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Eddy currents in flat two-layer superconducting cables
Cryogenics 35 (1995) 495-504 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 001 I -2275/95/$10.00
Eddy currents in flat two-layer superconducting cables A.A. Akhmetov,
K. Kuroda, K. Ono and M. Takeo
Research Institute of Superconductivity, 6-10-1, Higashi-ku, Fukuoka 812, Japan Received 27 December
Faculty of Engineering,
Kyushu University,
1994
The principle of minimum energy dissipation rate is applied for calculation of steady interstrand eddy currents in flat two-layer superconducting cables subjected to transverse time-dependent magnetic fields which are non-uniform along the cable. The resulting eddy current patterns and corresponding energy losses are found to depend strongly on whether the longitudinal profile of the magnetic field contains a significant harmonic term with a period equal to the cable twist pitch. The limitations imposed by the sample geometry and the ramp rate on the experimental confirmation of results are discussed. Keywords: superconducting
cable; eddy currents;
Interstrand coupling in superconducting cables under changing magnetic fields has attracted much attention over recent yearslm6. Nevertheless, due to its complexity, the problem is far from being fully understood. Here the influence of the intrinsic periodicity of the cable properties on the energy dissipation in flat two-layer cables subjected to time-dependent magnetic fields is discussed. Attention is paid mostly to the case of spatially non-uniform magnetic fields. The methods employed in Morgan’s work’ and successive projects8,9 allow us to avoid direct application of Maxwell’s equation while calculating the currents flowing between different layers of a cable. Instead, a simple set of Faraday’s equations can be written and easily solved in the case where the steady crossover (eddy) current distribution is uniform along the cable. However, generally speaking the latter condition cannot always be satisfied even if the magnetic field does not change along the cable. As was shown recently ‘O, the currents flowing through the crossover resistances between the strands are rather periodic, with a period equal to the cable pitch length. For uniform external magnetic fields, the amplitude of oscillation decreases with an increase in sample length”, and the crossover current distribution gradually approaches that given by Morgan7. However, uniform time-dependent magnetic fields are realized mostly in experimental set-ups’2m’4. The cables used in superconducting magnets are often exposed to periodic or pseudo-periodic magnetic fields”,“, and it is evident that, even in principle, the scheme’ cannot be used in these circumstances. Distribution of the crossover currents and the resulting energy dissipation should reflect both the magnetic field pattern and the cable pitch length.
energy loss
Basic equations According to the scheme of Morgan’ let us assume that a flat two-layer cable (see Figure la) is represented by the circuit shown in Figure lb and that the strands of the front layer have electrical contacts with the strands of the rear one but not between themselves. Elementary loops composed of adjacent strands inside the cable have four resistive contacts, while the loops located at the cable edges have only three; the outward comers of these loops are formed by the strands bending from one layer of the cable to the other. In a general case, currents flowing through the resistive contacts depend on the positions both along and across the
Figure la
Flat two-layer
Cryogenics
superconducting
1995 Volume
cable: general view
35, Number
8
495
Eddy currents
Figure lb
in flat SC cables: A.A. Akhmetov
Flat two-layer
superconducting
cable: equivalent
et al.
scheme
cable. Because of this, two indexes have to be assigned to refer properly to a particular contact. Let the network shown in Figure 1 b be formed by (N - 1) rows which are parallel to the cable axis, where N is the number of strands in the cable. Index n [ 1, N - 1] is designated to refer to the position across the cable, while index k [ 1, a] shows the position of a column along the cable with respect to an arbitrary point. Here, as in reference 10 the column is defined as a set of contacts located in a zigzag line across the cable width. Hence, the resistance of each contact is marked as r,,,, the current flowing through it as jn,k, the voltage drop on the corresponding resistance (eddy voltage) and the heat generation as as “&k9 where “,z,h =_h,krpc,kr g,2,k=,j,,,kv& The magnetic flux @,,k penetrating through each loop has the same indices as the resistance at the lefthand corner. The definitions made above allow us to compose (N - 1) Faraday’s equations for the loops belonging to the kth column of the cable. These equations differ slightly for n odd and n even as well as for the upper (n = N - 1) and lower (II = 1) loops of the cable. In the latter case “1.k
496
+
“l,k+l
-
“2,k
=
Cryogenics
6,
(1)
1995 Volume
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while inside the cable for n even and for n odd, correspondingly vn,k +
“,,k+l
“n,k
+
vn,k+l
“,+l.k+l -
-
“n+l,k
V,~l,k+l -
“,t-l.k
i
=
&,,,k
(2)
=
djE,,,
(3)
The form of the last equation depends on whether the number of strands N is odd or even. If N is even, then “N-l,k
+
“N-l,k+l
-
vN-2,k
-
~N~,,k
(4a)
while for N odd, one obtains VN-l,k
+
VN-l.k+l
-
VN-2,k+I
=
+N-Lk
(4b)
Equations (l)-(4) imply that electric fields created by the time-dependent magnetic flux are applied to the resistive contacts only. This assumption (as well as the results of references 7-l 1) is correct only if the cable is subjected to the time-dependent magnetic field for a long enough span of time compared to the characteristic times of relaxation of the currents in the cable. According to ref-
Eddy currents in flat SC cables: A.A. Akhmetov erences 17 and 18, the latter values depend on the cable length and can be extremely large. Let us now consider Equations ( l)-(4) as the set of N - 1 equations containing N - 1 unknowns, i.e. the voltage drops in the (k + 1)th column. This set can be solved easilyrO. One has for n = I ,2 “I,k+l = - “1.h + “Z,k+I
= -
“l,k
“2.k
+ “2.k
+
@,.k
-
“3.k
+ “4.k
+
V,+l.k
+
et al.
tives of the magnetic fluxes penetrating through the elementary loops of the kth column of the cable. Voltage drops in the (k + 2)th column of the cable are expressed through those in the (k + 1)th column in a similar manner
(Sal + %.k
+
+2,k
+
&,k
(5b)
For the voltage drops in the (k + s)th column, an arbitrary integer, one obtains
where s is
fornodd,3sn
I = “nm1.k
-
“n,k
&.,,.r
(5c)
forneven.4snsN-4 vn,k+I
=
V,,v2,k
-
V,t~l.k
“,,+2.h
+
&,,k
= “N-U
“N-l.k+l
+
&2,k
=
“N-2.h
v,,,h
+
and for n=N-2, “N-2.h+l
+
N-
-
&,k
v,,+I.k +
(5d)
&,+,.k
1
“N-M +
-
+ “N-2.k-
vN-l,k
+
&,,k
&,,k (5e)
vN-l.k
+
+N-I,,
tw
A set of Equations (5) is written for N even. In the case of N odd, the values of the voltage drop in the (li + 1)th position are expressed through the corresponding values in the kth position in a slightly different way. Uniform
=f’
where E is the identity matrix. By considering the properties of matrix A it was shown analytically in reference 10 that if the magnetic field and crossover resistances are uniform along the cable, then the crossover currents in the (k + N)th column of the cable are the same as in the kth column, i.e. the crossover currents are periodic, with a period equal to the cable pitch length. The proof in reference 10 has been given for N even, NJ2 odd and the magnetic flux distribution across the cable being symmetrical with respect to the cable centre line. A more general case is considered in the Appendix of the present work. Consider here the case when the magnetic flux penetrating the loops is periodic along the cable and repeats itself in every Lth column of the cable: Let, for example, the dependence of the flux derivatives @n,kon the column number k be written as
(10)
(N-p)&'
(ha)
I’
p= I
and n-l
v,,=nv,-~(n-p)&,,;25n5N-l ,,=I
(6b)
Solutions (6) are correct only in the case of an infinite cable subjected to a uniform time-dependent magnetic field for a long period.
Matrix
+ . . . +A&k+s2 + E&Q+~_, (9)
solutions
If the voltage drops in the kth and (k + 1)th columns of the cable are the same then the set of Equations (5) can be solved easily. Dependence of the voltages on the column number vanishes and, as was shown in references 7 and 10
v,
+ As-‘&+,
notation
and conditions
of periodicity
Matrix notation allows us to express Equations (5) in a symbolic form. If it is assumed that all values of the voltage drop in the kth column of the cable are represented by the column matrix V,, then Vk+, is expressed as V,+,=AV,+
;p,
(7)
where the matrix A consists of the coefficients of Equations (5) and the column matrix & represents the time deriva-
where C, depends only on n and L is the integer period which is measured in numbers of columns of the cable. Later the results obtained will be easily generalized to include arbitrary periods which are not equal to the integer of the columns. Let certain crossover voltages be given in the kth column of the cable. Numerical calculations reveal that, if L # N, the voltage drops on the crossover resistances of the columns are the same after every P steps, where P is the least common multiplier of N and L, i.e. the smallest integer which can be strictly divided by both N and L. An analytical proof of this statement is given in the Appendix. The results of calculations for N = 4, L = 11 and C,, = const. are shown in Figure 2~. Here and in the following text, eddy voltages v,,,~, eddy currents jn,k and energy loss g,,, are expressed in the natural units (nu.) of C, [see Equations (5) and ( 1O)], C,,/r,,k and c/r,,, correspondingly. The longitudinal profile of the magnetic flux is shown by the dashed line. Interference of oscillations with periods of both N and L is seen in the figure. We need to emphasize here that the distribution shown is intended to demonstrate the properties of periodicity only. It strongly depends on what eddy voltages in the kth column were taken as initial conditions. On the other hand, if L = N then the eddy voltage distribution becomes aperiodic. Eddy voltages do not repeat themselves after P steps anymore. An example for N = 4 = L is shown in Figure 2b. The same is true if the magnetic flux longitudinal variation G(k) is unharmonic.
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Eddy currents in flat SC cables: A.A. Akhmetov
et al. N-1.K G
=
c
(11)
&k
?l=l,k=l
as a certain function of the crossover voltages VI in the first column of the sample. Then N - 1 minimizing equations
F 3
si
-5
>’ 20
0
El B m-20 :: b
(12)
were used to calculate the column matrix VI and afterwards Equation (9) was used again to find the eddy voltages in the rest of the cable. It was found that both methods provide the same results. At the same time, the latter method is much more preferable in the case of large samples.
Column number, s
8 9 =9
ln=1.2,. .,N-l = o
Infinite cables
0
6
16
24
32
Column number, s Figure2 Crossover voltages in first row of four strand penetrated by a time-dependent magnetic flux which is oidal along the cable. The results are obtained by direct cation of Equations (5) in the cases of (a) L= 11 and (b)
cable sinusappliL=4
Now let the infinite cable be subjected to a time-dependent magnetic field which has a sinusoidal profile along the cable with an integer period L # N. Consider the interval of the cable between the columns k and (k + P). The crossover voltages are periodic with a period P, and the average heat generation per unit length of the cable g is the sum of the heat release in all crossover points G divided by P, i.e.
(13) In this case its Fourier decomposition necessarily includes the sinusoidal term with L = N, which provides an unperiodic contribution in the eddy voltage pattern.
or
g=f
Application of minimum rate (MEDR) principle
energy dissipation
Finite samples Without boundary conditions the set of Equations (5) is not closed. Indeed, it does not include Kirchhoff’s equations which are different for the resistive contacts inside the sample and at the sample edges. On the other hand, from the physical point of view it is evident that if the cable is long enough, boundary conditions cannot be responsible for the crossover voltage pattern. If one considers an infinite sample then another principle has to be applied to find the crossover voltage distribution. As was shown in reference 11 on a few specific examples, for samples of finite size subjected to a uniform time-dependent magnetic field the real crossover current (voltage) distribution provides the minimum total energy G dissipated per unit time among all the possible distributions satisfying Equations (5). In this work the calculations” were repeated for r,,, = r = const. and a few different magnetic flux rate &)n,k distributions in a few samples consisting of different numbers of strands N and columns K. First, calculations were done directly by applying the corresponding Faraday’s and Kirchhoff’s equations. To do this, eddy currents have been expressed through the currents in the strands. This leads to a set of K(N - 1) independent Faraday’s equations which have to be solved numerically to find the real eddy voltage distribution. As an alternative method, Equation (9) was used to express the sum of the heat generation in the crossover contacts of the cable
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Fr e,v,+, .S=rl
(14)
where Jk+s(l) is the row matrix which represents the crossover currents in the corresponding column of the cable. According to Equation (9), for given crossover resistances and rate of magnetic flux increase, the average heat generation can be symbolically expressed as a certain function of the voltage drops in the kth column g=F(Vlc)
(15)
For simplicity’s sake, let us consider the case of constant contact resistances r, k = r. One can use the principle of minimum energy dissipation rate (MEDR) to find the crossover voltage distribution in the kth column and consequently in the entire infinite cable subjected to a timedependent magnetic field which is cleanly sinusoidal along the cable or is composed of many sinusoidal terms except the one with the period L = N. Note that in long samples the average heat generation g is proportional to the total value G and we use (N - 1) additional equations
(16)
Then, using Equations
(9) and ( 14) it follows that
g = S[J~)J~+ P-l c (Jp) (A”)‘f’AsJk + J~‘(A”)“‘B, s=I
Eddy currents in flat SC cables: A.A. Akhmetov + B”‘A”J, + B”‘B s s s )] where superscript
(17)
(t) designates
transposition
and
Application of the MEDR principle in the form of Equations (16)-( 18) provides us with a set of equations in which the eddy currents in the kth column are expressed as a linear combination of the time derivatives of the magnetic fluxes penetrating through every elementary loop located in the kth to the (k + P - 1)th columns. The equations mentioned are too cumbersome to be given in the explicit form but can be easily solved numerically. In the following it is sometimes convenient to separate the factors influencing the eddy current distribution. Let the magnetic flux rate be set as &=
et al.
RIY,
-20 II
1, ” ’ 64 32 96 Column number, s
0
,I
(19)
where R is the rate of current increase in the magnet which provides the background time-dependent magnetic field and qk is the column matrix of corresponding geometrical factors. Then, the eddy currents in the kth column of the cable can be symbolically expressed as
J,i = RY( %+q)~o~qc~-,
(20)
where Y represents the corresponding of the geometrical factors V.
linear combinations
0
50 100 150 Column number, s
Figure 3 Crossover voltages in finite samples of eight strand cable calculated by means of the MEDR principle for L= 9 and (a) Ls = 108, fb) LS = 180
with a period L are present and the eddy voltage pattern does not depend on sample length. The dependence of the average dissipated energy g on sample length L, is shown in Figure 4~. Pseudo-oscillations
Results Uniform distribution infinite cable
1
of magnetic
flux along
2900
Sinusoidal distribution finite sample
of magnetic
1
1
I
a
’
’ 200
I
I 400
I
600
Sample length, L,
field along
The MEDR principle was applied to calculate eddy voltages in finite samples of different length LS = MP + Lo, where M=0,1,2 . . . and 0 < Lo < P, in the cases where longitudinal profiles of the magnetic field were sinusoidal with both L S N and L = N. The results are illustrated for N=8,L=9,L0=P12=36andM=landM=2inF’igures 3a and b, correspondingly. It is seen that oscillations with period L are modulated by oscillations with period P = LN, but unlike the results shown in Figure 2a there are practically no oscillations with a period equal to N. With increasing M, the amplitude of the oscillations with period P becomes smaller, while the amplitude of oscillations with period L approaches a constant value. On the other hand, it follows from calculations that if Lo = 0, i.e. the sample length is equal to the integral-fold of P, only oscillations
I
I
3100
If the magnetic flux penetrating the loops of the cable is homogeneous then the MEDR principle provides an eddy current distribution which is uniform along the cable. It coincides with the eddy current distribution obtained directly from Equation (6) written for the case where the crossover currents are the same at the left and right comers of every loop.
I
0
400
800
12 10
Sample length, L, Figure4 Dependence of average heat generation in the interstrand contacts of eight strand cable on sample length for (a) L=9 and (b) L=8
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Eddy currents in flat SC cables: A.A. Akhmetov
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with a period equal to L/2 appear superimposed on the larger ones with a period equal to P. Indeed, according to the previous results at N # L, the eddy voltage pattern and consequently g have to be the same at L, = P, 2P, 3P. . What is important, however, is that g has reached a nearly constant value at Ls of the order of a few times P. The situation changes drastically if the period of the magnetic field oscillations coincides with the cable pitch length. For different sample lengths the MEDR principle provides different eddy voltage patterns. In general, there are pseudo-oscillations which increase linearly from the sample centre to its ends. At the ends of long samples, the amplitude of the pseudo-oscillations reaches quite high values. Illustrations for L = N = P = 8 are given in Figures .5a for L, = 96 and in b for L, = 160. The average dissipated energy increases infinitely with lengthening of the sample if L = N, as seen in Figure 4b. Also, small superimposed pseudo-oscillations with a period Ll2 can be seen in the inset. Discounting these, the dependence of g on L, can be considered as an approximately square one. Sinusoidal distribution infinite cable
of magnetic
field along
Equations ( 16)-( 18) were solved numerically for a set of parameters which includes different numbers of cable strands N and the periods L # N of applied sinusoidal timedependent magnetic field. Then Equation (9) was used to find the eddy voltage (current) distribution throughout the cable. The eddy voltages in the first row of the cable are shown in Figure 6 as a function of the column number s for N = 4, L = 11. It is seen from a comparison of Figures 2~2,3a and b and 6 that application of the MEDR principle
I
I
I
I
0
64 32 Column number, s
96
0
60 Column number, s
160
100 3 ti ‘;? >’ 8 8 = P
0
5 B 3 6 -100
Figure 5 Crossover voltages calculated for the case where the period of the magnetic field spatial oscillations coincides with the cable pitch length for (a) Ls = 96 and (b) L, = 60
500
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Column number, s Figure6 Crossover voltages in an infinite four strand cable penetrated by a spatially oscillating time-dependent magnetic flux (---): -, exact solution; ., pseudo-uniform solution
to an infinite cable leads to suppression of eddy voltage oscillations with a period equal to both the cable pitch length and P. Instead, only the oscillations with a period equal to that of the external magnetic field are clearly seen. In other words, the crossover current distribution looks as if it is being determined only by the local value of the magnetic flux rate &. For comparison, the Morgan-like solution is shown in Figure 6 by the dotted line. It can be seen that the exact solution yields a significantly larger amplitude of oscillations than the approximate one. The difference between the solutions disappears as L - x and increases as L - N. The summations in Equation (17) have been performed on a cable interval P. Since for L #N the crossover voltages repeat themselves every P steps, exactly the same results were obtained while performing summations on any part of the cable which is the integral-fold of P. (This follows from the results of the previous section, also, as well as the fact that approximately the same eddy voltage pattern would be found if the summations were performed in any part of the cable which is much longer than P.) Formally speaking, also, in the case of L = N one can find the crossover current distribution in the kth column of the cable which minimizes the energy dissipated in the interval, for example, between the kth and the (k + P)th positions. However, the results will be different if obtained from any other interval, as can be seen in Figures 5a and b. Until now we have considered only those periods of magnetic field oscillation which exactly coincide with an integer of columns. Suppose now that L, is an arbitrary period. Using, for the summation in Equation (17), the interval P = uNL,, where u is the integer multiplier needed to make B an integer too (or nearly an integer), we obtain the exact (approximate) value of the array V, which satisfies the conditions of g minimization. TheAaccuracy of the approximation improves as the value of P approaches an integer. After the crossover voltage (current) distribution which minimizes the average energy dissipation per unit length of cable g in a given magnetic field has been found, Equation (14) allows us to calculate the value of g itself. The dependences of g on the period of the magnetic field variation L, are shown in Figure 7u and b for N = 4 and N = 14, respectively. In both cases C,, = C = const. It is seen that the energy loss in the cables increases as the period of magnetic field oscillation approaches the cable twist pitch. At L, = N the results diverge. As seen in Figures 7a and b L, - ~0 the average heat generation g approaches a constant value g,. An
Eddy currents in flat SC cables: A.A. Akhmetov
1
-1
\ / 1 14 28 42 Column number, s
0 Period of magnetic field oscillation, L,
I’
I’/
1
I
rl
\
-J 24980
Figure 7 Dependence of average heat generation in an infinite cable g on the period of the magnetic field oscillations L, for (a) N=4 and (b) N= 14
to that in the case of uniform
N(N“ - 1 )C
mag-
(21)
240r
can be used for its calculation. Arbitrary
periodic
distribution
of magnetic
field
Numerical solutions of Equations ( 16)-( 18) reveal that if the magnetic flux distribution along the cable is unharmanic, the resulting eddy voltage pattern consists of elementary ones which correspond to the terms of a Fourier decomposition of the magnetic flux longitudinal profle. Isolated non-uniformity in magnetic penetrating a finite sample
I 25000
25020
Column number, s
Period of magnetic field oscillation, La
g, =
5
,
1
expression similar netic field7,“’
et al.
flux
Consider now the problem of an unharmonic magnetic flux in more detail. Let the cable be of finite length L, S N with an isolated non-uniformity in magnetic flux ramp rate &J~,, at its centre. An example of a ‘bump’-like non-uniformity is shown in Figure 8a by the dashed line. Fourier decomposition of such non-uniformity leads to the appearance of a set of sinusoidal terms including that with a period L equal to the cable pitch length. Because of this, in the case of an infinite cable containing non-uniformity in every Lsth position, the solutions of Equations (16)-( 18) diverge. However, for single non-uniformity, application of the MEDR principle gives reasonable results. Figures 8a and b show the current distributions in the first and second rows of a four strand cable for two different
Figure 8 Eddy currents in samples of different length containing an isolated ‘bump’-like non-uniformity in the magnetic flux longitudinal profile for (a) L, = 56 and (b) L, = 5 x IO4
cable lengths Ls. It is seen in Figure 8a drawn for L, = 56 that the currents exhibit periodicity with a period N = 4 both in the non-uniformity and in those parts of the cable where the magnetic flux is uniform. If L, = 5 x 10’ (see Figure 8b), the crossover current oscillations become smaller in the uniform parts of the sample, while there is a noticeable increase in the variations in eddy currents in the non-uniformity.
Discussion Crossover current patterns In practice one never deals with infinite cables. It follows from calculations that in the case of harmonic longitudinal oscillations of the applied time-dependent magnetic field with a period L f N, the condition Ls 9 P, from a practical point of view, is equivalent to an infinite cable. The results presented in Figures 3a and b and 4a testify that one can consider the sample as practically infinite if its length reaches = 1OP. On the contrary, if the magnetic field is harmonic with a spatial period L = N, the results diverge when the MEDR principle or direct calculations are applied to different intervals of the cable. This means that the sample is never long enough to be called infinite. At the same time, estimation of the amplitude of the pseudo-oscillations in eddy voltages at the conductor ends can be based upon the sample length, as can be seen in Figures 5a and b. Equation (20) is a linear equation with respect to eddy voltages (currents). Combination of the harmonic terms with different periods in the longitudinal profile of the magnetic field provides superposition of the eddy voltage patterns caused by these terms. If the sample is long enough
Cryogenics
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Eddy currents in flat SC cables: A.A. Akhmetov
et al.
and the magnetic field distribution contains a significant harmonic term with a period L = N, it follows from the previous consideration that this term would be the main one determining the eddy current distribution. To explain the results shown in Figures 8a and b note that the periodicity in eddy voltages takes its origin not from the boundary conditions, which can be neglected” at Ls > lON, but from the presence of non-uniformity in the magnetic flux. Deviation of the eddy current distribution from a uniform one provides different contributions to the average dissipated energy from the uniform parts of the sample and in the non-uniformity. In the latter case the contribution is slightly larger. Because of this, in order to minimize the total energy G dissipated in a sample with short uniform parts, larger oscillations occur exactly in these parts. On the other hand, if the uniform parts are very long, as in the case illustrated in Figure 8b, the condition of G minimization leads to an increase in oscillations in the non-uniformity and to their decrease in the rest of the sample. For even larger LS the oscillations in the uniform parts would be almost completely suppressed. A similar situation has been treated in reference 10 by repeatedly applying Equations (5) to find eddy currents in subsequent columns of the sample starting from Morgan’s eddy current distribution at one of the sample ends. Nearly all the above (as well as Morgan’s basic model) is correct only if the (pseudo-) periodic patterns shown in the corresponding figures are flawlessly established. On the other hand, it is well known that if a magnetic field is applied to a cable at some instant to, for some time after this the screening currents are present mainly at the cable edges. Then, gradually, the currents diffuse inside the cable. Thus (pseudo-) periodic structures are created. This process takes a certain time, which is discussed in the next section. Correlation
time
In fact, (pseudo-) oscillations in the crossover currents with a period equal to the cable pitch length existing throughout the entire sample are due to some kind of current correlation, because both the ‘energy-based’ approach of this paper or direct calculations ‘Oimply that crossover currents in any column of the cable reflect the conditions existing far off. Figures 5 and 8 give quite clear examples. On the other hand, it is necessary to admit that establishing the correlation takes a certain time TVwhich has to be compared with the time of the magnetic field increase To. The latter term is different for various practical applications of superconductivity but rarely exceeds a few hours. It is not quite clear yet what the correlation time really is, though some analogies based on estimations are available. In references 19 and 20 the effect was observed in the performance of dipole magnetsI made of Rutherfordtype cables. It was found that the magnetic field and its main harmonics’6,‘9 oscillate along the magnet axis with a wavelength equal to the cable pitch length. At a low transport current, the periodic magnetic field patterns can persist without any significant decay for more than 12 h19. The explanation of the effect given in reference 18 is based on the assumption that during magnet energization slightly different currents are trapped in the strands, even at zero transport current in the cable. Current loops created by this consist of currents flowing along the cable through one set of strands and returning through another set of strands. The circuit is connected through crossover contact
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resistances and/or current leads. Since some effective inductance per unit length can be attributed to the strands”*‘*, current loops circulating in the cable do not disappear immediately after the magnet discharge. The decay time of the current loops is proportional to the second power of their length and in the case of the longest current loop it can be as large as a few days. Assuming that the results’9.20 properly reflect the characteristic time of the end-to-end correlation of the crossover currents throughout the cable and the explanation’* is pertinent to the effect under consideration, some rough estimations are possible. Let us suppose’* that the effective inductance of the strand is of the order of 10m8H m-‘, and that the transverse contact resistance between the strand and the rest of the cable R, is =lO-’ R m, which is consistent with the data of reference 21. It then follows18 that rc = 10 s if the sample length is -10 m, and T== 103s if the sample length is = 100 m. In turn, this means that for a 10 m long sample the eddy currents are correlated after 530 s, while it takes = 1 h to achieve correlation in a 100 m long sample. At the same time, if the cable pitch length, the period of the magnetic field variation and their difference are all = 10-l m, then already a few metre long sample (with a correlation time ~1 s) can be considered as practically infinite. Hence, given the same R, and effective inductances of the strands, the main results presented in Figures 3-6 can be observed in the experiments on such samples at times of magnetic field increase of the order of 10s. On the other hand, the inductive terms which have to be included in Equations (l)-(9) to describe unstationary processes22 if rr 5 r,, are a hindrance to establishing correlated patterns throughout cables with the same length as those in real windings. ‘Infinite’ energy loss per unit length of cable due to the harmonic term in the magnetic flux with a period L = N is a kind of mathematical abstraction. Indeed, the time needed to achieve correlation is proportional to the square of the sample length LS, and the ‘correlated’ ramp rate (i.e. the ramp rate at which rr > TV and the eddy current in the whole sample can be considered as being correlated) R, is proportional to the minus second power of the sample length. In turn, the average energy loss g, in the cable due to the eddy currents is proportional to the square of the ramp rate. (Subscript c is used here to emphasize the correlation. In fact, all previous results are related to cases of correlated eddy currents.) It follows that
gc
zgn(Ls),G L4s
go(Ls)
(22)
where go is a geometrical factor composed of the local column matrices !Pk [see Equations ( 19) and (20)]. It is seen in Figure 4b that go is nearly proportional to the square of LS. As a result, extension of the sample leads to such small ramp rates that g,, G, - 0 (if eddy currents are to correlate through the entire sample). Otherwise, rr < T=(R > R,) and there is no significant correlation between the eddy currents in different parts of the conductor and no significant increase in the eddy currents at the conductor ends. As a result, at finite ramp rates instead of divergence there is the maximum in energy loss when L = N. However, if LS % N is fixed and magnetic flux variation along the cable includes sufficient unharmonic terms then, when the ramp rate decreases, the value g/R2 should increase, while it is constant if no long distance correlation
Eddy currents in flat SC cables: A.A. Akhmetov is taken into account. In the authors’ opinion a corresponding experiment can be performed to prove indirectly the results of the present calculations. Also, short range (pseudo-) periodicity in eddy currents in non-uniformities similar to that shown in Figure 8b can, probably, take place at very short times of magnet energization. Quantitatively, we will consider time-dependent eddy currents in flat superconducting cables elsewhere.
10
11
12
Conclusions 13
Steady eddy currents in long flat two-layer superconducting cables penetrated by a time-dependent magnetic field, which is sinusoidal along the cable, exhibit a periodicity determined only by the magnetic field. The eddy current distribution is pseudo-periodic if the period of magnetic field oscillations exactly coincides with the cable pitch length and is periodic in the opposite case. In comparatively short samples, periodic eddy current distribution is modulated with a larger period common to the magnetic field oscillations and the cable pitch length. If the ramp rate of the magnetic flux is fixed, the energy loss per unit length of the cable increases as the period of the magnetic field longitudinal oscillations approaches the cable pitch length. Oscillations or pseudo-oscillations with a period equal to the cable pitch length appear in long cables when the longitudinal variations of the applied magnetic field are unharmonic. If the longitudinal distribution of the time-dependent magnetic field is periodic and contains a significant sinusoidal term with a period equal to the cable pitch length, then the average energy loss in the long cable due to interstrand eddy currents is not proportional to the square of the ramp rate.
14
15
16
17 18
19
20
21
22
Acknowledgements This work was supported in part by Kyushu Electric Power Co. Inc. One of the authors (A.A.) found it very useful to discuss many problems touched upon in this paper with Dr S. Takacs.
References Tacacs, S. Coupling losses in cables in spatially changing ax. fields Cr?/ogenics ( 1982) 22 661-665 Kwasnitza, K. and Horvath, I. Experimental evidence for an interaction effect in the coupling losses of the cabled superconductors Cryogenics (1983) 23 9-14 Fevrier, A. Losses in a twisted multifilamentary superconducting composite submitted to any space and time variations of the electromagnetic surrounding Cryogenics ( 1983) 23 185-200 Okada, T., Takahata, K. and Nishijima, S. Coupling losses in superconducting cables Adv Cryog Eng (1986) 32 763-769 Sumpson, M.D. and Collings, E.W. Influence of cable and twist pitch interactions on eddy currents in multifilamentary strands calculated using an anisotropic continuum model Proc ICMC Vol 40A (Eds Reed, R.P. Fickett, F.R., Summers, L.T. and Stieg, M.) Plenum Press, New York and London (1993) 5799588 Tactics, S. Coupling losses in inhomogeneous cores of superconducting cables Cryogenics (1992) 32 258-264 Morgan, G.H. Eddy currents in flat metal-tilled superconducting braids J Appl Phys (1973) 44 3319-3322 Sytnikov, V.E., Svalov, G.G., Akopov, S.G. and Peshkov, LB. Coupling losses in superconducting transposed conductors located in changing magnetic fields Cryogenics (1989) 29 926-930 Verweij, A.P. and ten Kate, H.H.J. Coupling currents in Rutherford
et al.
cables under varying conditions IEEE Tram Appl Supercond (1993) 3 146-149 Akhmetov, A.A., Devred, A. and Ogitsu, T. Periodicity of crossover currents in a Rutherford-type cable subjected to a time-dependent magnetic field J Appl Phys (1994) 75 3 176-3 I83 Akhmetov, A.A., Kuroda, K. and Takeo, M. Influence of sample geometry on amplitude of eddy current oscillation in Rutherford-type cables, paper presented at Applied Superconductivity Conf, Boston, MA, USA (Ott 1994) Sumiyoshi, F., Kanai, Y., Kawashima, T., Iwakuma, M. et al. Sweep-rate dependences of losses in aluminum-stabilized superconducting conductors for the Large Helical Device Applied Superconductiviryfor Nuclear Fusion Research Eds Miyahara, A. and Yamamoto, J., Elsevier, Amsterdam, The Netherlands (1993) 37 l-376 Kwasnitza, K. and Bruzzone, B. Measurement of end effects in the coupling losses of multifilament superconductors Proc ICEC IO Butterworths, Guildford, UK (1984) 606-609 Sumption, M.D., Callings, E.W., Scanlan R.M. and van Oort, J.M. VSM studies of interstrand eddy current coupling loss in a series of coated-strand cables, paper presented at Applied Superconductivity Conf, Boston, MA, USA (Ott 1994) Motojima, O., Akaishi, K., Fujii, K., Fujiwaka, S. et al. Physics and engineering design of the Large Helical Device Fusion Eng Design (1993) 20 3-14 Daum, C. and ter Avest, D. Three-dimensional computation of magnetic fields and Lorentz forces of an LHC dipole magnet using the method of image currents Proc MT- 11 (Eds Sekiguchi, T. and Shimamote, S.) Elsevier Applied Science, New York and London (1989) 302-307 Turck, B. Influence of the transverse conductance on current sharing in a two-layer superconducting cable Cryogenics (1974) 14 448-454 Akhmetov, A.A., Devred, A., Mints, R.G. and Schermer, R.I. Current loop decay in Rutherford-type cables Supercollider 5 Ed Hale, P. Plenum Press, New York, USA (1995) 443-446 Schmuser, P. Superconducting magnets for particle accelerators Proc AIP Conf Vol 249 (Eds Month, M. and Dienes, M.) American Institute of Physics, New York, USA ( 1992) 1099-I 158 Ghosh, A.K., Robins, K.E. and Sampson, W.B. Axial variations in the magnetic field of superconducting dipoles Supercollider 4 Ed Nonte, J. Plenum Press, New York, USA (1992) 765-772 Kovachev, V.T., Neal, MJ., Capone, D.W., Carr, WJ. and Swenson, C. Interstrand resistance of SSC magnets Cryogenics ( 1994) 34 813-820 Verweij, A.P. and ten Kate, H.HJ. Super coupling currents in Rutherford type of cables due to longitudinal non-homogeneities of dB/dt, paper presented at Applied Superconductivity Conf, Boston, MA, USA (Ott 1994)
Appendix For an N strand cable, N even, the matrix A of the rank (N - 1) has the symmetric form -110
0
0
o...o
0
0
0
0
0
-1
l-l
1 0
0
0
0
0
0
0
0
0
l-l
1 0
0 ... 0
0
0
0
0
0
0
l-l : ;
l-l ; ;
1 ... 0 ; ‘.. ;
0 f
0 f
0 f
0 f
0 i
0
0
0
0
0
0 .
l-l
0
0
0
0
0
0 .
0
0
0
0
0
0
0
0
0
0
0
A=;
l-l
1 0
0
1-l
1 0
0 ... 0
0
l-l
l-1
0 ... 0
0
0
0
(Al)
l-l
The corresponding matrix for N odd differs slightly. This difference does not affect the following consideration. Consider the characteristic equation for the matrix A pdh)=O,p,=I.-M(
Cryogenics
(AZ)
1995 Volume
35, Number
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503
Eddy currents in flat SC cables: A.A. Akhmetov
et al.
where A represents the eigen-values of the matrix A. Direct calculations performed for N = 3, 4, 5, 6 show that in these specific cases Equation (A2) can be written in the form AN-’ + AN-2+ . . + A* + A + 1 = 0
(A3)
To prove this statement for arbitrary N, we express the determinant pNtz through determinants of lower ranks. Applying double decomposition, it follows’” that PN+z(A)
=
(A2
+
1 )p,dA)
(A4)
-PN-AA)
Now, assuming that Equation (A3) holds in the case of matrices of both ranks N and (N - 2), one comes to the conclusion that the characteristic equation for the matrix of rank (N + 2) has the form of Equation (A3) also. All the roots of Equation (A3) are different, and application of Cayley-Hamilton’s theorem allow us to find the relation between different powers of matrix A. It follows that AN = E
9
AN-’
+ AN-2 +
.
.
.
+A2+A+E=0
If L is even and L # mN, then Vk+p= V, + 2(A”’
+AN-3 + . . . +A)&
+ 2(ANm2+AN4+
. . . +E)fk,
(A9)
where L-2 0,
=
c
L-1 +k+s>
a2
~0.2.4
=
c
&+s
(A101
SF1.3.5.
For a sinusoidal fl, and a2 are Lastly, when voltages in the
magnetic field with a spatial period L, both equal to zero, so that Vk+p= V,. P = L = mN, where m = 1, 2, 3 . . ., the eddy (k + P)th column can be expressed as mN-Nch-
V k+P= V, + gAN-O h=l
I
c
&+,
(All)
r=h-I,N+h-1,2N+h-I
We designate S(h) to the result for the inner summation in Equation (Al 1). For any given 1 ‘=rh 5 N and sinusoidal magnetic field
(As)
where 0 designates the zero matrix. Then, if the magnetic flux rate & does not depend on the position along the cable, Equation (9) for s = N accepts the form V k+N=ANVk+(AN-1+AN-2+...+A2+A+E)&=Vk (Ah) Now let Equation (10) represent the dependence of magnetic flux on position along the cable. For L odd and L f mN (m is an arbitrary integer), eddy voltages in the (k + P)th column are expressed through the eddy voltages in the kth column as
S(h) = C[sin(2rhlmN) + sin(2tiN + h)lmN) + . . + sin( 2_rr(mN - N + h)lmN)]
(A12)
where C is the column matrix which represents coefficients C, of Equation ( 10). Algebraic transformations allow us to rewrite the last equation in the form S(h) = C[cos( a)S, + sin( a)S,]
(A13)
where cY=2?--
h-N (A14)
mN
I’,,, = (AN)‘Tk + (AN-’ + AN-2 + . . +A2 +A + E)0 (A7) where
S, = sin[2til/m)]
=
+ . . . + sin(2n)
(A15)
and
L-l n
+ sin[2rr(2/m)]
c
(A81
&+s
S2 = cos[27r( l/m)] + cos[27r(2/m)l+.
c=a
. . + cos(2~) (A16)
It is clear from comparison that Vk+p = V,.
504
Cryogenics
of Equations
1995 Volume
(A5) and (A7) Both S, and S2 are equal to zero for any m except m = 1.
35, Number
8
Eddy currents in flat two-layer superconducting cables A.A. Akhmetov,
K. Kuroda, K. Ono and M. Takeo
Research Institute of Superconductivity, 6-10-1, Higashi-ku, Fukuoka 812, Japan Received 27 December
Faculty of Engineering,
Kyushu University,
1994
The principle of minimum energy dissipation rate is applied for calculation of steady interstrand eddy currents in flat two-layer superconducting cables subjected to transverse time-dependent magnetic fields which are non-uniform along the cable. The resulting eddy current patterns and corresponding energy losses are found to depend strongly on whether the longitudinal profile of the magnetic field contains a significant harmonic term with a period equal to the cable twist pitch. The limitations imposed by the sample geometry and the ramp rate on the experimental confirmation of results are discussed. Keywords: superconducting
cable; eddy currents;
Interstrand coupling in superconducting cables under changing magnetic fields has attracted much attention over recent yearslm6. Nevertheless, due to its complexity, the problem is far from being fully understood. Here the influence of the intrinsic periodicity of the cable properties on the energy dissipation in flat two-layer cables subjected to time-dependent magnetic fields is discussed. Attention is paid mostly to the case of spatially non-uniform magnetic fields. The methods employed in Morgan’s work’ and successive projects8,9 allow us to avoid direct application of Maxwell’s equation while calculating the currents flowing between different layers of a cable. Instead, a simple set of Faraday’s equations can be written and easily solved in the case where the steady crossover (eddy) current distribution is uniform along the cable. However, generally speaking the latter condition cannot always be satisfied even if the magnetic field does not change along the cable. As was shown recently ‘O, the currents flowing through the crossover resistances between the strands are rather periodic, with a period equal to the cable pitch length. For uniform external magnetic fields, the amplitude of oscillation decreases with an increase in sample length”, and the crossover current distribution gradually approaches that given by Morgan7. However, uniform time-dependent magnetic fields are realized mostly in experimental set-ups’2m’4. The cables used in superconducting magnets are often exposed to periodic or pseudo-periodic magnetic fields”,“, and it is evident that, even in principle, the scheme’ cannot be used in these circumstances. Distribution of the crossover currents and the resulting energy dissipation should reflect both the magnetic field pattern and the cable pitch length.
energy loss
Basic equations According to the scheme of Morgan’ let us assume that a flat two-layer cable (see Figure la) is represented by the circuit shown in Figure lb and that the strands of the front layer have electrical contacts with the strands of the rear one but not between themselves. Elementary loops composed of adjacent strands inside the cable have four resistive contacts, while the loops located at the cable edges have only three; the outward comers of these loops are formed by the strands bending from one layer of the cable to the other. In a general case, currents flowing through the resistive contacts depend on the positions both along and across the
Figure la
Flat two-layer
Cryogenics
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1995 Volume
cable: general view
35, Number
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495
Eddy currents
Figure lb
in flat SC cables: A.A. Akhmetov
Flat two-layer
superconducting
cable: equivalent
et al.
scheme
cable. Because of this, two indexes have to be assigned to refer properly to a particular contact. Let the network shown in Figure 1 b be formed by (N - 1) rows which are parallel to the cable axis, where N is the number of strands in the cable. Index n [ 1, N - 1] is designated to refer to the position across the cable, while index k [ 1, a] shows the position of a column along the cable with respect to an arbitrary point. Here, as in reference 10 the column is defined as a set of contacts located in a zigzag line across the cable width. Hence, the resistance of each contact is marked as r,,,, the current flowing through it as jn,k, the voltage drop on the corresponding resistance (eddy voltage) and the heat generation as as “&k9 where “,z,h =_h,krpc,kr g,2,k=,j,,,kv& The magnetic flux @,,k penetrating through each loop has the same indices as the resistance at the lefthand corner. The definitions made above allow us to compose (N - 1) Faraday’s equations for the loops belonging to the kth column of the cable. These equations differ slightly for n odd and n even as well as for the upper (n = N - 1) and lower (II = 1) loops of the cable. In the latter case “1.k
496
+
“l,k+l
-
“2,k
=
Cryogenics
6,
(1)
1995 Volume
35, Number
8
while inside the cable for n even and for n odd, correspondingly vn,k +
“,,k+l
“n,k
+
vn,k+l
“,+l.k+l -
-
“n+l,k
V,~l,k+l -
“,t-l.k
i
=
&,,,k
(2)
=
djE,,,
(3)
The form of the last equation depends on whether the number of strands N is odd or even. If N is even, then “N-l,k
+
“N-l,k+l
-
vN-2,k
-
~N~,,k
(4a)
while for N odd, one obtains VN-l,k
+
VN-l.k+l
-
VN-2,k+I
=
+N-Lk
(4b)
Equations (l)-(4) imply that electric fields created by the time-dependent magnetic flux are applied to the resistive contacts only. This assumption (as well as the results of references 7-l 1) is correct only if the cable is subjected to the time-dependent magnetic field for a long enough span of time compared to the characteristic times of relaxation of the currents in the cable. According to ref-
Eddy currents in flat SC cables: A.A. Akhmetov erences 17 and 18, the latter values depend on the cable length and can be extremely large. Let us now consider Equations ( l)-(4) as the set of N - 1 equations containing N - 1 unknowns, i.e. the voltage drops in the (k + 1)th column. This set can be solved easilyrO. One has for n = I ,2 “I,k+l = - “1.h + “Z,k+I
= -
“l,k
“2.k
+ “2.k
+
@,.k
-
“3.k
+ “4.k
+
V,+l.k
+
et al.
tives of the magnetic fluxes penetrating through the elementary loops of the kth column of the cable. Voltage drops in the (k + 2)th column of the cable are expressed through those in the (k + 1)th column in a similar manner
(Sal + %.k
+
+2,k
+
&,k
(5b)
For the voltage drops in the (k + s)th column, an arbitrary integer, one obtains
where s is
fornodd,3sn
I = “nm1.k
-
“n,k
&.,,.r
(5c)
forneven.4snsN-4 vn,k+I
=
V,,v2,k
-
V,t~l.k
“,,+2.h
+
&,,k
= “N-U
“N-l.k+l
+
&2,k
=
“N-2.h
v,,,h
+
and for n=N-2, “N-2.h+l
+
N-
-
&,k
v,,+I.k +
(5d)
&,+,.k
1
“N-M +
-
+ “N-2.k-
vN-l,k
+
&,,k
&,,k (5e)
vN-l.k
+
+N-I,,
tw
A set of Equations (5) is written for N even. In the case of N odd, the values of the voltage drop in the (li + 1)th position are expressed through the corresponding values in the kth position in a slightly different way. Uniform
=f’
where E is the identity matrix. By considering the properties of matrix A it was shown analytically in reference 10 that if the magnetic field and crossover resistances are uniform along the cable, then the crossover currents in the (k + N)th column of the cable are the same as in the kth column, i.e. the crossover currents are periodic, with a period equal to the cable pitch length. The proof in reference 10 has been given for N even, NJ2 odd and the magnetic flux distribution across the cable being symmetrical with respect to the cable centre line. A more general case is considered in the Appendix of the present work. Consider here the case when the magnetic flux penetrating the loops is periodic along the cable and repeats itself in every Lth column of the cable: Let, for example, the dependence of the flux derivatives @n,kon the column number k be written as
(10)
(N-p)&'
(ha)
I’
p= I
and n-l
v,,=nv,-~(n-p)&,,;25n5N-l ,,=I
(6b)
Solutions (6) are correct only in the case of an infinite cable subjected to a uniform time-dependent magnetic field for a long period.
Matrix
+ . . . +A&k+s2 + E&Q+~_, (9)
solutions
If the voltage drops in the kth and (k + 1)th columns of the cable are the same then the set of Equations (5) can be solved easily. Dependence of the voltages on the column number vanishes and, as was shown in references 7 and 10
v,
+ As-‘&+,
notation
and conditions
of periodicity
Matrix notation allows us to express Equations (5) in a symbolic form. If it is assumed that all values of the voltage drop in the kth column of the cable are represented by the column matrix V,, then Vk+, is expressed as V,+,=AV,+
;p,
(7)
where the matrix A consists of the coefficients of Equations (5) and the column matrix & represents the time deriva-
where C, depends only on n and L is the integer period which is measured in numbers of columns of the cable. Later the results obtained will be easily generalized to include arbitrary periods which are not equal to the integer of the columns. Let certain crossover voltages be given in the kth column of the cable. Numerical calculations reveal that, if L # N, the voltage drops on the crossover resistances of the columns are the same after every P steps, where P is the least common multiplier of N and L, i.e. the smallest integer which can be strictly divided by both N and L. An analytical proof of this statement is given in the Appendix. The results of calculations for N = 4, L = 11 and C,, = const. are shown in Figure 2~. Here and in the following text, eddy voltages v,,,~, eddy currents jn,k and energy loss g,,, are expressed in the natural units (nu.) of C, [see Equations (5) and ( 1O)], C,,/r,,k and c/r,,, correspondingly. The longitudinal profile of the magnetic flux is shown by the dashed line. Interference of oscillations with periods of both N and L is seen in the figure. We need to emphasize here that the distribution shown is intended to demonstrate the properties of periodicity only. It strongly depends on what eddy voltages in the kth column were taken as initial conditions. On the other hand, if L = N then the eddy voltage distribution becomes aperiodic. Eddy voltages do not repeat themselves after P steps anymore. An example for N = 4 = L is shown in Figure 2b. The same is true if the magnetic flux longitudinal variation G(k) is unharmonic.
Cryogenics
1995 Volume
35, Number
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497
Eddy currents in flat SC cables: A.A. Akhmetov
et al. N-1.K G
=
c
(11)
&k
?l=l,k=l
as a certain function of the crossover voltages VI in the first column of the sample. Then N - 1 minimizing equations
F 3
si
-5
>’ 20
0
El B m-20 :: b
(12)
were used to calculate the column matrix VI and afterwards Equation (9) was used again to find the eddy voltages in the rest of the cable. It was found that both methods provide the same results. At the same time, the latter method is much more preferable in the case of large samples.
Column number, s
8 9 =9
ln=1.2,. .,N-l = o
Infinite cables
0
6
16
24
32
Column number, s Figure2 Crossover voltages in first row of four strand penetrated by a time-dependent magnetic flux which is oidal along the cable. The results are obtained by direct cation of Equations (5) in the cases of (a) L= 11 and (b)
cable sinusappliL=4
Now let the infinite cable be subjected to a time-dependent magnetic field which has a sinusoidal profile along the cable with an integer period L # N. Consider the interval of the cable between the columns k and (k + P). The crossover voltages are periodic with a period P, and the average heat generation per unit length of the cable g is the sum of the heat release in all crossover points G divided by P, i.e.
(13) In this case its Fourier decomposition necessarily includes the sinusoidal term with L = N, which provides an unperiodic contribution in the eddy voltage pattern.
or
g=f
Application of minimum rate (MEDR) principle
energy dissipation
Finite samples Without boundary conditions the set of Equations (5) is not closed. Indeed, it does not include Kirchhoff’s equations which are different for the resistive contacts inside the sample and at the sample edges. On the other hand, from the physical point of view it is evident that if the cable is long enough, boundary conditions cannot be responsible for the crossover voltage pattern. If one considers an infinite sample then another principle has to be applied to find the crossover voltage distribution. As was shown in reference 11 on a few specific examples, for samples of finite size subjected to a uniform time-dependent magnetic field the real crossover current (voltage) distribution provides the minimum total energy G dissipated per unit time among all the possible distributions satisfying Equations (5). In this work the calculations” were repeated for r,,, = r = const. and a few different magnetic flux rate &)n,k distributions in a few samples consisting of different numbers of strands N and columns K. First, calculations were done directly by applying the corresponding Faraday’s and Kirchhoff’s equations. To do this, eddy currents have been expressed through the currents in the strands. This leads to a set of K(N - 1) independent Faraday’s equations which have to be solved numerically to find the real eddy voltage distribution. As an alternative method, Equation (9) was used to express the sum of the heat generation in the crossover contacts of the cable
498
Cryogenics
1995 Volume
35, Number
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Fr e,v,+, .S=rl
(14)
where Jk+s(l) is the row matrix which represents the crossover currents in the corresponding column of the cable. According to Equation (9), for given crossover resistances and rate of magnetic flux increase, the average heat generation can be symbolically expressed as a certain function of the voltage drops in the kth column g=F(Vlc)
(15)
For simplicity’s sake, let us consider the case of constant contact resistances r, k = r. One can use the principle of minimum energy dissipation rate (MEDR) to find the crossover voltage distribution in the kth column and consequently in the entire infinite cable subjected to a timedependent magnetic field which is cleanly sinusoidal along the cable or is composed of many sinusoidal terms except the one with the period L = N. Note that in long samples the average heat generation g is proportional to the total value G and we use (N - 1) additional equations
(16)
Then, using Equations
(9) and ( 14) it follows that
g = S[J~)J~+ P-l c (Jp) (A”)‘f’AsJk + J~‘(A”)“‘B, s=I
Eddy currents in flat SC cables: A.A. Akhmetov + B”‘A”J, + B”‘B s s s )] where superscript
(17)
(t) designates
transposition
and
Application of the MEDR principle in the form of Equations (16)-( 18) provides us with a set of equations in which the eddy currents in the kth column are expressed as a linear combination of the time derivatives of the magnetic fluxes penetrating through every elementary loop located in the kth to the (k + P - 1)th columns. The equations mentioned are too cumbersome to be given in the explicit form but can be easily solved numerically. In the following it is sometimes convenient to separate the factors influencing the eddy current distribution. Let the magnetic flux rate be set as &=
et al.
RIY,
-20 II
1, ” ’ 64 32 96 Column number, s
0
,I
(19)
where R is the rate of current increase in the magnet which provides the background time-dependent magnetic field and qk is the column matrix of corresponding geometrical factors. Then, the eddy currents in the kth column of the cable can be symbolically expressed as
J,i = RY( %+q)~o~qc~-,
(20)
where Y represents the corresponding of the geometrical factors V.
linear combinations
0
50 100 150 Column number, s
Figure 3 Crossover voltages in finite samples of eight strand cable calculated by means of the MEDR principle for L= 9 and (a) Ls = 108, fb) LS = 180
with a period L are present and the eddy voltage pattern does not depend on sample length. The dependence of the average dissipated energy g on sample length L, is shown in Figure 4~. Pseudo-oscillations
Results Uniform distribution infinite cable
1
of magnetic
flux along
2900
Sinusoidal distribution finite sample
of magnetic
1
1
I
a
’
’ 200
I
I 400
I
600
Sample length, L,
field along
The MEDR principle was applied to calculate eddy voltages in finite samples of different length LS = MP + Lo, where M=0,1,2 . . . and 0 < Lo < P, in the cases where longitudinal profiles of the magnetic field were sinusoidal with both L S N and L = N. The results are illustrated for N=8,L=9,L0=P12=36andM=landM=2inF’igures 3a and b, correspondingly. It is seen that oscillations with period L are modulated by oscillations with period P = LN, but unlike the results shown in Figure 2a there are practically no oscillations with a period equal to N. With increasing M, the amplitude of the oscillations with period P becomes smaller, while the amplitude of oscillations with period L approaches a constant value. On the other hand, it follows from calculations that if Lo = 0, i.e. the sample length is equal to the integral-fold of P, only oscillations
I
I
3100
If the magnetic flux penetrating the loops of the cable is homogeneous then the MEDR principle provides an eddy current distribution which is uniform along the cable. It coincides with the eddy current distribution obtained directly from Equation (6) written for the case where the crossover currents are the same at the left and right comers of every loop.
I
0
400
800
12 10
Sample length, L, Figure4 Dependence of average heat generation in the interstrand contacts of eight strand cable on sample length for (a) L=9 and (b) L=8
Cryogenics
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Eddy currents in flat SC cables: A.A. Akhmetov
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with a period equal to L/2 appear superimposed on the larger ones with a period equal to P. Indeed, according to the previous results at N # L, the eddy voltage pattern and consequently g have to be the same at L, = P, 2P, 3P. . What is important, however, is that g has reached a nearly constant value at Ls of the order of a few times P. The situation changes drastically if the period of the magnetic field oscillations coincides with the cable pitch length. For different sample lengths the MEDR principle provides different eddy voltage patterns. In general, there are pseudo-oscillations which increase linearly from the sample centre to its ends. At the ends of long samples, the amplitude of the pseudo-oscillations reaches quite high values. Illustrations for L = N = P = 8 are given in Figures .5a for L, = 96 and in b for L, = 160. The average dissipated energy increases infinitely with lengthening of the sample if L = N, as seen in Figure 4b. Also, small superimposed pseudo-oscillations with a period Ll2 can be seen in the inset. Discounting these, the dependence of g on L, can be considered as an approximately square one. Sinusoidal distribution infinite cable
of magnetic
field along
Equations ( 16)-( 18) were solved numerically for a set of parameters which includes different numbers of cable strands N and the periods L # N of applied sinusoidal timedependent magnetic field. Then Equation (9) was used to find the eddy voltage (current) distribution throughout the cable. The eddy voltages in the first row of the cable are shown in Figure 6 as a function of the column number s for N = 4, L = 11. It is seen from a comparison of Figures 2~2,3a and b and 6 that application of the MEDR principle
I
I
I
I
0
64 32 Column number, s
96
0
60 Column number, s
160
100 3 ti ‘;? >’ 8 8 = P
0
5 B 3 6 -100
Figure 5 Crossover voltages calculated for the case where the period of the magnetic field spatial oscillations coincides with the cable pitch length for (a) Ls = 96 and (b) L, = 60
500
Cryogenics
1995 Volume
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Column number, s Figure6 Crossover voltages in an infinite four strand cable penetrated by a spatially oscillating time-dependent magnetic flux (---): -, exact solution; ., pseudo-uniform solution
to an infinite cable leads to suppression of eddy voltage oscillations with a period equal to both the cable pitch length and P. Instead, only the oscillations with a period equal to that of the external magnetic field are clearly seen. In other words, the crossover current distribution looks as if it is being determined only by the local value of the magnetic flux rate &. For comparison, the Morgan-like solution is shown in Figure 6 by the dotted line. It can be seen that the exact solution yields a significantly larger amplitude of oscillations than the approximate one. The difference between the solutions disappears as L - x and increases as L - N. The summations in Equation (17) have been performed on a cable interval P. Since for L #N the crossover voltages repeat themselves every P steps, exactly the same results were obtained while performing summations on any part of the cable which is the integral-fold of P. (This follows from the results of the previous section, also, as well as the fact that approximately the same eddy voltage pattern would be found if the summations were performed in any part of the cable which is much longer than P.) Formally speaking, also, in the case of L = N one can find the crossover current distribution in the kth column of the cable which minimizes the energy dissipated in the interval, for example, between the kth and the (k + P)th positions. However, the results will be different if obtained from any other interval, as can be seen in Figures 5a and b. Until now we have considered only those periods of magnetic field oscillation which exactly coincide with an integer of columns. Suppose now that L, is an arbitrary period. Using, for the summation in Equation (17), the interval P = uNL,, where u is the integer multiplier needed to make B an integer too (or nearly an integer), we obtain the exact (approximate) value of the array V, which satisfies the conditions of g minimization. TheAaccuracy of the approximation improves as the value of P approaches an integer. After the crossover voltage (current) distribution which minimizes the average energy dissipation per unit length of cable g in a given magnetic field has been found, Equation (14) allows us to calculate the value of g itself. The dependences of g on the period of the magnetic field variation L, are shown in Figure 7u and b for N = 4 and N = 14, respectively. In both cases C,, = C = const. It is seen that the energy loss in the cables increases as the period of magnetic field oscillation approaches the cable twist pitch. At L, = N the results diverge. As seen in Figures 7a and b L, - ~0 the average heat generation g approaches a constant value g,. An
Eddy currents in flat SC cables: A.A. Akhmetov
1
-1
\ / 1 14 28 42 Column number, s
0 Period of magnetic field oscillation, L,
I’
I’/
1
I
rl
\
-J 24980
Figure 7 Dependence of average heat generation in an infinite cable g on the period of the magnetic field oscillations L, for (a) N=4 and (b) N= 14
to that in the case of uniform
N(N“ - 1 )C
mag-
(21)
240r
can be used for its calculation. Arbitrary
periodic
distribution
of magnetic
field
Numerical solutions of Equations ( 16)-( 18) reveal that if the magnetic flux distribution along the cable is unharmanic, the resulting eddy voltage pattern consists of elementary ones which correspond to the terms of a Fourier decomposition of the magnetic flux longitudinal profle. Isolated non-uniformity in magnetic penetrating a finite sample
I 25000
25020
Column number, s
Period of magnetic field oscillation, La
g, =
5
,
1
expression similar netic field7,“’
et al.
flux
Consider now the problem of an unharmonic magnetic flux in more detail. Let the cable be of finite length L, S N with an isolated non-uniformity in magnetic flux ramp rate &J~,, at its centre. An example of a ‘bump’-like non-uniformity is shown in Figure 8a by the dashed line. Fourier decomposition of such non-uniformity leads to the appearance of a set of sinusoidal terms including that with a period L equal to the cable pitch length. Because of this, in the case of an infinite cable containing non-uniformity in every Lsth position, the solutions of Equations (16)-( 18) diverge. However, for single non-uniformity, application of the MEDR principle gives reasonable results. Figures 8a and b show the current distributions in the first and second rows of a four strand cable for two different
Figure 8 Eddy currents in samples of different length containing an isolated ‘bump’-like non-uniformity in the magnetic flux longitudinal profile for (a) L, = 56 and (b) L, = 5 x IO4
cable lengths Ls. It is seen in Figure 8a drawn for L, = 56 that the currents exhibit periodicity with a period N = 4 both in the non-uniformity and in those parts of the cable where the magnetic flux is uniform. If L, = 5 x 10’ (see Figure 8b), the crossover current oscillations become smaller in the uniform parts of the sample, while there is a noticeable increase in the variations in eddy currents in the non-uniformity.
Discussion Crossover current patterns In practice one never deals with infinite cables. It follows from calculations that in the case of harmonic longitudinal oscillations of the applied time-dependent magnetic field with a period L f N, the condition Ls 9 P, from a practical point of view, is equivalent to an infinite cable. The results presented in Figures 3a and b and 4a testify that one can consider the sample as practically infinite if its length reaches = 1OP. On the contrary, if the magnetic field is harmonic with a spatial period L = N, the results diverge when the MEDR principle or direct calculations are applied to different intervals of the cable. This means that the sample is never long enough to be called infinite. At the same time, estimation of the amplitude of the pseudo-oscillations in eddy voltages at the conductor ends can be based upon the sample length, as can be seen in Figures 5a and b. Equation (20) is a linear equation with respect to eddy voltages (currents). Combination of the harmonic terms with different periods in the longitudinal profile of the magnetic field provides superposition of the eddy voltage patterns caused by these terms. If the sample is long enough
Cryogenics
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Eddy currents in flat SC cables: A.A. Akhmetov
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and the magnetic field distribution contains a significant harmonic term with a period L = N, it follows from the previous consideration that this term would be the main one determining the eddy current distribution. To explain the results shown in Figures 8a and b note that the periodicity in eddy voltages takes its origin not from the boundary conditions, which can be neglected” at Ls > lON, but from the presence of non-uniformity in the magnetic flux. Deviation of the eddy current distribution from a uniform one provides different contributions to the average dissipated energy from the uniform parts of the sample and in the non-uniformity. In the latter case the contribution is slightly larger. Because of this, in order to minimize the total energy G dissipated in a sample with short uniform parts, larger oscillations occur exactly in these parts. On the other hand, if the uniform parts are very long, as in the case illustrated in Figure 8b, the condition of G minimization leads to an increase in oscillations in the non-uniformity and to their decrease in the rest of the sample. For even larger LS the oscillations in the uniform parts would be almost completely suppressed. A similar situation has been treated in reference 10 by repeatedly applying Equations (5) to find eddy currents in subsequent columns of the sample starting from Morgan’s eddy current distribution at one of the sample ends. Nearly all the above (as well as Morgan’s basic model) is correct only if the (pseudo-) periodic patterns shown in the corresponding figures are flawlessly established. On the other hand, it is well known that if a magnetic field is applied to a cable at some instant to, for some time after this the screening currents are present mainly at the cable edges. Then, gradually, the currents diffuse inside the cable. Thus (pseudo-) periodic structures are created. This process takes a certain time, which is discussed in the next section. Correlation
time
In fact, (pseudo-) oscillations in the crossover currents with a period equal to the cable pitch length existing throughout the entire sample are due to some kind of current correlation, because both the ‘energy-based’ approach of this paper or direct calculations ‘Oimply that crossover currents in any column of the cable reflect the conditions existing far off. Figures 5 and 8 give quite clear examples. On the other hand, it is necessary to admit that establishing the correlation takes a certain time TVwhich has to be compared with the time of the magnetic field increase To. The latter term is different for various practical applications of superconductivity but rarely exceeds a few hours. It is not quite clear yet what the correlation time really is, though some analogies based on estimations are available. In references 19 and 20 the effect was observed in the performance of dipole magnetsI made of Rutherfordtype cables. It was found that the magnetic field and its main harmonics’6,‘9 oscillate along the magnet axis with a wavelength equal to the cable pitch length. At a low transport current, the periodic magnetic field patterns can persist without any significant decay for more than 12 h19. The explanation of the effect given in reference 18 is based on the assumption that during magnet energization slightly different currents are trapped in the strands, even at zero transport current in the cable. Current loops created by this consist of currents flowing along the cable through one set of strands and returning through another set of strands. The circuit is connected through crossover contact
502
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1995 Volume
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resistances and/or current leads. Since some effective inductance per unit length can be attributed to the strands”*‘*, current loops circulating in the cable do not disappear immediately after the magnet discharge. The decay time of the current loops is proportional to the second power of their length and in the case of the longest current loop it can be as large as a few days. Assuming that the results’9.20 properly reflect the characteristic time of the end-to-end correlation of the crossover currents throughout the cable and the explanation’* is pertinent to the effect under consideration, some rough estimations are possible. Let us suppose’* that the effective inductance of the strand is of the order of 10m8H m-‘, and that the transverse contact resistance between the strand and the rest of the cable R, is =lO-’ R m, which is consistent with the data of reference 21. It then follows18 that rc = 10 s if the sample length is -10 m, and T== 103s if the sample length is = 100 m. In turn, this means that for a 10 m long sample the eddy currents are correlated after 530 s, while it takes = 1 h to achieve correlation in a 100 m long sample. At the same time, if the cable pitch length, the period of the magnetic field variation and their difference are all = 10-l m, then already a few metre long sample (with a correlation time ~1 s) can be considered as practically infinite. Hence, given the same R, and effective inductances of the strands, the main results presented in Figures 3-6 can be observed in the experiments on such samples at times of magnetic field increase of the order of 10s. On the other hand, the inductive terms which have to be included in Equations (l)-(9) to describe unstationary processes22 if rr 5 r,, are a hindrance to establishing correlated patterns throughout cables with the same length as those in real windings. ‘Infinite’ energy loss per unit length of cable due to the harmonic term in the magnetic flux with a period L = N is a kind of mathematical abstraction. Indeed, the time needed to achieve correlation is proportional to the square of the sample length LS, and the ‘correlated’ ramp rate (i.e. the ramp rate at which rr > TV and the eddy current in the whole sample can be considered as being correlated) R, is proportional to the minus second power of the sample length. In turn, the average energy loss g, in the cable due to the eddy currents is proportional to the square of the ramp rate. (Subscript c is used here to emphasize the correlation. In fact, all previous results are related to cases of correlated eddy currents.) It follows that
gc
zgn(Ls),G L4s
go(Ls)
(22)
where go is a geometrical factor composed of the local column matrices !Pk [see Equations ( 19) and (20)]. It is seen in Figure 4b that go is nearly proportional to the square of LS. As a result, extension of the sample leads to such small ramp rates that g,, G, - 0 (if eddy currents are to correlate through the entire sample). Otherwise, rr < T=(R > R,) and there is no significant correlation between the eddy currents in different parts of the conductor and no significant increase in the eddy currents at the conductor ends. As a result, at finite ramp rates instead of divergence there is the maximum in energy loss when L = N. However, if LS % N is fixed and magnetic flux variation along the cable includes sufficient unharmonic terms then, when the ramp rate decreases, the value g/R2 should increase, while it is constant if no long distance correlation
Eddy currents in flat SC cables: A.A. Akhmetov is taken into account. In the authors’ opinion a corresponding experiment can be performed to prove indirectly the results of the present calculations. Also, short range (pseudo-) periodicity in eddy currents in non-uniformities similar to that shown in Figure 8b can, probably, take place at very short times of magnet energization. Quantitatively, we will consider time-dependent eddy currents in flat superconducting cables elsewhere.
10
11
12
Conclusions 13
Steady eddy currents in long flat two-layer superconducting cables penetrated by a time-dependent magnetic field, which is sinusoidal along the cable, exhibit a periodicity determined only by the magnetic field. The eddy current distribution is pseudo-periodic if the period of magnetic field oscillations exactly coincides with the cable pitch length and is periodic in the opposite case. In comparatively short samples, periodic eddy current distribution is modulated with a larger period common to the magnetic field oscillations and the cable pitch length. If the ramp rate of the magnetic flux is fixed, the energy loss per unit length of the cable increases as the period of the magnetic field longitudinal oscillations approaches the cable pitch length. Oscillations or pseudo-oscillations with a period equal to the cable pitch length appear in long cables when the longitudinal variations of the applied magnetic field are unharmonic. If the longitudinal distribution of the time-dependent magnetic field is periodic and contains a significant sinusoidal term with a period equal to the cable pitch length, then the average energy loss in the long cable due to interstrand eddy currents is not proportional to the square of the ramp rate.
14
15
16
17 18
19
20
21
22
Acknowledgements This work was supported in part by Kyushu Electric Power Co. Inc. One of the authors (A.A.) found it very useful to discuss many problems touched upon in this paper with Dr S. Takacs.
References Tacacs, S. Coupling losses in cables in spatially changing ax. fields Cr?/ogenics ( 1982) 22 661-665 Kwasnitza, K. and Horvath, I. Experimental evidence for an interaction effect in the coupling losses of the cabled superconductors Cryogenics (1983) 23 9-14 Fevrier, A. Losses in a twisted multifilamentary superconducting composite submitted to any space and time variations of the electromagnetic surrounding Cryogenics ( 1983) 23 185-200 Okada, T., Takahata, K. and Nishijima, S. Coupling losses in superconducting cables Adv Cryog Eng (1986) 32 763-769 Sumpson, M.D. and Collings, E.W. Influence of cable and twist pitch interactions on eddy currents in multifilamentary strands calculated using an anisotropic continuum model Proc ICMC Vol 40A (Eds Reed, R.P. Fickett, F.R., Summers, L.T. and Stieg, M.) Plenum Press, New York and London (1993) 5799588 Tactics, S. Coupling losses in inhomogeneous cores of superconducting cables Cryogenics (1992) 32 258-264 Morgan, G.H. Eddy currents in flat metal-tilled superconducting braids J Appl Phys (1973) 44 3319-3322 Sytnikov, V.E., Svalov, G.G., Akopov, S.G. and Peshkov, LB. Coupling losses in superconducting transposed conductors located in changing magnetic fields Cryogenics (1989) 29 926-930 Verweij, A.P. and ten Kate, H.H.J. Coupling currents in Rutherford
et al.
cables under varying conditions IEEE Tram Appl Supercond (1993) 3 146-149 Akhmetov, A.A., Devred, A. and Ogitsu, T. Periodicity of crossover currents in a Rutherford-type cable subjected to a time-dependent magnetic field J Appl Phys (1994) 75 3 176-3 I83 Akhmetov, A.A., Kuroda, K. and Takeo, M. Influence of sample geometry on amplitude of eddy current oscillation in Rutherford-type cables, paper presented at Applied Superconductivity Conf, Boston, MA, USA (Ott 1994) Sumiyoshi, F., Kanai, Y., Kawashima, T., Iwakuma, M. et al. Sweep-rate dependences of losses in aluminum-stabilized superconducting conductors for the Large Helical Device Applied Superconductiviryfor Nuclear Fusion Research Eds Miyahara, A. and Yamamoto, J., Elsevier, Amsterdam, The Netherlands (1993) 37 l-376 Kwasnitza, K. and Bruzzone, B. Measurement of end effects in the coupling losses of multifilament superconductors Proc ICEC IO Butterworths, Guildford, UK (1984) 606-609 Sumption, M.D., Callings, E.W., Scanlan R.M. and van Oort, J.M. VSM studies of interstrand eddy current coupling loss in a series of coated-strand cables, paper presented at Applied Superconductivity Conf, Boston, MA, USA (Ott 1994) Motojima, O., Akaishi, K., Fujii, K., Fujiwaka, S. et al. Physics and engineering design of the Large Helical Device Fusion Eng Design (1993) 20 3-14 Daum, C. and ter Avest, D. Three-dimensional computation of magnetic fields and Lorentz forces of an LHC dipole magnet using the method of image currents Proc MT- 11 (Eds Sekiguchi, T. and Shimamote, S.) Elsevier Applied Science, New York and London (1989) 302-307 Turck, B. Influence of the transverse conductance on current sharing in a two-layer superconducting cable Cryogenics (1974) 14 448-454 Akhmetov, A.A., Devred, A., Mints, R.G. and Schermer, R.I. Current loop decay in Rutherford-type cables Supercollider 5 Ed Hale, P. Plenum Press, New York, USA (1995) 443-446 Schmuser, P. Superconducting magnets for particle accelerators Proc AIP Conf Vol 249 (Eds Month, M. and Dienes, M.) American Institute of Physics, New York, USA ( 1992) 1099-I 158 Ghosh, A.K., Robins, K.E. and Sampson, W.B. Axial variations in the magnetic field of superconducting dipoles Supercollider 4 Ed Nonte, J. Plenum Press, New York, USA (1992) 765-772 Kovachev, V.T., Neal, MJ., Capone, D.W., Carr, WJ. and Swenson, C. Interstrand resistance of SSC magnets Cryogenics ( 1994) 34 813-820 Verweij, A.P. and ten Kate, H.HJ. Super coupling currents in Rutherford type of cables due to longitudinal non-homogeneities of dB/dt, paper presented at Applied Superconductivity Conf, Boston, MA, USA (Ott 1994)
Appendix For an N strand cable, N even, the matrix A of the rank (N - 1) has the symmetric form -110
0
0
o...o
0
0
0
0
0
-1
l-l
1 0
0
0
0
0
0
0
0
0
l-l
1 0
0 ... 0
0
0
0
0
0
0
l-l : ;
l-l ; ;
1 ... 0 ; ‘.. ;
0 f
0 f
0 f
0 f
0 i
0
0
0
0
0
0 .
l-l
0
0
0
0
0
0 .
0
0
0
0
0
0
0
0
0
0
0
A=;
l-l
1 0
0
1-l
1 0
0 ... 0
0
l-l
l-1
0 ... 0
0
0
0
(Al)
l-l
The corresponding matrix for N odd differs slightly. This difference does not affect the following consideration. Consider the characteristic equation for the matrix A pdh)=O,p,=I.-M(
Cryogenics
(AZ)
1995 Volume
35, Number
8
503
Eddy currents in flat SC cables: A.A. Akhmetov
et al.
where A represents the eigen-values of the matrix A. Direct calculations performed for N = 3, 4, 5, 6 show that in these specific cases Equation (A2) can be written in the form AN-’ + AN-2+ . . + A* + A + 1 = 0
(A3)
To prove this statement for arbitrary N, we express the determinant pNtz through determinants of lower ranks. Applying double decomposition, it follows’” that PN+z(A)
=
(A2
+
1 )p,dA)
(A4)
-PN-AA)
Now, assuming that Equation (A3) holds in the case of matrices of both ranks N and (N - 2), one comes to the conclusion that the characteristic equation for the matrix of rank (N + 2) has the form of Equation (A3) also. All the roots of Equation (A3) are different, and application of Cayley-Hamilton’s theorem allow us to find the relation between different powers of matrix A. It follows that AN = E
9
AN-’
+ AN-2 +
.
.
.
+A2+A+E=0
If L is even and L # mN, then Vk+p= V, + 2(A”’
+AN-3 + . . . +A)&
+ 2(ANm2+AN4+
. . . +E)fk,
(A9)
where L-2 0,
=
c
L-1 +k+s>
a2
~0.2.4
=
c
&+s
(A101
SF1.3.5.
For a sinusoidal fl, and a2 are Lastly, when voltages in the
magnetic field with a spatial period L, both equal to zero, so that Vk+p= V,. P = L = mN, where m = 1, 2, 3 . . ., the eddy (k + P)th column can be expressed as mN-Nch-
V k+P= V, + gAN-O h=l
I
c
&+,
(All)
r=h-I,N+h-1,2N+h-I
We designate S(h) to the result for the inner summation in Equation (Al 1). For any given 1 ‘=rh 5 N and sinusoidal magnetic field
(As)
where 0 designates the zero matrix. Then, if the magnetic flux rate & does not depend on the position along the cable, Equation (9) for s = N accepts the form V k+N=ANVk+(AN-1+AN-2+...+A2+A+E)&=Vk (Ah) Now let Equation (10) represent the dependence of magnetic flux on position along the cable. For L odd and L f mN (m is an arbitrary integer), eddy voltages in the (k + P)th column are expressed through the eddy voltages in the kth column as
S(h) = C[sin(2rhlmN) + sin(2tiN + h)lmN) + . . + sin( 2_rr(mN - N + h)lmN)]
(A12)
where C is the column matrix which represents coefficients C, of Equation ( 10). Algebraic transformations allow us to rewrite the last equation in the form S(h) = C[cos( a)S, + sin( a)S,]
(A13)
where cY=2?--
h-N (A14)
mN
I’,,, = (AN)‘Tk + (AN-’ + AN-2 + . . +A2 +A + E)0 (A7) where
S, = sin[2til/m)]
=
+ . . . + sin(2n)
(A15)
and
L-l n
+ sin[2rr(2/m)]
c
(A81
&+s
S2 = cos[27r( l/m)] + cos[27r(2/m)l+.
c=a
. . + cos(2~) (A16)
It is clear from comparison that Vk+p = V,.
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1995 Volume
(A5) and (A7) Both S, and S2 are equal to zero for any m except m = 1.
35, Number
8