vertical force, magnetic stiffness and damping for levitating type-ii superconductors

ELSEVIER

8 August 1996

Physica C 268 (1996) 46-52

PHYSICA

Vertical force, magnetic stiffness and damping for levitating type-II superconductors

Alvaro Sanchez *, Caries Navau Grup d'Electromagnetisme, Departament de F[sica, Universitat AutOnoma Barcelona, 08193 Bellaterra (Barcelona), Spain

Received 8 February 1996; revised manuscript received 19 June 1996

Abstract

The analytical expressions for the vertical force experienced by a superconducting cylinder situated above a cylindrical permanent magnet are derived within the framework of the critical-state model. The force, with an appropiate normalization, depends on a single parameter, the penetration field of the superconductor, which is proportional to its critical-current density. The results are applied to calculate the vertical magnetic stiffness and damping, from which the conditions for achieving a stable equilibrium of the system and the characteristics of the hysteretic losses associated to small displacements of the superconductor are discussed.

Keywords: Levitation; Applications of high-Tc superconductors; Magnetization

Magnetic levitation of superconductors is one of the most promising fields with regard to their technologi- cal applications at short term [ 1 ]. Big efforts are being done in various directions, such as improving the material properties of the superconductors for meet- ing the requirements for each particular application, or measuring the levitation forces in carefully designed experiments in order to gain information about the basic parameters that will allow the best device per- formance. These measurements have helped to under- stand the general features of the levitation, first studied by Brandt [2] , while at the same time have raised some questions which remain unclear up to now. Our work intends to clarify some of these questions, with a double aim: to provide a systematic framework for modeling superconducting levitation, and to derive the analytical expressions describing the vertical magnetic

* Corresponding author. Fax: + 34 3 581 1350; e-mail: [email protected].

force and two of the most important parameters, magnetic stiffness and damping, under some simple but useful approximations.

We will consider a type-II superconducting cylinder (SC) which is displaced over a cylindrical permanent magnet (PM) along their common axis, in the z direction. We will use cylindrical coordinates (r, ~b, z ), with the origin located at the center of the upper side of the PM. We assume that the PM has uniform magnetization .A,4 in the z direction. We will be only in- terested in the region close to the axis, where it can be demonstrated that the components of the H field created by the PM are such that Hz >> Hr, and H z is roughly independent of r. Then, the field in this region can be approximated by the value of the field on the axis, which has only z-component

M( z+b Hz = -~- v /a 2 + (z + b) 2

(1)

0921-4534/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved P11S0921-4534(96)00400-5

A. Sanchez, C. Navau/Physica C 268 (1996) 46-52

where a and b are the radius and thickness of the PM, respectively. For radius r < 0.2a the above approxi- mation yields a maximum error of 15%. The values of a, b, and .A4 are chosen for this work among the typical experimental values as 0.05 m, 0.025 m, and 8,000 A/m, respectively.

The SC is assumed to be in a critical state as described by the Bean's model [ 3 ] with neither equilibrium magnetization nor surface barrier. According to this model, for a zero-field cooled sample, any variation of the external field felt by the superconductor will generate a constant critical-current density at its surface. Further increasing of the field (for example, when descending the SC in the presence of the field of the PM) will result in a layer with current density Jc penetrating the SC inward, until the applied field is such that the sample is fully penetrated. This penetration field is Hp = JcR, where R is the radius of the SC.

Then, as long as R is small compared to a, and the thickness L of the SC is small compared to the vertical scale of magnetic field variation, as will be considered throughout this work, the magnetization resulting in the SC under the action of the magnetic field of the PM can be regarded as constant.

In general, the force over a distribution of currents J due to an external magnetic field H is

F = txo / J A HdV (2)

If the currents fill a volume V where the field is uniform enough to enable assigning them a mean magnetization M, and these currents have azimuthal direction, the expression for the force of Eq. (2) can be simplified yielding a force with only z-component f = _ __ given by

aHz Fz = tzoMz V (3)

cgz

Thus, within our approximations, the force between the SC and the PM can be described by Eq. (3), where M z and Hz are the z-components of the magnetization of the SC (which has a volume V = 7rR2L) and the field of the PM, respectively.

It is convenient to normalize Eq. (3) to use only dimensionless variables. In this way, we shall be able to describe the general behavior with the minimum number of parameters. Thus, we define the following variables, obviating the z subscript:

47

Fz f = txoTrR2H 2 ,

Z=zlL, h=;-/z//-/p, m=Mz/I-Ip, (4)

from which Eq. (3) becomes

f = mah/a~. (5)

We have derived the expressions corresponding to a zero-field cooled SC descending and then ascending along the axis of the PM, based on the magnetization of a superconductor in the critical state [4]. There are different cases, depending on whether the maximum applied field felt by the SC is smaller or larger than the penetration field. In the latter case (hm > 1 ), we have for the descending SC (increasing h)

f = - h - h 2 + ~ ( 0 < h < 1), (6)

1 Oh f . . . . (1 < h < h m ) . (7)

3OZ

Once the SC has descended until a particular height (corresponding to a maximum normalized field hm), if the SC is pulled upward, a new layer with supercurrents flowing with the opposite sense appears at the surface, keeping the flux and current distribution in the interior "frozen". In this case (decreasing h),

f = ( h ( - 1 + hm - ~h m ) l 2 q_ h2( lhm _ 1) _ l h 3

t 1 3 1 2 ah - g + h m + ]'~h m - ~h m)~-~

(hm > h > hm - 2), (8)

1 ah (hm - 2 > h > 0). (9)

38Z

If the maximum field is not enough to fully penetrate the sample with supercurrents (hm < 1), we have for descending SC

f = - ( h - h 2 + ~ h 3 ) Oh ( 0 < h < hm), (10) J aS

and when the SC is ascending

, 2 h2(¼hm ½) f = [ h ( - I -k- hm - ~h m) +

1 2 1 3 Oh + -- hm]

(hm > h > 0). (11)

48 A. Sanchez, C. Navau/Physica C 268 (1996) 46-52

0.0015

0.0010

0.0005

0.0000

- 0 . 0 0 0 5

I ll ........

~ ~ _ _ . . . . . S . .~ - .~

" " " . . . • " . . . ' " " "

" ' . . . . . . . . . . . . . . . . . . . "

. . . . I . . . .

H p = 2 H m

H p = H m

....... H p = H m / 2

- 0 . 0 0 1 o . . . . . . . . I . . . . I . . . . I . . . . 200 400 600 800 1000

Fig. 1. The normalized force f as function of normalized vea ica l distance ~, for the cases Hp = 2Hm (solid l ine), H o = Hm (dashed

l ine) , and H o = - ~ (dotted l ine) ; the descending branch corresponds, for each case, to the entirely positive curve.

0.0006 . . . . , . . . . , . . . . , . . . .

0.0004

0.0002

0.0000 -

- 0 . 0 0 0 2 . . . . I . . . . I . . . . I . . . . I . . . . 200 400 000 800 lO00

Fig. 2. Minor loops in the normalized ve~ical ~ r c e f ~ r the case Hp = Hm. B e ~ increment is &f = 100 ~ r NI minor loops.

With this normalization, for a given permanent magnet and a given thickness L, the force depends on a single paramater, the penetration field Hp. Fig. 1 shows the results for the three characteristic cases: when the

penetration field Hp is larger, equal or smaller than the maximum applied field H,n felt by the SC (when it is at the bottom position).

The three curves show a hysteretic behavior that has

A. Sanchez, C. Navau/Physica C 268 (1996) 46-52 49

been experimentally observed by several groups [5]. However, the presence of a maximum in the descending branch has only been occasionaly measured[6], and no explanation has been provided to our knowl- edge. With the model presented in this work, it is demonstrated that the maximum in the force arises from the minimum in the derivative of the field pro- duced by the PM. This can be easily illustrated for the simplest situation, when Hp << Hm; in this case, once the penetration field is reached, the normalized magnetization m in the Bean's critical-state model is saturated (m = - 1 / 3 ) , so after Eq. (5) the force is simply proportional to the field derivative. When the SC is not fully penetrated the dependence is not so obvious and the maximum of the force is shifted from the minimum of field derivative. In general, however, a key parameter for the position of this maximum is the relation b/a; the larger the ratio b/a, the closer the maximum to the origin. When the ratio b /a is suffi- ciently large, the maximum in the force may not even exist.

For practical purposes it is important to study the explicit dependence of the force on the value of the critical-current density Jc, which is not straightforward from Fig. 1, since f is the force Fz divided, among other things, by H 2, which is in turn proportional to

j2. The non-normalized expression for Fz for the descending curve when H z > H v is very simple:

I OHz 3 Fz =-g txocr- - z - -JcR L. (12)

" t~Z

F z is therefore proportional to Jc and depends upon the SC dimensions as shown. This general dependence is also valid for fully penetrated states in the ascending curve (with a positive sign for the force). The relation between F z and Jc is slightly more complicated for the non-penetrated cases but can be readily obtained from Eqs. (6), (8), (10), and (11).

We have extended the formulas for the force to derive the analytical expressions of minor loops which could appear, for example, as a result of a vibration of the SC. As an illustration of the general behavior, in Fig. 2 we show the results corresponding to the case Hp = Hm. A fixed increment A~ = 100 has been used for all minor loops.

An important parameter for characterizing the levitation is the magnetic stiffness. The vertical magnetic stiffness x is defined as the change in the vertical

force due to a change in the vertical position of the SC. Therefore, the stiffness at every point of the force curve corresponds to the initial slope of the minor loop started at this point. In our normalized quantities,

= - O f ~ O f . (13)

By applying this definition to the expression of the force [Eq. (5) ] and considering that according to the critical state model, any change of sign of the field sweeping rate will result in an induced current in the SC which will shield its interior, we obtain, for small displacements,

k = - ~mo---- ~ - Lo--~j ) A similar expression (only for the stiffness in the descending curve) was given in [ 8 ]. As it becomes clear from this equation, the stiffness depends on the magnetic history of the SC, since m does. Using Eqs. ( 6 ) - ( 11 ), and (14), we obtain the analytical expressions of the stiffness as functions of the vertical distance for both descending and ascending curves (results are shown in Fig. 3, for the case Hp = Hm). The stiffness is related to the stability of the levitation [ a positive (negative) stiffness means a stable (unstable) levitation]. Fig. 3 shows that the stiffness is positive for the descending curve until a maximum is reached and then decreases, passes through zero and becomes negative for values of z close to the upper side of the PM. The negative values for the stiffness can be understood by looking at the corresponding minor loops in Fig. 2: while for the rest of the curve, a perturbation in the SC results in a decrease of the vertical force, in a region near the PM the same perturbation pro- duces an increase of the force, and therefore a negative stiffness. This negative region, however, becomes smaller, and even dissapears with decreasing Hp. As for the ascending curve, after an initial negative region (for the same reason as above), the stiffness becomes positive, shows a maximum and eventually becomes negative again. The reason for the latter is also readily seen in Fig. 2: after reaching a certain vertical distance, the minor loops simply follow the force curve, meaning that the stiffness will tend to the slope of this curve, therefore becoming negative. This ean be understood by realizing that in this region the variation of the magnetic field created by the PM is small (situ-

- - ' ' ' ' I . . . . I . . . . I . . . . I . . . . -

I

o v

Z~

-1

0


I . . . . t . . . . I . . . . 1 . . . . 200 400 600 800 1000

Fig. 3. Normalized vertical magnetic stiffness ~" as function of normalized vertical distance Z, for the case Hp = Hm. The descending case corresponds to the upper curve and the ascending case to the lower one.

O.O00B

0 . 0 0 0 6

~ " 0.0004

0.0002

Hp=H m

=1oo

. . . . I . . . . I . . . . I ' '

0 200 I i i i i

0 . 0 0 0 0 4 0 0 6 0 0 8 0 0 t O 0 0

Fig. 4. Normalized vertical magnetic damping ~, as function of normalized vertical distance g, for the case Hp = Hm. The :~ increment is Af = 100.

ation that in the cri t ical-state mode l is equivalent to a

high value o f Jc ) , which yields a quasi- revers ible be-

havior, s ince the supercurrents that are being induced

penetrate only small distances.

The third important parameter necesary for charac-

ter izing a levi tat ion exper iment is the magnet ic vert ical damping "y, defined as the hysteret ic energy loss pro-

duced when cycl ing the SC along a minor loop; there-

A. Sanchez, (2. Navau/Physica C 268 (1996) 46-52 51

- I I . . . . I ""

3

O 2

1

0 , , , I , , , I . . . . I 0 2 4 6

f (10 -4)

Fig. 5. Normalized vertical magnetic stiffness k as function of normalized vertical force f for the descending curve for the case Hp = Hm.

fore, the damping will be proportional to the area of a force minor-loop of Af . In our normalized variables this area is equal to the area of a m(h) minor loop of width Ah = h(z-l) - h ( f2) , with fl and z2 being the vertical points at which the minor loop starts and ends. This area can be calculated analytically; when Ah < 1, the normalized damping is given by

" ~ = l ( A h ) 3 - - l ( A h ) 4 " (15)

Fig. 4 shows the calculated normalized magnetic damping as a function of f , with a fixed interval Af = fl -- f2 --- 100 for the case Hp = H,,, (actually, ~ means ~2 for the descending curve and fl for the ascending one; if instead we regarded f as the initial point at which the minor loop is started in either case, then the ascending curve would be shifted to the right by an amount of A f ) . It is important to notice that, for a fixed A f , the magnetic damping does not depends on the magnetic history. This can be understood by realizing that in the Bean's critical-state model, the shape of the m(h) minor loops with a fixed Ah interval depends only on the choice of Ah, and not on the particular value of h at which the minor loop is started. As a consequence, in the present case, force minor loops running between the same limits zl and z2 enclose the same area. The g-position of the maxi-

mum in the damping is related to that of the minimum in the derivative of the PM field. For small displacements (Af small; not shown), both points coincide to order (A f ) 2. The position for the maximum damping shifts to the left by increasing Af (as in Fig. 4).

Our model also explains other features observed in levitation experiments. In Refs. [9,1], it was found that the stiffness appeared to be directly proportional to the levitation force, although not for the whole range of forces and not for all cases. If fig. 5 we plot the calculated force and stiffness for the descending branch and the case Hm = lip and confirm that a roughly linear dependence is found for most of the range. The general dependence between the normalized stiffness and force can be obtained from Eqs. (5) and (14) as

[ l ( O h ) 82h/?~21 k = f ~Z Oh/O~ J"

(16)

Eq. (16) shows that a linear behavior is not always expected since the term inside brackects depends on z in general. However, when the particular experimental arrangement is such that this term is almost constant over a certain region, an approximate linear dependence of the stiffness upon the force will be found in this region.

In conclusion, we have introduced the relevant


equations describing vertical force, magnetic stiffness and damping for a type-II superconducting cylinder in the critical state situated over a cylindrical permanent magnet in its axis. With the appropiate normalization presented here, given a particular permanent magnet, the results for the force depend only on the penetration field of the superconductor. We have derived the analytical expressions for the levitation force for all the different stages and described its dependence on the superconducting parameters. These calculations have been extended to the case of minor loops in the force. Finally, we have studied the hysteretic behavior of the magnetic stiffness, discussing the regions in which has a negative value, and also derived the non- hysteretic curve for the magnetic damping. The model presented here may provide the adequate framework to address further important questions, such as the effect of the thickness of the superconductor in the levitation force, and the obtention of designs with op- timum stability-energy losses relation. Further work in these directions following this approach will be reported elsewhere.

References

[ 1 ] EC. Moon, Superconducting Levitation (Wiley, New York, 1994).

121 E.H. Brandt, Science 243 (1989) 349; Am. J. Phys. 58 (1990) 43.

131 C.P. Bean, Rev. Mod. Phys 36 (1964) 31; Phys. Rev. Lett 8. 250 (1962).

141 D.-X. Chen and R.B. Goldfarb, J. Appl. Phys. 66 (1989) 2489; D.-X. Chen, A. Sanchez, J. Nogu6s and J.S. Mufioz, Phys. Rev. B 41 (1990) 9510.

151 F.C. Moon, K.-C. Weng and P.-Z. Chang, J. Appl. Phys. 66 (1989) 5643; E-Z. Chang, F.C. Moon, J.R. Hull and T.M. Mulcahy, J. Appl. Phys. 67 (1990) 4358.

[61 B. Lehndorff, H.-G. KiJrchner, B. Liicke, H. Piel, Physica C 247 (1995) 280.

171 P. Sch6nhuber and EC. Moon, Appl. Supercond. 2 (1994) 523.

[81 J.R. Hull, T.M. Mulcahy, K. Salama, V. Selvamanickam, B.R. Weinberger and L. Lynds, J. Appl. Phys. 72 (1992) 2089.

[9] P.-Z. Chang, F.C. Moon, J.R. Hull and T.M. Mulcahy, J. Appl. Phys. 67 (1990) 4358.

We thank the DGICYT (project number PB93- 1249) and CIRIT (project number 1995SGR 00039) for financial support. C.N. acknowledges a grant from CUR (Generalitat de Catalunya).

vertical force, magnetic stiffness and damping for levitating type-ii superconductors

Documents