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Inter-Strand Coupling Losses in Superconducting Power Cables Without Coaxial Magnetic Shielding
S. Stenger, P. Manuel, F. Bouillault
To cite this version:
S. Stenger, P. Manuel, F. Bouillault. Inter-Strand Coupling Losses in Superconducting Power Cables Without Coaxial Magnetic Shielding. Journal de Physique III, EDP Sciences, 1996, 6 (12), pp.1775- 1783. �10.1051/jp3:1996213�. �jpa-00249557�
Inter-Strand Coupling Losses in Superconducting Power Cables Without Coaxial Magnetic Shielding
S. Stenger (~), P. Manuel (~) and F. Bouillault (~>*)
l~) Laboratoire de G6nie #lectrique de Paris, ESE, CNRS, UniversitAs Paris VI et XI, Plateau du Moulon, 91192 Gif sur Yvette Cedex, France
(~) #lectricit6 de France, DER, I
avenue du G6n6ral de Gaulle, 92141 Clamart Cedex, France
(Received 4 June 1996, revised 19 September1996, accepted 20 September 1996)
PACS.84.70.+p High-current and high-voltage technology: power systems; power
transmission lines and cables (including superconducting cables) PACS.85.25.Kx Superconducting wires, fibers and tapes
PACS.85.25.Am Superconducting device characterization, design and modeling
Abstract. This paper deals with computations of inter-strand coupling losses in supercon-
ducting power cables with Room Temperature Dielectric (RTD), in which no coaxial shielding
of magnetic field is provided. In three-phase circuits, magnetic fields produced by neighbouring phases induce in the third one significant currents through contact resistances between strands, owing to their helical winding around the cable axis with a pitch inversion between each layer.
Analytic expressions are first derived for a simple cable model, with only two strand layers (single "bilayer"), with or without taking internal screening effects into account. The validity limit of these computations is determined. Then a generalization to a real n-layer is proposed
and the development shows that a homogenization procedure can be performed on the radial coordinate, in some specific cases.
L'a~te~r principal de cet article est Stdphane Stenger, dispar~ accidentellement en ddcembre 19§$. Get article est ~n tdmoignage de ses compdtences scientifiq~es. Malgrd son bref sdjo~r
a~ LGEP, ses colldg~es ont p~ apprdcier ses grandes q~alitds scientifiq~es et h~maines.
Introduction
Studies of superconducting power transmission lines recently received a strong impetus from the new perspectives open by the discovery of high temperature superconductors. Owing
to the reduced refrigeration requirements to be satisfied at liquid nitrogen temperatures (as compared with previous conceptions cooled at liquid helium temperatures), new designs have been suggested. Among these, the so-called "Room Temperature Dielectric" (RTD) option [1]
is now considered as a major possible alternative to the more classical design which requires a Cryogenic Dielectric (CD).
(*) Author for correspondence le-mail: bouillault@lgep,supelec.fr)
Q Les #ditions de Physique 1996
1776 JOURNAL DE PHYSIQUE III N°12
ill Superconductor (77 Kl
@ Thermalinsulation
fl Dielectric (300Kl
Fig. 1. Cross section of a three phases superconducting power transmission line with room tem-
perature dielectric.
In the RTD conception, a thermal insulation is placed around each conductor of the (three phases) transmission line (Fig. 1). The interface between the thermal insulation and the dielectric is maintained at the conductor electric potential. An important consequence of such
a configuration is that it is no longer possible to shield the magnetic field produced by the conductors with superconducting coaxial screens, whereas this is the usual disposition in the CD option. Thus, in the RTD design, each conductor is submitted to the magnetic field resulting from the neighbouring phases, which is mainly a uniform AC transverse field with
elliptical polarization.
The main advantages of the RTD design is (I) to allow the use of a classical dielectric and
(it) to i-educe the requirement of superconducting materials by a factor two [2]. However, the lack of magnetic shielding results in additional losses in neighbouring parts of the system and in the conductors themselves. The purpose of this paper is to analyze the inter-strand coupling
losses produced inside the conductors in such a configuration. This loss term is produced by
induced currents which cross the contact resistances between the strands.
Well-kno~vn expressions are available for the estimation of coupling losses occurring in a
single strand constituted by superconducting filaments imbedded in a normal matrix, twisted
with a constant pitch p, and submitted to a transverse AC field [3-5]. This very common
configuration is not reproduced in conductors used for superconducting power cables. Indeed, mechanical and electromagnetic considerations favour in this case a double helix design in which neighbouring strand layers are wrapped with opposite twist pitches (Fig. 2).
Hysteretic losses and eddy currents inside strands and tapes wound in this configuration
have already been considered in previous studies [6]. However, to our knowledge, up to now, there is no such studies on coupling losses in power cables, although many articles analyse this
problem in different geometrical configurations, related to other applications [7-12]
An important point ~vhich further defines the framework of our study is the cross-section
shape assumed for the strands. The optimization of the critical current density in high tem-
perature superconductors requires a texture which, up to now, is obtained only in thin films
or flat tapes. We thus assume a rectangular cross-section with a small ratio between thickness
Fig. 2. Schematic drawing of the strands disposition in a cable conductor lone phase). For clarity's sake, only t~vo neighbouring layers and one strand (tape) per layer are drawn in full lines.
e and width I (Fig. 2). Multifilamentary tapes seem to be preferred in view of their better critical current behaviour under stress conditions, but this has no particular consequences for the inter-strand coupling losses.
To obtain the inter-strand coupling losses in superconducting power cables without coaxial magnetic shielding (RTD), we used a classical approach in which the e-m-f- induced in a
superconducting loop closing on contact resistances between strands is obtained (Paragr. 2).
A simple behaviour is obtained if screening effects by induced current can be neglected and if the cable is made of a single bilayer (Paragr. 3). For the low values of the resistance contact,
a more realistic behaviour is obtained if the reaction field is taken into account (Paragr. 3).
Then, we consider the extension to a n-bilayer cable (Paragr. 4). When the thickness of the tape is small enough, a homogenization procedure can be performed. In this case, an analytical
solution of the current distribution seems to be achieved.
1. General Equations
Let us define rn as the radius of the n~~ interface between two layers, we thus consider a closed path r (Fig. 3) which follows the strands at radii rn e/2 and rn + e/2 (where e is the layer thickness) over a half pitch p/2 between angular positions b and b + ir and is closed by t1&>o
elementary segments of length e, parallel to the radius, crossing the interface between two
adjacent layers.
The magnetic field created by two cables of a three phases circuit can be decomposed, near
the third cable, by two uniform fields, orthogonal in time and in space. So, we analyse no1&>, the magnetic behaviour of a cable, surrounded by an external field whose form is:
Bit) = Bm sin wtuy
where u~ is a unit vector along axis Oy. If the reaction field is not considered, the magnetic flux through r is given by:
#(rn, b,t)
= sin tot /~~~ ~~"~Bm cos ado = #o(rn) sin b sin tot Ii
~ 2ir
with:
40(Tnl " ~~f~ l~l
1778 JOURNAL DE PHYSIQUE III N°12
~x
p/2
< >
.' /
'_ .~ ~
/ IO f.
f~ ~ @
~ Z
/'II /. /.
/~
y
Fig. 3. The closed path r.
where p is the helix pitch.
Expression Ii) shows that only the radial component Bit) sin b of Bit) occurs in the expression of flux #.
Assuming that the superconducting strands are not saturated (the influence ofthis assumption will be given in a future paper), the electric field component along the strand local axis can be neglected. Then, we can deduce, from the Lenz law, the potential difference bV, appearing
at contact resistances 1 and 2 between the strands belonging to layers of radii rn e/2 and
rn + e/2 (Fig. 3):
2bV(rn,b,t)
=
-~~
= -w#o(rn) sinbcoswt (3)
dt
The radial current density Jr(rn,b,t) is then given by:
PsJr(Tn, b>t)
# 8+bV(Tn,b,t) (4)
where p~ IQ m~) is the equivalent contact resistance between the strands. The factor e+ has to be taken as +1, depending on the sign of the helix pitch (+p) of the internal layer at radius
r» e/2.
If the reaction field is not negligible, the current density can be obtained by a similar ex-
pression. In equation (3), flux # has just to be replaced by the total flux: this problem will be considered at the end of Paragraph 2.
2. Behaviour of a Single Bilayer Cable
To understand the influence of the main parameters on the value of losses, we first assume that the cable is first made of single bilayer In
= 1): neglecting the reaction field, a simple calculation gives the losses power per unit volume, averaged in space and in time:
with ri " R e where 2e is the thickness of the bilayer and R the radius of the cable.
This expression can be compared with the power Po obtained for a single strand composite in which all the filaments are twisted with the same pitch (with no inversion between adjacent layers and with a transverse resistivity equivalent to p~le) [3]. We then obtain:
$ = j (6)
For a thickness e of 0.3 mm and a cable radius of15 mm, typical of conductors assembled from OPIT (Oxide Powder In Tube) like tapes, this ratio is equal to 5000. Thus a very
large loss increase could result from the particular configuration of assembled conductors in
superconducting power cables.
However the currents Ii(b,t) and Ie(b,t) flowing through each strand of internal and external
layers (respectively) will generate a reaction field which can modify the flux through the surface S, delimited by the r boundary (Fig. 3). We first evaluate qualitatively the value po of the
contact resistance, for which, if p > po, the reaction field is negligible. At this point, it can be observed that only the axial component Jz of the current density will be at the origin of the radial component of the reaction field. We neglect currents transferred between tapes of the
same layer since e < Ii is the width of the strand), as well as corrections of order e/ri In this approximation, the following relations may be deduced from symmetry considerations:
11(~,t)
= ie(~,t)
Jz(b,t) " ~'~~~'~~ ~~~ ~ (7)
et JR e(b,t)
#
~~'~~~'~~ ~~~ ~
' ' et
where fl is the wrapping angle.
Jg~ and Jg~ are the b components of the current density in the external and extremal layers, respectively, and the sign of the second member of the last equation depends on the index or
e in its first member.
The current density Jz is obtained by writing the current conservation along an elementary
path of length ridb/ sin fl following one tape axis:
~
~~~ ~i~ ~ ~°~~~2~e ~' ~~~
We show below that the dependence of Jr is not modified by the reaction field. The Jz expression is thus the following
Jz(ri, @,t)
= Jzo cos cos wt (9)
with:
j ~L°P#o(ri)
~° 2p~ire
The field due to a thin current bilayer with a lineic density Jz (ri,9,t)2e is a dipolar field with the form:
Br(r', b,t)
= ~toJzo euy cos tot if r' < ri
Br(r',b,t) = ~toJzo e[sinbur cos bug] ~
,
~
cos tot if r' > ri (10)
r
~
1780 JOURNAL DE PHYSIQUE III N°12
Table I. Losses in si~percondi~ctor cable (Bm
= 0.01 T, f = 50 Hz, p = 0.16 m, ri "
0.015 m) (IBAD: Ion Beam Assisted Deposition, OPIT: Oxide Powder In Ti~be).
contact resistance ps losses (MW/m3)
OPIT tape (e = 300 ~tm) 10 po 0.0624
IBAD like YBaCUO tape (e = 50 ~tm) 10 po 0.374
OPIT tape (e = 300 ~tm) po " 7.68 (nQ m~) 0.312
IBAD like YBaCUO tape (e = 50 ~tm) po " 7.68 (nQ m~) 1.87
OPIT tape (e = 300 ~tm) po/10 0.0624
IBAD like YBaCUO tape (e = 50 ~tm) po/10 0.374
The reaction field is negligible if its modulus is small, compared to Bm. This condition is verified if the contact resistance p~ is much greater than po, given by:
For smaller values of contact resistance, the reaction field has to be taken into account. It is
easy to show that a current density Jz16, t) with a cosb angular dependence (as in Eq. (9)) is
still a solution of this new problem. Indeed, the magnetic reaction flux #r(ri, b,t) through r of a reaction field distribution such as (10) is given by:
#r(ri, b,t)
= ~toJzo(t)esinb ~~~~ (12)
7r
Besides, with such a b dependence of Jz16, t), the radial component of the interface current
density is again given by Jr16, t)
= (2ire/p) sin bJzoIt). Thus, the equation to be solved is now:
2p~J~i~, t) = -jj~(ri,~,t) + i~iri,~,t)j. i13)
Since the second member of this equation has a sinb angular dependence, the test functions
Jr16,t) and Jr16, t) are self consistent solutions of the problem. The complex amplitude Jr of the radial current can be obtained from equations (8),(12), and (13):
JrPs Ii + iii1 -
°~'i~~~~
Sin ~. i14)
With this expression of the current density, the losses become:
~~ i~eT ~ ~
~~~~~~'~'~~~~~~ ~j~~~~~
+ (~/Ps)~ ~~~~
From expression (15), we can conclude that the maximum value of losses are reached for a contact resistance ps equal to po. For this specific contact resistance po, the typical values of losses for two kinds of conductors are given in Table I.
The external field is indeed totally shielded when the inductive part of the impedance which appears in equation (14) is much higher than the resistive part. This is true if the contact
equivalent resistance is of low value when compared to po (Eq. (11)). In this case, the losses calculations per unit leads to:
Pr~ = ~~~~ (16)
p e~t~
From preliminary investigations, it appears that the amount of losses for p~
= po could con- stitute a sizable penalty for the efficiency of superconducting cables. Thus it appears that the
contact resistance between layers needs to be controlled in order to obtain either a regime with
low induced currents (p~ » po) or a quasi-diamagnetic regime (p~ < po). However a more precise evaluation of losses needs to take into account the magnetic coupling between the n
bilayers of a real cable. Afterwards, we give general indications allowing to make computations in this case.
3. Behaviour of a n-Bilayer Cable
The calculation of Paragraph 3 shows that the reaction field due to each thin, cylindrical bilayer
with a radius r is uniform for r' < r and bipolar for r' > r. The conclusion is that the total field radial component, defined by Art in complex notation, can be written, if the thickness of
a tape is small compared to the radius R of the cable, under the form:
An(r,b) = Bm sin b + /~ ~to
~~°~~'~
(~'
~
sin bdr' + /~~to ~~°~~'~ sin bdr', (17)
R~ 2 r
~, 2
where Ro is the internal radius of the cable winding.
It is assumed in equation (17) that the current density Jzo varies slowly along the radius.
Besides, the axial current can be obtained from the induction radial component. Indeed from equations (2) (3), (4), (8), we have:
$ ~~°irep~~~~~~~'~~' ~~~~
The integral equations giving the current distribution Jzo(r) or 7z(r) can be found from equations (11) and (12). For 7zo(r), it can bi written
as:
with
bo=fi.
p~low
The differential equation allowing us to calculate Jzo is obtained by deriving expression (19),
we find:
~dj'° ~ ~j1° ~ ~~°
(j jj
" ° ~~°~
The solution for Jz may be expressed in terms of Bessel functions, the coefficient of which
are obtained from the application of suitable boundary conditions. It is assumed in these
developments that all the quantities vary continuously along the radius. This approximation
1782 JOURNAL DE PHYSIQUE III N°12
is verified only in the case of a skin depth bo being greater than the strand thickness. The
contact resistance value should be much greater than pi-
pi =
~'ir~(~°~ (21)
For smaller values, the problem has to be expressed in a discretized form. Let us note that the discretization of integral (19) leads to a numerical solution by a simple matrix inversion. A side
case occurs when the skin depth bo is very small compared to the strand thickness e. In the
latter case, the whole field is shielded by the first bilayer, the study of which is similar to the
study of a 2e thick bilayer; yet this time, the reaction field has to be taken into consideration.
4. Conclusion
We have presented a computation of inter-strand coupling losses in superconducting power cables without coaxial magnetic shielding (RTD). A classical approach is used in which the e.m.f. induced in a superconducting loop closing on contact resistances between strands is
obtained from the time derivative of the magnetic flux through that loop (Lenz law). A simple
behaviour is obtained if screening effects by induced current can be neglected. However we find that such an approximation is only valid for high contact resistances between strands. Then,
we consider some particular cases which allow simplifying approximations and lead to an easy derivation of analytical expressions. An interesting limit is obtained if the layer thickness
goes to zero. This case allows an homogenization procedure to be performed on the radial
coordinate, and an analytical solution seems to be obtained.
The general conclusion of this work is that the contact resistances between superconducting
tapes need to be controlled in order to keep the coupling losses in RTD superconducting cables at an acceptable level.
References
iii Von Dollen D., Metra P. and Rahman M., Design concept of a room temperature dielectric HTS cable, Proc. American Power Conference (Chicago, 1993) p. 1206.
[2] Engelhardt J-S-, Dollen D-V- and Samm R., Applications considerations for HTSC power transmission cables, sth annual Conf. on Superc. and Appl., Buffalo (N.Y., 24-26 sept.
1991).
[3] Morgan G-M-, Theoretical behavior of a twisted multicore superconducting wire in a time
varying uniform magnetic field, J. Appt. Phys. 41 (1970) 3673.
[4] Carr W-S-. AC loss in a twisted filamentary superconducting wire, J. Appi. Phys. 45
(1974) 929.
[5] Ries G., AC losses in multifilamentary superconductors at technical frequencies, IEEE Trans. Magn. MAG13 (1977) 524.
[6] Forsyth E-B-, Energy loss mechanism of superconductors used in alternating current power transmission system, Science 242 (1988) 391.
[7] Mulder G-B-J- and Niessen E-M-J-, Coupling losses of multifilamentary superconductors having sevearl concentric regions and mixed matrix, IEEE Trans. Appt. Si~perc. 3 (1993)
142.
[8] Verweij A-P- and Ten Kate H-H-J-, Coupling currents in Rutherford cables under time
varying conditions, IEEE Trans. Appt. Si~perc. 3 (1993) 146.
[9] Takacs S., Coupling losses of superconducting cables, Cryogenics 24 (1984) 237.
[10j Takacs S., Coupling losses in inhomogeneous cores of superconducting cables, Cryogenics
32 (1992) 258.
[1Ii Takacs S., Yanagi N. and Yamamoyo J., Size effect in AC losses of superconducting cables,
IEEE Trans. Appt. Si~perc. 5 (1995) 2.
[12] Akhmetov A-A-, Kuroda K. and Takeo M., Influence of sample geometry on amplitude
of eddy current oscillation in Rutherford-type cable, IEEE Trans. Appt. S~tperc. 5 (1995)
725.
