direct current for different external magnetic flux density.
Jmc(Bext) and Jc(B) are different because – Jc(B) is a local law and Jmc(Bext) is a macroscopic law – B is not equal to Bext because of the self magnetic field BSF : ~ B ¼ ~ BSF þ ~ Bext Region 1
Region 2
5 4 J(A/mm2)
Several variation laws Jc(B) are presented in Table 1. The experimental tests are made using a cylindrical current lead of BiSCCO. The dimensions of this sample are: Rin = 3.8 mm, Re = 5 mm, tube section S = 33 mm2 and h = 11.7 cm. Without external magnetic field, the measured critical current is Imc0 = 96 A, with a 1 lV/cm criterion. To obtain the experimental curve Jc(B) of this tube, it was fed with direct current I and submitted to an external magnetic flux density Bext parallel to the axis [Oz] (Fig. 3). The sample voltage drop U versus I is measured for different external magnetic flux densities Bext (Fig. 4). From the measured critical current Imc(Bext) corresponding to the voltage drop equal to 1 lV/cm h (Fig. 4). We can ext Þ deduce the J mc ðBext Þ ¼ I mc ðB . Fig. 5 shows the curve S related to this function.
3 2 1 0
11
J c ðBÞ ¼ J cB J K0 J c ðBÞ ¼ 1 þ BjBjK
12
J c ðBÞ ¼
10
50
J c0 ðBj0 jBjÞ Bj0
Bean model [3]
0
0.005
0.01
0.015
0.02
B(T)
Kim model [7] Jc(B) Kim
Jc(B) Lin.
Jmc(Bext)
Linear model [5] Fig. 5. Measured Jc(Bext), Jc(B) with the Kim model and the linear model.
26
B. Douine et al. / Physica C 443 (2006) 23–28
When the value of Bext is sufficiently large, BSF becomes negligible and the values of Jmc are very close to Jc(B). Among the laws represented in Table 1, the Kim model is the most suitable to extrapolate Jmc(B). With the appropriate values of JK0 and BK, we propose the following expression of the Kim law J c ðBÞ ¼
4:6 106 jBj 1 þ 0:004
A=m2
0.005
B (T)
0.004
0.002 0.001
ð13Þ
r (m) 0.0038
As shown in Fig. 5, the values of Jc(B) provided by the Kim model are close to measurements for high values of B. An important discussion has to be set before going on. We have to find Ic in the region where the magnetic field is low, so we have to eliminate the self field effect. To do that we allow that the Kim law continues to be true near B = 0. To develop an analytical study in the previous region, we approximate the function by a linear one, with Jc0 = 4.6 A/mm2 and Bj0 = 7 mT J c ðBÞ ¼
4:6 106 ð0:007 jBjÞ A=m2 0:007
ð14Þ
Fig. 5 represents the different results of the previous approaches. The following part deals with the study of the flux density in the region 1. 4. Calculation of the magnetic flux density and current density penetration In this part, we present the calculations of the distributions of B(r, t) and J(r, t). The results provided by both the Bean model and the linear model are compared. For B(r, t) distribution with linear model, relation (8) is used
Bðr; tÞ ¼
rS ¼
0.003
Bj0 ðl0 J c0 r Bj0 Þ þ exp
l0 J c0 ðRe rÞ BJ 0
0.0042
0.0046
Rin
0.005
Re i(t) = Ip = 100A
i(t) = 50 A Bean model
Linear model
Linear model
Fig. 6. B(r) distributions with the Bean model and the linear model.
To compare the linear model and the Bean model for the B (r, t) distributions, the critical current density JcB has to be equal to Jc0 and so IpB = Ic = Jc0. S = 156 A. Fig. 6 represents B(r) for two values of current. For i(t) = 50 A the penetration is incomplete for both, the Bean model and the linear model. For i(t) = 100 A with the linear model there is complete penetration, so i(t) = Ip = 100 A. With the Bean model there is complete penetration for i(t) = Ip = Ic Æ S = 156 A. It follows that the magnetic flux density penetrates faster into the material with the linear model than with the Bean model. From B(r, t), the maximum penetration depth (Re rs) and Bmax(r) are deduced, where B(r, t) for i(t) = Imax
ðB2j0 þ l0 J c0 Re ðbe ðtÞ Bj0 ÞÞ
l0 J c0 r
Bj0 ð1 þ W ðX 1 ðt1 ÞÞÞ l0 J c0
rS < r < Re : Bmax ðrÞ ¼
Bj0 ðl0 J c0 r Bj0 Þ þ ðB2j0 þ l0 J c0 Re ðBe max Bj0 ÞÞ exp
6
l0 J c0 ðRe rÞ Bj0
ð15Þ
l0 J c0 r
ð0:007BÞ with J c ðBÞ ¼ 4:6100:007 ; J c0 ¼ 46 A=mm2 ;Bj0 ¼ 7 mT. From now, the current corresponding to Bean model full penetration current is named IpB, while the current corresponding to the linear model is named Ip. For a B(r, t) distribution with the Bean model [9], where Jc = JcB and IpB = Ic = JcB Æ S, there is
l0 J cB R2e R2e iðtÞ r 1 2 Bðr; tÞ ¼ 2 r Rin I pB
where Bemax = Bmax(r = Re). From Jc(B) and B(r, t), J(r, t) can be deduced 4:6 106 ð0:007 Bðr; tÞÞ A=m2 0:007 We represent J(r) for the two models at the same instant for i(t) = 50 A (Fig. 7). We understand that J(r) is not the same for the models because for linear model J(r) is weaker than for Bean model except where B(r) = 0. It follows that the current penetrates deeper in the case of linear model.
J ðr; tÞ ¼ J c ðBðr; tÞÞ ¼
B. Douine et al. / Physica C 443 (2006) 23–28
5. Influence of flux density on the full penetration current Ip 5.1. Analytical calculation of Ip This part shows the most important difference between the linear model and the Bean model. With the Bean model, the current corresponding to complete penetration of current density is IpB with IpB = Ic = JcB Æ S. The full penetration current Ip can be calculated with the linear model with
Bðr ¼ Rin ; iðtÞ ¼ I p Þ ¼ 0 ¼
Bj0 ðl0 J c0 Rin Bj0 Þ þ exp
27
ated by the tube (around 5 mT) is close to Bj0. Now, as shown in part 3, one has to use only the linear model for B much smaller than Bj0. So there is no sense to use this Ip formula (16) for very small values of Bj0, where B > Bj0. 6. Conclusion In this article the influence of Jc(B) on the current, electric field and magnetic field distributions in a superconductor tube fed by a current i(t), was studied. These
l0 J c0 ðRe Rin Þ Bj0
B2j0 þ l0 J c0 Re
l0 I p 2pRe
Bc0
l0 J c0 Rin
So
Ip ¼
2p Bj0 l0 J c0 ðRin Re Þ l J R B þ ðB l J R Þ exp ; c0 e j0 j0 c0 in 0 0 Bj0 l20 J c0
Ip is smaller than Ic0. 5.2. Ip variation with BJ0 I
Fig. 8 represents the ratio I c0p according to Bj0 for different internal radii for the same section. It shows that Ip is closer to Ic0 for large internal radii because both the magnetic flux density and the influence of Jc(B) are weaker. It also shows that Ip is close to Ic0 for large values of Bj0. On the other hand, Ip is much smaller than Ic0 for small values of Bj0. For these small values of Bj0, the self field cre-
ð16Þ
distributions were calculated using the linear model. The influence of Jc(B) is important for distributions of B(r, t), J(r, t) and E(r, t). It was found that the current penetrates deeper in the case of the linear model than for the Bean model. An analytical calculation of the influence of Jc(B) on the full penetration current Ip is given. For the Bean model, IpB = Ic = JcB Æ S, while for the linear model Ip < Jc0 Æ S. It follows that with the linear model there is full current penetration for smaller values of i(t) than for the
1.2 1
Ip/Ic
0.8 J (A/mm2)
4
0.6
3
0.4
2
0.2
1
0
0.0038
0.0042
0.0046
0.005
r (m)
0
0.005
0.01
0.015
Bj0(T) Linear model
Bean model
Fig. 7. Distribution of J(r) with the Bean model and the linear model for i(t) = 50 A.
Ip/Ico
Ip/Ico Rin1 Fig. 8. Ratio Ip/Ico versus Bj0.
Ip/Ico Rin2>Rin1
0.02
28
B. Douine et al. / Physica C 443 (2006) 23–28
Bean model. Thanks to these results, one is able to understand the relationship of the Ip variation with the variation of the Jc(B) characteristics. References [1] M.N. Wilson, Superconducting Magnets, Oxford Science Publications, 1983, 335 p. [2] B. Douine, D. Netter, J. Leveque, A. Rezzoug, AC losses in a BSCCO current lead: comparison between calculation and measurement, IEEE Trans. Appl. Supercond. vol. 12 (l) (2002). [3] C.P. Bean, Rev. Mod. Phys. (January) (1964) 31. [4] G. Fournet, A. Mailfert, J. Phys. tome 31 (April) (1970) 357.
[5] J.H.P. Watson, Magnetization of synthetic filamentary superconductors, the dependence of the critical current density on temperature and magnetic field, J. Appl. Phys. vol. 39 (7) (1968) 3406. [6] P. Vanderbemden, Determination of critical current in bulk high temperature superconductors by magnetic flux profile measuring methods, Ph.D. Thesis, 1999, U. Lg., 1999, 193 p. [7] Y.B. Kim, C.F. Hempstead, A.R. Strnad, Critical persistent currents in hard superconductors, Phys. Rev. Lett. 9 (1962) 306. [8] B. Douine, J. Leveque, D. Netter, A. Rezzoug, Calculation of losses in a HTS current lead with the help of the dimensional analysis, Physica C 399 (2003) 138. [9] J. Leveque, B. Douine, D. Netter, AC losses under self-field in a superconducting, in: A.V. Narlikar (Ed.), High Temperature Superconductivity 1, Materials, Springer-Verlag, 2004, p. 431.