Transport AC loss in high temperature
superconducting coils
Mark Ainslie
King’s College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
January 24, 2012
Declaration
The work presented in this dissertation was carried out at the Department of
Engineering, University of Cambridge, between January 2009 and January 2012
under the supervision of Dr. Tim Flack.
The author declares that, except for where specific reference is made to the
work of other authors or specifically indicated in the text, the contents of this
dissertation are his own work, and include nothing done in collaboration. This
report has not been previously submitted in part, or in whole, to any other
university for a degree, diploma or other qualification. This dissertation is within
the word limit of 65,000 words, including appendicies, bibliography, footnotes,
tables, and equations, and does not contain more than 150 figures.
Acknowledgements
There are a number of people who have contributed, either directly
or indirectly, to the work presented in this thesis and for their contri-
butions I am extremely thankful.
Firstly, I would like to express my utmost gratitude to my supervi-
sor, Dr. Tim Flack, for his advice, guidance and support throughout
the duration of my research. He taught me the skills required for
independent research and allowed me the freedom to explore my own
ideas, whilst providing appropriate guidance and criticism to ensure
that I carried out my research professionally and completed my work
on time.
I would like to also thank the members of the Bulk Superconduc-
tivity Group in the Department of Engineering at the University of
Cambridge for their support and welcoming me into their group. In
particular, group leader Prof. David Cardwell for support and advice
and extensive knowledge of superconductivity, Prof. Archie Camp-
bell for his informative discussions on superconductor modelling and
experimentation, Dr. John Durrell for his technical expertise and
informative discussions, and Mr. Tony Dennis for his technical assis-
tance.
I must also thank the EPEC Superconductivity Group in the Depart-
ment of Engineering for their advice, allowing access to experimental
equipment and assistance with experiments. In particular, Dr. Tim
Coombs, Dr. Zhiyong Hong, Dr. Weijia Yuan, and Dr. Ruilin Pei.
I attended a number of conferences and workshops during my PhD,
and I am extremely grateful to those who I have collaborated with or
who have provided comments and constructive criticism of my work:
Assoc. Prof. Frederic Sirois, Dr. Victor Rodriguez-Zermeno, and
Dr. Francesco Grilli for their assistance with numerical modelling;
Dr. Milan Majoros, Dr. Fedor Gomory, Dr. Stephen Ashworth, Dr.
Enric Pardo, and Prof. Pascal Tixador for their assistance with AC
loss measurements.
Prof. Harry Jones and Mr. Anthony Hickman at the University of
Oxford must also be thanked for winding the circular superconducting
pancake coil.
Finally, I must thank all of my friends and family for their love and
support. My parents, Harry and Christine, have supported me in all
of my pursuits and encouraged me to set my goals for achievement
high. My partner, Briony, has supported me wholeheartedly, and
without her help and encouragement, I would not have achieved all
that I have thus far. The emotional support required to complete a
PhD is often underestimated.
Abstract
In this dissertation, the problem of calculating and measuring AC
losses in superconducting coils is addressed, with a particular focus on
the transport AC loss of coils for electric machines. In order to model
the superconducting coil’s electromagnetic properties and calculate
the AC loss, an existing two dimensional (2D) finite element model
that implements a set of equations known as the H formulation, which
directly solves the magnetic field components in 2D, is extended to
model a superconducting coil, where the cross-section of the coil is
modelled as a 2D stack of superconducting coated conductors.
The model is also modified to allow the inclusion of a magnetic sub-
strate, which is present in some commercially available HTS wire. The
analysis raises a number of interesting points regarding the use of su-
perconductors with magnetic substrates. In particular, the presence
of a magnetic substrate affects the penetration of the magnetic flux
front within the coil and increases the magnetic flux density within
the penetrated region, both of which can increase the AC loss signifi-
cantly. In order to investigate these findings further, a comprehensive
analysis on stacks of tapes with weak and strong magnetic substrates
is carried out, using a symmetric model that requires only one quarter
of the cross-section to be modelled.
In order to validate the modelling results, an extensive experimental
setup is designed and built to measure the transport AC loss of a
superconducting coil using an electrical method based on inductive
compensation by means of a variable mutual inductance. Measure-
ments are carried out on the superconducting racetrack coil and it is
found that the experimental results agree with the modelling results
for low current, but some phase drift occurs for higher current, which
affects the accuracy of the measurement. In order to overcome this
problem, a number of improvements are made to the initial setup to
improve the lock-in amplifier’s phase setting and other aspects of the
measurement technique.
New measurements are carried out on a single, circular pancake coil
and the discrepancies between the experimental and modelling results
are described in terms of the assumptions made in the model and as-
pects of the coil that cannot be modelled. Using the original measured
properties of the superconducting tape, there is an order of magnitude
difference between the experiment and model. The properties of the
superconductor can degrade during the winding and cooling processes,
and a critical current measurement of the coil showed that the tape
critical current reduced from nearly 300 A, down to around 100 A.
Applying this finding to the model, the experimental and modelling
results show good agreement, and the difference in the slope of the
AC loss curve can be described in terms of the B-dependent critical
current dependency Jc(B) used in the model.
Finally, methods used to mitigate AC loss in superconducting wires
and coils are summarised, and the use of weak and strong magnetic
materials as a flux diverter is investigated as a technique to reduce
AC loss in superconducting coils. This technique can achieve a signif-
icant reduction in AC loss and does not require modification to the
conductor itself, which can be detrimental to the superconductor’s
properties.
List of publications
• M.D. Ainslie, J.H. Durrell, A.R. Dennis, S.P. Ashworth, A.M. Campbell,
T.J. Flack, ”Transport AC loss measurements of a YBCO-based supercon-
ducting pancake coil,” submitted for review (Supercond. Sci. Technol.),
2012
• M.D. Ainslie, T.J. Flack, A.M. Campbell, ”Calculating transport AC
losses in stacks of high temperature superconductor coated conductors with
magnetic substrates using FEM,” Physica C, Vol. 472, No. 1, p. 50-56,
2011
• M.D. Ainslie, V.M. Rodriguez-Zermeno, Z. Hong, W. Yuan, T.J. Flack,
T.A. Coombs, ”An improved FEM model for computing transport AC loss
in coils made of RABiTS YBCO coated conductors for electric machines,”
Supercond. Sci. Technol. 24 (2011) 045005 (8pp)
• M.D. Ainslie, W. Yuan, Z. Hong, R. Pei, T.J. Flack, T.A. Coombs, ”Mod-
eling and Electrical Measurement of Transport AC Loss in HTS-based Su-
perconducting Coils for Electric Machines,” IEEE Trans. Appl. Supercond.,
Vol. 21, No. 3, p. 3265-3268, 2011
• M.D. Ainslie, T.J. Flack, Z. Hong, T.A. Coombs, ”Comparison of First-
and Second-Order 2D Finite Element Models for Calculating AC Loss in
High Temperature Superconductor Coated Conductors,” COMPEL: The
International Journal for Computation and Mathematics in Electrical and
Electronic Engineering, Vol. 30, Iss. 2, p. 762-774, 2011
• M.D. Ainslie, Y. Jiang, W. Xian, Z. Hong, W. Yuan, R. Pei, T.J. Flack,
T.A. Coombs, ”Numerical analysis and finite element modelling of an HTS
synchronous motor,” Physica C, Vol. 450, No. 20, p. 1752-1755, 2010
• W. Yuan,M.D. Ainslie, W. Xian, Z. Hong, Y. Yan, R. Pei, T.A. Coombs,
”Theoretical and experimental studies on Jc and AC losses of 2G HTS
coils,” IEEE Trans. Appl. Supercond., Vol. 21, No. 3, p. 2441-2444, 2011
• R. Pei, R. Viznichenko, M.D. Ainslie, Z. Hong, W. Xian, L. Zeng, W.
Yuan, T.A. Coombs, ”The Ic Behavior of 2G YBCO Tapes under DC/AC
Magnetic Fields at Various Temperatures”, IEEE Trans. Appl. Supercond.,
Vol. 21, No. 3, p. 3226-3229, 2011
• Z. Hong, W. Yuan, M. Ainslie, Y. Yan, R. Pei, T.A. Coombs, ”AC Losses
of Superconducting Racetrack Coils in Various Magnetic Conditions,” IEEE
Trans. Appl. Supercond., Vol. 21, No. 3, 2466-2469, 2011
• Z. Hong, Z. Jin, M. Ainslie, J. Sheng, W. Yuan, T.A. Coombs, ”Numer-
ical Analysis of the Current and Voltage Sharing Issues for Resistive Fault
Current Limiter Using YBCO Coated Conductors,” IEEE Trans. Appl.
Supercond., Vol. 21, No. 3, p. 1198-1201, 2011
• W. Yuan, W. Xian, M. Ainslie, Z. Hong, Y. Yan, R. Pei, T.A. Coombs,
”Design and Tests of a Superconducting Magnetic Energy Storage (SMES)
coil,” IEEE Trans. Appl. Supercond., Vol. 20, No. 3, p. 1379-1382, 2010
• W. Yuan, A.M. Campbell, Z. Hong, M. Ainslie, T.A. Coombs, ”Compar-
ison of AC losses, magnetic field/current distributions and critical currents
of superconducting circular pancake coils and infinitely long stacks using
coated conductors,” Supercond. Sci. Technol. 23 (2010) 085011 (8pp)
Contents
Contents viii
List of Figures xi
List of Tables xvii
1 Introduction 1
1.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Superconductivity and theoretical overview 7
2.1 Introduction to superconductivity . . . . . . . . . . . . . . . . . . 7
2.1.1 Type I & type II superconductors . . . . . . . . . . . . . . 9
2.1.2 High temperature superconductors . . . . . . . . . . . . . 11
2.1.3 Commercial HTS wire . . . . . . . . . . . . . . . . . . . . 13
2.2 Modelling HTS behaviour . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Critical state models . . . . . . . . . . . . . . . . . . . . . 16
2.2.1.1 Bean model . . . . . . . . . . . . . . . . . . . . . 18
2.2.1.2 Superconducting strip model . . . . . . . . . . . 21
2.2.2 E-J power law . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Kim model (magnetic field dependency of Jc) . . . . . . . 23
2.3 AC Loss in HTS conductors . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 AC loss in superconductors . . . . . . . . . . . . . . . . . 27
2.3.2 AC loss types . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2.1 Magnetisation loss . . . . . . . . . . . . . . . . . 29
2.3.2.2 Transport current loss . . . . . . . . . . . . . . . 30
viii
CONTENTS
2.3.3 AC loss calculation using the CSM . . . . . . . . . . . . . 31
2.3.4 Analytical techniques . . . . . . . . . . . . . . . . . . . . . 33
2.3.4.1 Norris . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.4.2 Brandt . . . . . . . . . . . . . . . . . . . . . . . 35
3 Modelling HTS-based superconducting coils 36
3.1 Finite element method (FEM) models . . . . . . . . . . . . . . . . 37
3.1.1 The H formulation . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1.1 H formulation in cartesian coordinates . . . . . . 38
3.2 Existing coil models . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Test superconducting coil for modelling . . . . . . . . . . . . . . . 42
3.4 Artificial expansion technique for individual tapes . . . . . . . . . 47
3.4.1 Solver time and convergence comparison . . . . . . . . . . 50
3.4.2 AC loss comparison . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Artificial expansion vs. bulk approximation . . . . . . . . . . . . 57
3.6 Real thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6.1 Optimal number of mesh elements . . . . . . . . . . . . . . 61
3.6.2 Addition of magnetic substrate . . . . . . . . . . . . . . . 66
3.6.3 Modelling results . . . . . . . . . . . . . . . . . . . . . . . 67
3.6.4 Implication of results for motor performance . . . . . . . . 70
3.7 Detailed investigation on stacks with magnetic substrates . . . . . 75
3.7.1 Stack AC loss comparison of symmetric FEM and analyti-
cal models . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7.2 Stack AC loss comparison with and without magnetic sub-
strates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.7.3 AC loss in individual tapes for stacks with/without mag-
netic substrates . . . . . . . . . . . . . . . . . . . . . . . . 81
3.8 Summary of refinements . . . . . . . . . . . . . . . . . . . . . . . 88
4 AC loss measurement 90
4.1 Overview of techniques . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Proposed electrical method . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 Experimental results . . . . . . . . . . . . . . . . . . . . . 96
ix
CONTENTS
4.3 Improved electrical method . . . . . . . . . . . . . . . . . . . . . 98
4.3.1 New superconducting pancake coil . . . . . . . . . . . . . . 98
4.3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 99
4.3.3 Experimental results . . . . . . . . . . . . . . . . . . . . . 106
4.3.3.1 Voltage tap measurements . . . . . . . . . . . . . 110
4.3.3.2 Coil critical current measurement . . . . . . . . . 111
4.3.3.3 Possible sources of measurement error . . . . . . 114
4.3.4 Suggested future improvements . . . . . . . . . . . . . . . 116
5 AC loss mitigation 118
5.1 AC loss mitigation techniques . . . . . . . . . . . . . . . . . . . . 118
5.1.1 Striation into narrow filaments . . . . . . . . . . . . . . . . 119
5.1.2 Roebel transposition . . . . . . . . . . . . . . . . . . . . . 120
5.1.3 Twisted wires . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.4 Magnetic shielding/flux diverter . . . . . . . . . . . . . . . 122
5.2 Flux diverter analysis . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.1 Modelling results . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Conclusions 130
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Appendix 1 135
Appendix 2 138
References 140
x
List of Figures
2.1 Characteristics of superconductivity . . . . . . . . . . . . . . . . . 8
2.2 Critical magnetic field as a function of temperature for (a) type I
superconductors and (b) type II superconductors . . . . . . . . . 11
2.3 Critical temperatures and year of discovery for different supercon-
ductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Crystalline structure of YBCO . . . . . . . . . . . . . . . . . . . . 13
2.5 Overview of AMSC HTS wire manufacturing process . . . . . . . 15
2.6 SuperPower HTS wire configuration . . . . . . . . . . . . . . . . . 15
2.7 Superconducting slab in externally applied magnetic field example 19
2.8 Dependence of the internal magnetic field Bz(x), current density
Jy(x), and pinning force Fp(x) on strength of applied magnetic
field B0 for normalised applied fields given by (a) B0/µ0Jca = 1/2,
(b) B0/µ0Jca = 1, and (c) B0/µ0Jca = 2 using the Bean model . . 20
2.9 The power-law model from n = 1 (linear) to n = ∞ (Bean’s model) 24
2.10 AMSC data showing critical current change for parallel and per-
pendicular magnetic fields . . . . . . . . . . . . . . . . . . . . . . 26
2.11 SuperPower data showing critical current change for perpendicular
magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.12 Magnetisation loop for a full cycle of applied field . . . . . . . . . 29
2.13 Field profile and current density distribution in an infinite slab
exposed to a magnetic field . . . . . . . . . . . . . . . . . . . . . . 33
2.14 AC loss calculation using Norris’s equation for thin strip of finite
width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
xi
LIST OF FIGURES
3.1 FEM model of a high temperature superconductor using the H
formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Photo of the HTS PM synchronous motor rotor test rig . . . . . . 44
3.3 Photograph of an HTS PM synchronous motor stator coils . . . . 45
3.4 Dimensions of an HTS PM synchronous motor stator coil . . . . . 45
3.5 Critical current of the YBCO sample tape and coil . . . . . . . . 46
3.6 Mesh element for models A (triangular) and B (square) . . . . . . 49
3.7 Number of mesh elements for different YBCO layer thicknesses for
models A (represented by triangles) and B (represented by squares) 50
3.8 Triangular mesh solver times . . . . . . . . . . . . . . . . . . . . . 51
3.9 Square mesh solver times . . . . . . . . . . . . . . . . . . . . . . . 52
3.10 Total number of degrees of freedom for different YBCO layer thick-
nesses for all models. Triangles represent triangular meshes (model
A), squares represent square meshes (model B), and diamonds rep-
resent edge elements (model AE). The double solid line represents
second-order Lagrange elements and the double dashed line repre-
sents first-order Lagrange elements. . . . . . . . . . . . . . . . . . 53
3.11 Convergence behaviour for all models using solver time per degree
of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.12 AC loss calculation for all models for different YBCO layer thick-
ness compared with Norris’s analytical model (clockwise from top
left: I/Ic = 0.25, I/Ic = 0.5, I/Ic = 0.75, I/Ic = 0.9) . . . . . . . . 55
3.13 Comparison of current density distribution in superconductor with
Norris’s analytical model for models A2, AE and B2 for I/Ic = 0.5
for 2 and 20 µm YBCO layer thicknesses . . . . . . . . . . . . . . 57
3.14 Geometry and mesh for model of individual turns using artificial
expansion technique . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.15 Geometry and mesh for model using bulk approximation . . . . . 60
3.16 AC loss calculation for individual turns and bulk approximation
for constant Jc and Jc(B) dependence . . . . . . . . . . . . . . . . 60
3.17 Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.1 Ic . . . . . . . . . . . . 63
xii
LIST OF FIGURES
3.18 Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.3 Ic . . . . . . . . . . . . 64
3.19 Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.5 Ic . . . . . . . . . . . . 64
3.20 Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.7 Ic . . . . . . . . . . . . 65
3.21 Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.9 Ic . . . . . . . . . . . . 65
3.22 Model geometry and mesh using the actual thickness for the YBCO
layer and including the magnetic substrate . . . . . . . . . . . . . 69
3.23 Comparison of calculated AC loss for the four cases (Jc and Jc(B)
for inclusion/exclusion of magnetic substrate) . . . . . . . . . . . 70
3.24 Magnetic flux density profile of the magnetic field perpendicular
to the tape faces |By| without a magnetic substrate and with Jc(B)
at t = 15 ms (peak of applied current |I0| = 50 A) . . . . . . . . . 71
3.25 Magnetic flux density profile of the magnetic field perpendicular
to the tape faces |By| with a magnetic substrate and with Jc(B) at
t = 15 ms (peak of applied current |I0| = 50 A) . . . . . . . . . . 72
3.26 Comparison of AC loss for tapes at different locations within the
cross-section for Jc(B) models with and without a magnetic sub-
strate for an applied current I0 = 50 A . . . . . . . . . . . . . . . 73
3.27 Comparison of the superconducting coil transport AC loss (Jc(B)
models with and without the magnetic substrate) with an equiva-
lent copper coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.28 2D symmetric model geometry without a magnetic substrate and
using the actual thickness for the YBCO layer (shown is the 50
tape stack) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.29 Relative magnetic permeability µr(H) for a strongly magnetic sub-
strate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.30 Ferromagnetic substrate loss Qfe for a strongly magnetic substrate 77
3.31 Comparison of symmetric non-magnetic substrate stack model with
Norris (single tape) and Clem (infinite stack) models . . . . . . . 78
xiii
LIST OF FIGURES
3.32 Comparison of AC loss in a stack of 5 tapes with and without a
magnetic substrate (weak/strong) [NM = non-magnetic, WMS =
weakly magnetic substrate, SMS = strongly magnetic substrate;
TOTAL = total loss, SC = superconductor hysteretic loss, SUB =
ferromagnetic substrate loss] . . . . . . . . . . . . . . . . . . . . . 80
3.33 Comparison of AC loss in a stack of 50 tapes with and without a
magnetic substrate (weak/strong) . . . . . . . . . . . . . . . . . . 80
3.34 Comparison of AC loss in a stack of 150 tapes with and without a
magnetic substrate (weak/strong) . . . . . . . . . . . . . . . . . . 81
3.35 Comparison of AC loss in certain tapes (1/5, 3/5 and middle tapes)
within stack of tapes with a non-magnetic substrate (top figure), a
weakly magnetic substrate (middle figure), and a strongly magnetic
substrate (bottom figure) . . . . . . . . . . . . . . . . . . . . . . . 83
3.36 Magnetic flux penetration in tapes located at 1/5 between the top
(top figure) and centre (bottom figure) for the 20 tape stacks [NMS
= non-magnetic substrate, WMS = weakly magnetic substrate,
SMS = strongly magnetic substrate] . . . . . . . . . . . . . . . . . 84
3.37 Current density distribution in tapes located at 1/5 between the
top (top figure) and centre (bottom figure) for the 20 tape stacks
[NMS = non-magnetic substrate, WMS = weakly magnetic sub-
strate, SMS = strongly magnetic substrate] . . . . . . . . . . . . . 85
3.38 Magnetic flux penetration in tapes located at 1/5 between the top
(top figure) and centre (bottom figure) for the 100 tape stacks
[NMS = non-magnetic substrate, WMS = weakly magnetic sub-
strate, SMS = strongly magnetic substrate] . . . . . . . . . . . . . 86
3.39 Current density distribution in tapes located at 1/5 between the
top (top figure) and centre (bottom figure) for the 100 tape stacks
[NMS = non-magnetic substrate, WMS = weakly magnetic sub-
strate, SMS = strongly magnetic substrate] . . . . . . . . . . . . . 87
4.1 Schematic diagram of experimental setup for measuring transport
AC loss in superconducting coils electrically . . . . . . . . . . . . 93
xiv
LIST OF FIGURES
4.2 Compensation coil (variable mutual inductance) for proposed elec-
trical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Experimental results for the transport AC loss measurement of the
superconducting racetrack coil . . . . . . . . . . . . . . . . . . . . 97
4.4 Completed circular HTS pancake coil wound with SuperPower wire 99
4.5 Schematic diagram of new experimental setup for measuring trans-
port AC loss in superconducting coils electrically . . . . . . . . . 100
4.6 Main components of the new experimental setup for measuring
transport AC loss in superconducting coils: 1) lock-in amplifier, 2)
signal generator, 3) oscilloscope, and 4) power amplifier . . . . . . 101
4.7 Superconducting coil liquid nitrogen bath, compensation coil and
clamp meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.8 Superconducting coil submersed in liquid nitrogen, and its associ-
ated wiring and voltage taps . . . . . . . . . . . . . . . . . . . . . 102
4.9 Hand-wound Rogowski coil used to provide the lock-in amplifier
reference signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.10 Power loss measurement of 500 µΩ shunt resistor using two differ-
ent methods (lock-in amplifier and clamp meter) and setting the
lock-in amplifier phase using the Rogowski coil and shunt resistor 105
4.11 Calculated transport AC loss using a constant Jc and Jc(B) and
experimental results for four frequencies (39.93, 80.83, 120.1 and
158.2 Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.12 Superpower measured data for critical current Ic and n value for
5 m sections of tape for the 30 m spool of tape used to wind the
superconducting coil . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.13 Comparison of measured transport AC loss of entire coil and each
voltage tap for f = 80.9 Hz with modelling results for sections
corresponding to the voltage taps . . . . . . . . . . . . . . . . . . 111
4.14 Waveform of applied current for coil critical current measurement 112
4.15 Coil critical current measurement . . . . . . . . . . . . . . . . . . 113
4.16 Comparison of calculated transport AC loss using a modified Jc to
account for degradation of the superconducting tape and experi-
mental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
xv
LIST OF FIGURES
4.17 Coil critical current measurement after large quench . . . . . . . . 115
4.18 Voltage taps measurement after large quench . . . . . . . . . . . . 115
4.19 Relative in-phase (resistive) voltage error as a function of the ra-
tio between the inductive and resistive voltage components of the
measured signal for different phase errors . . . . . . . . . . . . . . 117
5.1 Striation of HTS coated conductor using laser ablation . . . . . . 119
5.2 Punched strands from SuperPower-manufactured YBCO coated
conductor (top) and an assembled YBCO Roebel cable (bottom) . 121
5.3 AC loss reduction using flux diverter using weak and strong mag-
netic materials of thickness 0.5 mm and 1 mm for stacks of tapes
with and without a weak magnetic substrate . . . . . . . . . . . . 125
5.4 Comparison of magnetic flux density profiles, including magnetic
flux lines, for a 50 tape stack (a) without and (b) with a flux
diverter (weak magnetic material) . . . . . . . . . . . . . . . . . . 126
5.5 Comparison of magnetic flux penetration in tapes located at the
1/5 point (top figure) and centre (bottom figure) for the 50 tape
stack (without a magnetic substrate) with and without a diverter 127
5.6 Peak diverter magnetic flux density for diverter of weak magnetic
material for thicknesses of 0.5 mm and 1 mm for stacks of 10 to
100 tapes with and without a weak magnetic substrate [NMS =
non-magnetic substrate, WMS = weakly magnetic substrate] . . . 128
5.7 Peak diverter magnetic flux density for diverter of strong magnetic
material for thicknesses of 0.5 mm and 1 mm for stacks of 10 to
100 tapes with and without a weak magnetic substrate [NMS =
non-magnetic substrate, WMS = weakly magnetic substrate] . . . 128
xvi
List of Tables
3.1 AMSC YBCO 344 tape properties . . . . . . . . . . . . . . . . . . 43
4.1 SuperPower SCS12050-AP wire specification . . . . . . . . . . . . 98
xvii
Chapter 1
Introduction
The annual world electricity consumption was estimated at 138 trillion kWh in
2006, and is estimated to reach almost 200 trillion kWh in the year 2030 [1].
Given that there is a finite quantity of fossil fuel remaining, and the world’s
population continues to grow, our existing methods for energy supply and usage
are clearly unsustainable. In developed industrialised nations, such as the UK and
the US, the industrial sector uses about one third of all energy consumed [2], and
approximately two thirds of this energy is consumed by electric motor drives [3].
Scientists and engineers can provide an important contribution to the reduc-
tion of energy consumption and its associated environmental pollution. Electrical
energy consumption can be reduced in electric machines in the following ways:
good housekeeping (for example, switching idling motors off), the use of variable-
speed drives, and the construction of electric motors and generators with better
efficiency [4]. Loss of electrical energy due to resistance to current flow, which
is prevalent in conventional machines, translates directly to wasted energy and,
therefore, to wasted economic resources.
Superconductivity offers zero to near zero resistance to electrical current when
cooled down to a particular cryogenic temperature. Consequently, the use of su-
perconducting materials can improve the overall electrical system efficiency. In
addition, superconducting materials are able to carry much larger current densi-
ties than conventional materials, such as copper. In electric machines, in particu-
lar, increasing the current and/or magnetic flux density increases the power den-
sity, which leads to reductions in both size and weight. The expected improved
1
performance and efficiency, as well as a smaller footprint in comparison with
conventional devices, has seen continued interest in introducing superconducting
materials to not only electric machines, but other electric power applications,
such as cables, superconducting fault current limiters (SFCL) and transformers.
Investigating and modelling the electromagnetic behaviour of superconductors
is crucial to the design of superconductor-based electrical devices. In order for
these devices to be cost- and performance-competitive with conventional devices,
the use of superconducting materials and the associated cooling system must be
shown to possess improved properties in comparison to its conventional counter-
part. Although lossless for DC (direct current), superconductors do experience
AC (alternating current) losses, which can be a significant problem in any de-
vice exposed to a time-varying current or magnetic field. Since superconductors
require operation at cryogenic temperatures, these AC losses increase the refrig-
eration load, which decreases the overall efficiency and increases the technological
complexity of the design.
In an electric machine, there are usually multiple superconductors in tape/wire
form wound into coils and interacting together in a complex magnetic environ-
ment. It is only in recent years that long lengths of wire have been available
commercially, which has made it possible to wind coils and cables for large scale
applications. Significant efforts have been made to understand the complex in-
teractions between multiple superconducting coated conductors in quite simple
geometries, but no reliable technique exists to both model and measure AC losses
in more complex geometries, such as superconducting coils. Superconducting coils
are found in a number of different applications, such as superconducting magnetic
energy storage (SMES) systems and transformers.
This dissertation aims to address the problem of calculating and measuring
AC losses in superconducting coils, with a particular focus on the transport AC
loss of coils for electric machines. In order to model the superconducting coil’s
electromagnetic properties and calculate the AC loss, a two dimensional (2D) fi-
nite element model that implements a set of equations known as the H formulation
has been developed, which directly solves the magnetic field components in 2D,
where the cross-section of the coil is modelled as a 2D stack of superconducting
coated conductors.
2
Firstly, in order to optimise the coil model in terms of accuracy of solution
and computational speed, an investigation is carried out on the artificial expan-
sion of the thickness of the YBCO layer and selecting an appropriate mesh type
and number of elements. This is done using a single tape, as the preliminary
groundwork for optimising more complex geometries. The artificial expansion
technique is then applied to the geometry of a superconducting racetrack coil of
an all-superconducting high-temperature superconductor (HTS) permanent mag-
net synchronous motor to model the individual turns of the coil, and the result
is compared with a model using a bulk approximation. The limitations of the
artificial expansion technique and its application to coils is discussed in detail.
A technique is then applied that allows the actual superconducting layer thick-
ness to be modelled without the associated problem of increased computation
time due to a large number of mesh elements, and a comprehensive study on
the minimum number of mesh elements required for accurate results is carried
out. The model is also modified to allow the inclusion of a magnetic substrate,
which is present in some commercially available HTS wire. The analysis raises a
number of interesting points regarding the use of superconductors with magnetic
substrates, and a comprehensive analysis on stacks of tapes with weak and strong
magnetic substrates is provided, using a symmetric model that requires only one
quarter of the cross-section to be modelled.
In order to validate the modelling results, an extensive experimental setup has
been designed to measure the transport AC loss of a superconducting coil using
an electrical method based on inductive compensation by means of a variable
mutual inductance. Electrical measurement methods are generally faster than
calorimetric methods and provide greater sensitivity, but a major problem when
applying this technique to a superconducting coil is the compensation of the
much larger inductive component of the coil’s voltage. Since the voltage related
to the superconducting coil’s AC loss, which is in-phase with the current, is
orders of magnitude less than the coil’s inductive voltage, which is 90◦ out of
phase with the current, the loss voltage is difficult to extract using conventional
measurement techniques. The variable mutual inductance is utilised to cancel the
inductive voltage, and is used in conjunction with a lock-in amplifier, which can
extract a signal with a known carrier wave where the signal-to-noise ratio is very
3
small. Measurements are carried out on the superconducting racetrack coil, as
well as a circular pancake coil, and the validity of the modelling and experimental
techniques is investigated in detail in regards to estimating a superconducting
coil’s transport AC loss.
The transport AC loss of a superconducting coil is found to be significantly
large, and this will reduce the efficiency of the device in which it is utilised, par-
ticularly when the loss is reflected back to room temperature by including the
refrigeration cost. Many existing AC loss mitigation techniques involve modifi-
cation of the HTS conductor itself, which can cause significant degradation of
the conductor properties. Therefore, an investigation is carried into the use of
weak or strong magnetic materials to manipulate the magnetic flux in a super-
conductor to reduce the AC loss, which is presented for stacks of tapes with and
without a (weak) magnetic substrate. The use of external magnetic materials as
a flux diverter can achieve a significant reduction in AC loss without modifying
the original conductor, and shows promise as a technique to mitigate AC losses
in practical superconducting coils.
An overview of the structure of the dissertation is provided in the follow-
ing section. A thorough literature review is presented at the beginning of each
chapter for the three main sections: modelling of HTS-based superconducting
coils, measurement of AC loss in HTS-based superconducting coils, and AC loss
mitigation.
1.1 Thesis outline
In Chapter 2, a brief introduction to superconductivity is presented, including
the main commercially available high temperature superconductor materials. The
underlying theoretical principles relating to the topics covered in this dissertation
are then presented, including the existing analytical techniques for modelling the
electromagnetic properties of HTS materials, followed by the mechanisms for and
analytical calculation of AC loss for these materials in simple geometries. This
theoretical background is referred to extensively in later chapters and forms the
basis for comparing the results using the finite element method in the following
chapter.
4
In Chapter 3, the modelling of HTS-based superconducting coils using the
finite element method is described in detail, including the evolution of the devel-
opment of the coil model as the research in this dissertation has progressed. Often
there is a compromise in computer modelling between the accuracy of the solution
and the computational time required, and a number of different models are com-
pared to examine the optimum parameters. Firstly, the numerical model used in
this thesis is described, which is based on solving a set of Maxwell’s equations in
2D implementing the H formulation using the commercial software package Com-
sol Multiphysics. The coil cross-section is modelled as the number of individual
turns in the coil, and an artificial expansion technique is investigated to improve
the computational speed of the model, which can require hundreds of thousands of
mesh elements. Different methods to approximate the critical current density Jc
are also discussed. A technique that allows the real superconducting layer thick-
ness to be modelled, using a mapped mesh, is then investigated, and the model
is modified to include the magnetic substrate present in some superconducting
tapes. This investigation raises some interesting points for further analysis, and
a detailed investigation on stacks of superconducting tapes with both weak and
strong magnetic substrates is carried out at the end of the chapter.
In Chapter 4, the measurement of AC loss in HTS-based superconducting
coils is described in detail, including an experimental setup that uses an electrical
technique to accurately measure the transport AC loss of a superconducting coil.
The experimental technique is based on the use of a lock-in amplifier to extract
the in-phase component of the superconducting coil voltage, which corresponds
to the AC loss voltage. In order to compensate for the coil’s large inductive
voltage, a variable mutual inductance is used. The technique is applied firstly
to measure the racetrack coil introduced in the previous chapter. It is found
that the experimental results agree with the modelling results for low current,
but some phase drift occurs for higher current, which affects the accuracy of the
measurement. In order to overcome this problem, a number of improvements are
made to the initial setup to improve the lock-in amplifier’s phase setting and other
aspects of the measurement technique, including the use of the signal generator’s
reference (TTL) output and a Rogowski coil to provide stable reference signals to
accurately set the reference phase of the lock-in amplifier, and new measurements
5
are carried out on a single, circular pancake coil. Discrepancies between the
experimental and modelling results are described in terms of the assumptions
made in the model and aspects of the coil that cannot be modelled. Finally,
some suggestions to improve the experimental setup in the future are presented.
In Chapter 5, methods used to mitigate AC loss in superconducting wires and
coils are summarised, and the use of weak and strong magnetic materials as a
flux diverter is investigated as a technique to reduce AC loss in superconducting
coils that does not require modification to the conductor itself, which can be
detrimental to the superconductor’s properties.
Chapter 6 summarises the contributions of the research completed thus far
and its implications, and discusses possible directions of research in this area in
the future.
6
Chapter 2
Superconductivity and
theoretical overview
This chapter aims to present a brief introduction to superconductivity, including
the main commercially available high temperature superconductor materials. The
underlying theoretical principles relating to the topics covered in this dissertation
are then presented, including the existing analytical techniques for modelling the
electromagnetic properties of HTS materials, followed by the mechanisms for and
analytical calculation of AC loss for these materials in simple geometries. This
theoretical background is referred to extensively in later chapters and forms the
basis for comparing the results using the finite element method in the following
chapter.
2.1 Introduction to superconductivity
Superconductivity has come a long way since first being discovered in 1911 by H.
Kamerlingh Onnes, one of the first professors in experimental physics at Leiden
University. Kamerlingh Onnes observed that the electrical resistance of metals
such as mercury, lead and tin disappeared completely from a finite value in a
small temperature range around a critical temperature, Tc, which is characteristic
of the particular material. His first discovery was the disappearance of solid
mercury’s resistance below a temperature of around 4 K [5], which he reached
7
using a refrigeration technique he designed just three years earlier. He received
the Nobel Prize in Physics in 1913 for his work regarding the properties of matter
at low temperatures.
Perfect conductivity was the first phenomenalogical characteristic of supercon-
ductivity; however, in 1933, perfect diamagnetism in these materials was discov-
ered by Meissner and Ochsenfeld, which meant that a superconductor completely
expelled an applied magnetic field except for a distance of λ, the penetration
depth [6]. This became known as the Meissner effect.
Therefore, to prove the existence of superconducting material, it was necessary
to observe two principal properties in the superconductor: the disappearance of
resistance and the complete expulsion of an applied magnetic field. A supercon-
ductor can be characterised by its critical temperature, Tc, its critical magnetic
field, Hc, and its critical current density, Jc, as shown in Figure 2.1. These pa-
rameters define the upper limits for the superconductivity in a material and can
be used to describe the state a superconductor is in (superconducting or normal)
for a given set of conditions [7]. The shaded volume in this figure corresponds to
the material being in its superconducting state.
Figure 2.1: Characteristics of superconductivity
8
By the end of the 1960s, a remarkably complete and satisfactory theoretical
picture of classic superconductors had emerged, based around the Meissner effect
[6], the London equations [8], Ginzburg-Landau theory [9], and BCS (Bardeen-
Cooper-Schrieffer) theory [10].
2.1.1 Type I & type II superconductors
For many years it was thought that the behaviour described above was inherent
to all superconductors, but in 1957 Alexei Alexeyevich Abrikosov published a
theoretical paper regarding another class of superconductors that may have dif-
ferent properties [11]. It is now realised that the apparently irregular properties
of certain superconductors are not just effects of impurities, but inherent features
of another class of superconductors, called ’Type II’ superconductors [12].
One of the characteristic features of ’Type I’ superconductors is the Meissner
effect (perfect diamagnetism) below Hc, which implies the existence of a surface
energy boundary between any normal and superconducting regions in the metal.
This plays an important role in determining a superconductor’s type [13]. Above
Hc the type I superconductor reverts back to the normal state, where the mag-
netic field fully penetrates the material. Type I superconductors are limited in
their current-carrying capacity due to the Meissner effect and Ampere’s law. Cur-
rent flow in a conductor is accompanied by a self-induced magnetic field, which
are both confined to the outer layer as the field is excluded from the interior by
the Meissner effect. According to Silsbee’s criterion of depairing current, a su-
perconductor loses its zero resistance when at any point on the surface the total
magnetic field strength (due to the transport current and applied magnetic field)
exceeds Hc [14]. The maximum current that can be carried by a Type I super-
conductor with a circular cross-section and radius r is given by Ic = 2pirHc [15].
For currents above this value, the self-induced magnetic field is large enough to
destroy the superconducting state [16]. This transformation of a superconducting
wire to the normal state when the current passing through it exceeds the critical
value is called the Silsbee effect [15]. The critical value Hc is dependent on the
temperature:
9
Hc = Ho(1− ( T
Tc
)2) (2.1)
Silsbee’s criterion mentioned above holds only for type I superconductors,
whereas for type II superconductors, the complete explusion of flux at H < Hc
does not take place [14]. Type I superconductors have a positive surface energy;
however, type II superconductors have a negative surface energy, which leads to
a ’mixed state,’ where there exists an upper and lower critical magnetic field,
Hc2 and Hc1, respectively. In the mixed state, the magnetic field penetrates
partially into the material in the form of vortices. These vortices (or flux tubes)
are small tubular regions of the order of the coherence length ξ (a length scale
that characterises superconducting electron pair coupling), each containing one
quantum of flux, which Abrikosov determined as Φ0 =
h
2e
= 2.1 × 10−15 Tm2.
The vortices form a periodic lattice called the Abrikosov vortex lattice. The
resistivity of the superconductor may be vanishing as long as the vortices are
pinned or trapped. As the external field increases towards Hc2, the size of the
superconducting region between the cores of the flux lines shrinks to zero, and
the superconductor shows a continuous transition towards the normal state [17].
For magnetic fields below Hc1, the material is in the superconducting state and
any magnetic field is expelled from the inside of the superconductor; above Hc2,
the material is in the normal state and the superconductivity is largely confined
to the surface of the material [18]. In this case, the number of vortices has reached
its maximum and no more vortices can be added.
Type II superconductors can carry larger amounts of current in higher mag-
netic fields in comparison with Type I superconductors because Hc2 can be hun-
dreds of times larger than Hc. For example, the strongest type I superconductor
(pure lead) has a critical field of around 80 mT (800 Gauss), whereas YBCO (Y-
Ba-Cu-O) has an Hc1 around 20 mT (200 Gauss) and an Hc2 as high as 100 T [19].
The critical field as a function of temperature for each type of superconductor is
shown in Figure 2.2.
There is also another defined field for type II superconductors, called the irre-
versibility field Hirr. For an applied magnetic field above this value, the vortices
10
Figure 2.2: Critical magnetic field as a function of temperature for (a) type I
superconductors and (b) type II superconductors [13]
begin to move, creating additional dissipation. Hirr is considered the practical
limit of type II superconductors and is an order of magnitude lower than Hc2 [18].
For YBCO, for example, the irreversibility field at 77 K is 5-6 T [20].
The group of type I superconductors is mainly comprised of metals and met-
alloids that show some conductivity at room temperature, including lead (Pb),
mercury (Hg), tin (Sn), indium (In), and aluminium (Al), and form part of an-
other group of superconductors called ’low temperature superconductors’ (LTS).
Except for the elements vanadium, technetium and niobium, the group of type II
superconductors is comprised of metallic compounds and alloys, and this group
includes some LTS, the high temperature superconductors (HTS) and magnesium
diboride (MgB2). Figure 2.3 shows a comparison of the different critical temper-
atures for many superconductors, as well as the year each superconductor was
discovered. The axis on the right hand side shows equivalent examples for these
temperatures. This dissertation focuses on HTS materials, in particular YBCO.
2.1.2 High temperature superconductors [22], [23]
High temperature superconductors were discovered in 1986, when Bednorz and
Muller discovered LSCO (La2−xSrxCuO2) [24], for which they were conferred the
fastest Nobel Prize ever awarded. These are layered materials dominated by
11
Figure 2.3: Critical temperatures and year of discovery for different superconduc-
tors [21]
copper oxide planes, called Perovskites, which have been discovered with Tcs of
over 100 K. HTS are generally defined as superconductors with a Tc higher than
around 23 - 30 K (30 K is the upper limit allowed by BCS theory, 23 K is the
1973 record that lasted until copper-oxide materials were discovered).
YBCO (YBa2Cu3O7) is the most famous of these HTS and was discovered
by Paul C. W. Chu and M-K. Wu in 1986 and 1987, respectively, and is known
as the second generation (2G) HTS - the second HTS to be used for making
conducting wires. It was the first material to superconduct above 77 K, the
boiling point of liquid nitrogen, and has consequently paved the way for a much
broader range of practical applications. All other materials discovered before this
became superconducting at temperatures near the boiling points of liquid helium
or hydrogen (4.2 K and 20 K, respectively), which are both more expensive and
difficult to obtain than liquid nitrogen. YBCO also appeals to researchers because
it is the cleanest and most ordered crystal - the crystalline structure of YBCO
12
is shown in Figure 2.4. Shown in the bottom left of this figure are the axes
(or planes) of the material. The crystal structure of YBCO is highly anisotropic,
with much higher conductivity within the CuO2 than perpendicular to the planes.
Thus, supercurrents flow only within the CuO2 (a-b) planes, i.e. left to right in
the figure, meaning the trapped field generated by these supercurrents is directed
along the c-axis [25].
Figure 2.4: Crystalline structure of YBCO [13]
2.1.3 Commercial HTS wire
There are two main companies that supply long lengths of YBCO-based HTS
tape/wire: American Superconductor (AMSC) [26] and SuperPower [27]. The
manufacturing techniques differ between the two, which results in a different
configuration for the final product. It is only in recent years that long lengths of
wire has been available commercially, which has made it possible to wind coils
and cables for large scale applications, such as electric machines, superconducting
magnetic energy storage (SMES) systems, transformers, and so on.
13
AMSC’s approach to manufacturing YBCO-based HTS wire is based on the
RABiTS/MOD (rolling assisted biaxially textured substrate/metalorganic depo-
sition) technology and an overview of this technology is shown in Figure 2.5. The
buffer layers (a 75 nm Y2O3 seed layer, a 75 nm YSZ barrier layer and a 75 nm
CeO2 cap layer) are deposited by high-rate reactive sputtering onto a metal alloy
(Ni-W) substrate, and the rare earth doped YBCO is coated onto the buffered
substrate. The YBCO is capped with an Ag layer, then oxygenated, and lami-
nated between two metallic stabliser strips, currently either brass or copper.
SuperPower’s approach is based on the IBAD/MOCVD (ion beam assisted
deposition/metal organic chemical vapour deposition) technology, which involves
sputtering a stack of buffer layers to introduce the biaxial texture for the YBCO
layer, which is deposited using MOCVD. A thin cover of silver is then sputtered to
provide electrical contact. Depending on the application, this is then electroplated
to completely surround the wire. The configuration of SuperPower’s YBCO-based
HTS wire is shown in Figure 2.6.
These two kinds of YBCO-based HTS wire are referred to in this dissertation
and the coils used for testing are wound with AMSC and SuperPower wire. The
terms wire, tape and coated conductor are used interchangeably. The large aspect
ratio of the tape and its crystalline structure makes this type of superconductor
highly anisotropic and the tape performance is affected greatly by magnetic fields
perpendicular to the tape’s wide face, i.e. perpendicular to the a-b plane.
2.2 Modelling HTS behaviour
High temperature superconductors (HTS) possess a number of unique proper-
ties that make them attractive for use in a range of engineering applications (for
examples, see Introduction). In order to optimise the design of a system that in-
cludes superconductors, it is necessary to predict the electromagnetic behaviour
of the superconductor [29]. The complexity of computing the quantitative elec-
tromagnetic properties of HTS materials is significantly increased because they
are characterised by a highly non-linear current-voltage relationship.
HTS models belong to two groups: microscopic models and macroscopic
models. Microscopic models aim to explain the properties of superconductors,
14
Figure 2.5: Overview of AMSC HTS wire manufacturing process [28]
Figure 2.6: SuperPower HTS wire configuration [27]
15
whereas macroscopic models use simplified descriptions of these properties to
predict the performance of superconducting devices [30]. Microscopic models are
based around the London model, and the BCS and Ginzburg-Landau theories;
macroscopic models are commonly based on critical state models, such as the
Bean [31] and Kim [32, 33] models. Other macroscopic models use a non-linear
current-voltage relationship, such as the E-J power-law [34], to model the super-
conductor as a non-linear conductor. Macroscopic models are of more interest to
engineers designing large-scale devices that use superconductors, and these will
be introduced in this chapter. This is because the computational effort required
to solve microscopic models in the context of such applications is prohibitive.
2.2.1 Critical state models
HTS materials are type II in nature, and magnetic flux entering a type II su-
perconductor does so in the form of discrete fluxons (or vortices). Cooper pairs,
i.e. super-electrons, flow around the fluxon to shield it from the superconducting
matrix. These fluxons always penetrate the sample initially from the edges of
the material. Their motion inward is impeded by irregularities in the material
microstructure, such as various lattice defects, non-superconducting precipitates,
grain boundaries, and dislocations, which are referred to as pinning sites [35].
Without these pinning sites, the magnetisation of a type II superconductor would
be reversible, and no magnetic field would be trapped within the superconductor.
The magnetic field is trapped due to the interaction force between the fluxon
from the pinning site, and is given by the Lorentz force. A fluxon can only pass
the pinning site if the Lorentz force is greater than the pinning force. A critical
state model (CSM) is often used to represent this behaviour of fluxons, which
predicts different ’operating modes’ for different situations.
Critical state models are based on the macroscopic behaviour of superconduct-
ing materials, derived from experimental observations of the relationship between
current density and magnetic field. In these models, the outer layer of the ma-
terial is said to be in a ’critical state’ for a low applied current and/or magnetic
field. The critical state occurs when an applied field exceeds a type II supercon-
ductor’s lower critical magnetic field Hc1. Magnetic flux vortices with circulating
16
superconducting shielding currents penetrate the material to shield the interior of
the material from the applied current/field. The vortices are pinned at locations
of defects in the crystal lattice of the material, and the depth of penetration de-
pends on the magnitude of the applied current/field. It should be noted here that
these defects are deliberately introduced into the superconductor, e.g. secondary
phase Y211 in the YBCO system.
For normal materials, current density and electric field are related by Ohm’s
law, but for superconductors, a different expression is required. The classical
CSM introduced by Bean has been successfully used to describe the Jc of type II
superconductors. The model comes from Bean’s studies of ferromagnetic materi-
als. It is important for two reasons: it introduces domain-like structure into the
current density, which seems to be retained even in modified models that allow
Jc to be dependent on magnetic field, and it greatly simplifies loss calculations.
The CSM model is used for the calculation of Jc from magnetic hysteresis loops
of classical type II superconductors - it provides approximate solutions for most
simple practical cases, even for those where the critical current density depends
on magnetic field [36].
The relevant Maxwell’s equations, with displacement current omitted, are
∇×H = J (2.2)
∇×E = −dB
dt
(2.3)
where B = µ0 H, which is a good approximation for practical applications
where Hc1 < H < Hc2 [37], and ∇ · J = 0 as there are no time-varying free charge
distributions.
Where there is a current flowing in the superconductor, the magnetic field
vortices experience a Lorentz force F = J × B. For a large enough Lorentz force,
the vortices become de-pinned and move in the direction of the force with velocity
v. This vortex movement induces an electric field E = B × v, and E is parallel
to B × (J × B). If B is perpendicular to J, which is always true for 2D models,
17
then E is parallel to J. Thus,
E = ρ(J)J (2.4)
where ρ(J) is a highly non-linear function for the region inside the material
and J = |J|. The equation above is analogous to Ohm’s law for conventional
materials. All CSMs state that the current density in the superconductor cannot
exceed the critical value Jc.
2.2.1.1 Bean model
Bean’s model [31] is the simplest of all CSMs and it states that the magnitude of
the superconductor’s current density takes values of either 0 or ±Jc, the critical
current density. It assumes a non-vanishing electric field with current density in
the direction of the electric field. Furthermore, the current density is only zero
in regions of the superconductor that have never experienced an electric field.
When the whole superconductor is penetrated with ±Jc, the superconductor is
in a critical state.
When applying an external field, the field begins to penetrate at the bound-
ary of the superconductor. The penetration depth depends on the value of the
external field and Jc. Without vortex pinning, Jc = 0, and any external field fully
penetrates the superconductor in the form of moving vortices. With vortex pin-
ning, a gradient of vortex density is maintained by the pinning, and this gradient
defines Jc.
Thus, we have two main assumptions for the Bean model:
1. The electric field E is parallel to the current density J
2. The critical current Jc flows wherever the material is in the critical state
J(x) = ± Jc if |E(x)| 6= 0
J(x) = 0 if |E(x)| = 0
If a superconductor is carrying an AC current, the current distribution at the
peak AC current is the same as it would be for the same value of DC current.
When the transport current or external field producing the shielding currents is
18
large enough, the current ’sheath’ reaches the centre of the superconductor. This
is called ”full penetration.” Losses differ below and above full penetration, and
depend on the direction of the applied fields.
As an example, a superconducting slab in an externally applied magnetic field
is considered in Figure 2.7. A superconducting slab of thickness 2a is oriented in
the y-z plane with an external magnetic field B0 applied in the z direction. The
induced shielding current density Jy flows in the y direction inside the front and
back faces [15]. The Bean model for this scenario for different states is shown
in Figure 2.8. In this figure, B∗ is the characteristic field [15] or full-penetration
field, and is given by B∗ = µ0Jca for a wire of radius a. The model can be applied
in the same way for transport current, and for both a magnetic field and transport
current, the individual solutions for the screening current and transport current
can be superposed.
Kim [32] and Anderson [33] modified this model to allow the current density
in the critical state to vary with the local magnetic field. This is discussed in
detail in the following section describing factors affecting Jc.
Figure 2.7: Superconducting slab in externally applied magnetic field example [15]
19
Figure 2.8: Dependence of the internal magnetic field Bz(x), current density Jy(x),
and pinning force Fp(x) on strength of applied magnetic field B0 for normalised
applied fields given by (a) B0/µ0Jca = 1/2, (b) B0/µ0Jca = 1, and (c) B0/µ0Jca
= 2 using the Bean model [15]
20
2.2.1.2 Superconducting strip model
Brandt [38, 39] realised that while the Bean model was applicable to long super-
conductors in a parallel field where demagnetisation effects were negligible, most
practical experiments used flat superconductors in a perpendicular field, for which
demagnetising effects are crucial. He produced an analysis of superconducting
strips in perpendicular magnetic fields and/or carrying transport currents, and a
summary of the results of the analysis are presented below. The superconducting
strip is a good approximation for the superconducting layer in a superconducting
coated conductors, and these results are often used as a basis to compare the
results of finite element method (FEM) models, which are described later in this
chapter.
For a strip of width 2a (x axis) and thickness d (y axis) in a perpendicular
magnetic field (perpendicular to the tape width) with magnitude H0, Brandt [39]
shows that the flux penetrates from the edges such that
J(x) =
2Jcd
pi
arctan(
√
a2 − b2
b2 − x2 (
x
a
)) for |x| < b
= Jcd for b < |x| < a (2.5)
where b is the pentration depth and is given by
b =
a
cosh(piH0
Jcd
)
(2.6)
The magnetic field strength along the tape is
H(x) = 0 for |x| < b
= Hc arctan(
√
x2 − b2
a2 − b2 (
a
|x|)) for b < |x| < a (2.7)
where Hc is the characteristic field, given by Hc =
Jcd
pi
.
The results are significantly different from that of Bean’s ’slab’ in that the flux
penetration in the Bean model is linear, whereas in the superconducting strip,
21
the flux penetration is initially quadratic. Additionally, the penetrating flux front
has a vertical slope, but in the Bean model it is constant and finite. When the
flux has partly penetrated and a critical state with J = Jc is established near the
edges of the strip, the current flow is over the entire width of the strip to shield
the central flux-free region, but in the Bean model the flux-free region is current
free. The screening current density is a continuous function with a vertical slope
at the flux front where it reaches Jc, but in the Bean model it is a piecewise
constant function.
For a transport current with magnitude I0, we have
J(x) =
2Jcd
pi
arctan(
√
a2 − b2
b2 − x2 ) for |x| < b
= Jcd for b < |x| < a (2.8)
where the total current I0 is
I0 = 2Jcd
√
a2 − b2 (2.9)
and b is the penetration depth, given by
b = a
√
1− (I0
Ic
)2 (2.10)
with critical current Ic = 2aJcd.
The magnetic field strength along the tape is
H(x) = 0 for |x| < b
=
Hcx
|x| arctanh(
√
x2 − b2
a2 − b2 ) for b < |x| < a (2.11)
where Hc is the characteristic field, given by Hc =
Jcd
pi
.
The AC loss calculation for each of these cases (perpendicular applied mag-
netic field and transport current) will be presented in a following section on AC
loss.
22
2.2.2 E-J power law
Bean’s model assumes a step relationship between the current density and elec-
tric field in the superconductor, based on the existence of a well-defined value for
the critical current density as a function of the magnetic field. This is applicable
to LTS and some HTS, for which Bean’s CSM has proven very successful [34].
However, there exist HTS materials where the critical current is ill-defined. An-
derson [33] proposed flux creep theory where this relationship is not discontin-
uous. In this theory, flux moves slowly, due to thermal activation, at currents
lower than the critical current, then an electric field appears and losses occur.
Rhyner [34] proposed the following E-J power law, which is commonly used to
model the non-linear behaviour
E = E0
(
J
J0
)n
(2.12)
This model fits well to the experimental I-V curves for DC for many HTS
materials, and n is a particular characteristic of the HTS dependent on the ma-
terial properties and its microstructure. The extreme cases of n = 1 and n =
∞ correspond to the linear Ohm’s law (E = ρ J) and Bean’s model (J is either
zero or Jc), respectively. n = 5 corresponds to a superconductor with strong flux
creep, n = 15 for weak flux creep, and n = 50 is the limiting value between HTS
and LTS values [37]. For n > 20, it becomes a good approximation of Bean’s
CSM model. The E-J relationship implied by Bean’s model (n = ∞) and the
power-law model are shown in Figure 2.9. Jc is the measured current for a given
electrical field, usually E0 = 10
−4 V/m.
2.2.3 Kim model (magnetic field dependency of Jc)
The magnitude of the critical current density Jc is fixed by the characteristics
of the particular superconductor, and depends on factors such as the type of
material, granularity, twinning, concentration of defect centres, and so on, which
can be referred to as ’internal’ factors [15]. As shown previously in Figure 2.1, the
critical current density Jc is dependent on temperature and magnetic field. The
I-V relationship of some type II superconductors also depends heavily on strain,
23
Figure 2.9: The power-law model from n = 1 (linear) to n = ∞ (Bean’s model)
which occurs when a superconductor is twisted or bent, as in superconducting
cables, motor/generator coils, and so on. These three factors - temperature,
magnetic field and strain - can be referred to as ’external’ factors. Material
scientists try to optimise the internal factors, whilst engineers try to optimise
the external factors. Here we introduce the magnetic field dependency of Jc that
will be used in this dissertation. Therefore, it assumed that the temperature of
the coil remains constant, i.e. the rate of heat production generated by losses
is exactly balanced by the rate of heat removal, no parts of the coil quench for
the currents applied, and that the superconductor properties are not affected by
strain.
Kim [32] and Anderson [33] showed that the critical current in type II super-
conductors exhibit a strong dependence on temperature, as well as local magnetic
field. This empirical relationship is given by
Jc(B) =
α(T)
B0 + B
=
Jc0(T)
1 + B
B0
(2.13)
where B0 is a constant dependent on the material, and Jc0 =
α(T)
B0
. A depen-
24
dence of α with temperature was also proposed:
α =
1
d
(a− bT) (2.14)
where d depends strongly on the physical microstructure of the material and
a
b
≤ Tc. The B-dependency of the critical current density and the power index n
can be written as [40]
Jc(B) =
Jc0
1 + |By|
B0
(2.15)
n(B) =
nc0
1 + |By|
B0
(2.16)
where only the y-component of B (parallel to the c-axis) is considered, as the
contribution to AC loss from a perpendicular field is much greater than that of
a transverse/parallel field [41]. Jc0 and nc0 are the critical current density and
power index in self-field, i.e. when there is no externally applied magnetic field.
B0 is obtained from the Ic-BDC experimental curve. This formulation leads to a
more accurate model of HTS electromagnetic behaviour, particularly for models
where both an applied current and magnetic field exist.
As described in [42] and [43], the Kim model above can be extended using the
factor k and a common denominator B0 to provide an equation for the critical
current density when the superconductor is subjected to a combination of parallel
and perpendicular magnetic field components.
Jc(|Bx|, |By|) = Jc0
1 +
√
k2|Bx|2+|By|2
B0
(2.17)
This was recently extended in [44] and [45] to obtain the angular and field
dependence of the critical current density, i.e. Jc(B, θ).
In this dissertation, a critical current density that is either constant or de-
pendent on the perpendicular magnetic field By is used. A regression can be
performed on the manufacturer-supplied data to find the coefficients k and B0.
For example, for the AMSC tape used in the stator coils of the motor described
previously B0 = 0.23 T, based on the data shown in Figure 2.10. Therefore, the
25
Jc can be described by
Jc(|Bperp|, 77K) = Jc0
1 + |Bperp|
0.23
(2.18)
For SuperPower tape, B0 = 0.12 T, based on the data shown in Figure 2.11.
Therefore, the Jc can be described by
Jc(|Bperp|, 77K) = Jc0
1 + |Bperp|
0.12
(2.19)
Figure 2.10: AMSC data showing critical current change for parallel and perpen-
dicular magnetic fields
26
Figure 2.11: SuperPower data showing critical current change for perpendicular
magnetic fields
2.3 AC Loss in HTS conductors
2.3.1 AC loss in superconductors
At low frequencies, typical of electrical power applications, resistance arises in
ordinary type II superconductors because of flux flow and flux creep. Type II
superconductors are of much more technological interest since they can carry
more current in larger magnetic fields. However, type II superconductors have
losses because electric fields can be produced inside [36].
As mentioned previously, superconductors have to meet several requirements
in order to compete with the presently used normal conductors, including a high
critical current and a low $/kAm price. In addition, the AC loss should be low
enough to justify the extra investment in the superconductor and the cooling
equipment. The decision to utilise superconductors in electrical power devices is
usually based on financial considerations: energy costs, superconducting material
costs, cooling system costs, and maintenance and reliability [46].
27
The AC loss in a superconductor is usually much lower than the resistive loss
in a normal conductor under the same circumstances. However, accurate calcula-
tion and measurement and, if possible, minimisation of the AC loss is technically
important because the AC loss is dissipated as heat in a low temperature envi-
ronment [46]. This dissipation leads to evaporation of the coolant or an increased
thermal load on the refrigeration/cooling system. The heat produced at low
temperatures translates to a much higher ’room temperature’ loss, as described
in Chapter 3. The use of HTS over LTS greatly reduces the problem of heat
removal, due to the higher operating temperature, but does not completely elim-
inate it. In the case of a superconducting machine, the superconductor and the
coil/windings need to be designed such that the size and weight gains realised by
using superconductors are not diminished by the requirement of a large cooling
system [47].
Calculating the AC loss of a superconductor allows for more detailed design
of applications that make use of superconductors, and by comparing experimen-
tal data with theoretical values, the completeness of that theory can be checked.
Each potential electrical power application of superconductivity needs to be eval-
uated separately as different devices see different time-varying currents and/or
magnetic fields. AC losses can generally be split into two categories - magnetisa-
tion loss and transport current loss - depending on the source that provides the
energy [48]. Both of these losses can be present in superconducting applications.
2.3.2 AC loss types
The two types of AC loss in superconductors are described below.
Magnetisation loss Qm is the power dissipated in the superconductor when
an alternating magnetic field, B, is applied to the superconductor. The energy
comes from source of the magnetic field.
Transport current loss Qt is the power that is delivered by the power
supply that enables a transport current, I, to flow through the superconductor.
The voltage, V, along the sample is a measure for the dissipated power.
The total power dissipated is Qtotal = Qm +Qt.
28
2.3.2.1 Magnetisation loss
There are three types of loss that make up the total magnetisation loss: hysteresis
loss, coupling loss and eddy current loss.
Hysteresis loss Hysteresis losses are a result of irreversibility caused by vortex
pinning [36]. These losses are called hysteresis losses because the flux that has
entered the superconductor does not leave precisely in the same manner by which
it entered due to this pinning. If one plots the magnetic induction, B, versus the
magnetic field, H, a hysteresis loop is obtained, which is traversed once per cycle.
The energy loss per cycle is proportional to the area of this loop, provided that
no transport currents are flowing [48]. Such hysteresis losses are dissipated as
heat. The loss becomes higher for stronger pinning; thus, the larger the critical
current of a hard type II superconductor, the larger are the hysteretic losses [35].
An example of a hysteresis (or magnetisation) loop is shown in Figure 2.12.
Figure 2.12: Magnetisation loop for a full cycle of applied field [49]
29
Coupling loss Coupling loss can be a significant problem in multifilamentary
conductors, such as BSCCO (Bi-Sr-Ca-Cu-O), which consists of multiple super-
conducting filaments within a silver sheath [48, 50]. It can also be a problem in
YBCO conductors if the tapes are striated into filaments (striation is discussed
in detail in Chapter 5 when discussing AC loss mitigation). An eddy current
induced by a varying magnetic field, flows partly through the superconductor
and also through the silver between the filaments. When currents flow from one
filament to another, they can couple the filaments together into a single large
magnetic system, which encounters a resistance along the current path through
the silver matrix. This ohmic loss in the metal matrix is often called the coupling
loss [48].
Eddy current loss When an external time-varying magnetic field penetrates
into a normal conductor, it induces a changing electric field, which in turn causes
currents to flow [36]. These are known as eddy currents. Due to eddy currents
in the tape, the ohmic energy dissipation can be significant if the magnetic field
is perpendicular to the tape [36]. At low frequencies the eddy-current loss can
be calculated for many conductor geometries. The basic approach to reducing
the eddy current losses is increasing the effective resistivity of the matrix. For
the calculations in this research, it is assumed (unless otherwise stated) that the
superconductor-related losses are dominant and other losses are ignored.
2.3.2.2 Transport current loss
Hysteresis loss When there is an alternating transport current flowing through
the superconductor, a hysteresis loss (similar to that described above for mag-
netisation loss) occurs. The self-field of the superconductor plays the role of the
applied field, and the energy of the self-field must be supplied by the source of
the transport current [48].
Flux flow loss When the transport current increases, more and more flux
lines are depinned and will move in the superconductor. The energy dissipated
associated with this process is called flux flow loss [48]. Initially, the self-field
30
dominates, but for increasing transport current, the flux flow loss contribution
becomes significant.
2.3.3 AC loss calculation using the CSM
The AC loss in a superconductor can be calculated using different methods; the
basic formulation is based on the Poynting vector (E × H). The energy loss per
unit volume per field cycle in J/cycle/m3 in a volume V enclosed by a surface S
is given by
Q =
1
V
∫ 1/f
0
∫
S
(E×H) · n dS dt (2.20)
Due to the non-linear voltage-current relationship, the magnetic behaviour of
the superconductor is hysteretic, just as in ferromagnetic materials. The mag-
netisation curve encloses an area that represents the magnetisation loss per unit
volume per field cycle. In this case, the loss can be described by an equation
that is derived from the above equation, where M is the magnetisation of the
superconductor.
Q =
∮
B
M · dB (2.21)
The AC loss can also be calculated using an electric method. An electric
field is induced by an applied time-varying magnetic field. A screening current
begins to flow and there is a local non-zero product of voltage and current. The
product E · J is integrated spatially over the conductor cross-sectional area and
with respect to time over the magnetic field cycle yields to give the loss (again
per unit volume per field cycle in J/cycle/m3).
Q =
1
S
∫ 1/f
0
∫
S
E · J dS dt (2.22)
The applied magnetic field is B = µ0 (H + M). If the value of the magnetic
field is taken at a considerable distance from the superconductor, where the in-
fluence of the screening currents in the superconductor is negligible, and B = µ0
H. If the applied magnetic field is larger than the lower critical magnetic field
31
Hc1, then the Meissner state can be ignored.
In order to solve Maxwell’s equations in superconductors with analytical meth-
ods, several assumptions need to be made. The flux penetration in a supercon-
ductor is described by the CSM [31] with the non-linear voltage-current relation
J = Jc
E
|E| if E 6= 0 (2.23)
This is the reason for a superconductor’s electromagnetic behaviour. The
current density induced in the superconductor by an alternating magnetic field is
Jc, the critical current density, irrespective of the value of the electric field. The
direction of the induced current depends on the direction of the last non-zero
electric field. In the special case, E = 0, the current density depends on history.
For example, J = 0 in regions where the electric field has been zero since the
conductor was cooled below Tc.
For an infinite slab (infinite length and height, but finite thickness), the mag-
netic field profile, following from Ampere’s law, is
∇×B = µ0J⇒ dBy
dx
= µ0Jz = µ0Jc (2.24)
where Jc has only a z component. The profile is one-dimensional in the cross-
section because the derivatives in the y and z directions are zero.
Figure 2.13 shows the field profile and current density distribution in a slab
for an applied magnetic field that is ramped up to a value Ba from a virgin
state. In the CSM description, a current density with magnitude Jc starts to flow
in the superconductor and the interior of the slab is shielded from the applied
magnetic field. An applied magnetic field that just penetrates to the centre of
the superconductor is called the ’full penetration field.’ The whole cross-section
is filled with current, from Ampere’s law (Bp = µ0Jcd). A further increase in the
applied field will result in a non-zero field, even in the centre of the slab. The
field profile will remain the same, but will shift upwards. The difference between
the field at the surface of the slab and in the centre remains Bp.
Bean’s model [31] gives the E-J characteristics for the superconductor and
assumes that the magnitude of the current density is a constant Jc. Since E and
32
J are parallel, equation 2.22 can be rewritten:
P =
∫
V
Jc · E dV (2.25)
An example of using Bean’s model to calculate the AC loss of a supercon-
ducting slab is provided in Appendix 1 for reference.
Figure 2.13: Field profile and current density distribution in an infinite slab
exposed to a magnetic field [48]
2.3.4 Analytical techniques
The equations presented here for transport current and magnetisation AC loss
can be used as a basis to check the accuracy of the AC loss calculation using
FEM techniques.
2.3.4.1 Norris
In 1969, Norris proposed an analytical method to estimate the AC loss for a self-
field [51]. This is based on the London model [8] that assumes idealised behaviour.
The resistance is assumed to rise very steeply as the current tries to increase above
33
the critical value and the resistance is such that the ohmic voltage drop exactly
balances the driving emf with the current density remaining constant. It assumed
that the current density is independent of the ambient magnetic field - although
it is well known that the critical current depends not only on the magnitude, but
also the direction of the field.
For a superconducting strip carrying a transport current with magnitude I0,
the transport AC loss can be computed as [51]
Pstrip [W/m] =
µ0I
2
cf
pi
[ (1− I0
Ic
) ln (1− I0
Ic
) + (1 +
I0
Ic
) ln (1 +
I0
Ic
)− (I0
Ic
)2 ]
(2.26)
A simple graph of the change in AC loss for a thin strip of finite width for
different frequencies and ratios of transport current to critical current is shown
in Figure 2.14.
Figure 2.14: AC loss calculation using Norris’s equation for thin strip of finite
width
34
2.3.4.2 Brandt
Brandt introduced a widely used method to compute AC losses in two dimensions,
which is based on solving Maxwell’s equations together with the E-J power law to
calculate the AC loss for a superconducting strip for a transport current and/or
a perpendicular external magnetic field [38, 39].
The equations for the current and magnetic field distributions were presented
earlier in this chapter (see Superconducting strip model). For a strip of width
2a and thickness d in a perpendicular magnetic field with magnitude H0, the
previous equations can be used to compute the magnetisation loss:
Pmag [W/m] = 4piµ0a
2H0Hcf [
2Hc
H0
ln cosh(
H0
Hc
)− tanh(H0
Hc
) ] (2.27)
For a transport current with magnitude I0, the previous equations can be used
to compute the transport loss:
Ptrans [W/m] =
µ0I
2
cf
pi
[ (1− I0
Ic
) ln(1− I0
Ic
) + (1 +
I0
Ic
) ln(1 +
I0
Ic
)− (I0
Ic
)2 ] (2.28)
which is the same as the Norris equation stated above.
35
Chapter 3
Modelling HTS-based
superconducting coils
In this chapter, the modelling of HTS-based superconducting coils using the finite
element method is described in detail, including the evolution of the development
of the coil model as the research in this dissertation has progressed. Often there
is a compromise in computer modelling between the accuracy of the solution and
the computational time required, and a number of different models are compared
to examine the optimum parameters. Firstly, the numerical model used in this
thesis is described, which is based on solving a set of Maxwell’s equations in 2D
implementing the H formulation using the commercial software package Comsol
Multiphysics. The coil cross-section is modelled as the number of individual turns
in the coil, and an artificial expansion technique is investigated to improve the
computational speed of the model, which can require hundreds of thousands of
mesh elements. Different methods to approximate the critical current density
Jc are also discussed. A technique that allows the real superconducting layer
thickness to be modelled, using a mapped mesh, is then investigated, and the model
is modified to include the magnetic substrate present in some superconducting
tapes. This investigation raises some interesting points for further analysis, and
a detailed investigation on stacks of superconducting tapes with both weak and
strong magnetic substrates is carried out at the end of the chapter.
36
3.1 Finite element method (FEM) models
Several numerical methods have been proposed to solve the critical state in super-
conductors, and analytical methods have been developed for simple geometries
and uniform field conditions, as described in the previous chapter. For more
complicated shapes and field conditions, numerical methods must be developed.
These numerical methods usually make use of the finite element method (FEM)
or the finite difference method to solve Maxwell’s equations, coupled with the
E-J power law, in 2D or 3D, and FEM is a popular technique for solving par-
tial differential equations (PDEs). These methods can be classified by the main
equations used: the A-V (based on the magnetic vector potential A) [37,52–54],
T-Ω (based on the current vector potential T) [55], E (based on the electric
field E) [56–58], and H (based on directly solving the magnetic field compo-
nents) [29, 59–61] formulations. Maxwell’s equations can be written in each of
these formulations and these formulations are equivalent in principle, but the
solutions of the corresponding PDEs can be very different [62].
3.1.1 The H formulation
The numerical model used in this thesis for modelling the electromagnetic be-
haviour of HTS is based on solving the set of Maxwell’s equations in 2D im-
plementing the H formulation using the software package Comsol Multiphysics,
version 3.5a. Methods based on the H formulation can converge more easily than
other methods and it is easy to impose boundary conditions related to the current
flowing in the superconductor(s) and/or externally applied magnetic fields.
The space is typically divided into two subdomains: the superconducting
region(s) and air. The addition of magnetic materials, e.g. the magnetic substrate
found in certain coated conductors, will be discussed in a later section. A set of
PDEs, sharing the same dependent variables, is defined in each subdomain. By
assuming that the constitutive law B = µ0 H is applicable in both the air and
superconducting regions, the relevant Maxwell’s equations are
∇× E = −dB
dt
= −µ0dH
dt
(3.1)
37
∇×H = J (3.2)
The E-J behaviour of the superconducting material is modelled assuming
• The electric field E is always parallel to the current density J
• The power-law relationship: E = E0
(
J
Jc(B)
)n−1
J
Jc
where E0 denotes the
threshold electric field used to define the critical current density Jc, usually
10−4 V/m
Using the power-law model is more suitable than the original Bean model
where n → ∞, and a voltage criterion E0 = 1 µV/cm and constant n = 21 is
assumed, which are typical values for melt-processed YBCO [63].
3.1.1.1 H formulation in cartesian coordinates
In this two dimensional model, the space is assumed to be infinitely long in the
z direction and the sample is assumed to consist of an infinitely long tape of
rectangular cross section w × d. The magnetic flux lies in the xy plane, and the
current density J flows in the z direction only. A visual description of the model
is shown in Figure 3.1.
Thus, Maxwell’s equations become
Jz =
dHy
dx
− dHx
dy
(3.3)
−dEz
dx
= −µ0dHy
dt
(3.4)
dEz
dy
= −µ0dHx
dt
(3.5)
Using these two PDEs defined above with the two dependent variables Hx and
Hy, and applying suitable boundary conditions, Comsol can be used to solve the
problem. The subdomain settings for ’PDE, General Form (g)’ in Comsol, which
is appropriate for non-linear PDEs, follows the following convention:
ea
d2u
dt2
+ da
du
dt
+▽ · Γ = F (3.6)
38
Figure 3.1: FEM model of a high temperature superconductor using the H for-
mulation
where u =
[
Hx
Hy
]
Since u is a vector of dependent variables, ea is a matrix and is known as the
mass coefficient. da is the damping coefficient and F is the source term. Γ is
known as the flux vector. The flux vector Γ, and ea, da and F, can be functions
of the spatial coordinates, the solution u, and the space and time derivatives of
u.
Combining this with the previous analysis of Maxwell’s equations, we find the
subdomain settings to be used in Comsol:
[
0 0
0 0
]
d2u
dt2
+
[
µ0 0
0 µ0
]
du
dt
+▽ ·
[
0 Ez
−Ez 0
]
=
[
0
0
]
A Dirichlet boundary condition is used at infinity, which corresponds to the
simplest case where transport current flows and there is no externally applied
magnetic field. The continuity equation n × (H1 −H2) = 0 is used at the
boundary between air and the superconductor. This is implemented in Comsol
by applying Neumann boundary conditions at the surface of the superconductor.
39
This implies that the tangential components of the magnetic field intensity are
conserved at either side of the material interface. An alternative way of expressing
this is
dHt1
dn
= 0,
dHt2
dn
= 0 (3.7)
where Ht1 and Ht2 are the tangential components of the magnetic field at the
surface of materials 1 and 2, respectively.
The PDEs are solved using Comsol, subject to these boundary conditions.
In order to set up these equations in Comsol, scalar expressions must be
defined. Firstly, Jz is defined by the expression ”d(Hy,x) - d(Hx,y)”. ρsc is defined
as ”ρsc =
E0
Jc
( |J|
Jc
)n−1”. The electric field Ez is defined for the superconductor
subdomain by the (subdomain) expression ρsc · Jz. Similarly, Ez is defined for the
air subdomain by the (subdomain) expression ρair · Jz where ρair is defined as a
constant (e.g. 2× 1014 Ωm).
A constraint must be placed on the current flowing in the superconductor
and this is done by defining a Subdomain Integration Variable Iint to ensure the
current flow is restricted to the superconductor subdomain. The current applied
can be either DC or AC and can be modified as described above for an applied
magnetic field. Thus, Iint =
∫ ∫
Jz dxdy =
∫
s
Jz ds where s is the cross-section
of the superconductor. This is then used as either a boundary or point setting
to constrain current flow to the superconductor; for example, Iint,subdomain = Iapp.
The current can be defined as a scalar expression as above (e.g. I0 sin(2pift) or
I0(1− e(− tτ ))) or defined explicitly in the boundary/point setting.
An external magnetic field can be applied to the tape by modification of
the Dirichlet boundary condition described in the previous section. For a trans-
verse applied magnetic field of constant magnitude, i.e. a DC field, the Dirichlet
boundary condition is modified such that Hx = Hext(1− e(−t/τ)) where Hext is the
magnitude of the field and the time constant τ is, for example, 0.02 s. Although
this function is clearly time-dependent, a step function may not be applied be-
cause of the boundary condition due to the initial conditions of the model (Hx
= Hy = 0) and that to apply a DC field instantaneously is non-physical. A
ramp function is also appropriate. For a parallel applied magnetic field of con-
40
stant magnitude, the Dirichlet boundary condition is modified for the y axis, i.e.
Hy = Hext(1− e(−t/τ)). An alternating magnetic field can be applied by modifying
the (1− e(−t/τ)) term to sin(2pift) where f is the frequency in Hertz.
Using the model described, the electromagnetic characteristics of a single su-
perconducting tape with an applied transport current and/or applied magnetic
field can be analysed for both constant and time-varying conditions.
3.2 Existing coil models
The cross-section of superconducting cables and coils is often modelled as a two-
dimensional stack of coated conductors, and these stacks can be used to estimate
the AC loss of a practical device. There are a number of examples in the literature
that, based on different assumptions and techniques, investigate different aspects
of stack problems, such as magnetisation and transport AC loss. A number of
techniques are based on the critical state model: [64, 65] use an analytical tech-
nique using the infinite1 stack approximation, [66–69] use a numerical technique
for an arbitrary stack, and [42, 70, 71] are based on variational formulations1.
To model such a stack in this research, the FEM model based on the H
formulation introduced previously is extended to allow for a multiple number of
tapes interacting together. Comparisons with experimental measurements of AC
loss in single and small numbers of tapes have shown the H formulation to be
accurate in predicting losses for simple geometries [62, 72, 73].
In the following sections of this chapter, different aspects of the coil model
are investigated, in order to produce an optimised model in terms of accuracy
of solution and computational speed. Firstly, the coil under examination for the
majority of this dissertation is introduced, then an investigation is carried out
on the artificial expansion of the thickness of the YBCO layer and selecting an
appropriate mesh type and number of elements. This is done using a single tape,
as the preliminary groundwork for optimising more complex geometries. Next,
the artificial expansion technique is applied to a coil geometry to model the in-
dividual turns of the coil, and the result is compared with a model using a bulk
1The term ’infinite’ here refers to the height of the stack, i.e. the number of coated con-
ductors in the stack.
41
approximation. A technique is then applied that allows the actual superconduct-
ing layer thickness to be modelled without the associated problem of increased
computation time due to a large number of mesh elements, and the model is
modified to allow the inclusion of a magnetic substrate. This analysis raises a
number of interesting points regarding the use of superconductors with magnetic
substrates, and a comprehensive analysis of stacks of tapes with weak and strong
magnetic substrates is provided at the end of this chapter, using a symmetric
model that requires only one quarter of the cross-section to be modelled.
3.3 Test superconducting coil for modelling
The superconducting coil under examination for the majority of this disserta-
tion is found on the stator of the all-superconducting HTS permanent magnet
synchronous motor (PMSM) designed by the EPEC Superconductivity Group at
Cambridge [41, 74–81].
The design of the motor is detailed in [76] and [77], and a photograph of the
completed test rig is shown in Figure 3.2. The rotor is made of 75 supercon-
ducting pucks, arranged in 15 columns of 5 pucks, which can be magnetised to
the equivalent of a four-pole permanent magnet. The stator consists of six HTS
armature windings, which are installed in slots made of a non-magnetic insulating
material. The entire HTS motor is to be cooled by liquid nitrogen to 77 K.
The armature winding in the HTS motor is made of six single flat-loop coils,
which are wound as flat racetrack pancake coils with a bend radius of several
centimeters. Double layers of HTS windings are stacked together to make one
racetrack winding. This maximizes the inductance of the winding within the
limited geometry [75]. The total number of turns per phase for the stator windings
is 200 turns (with 50 turns for each layer, 100 turns per coil, and two coils
per phase). Approximately 60 m of American Superconductor [26] type 344
2G coated conductor was used to wind each coil. The properties of the AMSC
YBCO 344 tape used in the stator coils are listed in Table 3.1. A photograph
of a manufactured coil is shown in Figure 3.3 and the dimensions are shown in
Figure 3.4.
The critical current of a YBCO tape sample and the whole coil were measured
42
by the DC pulse current measurement technique [76], and the results for the
sample and coil are shown in Figure 3.5. The critical current criteria used is
based on an electric field E0 = 1 µV/cm. For the YBCO tape sample, the length
between the voltage taps is 2 cm, so the measured critical current is 106 A. On
the other hand, the total length of the HTS coil is 60 m, so the measured critical
current of the coil is 51 A.
The decrease of the critical current is due to several factors. The first is the
self magnetic field generated by the HTS coil. Additionally, bending the tape
to construct the coil can degrade the tape’s performance. The dependence of
the critical current on the presence of a magnetic field will be discussed in detail
in the following chapter. Furthermore, the HTS tape used to wind the coil was
provided in 20 m lengths, while the whole length of the coil is 60 m. This leads
to at least two wire joints within the coil, which reduces its critical current due
to the finite resistance of these connections. When the critical current of the coil
was measured, there was no external magnetic field applied, so only the self-field
was present. There will also be a reduction in critical current due to the magnetic
field from the pucks on the rotor when the motor itself is run.
Table 3.1: AMSC YBCO 344 tape properties [76]
Average thickness 0.20 mm ± 0.02 mm
Width 4.35 mm ± 0.05 mm
Maximum width (bare) 4.4 mm
Minimum double bend diameter (RT) 30 mm*
Maximum rated tensile stress (RT) 150 MPa*
Maximum rated tensile strain (77 K) 0.3 %*
Maximum rated compressive strain (77 K) 0.3 %*
Length of single tape 20 m
Critical current at 77 K, self-field > 60 A
*95% Ic retention
43
Figure 3.2: Photo of the HTS PM synchronous motor rotor test rig [82]
44
Figure 3.3: Photograph of an HTS PM synchronous motor stator coils
Figure 3.4: Dimensions of an HTS PM synchronous motor stator coil
45
Figure 3.5: Critical current of the YBCO sample tape and coil [76]
46
3.4 Artificial expansion technique for individual
tapes
The geometry of the model is shown previously in Figure 3.1 and the specification
of the superconductor under analysis is listed in Table 3.1. The specification is
based on the material used in the superconducting racetrack coil shown in Figure
3.3. The critical current density Jc is given by
Jc =
Ic
w · dsc (3.8)
where Ic is the critical current of the tape measured experimentally, and w
and dsc are the width and thickness of the superconducting layer, respectively.
One simple way to model a coated conductor for a stack of tapes or coil
geometry is to approximate the geometry of the superconducting layer as the
whole tape thickness, i.e. 200 µm [66, 67]. Artificially increasing the supercon-
ducting layer can improve computational speed by reducing the number of mesh
elements required, making it easier to model more complex 2D geometries, but
the accuracy of results must not be compromised. The artificial expansion tech-
nique works because the dynamics of the flux penetration for this geometry is
essentially along only one axis. For a thin strip, which describes the layer of
superconducting material in a coated conductor, the flux front moves along the
x-axis in this geometry (see Figure 3.1), from the edge of the tape towards the
centre, for increasing values of current [38]. As long as a significantly large aspect
ratio is maintained, the coated conductor will still behave like an infinitely thin
strip, with the current distribution front moving along the x-axis. It will become
apparent where the model’s behaviour begins to deviate significantly from that
of an infinitely thin strip.
200 µm is assumed as the maximum thickness here, which is equal to the whole
conductor thickness, and the thickness is varied from this to 2 µm, where 1-2 µm is
the approximate real thickness of the YBCO layer. Keeping the width of the tape
constant (4.35mm) and the critical current Ic as 100 A, the critical current density
must be modified according to the thickness used. For a thickness of 200 µm, the
critical current density Jc is 1.15× 108 A/m2, which is known as the engineering
47
critical current density Je. For a thickness of 2 µm, Jc is 1.15× 1010 A/m2.
In order to simplify the analysis and allow comparison with Norris’s analyt-
ical equation introduced in the previous chapter, the critical current density is
assumed to be independent of applied magnetic field, i.e. Jc is constant. In real-
ity, the critical current depends not only on the magnitude, but also the direction
of the field, which includes the superconductor’s self field. Field dependence of
the critical current will be investigated later in this chapter. Furthermore, in
this analysis, only the superconductor layer is modelled and the presence of a
substrate (and its associated loss) is ignored. This, too, will be investigated later
in this chapter. Here we are concerned only with the hysteretic superconductor
loss, and the material surrounding the superconducting layer is represented by
air, i.e. a relative permeability of 1 and very low electrical conductivity. This
investigation is based on an analysis of transport AC loss only and this requires
appropriate settings for the boundary conditions. Since no externally applied
magnetic field exists, the boundary settings are Hx = Hy = 0 for a sufficiently
large surrounding air subdomain.
The computational effort required by the solver and the accuracy of the so-
lution is highly dependent on the selection of the mesh and its elements. The
first model (model A) in this section uses a triangular mesh, and the number of
the elements in the mesh is the smallest number possible, maintaining the same
aspect ratio for the elements and such that the mesh remains symmetric. The
motivation for using a small number of mesh elements is that complex geometries
will be easier to model, as modelling multiple turns in a superconducting coil
will require hundreds of thousands of mesh elements. This is done in Comsol by
setting the maximum element size of the mesh elements to be the same size as the
thickness of the superconducting layer to be meshed via the subdomain free mesh
parameters, i.e. for a superconducting layer thickness of 50 µm, the maximum
element size is 50 µm. The second model (model B) uses a single layer, square
mesh, which is achieved using the same setting. The meshes for the two models
are shown in Figure 3.6.
It should be noted that the number of mesh elements within the superconduc-
tor is inversely proportional to the thickness of the tape in order to maintain the
same aspect ratio for elements within the mesh. The number of mesh elements
48
Figure 3.6: Mesh element for models A (triangular) and B (square)
within the superconductor subdomain for different thicknesses for the two models
is shown in Figure 3.7, where model A is represented by triangles and model B
is represented by squares. Furthermore, the number of mesh elements in the air
subdomain for each model is identical for a particular thickness.
In order to make a comparison between the two different meshes and different
element types, the following terminology is used hereafter when describing the
different models:
• A1 = Triangular minimum symmetric, first-order Lagrange elements
• A2 = Triangular minimum symmetric, second-order Lagrange elements
• AE = Triangular minimum symmetric, edge elements
• B1 = Square single layer, first-order Lagrange elements
• B2 = Square single layer, second-order Lagrange elements
To implement edge elements, Comsol’s ”shvec” type was used. Comsol im-
plements two formulations for vector, or edge elements, known as ”shvec” and
”shcurl.” ”shvec” elements are only implemented for first-order triangular ele-
ments, whereas ”shcurl” elements are implemented for all element types within
Comsol. When this analysis was carried out using the same model settings as
49
Figure 3.7: Number of mesh elements for different YBCO layer thicknesses for
models A (represented by triangles) and B (represented by squares)
the other models, the ”shcurl” element either failed to converge to a solution,
or if it did, the convergence time was unacceptably long and resulted in non-
physical, spurious solutions. On the other hand, for first-order triangular ”shvec”
elements, the results were in good agreement with theory and also with the mod-
els where Lagrange elements were used. Hence, a mesh comprising square single
layer, edge elements has been omitted from the analysis. How to overcome this
problem is discussed in a following section on modelling the real thickness of the
superconducting layer.
3.4.1 Solver time and convergence comparison
One of the critical parameters for modelling coated conductors with high aspect
ratios is the computation time required to solve them. Figure 3.8 shows a compar-
ison of the computation time for models A1, A2 and AE for different I0/Ic ratios
(0.25, 0.5, 0.75 and 0.9) as the thickness of the tape is varied between 2 and 200
µm, where I0 is the amplitude of the input current. Figure 3.9 shows a com-
50
Figure 3.8: Triangular mesh solver times
parison of the computation time for models B1 and B2 for the same conditions.
Figures 3.8 and 3.9 show that the computation time increases as the thickness of
the superconductor decreases. The time to solve is an order of magnitude higher
when second-order elements are used instead of first-order elements for models A
and B, and the solver time for triangular edge elements is slightly longer than for
first-order elements.
When comparing the triangular mesh with the square mesh for the same
order of elements, the triangular mesh takes a few (approximately two to three)
times longer to solve. As shown previously in Figure 3.7, the number of mesh
elements in model B is four times less than model A. The computation time
cannot be estimated by the number of elements in the mesh alone, which in this
case would give the expectation that the square mesh (with less mesh elements)
would produce a faster solution.
A better measure of computation time is the number of degrees of freedom
51
Figure 3.9: Square mesh solver times
the model has [62], which is shown in Figure 3.10 - triangles represent Lagrange
elements in model A and squares represent model B. Edge elements (model AE)
are represented by diamonds. The double solid line represents second-order La-
grange elements and the double dashed line represents first-order Lagrange el-
ements. The number of degrees of freedom is related to both the number and
type of mesh elements used (first-order, second-order, and so on). The degrees
of freedom are comparable when using the same order of Lagrange elements, and
there are about four times as many degrees of freedom for second-order elements
as first-order. The number of degrees of freedom for edge elements is in between
the two. The solver times in Figures 3.8 and 3.9 correlate well with the degrees
of freedom in Figure 3.10.
A time-dependent solver is used to solve each model using Comsol’s default lin-
ear system solver (UMFPACK). The default time-stepping method uses variable-
order, variable-step-size backward differentiation formulae, and the time steps are
automatically selected by the solver. To give a quantitative analysis of the con-
52
Figure 3.10: Total number of degrees of freedom for different YBCO layer thick-
nesses for all models. Triangles represent triangular meshes (model A), squares
represent square meshes (model B), and diamonds represent edge elements (model
AE). The double solid line represents second-order Lagrange elements and the
double dashed line represents first-order Lagrange elements.
vergence of each model, Figure 3.11 shows the solver time divided by the degrees
of freedom. The second-order models require longer time steps than first-order,
meaning second-order models have worse convergence behaviour. The conver-
gence for the edge element model lies in between these two. When less current
is applied to the superconductor (e.g. 0.5 Ic instead of 0.9 Ic), the convergence
behaviour improves as the current front does not penetrate as far into the super-
conductor.
Here we have examined the model properties to optimise the computational
speed and convergence based on the type of elements and level of discretisation. In
the following section, the real quantity of interest, i.e. the AC loss, is calculated,
and the accuracy is compared for the different models.
53
Figure 3.11: Convergence behaviour for all models using solver time per degree
of freedom
3.4.2 AC loss comparison
The AC loss per unit length per cycle is calculated using the following equation
Q =
∫ 1/f
0
∫
S
E · J dS dt (3.9)
where 1/f is the period of the AC current of frequency f, S is the superconduc-
tor cross-section, and J and E are the critical current density and electric field
at each mesh node, respectively. This gives a loss with units of J/cycle/m.
Figure 3.12 shows the values calculated for the AC loss for all models for
YBCO layer thicknesses between 2 and 200 µm. The calculated AC loss in each
of the models is compared with the analytical model proposed by Norris, which
was introduced in the previous chapter. The analytical equation used for the
comparison is Equation 2.26 divided by the frequency f to give units of J/cycle/m.
For I0/Ic values between approximately 0.5 and 0.9, the calculated AC loss
value is similar to the analytical result for models A2, AE and B2. There is no
54
Figure 3.12: AC loss calculation for all models for different YBCO layer thickness
compared with Norris’s analytical model (clockwise from top left: I/Ic = 0.25,
I/Ic = 0.5, I/Ic = 0.75, I/Ic = 0.9)
55
significant variation for these currents over the whole range of thicknesses. For
I0/Ic = 0.25, there is a significant variation between the analytical result and the
finite element model, particularly when the YBCO layer thickness is large (−→
200 µm), where there is also variation between models A2, AE and B2. Similar
findings were reported in [83], but should not be a major concern as in practical
situations, the ratio I0/Ic should be maximised, i.e. −→ 1, in order to fully utilise
the superconductor’s properties, and the region of I0/Ic between 0.5 and 1 is of
primary importance.
For models A1 and B1, the AC loss is significantly lower than the analytical
model and deviates significantly as the thickness is increased. Therefore, the
first-order models do not produce results which can be relied upon to provide an
accurate estimation of AC loss. For these models, the current density profile does
not accurately represent the current flow as it should, and as a result, the electric
field profile is significantly distorted. For problems involving superconductors,
the errors in the current density profile should be minimised as the power-law
relationship with electric field creates a much larger error when calculating the
electric field and AC loss.
In Figure 3.13, the current density distribution within the superconductor is
shown for both models for a transport current I0 = 0.5 Ic for YBCO layer thick-
nesses of 2 and 20 µm. The average values of the current density for different
points along the tape width compares well with those calculated by the analyti-
cal model, although the current density distribution for triangular second-order
elements shows an increase between 2 and 20 µm thicknesses.
In terms of computational speed and accuracy of solution, model AE (edge
elements) performs the best of the models. Using first-order elements, whilst
faster, does not produce accurate results, and using second-order elements, whilst
producing accurate results, takes significantly longer to solve.
In terms of the artificial thickness expansion, the results show that when cal-
culating AC loss, the current density profile should not be significantly modified
by the expansion, such that any change in J affects the AC loss calculation. The
electric field is calculated using a power-law relationship, so errors in J are ampli-
fied. For the values of current of interest (I0/Ic between 0.5 and 1), the thickness
may be expanded up to 20-30 µm before the result begins to deviate appreciably.
56
Figure 3.13: Comparison of current density distribution in superconductor with
Norris’s analytical model for models A2, AE and B2 for I/Ic = 0.5 for 2 and 20
µm YBCO layer thicknesses
The ability to artificially expand the YBCO layer thickness will become even
more important when modelling complex device geometries, such as wound coils
or devices themselves. In the following section, the artificial expansion is applied
to a coil geometry to model the individual turns of the coil, and the result is
compared with the bulk approximation model.
3.5 Artificial expansion vs. bulk approximation
As described above, one method to improve the speed and convergence of the
model is to artificially increase the superconductor layer thickness. In this section
we model the cross-section of the racetrack coil using two models: 1) individual
turns using the artificial expansion technique, and 2) using a bulk approximation.
For the model using individual turns, a thickness of 50 µm is chosen, for which the
critical current density is Jc = 4.6× 108 A/m2. Although an expanded thickness
57
of 20-30 µm is ideal based on the conclusions from the preceding analysis of a
single tape, this would result in a number of mesh elements on the order of 2-
300,000, and an even greater number of degrees of freedom, which exceeds the
memory capacity of the computer being used. Even the 50 µm model with a
minimum symmetric mesh within the superconductor and a free triangular mesh
elsewhere results in over 100,000 mesh elements with around 175,000 degrees of
freedom.
Each layer is separated by 200 µm to account for the other layers of the su-
perconductor. For each superconducting layer, the minimum possible symmetric
mesh is used along with edge elements. For the bulk approximation, the same
critical current density is used, but for a geometry of one turn with thickness
n times 50 µm, effectively removing the non-superconducting area between the
tapes in the individual turns model. Hence, this approximation assumes that the
tapes couple electromagnetically such that the multiple tapes behave as a finite
superconducting slab carrying n times the current of each individual tape. The
geometry and mesh for the individual tapes and bulk models are shown in Figure
3.14 and 3.15, respectively.
For these two models, two critical current densities are compared: the constant
Jc value given above, and a B-dependent Jc, which is given by Equation 2.18.
Figure 3.16 shows the calculated AC loss for the four different cases: 1) individual
turns using a constant Jc, 2) individual turns using the Jc(B) dependence, 3) the
bulk approximation using a constant Jc, and 4) the bulk approximation using the
Jc(B) dependence.
The Jc(B) dependence increases the calculated AC loss, as the reduced Jc due
to the magnetic field results in further penetration into the stack. The calcu-
lated AC loss is also increased for the bulk approximation, but for a different
reason: the bulk approximation assumes that the individual turns are perfectly
electromagnetically coupled, but in reality the coupling between the turns is not
so simple. Thus, the bulk approximation will tend to overestimate the calcu-
lated loss. The individual turns model is more accurate, as long as the expanded
thickness remains within the limit specified earlier. The accuracy gained in com-
parison with a bulk approximation model is better than any time lost through a
small increase in computation time.
58
Figure 3.14: Geometry and mesh for model of individual turns using artificial
expansion technique
59
Figure 3.15: Geometry and mesh for model using bulk approximation
Figure 3.16: AC loss calculation for individual turns and bulk approximation for
constant Jc and Jc(B) dependence
60
3.6 Real thickness
It has been shown that the artificial expansion technique is a valid method to
improve the speed and convergence of the superconductor model. The number
of mesh elements can be reduced without significant loss of accuracy, provided
that the thickness of the superconducting layer is not artificially increased past
a certain limit (20-30 µm). However, a significant improvement was made in
[84, 85] by using large aspect ratio mapped meshes, hence reducing the number
of mesh elements in and between the coated conductors. A sparser mesh is used
between the superconducting layers, where it is not as critical to have a fine
mesh. Applying the findings of [84, 85], an overall reduction of about two orders
of magnitude in the number of elements can be achieved for the coil model in
comparison with the previous simulations, which used free meshes. When a free
mesh is created in Comsol, the number of mesh elements is determined from the
shape of the geometry and various mesh parameters, such as maximum element
size, element growth rate, mesh curvature factor, and so on. There are also
a number of predefined mesh sizes, ranging from extremely fine to extremely
coarse.
Accordingly, by using a mapped mesh, the actual thickness of the YBCO
layer can be modelled without a significant increase in computational time, also
allowing other layers of the coated conductor, such as the stabiliser layer and sub-
strate, to be implemented using accurate geometrical dimensions. In this section,
this technique is applied to model the coil geometry and the exact number of
elements required in the superconducting layer is investigated to strike a balance
between accuracy and computational speed. The magnetic substrate present in
the superconducting coil under investigation is added to the model, which re-
quires modification of the PDEs defined previously, which are only valid when
the relative permeability of the materials involved is µr = 1.
3.6.1 Optimal number of mesh elements
Previously it was mentioned that when using edge elements (Nedelec elements
implemented using the ”shcurl” element in Comsol) with square or rectangular
mesh elements, the model either failed to converge to a solution, or if it did, the
61
convergence time was unacceptably long and resulted in non-physical, spurious
solutions. To overcome this problem, which was due to the tolerance settings
of the solver, the relative and absolute tolerance settings of the solver are set
to lower values of 1e-5 and 1e-7, respectively. Previously, these settings were 0
and 1, which was appropriate in producing a fast and accurate solution for those
particular mesh/element types.
The relative tolerance specifies the largest acceptable solver error, relative to
the size of each state during each time step. If the relative error exceeds this
tolerance, the solver reduces the time step size. The acceptable error at each
time step is a function of both the relative and absolute tolerances. During each
time step, the solver computes the state values at the end of the step and also
determines the local error, which is the estimated error of these state values. If
the error is greater than the acceptable error for any state, the solver reduces the
step size and tries again.
The absolute tolerance specifies the largest acceptable solver error, as the
value of the measured state approaches zero. If the absolute error exceeds this
tolerance, the solver reduces the time step size. If the absolute tolerance is too
low, the solver might take too many steps around near-zero state values, and
therefore slow the simulation. If the simulation results do not seem accurate, and
the model has states whose values approach zero, the absolute tolerance may be
too large. Reducing the absolute tolerance forces the simulation to take more
steps around near-zero state values.
By lowering the relative and absolute tolerances as described above, the sim-
ulation performs much better, and fortunately this does not hinder the conver-
gence. Thus, square or rectangular edge elements can be utilised and the problems
described earlier can be avoided.
Figures 3.17 to 3.21 show a comparison of the AC loss calculation for a single
tape with mapped square/rectangular edge elements for I = 0.1, 0.3, 0.5, 0.7 and
0.9 Ic. The number of x and y elements correspond to the number of mapped ele-
ments along the x and y axes, respectively. As the base case for the comparison, a
very fine mesh of 2000 elements along the x-axis and 20 elements along the y-axis
is used, which takes a significantly long time to solve, but produces the most
accurate result. It can be seen that a mesh with as little as 100 elements along
62
the x-axis and a few elements along the y-axis can produce an accurate result,
except for small magnitudes of current, which is similar to the previous finding
with the artificial expansion technique. This is due to the nature of the flux
penetration for small currents/fields where the top/bottom losses are not domi-
nated by the edge losses, i.e. the perpendicular component of the field. Based on
these results, a mesh consisting of 100 elements along the x-axis and 4 elements
along the y-axis will be used in the superconducting layer for the coil model that
follows. However, for small currents, say 0.3 Ic and lower, the number of mesh el-
ements should be increased, in particular the number of elements along the y-axis.
Figure 3.17: Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.1 Ic
63
Figure 3.18: Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.3 Ic
Figure 3.19: Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.5 Ic
64
Figure 3.20: Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.7 Ic
Figure 3.21: Comparison of AC loss calculation for single tape with mapped
square/rectangular edge elements, I = 0.9 Ic
65
3.6.2 Addition of magnetic substrate
The modification of the equations used previously to allow the addition of the
magnetic substrate into the model is outlined below.
∇× E = −dB
dt
= −d(µ0µrH)
dt
(3.10)
∇×H = J (3.11)
−dEz
dx
= −µ0
(
dµr(H)
dt
Hy + µr(H)
dHy
dt
)
(3.12)
dEz
dy
= −µ0
(
dµr(H)
dt
Hx + µr(H)
dHx
dt
)
(3.13)
Therefore, the subdomain settings to be used in Comsol become
[
0 0
0 0
]
d2u
dt2
+
[
µrµ0 0
0 µrµ0
]
du
dt
+▽ ·
[
0 Ez
−Ez 0
]
=
[
−µ0 dµrdt Hx
−µ0 dµrdt Hy
]
which is solved iteratively by the solver.
To include the field dependence of the relative magnetic substrate of the Ni-
W ferromagnetic substrate of the RABiTS YBCO coated conductor, the fitting
function presented in [72] is used, which is based on the experimental results
published in [86]. The function is represented by [72]
µ(H) = 1 + 30600
(
1− exp(−( H
295
)2.5)
)
H−0.81 + 45exp
(
−( H
120
)2.5
)
(3.14)
where H is the amplitude of the magnetic field strength H =
√
H2x +H
2
y.
The superconductor hysteretic loss Qsc is calculated using the critical cur-
rent density and the electric field distribution across the cross-section of the
YBCO layer. For the model including the magnetic substrate, the additional
ferromagnetic loss Qfe is calculated using the fitting function presented in [72],
which is again based on the experimental results published in [86]. The loss,
in J/cycle/m−3, is calculated based on the maximum value of the magnetic flux
66
density seen by the substrate and is represented by [72]
Qfe(Bmax) =
{
4611.4B1.884max for Bmax ≤ 0.164
210(1− exp(−6.5Bmax)4) for Bmax > 0.164
(3.15)
3.6.3 Modelling results
The geometry of the model and its mesh is shown in Figure 3.22. Figure 3.23
shows the calculated AC loss for the following four cases:
1. Individual turns, without magnetic substrate, constant Jc
2. Individual turns, without magnetic substrate, Jc(B) (equation introduced
earlier)
3. Individual turns, with magnetic substrate, constant Jc
4. Individual turns, with magnetic substrate, Jc(B)
The substrate loss is indicated by the dotted lines. The Jc(B) dependence
increases the AC loss as the magnetic flux front penetrates further into the stack
of tapes along the y axis due to the reduced Jc from the edges inwards where the
magnetic field is strongest. The inclusion of the magnetic substrate in the model
also increases the AC loss, but not necessarily due to the ferromagnetic substrate
loss itself, and the reason for the loss increase is discussed in more detail below.
For the range of current investigated (5-50 A), the substrate is saturated (or
very close to saturation), and the loss in the superconductor layer is larger than
the substrate loss by an order of magnitude or higher for current in excess of
about 20 A (20% of the tape Ic, 40% of the coil Ic). In this range, the effect of
the substrate loss itself on the overall AC loss is minimal and can be neglected;
however, for low current, the substrate loss cannot be neglected, and can indeed
exceed the superconductor hysteretic loss for a suitably low current. Additionally,
the total AC loss for the model with the magnetic substrate and B-dependent Jc
can be seen to be the sum of the AC loss calculated in the constant Jc/substrate
and Jc(B)/no substrate models. It is also interesting to note that the results
67
shown in Figure 3.16 are in fact higher than when the actual thickness is used,
which is consistent with the results from the artificial expansion analysis.
Figures 3.24 and 3.25 show the magnetic flux density profiles of the magnetic
field perpendicular to the tape faces |By|, which has the greatest impact on the
AC loss due to the large aspect ratio of the coated conductor, for models excluding
and including the magnetic substrate, respectively, both with Jc(B). Without a
magnetic substrate, the magnetic flux front penetrates from the edges of the tape
towards the centre in a fairly consistent manner from the top tape in the stack
to the bottom. However, when the magnetic substrate is included, the flux front
changes shape to an almost triangular one, and penetrates furthest in the middle
of the stack and least in the top and bottom tapes. The same trend is observed
when a constant Jc is used, which is not included here.
The increase in loss can be attributed to (1) the increased penetration into
the middle of the stack, and (2) the higher magnetic flux density within the
penetrated region of the superconducting tapes. The higher magnetic flux density
in the superconductor is due to the presence of the magnetic substrate and the
increased penetration arises due to a local region of large permeability around
the middle of the entire stack at the peak of the input current - although the
magnetic field strength |H| in this region is relatively low in comparison to |Hmax|
for the whole stack (< 104A/m) according to the previous equation for the relative
permeability, the peak in this curve is around 400 A/m. As shown in Figure 3.23,
the contribution of Qfe is negligible.
Figure 3.26 shows a comparison of the AC loss for tapes at different locations
within the stack (here 1/4 refers to the 12th tape from the top of the stack and
3/4 refers to the 38th tape). The difference in the AC loss between tapes is
minimal for no magnetic substrate and a triangular distribution (with highest
loss in the middle tapes) can be observed when the magnetic substrate is present.
The overall loss is significantly higher in the latter case, when compared to the
former.
68
Figure 3.22: Model geometry and mesh using the actual thickness for the YBCO
layer and including the magnetic substrate
69
Figure 3.23: Comparison of calculated AC loss for the four cases (Jc and Jc(B)
for inclusion/exclusion of magnetic substrate)
3.6.4 Implication of results for motor performance
The AC losses must be kept to a very low level because the heat produced at
the low temperatures required to maintain superconductivity requires a certain
amount of refrigeration power. An important advantage of HTS materials is the
possibility of operating temperatures well above 20 K, which leads to a significant
reduction in the required refrigerator input power. The maximum theoretical
efficiency attainable is the Carnot efficiency, given by [87]
ηc =
Top
Tamb − Top (3.16)
where Top is the operating temperature of the refrigerator and Tamb is the
ambient temperature. The input power Pin required to remove a heat load dQ/dt,
considering an ideal, reversible cooling cycle, is [87]
Pin =
1
ηc
dQ
dt
(3.17)
70
Figure 3.24: Magnetic flux density profile of the magnetic field perpendicular to
the tape faces |By| without a magnetic substrate and with Jc(B) at t = 15 ms
(peak of applied current |I0| = 50 A)
Therefore, for a power of 1 W, an ideal, reversible refrigerator would consume
2.9 W at 77 K, 14 W at 20 K and 70.4 W at 4.2 K. In order to account for the
inefficiency of the cooler, as the efficiency of a real refrigerator is much smaller
than the Carnot efficiency, a multiplication factor should be introduced between
20 - 50 for operation at 77 K [88]. Large coolers are more efficient (say up to 20
or 30%), but efficient coolers are more expensive. For a small refrigerator able
to remove 1 W, the refrigerator efficiency is only a few percent of the Carnot
efficiency [87].
Here, a comparison is made between the superconducting coil’s transport
AC loss and the resistive loss of a similar coil wound with a room temperature
71
Figure 3.25: Magnetic flux density profile of the magnetic field perpendicular to
the tape faces |By| with a magnetic substrate and with Jc(B) at t = 15 ms (peak
of applied current |I0| = 50 A)
copper conductor. Although the ductility of copper allows for better distribution
of the coil windings within the stator of an electric machine, in order to simplify
the analysis, the same coil dimensions as the superconducting coil are assumed,
including the same length of conductor, i.e. 60 m. The resistivity of copper can
be calculated by R = ρl
A
where ρ = 1.68× 10−8 Ω m (at 20◦C), l = 60 m and
A = pi r2. Here r is the radius of the conductor required to carry a maximum
current (in rms) comparable to the critical current of the superconducting coil.
For example, AWG five wire (diameter 4.62 mm) carries a maximum 47 Arms
current [89]. Using the previous equation, the resistance is 0.0601 Ω. Therefore,
for I = 50 Apeak (35.35 Arms), P = 75.13 W; for I = 40 Apeak (28.28 Arms), P =
72
Figure 3.26: Comparison of AC loss for tapes at different locations within the
cross-section for Jc(B) models with and without a magnetic substrate for an
applied current I0 = 50 A
48.08 W; and so on.
A comparison of the simulation results and the equivalent copper coil is shown
in Figure 3.27. The superconducting coil AC losses are reflected to room tem-
perature by Pref = 20 PAC (at 77 K), which assumes a refrigerator efficiency of
around 15%, and is given in units of W, assuming a frequency of 50 Hz. This
shows that at least an order of magnitude reduction in the AC loss is required for
the coil to have comparable performance to that of copper at power frequencies
such as 50 Hz. Reducing the frequency of supply to the machine to a few Hz will
also reduce the loss by an order of magnitude, making it feasible for this type of
coil to be used in high torque, low speed applications, such as wind turbines.
A comparison can also be made between the transport AC loss reflected to
room temperature and the output power of the motor. In [90], the output power
of the machine was calculated as 4.52 kW for an electrical loading of 40 Apeak
and a trapped field in the rotor poles (comprising YBCO bulks) of 0.4 T. In the
73
Figure 3.27: Comparison of the superconducting coil transport AC loss (Jc(B)
models with and without the magnetic substrate) with an equivalent copper coil
machine design there are six stator coils, meaning the total reflected transport
AC loss of the superconducting PMSM is Ptotal = 6× 20× PAC. Therefore, for
40 Apeak, the total reflected loss is 1.59 kW (no substrate, Jc(B) model) and 4.14
kW (substrate, Jc(B) model), and these correspond to approximately 35% and
92% of the motor output power, respectively. For 20 Apeak, the output power was
calculated to be 2.26 kW [90], so the reflected loss (153 W - no substrate, 602
W - substrate) corresponds to approximately 7% and 27 % of the motor output
power, respectively.
From these results, it is clear that serious investigations need to be carried
out in the area of AC loss mitigation for all-superconducting machine design. For
example, shielding using superconducting or magnetic materials, or tape striation.
This topic will be explored further in Chapter 5. The magnetic flux density profile
for coils wound with a magnetic substrate raises some interesting points for further
analysis, which will be investigated in the following section.
74
3.7 Detailed investigation on stacks with mag-
netic substrates
In this section, a detailed investigation into the effect of a magnetic substrate on
the transport AC loss is carried out. The number of tapes in each stack is varied
from 1 to 150, and three types of substrate will be compared: non-magnetic,
weakly magnetic and strongly magnetic. The model makes use of symmetry, as
shown in Figure 3.28, where only one quarter of the stack is required by setting
appropriate boundary conditions. The left-most boundary condition is Hy = 0
(along the y-axis) and the bottom boundary condition is Hx = 0 (along the x-axis).
The constant Jc approximation is used in these models in order to simplify the
comparison between models and to compare the results with analytical models.
To model a weakly magnetic substrate, e.g. the Ni-W ferromagnetic substrate
of the RABiTS YBCO coated conductor, the same relative magnetic permeability
and ferromagnetic loss is used as described above. To include a strongly magnetic
Ni-alloy substrate, the following fitting function is used, which is based on the
experimental results published in [86]. The fitting function has been selected such
that it provides reasonable values for the permeability when the magnetic field
is either very small or very large. For the strongly magnetic substrate, there is
not a significant difference between the relative permeability at 77 K and that at
room temperature [86].
µ(H) = 1 + 120000
(
1− exp(−( H
70
)3.2)
)
H−0.99 (3.18)
For the strongly magnetic substrate, the additional ferromagnetic loss Qfe
is calculated using the following fitting function, which is again based on the
experimental results published in [86].
Qfe(Bmax) =
{
171.2B1.334max for 0.1 ≤ Bmax ≤ 1.53
375(1− exp(−(Bmax
1.407
)6.787)) for Bmax > 1.53
(3.19)
Figure 3.29 shows the experimental data and fitted function for the relative
magnetic permeability of a strongly magnetic substrate and Figure 3.30 shows
75
the substrate loss. The strongly magnetic substrate can be seen to saturate at
Bmax ≈ 1.7 T. The weakly magnetic substrate saturates at Bmax ≈ 0.2 T [72].
Figure 3.28: 2D symmetric model geometry without a magnetic substrate and
using the actual thickness for the YBCO layer (shown is the 50 tape stack)
76
Figure 3.29: Relative magnetic permeability µr(H) for a strongly magnetic sub-
strate
Figure 3.30: Ferromagnetic substrate loss Qfe for a strongly magnetic substrate
77
3.7.1 Stack AC loss comparison of symmetric FEM and
analytical models
Figure 3.31 shows the total transport AC loss per unit length for the symmet-
ric non-magnetic substrate stack model for a transport current with magnitude
varying from 10% to 90% of the critical current of a single coated conductor. In
order to compare the model with existing techniques, the result for the single
tape model is compared with Norris [51], and the Clem infinite stack model [66]
is utilised as the limiting value for a significantly large stack. The model agrees
well with the Norris equation for a single tape, as found previously in [29, 62],
for example, and as the number of tapes increases past 150, the AC loss calcu-
lation tends asymptotically towards the result for the Clem infinite stack model.
Similar graphs are presented in [70,71] with identical conclusions for other mod-
els based on the minimum magnetic energy variation (MMEV) method [70] and
the infinitely thin approximation based on a variational formulation of the Kim
critical-state model [71].
Figure 3.31: Comparison of symmetric non-magnetic substrate stack model with
Norris (single tape) and Clem (infinite stack) models
78
3.7.2 Stack AC loss comparison with and without mag-
netic substrates
Here the total AC loss per unit length of stacks of tapes is compared for tapes
with (for both weak and strong cases) and without a magnetic substrate. As
described previously, the total AC loss comprises the superconductor hysteretic
loss Qsc and the ferromagnetic loss Qfe. These separate components are shown on
each of the following figures. The transport current is varied between 10% and
90% of the critical current of the single tape. Figures 3.32, 3.33 and 3.34 show the
comparison of the total AC loss for a stack of 5, 50 and 150 tapes, respectively.
As shown in Figure 3.32, even for a stack with a relatively small number of
tapes, the superconductor hysteretic loss tends to dominate the ferromagnetic
loss. However, the hysteretic loss is different depending on the type of substrate
used. The presence of the magnetic substrate increases the loss due to increased
penetration of the magnetic field into the stack (particularly at the centre) and
higher magnetic flux density within the penetrated region. The stronger the
magnetism of the substrate, the higher the transport AC loss in the stack. In the
following section, the AC loss in certain locations of the stack is shown, which
confirms these findings. The results suggest that for practical applications, where
AC transport current is involved, superconducting coils should be wound where
possible using tapes with a non-magnetic substrate to reduce the total AC loss
of the coil.
79
Figure 3.32: Comparison of AC loss in a stack of 5 tapes with and without a
magnetic substrate (weak/strong) [NM = non-magnetic, WMS = weakly mag-
netic substrate, SMS = strongly magnetic substrate; TOTAL = total loss, SC =
superconductor hysteretic loss, SUB = ferromagnetic substrate loss]
Figure 3.33: Comparison of AC loss in a stack of 50 tapes with and without a
magnetic substrate (weak/strong)
80
Figure 3.34: Comparison of AC loss in a stack of 150 tapes with and without a
magnetic substrate (weak/strong)
3.7.3 AC loss in individual tapes for stacks with/without
magnetic substrates
Here the superconductor hysteretic loss per cycle per unit length for individual
tapes within each stack is evaluated to show how the presence of a magnetic
substrate affects the magnetic field penetration, which directly affects the AC
loss within the stack. The tape locations evaluated are the middle tape, and
tapes at positions one-fifth and three-fifths from the top to the middle tape; for
example, in the 100 tape stack, these correspond to the 10th, 30th and 50th tapes,
respectively. By symmetry, the loss in the tapes in the corresponding positions
mirrored from the middle tape is the same.
Figure 3.35 shows a comparison of the AC loss for non-magnetic (top), weakly
magnetic (middle) and strongly magnetic (bottom) substrates. It is clear that
the presence of a magnetic substrate increases the superconductor hysteretic loss
in the middle tape, where there is a higher localised magnetic flux density. There
is also an increase in loss at the 1/5 and 3/5 points, but the magnitude of this
increase varies between the weakly and strongly magnetic substrates, with the
81
presence of the strong magnetic substrate resulting in an increase in loss through-
out the entire stack. However, for significantly large stacks (100 tapes or more),
the middle and 3/5 point losses become comparable, regardless of whether the
magnetic substrate is weak or strong. The loss tends to be lowest in the tapes
towards the top of the stack for most cases.
In order to understand why the loss profile is as described; the magnetic
flux penetration can be investigated using a field profile, along with the current
density distribution across tapes. Figures 3.36 and 3.38 show the magnetic flux
penetration in the tapes at the 1/5 point and middle for the 20 tape and 100
tape stacks, respectively, and figures 3.37 and 3.39 show the current density
distribution across these tapes for the same stacks.
For the 20 tape stack, there is a clear increase in penetration into the stack
when a magnetic substrate is present, as well as a higher magnetic flux density.
The strongly magnetic substrate results in further penetration at the 1/5 posi-
tion as well. However, for the weakly magnetic substrate, the flux penetration
at this position is similar to the case where no magnetic substrate is present,
but with slightly less penetration and a higher slope. This corresponds to the
triangular flux penetration characteristic observed previously. The current den-
sity distributions in Figure 3.37 show an increased current density at the edges
in the central tape when a magnetic substrate is present, which corresponds to
the characteristics of the magnetic flux penetration described above. This higher
current density results in a higher electric field, which increases the AC loss.
For the 100 tape stack, the magnetic flux penetrates a similar distance into
the stack at the centre for all substrates (non-magnetic and magnetic), but the
presence of the magnetic substrate results in a higher slope. The similar pene-
tration distance may be due to the local magnetic flux density approaching the
irreversibility field. This results in the slightly higher current density distribution
at the edges, as shown in Figure 3.39. For the tape at the 1/5 position, there
is slightly increased penetration with a higher slope for the strongly magnetic
substrate, indicating a higher magnetic flux density and further penetration into
the stack overall.
82
Figure 3.35: Comparison of AC loss in certain tapes (1/5, 3/5 and middle tapes)
within stack of tapes with a non-magnetic substrate (top figure), a weakly mag-
netic substrate (middle figure), and a strongly magnetic substrate (bottom figure)
83
Figure 3.36: Magnetic flux penetration in tapes located at 1/5 between the top
(top figure) and centre (bottom figure) for the 20 tape stacks [NMS = non-
magnetic substrate, WMS = weakly magnetic substrate, SMS = strongly mag-
netic substrate]
84
Figure 3.37: Current density distribution in tapes located at 1/5 between the
top (top figure) and centre (bottom figure) for the 20 tape stacks [NMS = non-
magnetic substrate, WMS = weakly magnetic substrate, SMS = strongly mag-
netic substrate]
85
Figure 3.38: Magnetic flux penetration in tapes located at 1/5 between the top
(top figure) and centre (bottom figure) for the 100 tape stacks [NMS = non-
magnetic substrate, WMS = weakly magnetic substrate, SMS = strongly mag-
netic substrate]
86
Figure 3.39: Current density distribution in tapes located at 1/5 between the
top (top figure) and centre (bottom figure) for the 100 tape stacks [NMS =
non-magnetic substrate, WMS = weakly magnetic substrate, SMS = strongly
magnetic substrate]
87
3.8 Summary of refinements
In this chapter, the modelling of HTS-based superconducting coils using the fi-
nite element method was described in detail. The numerical model is based on
solving a set of Maxwell’s equations in 2D implementing the H formulation using
the commercial software package Comsol Multiphysics. The coil cross-section is
modelled as the number of individual turns in the coil. A number of different as-
pects of the model were investigated in order to deduce the optimum parameters
of the model in terms of computational time and accuracy. A summary of the
parameters discussed, as well as comments on other aspects not described above,
are outlined below.
1. Mesh refinement
A refined mesh can result in a much longer computational time, but
higher stability and a more accurate result. A combination of different kinds
of meshes (free and mapped) and using a finer mesh for subdomains where
the accuracy of the solution is most critical, i.e. the superconducting layer,
can be advantageous. A solution using as fine a mesh as possible within
the limits of the computer’s memory can be used to verify the solution for
coarser meshes.
2. First order edge elements
The use of first order edge elements results in a faster solution with
higher stability and accuracy than using the default Lagrange elements in
Comsol.
3. Artificial expansion of tape thickness
As long as the tape thickness is not artificially increased above a cer-
tain limit, the artificial expansion technique can provide an accurate solu-
tion with a reduction in the number of mesh elements required, which can
significantly reduce the computation time. This idea works because the
dynamics of the flux penetration for the tape geometry is essentially along
one axis.
88
4. Critical current density approximations
A constant Jc approximation for the critical current density of the su-
perconductor provides a simplified model that can be compared with ex-
isting analytical models. The Kim model can be adapted to model the
dependence of Jc on the magnitude and direction of the magnetic field us-
ing manufacturer-supplied or measured experimental data. This results in
a more accurate representation of the superconductor’s properties, but re-
sults in a longer computational time since the model requires additional
calculations each time step.
5. E-J power law n value
Using a small n value in the E-J power law can result in a faster so-
lution, since the material behaves more like a linear material. However,
the n value needs to remain within certain limits to adequately represent
the superconductor properties since lowering the n value would cause an
increase in flux creep. Using a very high n value allows comparison of the
results with Bean-like models, but this can cause stability problems as very
small changes in magnetic field can result in large variations in the electric
field, as described in [56].
6. Magnitude of applied current/field
In general, a larger magnitude of applied current/field results in a longer
computational time, and for small magnitudes of current/field a more re-
fined mesh is required along the y-axis since the current density distribution
along the top and bottom of the tape can be of a similar magnitude as the
edges.
7. Magnetic substrates
The underlying equations of the simple superconductor model can be
modified to allow the inclusion of a magnetic substrate, which is present in
some superconducting tapes. Including the relatively permeability of such
substrates increases the number of calculations per time step, but more
accurately represents the whole tape properties.
89
Chapter 4
AC loss measurement
In this chapter, the measurement of AC loss in HTS-based superconducting coils
is described in detail, including an experimental setup that uses an electrical tech-
nique to accurately measure the transport AC loss of a superconducting coil. The
experimental technique is based on the use of a lock-in amplifier to extract the
in-phase component of the superconducting coil voltage, which corresponds to the
AC loss voltage. In order to compensate for the coil’s large inductive voltage, a
variable mutual inductance is used. The technique is applied firstly to measure
the racetrack coil introduced in the previous chapter. It is found that the exper-
imental results agree with the modelling results for low current, but some phase
drift occurs for higher current, which affects the accuracy of the measurement. In
order to overcome this problem, a number of improvements are made to the initial
setup to improve the lock-in amplifier’s phase setting and other aspects of the mea-
surement technique, including the use of the signal generator’s reference (TTL)
output and a Rogowski coil to provide stable reference signals to accurately set
the reference phase of the lock-in amplifier. New measurements are then carried
out on a single, circular pancake coil. Discrepancies between the experimental and
modelling results are described in terms of the assumptions made in the model and
aspects of the coil that cannot be modelled. Finally, some suggestions to improve
the experimental setup in the future are presented.
90
4.1 Overview of techniques
In order to validate the modelling results of the preceding chapter, it is necessary
to establish experimental measurement techniques, and there exist a number of
different techniques to measure the AC loss in superconductors. There are a
number of reasons why the AC loss should be measured: to better understand
the physical mechanism(s) of the loss; to investigate AC loss mitigation methods
for the wires themselves, as well as at a device design level; to appropriately
judge whether a cable/coil/device will be thermally stable; and to estimate the
cooling power required, and hence design an appropriate cryogenic system. In
this chapter, the measurement of transport AC loss in a superconducting coil is
investigated.
AC loss measurement techniques can be divided into three main groups:
calorimetric, electrical and magnetisation measurements. These are summarised
below.
• Calorimetric
The dissipated heat (AC loss) of the superconductor is determined in-
directly by measuring the temperature rise or the amount of gas boil-off.
• Electrical
Based on measurement of the component of the voltage in-phase with
the current to determine the AC loss, which is seen as a ’resistive’ power.
• Magnetisation
Losses can be determined from the hysteresis loop in the magnetisation
curve of the superconductor [91], which can be achieved by integrating
signals from pick-up coils wound around or Hall probes placed close to the
sample [92].
Measurement techniques are well established for measuring transport and
magnetisation AC loss in short samples of wire, and for power applications (usu-
ally 50 or 60 Hz), the Hall probe and pick-up coil techniques are most suitable for
measuring the magnetisation AC loss, whereas a four-point measurement tech-
nique using voltage taps is commonly used to measure the transport AC loss [48].
91
It is difficult to utilise calorimetric methods for short samples (approximately up
to 0.1 m in length) because liquid nitrogen has a large evaporation heat and the
heat capacity of materials used at this temperature is increased, compared with
lower temperatures, such as that of liquid helium [48]. In [93,94], the calorimetric
and electromagnetic methods are compared for measuring the transport AC loss
of short samples and it is shown that both approaches yield the same results.
More recently, it has been shown these approaches also yield good agreement
when measuring the transport AC loss of HTS pancake coils [95].
The calorimetric and electrical methods are best suited to coil AC loss mea-
surements, as it is difficult to utilise the pick-up coils required for the magnetisa-
tion method with a complex shape, such as a coil. Calorimetric methods are ad-
vantageous for AC loss measurements in complex electromagnetic environments,
such as those found in real superconductor-based devices [93], i.e. a combination
of an AC or DC applied magnetic field with an AC or DC transport current.
However, calorimetric methods have a number of drawbacks in comparison to
electrical methods, including the long time to obtain a stationary regime, the
extensive calibration required due to heat losses other than the superconductor
AC loss, and the fact that individual components of loss cannot be measured [96].
Electrical methods are generally faster and provide greater sensitivity [97], but a
major problem when applying this technique to a superconducting coil is the com-
pensation of the much larger inductive component of the coil’s voltage compared
to the in-phase component.
In the following section, an electrical method is proposed to measure the
transport AC loss of HTS-based superconducting coils, and the measurement
results are compared with the modelling results.
4.2 Proposed electrical method
The experimental setup is shown in Figure 4.1 and is based on a combination
of a modified existing experimental setup used for measuring the transport AC
loss of superconducting tapes [80] with a compensation coil to cancel the large
inductive component of the superconducting coil voltage, which can be orders of
magnitude higher than the in-phase AC loss component [96]. The compensation
92
Figure 4.1: Schematic diagram of experimental setup for measuring transport AC
loss in superconducting coils electrically
coil needs to provide a high enough compensating voltage with low phase shift
and low noise.
Power is supplied via two 400 W KEPCO power supplies connected in parallel
(one master, one slave) to provide up to ±20 V, ± 40 A. The output of the power
supplies is controlled by the internal oscillator of the lock-in amplifier, which also
acts as the lock-in amplifier’s reference signal.
The lock-in amplifier can extract a signal with a known carrier wave where
the signal-to-noise ratio (SNR) is very small. Lock-in amplifiers use a technique
known as phase sensitive detection (PSD) [98] to single out the component of
the signal of interest at a specific frequency and phase. It uses mixing, through a
frequency mixer, to convert the signal’s phase (in reference to the reference signal)
and amplitude to a voltage signal, which is then displayed on the lock-in amplifier.
To recover a signal at a low SNR requires a strong, clean reference signal at the
93
same frequency as the input signal, and this reference can be internal (e.g. using
the lock-in amplifier’s internal oscillator to drive the amplifier) or external.
The current flowing in the circuit is measured by a current transducer con-
nected to an Agilent multimeter, and the input signal to the lock-in amplifier is
the compensated superconducting coil voltage. The critical current, measured
in [77] and shown in Figure 3.5, is approximately 50 A. The coil is contained
within a polystyrene liquid nitrogen bath, and is connected in series with the
power supplies and the compensation coil. The signal from the voltage taps at
either end of the superconducting coil is connected in series, but with opposite
polarity, to the secondary of the compensation coil.
A compensation coil is used, known as inductive compensation, rather than
capacitive compensation (for examples, see [48,99]) to avoid the effects of higher
harmonics, which make capacitive compensation very frequency sensitive. It is
much easier to design a variable mutual inductance, and any higher harmonics
will induce a voltage in the compensation coil in the same way as in the super-
conducting coil. A voltage divider is not used to reduce the superconducting coil
signal [48, 99] as this may introduce a phase shift and reflection [100], so a large
mutual inductance must be provided. As thin a wire as possible is used for both
the primary and secondary coils to reduce the effect of any induced eddy currents
in the windings due to time-varying fields and, for the same reason, no magnetic
materials are used in the construction of the coils. The variation of the mutual
inductance is achieved by displacing the secondary coils relative to the primary,
whose central axes are parallel to each other, which changes the amount of flux
linked between the coils.
The compensation coil was designed with a mutual inductance higher than
the estimated 3 mH of the racetrack coil. Assuming perfect flux linkage between
the primary and secondary coils, the compensation coil mutual inductance was
calculated to be 9.4 mH, using
M = µ0N1N2
pi
2
r22
r1
(4.1)
where r1 and r2 are the radii of the primary and secondary coils, respectively.
This assumes perfect flux linkage between the primary and secondary coils, which
94
in practice is not achieved (see Experimental results). A derivation of this equa-
tion is provided in Appendix 2.
The primary coil is wound with 2.18 mm diameter enamelled copper wire with
N1 = 20 turns and is 5 cm long. The secondary coil is wound with 0.314 mm
diameter enamelled copper wire with N2 = 3000 turns and is also 5 cm long.
Thicker wire was used on the primary coil to enable large currents (up to 50 A)
through this coil. Plastic plumbing pipe was used as the former for the primary
(5 inch pipe) and secondary (4 inch pipe) coils. The compensation coil is shown
in Figure 4.2.
Figure 4.2: Compensation coil (variable mutual inductance) for proposed electri-
cal method
The superconducting coil voltage is compensated such that
vsc − vcomp = vAC (4.2)
where vAC is the AC loss component. The following equation shows the com-
ponents of the above equation in more detail:
(
iRsc + Lsc
di
dt
)
−Mdi
dt
= vAC (4.3)
95
Therefore, for complete compensation Lsc = M. For an input current i(t) =
I0sin(ωt), the inductive voltage across the superconducting coil is vL = ωLscI0cos(ωt).
The resistance of the compensation coil’s secondary voltage can be ignored due
to the large input impedance of the lock-in amplifier (10 MΩ).
The measured AC loss in J/cycle is calculated using the following equation.
Q =
VrmsIrms
f
(4.4)
where Vrms is the measured voltage across the coil ends, Irms is the transport
current flowing through the coil, and f is the frequency of the supply.
4.2.1 Experimental results
The range of the mutual inductance of the compensation coil was measured for
various displacement levels of primary versus secondary coil. Using a function
generator (50 Hz signal) and an oscilloscope, the mutual inductance was measured
to vary between 6.6 mH (no displacement) and 1 mH (9 cm displacement).
The inductance of the superconducting coil was measured by applying a small
current (0.6 A) after cooling the coil in the liquid nitrogen bath, and measuring
the coil voltage directly with the lock-in amplifier. At such a low current, the
superconducting coil voltage is expected to be predominately inductive, but it
is important that the current remains low for this measurement as the lock-in
amplifier may be damaged by the high inductive voltage presented at the input
for higher current. A frequency of 41.7 Hz was used to avoid using a frequency
too close to the mains frequency. The inductance was measured as 2.74 mH, close
to the estimated 3 mH.
The compensation coil was then connected to the lock-in amplifier (without
the superconducting coil) and was adjusted until a voltage equal to the mea-
sured superconducting coil voltage was displayed on the lock-in amplifier. The
superconducting and compensation coil signals were connected in series, ensuring
opposite polarities to cancel the inductive component and achieve the required
compensation. The setting of the phase of the lock-in amplifier was performed
using its internal reference, which also provides the control signal to the power
supply. The phase setting was performed initially, and the same phase was used
96
for subsequent measurements. The current transducer is used only for the pur-
pose of measuring the magnitude of the current flowing in the circuit and does
not provide a reference to the lock-in amplifier. The experimental results for two
sets of measurements are shown in Figure 4.3 and these results are compared with
the model with Jc(B) and including the magnetic substrate. The power supply
described above was able to provide up to approximately 25 A.
Figure 4.3: Experimental results for the transport AC loss measurement of the
superconducting racetrack coil
In analytical models based on the critical state approximation, the AC loss has
a cubic or quartic dependence [51,67,101]; however, for currents greater than 10 A,
the measured AC loss tends towards a square law relationship, suggesting that the
actual phase had drifted from the original phase setting. As the transport current
increases, the AC loss voltage increases at a faster rate than the inductive voltage,
which increases linearly, and this results in a phase shift from the original setting,
i.e. phase drift. If the phase between the control voltage (i.e. the oscillator output
97
of the lock-in amplifier) and the current does not change, then one phase setting
is accurate, but this is not always the case, as found here. In the following section,
a method to improve the phase setting in order to obtain more accurate results
over a wider range of currents is suggested.
4.3 Improved electrical method
In this section, an improved electrical method is proposed in order to overcome
the phase drift observed above, and to improve other aspects of the experimental
setup. To avoid phase drift, the phase needs to be adjusted each time the current
is varied to ensure that it is set correctly. By using a purely resistive or inductive
reference signal, this can be achieved. A different superconducting coil, which
is described below, is used for these experiments as the previous racetrack coil
has now been installed in the all-superconducting HTS PMSM for its electrical
testing.
4.3.1 New superconducting pancake coil
The circular HTS pancake coil to be tested is shown in Figure 4.4 and was wound
at the University of Oxford. The wire specification is listed in Table 4.1. The total
length of wire used is 29.55 m, resulting in 81.5 layers of silk ribbon interleave,
vacuum impregnated with epoxy resin. The inner diameter of the coil is 100 mm
and the outer diameter is 130 mm. There are voltage taps located 150 mm from
the end of each coil and at intervals of approximately 10 m within the coil. The
estimated inductance of the coil is 1 mH.
Table 4.1: SuperPower SCS12050-AP wire specification
Thickness 0.097 mm
Width 12.01 mm
Average critical current at 77K, self-field 295 A
Minimum critical current at 77K, self-field 269 A
98
Figure 4.4: Completed circular HTS pancake coil wound with SuperPower wire
4.3.2 Experimental setup
A schematic of the new experimental setup is shown in Figure 4.5 and the main
components of the setup are shown in Figure 4.6. The main components shown
are a lock-in amplifier, a signal generator to provide the sinusoidal waveform
input, an oscilloscope to examine signals in real time, and a power amplifier. The
liquid nitrogen bath for the superconducting coil, the compensation coil and the
clamp meter are shown in Figure 4.7. The superconducting coil submersed in
liquid nitrogen, and its associated wiring and voltage taps, are shown in Figure
4.8.
The compensation coil is the same one used above, but a number of turns
have been taken off the primary coil to reduce its total mutual inductance, as the
new coil under test has an inductance approximately one third of the previous
racetrack coil. The transformer has a turns ratio of 115:44 and is used to match
99
Figure 4.5: Schematic diagram of new experimental setup for measuring transport
AC loss in superconducting coils electrically
the power amplifier to the load.
In order to provide a reference signal to the lock-in amplifier, a Rogowski coil
is used, which is shown in Figure 4.9. This is a crucial part of the experimental
setup, which provides a purely inductive signal 90◦ out of phase with the cir-
cuit current, but at the same frequency, that can be used to accurately set the
phase of the lock-in amplifier. A resistor of known value could also be used, but
since all resistors have some inductance this may pose problems when setting
the phase. Additionally, a Rogowski coil is largely immune to electromagnetic
noise if wound correctly with evenly spaced windings. A shunt resistor of 500 µΩ
has been provided in the circuit as an additional reference signal (instead of the
Rogowski coil), but also as an additional means to measure the current (instead
of the clamp meter). Shielded twisted pair cable is used to connect the Rogowski
coil to the lock-in amplifier; this type of cable is less susceptible to electrical in-
terference from nearby equipment and wires, and the signal is less likely to cause
interference itself. Shielded twisted pair cable is also used for the connection
of the compensated superconducting coil voltage signal to the lock-in amplifier.
100
Figure 4.6: Main components of the new experimental setup for measuring trans-
port AC loss in superconducting coils: 1) lock-in amplifier, 2) signal generator,
3) oscilloscope, and 4) power amplifier
101
Figure 4.7: Superconducting coil liquid nitrogen bath, compensation coil and
clamp meter
Figure 4.8: Superconducting coil submersed in liquid nitrogen, and its associated
wiring and voltage taps
102
Instead of using the internal reference of the lock-in amplifier as done previously,
the reference (TTL) output of the signal generator is used, which provides a clean
5 V square-wave reference signal to the lock-in amplifier at the same frequency as
the circuit current. The phase setting of the lock-in amplifier is one of two major
problems that can arise when using a lock-in amplifier to measure AC loss [102],
and the use of the Rogowski coil in combination with the compensation coil and
TTL reference significantly improves the accuracy of the phase setting.
Figure 4.9: Hand-wound Rogowski coil used to provide the lock-in amplifier ref-
erence signal
The other major problem that can arise is the presence of a common mode
voltage vcm at the lock-in amplifier input [102]. Since the operational amplifiers
present at the input stage of all lock-in amplifiers are not ideal, the output of
these operational amplifiers is given by
vout = A · (v1 − v2) + vcm · CMRR (4.5)
where A is the gain of the amplifier, v1 and v2 are the input voltages, vcm is the
common mode voltage v1+v2
2
, and CMRR is the common mode rejection ratio, a
103
factor that specifies the degree of cancellation of the common mode voltage [102].
The lock-in amplifier used here is the Signal Recovery Model 7225 DSP lock-
in amplifier, which has a CMRR of > 100 dB at 1 kHz. The common mode
voltage can also be reduced by grounding the circuit near the coil and ensuring
that the entire measurement circuit only has one common ground point [103].
As shown in Figure 4.5, the circuit is grounded between the superconducting
coil and shunt resistor; the ground wire is connected via a 1 kΩ resistor to the
lock-in amplifier chassis, the transformer aluminium housing and the outside of
the transformer. The use of the differential input (A-B) of the lock-in amplifier
provides a balanced input, and combined with the use of the GND/AC setting,
which connects the input terminals to chassis ground, avoids the problem of
’hidden,’ multiple grounds that can arise with some lock-in amplifiers [103]. This
setting also makes the signal channel input AC coupled.
Another important setting for the lock-in amplifier is the type of input device.
For this, the ’FET’ device is selected. The FET device provides the largest input
impedance possible (10 MΩ), whereas the other option (Bipolar) provides only
10 kΩ. When the Bipolar setting is used, the lower input impedance can draw
some current from the signal source, which affects the phase of the measured
signal. For example, the secondary coil of the compensation coil has a relatively
high impedance (much greater than the superconducting coil and approximately
230 Ω), and using the Bipolar setting a small in-phase component is measured,
whereas using the FET setting, a purely inductive voltage is measured. The
combined use of the FET setting and twisted pair cable of small wire diameter
for the secondary coil satisfies the requirements of low phase shift and low noise
for the compensation coil.
A dummy load, a 0.5 Ω resistive load that has a similar impedance to the
superconducting coil at 50 Hz, is used to test the phase setting by comparing
the V
2
R
loss using the voltage measured by the lock-in amplifier across a small
(500 µΩ) shunt resistor connected in series with the dummy load and the I2R loss
using the current measured by a clamp meter. It was found that these results
are consistent, as shown in Figure 4.10, implying that the phase was correctly
set when using both the Rogowski coil and shunt resistor signals as the reference.
The dummy load was also used to test the maximum voltage/current provided
104
Figure 4.10: Power loss measurement of 500 µΩ shunt resistor using two different
methods (lock-in amplifier and clamp meter) and setting the lock-in amplifier
phase using the Rogowski coil and shunt resistor
by the setup, which was found to be 30 V/60 A.
The steps involved in the measurement are as follows.
1. Connect the Rogowski coil signal to inputs A and B of the lock-in amplifier,
which is set to differential (A-B) mode
2. Measure the voltage across the Rogowski coil and set the phase of the lock-
in amplifier to the measured phase ± 90◦ (since the Rogowski is 90◦ out of
phase with the circuit current)
3. Connect the compensated* superconducting coil voltage to inputs A and B
of the lock-in amplifier
4. If the phase is set correctly, the reading ’X’ on the lock-in amplifier is the
AC loss voltage
5. Increase the current, and repeat steps 1-4 for each measurement point
105
* Compensation is carried out by measuring the combined superconducting coil
and compensation coil voltage when applying a small current to the coil after
setting the phase with the Rogowski coil and minimising the inductive component
(’Y’ reading) on the lock-in amplifier by adjusting the compensation coil. If the
phase is set correctly, adjusting the compensation coil will change the ’Y’ reading,
but not affect the ’X’ reading, i.e. in-phase, ’resistive’ component. If the ’X’
reading does change, the phase is incorrect and must be set again.
4.3.3 Experimental results
Figure 4.11 shows the experimental results for the transport AC loss measure-
ment for four frequencies (39.93, 80.83, 120.1 and 158.2 Hz) indicated by the
transparent symbols, as well as the calculated transport AC loss using a constant
Jc (dashed blue line) and Jc(B) (red line). Included in the figure are additional
measurements taken at 80.5 Hz with the coil raised from the floor and the com-
pensation coil located further from the coil, which will be discussed later in this
section.
Figure 4.11: Calculated transport AC loss using a constant Jc and Jc(B) and
experimental results for four frequencies (39.93, 80.83, 120.1 and 158.2 Hz)
106
The transport AC loss in J/cycle for each of the frequencies measured is fairly
consistent, which is expected for a hysteretic loss. The losses obey an almost
cubic law, which is also expected, as described earlier in this chapter. However,
the magnitude of the loss is about an order of magnitude higher than predicted
by modelling for the coil at low current (10 A), and tends towards the modelled
loss for increasing current. The reasons for the discrepancy between the measured
results and the model are outlined in the following discussion.
Although the model can account for a reduction in Jc due to magnetic field,
which is discussed below, it does not account for a number of other factors, such as
any reduction due to bending strain on the tapes, mechanical stresses during the
winding process or thermal stresses when the coil is cooled. In regards to the first
two factors, if the coil winding process applies higher tension levels than those
used by the manufacturer, there is potential to induce mechanical defects in the
wire, which will result in a degraded Jc. In regards to the third factor, Siemens
recently presented [104] results on coil winding techniques for coils wound with
2G HTS wire. A number of racetrack coils were wound using both American
Superconductor and Superpower wire, and it was found that most of the coils
had a critical current Ic much lower than the wire Ic, except for a couple of coils.
The Ic for some coils degraded with subsequent repeated measurements of the I-V
curve, which has been reported elsewhere [99, 105], and the worst coils degraded
with each thermal cycle, i.e. each time the coil was cooled down. The likely
cause of this degradation was suggested to be how the inner tape is bonded to
the former, as when the coil is cooled down, the coil former shrinks more than
the tape, which increases the radial tensile stress on the tape and can cause
tape delamination. In [105], the authors present evidence for this in the form of
images of the microstructure of a fractured surface taken by a scanning electron
microscope (SEM).
The solder joint between the superconductor and the current lead contacts
(and solder joints between tape lengths in larger coils) has also been identified as
a cause of Jc degradation [106]. The model assumes a perfect current connection
between the current leads and the superconductor, and heat propagating from
the current contacts could reduce Jc locally, without quenching the entire coil,
which would cause a transition to the normal state and dramatically increase the
107
measured loss.
There may also exist localised heat spots within the coil where heat cannot
dissipate well. In regards to these potential ’hot spots,’ a visual inspection of the
coil in the liquid nitrogen bath while current is applied showed a fairly uniform
boil-off around the coil, except at the current contacts where there is a large
amount of boil-off. There was no conclusive evidence that any particular section
of the coil was weak, resulting in a larger localised boil-off. Research into coil
winding techniques, including types of and ways to make current contacts, types
of coil impregnation, and current leads to connect to a superconducting coil,
continues to be carried out by a number of research groups worldwide.
The model therefore represents the best case scenario and assumes a per-
fectly wound coil, which is currently quite difficult to achieve in practice. Non-
uniformities in Jc within the coil can alter the power exponent of the AC loss
curve, which has been found for single tapes [107] and for tape-wound coils [108].
In [107], the authors investigate the transport AC loss in a single tape versus
Norris’s strip model and find that deviation of this loss from Norris’s model is
mainly caused by degradation of the critical current distribution, particularly at
the edges, which results in significant deviation between the experimental results
and the model for currents much less than the Ic of the tape. The 30 m spool of
Superpower tape used to wind the coil has some variation in Ic from the manu-
facturing process, as shown in Figure 4.12, varying between approximately 270
A and 310 A for 5 m sections of tape. The n value of the tape also varies from
about 22 to 27 between these sections. Although it is possible to achieve good
agreement between AC loss measurements and FEM simulations for single tapes
and small stacks of tapes [109] without any adjustable parameters, i.e. using only
the Ic and n values as inputs, this may not be possible with a coil. It is possible
to include in the model Jc values for certain sections of tape that relate to the
inhomogeneity of critical current in certain sections or sections with degraded Jc,
if these are known.
As observed in the modelling chapter, and as shown in Figures 4.3 and 4.11,
the approximation used for Jc can have a large effect on the magnitude of the
calculated AC loss, as well as the power exponent of the AC loss curve, particu-
larly when comparing the constant Jc approximation with Jc(B). The constant Jc
108
Figure 4.12: Superpower measured data for critical current Ic and n value for 5
m sections of tape for the 30 m spool of tape used to wind the superconducting
coil
approximation assumes no magnetic field dependence and has the lowest power
exponent, and using a B-dependent Jc results in an increasing power exponent.
The Jc(B) used in the model is taken from manufacturer data for a representative
tape, but not for the actual tape in the spool used to wind the coil. For improved
accuracy for Jc(B), a short sample of tape from the spool could be measured in
different magnetic fields (and even for different angles of applied magnetic field)
at 77 K before winding. Due to the dynamics of the flux penetration, i.e. mainly
along the x axis, the component of loss due to the perpendicular field By should
be much higher than any effect from Bx, but this may not be true for all coil
geometries.
Although the power amplifier was able to provide up to 60 A to the resistive
dummy load, it was found that the amplifier did not perform well when the load
was changed to the superconducting coil, most likely due to the inductive nature
of this load and its poor power factor. The dummy load was added in series with
109
the superconducting coil, which provided a load with a much better power factor,
but this increased the overall impedance of the load, given by
Z = 0.5 + jωL = 0.5 + j2pifL [Ω] (4.6)
where f is the frequency of supply and L is the inductance of the coil. The
inductance of the coil was measured to be 1.04 mH (cf. estimated value of 1
mH). This increased load impedance meant that the circuit could supply up to
only approximately 30 A. However, without knowledge of the Ic of the coil, the
percentage of coil Ic that this maximum value of current supplied corresponds to
cannot be determined. In both [107] and [110], it was found that the measurement
results converged on the modelling results for current close to Ic, but deviated
somewhat for current much less than Ic. According to the model, the Ic of the
coil should be around 233 A, which was calculated using the model used in the
comparison in Figure 4.11, based on complete penetration of the perpendicular
magnetic field to the centre of any tape in the stack. This model uses the Jc(B)
relationship described previously and assumes that all tapes have an Ic of 300 A.
An Ic measurement of the coil will provide information on the overall Jc of the
coil, and measurement of the voltage across the voltage taps will provide detailed
information on the sections where Jc may be degraded the most. This will be
presented later in this section.
4.3.3.1 Voltage tap measurements
These voltage taps were utilised in the transport AC loss measurement to mea-
sure the loss in certain sections of the coil. Figure 4.13 shows a comparison of
the measured transport AC loss of the entire coil and each voltage tap for f =
80.9 Hz with the modelling results for sections corresponding to the voltage taps.
v1 corresponds to the voltage tap on the outermost turn, v2 and v3 correspond
to voltage taps located at positions 10 m and 20 m from the outermost turn,
respectively, and v4 corresponds to the voltage tap on the innermost turn. The
measured results indicate that the largest loss occurs between the innermost volt-
age taps (v3−v4) and the smallest loss occurs between the outermost voltage taps
(v1−v2). The modelling results indicate that the largest loss is expected between
110
Figure 4.13: Comparison of measured transport AC loss of entire coil and each
voltage tap for f = 80.9 Hz with modelling results for sections corresponding to
the voltage taps
the middle voltage taps (v2− v3) and the smallest between the innermost voltage
tapes (v1 − v2). Therefore, the major difference between the experimental and
modelling results is a larger loss measured for the innermost voltage tap, which
may correspond to the problem described above in relation to stress on the inner
turn and perhaps the joint between the inner turn and the current contact.
4.3.3.2 Coil critical current measurement
In order to account for degradation of the tape Jc within the coil in the model
to accurately estimate the AC loss, the critical current measurement should be
performed first. However, there is a risk of quenching the coil with this type of
measurement and irreversible damage to the coil can occur. The waveform of
the applied current for this measurement is shown in Figure 4.14, where Istep is
the magnitude of the step change in current and ts is the settling time between
each step change. Figure 4.15 shows the first set of critical current measurements
111
for Istep = 1 A. Measurements 1 and 2 were carried out with ts = 1 s, and
measurements 3 and 4 were carried out with ts = 2 s. Using the voltage criterion
of 1 µV/cm and the length of tape used to wind the coil l ≈ 30 m, the voltage
across the coil when carrying its critical current is defined as 30 mV, which
corresponds to approximately 60 A in Figure 4.15. This is significantly less than
the predicted critical current (233 A) based on the model above, which uses
the original critical current of the tape, assuming no degradation. The curve
continues to increase until approximately 100 A, where there is a large increase
in voltage due to thermal runaway, which is more prevalent when ts = 2 s, as
there is more heat generated for each step in this case. It should also be noted
that the critical current did not degrade with each measurement.
Figure 4.14: Waveform of applied current for coil critical current measurement
These results indicate that the Jc of the tape has degraded during the winding
and/or cooling process, and the coil critical current data can be utilised in the
model to more accurately calculate the AC loss. Since the critical current of
the coil is approximately 60 A, an average critical current for the tape can be
estimated as 100 A, taking into account that the critical current of the coil is
reduced in comparison to the tape due to the larger coil self-field and the B-
dependent Jc (cf. the coil critical current is approximately 233 A using a tape
critical current of 300 A). A comparison of the calculated transport AC loss using
this modified Jc to account for degradation of the superconducting tape and the
experimental results presented earlier is shown in Figure 4.16. The model shows
good agreement with the experimental results when the model is modified to
112
Figure 4.15: Coil critical current measurement
account for Jc degradation. It should be noted here that the coefficient relating
to the B-dependence of Jc has not been modified here, which has an effect on
the power exponent of the AC loss curve. Since the Jc of the tape has degraded
significantly, it is not unreasonable to expect that this coefficient has also changed,
but this would be difficult to measure for the wound coil.
In order to assess which area (or areas) of the coil has degraded, the voltage
taps were then utilised, but a large quench of the coil occurred during the first
measurement. The critical current of the coil was re-measured, and the results
are shown in Figure 4.17. Some damage to the coil occurred during the quench,
which resulted in the critical current reducing to approximately 30 A. The voltage
across each of the voltage taps was then measured in order to deduce where the
damage occurred and where the weakest part of the coil is located. Figure 4.18
shows the voltage measured across each set of voltage taps: the outermost taps
(v1−v2), middle taps (v2−v3), and the innermost taps (v3−v4). A large voltage is
observed across the innermost taps, present even at even low current, suggesting
the damage occurred in the area closest to the inner turn. The voltage across
the other taps remains low, although the voltage between the middle taps does
113
Figure 4.16: Comparison of calculated transport AC loss using a modified Jc to
account for degradation of the superconducting tape and experimental results
increase more as the current is increased past approximately 60 A. This suggests
the weakest part of the coil is closest to the inner turn, providing evidence for
the argument above. This result is consistent with the AC loss measured across
the voltage taps shown in Figure 4.13. One last coil critical current measurement
was carried out after measuring the voltage taps, and no further degradation of
the coil was observed.
4.3.3.3 Possible sources of measurement error
Although as much care as possible was taken in regards to sources of measurement
error, there may be small sources of loss within the experimental environment that
could contribute to the losses measured. These losses can be attributed to mag-
netic losses (proportional to frequency) and eddy current losses (proportional to
frequency-squared). The measured voltage signals increase in proportion to fre-
quency, which results in the consistent J/cycle loss for different frequencies shown
in Figure 4.11. Both magnetic losses and the hysteretic loss in the superconduct-
ing coil are proportional to frequency, so it is hard to determine from the voltage
114
Figure 4.17: Coil critical current measurement after large quench
Figure 4.18: Voltage taps measurement after large quench
115
signal whether there is a significant magnetic loss present. However, measures
were taken to ensure that any magnetic or conductive materials (for example,
the transformer, which has a large magnetic core, and the power amplifier, which
has a magnetic toroidal transformer inside) were located at a significant distance
away from the superconducting coil and compensation coil. A concern was raised
that if there is magnetic material present in the floor of the room housing the
experiment (for example, steel reinforcement), then this can affect the measure-
ment, since the coil is close to the floor. In order to check this potential problem,
the liquid nitrogen bath was raised 17 cm from the floor using a plastic box, and
measurements were carried out at f = 80.5 Hz. This measurement was consistent
with the previous measurements, satisfying this concern.
4.3.4 Suggested future improvements
In order to improve measurements in the future, the following should be consid-
ered.
• Two compensation coils
It is difficult to achieve exact compensation of the inductive voltage
using the large compensation coil used in these experiments. An additional,
smaller compensation coil could be introduced to provide fine-tuning of the
compensation. Although exact compensation is not necessary, it would
improve the accuracy of the measurement, since the measurement error of
the in-phase voltage depends strongly on the inductive and resistive voltage
component ratio of the sample [102]. More accurate compensation increases
the tolerance of any phase error that may exist in the measurement of the
reference phase. For example, if the inductive voltage is two orders of
magnitude higher than the resistive voltage for an uncompensated sample,
then a phase error of 0.1◦ will result in 0.1745% of the inductive voltage
being added to the resistive voltage [102]. Figure 4.19 shows the calculated
relative in-phase (resistive) voltage error as a function of the ratio between
the inductive and resistive voltage components for phase errors of 0.05, 0.1
and 0.5◦.
116
Figure 4.19: Relative in-phase (resistive) voltage error as a function of the ratio
between the inductive and resistive voltage components of the measured signal
for different phase errors
• Two lock-in amplifiers
The existing setup requires that the phase setting be performed first
using the Rogowski coil, then the AC loss measured. By using two lock-in
amplifiers - one to set the phase and one to measure the loss - the swapping
of the Rogowski coil signal and the compensated superconducting coil signal
wires, which is currently required for each measurement point, could be
avoided.
• Isolated experimental environment
As described above, the measurement can be sensitive to nearby con-
ducting and magnetic materials, and carrying out the experiments in an
isolated room free of these materials, other than those necessary for the ex-
perimental setup itself, would improve the accuracy of the setup. A larger
room, where the superconducting coil and compensation coil can be isolated
at a large distance from other equipment, would also be recommended.
In the following chapter, an investigation on a method to reduce transport
AC losses in superconducting coils is carried out, using magnetic materials as a
flux diverter.
117
Chapter 5
AC loss mitigation
In this chapter, methods used to mitigate AC loss in superconducting wires and
coils are summarised, and the use of weak and strong magnetic materials as a flux
diverter is investigated as a technique to reduce AC loss in superconducting coils
that does not require modification to the conductor itself, which can be detrimental
to the superconductor’s properties.
5.1 AC loss mitigation techniques
As seen in previous chapters, the AC loss of a superconducting coil is significantly
large, and this will reduce the efficiency of the device in which it is utilised, par-
ticularly when the loss is reflected back to room temperature by including the
refrigeration cost. In order to improve the efficiency of practical superconducting
devices, the AC loss needs to be reduced, and a number of groups have been in-
volved in research on possible AC loss mitigation methods. There exist methods
to reduce AC loss through improved material manufacturing techniques, which
can improve pinning, grain structure/boundaries, and so on, but this is the do-
main of materials scientists. Here the discussion is limited to existing methods
to reduce AC loss of already manufactured, i.e. ’off-the-shelf’, HTS conductors,
and these methods are summarised below.
118
5.1.1 Striation into narrow filaments
This method involves the HTS conductor being striated into narrow filaments,
which reduces the hysteretic loss, since hysteretic loss is proportional to the width
of the tape [111–113]. The conductor can be striated using a slitter machine
[114], which cuts the conductor into separate filaments, or with laser ablation
[115, 116], photolithography or wet etching [111]. An example of tape striation
using laser ablation is shown in Figure 5.1. However, striated tapes are vulnerable
to localised defects [116], which can impede the flow of current through a filament,
so uniformity of the properties of individual filaments is extremely important
[117]. This problem can be overcome either by covering the filaments with a thick
normal metal layer of low resistivity or by making a network of superconducting
bridges, in such a way that allows current sharing between filaments [116]. For
mechanical cutting, degradation of Jc can occur at the edges of the filaments and
can depend on the cutting process used [114]. However, the original tape itself
may suffer from non-uniformities, from which the striated tape is prepared [117].
Whilst striation reduces the hysteretic loss in comparison with a single tape, it
introduces a new loss in the form of a coupling loss due to the coupling of separate
filaments [116,118] and this loss can be significantly more than the self-field loss
for an uncoupled tape carrying the same current [119]. Research continues in this
area [120–123].
Figure 5.1: Striation of HTS coated conductor using laser ablation [116]
119
5.1.2 Roebel transposition
The concept of Roebel transposition was first introduced by Ludwig Roebel in
1914 in his patent application to reduce AC losses in copper cables for genera-
tors, and the design is particularly suitable for AC windings [124]. In relation
to superconducting wires, the Roebel concept has been applied already to NbTi
cables [125] for use in the International Energy Agency (IEA) Large Coil Task
(LCT), which was an international collaboration between the United States, EU-
RATOM, Japan, and Switzerland to develop large superconducting magnets for
fusion reactors [126], and to BSCCO-2223 for use in a transformer [127]. Re-
cent research has seen the application of the Roebel concept to YBCO coated
conductors; for example, [124, 128, 129]. The YBCO conductor must be cut or
punched into shape before winding, and an example of cut/punched filaments and
an assembled cable is shown in Figure 5.2. Various groups have shown promising
results, such as measurement of reduced AC loss [129] and accurate prediction
of a cable’s critical current from the Jc(B) dependence of a single tape [44], and
research continues in this area [130–136].
5.1.3 Twisted wires
The large current that can be carried by a superconductor generates a large
self-field, and in the past superconducting wires have been twisted to avoid flux
linkage between the filaments. Ideally the wires would be fully transposed, where
each wire swaps places with every other wire along the length, so that averaged
over the length, no net mutual flux linkage occurs [137]. Many studies have been
carried out on the AC loss of twisted multifilamentary superconductors, most
extensively for low temperature superconductors (NbTi and Nb3Sn, for exam-
ple) in [138–140], but also for 1G HTS (BSCCO) in [141–143] and for MgB2
in [144–146]. Indeed, one particular type of twisted multifilamentary supercon-
ductor configuration - the Rutherford cable - has been highly successful, and
has been used in all particle accelerators to date [137]. However, twisting the
filaments severely will damage the microstructure, the evolution of texture, and
eventually decrease the critical current [141]. Due to the nature of the geometry
of the high temperature superconductors, having a large aspect ratio, the damage
120
Figure 5.2: Punched strands from SuperPower-manufactured YBCO coated con-
ductor (top) and an assembled YBCO Roebel cable (bottom) [124]
can be even greater than with other superconductors. A conceptual approach to
the ”ultimate low AC loss YBCO superconductor” is presented in [147], where
a fully transposed YBCO tape approximating a Rutherford cable has been con-
ceived. When compared with the AC coupling loss in a flat twisted tape, the loss
can be reduced by as much as a factor of 20 [147]. However, since the Ruther-
ford configuration requires a tape edge turnaround for the YBCO current path,
a more elaborate analysis is required of the YBCO material in these locations, as
there will most likely be disrupted grain orientations [147], which would signifi-
cantly affect the current path and the critical current density. The current path
turnarounds on both tape edges will see a significantly reduced Jc due to YBCO
Jc limits on the c-axis [147].
121
5.1.4 Magnetic shielding/flux diverter
Magnetic materials can be used to manipulate the magnetic flux in a supercon-
ductor to reduce the AC loss, and different terms have been used in the liter-
ature for this: magnetic shielding/screening [119, 148, 149] and (magnetic) flux
diverter [43, 150, 151].
In [148, 149], a magnetic cover is used for individual filaments in a striated
YBCO conductor in an attempt to decouple the filaments and reduce losses. The
former reference refers to filaments that are totally enclosed by the magnetic
material, and the latter refers to filling only the slits between the filaments with
a magnetic material. Both configurations result in a similar reduction in AC loss;
however, the latter is much more practical.
In [119], a magnetic cover is used around BSCCO-2223 multifilimentary tapes,
and it was found that this can actually increase the AC loss in a single tape, but
that it can be an effective screening material to decouple multiple tapes. An
iron sheath around a BSCCO-2223 tape in [152] also resulted in a significant
increase in AC loss (three orders of magnitude), but in addition an increase in
the critical current density was reported. However, in [150], a magnetic material
(nickel) is used in a C-shape to cover only the edges of BSCCO-2223 tapes, in
contrast to the covers [119, 152], which covered the entire outside of the tape.
This resulted in a substantial reduction in AC loss, indicating that a number of
factors, such as the shape and location of the magnetic material, plays a role in
whether there is an increase or decrease in loss. The work in [150] was extended
to YBCO coated conductors in [43], where a horse-shoe cover of ferromagnetic
material is applied to the edges of the conductor. It is observed that the loss in the
superconductor is significantly less than the Norris strip model, but the additional
loss incurred in the cover must be taken into account, which can be higher than
the superconductor loss itself. The authors also mention that the material used
(nickel) is far from a low loss ferromagnetic material, stressing the importance of
material selection. In [153], the authors find that an AC loss reduction can also
be achieved using ferromagnetic covers on the edges of multiple superconducting
tapes in a stack. Ferromagnetic diverters have also been investigated for their
application in power transmission cables to improve the magnetic flux distribution
122
for a given cable geometry [154] and in a synchronous generator [155]. There
is a minimal number of studies on the effect of flux diverters on AC loss in
superconducting coils, but one particular study on the use of a flux diverter in a
YBCO-based superconducting coil has shown a reduction in AC loss without any
change in the critical current [151], which is a promising result for this technique.
5.2 Flux diverter analysis
The first three techniques, and some magnetic shielding techniques, involve mod-
ification of the HTS conductor itself. If these modifications are not carried out
precisely, significant degradation of the conductor properties can occur. The use
of external flux diverters modifies the magnetic flux profile of the conductor(s) in
order to achieve a reduction in AC loss without modifying the original conductor.
Hence, the conductor can be used ’as is’ (off the shelf). In this section, a study
of the use of flux diverters to reduce AC loss is carried out for stacks of tapes
with and without a (weak) magnetic substrate. Extending this kind of numeri-
cal modelling in the future, it will be possible to find an optimal geometry and
location for the diverter to achieve the greatest reduction in loss, and different
magnetic materials can be used to deduce their effect on the loss of the coil.
5.2.1 Modelling results
Figure 5.3 shows a comparison of the reduction in AC loss using a flux diverter
using weak and strong magnetic materials of thickness 0.5 mm and 1 mm for
stacks of tapes with and without a magnetic substrate. The flux diverter is
placed along the right-hand edge of the stack of tapes, and by symmetry, would
be present on the opposite side. The loss is given as a percentage of the original
loss calculated for the models with no flux diverter present. It is apparent that the
use of a flux diverter achieves a reduction in the AC loss of a stack of tapes, which
is particularly pronounced for smaller stacks. The greatest AC loss reduction
occurs when the thicker (1 mm), strong magnetic material is used, and this will
be discussed in the following section. There is a pronounced decrease in the AC
loss for stacks of tapes with a magnetic substrate, which is a promising result
123
as it was shown previously that the presence of a magnetic substrate in the
superconducting tape causes an increase in AC loss.
5.2.2 Discussion
Figure 5.4 shows a comparison of the magnetic field density profiles, including
magnetic flux lines, for a 50 tape stack (without a magnetic substrate) with and
without a flux diverter (weak magnetic material). The same scale is used for
both figures, ranging from 0 T (dark blue) to 0.567 T (red), and the density of
magnetic flux lines is the same. By comparing the magnetic flux lines between
the two, it can be observed that the flux diverter attracts magnetic flux towards
it and changes the distribution of the field lines, which are densely packed within
the magnetic material. Figure 5.5 shows the difference in the penetration of the
perpendicular component of the magnetic field into the stack with and without
a flux diverter for a 50 tape stack, for tapes at the 1/5 point (top) and centre
(bottom).
For the flux diverter to work well, it must be within the saturation limit of the
magnetic material used. When the magnetic material is saturated, any increase
in the external magnetising field H, i.e. the field from the stack of tapes, cannot
increase the magnetisation of material any further, so the total magnetic flux
density B levels off. Hence, it ceases to act as a diverter of flux. Figures 5.6 and
5.7 show comparisons of the peak diverter magnetic flux density for diverters of
weak and strong magnetic materials, respectively, for thicknesses of 0.5 mm and 1
mm for stack of 10 to 100 tapes with and without a weak magnetic substrate. The
thick dashed line indicates the saturation magnetic flux density of the material. It
is clear that the weak magnetic material is unsuitable as a flux diverter as for all
cases the magnetic flux density exceeds the material’s saturation limit. This also
explains why the weak diverter performs worse than the strong diverter in respect
to reducing AC loss; the strong magnetic material saturates for the 50 and 100
tape stacks, but the higher saturation limit reduces the AC loss in comparison to
the weak magnetic material. Therefore, the ideal flux diverter material will have
a high saturation field, as well as a low remanent field, which reduces the size of
its hysteresis loop, resulting in lower loss in the material for each AC cycle.
124
Figure 5.3: AC loss reduction using flux diverter using weak and strong magnetic
materials of thickness 0.5 mm and 1 mm for stacks of tapes with and without a
weak magnetic substrate
125
Figure 5.4: Comparison of magnetic flux density profiles, including magnetic flux
lines, for a 50 tape stack (a) without and (b) with a flux diverter (weak magnetic
material)
126
Figure 5.5: Comparison of magnetic flux penetration in tapes located at the 1/5
point (top figure) and centre (bottom figure) for the 50 tape stack (without a
magnetic substrate) with and without a diverter
127
Figure 5.6: Peak diverter magnetic flux density for diverter of weak magnetic
material for thicknesses of 0.5 mm and 1 mm for stacks of 10 to 100 tapes with
and without a weak magnetic substrate [NMS = non-magnetic substrate, WMS
= weakly magnetic substrate]
Figure 5.7: Peak diverter magnetic flux density for diverter of strong magnetic
material for thicknesses of 0.5 mm and 1 mm for stacks of 10 to 100 tapes with
and without a weak magnetic substrate [NMS = non-magnetic substrate, WMS
= weakly magnetic substrate]
128
One shortcoming of this model is that the magnetic material model does not
account for the hysteresis loop of the magnetic material, i.e. when the field
is removed, the material returns to the virgin magnetisation state rather than
having some remanent magnetisation. This is adequate for the AC loss calcula-
tion described previously, as the model is combined with experimental results to
calculate the total loss. For complete and accurate representation of magnetic
materials, hysteresis loops need to be implemented in the model, which is difficult
to achieve in finite element modelling. However, this would allow the ferromag-
netic losses to be calculated using the model, rather than relying on experimental
data for this calculation.
In conclusion, there is significant promise in using magnetic materials as a
flux diverter to significantly reduce the AC loss in superconducting coils, al-
though further research is necessary to prove these results experimentally. At the
cryogenic temperatures at which the superconductor operates, there is less of a
restriction on the number of magnetic materials that could be utilised for this
purpose, as operation would occur at temperatures well below the Curie point,
the temperature at which a ferromagnetic material becomes paramagnetic and
loses its magnetism. Ideally the magnetic material will have a high saturation
field (to reduce the losses more and to work for a larger range of coils) and a low
remanent field (to reduce the ferromagnetic loss in the diverter itself).
129
Chapter 6
Conclusions
6.1 Conclusions
In this dissertation, the problem of calculating and measuring AC losses in su-
perconducting coils, with a particular focus on the transport AC loss of coils
for electric machines, is addressed. In an electric machine, and indeed in other
superconductor-based devices, such as SMES systems and transformers, there
are usually multiple superconductors in tape/wire form wound into coils and in-
teracting together in a complex magnetic environment. In order to assess the
performance of such devices, it is crucial to have reliable techniques to model and
measure the AC losses in complex geometries, since these AC losses increase the
refrigeration load, which decreases the overall efficiency and increases the tech-
nological complexity of the design. The main contributions of this dissertation
are highlighted below.
In order to model the superconducting coil’s electromagnetic properties and
calculate the AC loss, an existing two dimensional (2D) finite element model that
implements a set of equations known as the H formulation, which directly solves
the magnetic field components in 2D, was extended to model a superconducting
coil, where the cross-section of the coil is modelled as a 2D stack of supercon-
ducting coated conductors.
Firstly, the artificial expansion of the thickness of the YBCO layer was inves-
tigated using a single tape, as the preliminary groundwork for optimising more
complex geometries, which can require hundreds of thousands of mesh elements.
130
It was found that the thickness may be expanded in the model up to 20-30 µm be-
fore the calculated result begins to deviate appreciably from the analytical model
used for comparison. The use of edge elements provides the best compromise
between the computation time required to solve the model and the accuracy of
the solution.
The artificial expansion technique was applied to the geometry of a supercon-
ducting racetrack coil of an all-superconducting high-temperature superconduc-
tor (HTS) permanent magnet synchronous motor to model the individual turns
of the coil. The result is compared with a model using a bulk approximation,
which assumes that the tapes couple electromagnetically such that the individ-
ual tapes behave as a finite superconducting slab carrying n times the current of
each individual tape, where n is the number of tapes. The artificially-expanded,
individual tapes model is more accurate than the bulk approximation, as long as
the expanded thickness remains within the limits specified above.
A technique, which uses large aspect ratio mapped meshes, is then applied
to allow the actual superconducting layer thickness to be modelled without the
associated problem of increased computation time due to a large number of mesh
elements. In combination with a sparser mapped mesh between the superconduct-
ing layers, an overall reduction of about two orders of magnitude in the number
of mesh elements was achieved.
The model was modified to allow the inclusion of a magnetic substrate, which
is present in some commercially available HTS wire. The analysis raised a num-
ber of interesting points regarding the use of superconductors with magnetic
substrates. In particular, the presence of a magnetic substrate affects the pen-
etration of the magnetic flux front within the coil and increases the magnetic
flux density within the penetrated region, both of which can increase the AC loss
significantly. The effect of the substrate loss itself on the overall AC loss can, in
general, be neglected, except for a suitably low current. In order to investigate
these findings further, a comprehensive analysis on stacks of tapes with weak
and strong magnetic substrates was carried out, using a symmetric model that
requires only one quarter of the cross-section to be modelled.
In order to validate the modelling results, an extensive experimental setup
was designed and built to measure the transport AC loss of a superconducting
131
coil using an electrical method based on inductive compensation by means of a
variable mutual inductance. The variable mutual inductance is utilised to cancel
the inductive component of the superconducting coil’s voltage, which is 90◦ out
of phase with the current and much larger than the AC loss voltage, which is
in-phase with the current. This is used in conjunction with a lock-in amplifier,
which can extract a signal with a known carrier wave where the signal-to-noise
ratio is very small.
Measurements were carried out on the superconducting racetrack coil and it
was found that the experimental results agree with the modelling results for low
current. However, some phase drift occurs for higher current, which affects the
accuracy of the measurement. In order to overcome this problem, a number of
improvements were made to the initial setup to improve the lock-in amplifier’s
phase setting and other aspects of the measurement technique, including the
use of the signal generator’s reference (TTL) output and a Rogowski coil to
provide stable reference signals to accurately set the reference phase of the lock-
in amplifier.
New measurements were carried out on a single, circular pancake coil and the
discrepancies between the experimental and modelling results were described in
terms of the assumptions made in the model and aspects of the coil that cannot
be modelled. Using the original measured properties of the superconducting tape,
there is an order of magnitude difference between the experiment and model. The
properties of the superconductor can degrade during the winding and cooling
processes, and a critical current measurement coil showed that the tape critical
current reduced from nearly 300 A, down to around 100 A. Applying this finding
to the model, the experimental and modelling results showed good agreement,
and the difference in the slope of the AC loss curve can be described in terms of
the B-dependent critical current dependency Jc(B) used in the model. Accurate
information on the superconductor properties is crucial for estimating the AC
loss. The utilisation of voltage taps can provide more information on different
regions of the coil and identify areas of weakness, i.e. areas of significant Jc
degradation.
Finally, methods used to mitigate AC loss in superconducting wires and coils
are summarised, and the use of weak and strong magnetic materials as a flux
132
diverter is investigated as a technique to reduce AC loss in superconducting coils.
This technique can achieve a significant reduction in AC loss and does not require
modification to the conductor itself, which can be detrimental to the supercon-
ductor’s properties.
6.2 Future research
The analyses presented in this dissertation provide a number of interesting results
which, combined with addressing some shortcomings in the present modelling and
experimental setup, will form the basis for a number of fundamentally interlinked
and exciting research topics to be carried out in the future. These are outlined
below.
• Modelling technique:
One shortcoming of the model is that the modelling of magnetic mate-
rials does not account for the hysteresis loop of the material. In the current
model, when the field is removed, the material returns to its virgin mag-
netisation state rather than having some remanent magnetisation. This is
adequate for the AC loss calculation described in this disseration, as the
model is combined with experimental results to calculate the total loss.
However, implementing hysteresis loops for magnetic materials would allow
ferromagnetic losses to be calculated directly and would alleviate the need
to use experimental data.
For some coil geometries and magnetic environments, the infinitely long
approximation may not provide an accurate solution, and axisymmetric
modelling of the coil in 2D cylindrical coordinates or extending the model to
3D would provide information on cases where it is applicable and where it is
not, for most practical coil geometries. This may be particularly important
when calculating the magnetisation AC loss of a superconducting coil due
to an applied magnetic field. A 3D model would also allow more complex
superconducting coils, such as saddle coils, to be analysed.
There are a number of approximations that can be made in regards
133
to the critical current density Jc, and in this dissertation, a constant Jc
approximation and a Jc(B) approximation that assumes the perpendicular
component dominates the suppression of Jc are used. In reality, there are
components at various angles to the tape face, i.e. Jc(B, θ), and if detailed
information on this relationship is provided (either by the manufacturer or
by measuring directly), this would improve the modelling further.
• Experimental technique:
The transport AC loss of a superconducting coil is the focus of this
dissertation, but the modelling and experimental setup could be extended
to calculate and measure the magnetisation AC loss, which is crucial for the
design of a superconducting electric machine, where a superconducting coil
may be subjected to a combination of a transport current and an external
magnetic field.
In the final chapter of this dissertation, an investigation on the use
of magnetic materials as a flux diverter was presented as a technique to
reduce AC loss in superconducting coils. The results of this investigation
show significant promise for this technique, but further research is required
to prove these results experimentally.
134
Appendix 1
AC loss calculation example using a superconducting slab
In this section, an example of an AC loss calculation for a superconducting
slab is given [156]. The superconducting slab is infinitely long in the y and z
directions, and has a finite width in the x direction. According to Bean’s model,
transport current begins to penetrate from the edge of a slab. If a coordinate
system is chosen such that 0 corresponds to the point of full penetration for
the maximum transport current and s corresponds to the surface of the slab.
Ampere’s law states that
∇×H = J (1)
and for only one dimension, the equation reduces to
dH(x)
dx
= J(x) (2)
Bean’s model:
dH(x)
dx
= Jc (3)
Integrating both sides and assuming that the field is zero at the penetration
point:
H(x) = Jcx (4)
135
Similarly, for the minimum transport current, -I, the field is
H(x) = −Jcx (5)
The enclosed flux per unit length in the ’positive’ half of the slab for currents
-I and I can be determined by
φ(x, 0) =
∫ x
0
B(x, 0) dx =
∫ x
0
µ0H(x, 0) dx =
∫ x
0
−µ0Jcx dx = −µ0Jcx
2
2
(6)
Similarly,
φ(x,
T
2
) =
µ0Jcx
2
2
(7)
The total AC loss per cycle of current per unit area of the slab can be deter-
mined using equation 2.25:
Q
S
= 2
∫ T
0
∫ p
0
E · Jc dx dt = 4Jc
∫ T
2
0
∫ p
0
Edx dt (8)
where p is the penetration depth. E = 0 in the slab when J = 0, so the electric
field can be solved using Faraday’s law:
Q
S
= 4Jc
∫ T
2
0
∫ p
0
dφ(x, t)
dt
dx dt (9)
Changing the order of integration:
136
QS
= 4Jc
∫ p
0
∫ T
2
0
dφ(x, t)
dt
dt dx
= 4Jc
∫ p
0
(φ(x,
T
2
)− φ(x, 0)) dx
= 4Jc
∫ p
0
µ0Jcx
2 dx
= 4µ0J
2
c
∫ p
0
x2 dx
=
4
3
µ0J
2
cs
3 (10)
To investigate this equation numerically, the following assumptions can be
made:
• Slab width = 1 mm
Thus, s = 0.5 mm
• Jc = 1× 105 A/m2
Thus, critical current of the slab per unit length along the y axis is 50
A/m
• Transport current of 40 A/m
Thus, the penetration depth, p, is
p =
I
Ic
s = 4× 10−4m (0.4 mm) (11)
This gives a total AC loss of
Q
S
=
4
3
µ0J
2
cp
3 ≈ 1.07× 10−6 [J/cycle/m2] (12)
137
Appendix 2
Derivation of equation for mutual inductance of compen-
sation coil
The equation for the mutual inductance of the compensation coil is derived
as follows.
The magnetic field from a current element in the Biot-Savart law is given by
dB = µ0
IdL× r
4pir21
(13)
Therefore,
dB = µ0
IdLsinθ
4pir21
(14)
which, in the case of a circular current loop of radius r1, becomes
B =
µ0I
4pir21
∮
dL =
µ0I
4pir21
2pir1 =
µ0I
2r1
(15)
In the case of a circular loop of N1 turns, i.e. the compensation coil primary
coil, this becomes
B =
µ0N1I
2r1
(16)
For a loop of N2 turns placed inside of this coil, i.e., the compensation coil
secondary coil, of a small enough radius such that B is assumed constant, the
138
mutual inductance is defined as
M =
N2Φ21
I
(17)
where Φ21 is the flux linked by the secondary coil and is given by
Φ21 = Φ1
A2
A1
(18)
where Φ1 = B · A1, which results in Φ21 = B · A2.
The mutual inductance is then
M =
N2BA2
I
(19)
where A2 = pir
2
2 and B =
µ0N1I
2r1
, which gives
M = µ0N1N2
pi
2
r22
r1
(20)
139
References
[1] Energy Information Administration, U.S. Department of En-
ergy. International Energy Outlook 2009. [Online]. Available:
http://www.eia.doe.gov/oiaf/ieo/ 1
[2] U.S. Energy Information Administration, “Annual Energy Review 2010:
Energy Consumption Estimates by Sector Overview,” 2010. 1
[3] ABB, “ABB drives and motors for improving energy efficiency,” 2010. 1
[4] C. McNaught, “Running smoothly: making motors more efficient,” IEE
Review, vol. 39, no. 2, pp. 89–91, 1993. 1
[5] H. K. Onnes, Comm. Phys. Lab. Univ. Leiden, pp. 119,120,122, 1911. 7
[6] W. Meissner and R. Ochsenfeld, Naturewiss, vol. 21, p. 787, 1933. 8, 9
[7] M. Tinkham, Introduction to Superconductivity, 2nd ed. Dover Publica-
tions, 1996. 8
[8] F. and H. London, “The electromagnetic equations of the supraconductor,”
Proc. Roy. Soc., vol. 149, no. 866, pp. 71–88, 1935. 9, 33
[9] V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz., vol. 20, p. 1064,
1950. 9
[10] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Microscopic theory of
superconductivity,” Phys. Rev., vol. 106, pp. 162–164, 1957. 9
140
REFERENCES
[11] A. A. Abrikosov, “On the magnetic properties of superconductors of the
second group,” Sov. Phys. JETP-USSR, vol. 5, no. 6, pp. 1174–1183, 1957.
9
[12] A. C. Rose-Innes and E. H. Rhoderick, Introduction to Superconductivity,
revised 2nd ed. Pergamon Press, 1994, pp. 183–194. 9
[13] Laboratory of Advanced Energy Systems, Helsinki University of
Technology. High-Temperature Superconductivity. [Online]. Available:
http://tfy.tkk.fi/aes/AES/projects/prlaser/supercond.htm 9, 11, 13
[14] A. A. Golubov, Handbook of Applied Superconductivity. IOP Publishing,
1998, vol. 1, ch. A1: The evolution of superconducting theories, pp. 3–36.
9, 10
[15] C. P. Poole, Superconductivity, 2nd ed. Elsevier Ltd., 2007. 9, 19, 20, 23
[16] A. C. Fischer-Cripps, The Material Physics Companion. Taylor & Francis,
2008, p. 192. 9
[17] G. Fuchs and L. Schultz, Concise Encyclopedia of Magnetic and Supercon-
ducting Materials, 2nd ed. Elsevier, 2005, ch. Superconducting Permanent
Magnets: Principles and Results, pp. 1166–1174. 10
[18] P. Lee, Ed., Engineering Superconductivity. John Wiley & Sons, 2001. 10,
11
[19] Oak Ridge National Laboratory. Fundamen-
tals of Superconductors. [Online]. Available:
http://www.ornl.gov/info/reports/m/ornlm3063r1/pt3.html 10
[20] D. A. Cardwell, “Processing and properties of large grain (RE)BCO,” Ma-
terials Science and Engineering: B, vol. 53, no. 1-2, pp. 1–10, 1998. 11
[21] Coalition for the Commercial Applications of Super-
conductivity (CCAS). Superconductivity. [Online]. Available:
http://www.ccas-web.org/superconductivity/ 12
141
REFERENCES
[22] D. U. Gubser, “Superconductivity: An emerging power-dense energy-
efficient technology,” IEEE Trans. Appl. Supercond., vol. 14, no. 4, pp.
2037–2046, Dec 2004. 11
[23] Hoffman Lab. Superconducting Cuprates. [Online]. Available:
http://hoffman.physics.harvard.edu/materials/CuprateIntro.php 11
[24] J. G. Bednorz and K. A. Muller, “Possible high Tc superconductivity in the
Ba-La-Cu-O system,” Zeitschrift fur Physik B Condensed Matter, vol. 64,
no. 2, pp. 189–193, 1986. 11
[25] B. Oswald, Concise Encyclopedia of Magnetic and Superconducting Mate-
rials, 2nd ed. Elsevier, 2005, ch. Superconducting Permanent Magnets:
Potential Applications, pp. 1164–1166. 13
[26] American Superconductor. American Superconductor. [Online]. Available:
http://www.amsc.com 13, 42
[27] SuperPower Inc. HTS Materials Technology. [Online]. Available:
http://www.superpower-inc.com/ 13, 15
[28] X. Li et al., “The development of second generation HTS wire at American
Superconductor,” IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 3231–
3235, 2009. 15
[29] Z. Hong, A. M. Campbell, and T. A. Coombs, “Numerical solution of crit-
ical state in superconductivity by finite element software,” Supercond. Sci.
Technol., vol. 19, pp. 1246–1252, 2006. 14, 37, 78
[30] R. Pecher, M. D. McCulloch, S. J. Chapman, L. Prigozhin, and C. M.
Elliott, “3D-modelling of bulk type-II superconductors using unconstrained
H-formulation,” in Proceedings of the 6th EUCAS, 2003, pp. 1–11. 16
[31] C. P. Bean, “Magnetization of hard superconductors,” Phys. Rev. Lett.,
vol. 8, p. 250, 1962. 16, 18, 32
142
REFERENCES
[32] Y. B. Kim, C. F. Hempstead, and A. R. Strnad, “Critical persistent currents
in hard superconductors,” Phys. Rev. Lett., vol. 9, no. 7, pp. 306–309, 1963.
16, 19, 24
[33] P. W. Anderson, “Theory of flux creep in hard superconductors,” Phys.
Rev. Lett., vol. 9, no. 7, pp. 309–311, 1963. 16, 19, 23, 24
[34] J. Rhyner, “Magnetic properties and AC-losses of superconductors with
power law current-voltage characteristics,” Physica C, vol. 212, pp. 292–
300, 1993. 16, 23
[35] A. A. Golubov, Handbook of Applied Superconductivity. IOP Publishing,
1998, vol. 1, ch. A2: Type II superconductivity, pp. 37–52. 16, 29
[36] W. J. Carr, AC loss and macroscopic theory of superconductors, 2nd ed.
CRC Press, 2001. 17, 27, 29, 30
[37] S. Stavrev et al., “Comparison of numerical methods for modeling of su-
perconductors,” IEEE Trans. Mag., vol. 38, no. 2, pp. 849–852, Mar 2002.
17, 23, 37
[38] E. H. Brandt and M. Indenbom, “Type-II superconductor strip with current
in a perpendicular magnetic field,” Phys. Rev. B, vol. 48, no. 17, pp. 12 893–
12 906, 1993. 21, 35, 47
[39] E. H. Brandt, “Superconductors of finite thickness in a perpendicular mag-
netic field: strips and slabs,” Phys. Rev. B, vol. 54, pp. 4246–4264, 1996.
21, 35
[40] N. Nibbio, S. Stavrev, and B. Dutoit, “Finite Element Method simulation
of AC loss in HTS tapes with B-dependent E-J power law,” IEEE Trans.
Appl. Supercond., vol. 11, no. 1, pp. 2631–2634, Mar 2001. 25
[41] Q. Jiang, M. Majoros, Z. Hong, A. M. Campbell, and T. A. Coombs, “De-
sign and AC loss analysis of a superconducting synchronous motor,” Su-
percond. Sci. Technol., vol. 19, pp. 1164–1168, 2006. 25, 42
143
REFERENCES
[42] J. Souc, E. Pardo, M. Vojenciak, and F. Gomory, “Theoretical and experi-
mental study of AC loss in high temperature superconductor single pancake
coils,” Supercond. Sci. Technol., vol. 22, no. 015006, 2009. 25, 41
[43] F. Gomory, M. Vojenciak, E. Pardo, M. Solovyov, and J. Souc, “AC losses
in coated conductors,” Supercond. Sci. Technol., vol. 23, no. 034012, 2010.
25, 122
[44] M. Vojenciak, F. Grilli, S. Terzieva, W. Goldacker, M. Kovacova, and
A. Kling, “Effect of self-field on the current distribution in Roebel-
assembled coated conductor cables,” Supercond. Sci. Technol., vol. 24, no.
095002, 2011. 25, 120
[45] E. Pardo, M. Vojenciak, F. Gomory, and J. Souc, “Low-magnetic-field de-
pendence and anisotropy of the critical current density in coated conduc-
tors,” Supercond. Sci. Technol., vol. 24, no. 065007, 2011. 25
[46] M. P. Oomen, “AC loss in superconducting tapes and cables,” Ph.D. dis-
sertation, University of Twente, Enschede, The Netherlands, 2000. 27, 28
[47] P. N. Barnes, M. D. Sumption, and G. L. Rhoads, “Review of high power
density superconducting generators: present state and prospects for incor-
porating YBCO windings,” Cryogenics, vol. 45, pp. 670–686, 2005. 28
[48] J. J. Rabbers, “AC loss in superconducting tapes and coils,” Ph.D. disser-
tation, University of Twente, Enschede, The Netherlands, 2001. 28, 29, 30,
33, 91, 92, 94
[49] Z. Hong, A. M. Campbell, and T. A. Coombs, “Computer modeling of
magnetisation in high temperature bulk superconductors,” IEEE Trans.
Appl. Supercond., vol. 17, no. 2, pp. 3761–3764, Jun 2007. 29
[50] M. M. Farhoudi, “AC loss in Ag/Bi-2223 tapes in AC field,” Master’s thesis,
University of Wollongong, Wollongong, Australia, 2005. 30
[51] W. T. Norris, “Calculation of hysteresis loss in hard superconductors car-
rying ac: isolated conductors and edges of thin sheets,” J. Phys. D: Appl.
Phys., vol. 3, pp. 489–507, 1969. 33, 34, 78, 97
144
REFERENCES
[52] G. Barnes, M. McCulloch, and D. Dew-Hughes, “Computer modelling of
type II superconductors in applications,” Supercond. Sci. Technol., vol. 12,
pp. 518–522, 1999. 37
[53] L. Prigozhin, “Analysis of critical-state problems in type-II superconduc-
tivity,” IEEE Trans. Appl. Supercond., vol. 7, pp. 3866–3873, 1997. 37
[54] A. M. Campbell, “A direct method for obtaining the critical state in two
and three dimensions,” Supercond. Sci. Technol., vol. 22, no. 034005, 2009.
37
[55] N. Amemiya, K. Miyamoto, S. Murasawa, H. Mukai, and K. Ohmatsu,
“Finite element analysis of AC loss in non-twisted Bi-2223 tape carrying
AC transport current and/or exposed to DC or AC external magnetic field,”
Physica C, vol. 310, pp. 30–35, 1998. 37
[56] J. K. Sykulski, R. L. Stoll, A. E. Mahdi, and C. P. Please, “Modelling
HTc superconductors for AC power loss estimation,” IEEE Trans. Magn.,
vol. 33, no. 2, pp. 1568–1571, 1997. 37, 89
[57] J. K. Sykulski, M. Rotaru, and R. L. Stoll, “Highly non-linear field diffusion
in HTC superconducting tapes,” COMPEL, vol. 18, no. 2, pp. 215–224,
1999. 37
[58] ——, “2D modeling and field diffusion and AC losses in high temperature
superconducting tapes,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1178–1182,
2000. 37
[59] K. Kajikawa, “Numerical evaluation of AC losses in HTS wires with 2D
FEM formulated by self magnetic field,” IEEE Trans. Appl. Supercond.,
vol. 13, no. 2, pp. 3630–3633, 2003. 37
[60] R. Brambilla, F. Grilli, and L. Martini, “Development of an edge-element
model for AC loss computation of high-temperature superconductors,” Su-
percond. Sci. Technol., vol. 20, pp. 16–24, 2007. 37
145
REFERENCES
[61] F. Sirois, M. Dione, F. Roy, F. Grilli, and B. Dutoit, “Evaluation of two
commercial finite element packages for calculating AC losses in 2-D high
temperature superconducting strips,” J. Phys.: Conf. Ser., vol. 97, no.
012030, 2008. 37
[62] F. Grilli, R. Brambilla, and L. Martini, “Modeling High-Temperature Su-
perconducting Tapes by Means of Edge Finite Elements,” IEEE Trans.
Appl. Supercond., vol. 17, no. 2, pp. 3155–3158, 2007. 37, 41, 52, 78
[63] P. Vanderbemden, Z. Hong, T. A. Coombs, S. Denis, M. Ausloos,
J. Schwartz, I. B. Rutel, N. H. Babu, D. A. Cardwell, and A. M. Campbell,
“Behaviour of bulk high-temperature superconductors of finite thickness
subjected to crossed magnetic fields: experiment and model,” Phys. Rev.
B, vol. 75, no. 174515, 2007. 38
[64] Y. Mawatari, “Critical state of periodically arranged superconducting-strip
lines in perpendicular fields,” Phys. Rev. B, vol. 54, pp. 13 215–13 221, 1996.
41
[65] K.-H. Muller, “Self-field hysteresis loss in periodically arranged supercon-
ducting strips,” Physica C, vol. 289, no. 1-2, pp. 123–130, 1997. 41
[66] J. R. Clem, J. H. Claassen, and Y. Mawatari, “AC losses in a finite Z
stack using an anisotropic homogeneous-medium approximation,” Super-
cond. Sci. Technol., vol. 20, pp. 1130–1139, 2007. 41, 47, 78
[67] W. Yuan, A. M. Campbell, and T. A. Coombs, “A model for calculating the
AC losses of second-generation high temperature superconductor pancake
coils,” Supercond. Sci. Technol., vol. 22, no. 075028, 2009. 41, 47, 97
[68] ——, “ac losses and field and current density distribution during a full cycle
of a stack of superconducting tapes,” J. Appl. Phys., vol. 107, no. 093909,
2010. 41
[69] W. Yuan, A. M. Campbell, Z. Hong, M. D. Ainslie, and T. A. Coombs,
“Comparison of AC losses, magnetic field/current distributions and critical
currents of superconducting circular pancake coils and infinitely long stacks
146
REFERENCES
using coated conductors,” Supercond. Sci. Technol., vol. 23, no. 085011,
2010. 41
[70] E. Pardo, “Modeling of coated conductor pancake coils with a large number
of turns,” Supercond. Sci. Technol., vol. 21, no. 065014, 2008. 41, 78
[71] L. Prigozhin and V. Sokolovsky, “Computing AC losses in stacks of high-
temperature superconducting tapes,” Supercond. Sci. Technol., vol. 24, no.
075012, 2011. 41, 78
[72] D. N. Nguyen, S. P. Ashworth, J. O. Willis, F. Sirois, and F. Grilli, “A new
finite-element method simulation model for computing AC loss in roll as-
sisted biaxially textured substrate YBCO tapes,” Supercond. Sci. Technol.,
vol. 23, no. 025001, 2010. 41, 66, 67, 76
[73] D. N. Nguyen, J. Y. Coulter, J. O. Willis, S. P. Ashworth, H. P. Kraemer,
W. Schmidt, B. Carter, and A. Otto, “AC loss and critical current char-
acterization of a noninductive coil of two-in-hand RABiTS YBCO tape for
fault current limiter applications,” Supercond. Sci. Technol., vol. 24, no.
035017, 2011. 41
[74] Y. Jiang, R. Pei, Q. Jiang, Z. Hong, and T. A. Coombs, “Control of a
superconducting synchronous motor,” Supercond. Sci. Technol., vol. 20, pp.
392–396, 2007. 42
[75] Y. Jiang, R. Pei, Z. Hong, J. Song, F. Fang, and T. A. Coombs, “Design
and control of a superconducting permanent magnet synchronous motor,”
Supercond. Sci. Technol., vol. 20, pp. 585–591, 2007. 42
[76] Y. Jiang, R. Pei, Z. Hong, Q. Jiang, and T. A. Coombs, “Design of an HTS
motor,” J. of Phys.: Conf. Ser., vol. 97, no. 012123, 2008. 42, 43, 46
[77] Y. Jiang, R. Pei, W. Xian, Z. Hong, and T. A. Coombs, “The design, mag-
netization and control of a superconducting permanent magnet synchronous
motor,” Supercond. Sci. Technol., vol. 21, no. 065011, 2008. 42, 94
147
REFERENCES
[78] Y. Jiang, R. Pei, W. Xian, Z. Hong, W. Yuan, R. Marchant, and T. A.
Coombs, “Magnetization process of an HTS motor and the torque ripple
suppression,” IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 1644–1647,
2009. 42
[79] W. Xian, W. Yuan, and T. A. Coombs, “Numerical assessment of efficiency
and control stability of an HTS synchronous motor,” J. of Phys.: Conf.
Ser., vol. 234, no. 032063, 2010. 42
[80] R. Pei, A. Velichko, M. Majoros, Y. Jiang, R. Viznichenko, Z. Hong,
R. Marchant, A. M. Campbell, and T. A. Coombs, “Ic and AC loss of
2G YBCO tape measurement for designing and fabrication of an HTS mo-
tor,” IEEE Trans. Appl. Supercond., vol. 18, no. 2, pp. 1236–1239, Jun
2008. 42, 92
[81] R. Pei, A. Velichko, Z. Hong, Y. Jiang, W. Yuan, A. M. Campbell, and
T. A. Coombs, “Numerical and experimental analysis of Ic and AC loss
for bent 2G HTS wires used in an electric machine,” IEEE Trans. Appl.
Supercond., vol. 19, no. 3, pp. 3356–3360, 2009. 42
[82] Y. Jiang, “The design and control of a superconducting motor,” Ph.D.
dissertation, University of Cambridge, Cambridge, UK, 2009. 44
[83] R. Brambilla, F. Grilli, L. Martini, and F. Sirois, “Integral equations for the
current density in thin conductors and their solution by the finite-element
method,” Supercond. Sci. Technol., vol. 21, no. 105008, 2008. 56
[84] V. M. Zermeno-Rodriguez, M. P. Sorensen, N. F. Pedersen, N. Mijatovic,
and A. Abrahamsen, “Fast 2D simulation of superconductors: a multiscale
approach.” 61
[85] V. M. Zermeno-Rodriguez, N. Mijatovic, C. Traeholt, T. Zirngibl, E. Seiler,
A. B. Abrahamsen, N. F. Pedersen, and M. P. Sorensen, “Towards faster
FEM simulation of thin film superconductors: a multiscale approach,”
IEEE Trans. Appl. Supercond., vol. 21, no. 3, pp. 3273–3276, 2011. 61
148
REFERENCES
[86] D. Miyagi, Y. Yunoki, M. Umabachi, N. Takahashi, and O. Tsukamoto,
“Measurement of magnetic properties of Ni-alloy substrate of HTS coated
conductor in LN2,” Physica C, vol. 468, pp. 1743–1746, 2008. 66, 75
[87] S. Kapa and P. Capper, Eds., Springer Handbook of Electronic and Photonic
Materials, 1st ed. Springer, 2007, p. 1195. 70, 71
[88] S. P. Mehta, N. Aversa, and M. S. Walker, “Tranforming transformers,”
IEEE Spectrum, vol. 34, no. 7, pp. 43–49, 1997. 71
[89] PowerStream Technologies. American Wire Gauge table and AWG
Electrical Current Load Limits with skin depth frequencies. [Online].
Available: http://www.powerstream.com/Wire Size.htm 72
[90] M. D. Ainslie, Y. Jiang, W. Xian, Z. Hong, W. Yuan, R. Pei, T. J. Flack,
and T. A. Coombs, “Numerical analysis and finite element modelling of an
HTS synchronous motor,” Physica C, vol. 450, no. 2, pp. 1752–1755, 2010.
73, 74
[91] J. Evetts, Ed., Concise Encyclopedia of Magnetic & Superconducting Ma-
terials, 1st ed. Pergamon Press, 1992, pp. 19–22. 91
[92] C. Schmidt, K. Itoh, and H. Wada, “Second VAMAS a.c. loss measurement
intercomparison: a.c. magnetization measurement of hysteresis and cou-
pling losses in NbTi multifilamentary strands,” Cryogenics, vol. 37, no. 2,
pp. 77–89, 1997. 91
[93] P. Dolez, M. Aubin, W. Zhu, and J. Cave, “A comparison between ac
losses obtained by the null calorimetric and a standard electrical method,”
Supercond. Sci. Technol., vol. 11, pp. 1386–1390, 1998. 92
[94] A. E. Mahdi et al., “Thermometric measurements of the self-field losses
in silver sheathed PbBi2223 multifilimentary tapes,” IEEE Trans. Appl.
Supercond., vol. 7, no. 2, pp. 1658–1661, 1997. 92
[95] J.-H. Kim, C. H. Kim, G. Iyyani, J. Kvitkovic, and S. Pamidi, “Trans-
port AC loss measurements in superconducting coils,” IEEE Trans. Appl.
Supercond., vol. 21, no. 3, pp. 3269–3272, 2011. 92
149
REFERENCES
[96] J. Kokavec, I. Hlasnik, and S. Fukui, “Very sensitive electric method for
AC measurement in SC coils,” IEEE Trans. Appl. Supercond., vol. 3, no. 1,
pp. 153–155, 1993. 92
[97] H. Okamoto, F. Sumiyoshi, K. Miyoshi, and Y. Suzuki, “The nitrogen boil-
off method for measuring AC loss in HTS coils,” IEEE Trans. Appl. Super-
cond., vol. 16, no. 2, pp. 105–108, 2006. 92
[98] B. Armen. Phase sensitive detection:
the lock-in amplifier. [Online]. Available:
http://www.phys.utk.edu/labs/modphys/Lock-InAmplifierExperiment.pdf
93
[99] W. Yuan, “Second-generation high-temperature superconducting coils and
their applications for energy storage,” Ph.D. dissertation, University of
Cambridge, Cambridge, UK, 2010. 94, 107
[100] H. Daffix and P. Tixador, “Electrical AC loss measurements in supercon-
ducting coils,” IEEE Trans. Appl. Supercond., vol. 7, no. 2, pp. 286–289,
1997. 94
[101] M. P. Oomen, “AC Losses in HTS Conductors, Cables and Coils,” in Flux
Pinning and AC Loss Studies on YBCO Coated Conductors, 1st ed., M. P.
Paranthaman and V. Selvamanickam, Eds. Nova. 97
[102] L. Jansak, “ac self-field loss measurement system,” Rev. Sci. Instrum.,
vol. 70, no. 7, pp. 3087–3091, 1999. 103, 104, 116
[103] S. Ashworth, Los Alamos National Laboratory, “Measurement of AC losses
in HTS conductors,” ASC 2008 Short Course: Testing of superconducting
wires and coils. 104
[104] M. P. Oomen, W. Herkert, D. Bayer, W. Nick, and T. Arndt, “Manufactur-
ing and test of 2G-HTS coils for rotating machines: Challenges, conductor
requirements, realisation,” 2011, EUCAS 2011 conference invited presenta-
tion. 107
150
REFERENCES
[105] T. Takematsu, R. Hu, T. Takao, Y. Yanagisawa, H. Nakagome, D. Uglietti,
T. Kiyoshi, M. Takahashi, and H. Maeda, “Degradation of the performance
of a YBCO-coated conductor double pancake coil due to epoxy impregna-
tion,” Physica C, vol. 470, no. 17-18, pp. 674–677, 2010. 107
[106] M. Sugano, T. Nakamura, K. Shikimachi, N. Hirano, and S. Nagaya, “Stress
tolerance and fracture mechanism of solder joint of YBCO coated conduc-
tors,” IEEE Trans. Appl. Supercond., vol. 17, no. 2, pp. 3067–3070, 2007.
107
[107] D. Miyagi and O. Tsukamoto, “Characteristics of AC transport current
losses in YBCO coated conductors and their dependence on distributions of
critical current density in the conductors,” IEEE Trans. Appl. Supercond.,
vol. 12, no. 1, pp. 1628–1631, 2002. 108, 110
[108] F. Grilli and S. P. Ashworth, “Measuring transport AC losses in YBCO-
coated conductor coils,” Supercond. Sci. Technol., vol. 20, pp. 794–799,
2007. 108
[109] ——, “Quantifying AC losses in YBCO coated conductor coils,” IEEE
Trans. Appl. Supercond., vol. 17, no. 2, pp. 3187–3190, 2007. 108
[110] S. Ashworth, D. Nguyen, J.-H. Kim, C. H. Kim, and S. Pamidi, “Exper-
imental and numerical studies for ac losses in IBAD and RABiTS YBCO
coils,” 2011, 2011 CEC/ICMC conference presentation. 110
[111] M. Polak, L. Krempasky, S. Chromik, D. Wehler, and B. Moenter, “Mag-
netic field in the vicinity of YBCO thin film strip and strip with filamentary
structure,” Physica C, vol. 372-376, pp. 1830–1834, 2002. 119
[112] M. Majoros, R. I. Tomov, B. A. Glowacki, A. M. Campbell, and C. E.
Oberly, “Hysteretic losses in YBCO coated conductors on textured metallic
substrates,” IEEE Trans. Appl. Supercond., vol. 13, pp. 3626–3629, 2003.
119
[113] N. Amemiya, K. Yoda, S. Kasai, Z. Jiang, G. A. Levin, P. N. Barnes, and
C. E. Oberly, “AC loss characteristics of multifilamentary YBCO coated
151
REFERENCES
conductors,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, p. 1637, 2005.
119
[114] N. Amemiya, Q. Li, R. Nishino, K. Takeuchi, T. Nakamura, K. Ohmatsu,
M. Ohya, O. Maruyama, T. Okuma, and T. Izumi, “Lateral critical current
density distributions degraded near edges of coated conductors through
cutting processes and their influence on ac loss characteristics of power
transmission cables,” Physica C, vol. Article in press, 2011. 119
[115] C. B. Cobb, P. N. Barnes, T. J. Haugan, J. Tolliver, E. Lee, M. Sumption,
E. Collings, and C. E. Oberly, “Hysteretic loss reduction in striated YBCO,”
Physica C, vol. 382, no. 1, pp. 52–56, 2002. 119
[116] M. Polak, E. Usak, L. Jansak, E. Demencik, G. A. Levin, P. N. Barnes,
D. Wehler, and B. Moenter, “Coupling losses and transverse resistivity
of multifilament YBCO coated superconductors,” J. Phys.: Conf. Ser.,
vol. 43, pp. 591–594, 2006. 119
[117] M. Polak, J. Kvitkovic, P. Mozola, P. N. Barnes, and G. A. Levin, “Char-
acterization of Individual Filaments in a Multifilamentary YBCO Coated
Conductor,” IEEE Trans. Appl. Supercond., vol. 17, no. 2, pp. 3163–3166,
2007. 119
[118] N. Amemiya, S. Kasai, K. Yoda, Z. Jiang, G. A. Levin, P. N. Barnes, and
C. E. Oberly, “AC loss reduction of YBCO coated conductors by multifila-
mentary structure,” Supercond. Sci. Technol., vol. 17, pp. 1464–1471, 2004.
119
[119] M. Majoros, B. A. Glowacki, and A. M. Campbell, “Transport ac losses
and screening properties of Bi-2223 multifilamentary tapes covered with
magnetic materials,” Physica C, vol. 338, pp. 251–262, 2000. 119, 122
[120] S. Terzieva, M. Vojenciak, F. Grilli, R. Nast, J. Souc, W. Goldacker,
A. Jung, A. Kudymow, and A. Kling, “Investigation of the effect of striated
strands on the AC losses of 2G Roebel cables,” Supercond. Sci. Technol.,
vol. 24, no. 045001, 2011. 119
152
REFERENCES
[121] G. Majkic, I. Kesgin, Y. Zhang, Y. Qiao, R. Schmidt, and V. Selvaman-
ickam, “Ac loss filamentization of 2G HTS tapes by buffer stack removal,”
IEEE Trans. Appl. Supercond., vol. 21, no. 3, pp. 3297–3300, 2011. 119
[122] B. W. Han, S. H. Park, W. S. Kim, J. K. Lee, S. Lee, S. B. Byun, C. Park,
B. W. Lee, K. Choi, and S. Hahn, “Non-contact measurement of current
distribution in striated CTCC,” IEEE Trans. Appl. Supercond., vol. 20,
no. 3, pp. 1952–1955, 2010. 119
[123] M. Polak, S. Takacs, P. N. Barnes, and G. A. Levin, “The effect of resis-
tive filament interconnections on coupling losses in filamentary YBa2Cu3O7
coated conductors,” Supercond. Sci. Technol., vol. 22, no. 025016, 2009. 119
[124] W. Goldacker, A. Frank, R. Heller, S. I. Schlachter, B. Ringsdorf, K.-R.
Weiss, C. Schmidt, and S. Schuller, “ROEBEL Assembled Coated Conduc-
tors (RACC): Preparation, Properties and Progress,” IEEE Trans. Appl.
Supercond., vol. 17, no. 2, pp. 3398–3401, 2007. 120, 121
[125] D. S. Beard, W. Klose, S. Shimamoto, and G. Vecsey, “The IEA Large Coil
Task; Development of Superconducting Toroidal Field Magnets for Fusion
Reactors,” Fus. Eng. and Design, vol. 7, 1988. 120
[126] M. S. Lubell et al., “The IEA Large Coil Task Test Results in IFSMTF,”
IEEE Trans. Magn., vol. 24, no. 2, pp. 761–766, 1988. 120
[127] M. Leghissa, V. Hussennether, and H.-W. Neumuller, “kA-class high-
current HTS conductors and windings for large scale applications,” Ad-
vances in Science and Technology, vol. 47, pp. 212–219, 2006. 120
[128] W. Goldacker, R. Nast, G. Kotzyba, S. I. Schlachter, A. Frank, B. Rings-
dorf, C. Schmidt, and P. Komarek, “High current DyBCO-ROEBEL As-
sembled Coated Conductor (RACC),” J. of Phys.: Conf. Ser., vol. 43, pp.
901–904, 2006. 120
[129] N. J. Long, R. Badcock, P. Beck, M. Mulholl, N. Ross, M. Staines, H. Sun,
J. Hamilton, and R. G. Buckley, “Narrow strand YBCO Roebel cable for
lowered AC loss,” J. of Phys.: Conf. Ser., vol. 97, no. 012280, 2008. 120
153
REFERENCES
[130] K. P. Thakur, Z. Jiang, M. P. Staines, N. J. Long, R. A. Badcock, and
A. Raj, “Current carrying capability of HTS Roebel cable,” Physica C, vol.
471, no. 1-2, pp. 42–47, 2011. 120
[131] Z. Jiang, K. P. Thakur, M. Staines, R. A. Badcock, N. J. Long, R. G.
Buckley, A. D. Caplin, and N. Amemiya, “The dependence of AC loss
characteristics on the spacing between strands in YBCO Roebel cables,”
Supercond. Sci. Technol., vol. 24, no. 065005, 2011. 120
[132] L. S. Lakshmi, N. J. Long, R. A. Badcock, M. P. Staines, Z. Jiang, K. P.
Thakur, and J. Emhofer, “Magnetic and transport AC losses in HTS Roebel
cable,” IEEE Trans. Appl. Supercond., vol. 21, no. 3, pp. 3311–3315, 2011.
120
[133] Z. Jiang, R. A. Badcock, N. J. Long, M. Staines, K. P. Thakur, L. S. Lak-
shmi, A. Wright, K. Hamilton, G. N. Sidorov, R. G. Buckley, N. Amemiya,
and A. D. Caplin, “Transport AC loss characteristics of a nine strand YBCO
Roebel cable,” Supercond. Sci. Technol., vol. 23, no. 025028, 2010. 120
[134] N. J. Long, R. A. Badcock, K. Hamilton, A. Wright, Z. Jiang, and L. S.
Lakshmi, “Development of YBCO Roebel cables for high current transport
and low AC loss applications,” J. of Phys.: Conf. Ser., vol. 234, no. 022021,
2010. 120
[135] L. S. Lakshmi, M. P. Staines, R. A. Badcock, N. J. Long, M. Majoros,
E. W. Collings, and M. D. Sumption, “Frequency dependence of magnetic
ac loss in a Roebel cable made of YBCO on a Ni-W substrate,” Supercond.
Sci. Technol., vol. 23, no. 085009, 2010. 120
[136] Z. Jiang, M. Staines, R. A. Badcock, N. J. Long, and N. Amemiya, “Trans-
port AC loss measurement of a five strand YBCO Roebel cable,” Supercond.
Sci. Technol., vol. 22, no. 095002, 2009. 120
[137] M. N. Wilson, “Lecture 3: Cables and quenching,” Superconducting Mag-
nets for Accelerators, JUAS, Feb 2006. 120
154
REFERENCES
[138] D. Ciazynski, B. Turck, J. L. Duchateau, and C. Meuris, “AC losses and
current distribution in 40 ka NbTi and Nb3Sn conductors for NET/ITER,”
IEEE Trans. Appl. Supercond., vol. 3, no. 1, 1993. 120
[139] M. Sumption, R. M. Scanlan, and E. W. Collings, “Coupling current control
in Rutherford cables wound with NbTi, Nb3Sn, and Bi:2212/Ag,” Physica
C, vol. 310, pp. 291–295, 1998. 120
[140] M. N. Wilson, “NbTi superconductors with low ac loss: A review,” Physica
C, vol. 48, pp. 381–395, 2008. 120
[141] J. Yoo, J. Ko, H. Kim, and H. Chung, “Fabrication of Twisted Multi-
filamentary BSCCO 2223 Tapes by Using High Resistive Sheath for AC
Application,” IEEE Trans. Appl. Supercond., vol. 9, no. 2, 1999. 120
[142] Z. Jiang, N. Amemiya, N. Ayai, and K. Hayashi, “AC Loss Measurements
of Twisted and Non-Twisted BSCCO Tapes in Transverse Magnetic Field
With Various Directions,” IEEE Trans. Appl. Supercond., vol. 13, no. 2,
2003. 120
[143] Z. Jiang, N. Amemiya, T. Nishioka, and S.-S. Oh, “AC loss measurements of
twisted and untwisted BSCCO multifilamentary tapes,” Cryogenics, vol. 45,
no. 1, pp. 29–34, 2005. 120
[144] A. Kawagoe, F. Sumiyoshi, Y. Fukushima, Y. Wakabayashi, T. Mito,
N. Yanagi, M. Takahashi, and M. Okada, “Critical Currents and AC Losses
in MgB2 Multifilamentary Tapes With 6 Twisted Filaments,” IEEE Trans.
Appl. Supercond., vol. 19, no. 3, pp. 2686–2689, 2009. 120
[145] A. Malagoli, C. Bernini, V. Braccini, C. Fanciulli, G. Romano, and M. Vi-
gnolo, “Fabrication and superconducting properties of multifilamentary
MgB2 conductors for AC purposes: twisted tapes and wires with very thin
filaments,” Supercond. Sci. Technol., vol. 22, no. 105017, 2009. 120
[146] M. Polak, E. Demencik, I. Husek, L. Kopera, P. Kovac, P. Mozola, and
S. Takacs, “AC losses and transverse resistivity in filamentary MgB2 tape
with Ti barriers,” Physica C, vol. 471, pp. 389–394, 2011. 120
155
REFERENCES
[147] C. E. Oberly, B. Razidlo, and F. Rodriguez, “Conceptual Approach to
the Ultimate Low AC Loss YBCO Superconductor,” IEEE Trans. Appl.
Supercond., vol. 15, no. 2, pp. 1643–1646, 2005. 121
[148] M. Majoros, B. A. Glowacki, and A. M. Campbell, “Modelling of the In-
fluence of Magnetic Screening on Minimisation of Transport AC Losses in
Multifilamentary Superconductors,” IEEE Trans. Appl. Supercond., vol. 11,
no. 1, pp. 2780–2783, 2001. 122
[149] M. Majoros, M. D. Sumption, and E. W. Collings, “Transport AC Loss
Reduction in Striated YBCO Coated Conductors by Magnetic Screening,”
IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 3352–3355, 2009. 122
[150] F. Gomory, M. Vojenciak, E. Pardo, and J. Souc, “Magnetic flux penetra-
tion and AC loss in a composite superconducting wire with ferromagnetic
parts,” Supercond. Sci. Technol., vol. 22, no. 034017, 2009. 122
[151] E. Pardo, J. Souc, and M. Vojenciak, “AC loss measurement and simulation
of a coated conductor pancake coil with ferromagnetic parts,” Supercond.
Sci. Technol., vol. 22, no. 075007, 2009. 122, 123
[152] P. Kovac, I. Husek, T. Melisek, M. Ahoranta, J. Souc, J. Lehtonen, and
F. Gomory, “Magnetic interaction of an iron sheath with a superconductor,”
Supercond. Sci. Technol., vol. 16, pp. 1195–1201, 2003. 122
[153] S. Safran, F. Gomory, and A. Gencer, “AC loss in stacks of Bi-2223/Ag
tapes modified with ferromagnetic covers at the edges,” Supercond. Sci.
Technol., vol. 23, no. 105003, 2010. 122
[154] M. Vojenciak, J. Souc, and F. Gomory, “Critical current and AC loss anal-
ysis of a superconducting power transmission cable with ferromagnetic di-
verters,” Supercond. Sci. Technol., vol. 24, no. 075001, 2011. 123
[155] H. Wen, W. Bailey, K. Goddard, M. Al-Mosawi, C. Beduz, and Y. Yang,
“Performance Test of a 100 kW HTS Generator Operating at 67 K-77 K,”
IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 1652–1655, 2009. 123
156
REFERENCES
[156] L. Rostila, Tampere University of Technology, “Introduction to AC losses,”
2008, ESAS Summer School on Superconductivity, Finland. 135
157