Eddy Current Losses at Cryogeni Temperature

8/11/2019 Eddy Current Losses at Cryogeni Temperature

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IEEE TRANSACTIONS ON MAGNETICS, VOL.

29

NO. 3, MAY 1993 2095

Eddy

Current Losses at Cryogenic

Temperatures

V .

Sokolovsky,

V .

Meerovich, and M. Slonim

Absfrucf-The present paper analyses e5wt of thermal pro-

cesses on eddy-current losses in construction elements of

cryogenic and superconducting devices. Maxwells equations

coupled with heat-conduction equation are solved with taking

into account the dependence

of

resistivity, heat capacity and

heat-transfer coefficient on temperature. Analysis of lossesasa

function of magnetic field, frequency and geometry factors is

given for the case of thin strip in a uniform magnetic field. It

is shown that losses calculated with taking into account the

thermal processes may di5er from those obtained at constant

temperature.

INTRODUCTION

NE of the problems arising at design of cryogenic

nd superconducting devices is calculation of eddy

current losses in constructional elements (CE), Le., in

cryostat walls, radiation shields, etc.

[1]-[4].

The accu-

rate determination of these losses is required for the cool-

ing system construction and for choosing the power of

cryogenic refrigerators.

The main feature which must be taken into account at

cryogenic temperatures is the strong dependence of the

metals resistivity

p

on temperature

T .

Thus, the resistiv-

ity p may change by several times when the temperature

varies by

20 K

[5]. The temperature of CE may vary con-

siderably even within the magnetic field period because

the heat capacity of metals at 4.2 K is approximately two

orders of magnitude lower than that at room temperature.

Also, the boiling crisis in the coolant causes the reduction

of the heat-transfer coefficient and the temperature in-

crease [5].

Usually eddy current losses are calculated at constant

resistivity corresponding to the selected temperature. The

effect of temperature on resistivity is taken into account

with the use of correction coefficients [11, [6].

This work is devoted to the analysis of the effect of

thermal processes on eddy-current losses in CE of devices

operating at cryogenic temperatures. As is done in induc-

tion heating problems

[7], we will use an approach

founded on the coupled solution of electromagnetic and

thermal problems.

STATEMENT

F

THE

PROBLEM

Generally, eddy current losses, temperature and, con-

sequently, resistivity are non-permanent in a volume of

Manuscript received July 7 , 1992; revised January 15, 1993.

The authors are with the Department of Electrical and Computing En-

IEEE Log Number

9208216 .

gineering, Ben-Gurion University of the Negev , Beer-Sheva, Israel.

CE. In metals whose resistivity

p

varies from point to

point, free space charge

u

may exist and Maxwells equa-

tion set takes the form:

r o t H =

j ;

div

B

= 0;

(3)

div E

= U / E ~

4)

where

2

2 ar: the elect@ and magnetic field inte sities,

respectively (E =

p j

; i is the current density; B is the

magnetic flux density

B =

p s ;

p a

is the magnetic

permeability,

E,

is the dielectric constant,

t

is the time.

The space charge density is described by the expres-

sion:

u =

E j

gradp.

5)

From

(5)

it follows that

u

= 0 if grad

p

=

0.

This as-

sumption is used for most electromagnetic field problems.

5 should be noted that, even for great values of grad p

j

grad p = 0 and hence

u

=

0

are satisfied when

all

the

current lines are parallel or if one can neglect the propa-

gation of heat in CE.

The problem now is to solve (1)-(5) coupled with the

heat-conduction equation:

6)

aT

C

-

t

= div

X

grad

T )

+

W,

where

C

is the specific heat capacity;

X

is+the thermal

conductivity,

T

is the temperature;

W = I

E I 2 / p is the

loss power per unit volume.

In (1)-(6)

C, , p

are functions of temperature and con-

sequently of time and

of

point-to-point variation.

The boundary conditions for Maxwells equations are

usual conditions for the magnetic and electric fields at a

conductor-dielectric interface. The type of boundary con-

ditions for (6) is defined by the device construction, the

cooling method and the form of the examined CE.

Losses in a Thin Strip

To analyze the influence of thermal processes on eddy

current losses, we will consider, for example, a thin strip

(Fig.

1)

in a uniform magnetic field normal to its surface

Hz

=

-H , sin

at a =

271-5

f

s the frequency). This

simple problem is a model for a number of practical cases.

0018-9464/93 03.00 1993 IEEE

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TRANSACTIONS

ON

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VOL. 29

NO.

3 ,

MAY 1993

boiling crisis starts;

T

= ( T - T q ) / ( T c- T J ; 8 = w t ; Q

=

c,wd/h, .

In the quasi-steady state regime, the initial condition

for ( 9 ) s chosen as

Ho

T e =

0

=

7 e

= r .

(10)

The average loss for a unit of the strip's surface to be

divided by

h,

(Tc-Tq)

is given by the expression:

td

Fig.

1 A

thin strip in a uniform magn etic

field

normal

to

its surface.

Under the above-described assumptions, the losses per

unit of area of liquid nitrogen cooling (77 K) copper strip

were Calculated. The values of c1 T) ,

p I

7 ) and

h i

7 ) were

taken from data-tables for copper. The function

h l

7 ) was

We make the assumption that the strip is sufficiently hin

that one can neglect eddy current reaction. Then the elec-

tric field intensity E is defined by the solution of the equa-

tions given by

for

7

;

1.92

-

0.927

for 1

2 .

With this approximation,

E

does not depend on re&

This

form

approximates

to

the

q T )

tivity and hence on temperature. In the thin strip, the elec-

tric field has only component E,,. Therefore, the solution

of (7) is of the form:

E, =

wpaxH,

cos ut. ( 8 )

with sufficientaccuracy*

In Fig. 2 a)- c) are given the calculated results of loss

densities as functions of the magnetic field intensity, fre-

quency and strip thickness in normalized units 8, 3

2,

respectively). The values k =

1,

d = 1,

6

= were

chosen, corresponding to 6 = 0.5 and Q = 0.5. In the

same figure the calculated results without taking into ac-

count the thermal processes are plotted (dotted curves).

Solving the heat-conduction

(6)

we will make the fol-

-the temperature is constant in the strip's thickness;

lowing assumptions:

-one can neglect heat propagation along axis

n;

-the thermal flow density

q

from the strip's surface is

DISCUSSION

supposed to be given in the form:

Influence of Boiling Crisis

=

h ( T -

Tp)

where

Tq

is the temperature of coolant;

h

=

h ( T )

is the

heat-transfer coefficient.

The two former assumptions are valid under following

conditions:

hd x


3/4

SOKOLOVSKY et a [ . :

EDDY

CURRENT

LOSSES AT CRYOGENIC

TEMPERATURES

. 1.0-

2097

S /

/

/

/

B

/

I

0.

/

/L

0.5 1.0 1.5

0L c -

2 .

(C)

Fig. 2. Loss density versus magnetic field (a), frequency (b) , strip thick-

ness (c). Solid curves correspond to the calculation which takes into ac-

count thermal processes. Dotted curves were obtained at constant temper-

ature.

temperature. If

7

< 73 the stable state is attained at point

2;

if

7

>

73

it will be at point

4.

As a result with increase

of magnetic field, the losses change in accordance with

curve A-B and with a sharp transition to point

C.

With

decrease of field, the losses change in accordance with

curve C - D and with a transition to point A .

For real conditions, the heat propagation in the

CE

and

in the coolant may lead to smoothing of the hysteretic part

of the curve and to a smooth curve trend in sections B-C

and

D - A .

The total loss in the unit of the strip length is defined

as the integral of the loss density over strip width 1 The

curves have a smoother shape. The explanation of these

results is that because of the unequal distribution of the

2, 2 3

Fig. 3 . Determination of thermal equilibrium points taking account of

coolant boiling crises.

p

I

0.4

I

f

/

/

Fig.

4.

Total losses in the strip length unit versus magnetic field. Solid

curve corresponds to the calculation which takes into account thermal pro-

cesse s; dotted curve was obtained at constant temperature.

The

Functional Dependence Analysis

In Fig.

2

it will be seen that for two values of 8,G 2,

the losses are proportional to H2 , G2, 2. However, the

losses dependence on 8 and ij is practically linear to the

right of region

A-B-C-D.

To explain these results, we

will analyze the heat-conduction equation solution on the

assumption that and

h

are constant and that the resistiv-

ity is given by the linear function

of

temperature:

p 1 = 1 + y 7

where y is a numerical coefficient (for copper and alu-

minum in nitrogen

y

=

0.4).

The solution of (9) in the form of a series is

7

= + 7

, e )

+

where

T o = [ l 4py)12 11/27

7

=

p

sin

28

+

J /

4 ~ *

1

+

y

7~~

+

1

+

2y

T,)~]

3 / 2

1 1)

temperature

and

losses x, the

crisis at

dif-

J is the phase differencebetween losses and temperamre;

ferent points is attained at different values of parameters

H, w and d . For example, in Fig. 4we have plotted the

curve of total loss (in a unit of strip length) as a function

7o is the mean temperature.

From

11) for 4py


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2098 IEEE TRANSACTIONS ON MAGNETICS, VOL.

29,

NO. 3, MAY

1993

In this case the temperature and resistivity,do not vary

appreciably, and eddy-current losses are proportional to

w2H2d.

In the other limiting case when 407 >> 1 we

obtain:

7

=

m =

P O U

+

6 Po

Jpr

and the losses are proportional to wH . In Fig. 2 and

Fig.

4,

one can see these two limiting cases.

Note that the losses, to a great degree, depend on the

coefficient

@

which depends on H nd

w

in the same mea-

sure. This explains the near trend of the loss curves as

functions ofH nd

w .

The small difference arises from the

fact that parameter Q depends on frequency too.

CONCLUSION

The analysis performed shows the strong influence of

thermal processes on the eddy-current losses in

a CE

of

cryogenic and superconducting devices. To calculate these

losses, the approach based on coupled solution of the

electromagnetic and thermal problems may be used. The

eddy-current losses in a

CE

of cryogenic and supercon-

ducting devices, calculated taking into account the ther-

mal processes, differ by several times from those obtained

at constant temperature. This difference attains about 40%

at

= 0.6

and nearly

300

at

i = 1.0

as one can see

from Fig. 4 . Also, the boiling crises of the coolant leads

to a hysteresis in the loss density dependence on magnetic

field and freauencv.

[4] U. N. Vershinin,

V .

M. Meerovich, I . E. Naumkin, N. L. Novikov,

and V.

L.

Sokolovsky, A comparative analysis of nonlinear reactors

with shields from low-temperature and high-temperature about 90K

superconductors, Electrical Technology. USSR

UK),

no. 1, pp.

1-9, January 1989.

[ 5 ]

V.

G. Fastovsky, U.

V.

Petrovsky, and A. E. Rovinsky, Cryogenic

technique, Mosc ow, 1974, in Russian).

[6] H. Tsuboi and K. Kenisu, Eddy current analyses of the thin plates

taking into account of the source curre nt distributions and i ts experi-

mental verifications,

ZEEE

Trans. Magn ., vol. 27, no.

5

pp. 4020-

4023, May 1991.

[7]

R. C. Mesquita and J. P. A Bastos, 3-D finite element solution of

induction heating pr oblem s with efficient time-stepping,

EEE

Trans.

Magn. , vol. 27, no. 6, pp. 4065-4068, June 1991.

Vladimir

Sokolovsky received the M.Sc. and Ph.D. degrees from the Uni-

versity of Novosibirsk USSR) in 197 3 and from the Electrotechnical In-

stitute of Novosibirsk in 1987, respectively.

He has part icipated in the programs on development and de sign super-

conducting e lectrical machines, current l imiting devices, magnetic energy

storage, cables. The main field of his scientific activity is theoretical and

experimental research of electromagnetic and thermal processes at low

temperatures. Since 199 2, he has been working at the Ben-Gurion Univer-

si ty, Israel on the research project of development of superconducting cur-

rent-limiting de vices for powe r systems.

Vitor Meerovich

received the M.Sc. degree in physics from the University

of Novosibirsk USSR) in 1970 and the Ph.D. degree in electrical engi-

neering from the All-Union E lectrotechnical Insti tute Moscow, USSR) in

1987.

He has worked more than 15 years in the field of applied superconduc-

tivity and in development of superconducting power equipment. He has

participated in the USSR program on the development of current-limiting

devices and superconducting magnetic energy storage. Since 199 2, he has

been working in the field of the h igh-temperature superconductivity at the

Ben-Gurion University, Israel .

1 ----a

~~

REFERENCES

Takao Takahashi, Numerical analysis of eddy current problems in-

volving saddle shaped coils in superconducting MRI magne ts,

IEEE

Trans. Magn.,

vol. 27, no.

5

pp.

3396-3999,

May 1991.

H. Brechna, Superconducting Magnet Systems, Springer-Verlag, Ber-

lin, 1973.

I. A. Glebow, J . B. Danilevish, and V. N. Shachtarin, Turbogener-

atom using superconductivity, Leningrad , Science, 1981 in Rus-

sian).

M.

Slonim

biography not available at t ime of pu blication.

Documents

Eddy Current Losses at Cryogeni Temperature