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ComparisonofACLossesinMultilayerSuperconductingPowerTransmissionCablesARTICLEJUNE2015

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Vol 65, No. 7;Jul 2015

207 Jokull Journal

Comparison of AC Losses in Multilayer Superconducting

Power Transmission Cables

M. Erdogan1, F. Inanir 2, *

1 Department of Physics, Faculty of Arts and Sciences, Namk Kemal University, 59030, Tekirda / Turkey 2 Department of Physics, Faculty of Arts and Sciences, Yildiz Technical University, 34220, stanbul/Turkey.

Abstract

Alternating-current losses in long straight cables with tangential superconducting strips

with the number of layers 1, 2, 3 and 5 are calculated through finite-element method

simulations for various amplitudes of the applied current currents. Alternating current

losses in 1, 2 and 3-layer cables are found to be close to each other for amplitudes greater

than half the critical current in a strip, whereas a discrepancy is observed at small

amplitudes where the loss generally increases with the number of layers. The

corresponding loss in the 5-layer cable is always smaller than in the other cables, where

the discrepancy is 60% and 13% with respect to a 1-layer cable at small and large

amplitudes, respectively. The loss distribution in the radial direction of the 5-layer cable

is such that the inner layers experience higher losses, while the innermost layer is least

affected due to screening of the normal component of the magnetic vector field. Current

flows primarily along the edges of the superconductors, whereas almost constant current

flows in the interiors of the superconductors on any layer.

1. Introduction

Superconducting cables are currently drawing attention due to their capability to transmit larger

currents with low rate of energy loss than conventional cables. It is important to reduce

alternating-current (AC) losses in these cable to be able to manufacture them commercially.

Numerous works have been carried out to understand the physics of AC loss mechanisms in

superconducting cables [1-9]. Elucidation of these mechanisms related primarily to the geometry

of the windings, is not straightforward, as the superconductors are wound around a core in

practice, which introduces extra complexity into the problem. However, this is generally omitted

in calculations in which infinitely-long straight cables are assumed.

The AC losses in superconductors are qualitatively well understood. Hysteresis, coupling and

matrix vortex eddy current losses are well-known and several methods have been proposed to reduce these losses [10]. To reduce hysteresis losses, superconductors are generally produced as

small filaments or thin layers [11]. The coupling losses can be reduced by twisting the

superconducting filaments, by increasing matrix resistance or through supporting the filaments

by high-resistance barriers [11-13]. These barriers provide small matrix resistance and, thus,

better conductor stability is achieved [12, 14]. In the cases requiring transmission of large

currents, cables comprising superconducting wires or the tapes must be used. Similar to those in

individual wires, losses in such cables must be reduced, as well. The conductors in cables are

wound on a conducting core by twisting, or the same methods used in conductors are implemented

on cables to reduce hysteresis and coupling losses.

AC losses in single- and multi-layer superconducting cables may be influenced by the

arrangement and geometry of the superconducting strips in different ways. For instance,

calculations show that the lateral spacing between adjacent groups of strips has significant impacts

on the AC losses in single-layer cables, while the critical current density (JC) distribution is more

influential in multilayer cables [15]. Moreover, smaller AC losses in 1, 2 and 4-layer cables

Vol 65, No. 7;Jul 2015

208 Jokull Journal

constructed by 2 mm-wide conductors than compared to standard 4 mm-wide conductors are

reported in FEM analyses [16].

In this work, influences of the spacing between adjacent tapes and the lateral distribution of the

critical current density flowing along the strips on AC losses in 1, 2, 3 and 5-layer cables of 4 mm

width are investigated via the Finite-Element Method (FEM) [15]. How the applied AC current

amplitude affects the losses in each configuration is investigated in detail.

2. Cable Properties and Computational Methods

Electromagnetic analyses of a group of multi-layer power transmission cables with different

number of layers are carried out and their AC losses are compared. The cables depicted in Fig.

1(a)-(d) lay on the xy plane and are infinitely long along z axis. The current flows along z direction.

Each superconducting strip is wSC=4.0 mm wide and hSC=2.0 m high, whereas the inter-layer spacing is d=1 mm in the multilayer cables, as depicted in Fig. 1(e). All cables, in which each

layer comprises N=15 strips around the core, are wound up on a cylindrical copper core with

DC=20 mm diameter. Twisting of the superconductors around the core along the z direction is

ignored. The cross-sections of 1, 2, 3 and 5-layer cables on the xy plane are considered in

computations. Other physical and scaling parameters adopted in computations are listed in Table

1.

Fig. 1: (Color online) Geometrical models of the 1 (a), 2 (b), 3 (c) and 5 (d) layer cables, as well

as the computational domain (e) considered in FEM analyses. Superconducting strip dimensions

are not drawn to scale for clarity.

Computations are carried out through stationary FEM analyses as implemented in the AC/DC

module of the commercial COMSOL MultiPhysics Package. The computational model assumes

two distinct sub domains, vacuum and the superconductors, as shown in Fig. 1(e). Magnetic

response of the core material is ignored in computations. The computational domain in Fig. 1(e)

is pie-shaped due to the fact that each cable possesses discrete rotational symmetry by =24 (2/15). The radius of the computational domain is RCD=25 cm and the vacuum layer is not drawn to scal in Fig. 1(e) for compactness.

Table 1. Physical and scaling parameters used in FEM analyses of the cables in Fig. 1.

Vol 65, No. 7;Jul 2015

209 Jokull Journal

Quantity (Unit) Explanation Value

IC (A) The critical current in superconductor 80

Imax (A) The maximum applied current 75

An The scaling parameter of the vector potential 8.310-8 f (s-1) The frequency of the applied current 50

For the sake of simplicity and to carry out better qualitative analysis of cable data, the field,

temperature and position dependences of the critical current are ignored. This provides us with

the possibility of comparing numerical results in this work with analytical results [17]. Under

these conditions, the superconducting current on each superconductor is given by [18, 19];

( , )

( , )

( , ) tanh

z

s c

c

A x yx y

tJ x y JE

(1)

where Jc is the critical current density, Ec is the electric field scaling parameter which is generally

taken as 10-4 V/m, zA and are vector and scalar potentials respectively. An iterative method

is implemented to equalize the current on each layer.

To obtain reliable results from FEM analyses, proper boundary conditions should be introduced.

In this work, the vector potential of a conductor lying on the xy plane and carrying a uniform Jz

current density which flows along z direction is assumed [20, 21]. Under this conditions, the

vector potential of the superconducting strips lying in the region defined by x(x1,x2) and

y(y1,y2) is given by

2 201 1 1 1

2 2

1 2 1 2

2 2

2 1 2 1

2 2

2 2 2 2

2 211 1

1

( , ) { ( ) ( ) ln ( ) ( )4

( ) ( ) ln ( ) ( )

( ) ( ) ln ( ) ( )

( ) ( ) ln ( ) ( )

( ) arctan ( ) arc

zz

JA x y x x y y x x y y

x x y y x x y y

x x y y x x y y

x x y y x x y y

y yx x x x

x x

2

1

2 21 22 2

2 2

2 21 21 1

1 1

2 21 22 2

2 2

tan

( ) arctan ( ) arctan

( ) arctan ( ) arctan

( ) arctan ( ) arctan

y y

x x

y y y yx x x x

x x x x

x x x xy y y y

y y y y

x x x xy y y y

y y y y

(2)

where 0 is the permeability of vacuum. The AC losses in superconductors are evaluated as

1/

0

f

SC

S

Q f dSdt J E (3)

where f is the frequency of the transport current, J is current density, E is electric field inside the

superconducting layer and S is the cross-sectional area of superconducting layer.

Vol 65, No. 7;Jul 2015

210 Jokull Journal

3. Results and Discussion

The AC losses in single and multilayer cables are compared in Fig. 2 for different current

amplitudes. Equation (3) was used to obtain the curves in Fig. 2. For better qualitative explanation

of the data in Fig. 1, the maximum current (Imax) is normalized by the critical current density (IC)

and the loss (QSC) is normalized to IC2. The critical current through the superconductor is given

by

C C SCI J dA (4)

Fig. 2 shows that the similar losses are observed at current amplitudes greater than IC/2for layer

numbers up to 3. On the other hand QSC is considerably smaller for the 5-layer cable at such

amplitudes. For instance, QSC is approximately 13% smaller in 5-layer cable than in 1-layer cable

at amplitudes close to IC. In contrast, for Imax less than IC/2, a discrepancy which increases as the

curren amplitude decreases is clearly observed. For such amplitudes, the loss generally decreases

with increasing layer number. For Imax/IC=1/16, for instance, the normalized loss is 1.4410-9,

1.3010-9, 1.1610-9 and 8.1110-10 for the 1, 2, 3 and 5-layer cables, respectively. At this

amplitude, the loss in the 5-layer cable is approximately 60% smaller than the one in the 1-layer

cable.

Fig. 2: (Color online) AC losses in single and multi-layer superconducting cables with respect to

normalized amplitudes of the applied current (Imax). The insets depicts variation around Imax=IC/2.

The reason for decreasing of AC loss as the layer number increases is qualitatively described in

Fig. 3, where the cross sections of magnetic equipotential lines for the four investigated cable

types in Fig. 1 are depicted. In the case of mono layer, there is a considerable amount of

penetration of radial component of magnetic intensity, in both inner and outer regions of the layer.

The magnetic field lines around the innermost layer become more and more sparse as the number

of layers is increased, where the innermost layer in the 5-layer cable is exposed only to its self-

Vol 65, No. 7;Jul 2015

211 Jokull Journal

field, as seen in Fig. 3(d). The current flow in the superconductor is expected to penetrate deeper

as the normal component of the magnetic field is increased.

Fig. 3: (Color online) Distribution of the equipotential lines of the magnetic vector field (Az) in

(a) 1, (b) 2, (c) 3 and (d) 5-layer cables. Computations in all cases correspond to Imax/IC=1/8.

The above discussions suggest that a layer-by-layer analysis for AC loss in the 5-layer cable is

required for a thorough understanding. Variation of AC loss with respect to Imax in each layer of

the cable are plotted in Fig. 4. The loss is the innermost layer (layer 1) is clearly the smallest at

all Imax values. The order of layers with increasing loss at any current amplitude is 1, 5, 2, 4 and

3. The loss in layer 1 is approximately 50% smaller than the loss in layer 3 (mid-layer) at around

Imax=IC/2, where the discrepancy decreases at both higher and lower amplitudes. Another point

that draws attention in Fig. 4 is that the AC loss exhibits the same trend for all current amplitudes.

Vol 65, No. 7;Jul 2015

212 Jokull Journal

Fig. 4: (Color online) Variation of AC loss in each layer of the 5-layer cable with respect to

applied current amplitude. The insets depicts variation around Imax=IC/2.

The reason for the fact that the loss in the innermost layer is the smallest one can be attributed to

electromagnetic screening of the outer layers, leading to exposition of this layer to the normal

component of the magnetic field in a much smaller extent than the outer layers. In fact, the

conductors are exposed to magnetic fields due to currents in both inner and outer layer conductors.

Due to the spacing between two neighboring superconductors along the radial direction, the

normal component of the magnetic field to the superconductor becomes more prominent and leads

to penetration of current more into this layer. Fig. 5 indicates that, the higher current densities

close to the critical current density (JC) the deeper it penetrates.

Fig. 5: (Color online) Distribution of the current density in each layer along x direction in the 5-

layer cable for Imax/IC=1/8. The inset depicts a close-up view of the distribution around the edge

of a superconducting strip.

4. Conclusion

Vol 65, No. 7;Jul 2015

213 Jokull Journal

Numerical analyses of alternating-current losses in single- and multi-layer cables composed of 15

rectangular superconducting strips wound around a core via the finite-element method show that

the similar losses in cables with up to 3 layers are experienced for current amplitudes greater than

half of the critical current in a strip. In contrast, the loss decreases as the number of layers is

increased for amplitudes less than half of the critical current. A considerably smaller loss in a 5-

layer cable than 1, 2, and 3-layer cables is observed at any amplitude. The innermost layer of a 5-

layer cable suffers least loss due to screening of magnetic field by the outer layers. Current

distribution in strips on the layers of a 5-layer cable is such that a large current close to the critical

current flows in the vicinity of the superconductors edges, while a smaller but stabilized current flows in the interiors.

Acknowledgments

This work is supported by the Scientific and Technological Research Council of Turkey

(TBTAK) under the grant number 110T876.

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