A.c. magnetic properties of YBaCuO bulk superconductor in high Tc superconducting levitation

CIyogenics 35 (1995) 243-241

0 1995 Elswier Science Limited F’ritttedin Great Britain. AU tights tescrved 001 l-2275/95/$10.00

A.c. magnetic properties of YBaCuO bulk superconductor in high Tc superconducting levitation M. Uesaka, A. Suzuki, N. Takeda, Y. Yoshida and K. Miya Nuclear Engineering Research Laboratory, University Shirakata, Tokai, Naka, Ibaraki, 319-11 Japan Received 29 August

1994; revised 30 October

of Tokyo, 2-22 Shirane-

1994

High T, melt-processed YBaCuO bulk superconductors have been utilized for the development of high T, superconducting magnetic bearings and flywheels. In such systems the superconductors are exposed to an a.c. magnetic field with frequencies of 100 Hz to 10 kHz during rotation of the rotor where the permanent magnet rings are installed. The a.c. magnetic field is caused by inhomogeneity of the magnetic field generated by the magnet in the azimuthal direction. Here the decay of rotational speed, termed rotational loss, becomes a very serious technical problem. In this paper, we analyse the a.c. magnetic properties of high T, superconductors in the above frequency range using a fundamental experiment and numerical simulation based on the flux flow and creep model and, in addition, we elucidate the mechanism of energy dissipation which causes the rotational loss.

Keywords: a.c. magnetic properties; flux flow and creep model; high T, superconducting flywheel

The application of high T, melt-processed YBaCuO superconductors is proposed in many engineering systems such as flywheeW2, magnetic bearings3, magnetic field design in fusion reactors and magnetic shielding. In such applications, it is important to elucidate the magnetic properties of high T, superconductors in a.c. magnetic fields and to evaluate the electromagnetic forces involved. Magnetomechanical characteristics can be stmnnarized according to the frequency of the applied magnetic field, as shown in Figure 1. Static characteristics such as static levitation force, hysteresis and stability can be analysed adequately using the critical state mode14, while dynamic but slow characteristics such as relaxation, dynamic force and damp-

ing (< 100 Hz), where the critical state model is not applicable, can be analysed successfully employing the flux flow creep models. However, the magnetic properties and the mechanisms of energy dissipation in an a.c. magnetic field of 100 Hz to 10 kHz, which are encountered with the a.c. loss of a superconducting magnet or the rotational loss (decay of rotational speed) of a high T, superconducting flywheel, have not yet been elucidated. Here we propose that the constitutive relation between the current density J and the electric field E based on the flux flow and creep model (FFC model) is applicable to the analysis for an MPMG-processed YBaCuO bulk superconductofl in an a.c. magnetic field in the above frequency range. In addition, the energy dissipation mechanisms are studied.

1.StroqLevitatimForce 2. Hysteresis

Modelling

3. sabiliw 4. Relaxation of Force

7. Rotstional Lass 8. Surface Impedance

of high 7” superconductors

I

I

I

..._......___.

_. / TwoFluid

61MHZ

Model

GHZ 1,

Figure 1 Magnetomechanical characteristics, frequency of applied a.c. magnetic field

modellings

and

Flux flow and creep refer to the dynamics of fluxoids (quantized magnetic flux) which are initially pinned by pinning centres in a superconductor. The former refers to the movement of fluxoids due to the Lorentz force, which surpasses the pinning force, while the latter refers to the movement of fluxoids which are thermally activated. As the fluxoids move, normal electrons existing in the fluxoids move together and a transverse electric field is induced in the superconductor7. The above situation is shown sche-

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Ax. magnetic properties of YBaCuO bulk superconductor: M. Uesaka et al. matically in Figure 2, where the Lorentz force is expressed as the potential which has a constant slope, and d and V are the radius and volume of the pinning centre, respectively. The constitutive relation between the induced electric field E and the current density J is expressed as follows’ ,=2&sinh($$exp(-$)

(ii) CrHM

forJIJ,(creepregion) U

for J > J, (flow region)

(0

L

stale modal

J

(1) where: J, and EC are the critical density and electric field, respectively: pf is the flow resistivity; 8 is the temperature; U, is the pinning potential; and k is the Boltzmann constant. The J-E constitutive relation expressed in Equation (1) is schematically depicted in Figure 3(i)-(I). Here, the product of J and E expresses the energy dissipation due to the collision between the normal electrons and molecules of the superconductor. For better understanding, we approximately decompose this non-linear constitutive curve to two curves, as shown in Figures 3(C) and 3( iii). In other words, the total currents are approximately decomposed to the superconducting current governed by flux pinning and the normal current induced by flux flow in parallel (and where flux creep is neglected for simplicity). The former and latter curves correspond to the critical state model and Ohm’s law, respectively. Therefore, if the Maxwell equations are solved so as to satisfy the J-E constitutive relation, both the non-linear magnetic behaviour of the superconductor governed by flux pinning and the normal conductivity attributed to flux motion can be simulated. On the other hand, it is well known that a superconductor in a d.c. magnetic field exhibits a weaker pinning force and flux flow occurs more easily. Therefore, the curve of the

w

W

(iii) Ohm’s law

J

Figure 3 J-E constitutive relation based on the flux flow and creep model (i). Relation (1) is approximately broken down to the critical state model (ii) and Ohm’s law (iii), where flux creep is neglected for simplicity. The variation of the flux flow and creep model under a d.c. magnetic field (2) and a.c. magnetic field (3) is also depicted schematically

relation is changed as shown in Figure 3(i)-(2). In order to take into account the dependence of J, and pf on the applied magnetic field, the Matsushita model9

J-E

J, = J,&

(2)

and Bardeen-Stephan model” Pf = Pfc4

(3)

d

were adopted, respectively. Moreover, in a.c. magnetic fields, it is speculated that the boundary of the flow and creep region is obscure, as shown in Figure 3(i)-(3), since it becomes very difficult to distinguish between flux flow and creep. This fact was observed for a Pb-Bi foil in 60 Hz a.c. magnetic field’ l. According to the flux dynamics in a mixed state type II superconductor, the equation of motion of a fluxoid can be written as follows’*

/-G-Gq

mt + TV + ksvdt = J40

a) J=O

Pinning center

0

U

Fluxoid

e, : superconducting electrons en: normal conducting *

b)

O
U = U. - JBVd

cl

J=Jc

(4)

where: m is the mass of the normal elections moving together with the fluxoid; v is the velocity of the fluxoid; n is the friction coefficient; k is the spring constant ascribed to pinning; J is the macroscopic current density (see Figure 2); and +. is the flux of the fluxoid, which is equal to h/2e (h = Planck’s constant; e = electron charge). The first inertial term is often neglected because of its small contribution’*. By multiplying Eq uation (2) by the macroscopic magnetic field B and introducing the induced electric field E, which is defined by vB, we can deduce the relation between E and J as follows qE + k_fEdt = JB+,,

(5)

A.c. analysis gives tbe following complex form Figure 2 Schematic drawing of fluxoid dynamics in flux creep and flow phenomena

244

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J=(&-i&-E

(6)

A.c. magnetic properties where o is the frequency. The following complex electrical resistivity can also be defined E=(p’+ip’)J

of YBaCuO butk superconductor:

M. Uesaka et al.

Vol

(7)

where the superscripts r and i imply the real and imaginary parts, respectively. The complex electrical resistivity comes from the shielding effect due to flux pinning and motion; here p’ stands for the dissipative process due to collisions between normal electrons and molecules of the superconductor during flux motion. Time(SCQLsjdiv)

Experimental details To confirm the existence of flux flow and the resultant resistance, cryogenic eddy current testing (Cryo-ECT) was carried out as shown in Figure 4. The principle of this experiment can be explained as follows. A.c. current is supplied to the exciting coil and the magnetic flux change due to the superconducting shielding current can be obtained by measuring the voltage of the coil. For a better understanding of the phenomenon we assume that the superconductor and exciting coil can be regarded as magnetically coupled electric circuits. The circuit equations become Exciting coil:

R,I, + LJ, +

Superconductor:

Mi,= v

R;I, + (I!& + R:)i, + Mi, = 0

(10)

where the symbol 1 1 represents the equivalent value of a sinusoidal signal. We measured the voltage signals from the coil without and with the superconductor and subtracted

n

1

1

-____

Superconductor MPMG-YBaCuO

Pr-type

Experimental

I

I

I

I

I, (mA) Figure 6 OT

Measured

IM&/ol versus l/,1 curves for a bias field of

the former from the latter, for I111= 100 mA, 500 Hz, 0 T, for example, as shown in Figure 5, and obtained IMtzlwl versus 11~)curves for different frequencies of exciting current. A praseodymium-type permanent magnet (B, = 1.5 T at 77 K, 18 mm in diameter, 18 mm in height) was placed below the superconductor when we formed the bias field of -0.4 T at the lower surface of the superconductor. Information about the resistance of the superconductor can be obtained by investigating the dependence of the slope of the curves on the frequency w, the exciting coil current 1~~1 and the bias magnetic field. Figures 6 and 7 show the versus 11~)curves for bias fields of 0 and 0.4 T,

m

bermonent

Figure 4

I

Digital Multimeter

Exciting CotI

7,

!

(9)

M W21 ~ 6J = [(L2 + Ri2)*+ (R;/w)‘]* lb1

s

50.

(8)

where: V is the applied voltage; R is the resistance; L is the self-inductance; and M is the mutual inductance. The subscripts 1 and 2 denote the exciting coil and superconductor, respectively. The superscripts r and i denote the real and imaginary parts of the electrical resistance, respectively. A.c. analysis gives the following relation

____

Figure5 Measured a.c. voltage signals from cryo-ECT probe with and without superconductor and the difference for \/,I = 100 mA, 500 Hz, 0 T

‘-IB 918x25

0

50

Magnet 0 12 x I4

100 I,

150

200

(mAI

Br=l 5T

set-up of cryo-ECT

Figure 7 0.4 T

Measured

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IMi,ol

versus l/,1 curves for a bias field of

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Ax. magnetic properties of YBaCuO bulk superconductor:

M. Uesaka et al.

respectively. From the above results, the following physical issues are deduced: the dependence of the slope of the curves on frequency o marks the existence of the real part of the electrical resistance; (b) the change of slope clearly appearing in the case of 500 Hz marks the change of the real part of the electrical resistance due to the transition between flux creep and flux flow; and (cl the decrease of the IM/Jol values due to the bias field marks the degradation of the critical current density, namely the pinning potential.

(4

Numerical

analysis

Here, in order to determine the electromagnetic field, we solve the Maxwell equations VxE=-8,-k,

(11)

VxH=J

B=fl

e

:Ba

.

: cu

0,o : 0

(12)

where E0 and B, stand for the magnetic fields due to the exciting coil and the superconductor, respectively, and the dot above these terms represents the partial time derivative. The following B-H constitutive relation for a vacuum is used

Figure8 Crystallographic structure of YBaCuO and thin plate approximation

&En

+ls=-n.VxE-n.k,

-

where J=VTxn

J=VxT

(17)

(14)

where T is analogous to the magnetization vector M of a magnetic material. Since the magnetic property of the superconductor is to be expressed by the above current density J, the permeability of a vacuum h can be applied to regions not only of air but also of superconductor. The Maxwell equations emerge in the following form

(15) where R = Irf - rsl

and r, and r, are the field point and source point, respectively, and II is the surface normal. The second term in the left-hand side calculates the self magnetic field generated by the superconducting shielding current. Here, we take into account the crystallographic anisotropy of J, in MPMG-YBaCuO, as depicted in Figure 8. J, parallel to the c-axis was found to be almost one-third of that parallel to the u-b plane l4. Thus, we assume that the bulk superconductor is a lamination of thin plates parallel to the u-b plane and that J, across the plates is negligible. The triangular mesh structure is also depicted in the figure. This assumption reduces the degrees of freedom from the three-dimensional vector T to the scalar T which is the normal component. Finally, the governing equation becomes

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1995 Volume

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(16)

d

(13)

The following current vector potential is introduced13

246

@ :Y

The superconductor is assumed to be single-grained. The above equation was solved by the finite element method with the iterative calculation technique, so that the nonlinear J-E constitutive relation implied by Equations ( 1), (2) and (3) is satisfied. In particular, in order to avoid V( 1lR) singularity, the r-0 integral calculus in two-dimensional polar coordinatesi was applied to the integral of the second term in the right-hand side, which calculates the self-field. Thus, the precision of calculation was much improved so that the a.c. magnetic properties can be analysed even if the shielding currents flow in a very superficial layer of the superconductor. The numerical analysis for the IM&lwl versus lZ,l relation was carried out for the following cases with the critical state and FFC models: (i) (ii) (iii) (iv)

B B B B

= = = =

0 T, 500 Hz to 3 kHz, critical state model

0.4 T, 500 Hz, FFC model 0.4 T, 3 kHz, FFC model 0 T, 500 Hz, FFC model

The results are shown in Figure 9. The same dependence of J, or B as written in Equation (2) was used for the critical state model and the following physical parameters were adopted, which are quite close to the values used in the previous study on the dynamic levitation force analysis5: J,, = 8.0 x 106 A me2, pf = 7.0 x 1W9 K?m, U, = 96 meV and EC = 100 I.LVm-l. The results based on the critical state model for 500 Hz

A.c. magnetic

properties

of YBaCuO bulk superconductor:

M. Uesaka et al.

(iv)FFCmodel

(I 1Criticalstate model /EOOHz,B=OT)l5OOHz-3kHz) /

20 ;; x

> 4

)... :” (III) FFC model (3kHz,B=0.4T)-

-

0

50

100 I,

Figure9 Calculated critical state models

IM&/ol

150

200

250 “0

(mA)

1

0.5

1.5

2

2.5

Frequency

versus [/,j curves by the FFC and

Figure 10 Calculated a.c. energy quency of applied magnetic field

3

3.5

4

f [kHzl loss as a function

of fre-

to 3 kHz are expressed

by the only straight line (i), which indicates that the calculated relation is independent of frequency. Therefore, it is obvious that the critical state model cannot describe the a.c. magnetic property in the frequency domain of 500 Hz to 3 kHz observed in the experiment. When we compare curve (ii) with (iii), the difference in the slope according to the different frequencies is clear and is ascribed to the real part of the electrical resistance due to the flux creep and flow. Curve (ii) corresponds to the lowest curve in Figure 7 in the experiment. From the above two curves, it was confirmed by both calculation and experiment that the computation simulates the change of slope due to the transition between flux creep and flux flow. The fact that curve (ii) is lower than (iv) indicates that the shielding effect is degraded due to the d.c. bias magnetic field. This is caused by the decrease of J, and the increase of pr due to the d.c. bias magnetic field. Thus, the experimental facts (a), (b) and (c) which validate the FFC model were also confirmed by the calculation. However, it can be found in Figures 6, 7 and 9 that the calculated amplitudes of llMi2/01 are always smaller than the experimental values. The calculation should give larger values than the experiment because the superconductor is assumed to be ideally single-grained in the calculation, while it is really multi-grained. The multi-grained structure, where the shielding current is localized in each grain, reduces the magnetic flux change at the coil even if J, is the same. This contradiction could be caused by still insufficient precision of integration in the second term in the left-hand side of Equation (16), which calculates the selffield. Further improvement is under consideration. Finally, the energy dissipation was calculated and plotted as a function of frequency for II = 125 mA, B = 0 T, as shown in Figure 10. If we suppose that the source of energy dissipation is magnetic hysteresis loss or Joule loss, its dependence on frequency should be to the first or second power, respectively. As explained before, the J-E constitutive relation based on the FFC model implies features of both hysteretic magnetization and normal conductivity. Hence the dependence should be an intermediate between the first and second powers. The energy dissipation versus frequency curve validates the above prediction.

Conclusions The conclusions follows:

deduced in this study are summarized

as

It was ascertained by cryogenic eddy current testing and numerical analysis that it is not the critical state model but the FFC model which is applicable for analysing the a.c. magnetic properties of MPMG-YBaCuO in the frequency domain 500 Hz to 3 kHz. A decrease in critical current density and increase in flow resistance in the FFC model due to an applied d.c. bias magnetic field were observed in the same frequency domain both in the experiments and calculations. It was found by numerical analysis that the energy dissipation in an a.c. magnetic field has two contributions, namely hysteretic diamagnetization and normal conductivity, so that its dependence on frequency is an intermediate between the first and second powers.

Acknowledgements The authors would like to thank Dr Masato Murakami for supplying the MPMG-YBaCuO superconductors and helpful discussion and Dr Hiroki Tokuhara for supplying the praseodymium-type permanent magnets.

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2

Higasa, H., Ishikawa, F., Kawauchi, N., Yokoyama, S. ef al. Adv Supercond (1993) 6 1249

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Yoshida, Y., Uesaka, M. and Miya, K. J Appl Elec Magn in Mater

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(1991) 30 L342 10 Duzer, T.V. and Turner, C.W. Principles of Superconductive Devices and Circuits (1981) Elsevier-North Holland, Amsterdam, The Netherlands 11 Sakamoto, N. and Yamafuji, K. Jpn J Appl Phys (1977) 26 1663 12 Portis, A&L Electmdymmics of High-Temperature Superc~rs: Lecture Notes in Physics Vol 48, World Scientific, Siiapore (1993) 13 Hashizmne, H., Sugiura, T., Miya, K., Ogasawara, T. Cryogenics (1991) 31 601 14 Murakami, M., Gotoh, S., Fujino, H., Tanaka, S. Supercond Sci Technol (1991) 4 S43 15 Utamura, M., Koizumi, M. Nucl Sci (1985) 22 733

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