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Electronic Theses, Treatises and Dissertations The Graduate School

2007

Alternating Current Loss Characteristicsin (Bi,Pb)#Sr#Ca#Cu#O## andYBa#Cu#O[subscript 7-#] SuperconductingTapesHau T. B. Nguyen

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THE FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

ALTERNATING CURRENT LOSS CHARACTERISTICS IN (Bi,Pb)2Sr2Ca2Cu3O10 AND

YBa2Cu3O7-δ SUPERCONDUCTING TAPES

By

DOAN NGOC NGUYEN

A Dissertation submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Degree Awarded: Summer Semester, 2007

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The members of the Committee approve the Dissertation of Doan N Nguyen defended on May 31, 2007.

Justin Schwartz

Professor Co-Directing Dissertation

Gregory Boebinger Professor Co-Directing Dissertation Timothy M. Logan Outside Committee Member Linda Hirst Committee Member Oskar Vafek Committee Member

Approved: David Van Winkle, Chair, Department of Physics

Joseph Travis, Dean, College of Arts and Sciences The Office of Graduate Studies has verified and approved the above named committee members.

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This dissertation is dedicated to my parents and to my wife, Hau T.B. Nguyen.

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ACKNOWLEDGEMENTS

This work has been partially supported by the Office of Naval Research and the Department of

Energy through the Center for Advanced Power Systems.

I would like to express sincere appreciation to my advisor, Professor Justin Schwartz for being

an excellent advisor throughout my graduate study at Florida State University. This dissertation

would not be complete without his generous support, scientific guidance and encouragement. I

would like to thank my former co-advisor, Professor Jack Crow and my current co-advisor,

Professor Gregory Boebinger for their support and valuable advice. I would especially like to

express my gratitude to Dr. Sastry Pamidi for his helpful discussions and financial support for

my research. Without his help, this dissertation would not be complete. I also would like to take

this opportunity to thank to David Knoll, John Hauer, Steve Rainer, Danny Crook and other

support staff at the Center for Advanced Power Systems, National High Magnetic Field

Laboratory, Department of Physics for their willingness to help me every time I ask for. IGC

SuperPower and American Superconductor Corporation are acknowledged for supplying

superconducting samples used in this work.

Furthermore, I would like to thank my colleagues and group members: Ulf Trociewitz, Xiaorong

Wang, Guomin Zhang for their help in many different ways. Thanks also go to my Vietnamese

friends in Tallahassee for their friendship and encouragement.

I would also like to thank the members of my Ph.D. committee: Professor Linda Hirst, Professor

Oskar Vafek and Professor Timothy Logan for their critical review of this work. I also thank all

others who are not acknowledged by name but helped me to complete this work.

Lastly, from the bottom of my heart, I would like to take this opportunity to express my love and

appreciation to my wife, Hau Nguyen, who always stands by and supports me. I also would like

to thank my parents and my sister for their continuous support and encouragement.

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TABLE OF CONTENTS

List of Tables…… ........................................................................................................... ix

List of Figures………...................................................................................................... x

Abstract………................................................................................................................ xix

1. INTRODUCTION ....................................................................................................... 1

1.1. Brief history of superconductivity ................................................................... 1

1.2. Basic physical properties of superconductors.................................................. 3

1.3. Bi-2223 and YBCO high temperature superconductors .................................. 5

1.3.1. Bi-2223 superconducting tapes............................................................... 6

1.3.2. YBCO superconducting tapes................................................................. 7

1.4. Application of HTS conductors ....................................................................... 10

1.5. AC loss in HTS conductors ............................................................................. 11

1.6. Aim and outline of the thesis ........................................................................... 12

2. BASIC CONCEPTS AND THEORIES FOR AC LOSSES IN HTS CONDUCTOR 15

2.1. Basic concepts and theories of superconductivity ........................................... 15

2.1.1. The London (penetration) theory............................................................ 15

2.1.2. The Ginsburg-Landau (G-L) Theory ...................................................... 17

2.1.3. BCS theory.............................................................................................. 19

2.1.4. The mixed state and motion of vortices in Type-II superconductors.. 19

2.1.5. Critical current density in HTS conductor ........................................... 21

2.1.6. Critical state model and the power-law for HTS conductor................ 22

2.2. Theory for AC loss in HTS tapes..................................................................... 26

2.2.1. AC loss mechanisms ............................................................................... 27

2.2.1.1. Hysteresis losses in superconductors.......................................... 27

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2.2.1.2. Eddy current and coupling losses ............................................... 28

2.2.1.3. Ferromagnetic loss ...................................................................... 31

2.2.1.4.Flux flow and resistive loss ......................................................... 32

2.2.2. AC loss models for a slab ....................................................................... 32

2.2.2.1. An infinite slab in parallel AC magnetic field............................ 32

2.2.2.2. Infinite slab carries an AC current in a parallel magnetic field .. 34

2.2.3. AC loss models for an ellipse tape.......................................................... 36

2.2.3.1. AC loss in an ellipse tape carrying AC transport current ........... 37

2.2.3.2. AC loss in an ellipse tape in AC applied field............................ 38

2.2.4. AC loss in a thin strip HTS tape ............................................................. 45

2.2.4.1. Strip conductor carrying a transport current ............................... 46

2.2.4.2. Strip conductor in a perpendicular AC field............................... 50

2.2.4.3. Total AC loss in a strip ............................................................... 53

3. NUMERICAL CALCULATIONS .............................................................................. 57

3.1. Introduction...................................................................................................... 57

3.2. Brandt model.................................................................................................... 58

3.3. Numerical calculation for a YBCO coated conductor ..................................... 61

3.3.1. AC loss in the HTS layer ........................................................................ 61

3.3.2. Eddy current loss .................................................................................... 65

3.3.3. Ferromagnetic loss .................................................................................. 67

3.4. Effect of the initial condition ........................................................................... 68

3.5. Electrodynamics and AC loss in HTS tapes with rectangular cross-section ... 70

3.5.1. Electrodynamics...................................................................................... 70

3.5.2. AC loss in rectangular tapes ................................................................... 74

3.6. Electrodynamics and AC loss in YBCO tapes................................................. 80

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3.6.1. Conductor in perpendicular magnetic field ............................................ 80

3.6.2. Conductor with a transport current ......................................................... 84

3.6.3. Current carrying conductor in a perpendicular applied magnetic field .. 86

3.6.4. AC loss and the critical current distribution ........................................... 87

3.7. Chapter summary ............................................................................................. 90

4. AC LOSS MEASUREMENTS.................................................................................... 91

4.1. Introduction...................................................................................................... 91

4.2. Measurement principle of the electromagnetic method................................... 92

4.2.1. Transport current loss measurement ....................................................... 95

4.2.2. Magnetization loss measurement............................................................ 95

4.2.3. Total AC loss measurement .................................................................... 99

4.3. Experimental setup .......................................................................................... 100

4.3.1. Electrical setup........................................................................................ 100

4.3.2. Magnet .................................................................................................... 103

4.4. Variable temperature measurement by the electromagnetic method............... 105

4.4.1. Measurement design ............................................................................... 105

4.4.2. Measurement procedure.......................................................................... 109

. 4.5. Errors in the electromagnetic method.............................................................. 111

4.5.1. Phase error .............................................................................................. 111

4.5.2. Error from common mode signal ............................................................ 112

4.5.3. Error from the voltage loop geometry .................................................... 113

4.5.3. Error from the empty coil effect ............................................................. 113

4.6. Calorimetric method ........................................................................................ 113

4.6.1. Measurement procedure.......................................................................... 114

4.6.2. Error and limitation of the calorimetric method ..................................... 117

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5. AC LOSS CHARACTERISTICS IN (Bi,Pb)2Sr2Ca2Cu3O10 AND YBa2Cu3O7-δ TAPES…………… ................................................................................................... 119

5.1. Sample specification ........................................................................................ 119

5.2. Field dependence of Jc and n-value ................................................................. 121

5.3. Comparison between calorimetric and electromagnetic methods ................... 124

5.4. Self-field AC loss in HTS tapes....................................................................... 127

5.4.1. AC loss components in the self-field loss............................................... 127

5.4.2. Self-field AC loss in a DC magnetic field .............................................. 130

. 5.4.3. Frequency dependence of self-field AC loss .......................................... 137

5.5. Magnetization loss at 77 K .............................................................................. 139

5.6. Total AC loss at 77 K ...................................................................................... 144

5.6.1. Total AC loss in a Bi-2223 tape.............................................................. 144

5.6.2. Total AC loss in YBCO tapes................................................................. 146

. 5.6.3. Dependence of phase difference on AC losses ....................................... 150

5.7. Temperature dependence of AC loss characteristics ....................................... 154

5.7.1. Temperature dependence of Ic and n-value............................................. 154

5.7.2. Self-field AC loss at variable temperatures ............................................ 155

. 5.7.3. Magnetization at variable temperature ................................................... 159

5.7.4. Total AC loss in variable temperatures................................................... 163

6. SUMARY AND FUTURE WORK ............................................................................. 168

6.1 Primary finding ................................................................................................. 168

6.2. AC loss reduction solutions ............................................................................. 172

6.3. Future work...................................................................................................... 173

REFERENCES ............................................................................................................ 175

BIOGRAPHICAL SKETCH .......................................................................................... 186

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LIST OF TABLES Table 1.1: Superconducting transition temperature records through years [9] ............... 1

Table 4.1: Comparison between electromagnetic and calorimetric methods .................. 92

Table 5.1: Specifications of the Bi-2223 samples ........................................................... 120

Table 5.2: Specifications of the YBCO samples ............................................................. 120

Table 6.1: Measurement capabilities for total AC loss.................................................... 168

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LIST OF FIGURES

Figure 1.1. Resistivity as a function of temperature for a superconductor and a normal conductor ……………………………………………………………………… 1

Figure 1.2. Internal magnetic field Bi and magnetization μ0M as functions of applied magnetic field Ba for type-I (a) and type-II superconductors [9]……………… 3

Figure 1.3. Critical surface limited by three critical parameters, Jc, Bc, and Tc of type II superconductors……………………………………………………………… 4

Figure 1.4. Unit cell of Bi-2223 (a) and YBCO (b)……………………………………. 5

Figure 1.5. Cross-section of a multi filamentary Bi-2223 tape with silver-alloy sheath and matrix……………………………………………………………………… 7

Figure 1.6. Architecture of a YBCO coated conductor………………………………... 8

Figure 1.7. Dependence of the irreversibility magnetic field (red lines) and the upper critical field (grey lines) of Bi-2223, YBCO, MgB2 and other low temperature superconductors [13]…………………………………………………………… 10

Figure 2.1. Exponential decay of the magnetic flux inside a superconductor…………. 16

Figure 2.2. Illustrations of “super-electron” density and magnetic field profile inside type-I (a) and type-II superconductors (b). In a type-II superconductor, a vortex state is formed….……………………………………………………….. 18

Figure 2.3. Typical I-V curves in superconductors, technical critical current is determined when electric field in the conductor reach the criterion 104 V/m… 21

Figure 2.4. Sketch of an “infinite” slab conductor (infinite length and height) placed in xyz-coordinates. External magnetic field is applied parallel to the conductor (in y-direction)…………………………………………………………………. 22

Figure 2.5. Magnetic field and current profiles in an infinite slab when it is exposed in a parallel applied field. Applied field increases from zero to penetration field Bp (a), then increases to peak field B0 (b), decreases from B0 to B0 - 2Bp (c) and then continues decreasing to – B0 (d)………………………………….. 24

Figure 2.6. Eddy and coupling currents in a two-filament conductor when it is placed in a time varying perpendicular field…………………………………………... 28

Figure 2.7. Coupling current in a twisted two-filament conductor……………...……... 29

Figure 2.8. Rectangular (a) and hollow rectangular (b) cross-sections………………... 30

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Figure 2.9. Magnetization loop for an infinite slab in parallel applied magnetic field with the peak values B0. The solid line loop represents B0 > Bp and the dashed line loop represents for B0 < Bp………………………………………………… 31

Figure 2.10. Dependence of reduced loss function in an infinite slab on normalized parallel magnetic field at various transport currents.…………………………... 35

Figure 2.11. A superconducting tape with an elliptical cross-section placed in xyz-coordinates. Transport current flows along the conductor in the z-direction….. 37

Figure 2.12. Current distribution in an elliptical tape when transport current in its descending half-period…………………………………………………………. 38

Figure 2.13. Current distribution in superconducting wire with circular cross-section when applied magnetic field change from zero to a small field Ba……………. 40

Figure 2.14. Normalized magnetization as a function of the normalized applied magnetic field for an elliptical tape with different values of aspect ratio……... 42

Figure 2.15. Penetration field as a function of aspect ratio α ………………………… 43

Figure 2.16. The magnetization loss, Qm, as a function of the normalized magnetic field for elliptical tapes with different aspect ratio…………………………….. 44

Figure 2.17. The loss ratio between magnetization loss in perpendicular and parallel magnetic field, q = Q⊥/Q//, as a function of applied magnetic field for the tape with 1.0=α and 01.0=α ……………………………………………………. 45

Figure 2.18. Thin HTS strip with a metal strip placed in xyz-coordinates. Magnetic field is applied in the x-direction and the transport current flows in the z-direction………………………………………………………………………... 46

Figure 2.19. Current and magnetic profiles in a thin strip conductor when the transport current is in a decreasing half-period, reducing from the positive peak, I0 = 0.95Ic, to the negative peak. The “snapshots” are taken at times such that I(t)/Ic = 0.95, 0.5, 0, -0.5, -0.8 and -0.95 [65]…………………….. ……… 48

Figure 2.20. Reduced self-field loss ( )iqe , ( )iqs and ( )if as a function of the

normalized transport current i………………………………………………….. 49

Figure 2.21. Current and magnetic field profiles in a thin strip conductor when the applied field is in a decreasing half-period, reducing from the positive peak, B0 = 2Bc to the negative peak. The “snapshots” are taken at times such that B(t)/Bc = 2, 1, 0, -1, and –2 [65]……………………………………………….. 51

Figure 2.22. Reduced loss functions ( )ig and ( )ie as a function of the normalized applied magnetic field………………………………………………………….. 52

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Figure 2.23. Normalized total loss in a thin strip HTS conductor as a function of the normalized magnetic field for different transport currents. The dashed line indicates the boundary between high current-low field and low current-high field regimes………………………………………………...…………………. 55

Figure 3.1. A rectangular tape with applied magnetic field and transport current in xyz-coordinates …………………………...…………………………………… 59

Figure 3.2. YBCO conductor with the substrate and stabilizer in xyz-coordinates (not to scale……………………………………………………………………...….. 62

Figure 3.3 An imagined loop on the stabilizer formed by the negative edge and a line at yi position (not to scale)……………………………………………………... 65

Figure 3.4. Calculated electric field along the conductor with perpendicular field Ba = 15.6 mT and transport current It = 75A ……………………………………….. 68

Figure 3.5. Calculated transport losses and total losses obtained from the first, second, third and forth half-periods. For comparison, experimental results of transport losses and total losses are also plotted. Ba = 15.6 mT.……………………….... 69

Figure 3.6. Current density profile captured at (a) ωt = π/2 and (b) ωt = π in 4 mm x 0.25 mm tape when it carries a transport current of 80 A …………………….. 70

Figure 3.7. Current profile in 4 mm x 0.25 mm tape (a, b, c, d) and in square wire (e, f) for perpendicular applied magnetic field of 30 mT ………………………… 71

Figure 3.8. Current profile in 4 mm x 0.25 mm when magnetic field of 30 mT is applied at angle θ = 60°………………………………………………………... 73

Figure 3.9. Current profile in the tape in the high current – low field regime with B = 5 mT, I = 80 A (a, b) and the low current – high field regime with B0 = 30 mT, I0 = 50 A (c, d, e, f)…………………………………………………………….. 74

Figure 3.10. Transport loss in rectangular tapes with different aspect ratios as a function of current amplitude………………………………………………….. 75

Figure 3.11. Numerically calculated magnetization loss of rectangular tapes in perpendicular applied field. Analytical results for elliptical tape of thickness 0.25 mm and width 4 mm is also plotted ……………………………………… 76

Figure 3.12 Magnetization loss in the 4 mm x 0.25 mm tape with applied magnetic field at different angles.………………………………………………………... 77

Figure 3.13. Magnetization loss as a function of the perpendicular field component for different field angles (tape 4 mm x 0.25 mm)……………………………… 77

Figure 3.14. The loss ratio Q (θ = 90°)/Q(θ = 0°) for two tapes: 2.5mm x 0.4mm (α = 1/6.25) and 4 mm x 0.25 mm (α = 1/16)…………………………..…… …….. 78

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Figure 3.15. Dependence of transport and magnetization loss in 4mm x 0.25mm tape on field angle and B = 30 mT.……………………………………..…………... 79

Figure 3.16. Transport and magnetization loss in square wire for several field orientations.…………………………………………………………………….. 80

Figure 3.18. (a) Current density J(y), (b) magnetic field B(y) and (c) eddy current density Je(y) are plotted for the seven times indicated in Fig. 3.17. In (a) and (b), analytical results (solid lines) are plotted for the positive half of the conductor and numerical result (dashed lines) are plotted everywhere. (c) shows only numerical results……………………………………..……………. 82

Figure 3.19. Numerical results of Qm and Qe for frequency of 51 Hz (dotted lines with open symbols) and 151 Hz (dotted lines with filled symbols). Corresponding analytical results (solid lines) are also plotted.…………..……………………. 83

Figure 3.20. (a) Current density J(y), (b) magnetic field B(y) and (c) eddy current density Je(y) are plotted for the seven times indicated in Fig. 3.17. In (a) and (b), analytical results (solid lines) and numerical results (dashed lines) are shown. (c) shows only numerical results ………….…………………………... 85

Figure 3.21. Numerical results (open symbols) of transport loss Qt, and eddy current loss Qe at 51 Hz. Corresponding analytical results for Qt and Qe (filled symbols) are also plotted………………………………………………………. 86

Figure 3.22. The total loss Qs in the superconducting layer is plotted for I/Ic = 0.1, 0.3, 0.5, 0.7 and 0.9. Numerical (open symbols) and analytical results (solid lines) are seen.………………………………………………………………...... 87

Figure 3.23. Elliptical cross-section of a HTS tape in xy plane……………………….. 88

Figure 3.24. Normalized critical current distribution Jc(y)/ Jc0 as a function of y/a for three different Jc(y) distributions ………………………….……………….….. 88

Figure 3.25. Numerical results of the self-field loss as a function of transport current for different Jc(y) distributions. Analytical results obtained from Norris model for elliptical and strip tapes are also plotted………………………………….... 89

Figure 3.26. Numerical results of the magnetization loss as a function of magnetic field for different Jc(y) distributions …………………………………………... 90

Figure 4.1. Sketch of a HTS conductor with a voltage loop to measure transport AC loss……………………………………………………………………………... 93

Figure 4.2. A thin tape HTS of width 2a and an in-plane pick-up coil of width 2W…. 95

Figure 4.3. Dependence of C of pick-up coils with different widths on the applied magnetic field, simulated by Brandt’s equations for a thin strip conductor...…. 97

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Figure 4.4. Calibration factor C calculated by Brandt equations for different values of applied magnetic field. Dependence of C on the pick-up coil width calculated by uniform magnetization approximation (the solid line) is also plotted…….... 98

Figure 4.5. An HTS thin tape of width 2a and a double pick-up coil.……………….… 99

Figure 4.6. Pick-up coil arrangement and a figure “8” shaped voltage loop for total ac loss measurement.………………………….…………………………….…….. 100

Figure 4.7. Electrical setup for total AC loss measurement………………….….…….. 102

Figure 4.8. Picture of the large double-helix magnet and magnetic field configuration of two helically wound coils at opposite tilt angles……………………….…… 104

Figure 4.9. Sketch of the cryostat with the sample holder and magnet for the variable temperature measurement.……………..……………………...……………….. 106

Figure 4.10. Photo and sketch of the sample holders.………………………..….…….. 107

Figure 4.11. Sample temperature profile measured by seven thermocouples T1 to T7 mounted along the sample.……………………….……………………...…….. 108

Figure 4.12. Temperature along the tape when it carries AC transport currents in an AC perpendicular applied field of 50 mT, f = 51 Hz.………………………….. 110

Figure 4.13. Temperature versus times for three locations along the tape during AC loss measurements at I0 = 272 A, B0 = 50 mT and f = 51 Hz …...…………….. 111

Figure 4.14. Arrangement of sample holder for the calorimetric method.…………….. 114

Figure 4.15. Temperature along the tapes during AC loss measurements at I = 272 A, B = 50 and f = 51 Hz…………………………………………………………… 115

Figure 4.16. Temperature rise on the sample when it carries an AC transport current with amplitude varying from 20 A to 193 A in an AC perpendicular magnetic field B0 = 10.4 mT, f = 51 Hz.………………………….…………………...…. 116

Figure 4.17. Calibration curve before and after compensating effect of heat generated in the current leads………………………………..……………………………. 117

Figure 5.1. Normalized critical current Ic(B)/Ic(0) of samples Y1 and Y3 as a function of perpendicular DC magnetic field. ………………………………………..… 121

Figure 5.2. Normalized n-value, n(B)/n(0), of sample Y1 and Y3 as a function of the perpendicular DC magnetic field ……………………………...………….…… 122

Figure 5.3. Normalized critical current and n-value of samples Y1 and Y3 as a function of parallel DC magnetic field ………………………………...……… 124

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Figure 5.4. Self-field power loss in sample Y2 measured by the electromagnetic and calorimetric methods at frequencies 91 Hz and 111 Hz………………..……… 125

Figure 5.5. Magnetization loss in sample B1 measured by the electromagnetic and calorimetric methods at 51 Hz……………..…………………………………... 125

Figure 5.6. The total AC loss in sample Y2 measured by the electromagnetic (open symbols) and calorimetric (filled symbols) methods at 51 Hz……………..….. 126

Figure 5.7. Self-field loss in Bi-2223 samples B1, B2 and B3 measured by the EM method at 51 Hz. Analytical results obtained from Norris’ model for a thin strip and an elliptical tape are also plotted..……...…………………………….. 127

Figure 5.8. Voltage loop configurations to measure the total self-field loss (V1), hysteresis loss (V2) and resistive, flux-flow losses (V3)……………………….. 128

Figure 5.9. The self-field loss components in sample B2 measured from the three voltage loops shown in figure 5.8…………………..………………………….. 129

Figure 5.10. The self-field loss in YBCO samples Y1, Y3 measured by the EM method at 51 Hz. Analytical results obtained from Norris’ model for a thin strip and an elliptical tape are also plotted………...……………………..…….. 129

Figure 5.11. The self-field AC loss in sample Y1 when the perpendicular DC applied magnetic field increases from 0 to 90 mT……………………………...……… 130

Figure 5.12. Self-field AC loss in sample Y1 when parallel DC applied magnetic field increases from 0 to 50 mT …………………………………………………….. 131

Figure 5.13. Self-field loss in sample Y1 as a function of the parallel DC magnetic field at different AC transport current amplitudes.…………………………….. 132

Figure 5.14. Experimental data for dependence of ferromagnetic loss in the substrate on amplitude of parallel DC field [58], the fitting curve from Eqt. 5.9 is also plotted………………………………………………………………………….. 133

Figure 5.15. The self-field loss in sample Y1 measured by the EM method at 51 Hz and calculated AC loss components: the self-field loss in the YBCO layer and the ferromagnetic loss in the substrate…………….…………………………… 134

Figure 5.16. The self-field AC loss in sample Y3 when perpendicular DC applied magnetic field increases from 0 to 70 mT.…………………………………….. 135

Figure 5.17. The self-field AC loss in sample Y3 when parallel DC applied magnetic field increases from 0 to 50 mT.………………………………....…………….. 135

Figure 5.18. Normalized self-field AC loss in sample Y1 when perpendicular DC applied magnetic field increases from 0 to 70 mT.…………….………...…….. 136

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Figure 5.19. Normalized self-field AC loss in sample Y1 when parallel DC applied magnetic field increases from 0 to 50 mT.…………………………………….. 136

Figure 5.20. Self-field AC loss in sample B2 at different frequencies from 51 Hz to 2500 Hz………………………………………………………………………… 138

Figure 5.21. Self-field AC loss in sample Y3 at several frequencies.……….….…….. 139

Figure 5.22. Magnetization loss in sample B1 measured by the CM for different orientation of applied magnetic field. Numerical results are also plotted ......… 140

Figure 5.23. Normalized magnetization loss in YBCO samples Y1 and Y3. The normalized loss function given by Brandt’s equation is also plotted.….…...…. 141

Figure 5.24. Magnetization loss in sample Y1 measured by EM method at 51 Hz. The numerically calculated results for the magnetization AC loss components: hysteresis loss in YBCO layer, ferromagnetic loss in the substrate and the eddy current loss are also plotted.…….…………………………………..……. 142

Figure 5.25. Magnetization loss in sample Y3 measured by EM method at different frequencies, from 51 Hz to 550 Hz …………………………...……..………… 143

Figure 5.26. Magnetization loss in sample Y3 measured by EM method at 51 Hz and 200 Hz. The eddy current loss in the stabilizer at 200 Hz is also plotted...……. 144

Figure 5.27. Transport AC loss component as a function of transport current at different values of perpendicular applied field. The numerical results obtained from a rectangular uniform tape and a thin strip tape with elliptical Jc distribution are also depicted………………………………………….……….. 145

Figure 5.28. The total AC loss as a function of transport current at different values of perpendicular applied field. The numerical results obtained from a rectangular uniform tape and a thin strip tape with elliptical Jc distribution are also depicted…………………………………………………………………..…….. 146

Figure 5.29. The numerical and experimental results of the transport AC loss components in sample Y3 as a function of perpendicular AC field at different transport current values……………………………………………………...…. 147

Figure 5.30. The numerical and experimental results of the magnetization AC loss components in sample Y3 as a function of perpendicular AC field at different transport current values………………………………………………………… 147

Figure 5.31. The numerical and experimental results of the total AC loss in sample Y3 as a function of perpendicular AC field at different transport current levels 148

Figure 5.32. The numerical and experimental results of the transport AC loss components in sample Y1 as a function of perpendicular AC field at different transport current levels………………………………………………………… 149

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Figure 5.33. The numerical and experimental results of the total AC loss in sample Y1 as a function of perpendicular AC field at different transport current levels 149

Figure 5.34. The experimental results of the total AC loss in sample B1 as a function of the transport current and the phase difference Δϕ, B0 = 31.5 mT and f = 51 Hz………………………………………………………………….…………… 151

Figure 5.35. The numerical results of the total AC loss in sample B1 as a function of the transport current and the phase different Δϕ, B0 = 31.5 mT and f = 51 Hz... 151

Figure 5.36. Numerical (the lines) and experimental (the symbols) data of normalized total loss Qt(Δφ )/Qt(Δφ = 0°) in sample B1 as a function of the phase different Δϕ, Ba = 31.5 mT and f = 51 Hz……………………..………………. 152

Figure 5.37. Numerical magnetization loss in sample B1 as a function of the phase different Δϕ, B= 31.5 mT and f = 51 Hz…….………………………………… 153

Figure 5.38. Numerical transport loss in sample B1 as a function of the phase different Δϕ, B= 31.5 mT and f = 51 Hz.……………………………………… 154

Figure 5.39. The DC I-V curves measured in sample Y4 at different temperature……. 155

Figure 5.40. The temperature dependent critical current of sample B2 and sample Y4. 155

Figure 5..41. Temperature dependent self-field loss in sample B2 as a function of the transport current………………………………………………………………... 156

Figure 5.42. Temperature dependent self-field loss in sample Y4 as a function of the transport current………………………………………………………………... 157

Figure 5.43. The normalized self-field loss in sample B2 as a function of the transport current measured at several temperatures……..……………………………….. 158

Figure 5.44. The normalized self-field loss in sample Y4 as a function of the transport current measured at several temperatures…………...……………………...….. 158

Figure 5.45. Magnetization loss in sample Y4 as a function of the temperature and applied magnetic field………………………..……………………………….... 159

Figure 5.46. Magnetization loss in sample Y4 as a function of applied magnetic field. The corresponding analytical results obtain from Brandt equation are also plotted.…………………………………………………………………………. 160

Figure 5.47. Temperature dependent magnetization loss in sample Y4 when magnetic field B0 = 20 mT is applied………………………………………………….… 161

Figure 5.48. Temperature dependence of the transport and magnetization critical current of sample Y4……………………………...……………………………. 162

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Figure 5.49. Temperature dependence of the transport and magnetization critical current of sample Y4…………………………………...…………………...….. 163

Figure 5.50. The experimental and numerical transport loss component in sample Y4 at 45 K………………………………………………………………………..... 164

Figure 5.51. The experimental and numerical magnetization loss component in sample Y4 at 45 K……………………………………………………………... 164

Figure 5.52. Experimental total AC loss in sample Y4 at temperature several temperatures from 45 K to 77 K, B0 = 10 mT, f = 51 Hz……….……………… 165

Figure 5.53. Experimental total AC loss in sample Y4 at several temperatures from 45 K to 77 K, B0 = 50 mT, f = 51 Hz……………………………….……………... 166

Figure 5.54. Total AC loss in sample Y4 measured at 45 K for several of magnetic fields……………………………………………………………………………. 166

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ABSTRACT

Alternating current (AC) loss and current carrying capacity are two of the most crucial

considerations in large-scale power applications of high temperature superconducting (HTS)

conductors. AC losses result in an increased thermal load for cooling machines, and thus

increased operating costs. Furthermore, AC losses can stimulate quenching phenomena or at

least decrease the stability margin for superconducting devices. Thus, understanding AC losses is

essential for the development of HTS AC applications.

The main focus of this dissertation is to make reliable total AC loss measurements and interpret

the experimental results in a theoretical framework. With a specially designed magnet, advanced

total AC loss measurement system in liquid nitrogen (77 K) has been successfully built. Both

calorimetric and electromagnetic methods were employed to confirm the validity of the

measured results and to have a more thorough understanding of AC loss in HTS conductors. The

measurement is capable of measuring total AC loss in HTS tapes over a wide range of frequency

and amplitude of transport current and magnetic field. An accurate phase control technique

allows measurement of total AC loss with any phase difference between the transport current and

magnetic field by calorimetric method. In addition, a novel total AC loss measurement system

with variable temperatures from 30 K to 100 K was successfully built and tested. Understanding

the dependence of AC losses on temperature will enable optimization of the operating

temperature and design of HTS devices.

As a part of the dissertation, numerical calculations using Brandt’s model were developed to

study electrodynamics and total AC loss in HTS conductors. In the calculations, the

superconducting electrical behavior is assumed to follow a power-law model. In general, the

practical properties of conductors, including field-dependence of critical current density Jc, n-

value and non-uniform distribution of Jc, can be accounted for in the numerical calculations. The

numerical calculations are also capable of investigating eddy current loss in the stabilizer and

ferromagnetic loss in the substrate of YBa2Cu3O7-δ (YBCO) coated conductor.

AC loss characteristics and electrodynamics in several (Bi,Pb)2Sr2Ca2Cu3Ox (Bi-2223) and

YBCO tapes were studied experimentally and numerically. It was found that AC loss behavior

xx

in HTS tapes is strongly affected by the sample parameters such as cross-section, structure,

dimensions, critical current distribution as well as by operation parameters including

temperature, frequency, the phase difference between transport current and magnetic field, the

orientation of magnetic field. The Ni-5%W substrate in YBCO conductors generates some

ferromagnetic loss but this loss component is significantly reduced by a small parallel DC

magnetic field. At a given AC magnetic field B0, there is a temperature Tmax at which the

magnetization loss is maximum. The design of HTS devices needs to be optimized to avoid

operating at that temperature. In general, the total AC loss in HTS tapes is still high for many

power device applications, especially for those that present a rather high AC applied magnetic

field. The development of low loss conductors is therefore crucial for HTS large-scale

applications.

1

CHAPTER 1

INTRODUCTION

1.1 Brief history of superconductivity

Superconductors are materials that carry current without resistance when they are cooled below

the critical temperature Tc. Therefore, a superconductor can carry a high electrical current

without dissipation. In the normal state, the resistivity of superconductors decreases as the

temperature decreases. However, when the temperature reaches the critical temperature Tc, the

resistivity of superconductor suddenly drops to zero as seen in Fig. 1.1. At Tc, a phase transition

from normal state to superconducting state occurs.

Fig. 1.1. Resistivity as a function of temperature for a superconductor and a normal conductor

Superconductor

Temperature (K)

Res

isti

vity

Normal conductor

Tc

2

The first superconductor, mercury, with Tc = 4.15 K, was discovered by the Dutch physicist

Kamerlingh Onnes in 1911 [1]. Seventy-five years later, the known superconducting material

holding the record for the highest critical temperature was Nb3Ge, Tc = 23.2 K [2]. All these

“low temperature superconductors” are usually cooled with liquid helium. Such low transition

temperatures make applications of these superconductors expensive and limited.

On April 17, 1986, J. G. Bednorz and K.A. Muller reported their discovery of superconductivity

at 30 K in the Ba-La-Cu-O system [3]. Soon after that, other materials with transition

temperatures above the boiling point of liquid nitrogen (77 K), such as, YBa2Cu3Ox (YBCO),

Bi2Sr2Ca2Cu3Ox (Bi-2223) and Bi2Sr2Ca2Cu1Ox (Bi-2212) [4-7] were discovered. These

discoveries initiated the era of high-temperature superconductivity. Recently, superconductivity

in another class of material, magnesium diboride (MgB2), with critical current of 39 K was

discovered [8]. The main superconductors along with their critical temperature and year of

discovery are summarized in Table 1.1 [9].

Table 1.1. Superconducting transition temperature records through years [9]

Material Tc (K) Year

Hg Pb Nb NbN0.96 Nb3Sn Nb3(Al0.75Ge0.25) Nb3Ga Nb3Ge BaxLa5-xCu5Oy

(La0.9Ba0.1)2Cu4-δ at 1 GPa

YBa2Cu3O7-δ Bi2Sr2Ca2Cu3O10 Tl2Ba2Ca2Cu3O10 Tl2Ba2Ca2Cu3O10 at 7 GPa

HgBa2Ca2Cu3O8+δ

HgBa2Ca2Cu3O8+δ at 25 GPa Hg0.8Pb0.2Ba2Ca2Cu3Ox

HgBa2Ca2Cu3O8+δ at 30 GPa MgB2

4.1 7.2 9.2 15.2 18.1 20-21 20.3 23.2 30-35 52 95 110 125 131 133 155 133 164 39

1911 1913 1930 1950 1954 1966 1971 1973 1986 1986 1987 1988 1988 1993 1993 1993 1994 1994 2001

3

1.2. Basic physical properties of superconductors

There are two essential characteristics distinguishing superconductors from other materials, zero

electrical resistance and the Meissner effect. Zero resistance means there is no voltage drop

along a superconductor when it carries a transport current. The Meissner effect means that

induced surface currents in the superconductor shield the interior from an external magnetic field

(diamagnetism).

Superconductors are sorted into two types, type I and type II which are distinguished by their

magnetic properties. Type I superconductors show perfect diamagnetism at applied magnetic

field lower than the critical field Bc, and are normal for B > Bc. In this type of superconductors, it

is not energetically favorable for the formation of a mixed state that is composed by both

superconducting and normal phases. The magnetization, μ0M, and internal field Bin of an ideal

type-I superconductors are shown in Fig. 1.2(a). The small critical field Bc of type-I

superconductors limits their potential of applications.

The magnetization behavior of type-II superconductors is shown in Fig. 1.2(b). Perfect

diamagnetic behavior is found in the Type-II superconductors when B < Bc1, the lower critical

Fig. 1.2. Internal magnetic field Bi and magnetization μ0M as functions of applied magnetic fieldBa for type-I (a) and type-II superconductors [9]

4

field. When Bc1 < B, magnetic flux penetrates the conductor in form of “flux lines” and the

conductor exists in a mixed state. A flux-line consists of quantized vortices, or fluxons, which

are surrounded by superconducting shielding current. Discussions about vortex and mixed state

in type II superconductors will be given in more detail in chapter 2. When the applied magnetic

field increases, the flux line density increases. When applied field reaches the upper critical field,

Bc2, the vortices overlap and the magnetic field outside and inside the conductor are the same.

For B > Bc2, the conductor is in the normal state. Thus, between Bc1 and Bc2, type-II

superconductors are in the mixed state with partial flux penetration and most applications of

superconductors happen in this state. In many cases, applications are limited by a lower

characteristic field, the irreversibility field, B*(T). At magnetic fields stronger than B*(T) the

critical-current density is nearly zero due to the flowing motion of the flux (flux flow

phenomenon). The irreversibility field is not an intrinsic property of the material. Introducing

more and better pinning centers that hinder the penetration of flux lines into superconductors can

increase B*(T). The upper critical field is technically relevant as an upper limit of the

irreversibility field.

Bc

Magnetic field, Bc

Temperature, Tc

Superconducting region

Current density

Jc

Jc

Fig. 1.3. Critical surface limited by three critical parameters, Jc, Bc, and Tc of type IIsuperconductors

5

Basically, the superconductivity of a certain material exists in limitations of three parameters,

including temperature, magnetic field and density of electric current. These parameters form a

critical surface that separates the superconducting and non-superconducting states as seen in Fig.

1.3.

1.3. Bi-2223 and YBCO high temperature superconductors

High temperature superconducting (HTS) conductors Bi2Sr2Ca2Cu3Ox (Bi-2223) with Tc = 110 K

and YBa2Cu3Ox (YBCO) with Tc = 95 K are now available in form of long length tapes [10-13].

Similar to other high temperature superconductors, Bi-2223 and YBCO are type-II

superconductors with considerable high upper critical fields, Bc2. Those conductors, therefore,

Fig. 1.4. Unit cell of Bi-2223 (a) and YBCO (b)

6

are the most potential candidates for the large-scale applications such as magnet applications

(e.g. MRI, NMR) or power applications (e.g. HTS motors, HTS transmission cables, HTS fly-

wheel, SMES, fault current limiter, generators…) [14-18]

The unit cell of Bi-2223 and YBCO are shown in Fig.1.4. Both these materials have copper-

oxide planes alternating with Cation-oxygen planes of other elements like Bi, Sr, Ca, Ba and Y.

Bi-2223 has a tetragonal, layered, orthorhombic perovskite structure composed of two charge-

reservoir layers (Bi-O, Sr-O) sandwiching three CuO2 planes while YBCO has a layered,

orthorhombic perovskite structure consisting of two charge-reservoir layers (Cu-O, Ba-O2)

sandwiching with two CuO2 planes. The superconductivity is expected to take place mainly in

the CuO2 plane (ab-plane). Thus, the materials are anisotropic, the superconducting properties in

the ab-direction differ from properties in c-direction.

1.3.1. Bi-2223 superconducting tapes

Bi-2223 tapes produced by the power-in-tube method [11] are now available in thousand-meter

production lengths with high uniformity and good mechanical and electrical stability. Therefore

it is being commercialized for large-scale power applications. The precursor materials are ground

to a fine powder and packed in a metal tube. The tube is drawn down to form a small round wire

and then rolled into a flat tape. To make a multi filamentary tape, the small round wires are

bundled and packed into a silver (Ag) tube and then drawn and rolled. The superconducting

ceramic is formed by a chemical reaction during a heat treatment process. Therefore the metal

tube must be oxygen permeable and not reacting with the ceramic or oxygen up to 1000 C. Ag

and Ag-alloys are typically chosen materials. The superconductor inside the tube is granular and

the critical current is mainly determined by the density and quality of the links between the

grains. The grains of Bi-2223 are flat platelets, much thinner in the c-direction than in the ab-

directions. Thus, the rolling procedure in the conductor fabrication helps to texture the material.

The ab-planes of Bi-2223 grains are mostly oriented in the plane of the tape. The texturing is

essential for establishing the links between the grains and yields high critical current of Bi-2223

tapes.

Bi-2223 conductor is fabricated with a multi filamentary structure to improve electrical and

mechanical performance and to reduce AC losses. A cross-section of a Bi-2223 multi-

7

filamentary tape manufactured by American Superconductor Corporation (AMSC) is shown in

Fig. 1.5 [11]. The Bi-2223 filaments are black and the Ag-alloy matrix and sheath material is

white. The tape width 2a is typically 2-5 mm and the tape thickness 2b is 0.2-0.3 mm. State-of-

the-art Bi-2223 tapes have a critical current of 100-200 A at 77 K in self-field, in unit lengths of

thousand meters [11,13]

1.3.2. YBCO superconducting tapes

YBCO conductors are formed by a very different process than Bi-2223. The grains of YBCO are

cubic and the material shows no texture after rolling. Therefore, YBCO conductors fabricated by

PIT method have very limited critical currents and must be prepared by different methods. Two

main techniques of YBCO fabrication are RABiTSTM (Rolling Assisted Biaxially Textured

Substrate) [19-20] and IBAD (Ion Beam Assisted Deposition) [21-22]. In 2006, American

Superconductor Cooperation (AMSC) reported a 100 m long, high quality YBCO tape produced

by RABiTSTM/MOD technology (MOD is an acronym for Metal-Oxide Deposition) [10, 12].

Similarly, IBAD conductors in lengths over 300 m are reported by IGC Superpower [12]. The

length of coated conductors is anticipated to continue increasing and will be soon available in

industrial length.

Fig. 1.6 shows one example of a YBCO coated conductor architecture produced by

RABiTSTM/MOD. The conductor consists of a multi layer structure as follows:

• Ni-alloy substrate layer (thickness ~ 50 μm)

Fig. 1.5. Cross-section of a multi filamentary Bi-2223 tape with silver-alloy sheath and matrix

2b

2a

8

• One or several thin buffer layer(s) (thickness < 100 nm)

• Thin YBCO layer (thickness ~ 1 μm)

• Thin silver cap layer (thickness < 10 μm)

In addition, copper stabilizer is often added either on top of the silver cap layer or surrounding

the entire conductor to enhance the electrical stability and mechanical strength.

The substrate architecture serves as the carrier for the YBCO conductor. Lattice match and

suitable texturing are two prerequisites for an effective substrate. Many other factors must also

be considered, including thermal expansion coefficient, chemical compatibility, high-

temperature stability, magnetic properties, mechanical properties (ductility and strength) and cost

effectiveness for long length production [23].

The RABiTSTM technique involves the deformation of a Ni-alloy ingot via a rolling followed by

annealing to develop the desired texture with the Ni-alloy. Ease of deformation, oxidation

resistance, and other physical and chemical properties make Ni and its alloys suitable substrates

for RABiTSTM process. Ni-W textured tapes with improved yield strength and reduced

ferromagnetism (relate to pure Ni) have been developed and are now available in long lengths

[10-11].

Ni-alloy

YBCOSilver

Ni-alloy

YBCOSilver

Fig. 1.6. Architecture of a YBCO coated conductor

9

The IBAD technique uses a non-textured polycrystalline metallic substrate or even on an

amorphous substrate [23]. The range of potential materials is therefore rather large. Many types

of substrates include various Ni-based alloys, Hastelloy X, Inconel 601, Rene 41, and stainless

steel SS304 were studied for the substrate of YBCO conductors fabricated by the IBAD method

[24-29]. Hastelloy is the most popular choice [10-12].

Buffer layers must provide a textured template for YBCO growth while also providing a reaction

barrier between the YBCO and the Ni-alloy. Buffer layers must also have closely matched

thermal expansion coefficient with the substrate and superconducting layer to avoid cracking

during thermal cycling. The buffer layer textures must be biaxial with suitable lattice parameters

and it must not react chemically with YBCO. Metallic materials such as Ag and Pd and ceramics

such as yttrium stabilized zirconate (YSZ), CeO2, MgO, LaAlO3 and LaMnO3 have been

investigated [22, 23, 30]. At present, IBAD tapes with MgO or LaMnO3/MgO buffers [13] and

the RABiTSTM with CeO2/YSZ/ CeO2 buffers exhibit excellent electromechanical properties and

have been produced in long length at IGC Superpower and AMSC, respectively [10-11].

Consider the superconducting layer of YBCO coated conductors. A textured substrate obtained

from either the RABiTSTM process or the IBAD process provides a starting template over which

the epitaxial YBCO layer is deposited using various candidate options, such as metal organic

chemical vapor deposition (MOCVD), metal organic deposition (MOD), pulsed laser deposition

(PLD) and pulsed electron deposition (PED). In general, thicker YBCO layers tend to exhibit

less epitaxy, more porosity and thus smaller critical current density. Currently, the YBCO layer

thickness is usually around 1μm to obtain the optimized critical current [10, 13, 31]. Numerous

research projects have been carried out to reduce disoriented grains, to optimize the process for

YBCO nucleation and growth and to improve tapes. YBCO tape with high critical current

density up to 600 A/cm-width has been reported [12].

Coated conductors have many advantages over Bi-2223 tapes. Despite the relative ease of

fabrication, the production cost of Bi-2223 tapes is much greater than YBCO coated conductors

because they utilize a large amount of silver. In addition, coated conductors have better electrical

performance at higher temperatures, particularly at intermediate and high magnetic field. Bi-

2223 compounds have limited applications in high magnetic field at 77 K due to their

10

intrinsically low B* as shown in Fig. 1.7 [13]. Dependence of irreversibility field on temperature

is shown for Bi-2223 and YBCO as well as some other low temperature superconductors. At 77

K, B* of Bi-2223 is only about 0.2 T, well below its upper critical field, Bc2(77K), which is 50 T

for B parallel to the c-axis. As seen in figure 1.7, B*(77K) of YBCO tape is much higher, about 7

T. Recently, some research groups discovered that doping YBCO conductor with nano-particles

increases magnetic flux pinning and therefore make a strong improvement of current carrying

capacity of YBCO wire in high magnetic field [32].

1.4. Application of HTS conductors

With availability in industrial lengths, Bi2223 and YBCO conductors are applicable for many

large-scale alternating current (AC) applications such as transformers, power transmission

cables, generators, motors, fault current limiters, gradient coils in MRI systems. In general, the

main benefits of HTS devices are:

Fig. 1.7. Dependence of the irreversibility magnetic field (red lines) and the upper critical field(grey lines) of Bi-2223, YBCO, MgB2 and other low temperature superconductors [13]

Mag

neti

c fi

eld

(T)

11

• Compact size and weight: The small size of HTS conductors will create small HTS

devices. This will provide critical benefits for transportation applications, such as HTS

motors for ship propulsion [33, 34].

• Efficiently, with developments of low loss HTS tapes with high critical current density,

HTS devices are expected to be more efficient than the conventional counterparts [13,

34].

• Low production cost: The price of HTS conductors has dropped the past few years. HTS

devices are therefore envisioned to have lower and lower production cost because of

savings in materials and labor.

• Lower operating cost: HTS conductors can replace some other low temperature

conductors in the current generation of superconducting magnets (e.g. MRI or NMR

systems) which are operating in expensive liquid He.

• Lower environment hazard: HTS transformers do not use oil for dielectric, therefore there

will be no leak of oil to environment.

Large-scale applications of HTS conductors are usually operated at line frequency (50 or 60 Hz).

However there are also many HTS transformers or generators for special applications operating

at other frequencies. Gradient coils in MRI systems and magnet coils for maglev train demand a

frequency up to few kHz.

1.5. AC loss in HTS conductors

AC loss and current carrying capacity are two of the most crucial considerations in large-scale

AC applications of HTS conductors such as superconducting motors and generators, SMES,

transformers, and power transmission cables. Superconductors carry DC electrical current lower

than critical current with negligible loss. Heat dissipation occurs, however, when a HTS is

exposed to an AC magnetic field or carries an AC transport current. AC loss generated by

transport current is known as “transport loss” while the loss caused by an external AC magnetic

field is called “magnetization loss”. Those loss energies are delivered by the power supplies of

the transport current and the magnetic field. The dissipation is particularly important since

12

applications of HTS conductors operate at cryogenic temperatures ranging from 4.2 K to 77 K.

AC loss results in an increased thermal load for cooling machines, and thus increased operating

costs. In practical applications, the maximum current densities for copper conductors are in the

range from 100 to 400 A/cm2. With copper resistivity of 2x10-8 Ωm, the resistive loss is 20 – 80

W/kAm. Since superconductors work at cryogenic temperature, a cooling penalty must be taken

into account. At 77 K, the cooling penalty is between 10 and 20. Therefore, the AC loss of

superconductor at 77 K should be less than 5 W/kAm in order to compete to copper.

In addition, AC losses either stimulate quenching phenomena or at least decrease the stability

margin for conductors. This issue is particularly significant for YBCO conductors because of its

complicated structure. Understanding and minimizing losses are therefore critical for

development of AC applications of HTS materials.

AC losses in HTS tapes complicatedly depend both on the real parameters of HTS tapes (e.g.

critical current distributions, architecture, dimensions, cross-section…) and on the practical

parameters of working environments (e.g. frequency, phase, and amplitude of transport current

and/or magnetic field, working temperature…)

1.6. Aim and outline of the thesis

This Ph.D. dissertation focuses on experimental and numerical studies of AC losses in both Bi-

2223 and YBCO conductors. The aims of this dissertation are:

1. To build accurate, advanced measurement systems to study total AC loss of HTS

conductors. Total AC losses of HTS tapes need to be studied in circumstances similar to

those in their practical applications. Both electromagnetic and calorimetric methods are

employed to elucidate relations between the total AC losses in HTS tapes and all practical

parameters such as frequency, temperature, phase difference between transport current

and applied magnetic field, and orientation of magnetic field. As a part of this

dissertation, a novel experiment setup was built to study the total AC losses in variable

temperatures between 30 K to 100 K.

13

2. To develop numerical models to study dynamics of electromagnetic behavior as well as

the total AC losses in HTS conductors. The numerical results not only are used to

compare with those obtained from experiments but also to help study relationships

between loss behavior of HTS wires and its cross-section, structure, and critical current

distributions. In addition, numerical analysis is used to study AC losses of HTS in

extreme conditions that are difficult to be created experimentally, such as in rather high

AC magnetic field at high frequency.

3. To employ both experiments and numerical calculations to study AC loss in Bi-2223 and

YBCO HTS samples. The results on AC loss behavior of HTS tapes and relationships

between AC loss and the tape parameters will suggest solutions for development of low-

loss HTS tapes. Studying the dependence of AC losses on the parameters of practical

applications will permit optimization of the design and operating temperatures of HTS

devices.

With the above aims, the dissertation is structured as follows:

Chapter 1 gives a short introduction about superconductivity, Bi-2223 and YBCO conductors,

AC loss in HTS conductors and the aim of this dissertation

Chapter 2 gives an overview of the basic concepts of high temperature superconductors and the

available theories of AC loss in HTS tapes. In this chapter, the concepts of the vortiex state,

critical state model, and engineering power law for HTS tape are presented. Also, analytical

models to calculate hysteresis, flux flow, eddy (coupling) current losses are discussed for both

Bi-2223 and YBCO conductors.

Chapter 3 presents numerical calculations electromagnetic behavior and AC losses of HTS tapes.

The eddy current loss in the stabilizer and ferromagnetic loss in the substrate of YBCO are also

modeled. The relation between AC losses and tape parameters including critical current density,

tape architecture, and tape dimensions are investigated. The results obtained from numerical and

analytical calculations are compared for some specific cases.

Chapter 4 describes experimental setups. The losses generated in a HTS tape carrying a DC/AC

transport current in a DC/AC applied magnetic field are measured by both electromagnetic and

14

calorimetric methods. Magnet design, electrical setup, error estimation and measuring

procedures to obtain accurate results are discussed in detail. The theoretical approach to estimate

the calibration factor in magnetization loss measurement is presented in this chapter.

Chapter 5 presents measured results of AC losses for both Bi-2223 and YBCO tapes. The

comparison between experimental and calculated results are also given and discussed. The

dependence of AC loss in Bi-2223 and YBCO conductors on their tape parameters and on

parameters of practical applications is presented. Conclusions and recommended further studies

are presented in chapter 6. Solutions to develop low loss conductors and optimization of

operating parameters for HTS devices are also suggested here.

15

CHAPTER 2

BASIC CONCEPTS AND THEORIES FOR AC LOSSES

IN HTS CONDUCTORS

2.1 Basic concepts and theories of superconductivity

2.1.1 The London (penetration) theory

The first phenomenological theory of superconductivity was proposed by the London brothers,

Fritz and Heinz [35]. They introduced a characteristic parameter for a superconductor, the

London penetration depth L, which was assumed to be independent of position and defined

as:

20 enm sL μλ = (2.1)

where ns is the number of “super-electrons” per unit volume and m and e are the mass and the

electric charge of the super-electrons, respectively. Using classical electromagnetic laws, the

relation between electric field, electric current and magnetic field of a superconductor can be

expressed in the following equations (London equations):

EEJ

20

2 1

L

ss

m

en

t λμ==

∂∂

(2.2)

16

BJ

20

1

L

s

t λμ−=

∂×∇

(2.3)

Combining those equations with Ampere’s Law of static electromagnetism, the magnetic field

inside a superconductor must obey:

2

2

LλB

B =∇ and 22

L

JJ

λ=∇ (2.4)

The solutions of the above equations depend on the boundary and the shape of the conductors.

For one-dimensional superconductors, the general solution can be written as:

)/exp()( La xBxB λ−= (2.5)

where Ba is the applied magnetic flux density.

Fig. 2.1. Exponential decay of the magnetic flux inside a superconductor

17

As seen in Fig. 2.1, the magnetic field inside the superconductor decays exponentially. At the

position of the penetration depth, x = L, the magnetic flux density has decreased by 63%.

The London equations have been suggested in order to approximately describe the behavior of a

type-I superconductor in an applied magnetic field. Hence, they are quite simple but give good

results in most cases of type-I conductors. The results of London theory can be easily obtained in its

latter and more general alternatives, the Ginsburg-Landau theory [36].

2.1.2 The Ginsburg-Landau (G-L) Theory

The Ginsburg-Landau (G-L) theory [36] used quantum mechanics instead of classical

electromagnetism to develop a better description of superconductivity. In G-L theory, the state of

super-electrons is described by a pseudo-complex wave function: )exp()( φψψ ix = . The local

density of superconducting electrons is then expressed by 2

)(xns ψ=

The advantage of G-L theory is that it can be used to analyze the intermediate state of the

superconductor, where both superconducting and normal states coexist in the presence of an

applied magnetic field B ~ Bc. The central problem is to find the wave function )(xψ and vector

potential A. To do so, the theory starts with expressing the Gibbs free energy in terms of the

wave function and vector potential. Minimizing the Gibbs energy by requiring its derivatives

with respect to the wave function and vector potential to be equal to zero results in the Ginsburg-

Landau equations. In most general cases, those equations must be solved numerically to obtain

)(xψ and A.

The Ginzburg-Landau theory introduced another characteristic parameter of superconductors, the

coherence length ξ. It characterizes the distance over which the density of super-electron reduces

from its full bulk value (superconducting state) to zero (normal state) while the penetration depth

L is understood as the width of an induced vortiex within which most of the flux is confined.

G-L theory showed that the relation between the London penetration depth and the coherence

length can distinguish between type-I and type-II superconductors. The Ginsburg-Landau ratio is

defined as:

= L(T)/ξ(T) (2.6)

18

If the penetration depth is larger than coherence length, the applied magnetic field can penetrate

inside the conductor to a distance of several coherence lengths. As the consequence, there will be

a high density of super-electrons in a relative high magnetic field near the interface (Fig. 2.2(b)).

Therefore, vortices are formed by circulating super-currents. The density of super-electrons

inside the core of vortices which has a radius of ξ (Fig. 2.2(b)) will be zero (normal state). Thus,

a superconductor with these characteristics is type-II since the formation of vortices creates a

mixed state, where both the normal phase (in the core of vortices) and superconducting phase co-

exist. On the other hand, a long coherence length prevents the super-electron density from rising

quickly enough to form a vortex (Fig. 2.2(a)). Thus, type-I superconductors is adopted to have

21≤κ .

Fig. 2.2. Illustrations of “super-electron” density and magnetic field profile inside type-I (a) andtype-II superconductors (b). In a type-II superconductor, a vortex state is formed.

19

2.1.3 BCS theory

Although G-L theory succeeded in explaining many primary superconductor properties, it does

not elucidate the microscopic origin of superconductivity. In 1957, Bardeen, Cooper and

Schrieffer developed a microscopic theory, known as BCS theory, that explained quantitatively

many important properties of superconductor in terms of quantum mechanics [37]. One of the

main points of BCS theory is the prediction of the formation of electron-electron pairs, called

Cooper pairs, in the superconducting state. Two electrons with the same electric charge of –e

will typically repulse each other because of the electrostatic force. Cooper predicted,

however, that there is an attractive force between them created by the mediation of phonons.

At a certain temperature, the creation of the Cooper pairs lowers the energy of the system.

Therefore, in the superconducting state, electrons continue to form Cooper pairs until the

system reaches the ground state.

The G-L theory can be derived from BCS theory with the assumption that the charge and the

mass of the super-electrons are the natural mass and the charge of the Cooper pair, 2me and –

2e, respectively. With the discovery of heavy fermion and copper-oxide HTS

superconductors, it is not clear if BCS theory is still suitable to explain the origin of

superconductivity of the new class of materials.

2.1.4 The mixed state and motion of vortices in Type-II superconductors

For HTS conductors, the lower critical field, Bc1, is very small. Almost all applications of HTS

superconductor therefore, happen in the mixed state (Bc1 < Ba < Bc2), where both supeconducting

and normal phases co-exist. It was foreseen by Abrikosov [38] that the magnetic field would

penetrate in a regular array of flux tubes or vortices which have the quantum flux of Φ0 = h/(2e) =

2.6678.10−15 Wb (h = Planck’s constant, e = electron charge). A change in the applied field must

consequently cause a variation in the density of these flux tubes. When the magnetic field

increases, the density of the vortex, Φ0 also increases and they eventually overlap, making the

vortex-vortex nearest neighbor distance less than the penetration depth. For HTS

superconductors, L is much larger than ξ. Thus strong overlapping is observed in the mix state

of HTS conductor and the magnetic flux presents mainly in the superconducting region rather

than in the actual core.

20

There are two primary forces acting on vortices in the mixed state of a type-II superconductor:

the Lorentz force and the pinning force. When a vortex Φ0 is in a region of current density J, the

Lorentz force is given by:

0ΦJF ×=L (2.7)

The Lorentz force can pull the vortices to move in the direction of the force. As vortices move,

they induce an electric field:

νBE ×= (2. 8)

where ν is the average velocity of vortices. Since developed electric field and transport current

are not perpendicular, heat is dissipated:

EJ=P (2. 9)

In fact, many vortices in type-II superconductors are pinned by pinning centers and hinder the

motion of nearby vortices. The pinning force, Fp, is the short-range force that holds the core of a

vortex in 1 place, thus reducing the vortex motion. There are many types of pinning centers in

superconductors such as defects of the lattice, oxygen vacancies or grain boundaries [39-40].

They can also be created by irradiation.

The pinning center is responsible for increasing the irreversibility field of type-II

superconductors. In principle, when the Lorentz force, which is proportional to the current

density J, is less than the pinning force, the vortices do not move. When the Lorentz force is

much higher than pinning force, the flux motion occurs and the conductor will be no longer a

superconductor. The transport capability of the superconductor is therefore linked to the strength

of the pinning centers. As a consequence, an ideal superconductor without defects acting as

pinning centers would have critical current density Jc = 0. Vortex motion will be discussed in

more detail in section 2.1.6 of this chapter.

21

2.1.5 Critical current density in HTS conductor

In principle, the critical current density is defined as the maximum current that a HTS

conductor can carry without resistance. When a transport current flows in a HTS tape, it will

create some magnetic field around the tape, called self-field. The self-field penetrates into the

tape in form of vortices. As the current increases, the vortex density will increase and the

Lorentz force acting on the vortices also increases. When the Lorentz force is greater than

pinning force, the vortices move and some heat is generated as discussed in the previous

section. As a consequence, the zero resistance state no longer exists and some voltage drop

along the conductor can be detected. The critical current is usually measured by a four-probe

technique which measures the voltage drop along the conductor when it carries a transport

current. In practice, the critical current is determined by an electrical field criterion which is

usually chosen to be 10-4 V/m (1 μV/cm) or 10-5 V/m (0.1 μV/cm).

The critical current density for a given HTS conductor depends on applied magnetic field and

temperature. The typical I-V characteristic of a HTS conductor is shown in Fig.2.3. More

details about the I-V relationship of HTS conductor will be discussed in the next section.

Fig. 2.3. Typical I-V curves in superconductors, technical critical current is determined whenelectric field in the conductor reach the criterion 10-4 V/m

E= 10-4 V/m

22

2.1.6 Critical state model and the power-law for HTS conductor

The critical state (CS) model, also known as the Bean model, was first proposed in 1962 [41,

42]. This model is simple but very useful for describing the current and magnetic field

distributions inside a superconductor. The main idea of the CS model is the assumption that

the current density in a superconductor only accepts two values, zero (in the region of perfect

diamagnetism) or critical current density, Jc (in mixed state region). The magnetic field inside

the conductor is then given by:

cJ0μ=×∇ B (2.10)

To demonstrate this model in detail, we apply it for an infinite slab with infinite length and

height, thickness of 2d as shown in Fig. 2.4., an AC magnetic field Ba(t) = B0sin(ωt) is applied

parallel to the conductor, i.e. along the y-axis. In this case, the problem is one-dimensional and

relatively simple. Using Eqt (2.10), the penetrated magnetic flux inside the conductor must have

a constant slope:

Fig.2.4. Sketch of an “infinite” slab conductor (infinite length and height) placed in xyz-coordinates. External magnetic field is applied parallel to the conductor (in y-direction)

23

c

a Jdx

xdB0

)(μ±= (2.11)

The current density J = (Jc,0,0) and magnetic field profile Ba = (0,Ba,0) inside the conductor are

depicted in Fig. 2.5. Starting from t = 0, the applied magnetic field increases and penetrates

deeper and deeper in the conductor as time t increases. This flux penetration causes shielding

current density of Jc as seen in Fig. 2.5(a). Near the center, where there is no presence of

penetrated magnetic field, the screening current is zero. The slab is fully penetrated when applied

magnetic field increases to the penetration field Bp (Fig. 2.5(b)):

dJBtB cpa 0)( μ== (2.12)

If the applied magnetic field continues increasing to the maximum value, B0, the shielding

current density and the slope of the internal field remain unchanged (Fig. 2.5(b)). After reaching

its peak value, the applied magnetic field starts its descending half-period. As a consequence, the

current density has to change its sign, starting from the edge to the center, as shown in Fig. 2.5(c)

to satisfy the requirement that the absolute value of the slope of the internal magnetic field is a

constant (Eqt. 2.11). When Ba(t) = B0 - 2Bp, the full penetration, again, is obtained (Fig.2.5(d)).

When applied magnetic field decreases from B0 - 2Bp to - B0, the current distribution and the

slope of the magnetic field profile are unchanged. The ascending half-period will happen in a

similar way as the descending half-period.

Thus, in the Critical State (CS) model, vortices in a type II-superconductor exist only in two

states, completely pinned (current density is Jc) and flux-flow (current density is zero). In fact,

the transition between superconducting state and normal state is not as abrupt as described in the

CS mode. Before flux-flow begins, vortices have already started moving due to the impact of

thermal activation and small Lorentz force. This motion of vortices is known as flux-creep.

Numbers of moving vortices and the motion speed increase when the temperature and/or Lorentz

force (or current density) increases.

24

Fig. 2.5. Magnetic field and current profiles in an infinite slab when it is exposed in a parallelapplied field. Applied field increases from zero to the full penetration field Bp (a), then increases to the peak field B0 (b), decreases from B0 to B0 - 2Bp (c) and then continues decreasing to – B0

(d).

25

The flux vortices jump between the pinning centers due to thermal activation [43]. The hopping

rate out of a pinning center with potential well of the depth U0 is:

R = R0 exp(−U0/kT), (2.13)

where R0 is the frequency of the hop attempts, k is the Boltzmann constant and T is the

temperature. For low-Tc superconductors, the ratio U0/kT is quite large, so that the vortices are

confined in the pinning centers up to temperatures close to Tc. This is not the case for high-Tc

materials. The hopping rate is considerable at temperatures quite below the Tc. Thus thermally

activated motion of the vortices is more significant for HTS conductors.

Since there is no preferred direction, thermally activated vortex motion of the vortices is not an

effective movement. In the presence of a small external driving force, e.g. Lorentz force, vortex

motion will have a preferred direction. With the impact of a Lorentz force fL, the potential barrier

decreases by ΔW in the direction of the Lorentz force and increases by ΔW in the opposite

direction. The hopping rates in the same and the opposite directions of fL are then given by:

R+ = R0 exp((−U0 +ΔW)/kT) (2.14)

R− = R0 exp((−U0 − ΔW)/kT) (2.15)

The unequal hopping rates as expressed above result in the effective motion of the vortices in the

direction of the fL with the rate:

R = 2R0 exp(−U0/kT) sinh(ΔW/kT) (2.16)

When ΔW/kT << 1, the flux motion corresponds to the flux-creep regime and when ΔW/kT >> 1

the flux motion is in the limit of flux flow. Thus, based on this theory of thermally activated flux-

creep, the electric field along the conductor will increase gradually when the transport current

increases, as shown in Fig 2.3. The electric field per unit length created by the flux creep

mechanism can be written as [43, 44]:

E = E0 exp(−U/kT) (2.17)

26

where U is the effective energy barrier. Assuming that this potential energy decreases with the

logarithm of the carried current, one has U = U0 ln(J/Jc). When J = Jc, flux-flow start happens,

thus U = 0. With this assumption, E-J characteristics for HTS conductor can be described by a

non-linear power law [44]:

E = Ec(J/Jc)n (2.18)

Ec is known as the electrical field criterion, which is typically to be equal to 10-4 V/m or 10-5

V/m. The power index n = U0/kT (“n-value”) is also an important parameter of a

superconductor which characterizes the sharpness of the transition from the superconducting

state to the normal state. In the framework of the power-law description of the superconducting

transition, the critical state model corresponds to a power index n = ∞. The CS model is therefore

less accurate to describe the E-J properties of superconductor, but it is quite useful for generating

analytical models for studying electromagnetic behavior and AC loss in HTS tapes.

Equation 2.18 applies for local current density, but it is also generalized for a whole HTS

tape to describe its I-V relations as shown in Fig. 2.3:

E = Ec(I/Ic)n (2.19)

The critical current Ic is generally the integral of critical current density over the cross-section

of a HTS tape. For a Bi-2223 and YBCO tape, n-value ranges from 20 to 40. The critical

current and the n-value of a superconductor decline as applied magnetic field and temperature

increase (Fig. 2.3).

2.2 Theory for AC loss in HTS tapes

AC losses generated in a HTS tape are contributed by two main sources, magnetization loss and

transport loss, with different mechanisms. Magnetization loss is caused by an AC magnetic field

and transport loss is caused by an AC transport current. The mechanisms of AC loss originate

from both the non-linear electromagnetic behavior of the superconductor and ohmic properties of

normal conductors forming the tape. In general, the loss mechanisms are superconducting

27

hysteresis, eddy current, coupling current, flux-flow, normal resistance and ferromagnetic

hysteresis. Various AC loss models for different loss components in HTS tapes with different

tape cross-sectional shapes have been reported in the literature. In this section, an overview of

AC loss mechanisms and analytical models for AC losses calculations in superconducting tapes

with typical cross-sections will be presented.

2.2.1 AC loss mechanisms

2.2.1.1 Hysteresis losses in superconductors

An AC applied magnetic field will induce screening currents in HTS conductors as a

consequence of the Meissner effect. This magnetic field also results in flux penetration into the

conductor. From the microscopic point of view, when an AC field is applied, the oscillating

motion of vortices through the area with screening currents results in losses in the same way as

discussed in section 2.1.3.

From the macroscopic point of view, when the applied magnetic field changes in time, the flux-

line pattern and internal magnetic field profile also change. The time varying magnetic field

inside the material induces an electric field E according to Faraday’s law: /dtdBE −=×∇ . The

electric field E will create screening currents in the material and these screening currents

determine the magnetic field distribution inside the superconductor according to Ampere’s law:

JB 0μ−=×∇ . As a consequence, heat is generated by screening currents with a local power

density given by EJ.

The motion of vortices in a superconductor with pinning is irreversible and depends on the

magnetic history of the sample. As seen in Fig. 2.5, there is still flux trapped inside the conductor

even when the external magnetic field returns to zero. This trapped flux is the result of hysteresis

phenomena. Thus, the losses generated in the HTS conductor in an AC magnetic field are

hysteretic in nature and are proportional to the frequency of the applied field. This loss can be

understood as the energy required for depinning and moving the flux lines inward and outward,

which is a dissipative process as described in the previous section. The hysteresis loss in a HTS

tape can be caused either by an external AC magnetic field (magnetization loss) or by the self-

field which is created by the AC transport current carried in the conductor (self-field loss).

28

2.2.1.2 Eddy current and coupling losses

In multi filamentary conductors, the filaments are coupled by an alternating magnetic field.

Consider a conductor with two filaments separated by a silver layer as shown in Fig. 2.6.

According to Faraday’s law, there will be an induced voltage around a closed loop due to

changing magnetic field. That voltage will drive a current around the loop as shown in Fig. 2.6.

Since there is no resistance in the superconducting filament, the current will run in those

filaments for a long distance before crossing through the silver matrix and generating loss. Thus,

the induced voltage is proportional to the length of the loop. If the length of the loop is greater

than critical length, the induced current in the filament will saturate at the critical current. This

phenomenon is known as full coupling in superconducting wire. In this case, the magnetic

moments and AC loss of multi filamentary conductors are the same as mono filamentary

conductors with the same critical current and cross-sectional area. Due to coupling eddy current

losses, AC losses in fully coupled, multi filamentary wires are higher than in mono filamentary

conductors. With the structure as shown in Fig. 2.6, the critical length is determined by [45]:

024

B

dJL

fcm

c ωρ

= (2.20)

Fig. 2.6. Eddy and coupling currents in a two-filament conductor when it is placed in a timevarying perpendicular field

29

where Jc and df are critical current density and the thickness of the superconducting filament, mρ

is the resistivity of the matrix materials, and B0 and ω are amplitude and frequency of the

applied magnetic field. Thus, the critical length is independent of the distance between the

superconducting filaments.

A twisted conductor, as shown in Fig. 2.7, reduces the length of the closed loop and therefore

reduces eddy current coupling losses. The effective length of the closed loop in a twisted multi-

filamentary tape is one-half of the filament-twist pitch Lp. Since the coupling loss is proportional

to Lp2/ρm , it can be reduced by increasing the resistivity of the matrix material and decreasing the

twist-pitch. A detailed study of coupling loss in Bi-2223 can be found in numerous published

papers [46-49].

HTS tapes are usually covered by metal layers, for example Ag sheath in Bi-2223 conductor and

Ag cap and/or Cu stabilizer in YBCO coated conductor. The eddy currents induced in a normal

metal sheath also result in Ohmic losses. At low frequencies, the eddy current losses due to

normal metal sheath can be calculated for different conductor geometries by using the moment of

inertia of the conductor cross-section [50].

Fig. 2.7. Coupling current in a twisted two-filament conductor

30

If ρ is the resistivity of the metal material, then the skin depth of the eddy currents in a

sinusoidal magnetic field with frequency f is given by [50]:

f0πμρδ = (2.21)

For silver, the skin depth is about 3.8 mm at 77 K and a frequency of 50 Hz. A normal HTS tape

has thickness of 0.1 to 0.3 mm and the width of 3 mm to 10 mm. So the conductor width is

comparable to the eddy current skin depth. As a consequence, in an applied magnetic field

perpendicular to the wide surface of the tape, the eddy current should be considered. The power

loss Pe (in W/m) due to eddy currents is [50]:

ρπ

S

JBfP

yM

e

,0222

= (2.22)

where JM,,y is the moment of inertia around the y-axis (which is parallel to the direction of the

magnetic field) of the cross-section of the metal and S is the cross-sectional area of the metal.

In YBCO coated conductor, the stabilizer layer is usually a copper strip with rectangular cross-

section (Fig. 2.8(a)). For a rectangular cross section with dimensions shown in Fig. 2.8(a), the

moment of inertial is given by:

2b

2a 2a

2b2bi 2ai

B

y

x(a) (b)

2b

2a

2b

2a 2a

2b2bi 2ai

2a

2b2bi 2ai

B

y

xB

y

x(a) (b)

Fig. 2.8. Rectangular (a) and hollow rectangular (b) cross-sections

31

12

28 3

,

baJ yM = (2.23)

Bi-2223 tapes are usually surrounded with silver sheath. Thus, it is necessary to consider a metal

tube with cross-section of a hollow rectangular as shown in Fig. 2.8(b). The moment of inertial

for a hollow rectangular cross-section is:

12

2828 33

,ii

yM

babaJ

−= (2.24)

The eddy current power loss is proportional to the frequency square. Thus, it must be specially

considered in high frequency applications, such as in MRI gradient coils or in the HTS coils of a

maglev train.

Equation 2.22 calculates the eddy current power loss caused by a uniformly external magnetic

field only. In fact, the local magnetic field on the surface of the metal region must be the

superposition of uniform external field and the self-field. More accurate analytical and numerical

calculations of eddy current loss for stabilizer in YBCO conductors will be given in section

2.2.2.3 of this chapter and section 3 of chapter 3.

2.2.1.3 Ferromagnetic loss

In Bi-2223 tapes, the silver matrix and sheath are mechanically weak and reinforcement is often

required. Therefore, Fe, Ni or their alloys can be used as a reinforcing sheath of Bi-2223 tape

[51-52]. Also, most YBCO coated conductors are manufactured with Ni or Ni alloy substrates.

Utilizing these ferromagnetic materials will result in ferromagnetic loss in AC environment. The

loss per volume of a ferromagnetic material in an applied magnetic field higher than its

saturation field is given by:

fMHP SCFe 04μ= (2.25)

where CH is the coercive field and SM is saturation magnetization of the material. With Ni, the

saturation magnetization is about 0.61 T and coercive field of 6400 A/m [53], the power

ferromagnetic loss is estimated to be in the order of 104 J-m3 per cycle. The cross-section of the

substrate of YBCO conductor is about 50 μm x 1 cm. Thus the ferromagnetic loss per 1 m long

32

in YBCO tape contributed by the Ni substrate is in the order of 0.005 J/cycle, which is

comparable with the superconducting loss. Recently, some publications reported that the effect

of ferromagnetic loss in substrate layer of YBCO coated conductors may be significant in

comparison to transport self-field loss [54-57]. The ferromagnetic loss can be reduced by

alloying Ni with W or Cr, which reduces both CH and SM [58-59]. In some IBAD YBCO

conductors which use Hastelloy for substrate, the ferromagnetic loss is significantly reduced. A

numerical model to calculate ferromagnetic loss in the substrate of YBCO conductor will be

given in chapter 3.

2.2.1.4. Flux flow and resistive loss

As discussed earlier, when transport current is close to or higher than Ic, Lorentz force is

dominant and stimulates flux to jump between pinning centers. By then, flux-flow results and the

electric field inside the conductor increases dramatically as a consequence of power law V-I

characteristic of HTS conductor. With considerable voltage drop along the conductor, some of

the transport current will be “shared” by metal materials (Ag matrix, stabilizers, substrate…).

Therefore some resistive loss will be generated in metal layers. Both resistive loss and flux flow

loss are frequency independent.

2.2.2. AC loss models for a slab

A Bi-2223 tape with rectangular cross section can be treated as a slab. Based on the critical state

model, some analytical models have been derived to calculate AC loss of an infinite

superconducting slab in some specific cases.

2.2.2.1. An infinite slab in parallel AC magnetic field

In this section, the Critical State Model is illustrated to calculate AC loss in an infinite slab. In

parallel magnetic field, if the width of the slab is much larger than its thickness, the slab can be

considered as “infinitely high” with acceptable error. In this case, the dependence of the internal

magnetic field and current density inside the conductor on the applied field are depicted in detail

in section 2.1.4 and by Fig. 2.5

33

The magnetization loss can be calculated from the virgin magnetization curve of the conductor

when the applied magnetic field Ba starts to increase from zero to its peak B0. When the applied

magnetic field increases from zero to the full penetration field, cp dJB 0μ= , the magnetization M

per unit length and height of the conductor is a function of the penetration depth p [60]:

ca JBdp 0/ μ−= and ( )22

2pd

d

JM c −−= (2.26)

For applied field higher than Bp, the conductor is in the saturated state with its saturation

magnetization 2/cs dJM = and carries the screening current at the density of Jc in one half and

–Jc in the other half as discussed in section 2.1.4. Figure 2.9 sketches the virgin magnetization

curve and magnetization loops for an infinite slab in parallel field pBB >0 (the solid line) and

pBB <0 (the dashed line). The direction of the arrows in the magnetization loop interprets how

the applied magnetic field changes. The magnetization AC loss for this case can be calculated

Fig. 2.9. Magnetization loop for an infinite slab in parallel applied magnetic field with the peakvalues B0. The solid line loop represents B0 > Bp and the dashed line loop represents for B0 < Bp

34

from the area of the magnetization loop or directly from the virgin solution of the magnetization

M(Ba) by the following equation [61]:

[ ] a

B

am dBBMBMQ ∫ −=0

0

0 )(2)(4 (2.27)

Hence, the magnetization loss of an infinite slab in parallel applied magnetic field per unit

volume, per cycle is given in terms of the normalized magnetic field pBB /0=β :

3

2)(

0

20

0

βμB

BQm = , for 1/0 <= pBBβ (2.28)

⎥⎦

⎤⎢⎣

⎡−=

20

20

03

212)(

ββμB

BQm , for 1/0 >= pBBβ (2.29)

In some situations, it is useful to use dimensionless quantity, called the loss function

20 2/)( PBQμβ =Γ which can reduce Eqts.(2.28) and (2.29) to:

3)(

3ββ =Γ , for 1/0 <= pBBβ (2.30)

3

2)( −=Γ ββ , for 1/0 >= pBBβ (2.31)

Thus, for small applied magnetic fields, the magnetization is proportional to B03 while in the high

magnetic field limit, the magnetization loss linearly increases with B0.

2.2.2.2 Infinite slab carries an AC current in a parallel magnetic field

Based on the CS model, the total power loss in a parallel infinite slab when it carries a sinusoidal

transport current in an in-phase sinusoidal magnetic field, was calculated by Carr [62]. The

sinusoidal time dependences of the transport current )(tI and applied magnetic field )(tB are

assumed as )sin()( 0 tItI ω= and )sin()( 0 tBtB ω= , respectively. The normalized current is

defined as i = I0 / Ic and normalized magnetic field is defined as β = B0 / Bp, where Bp is the

35

penetration field at zero transport current. Then, the reduced total power loss

20 2/),(),( PBiQi βμβ =Γ per unit volume, per cycle is given in three cases:

23

3),( ββ i

ii +=Γ , β ≤ i ≤ 1 (2.32)

βββ 23

3),( ii +=Γ , i ≤ β ≤1 (2.33)

( ) ( ) ⎥⎦

⎤⎢⎣

⎡−−

−−

−+−−+=Γ

2

32

2232

)(

)1(4

)1(6123

3

1),(

i

ii

i

iiiii

ββββ , β >1 and i <1 (2.34)

This model cannot be applied for transport current higher than critical current, i.e. i > 1. Eqt.

(2.33) reduces to Eqt. (2.30) when the transport current is zero. There is no separate expression

for the transport-current loss or magnetization loss components of the total loss generated by in-

phase AC transport current and magnetic field. However, we can have a rough estimation of the

Fig. 2.10. Dependence of reduced loss function Γ(β,i) in an infinite slab on normalized parallelmagnetic field β at various transport currents

36

contribution of these loss components in the total loss when the normalized magnetic field β is

much larger than 1.

For 1>>β , only the first term in Eqt. (2.34) is significant. As a consequence, the total AC loss

for high magnetic field at any transport current I0 < Ic is less than 4/3 times the loss at zero

current. This means the magnetization loss component significantly contributes to the total loss

when the applied magnetic field is rather higher than the full penetration field and transport

current is smaller than Ic. From Eqt. (2.32), the self-field loss in zero field also can be derived by

letting 0=β . Thus, the self-field loss of an infinite slab is proportional to the third power of the

transport current. Fig. 2.10 depicts the dependence of the reduced total AC loss of an infinite slab

on the normalized magnetic field for different values of transport current, i = 0.0, 0.2, 0.4, 0.6,

0.8 and 1.0.

This model is applicable only when the transport and magnetic field are in-phase. No analytical

model is known for the situation that the transport current and magnetic field are out-of-phase.

Also, there is no analytical expression for AC loss in the lab when magnetic field is applied

perpendicular to its wide surface. Numerical calculations presented in chapter 3 solve these

problems for a slab conductor with a finite rectangular cross-section.

2.2.3 AC loss models for an ellipse tape

Bi-2223 tapes are usually fabricated by the power-in-tube process. In this process, the rolling

process to make a flat tape from a round wire is likely to result in Bi-2223 tapes with elliptical

cross-section. This section will provide an overview of the available analytical models to

calculate AC loss of elliptical tapes.

The elliptical cross-section of a tape is characterized by its major and minor axes. Assuming that

we have a tape with elliptical cross-section placed in xyz-coordinates as shown in Fig. 2.11, the

semi-axis in x-direction is a and the semi-axis on y-direction is b. The aspect ratio of the cross-

section is defined as ab /=α . This aspect ratio will be an important parameter when calculating

the magnetization loss. The transport current always is in the z-direction.

37

2.2.3.1 AC loss in an ellipse tape carrying AC transport current

Using conformal mapping transformations and the critical state model, Norris proposed a model,

known as the Norris model, to calculate AC loss of an elliptical tape carrying an AC transport

current with amplitude less than critical current [63]. This loss is also known as the self-field loss

of the tape. Assume that an elliptical tape with the semi-axis a carries an AC current

)2sin()( 0 ftItI π= with I0 < Ic. When the transport current reaches its peak value, +I0, the current

penetrates in the conductor to an ellipse of semi-axis of a0 which has the same aspect ratio as the

tape cross-section (Fig. 2.12). Thus, the conductor is separated into two regions, the inner

current-free core region and the outer region with current density of +Jc. When the transport

current is in the descending period, the current changes its direction, starting from the tape

surface. As a consequence, the tapes is composed of three regions with current density of -Jc, +Jc

and 0, separated by similar ellipses of semi- axis a1, a0 as seen in Fig. 2.12. Using this current

distribution mapping, the self-field AC loss is given by [63]:

y

x

z

b

-ba-a

Iy

x

z

b

-ba-a

I

Fig. 2.11. A superconducting tape with an elliptical cross-section placed in xyz-coordinates.Transport current flows along the conductor in the z-direction.

38

( ) ( ) ( )[ ]2/1ln1 20000

20

cccc

c

t IIIIIIIII

Q −+−−=π

μ (2.35)

Employing the normalized current i = I0/Ic, the self-field loss can be rewritten as:

( ) ( ) ( ) 2/1ln1)( with , )( 22

0 iiiiiqiqI

Q ee

c

t −+−−==π

μ (2.36)

This equation is applicable for the case that I0 < Ic. In this model, the loss depends only on the

critical current of the tape, not on the thickness or the aspect ratio between the major and minor

axes of the tape. Therefore, it can be applied for a round superconducting wire.

2.2.3.2 AC loss in an ellipse tape in AC applied field

Recently, a general analytical model to calculate magnetization AC loss in an elliptical tape was

given by Ten Haken et al [64]. In this model, the magnetization AC loss in an infinite long

elliptical tape with any aspect ratio was investigated. Assuming that a HTS tape with elliptical

cross section of width 2a and height 2b is placed in a xyz-coordinates as seen in Fig. 2.11. An

a0 a1 a

x

y

a0 a1 a

x

y

Fig. 2.12. Current distribution in an elliptical tape when transport current in its descending half-period.

39

AC magnetic field with amplitude B0 is applied in the y-direction. When the aspect

ratio 1/ >= abα , we have a HTS tape with height greater than the width. Hence, the magnetic

field is parallel to the conductor in this case. On the other hand, if α < 1, we have the magnetic

field perpendicular to the tape. If 1=α , we have a round wire and the properties of the tape is

independent of the direction of the magnetic field.

Similar to the calculation of magnetization loss in an infinite slab, the virgin magnetization curve

has to be constructed to calculate the hysteresis loss by using Eqt. (2.27). In order to do that,

firstly, the magnetization Mp and the full penetration field Bp are calculated for a fully penetrated

ellipse. The slope of the virgin magnetization curve M(Ba) in the limit of very small applied field

Ba is then derived. These conditions are used as constraints for an approximation of the entire

virgin magnetization curve.

The full penetration field was obtained when the conductor current distribution included the left

half of the conductor filled with current of density Jc and the right half filled with current of

density -Jc. The full penetration field has the same magnitude as the magnetic field created by the

screening current at the very center of the conductor. Hence it can be calculated from the integral

form of the Biot–Savart law [64]:

1arctan1

1arctan1

2)( 2

2,2

2

0 −−

=−−

= αα

ααα

απ

μα cP

cP B

aJB

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

−+

−=

2

2

2,11

11ln

12 α

α

γ

γcPB (2.37)

where πμ /2 0, aJB ccP = is the full penetration field of a superconducting wire with circular

cross-section of radius a. With the current distribution as described above, the magnetization of

the conductor when it is fully penetrated is given by [30]

π3

4 aJM c

p = (2.38)

40

At small magnetic field, the current must distribute in the conductor to generate a perfect

screening field. The current distribution inside an elliptical tape in a small applied magnetic field

was generalized from the model of a circular wire [64]. In the context of the Critical State model,

the screening current distribution in a circular conductor was proven to follow the “sinθ” law

[11]. Assume that we have a circular wire with cross sectional radius of R as shown in Fig 2.13,

and that the magnetic field is applied parallel to y-axis.

For a circular wire, in the virgin state, when the magnetic field increases from 0 to small field B0,

the flux will start penetrating the conductor and separating into three regions: regions I and III

with current density ±Jc and the current-free region II (Fig. 2.13). The surface thickness of region

I and III satisfy the “sinθ” relation:

θsin0rr = (2.39)

As a consequence, at small magnetic field, the surface current density is:

θθθ sin).0()( == JJ , where cJrJ 0)0( ≈=θ . (2.40)

I II

θ

III

r0

r

y

-Jc+Jc

J= 0

B

I II

θ

III

r0

r

y

I II

θ

III

r0

r

y

-Jc+Jc

J= 0

B

Fig. 2.13. Current distribution in superconducting wire with circular cross-section when appliedmagnetic field change from zero to a small field Ba

41

This model was generalized for a conductor with elliptical cross-section of any aspect ratio α.

Details are found in [30]. After some calculations, shielding magnetic field at the center of the

conductor, Bc and magnetization, )( aBM was calculated. Since at small field the conductor must

be perfectly shielded, Bc = -Ba. As a consequence, the slope of magnetization curve in small

applied field Ba is calculated. Assume that the normalized magnetic field and magnetization are

denoted as pa BB /=β and pMMm )()( ββ = , respectively. Then, the slope of the

normalized virgin magnetization curve is given by [64]:

1arctan1

1

2

3)( 2

20 −−

+=

∂∂

= αα

αββm

m (2.41)

Thus, at small magnetic field, the slope of the virgin magnetization curve is independent of the

field and depends only on the conductor aspect ratio. The imaginary part in the square root in

Eqt. (2.41) can be avoided by a similar substitution as seen in Eqt. (2.37) with the constraints in

the limit to zero field and penetration field:

0)0( =m , 1)0( −=m , 0

0)(

)(m

m=

∂∂

→βββ

and 0)(

)(

1

=∂

∂

>βββm

(2.42)

the reduced virgin magnetization curve can be approximated as follow [30]:

( ) 11)( 0 −−= −mm ββ for 10 <≤ β and 1)( −=βm for β≤1 (2.43)

In the derivation of the virgin magnetization curve, the physics background of the curve in the

intermediate field regime is neglected. The above analytical results were compared to the virgin

magnetization curve calculated numerically using an optimization method, which optimizes the

current distribution inside the conductor in order to minimize the field in the current free region

in the conductor. A maximum variation between Eqt. (2.43) and numerical results is found to be

about 5% for α ranging from 0.0001 to 100. The virgin magnetization curves calculated by Eqt.

(2.43) were plotted against various values of α in Fig. 2.14. As observed in the figure Fig. 2.14,

when 10>α , the magnetization curve is almost independent of α and saturated at the field very

close to the penetration field ( 1≈β ).

42

Inserting Eqt. (2.43) into Eqt. (2.27) the reduced magnetization loss is obtained [64]:

( ) ( )⎥⎦

⎤⎢⎣

⎡+−+−−

−= − 1)1(1)1(

1

24)( 001

0

mm

mq ββββ , 0 ≤ β ≤ 1 (2.44)

⎥⎦

⎤⎢⎣

⎡−

−=01

24)(

mq ββ , β> 1 (2.45)

The actual magnetization loss per cycle per unit volume of the tape is given by:

)()()( βα qBMBQ ppam = (2.46)

Fig. 2.14. Normalized magnetization as a function of the normalized applied magnetic field for an elliptical tape with different values of aspect ratio

43

This model is used to calculate AC loss for elliptical tapes with the same critical current density

and cross-sectional area but different aspect ratios. Assume that the cross-section of the tapes is a

x b = 0.4 mm2 and critical current density is typically 100 A/mm2 (or Ic ~ 120.5 A). These are

typical parameters for real tapes. Full penetration field is an important parameter since it

determines the normalized field β . The dependence of full penetration field on aspect ratio is

depicted in Fig. 2.15. As observed in that figure, the tape with 5.0≈α has the highest

penetration field of mT 55≈pB , thus that tape is the most difficult to saturate.

The magnetization loss per unit length, per cycles in the tapes with different values of aspect

ratio are plotted in Fig. 2.16. With the same cross-sectional area, the loss, in general, increases

when the aspect ratio decreases. The smaller the aspect ratio is, the wider the tape is, and thus the

larger surface to “see” the magnetic field or the higher magnetization loss. However, for field

less than 0.01 T, the magnetization loss increases when α increases from 10 to 100. This is

explained due to the strong decrease in the tape penetration field in parallel applied magnetic

field when α increases, as seen in Fig. 2.15

Fig. 2.15. Penetration field as a function of aspect ratio α

44

In fact, the pair of tapes with 1.0=α and 10=α have exactly same cross-section, but one tape

is placed in perpendicular field and the other is in parallel field. This is also true for the pair of

tapes with 01.0=α and 100=α . Thus, from the above results, we can obtain the ratio between

AC loss in perpendicular and parallel fields, q = Q⊥/Q// for two tapes with 1.0=α and 01.0=α .

Figure 2.17 plots that loss ratio as a function of applied magnetic field. At small magnetic field, q

oscillates around α/1 and becomes asymptotic to α/1 when the magnetic field is high enough.

Therefore, for HTS conductor with a very small aspect ratio, the magnetization loss in parallel

magnetic field can be ignored in comparison to magnetization loss in perpendicular field.

Fig. 2.16. The magnetization loss, Qm, as a function of the normalized magnetic field forelliptical tapes with different aspect ratio

45

2.2.4. AC loss in a thin strip HTS tape

As mentioned earlier in chapter 1, YBCO coated conductors consist of a very thin

superconducting layer. These conductors, therefore, have very high aspect ratio, ranging from

103 to 104 and can be treated as a thin strip with negligible thickness. The calculations for

electromagnetic behavior in a thin strip become a one-dimensional problem and are performed

across the width of the conductor. In this section, AC loss in the superconducting layer and eddy

current loss in a thin metal strip (e.g. Cu stabilizer) of YBCO coated conductors will be

discussed. Figure 2.18 illustrates a thin YBCO strip of width 2a placed in xyz-coordinates. A

metal stabilizer with thin thickness, dm, is also sketched in the figure. With negligible thickness,

the current density used in calculations is the sheet current density, i.e. the current per unit width

Fig. 2.17. The loss ratio between magnetization loss in perpendicular and parallel magnetic field,q = Q⊥/Q//, as a function of applied magnetic field for the tape with 1.0=α and 01.0=α

46

of the tape, aIJ cc 2/= , and only the perpendicular magnetic field is considered. The transport

current flows in the z-direction.

2.2.4.1. Strip conductor carrying a transport current

Self-field loss in a thin strip superconductor was first derived by Norris [29]. From the current

distribution inside the tape when the transport current is at peak value, Norris showed that the

self-field loss was given by [63]:

( ) ( ) ( ) ( ) ( )[ ]200000

20 1ln11ln1 ccccc

c

t IIIIIIIIIII

Q −+++−−=π

μ (2.47)

Employing the normalized current i = I0/Ic, the self-field loss can be rewritten as:

( ) ( ) ( ) ( ) ( )22

0 1ln11ln1)( with , )( iiiiiiqiqI

Q ss

c

t −+++−−==π

μ (2.48)

z

x

y

Metal strip

HTS layer

dm

a-a

I

B

z

x

y

Metal strip

HTS layer

dm

a-a

I

B

Fig. 2.18. Thin HTS strip with a metal strip placed in xyz-coordinates. Magnetic field is appliedin the x-direction and the transport current flows in the z-direction

47

Later, the time dependent current density, ),( tyJ , and magnetic field distribution, ),( tyB , in a

thin strip carrying an AC transport current ( )tItI ωsin)( 0= was calculated explicitly by Brandt

and Indenbom [65], and Zeldov et al. [66]. ),( tyJ and ),( tyB were derived from virgin

distributions, ),( IyJ and ),( IyB , which are the distribution of current density and magnetic

field when the tape carries a DC transport current I. By introducing new variables

( ) 2/1221 cIIab −= , πμ /0 cc JB = , the virgin distribution )(yJ , )(yB are given by:

⎪⎩

⎪⎨

⎧

<<

<⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=

aybJ

byyb

baJ

JIyJ

c

c

c

,

,arctan2

),,(

2/1

22

22

π (2.49)

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

>⎟⎟⎠

⎞⎜⎜⎝

⎛−−

<<⎟⎟⎠

⎞⎜⎜⎝

⎛−−

<

=

ayby

ba

y

yB

aybba

by

y

yB

by

JIyB

c

c

c

,arctanh

,arctanh

,0

),,(

2/1

22

22

2/1

22

22

(2.50)

Consider a half–cycle when the transport current reduces from its positive peak value, I0, to its

negative peak value, - I0 when an AC transport current is applied to the tape. The time evolution

of current and magnetic field profiles are expressed by the following linear superposition [31]:

)2),(,(),,()),(,( 00 ccc JtIIyJJIyJJtIyJ −−=↓ (2.51)

)2),(,(),,()),(,( 00 ccc JtIIyBJIyBJtIyB −−=↓ (2.52)

Assume that the peak current, I0 = 0.95Ic , Fig. 2.19 illustrates the current and magnetic field

profiles in the decreasing half-cycle at times such that I(t)/Ic = 0.95, then I(t)/Ic = 0.5, 0, -0.5, -

0.8, and -0.95 [65].

48

With the distribution ),( tyJ and ),( tyB as shown in Fig. 2.19, Norris’s results (Eqt. 2.47) for

self-field loss are reproduced [65]. The loss can be calculated from the integral for a half-cycle:

∫∫=a

bT

dytyJtyEdtQ ),().,(42/

(2.53)

Since the current and electric field at the center of the tape must be zero,

∫==y

duuBdt

dty

dt

dtyE

00

)(1

),(),(μ

φ (2.54)

As seen in Fig. 2.19, the magnetic distribution at any position inside the tape, ),( tyB , is “frozen”

until the current density at that position reaches the critical state, i.e. cJtyJ −=),( . Thus, in the

Fig. 2.19. Current and magnetic profiles in a thin strip conductor when the transport current is ina decreasing half-period, reducing from the positive peak, I0 = 0.95Ic, to the negative peak. The “snapshots” are taken at times such that I(t)/Ic = 0.95, 0.5, 0, -0.5, -0.8 and -0.95 [65].

49

Norris model, only the current distribution at the time when the transport current is at its peak

values is needed to obtain the self-field loss [63].

In [67], Müller used magnetic field profiles given by Brandt et al to derive a formula to calculate

eddy current loss dissipated in the metal strip. If ρ and dm are the resistivity and thickness of the

metal strip, respectively (Fig. 2.18), the eddy current loss per unit length, per cycle is given by:

( )ifIad

Q c

m

e

2

2

208

ρω

πμ

= (2.55)

where

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−+

++−

−−−−= ∫2

2

0

2

1

1ln

8

1

1

1ln

211

u

u

u

uuuuxuduif

i

(2.56)

Fig. 2.20. Reduced self-field loss ( )iqe , ( )iqs and ( )if as a function of the normalized transport

current i

50

Figure 2.20 is the comparison between the three reduced loss functions: ( )iqe for self-field loss

in an elliptical tape (Eqt. 2.36), ( )iqs for self-field loss in a thin trip (Eqt. 2.48) and ( )if for

eddy current loss in the metal layer in a HTS strip tape (Eqt. 2.56). The self-field loss in an

elliptical tape is higher than that of a thin strip conductor with the same critical current. At small

current, I0 << Ic, the loss curve for an elliptical tape is proportional to I03 while the loss curve for

a strip tape is proportional to I04.

2.2.4.2. Strip conductor in a perpendicular AC field

Brandt et al also constructed current and magnetic field profiles when a thin strip tape is exposed

to a perpendicular AC magnetic field, tBtB ωsin)( 0= [65]. The virgin solutions for current and

magnetic field distribution inside the tape when a DC perpendicular field B(t) increases from

zero to its peak values B0 are given by the following equations:

⎪⎪⎩

⎪⎪⎨

⎧

<<

<−

=ayb

y

yJ

byyb

cyJ

yJc

c

,

,)(

arctan2

)(2/122π

(2.57)

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

>−

<<−

<

=

ayby

ycB

aybyc

byB

by

yB

c

c

,)(

arctanh

,)(

arctanh

,0

)(

2/122

2/122

(2.58)

Parameters b and c in Eqts. 2.57 and 2.58 are defined as:

)/)(cosh( cBtB

ab = (2.59)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−=

cB

tB

a

bac

)(tanh

)( 2/122

(2.60)

51

Similar to the previous case, the current density and magnetic field when the applied field B(t)

decreases from its peak value B0 to the negative peak –B0 are found from the following

superposition:

)2),(,(),,()),(,( 00 ccc JtBByJJByJJtByJ −−=↓ (2.61)

)2),(,(),,()),(,( 00 ccc JtBByHJByHJtByH −−=↓ (2.62)

Figure 2.21 illustrates the current and magnetic field profiles in the decreasing half-cycle at time

such that B(t)/Bc = 2, 1, 0, -1 and -2 (the peak value of the applied magnetic field is B0 = 2 Bc).

Magnetization of the tape was derived from the current profile:

Fig. 2.21. Current and magnetic field profiles in a thin strip conductor when the applied field isin a decreasing half-period, reducing from the positive peak, B0 = 2Bc to the negative peak. The“snapshots” are taken at times such that B(t)/Bc = 2, 1, 0, -1, and –2 [65].

52

∫−

=a

a

dytyJytM ),(.)( (2.63)

Then, the magnetization loss of the tape per cycle per unit length can be calculated by integrating

the hysteresis magnetization loop [65]:

⎟⎟⎠

⎞⎜⎜⎝

⎛== ∫

c

aamB

BgB

adBBMQ 02

00

24)(

μπ

(2.64)

where

( ) ( ) ⎥⎦⎤

⎢⎣⎡ −= xx

xxxg tanhcoshln

21 (2.65)

Based on current and magnetic field distributions calculated by Brandt, Müller estimated the

eddy current loss in the metal strip in this situation as [67]:

Fig. 2.22. Reduced loss functions ( )ig and ( )ie as a function of the normalized applied magneticfield.

β = x = B0/Bc

53

⎟⎟⎠

⎞⎜⎜⎝

⎛=

c

Ce

B

BeB

adQ 02

0

2

03

8

ρω

πμ (2.66)

where

( ) ⎥⎦⎤

⎢⎣⎡ +−−= ∫ uu

uxuduxe

x

320

2

cosh

2

cosh

31 (2.67)

Figure 2.22 compares the two reduced loss functions, ( )βg for magnetization loss in a thin HTS

strip (Eqt. 2.65) and ( )βe for eddy current loss in metal layer (Eqt.2.67), when a HTS strip tape

is exposed to a perpendicular magnetic field. As seen in the figure, ( )βg first increases with

increasing β but it decreases when β continues to increase above 1.8. The function ( )βe

increases with increasing β, but with gradually decreasing slope.

2.2.4.3 Total AC loss in a strip

Based on Brandt’s explicit solutions for current density and magnetic field profiles for a thin

strip when both transport current and in-phase magnetic field are applied, Schönborg derived

analytical formulas to calculate the total AC in this case [68]. The current density and magnetic

field distribution were constructed from a appropriate superposition of the “current only” and

“field only” solutions obtained in the previous cases to ensure that the current density at the

edges of the tape must be equal to ±Jc. Depending on the sign of the current density, the

calculation is separated into two cases, the high current-low field and the low current-high field

regimes. The boundary between those regimes is determined from the relative relationship

between i = I0/Ic and ( ) ( )cBB0tanhtanh =β .

If ( )βtanh<i , it is the low current-high field regime and the total loss per cycle, per unit length

is given by [68]:

54

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦⎤

⎢⎣⎡ −−+−+×

⎥⎦⎤

⎢⎣⎡ −−−−+−

⎟⎠⎞⎜

⎝⎛ −−−−++

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++×

⎟⎠⎞⎜

⎝⎛ −−−−++

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛ +×

⎟⎠⎞⎜

⎝⎛ −−−+−++−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−−+

+−+

=

−−

−−

−

20

20

20

20

20

20

20

20

22

02

02

02

0

0

010

0

010

20

20

20

20

0

01

0

01

20

200

20

200

20

20

20

20

20001

20

)1()1(

)1()1(2

1

)1()1(4

1

1cosh)1(

1cosh)1(

)1()1(2

1

1cosh

1cosh

)1()1()1()1(4

1

)1()1(

)1)(1(coth2

apap

apap

apap

a

pp

a

pp

apap

a

p

a

p

appapp

apap

app

IQ c

tt πμ

(2.68)

In the above equation: ( )βcosh1 20 ia −= and ( )βtanh0 ip =

When ( )βtanh>i , it is the high current-low field regime and the total loss per cycle, per unit

length, is given by the following equation [68]:

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞⎜

⎝⎛ −−−−+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++×

⎟⎠⎞⎜

⎝⎛ −−−−++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ +×

⎟⎠⎞⎜

⎝⎛ −−−+−++−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−−+

+−+−

=

−−

−−

−

22

02

02

02

0

0

010

0

010

20

20

20

20

0

01

0

01

20

200

20

200

20

20

20

20

20001

20

)1()1(4

1

1cosh)1(

1cosh)1(

)1()1(2

1

1cosh

1cosh

)1()1()1()1(4

1

)1()1(

)1)(1(coth2

apap

a

pp

a

pp

apap

a

p

a

p

appapp

apap

app

IfQ c

tt πμ (2.69)

55

In the boundary between two regimes, i.e. ( )βtanh=i , the two equations can be reduced into

one equation:

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−++=

0

020

200

20 1

cosh)1()1(4 a

papp

IP c

πμ

(2.70)

Equations 2.68 reduces to Eqt. 2.64 when the transport current I0 = 0 and Eqt. 2.69 reduces to

Eqt.2.48 when the applied field B0 = 0.

The total loss given by Eqts. (2.68) and (2.69) can be normalized to a function of i and β:

( )πμβ 20/),( cttn IQiQ = . The normalized total loss is plotted against β for different values of the

0.00001

0.0001

0.001

0.01

0.1

1

10

100

0.1 1 10

β = B0/Bc

No

rma

lize

d t

ota

l lo

ss

Qn

Decreasing current, I/Ic =0, 0.2, 0.4, 0.6, 0.8 and 1

Low current-high filed regime

High current-low field regime

Fig. 2.23. Normalized total loss in a thin strip HTS conductor as a function of the normalizedmagnetic field for different transport currents. The dashed line indicates the boundary betweenhigh current-low field and low current-high field regimes.

56

transport current in Fig. 2.23. In the figure, the dashed line indicates the boundary between high

current-low field and low current-high field regimes.

Based on Eqts. 2.36 and 2.48, the self-field loss in an elliptical or a thin strip HTS tape is

independent of the tape width. For a given sheet critical current density, the magnetization in a

HTS thin strip, however, is proportional to square of the tape width as predicted in Eqt. 2.64. As

suggested by Eqts. 2.55 and 2.67, the eddy current loss in the metal strip is proportional to its

width and thickness, but reversely proportional to the resistivity of the metal material.

The hysteresis superconducting total losses (Eqts. 2.68 and 2.69) were calculated from the

current density and magnetic field distribution when the transport current and/or magnetic field

are at their peak values and the current density in the penetrated region is constant, J = Jc. This is

not true for calculation of the eddy current loss. The current density in a metal strip does not

follow critical state model and will change in time, hence the calculation of eddy current is more

complicated and there is no available model for eddy current loss in this case. The numerical

calculation for eddy current loss will be presented in more detail in the next chapter.

57

CHAPTER 3

NUMERIAL CALCULATIONS

3.1 Introduction

In the previous chapter, several analytical models to estimate AC loss in HTS tapes for some

specific cases have been presented. All analytical calculations, however, are based on the critical

state model. In fact, the superconducting properties of HTS conductors are better described by a

power law as discussed in chapter one.

There are many attempts to calculate electrodynamics and AC loss in HTS tapes numerically

using nonlinear power-law characteristics. In general, numerical models allow taking into

consideration the following practical properties of HTS conductors to produce more precise

computation of AC loss and electrodynamics:

• Non-linear power-law characteristics

• Field dependence of critical current and n-value

• Non-uniform critical current distribution

• Various shapes of cross-section of the sample

58

• Conductor with complicated structure, for example, twisted tapes, multi-filamentary

tapes, YBCO tape with stabilizer and substrate

• Phase difference between transport current and applied magnetic field

The finite element method (FEM) is intensively used to evaluate AC loss in HTS conductors [69-

77]. FEM calculations can be based on home-made code [69-71] or used commercial FEM

packages, such as Flux 2D, Console [72-77]. FEM can be applied to complicated structures, for

example, HTS cables, stack of HTS tapes, or twisted HTS tape. In this method, a fine mesh is

created in a boundary containing the sample and all the calculations are performed at the nodes

of the mesh. For a HTS tape with a high aspect ratio, for instance YBCO tape, the width is 103 to

104 times larger than the thickness. A fine mesh with a large number of nodes is therefore needed

for accurate calculations [76]. In this case, FEM calculations consume a large amount of

computing time and memory [71,76]. In the FEM method, the thin strip is therefore

“approximated” by increasing its thickness and thus decreasing artificially the aspect ratio of the

conductor by a factor of 10 to 1000 [76].

Electrodynamics and AC loss in HTS tape can also be numerically calculated by finite difference

methods based on magnetic energy minimization [78-80] or based on solving Poisson’s

equations for a cross-section of the tape [81-85]. In this dissertation, the latter technique is

employed to simulate the AC electrodynamics and AC loss in HTS tapes. This model is also

known as the Brandt model [81] for calculating electric current density of a rectangular HTS tape

in an applied magnetic field. In this study, the Brandt model is applied to the electrodynamics

and total AC loss in HTS tapes with different cross-sectional shapes. With a variable

substitution, this calculation method can provide accurate results for YBCO tapes that are treated

as thin strips. Models for estimating the eddy current in the stabilizer and ferromagnetic loss are

also proposed. The numerical calculations are a useful tool to explain and compare with the

experimental results as presented in chapter 5.

3.2. Brandt model

Sinusoidal time dependences of the transport current )(tI and applied magnetic field )(tB are

assumed as: )2sin()()( 0 ftItItI a π== and )2sin()()( 0 ϕπ Δ+== ftBtBtBa . The Brandt model

59

[81] solves Poisson’s equations in integral form for a superconducting tape with a rectangular

cross-section. The superconductivity in the tape was assumed to follow power-law

characteristics:

)(

)(

Bn

c

cBJ

JEE ⎟⎟

⎠

⎞⎜⎜⎝

⎛= (3.1)

where Ec is the critical electric field criterion used to determine the critical current. Ec is

conventionally chosen to be 1 μV/cm. For simplicity, a conductor with a rectangular cross-

section was considered. The sample was placed along the z-axis with its cross-section in the x-y

plane. Field angle θ is defined as the angle between the magnetic field and y-axis. The twist pitch

of the conductor is sufficiently long so that the filaments are assumed to be fully coupled.

Poisson’s equation for the distribution of current is given by

),(),( 02 yxJyxAJ μ−=∇ (3.2)

where AJ(x,y) is the z-component of the vector potential produced by current flowing inside the

conductor and 0μ is the vacuum permeability. If the applied field is in the x-y plane, Ba = (Bax,

Bay, 0), then the solution for Poisson’s equation becomes: [81-82]

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛= −∫ axayz

Bn

c

zc

s

z ByBxBJ

JEQrdJ &&& ''

)(

)'r()r',r('

1)r(

)(

12

0

ϕμ

(3.3)

B

x

y

z

I

θB

x

y

z

I

θ

Fig. 3.1. A rectangular tape with applied magnetic field and transport current in xyz-coordinates

60

In Eqt. (3.3), )',( rrQ is the integral kernel and defined as [81-82]

π2

)/ln()',( 0r

Qr'r

rr−

= (3.4)

where 0r is an arbitrary constant length chosen to be much larger than the tape cross-section (1

m, for example). Integrating the current distribution over the whole conductor cross-section gives

the transport current flowing in the wire, thus

∫==S

rdJftItI 20 )()2sin()( rπ (3.5)

Equations (3.3) and (3.5) can be solved numerically by changing them into discrete forms by

creating a net of N rectangular cells for the cross-section of the sample:

∑=

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛ΔΔ

Δ+=Δ+

N

j

axjayjz

Bn

c

j

cijii ByBxBJ

JEQ

yx

ttJttJ

1

)(

1

0 )()()( &&ϕ

μ, i = 1…N (3.6)

[ ])(2sin)(1

ttfIttyJx t

N

i

i Δ+=Δ+ΔΔ∑=

π (3.7)

In Eqt. (3.6), 1−ijQ is the inversed matrix of matrix ijQ . Matrix element ijQ can be understood as

mutual ( )ji ≠ and self ( )ji = inductance between the cells. It is worth noting that

),( jiij QQ rr= diverges as ji rr −ln( ) when ir approaches jr . For practical purposes, it was

proven that good accuracy is obtained by approximating ji rr −ln( ) in diagonal elements of the

matrix ijQ by ])ln[(2

1 22 δ+− ji rr where dxdy015.02 =δ if dydx ≈ [81, 60].

If the field dependence of critical current and n-value are taken into consideration for calculating

the eddy current loss and ferromagnetic loss, the field distribution must be calculated. In this

case, the self-field should be taken into account and magnetic field distributions after each time

step are:

61

j

N

j

ijxaix JXBB ∑=

+=1

,, , with yxXji

ij

ij ΔΔ−

−=rrπ

θμ

2

sin0 (3.8)

j

N

j

ijyaiy JYBB ∑=

+=1

,, , with yxYji

ij

ij ΔΔ−

=rrπ

θμ

2

cos0 (3.9)

where ijθ is the angle between vector ji rr − and the positive direction of the x-axis. The

matrices ijX and ijY were calculated from the conductor geometry, so they were calculated only

once to reduce computing time. From the calculated current distribution, the total AC loss per

unit length, per cycle, is the sum of the magnetization loss and transport loss given by

dSdtJBJ

JEdSdtJEQQQ z

Bn

c

zc

f S

z

f S

ztmtotal

)(

/1/1 )( ⎟⎟⎠

⎞⎜⎜⎝

⎛==+= ∫ ∫∫ ∫ (3.10)

The transport loss also can be calculated by using

dSdtJQf S

zzt ϕ∫ ∫ ∇−=/1

(3.11)

The magnetization loss was then derived by subtracting transport losses Qt from total AC loss

Qtotal.

To apply this model for a HTS tape with an arbitrary cross-sectional shape (such as an elliptical

cross-section), the variation of the thickness along the width of the tape b = b(y) must be

considered in the calculations.

3.3. Numerical calculation for a YBCO coated conductor

3.3.1. AC loss in the HTS layer

As presented in chapter 1, the typical coated conductor architecture consists of a substrate (with

the thickness of 50-100μm) which is often ferromagnetic, one or more buffer layers, the HTS

layer, a thin Ag layer and Cu stabilizer layer. The buffer layers are typically thin insulating

oxides and thus neither carry current nor generate heat. The Ag layer is also usually thin (~ 5-

62

10μm) in comparison to the Cu stabilizer (50-100μm), while the substrate has very high

electrical resistivity. Therefore, the eddy current dissipation in the buffer, Ag and substrate can

typically be ignored. If the Ag-layer thickness is sufficiently large to be considered in the

calculation, an effective stabilizer thickness can be introduced to calculate the eddy current loss

for both the Ag and stabilizer layers. Thus, the total conductor AC loss, Qtt, is the sum of the

various loss components from the HTS layer (Qs), substrate (ferromagnetic loss, Qf) and

stabilizer (eddy current loss, Qe)

festt QQQQ ++= (3.12)

sQ is the sum of transport loss, Qt, and magnetization loss, Qm, ( mts QQQ += ) generated in the

HTS layer. Understanding the contribution of each loss component to the total loss is an

important step for reducing the AC loss of coated conductor.

Figure 3.2 shows a sketch of a YBCO tape with its three loss-generating layers: substrate, HTS

layer and stabilizers. For YBCO conductor, the thickness m 1μ≈d and the width

d mm 10 - 3 >>≈a , the conductor therefore can be treated as a one-dimensional thin-film with a

sheet critical current density Jc(y). In this limit, Eqts. (3.3) and (3.5) transform to

Fig. 3.2 YBCO conductor with the substrate and stabilizer in xyz-coordinates (not to scale)

63

∫−

=−∇+⎟⎟⎠

⎞⎜⎜⎝

⎛ a

a

zxz

Bn

c

z

z

zc tuJuyduQtByt

BJ

tyJ

JE ),(),()()(

)(

),(0

)(

&& μϕJ

(3.13)

∫−

==a

a

z dytyJftItI ),()2sin()( 0 π (3.14)

Equations (3.13) and (3.14) can be solved numerically by transforming them into discrete forms.

In Eq. (3.13), however, the kernel π2/ln),( uyuyQ −= has a logarithmic singularity at y = u .

This problem is overcome by substituting for y and u an odd function y = y(s) = u(s) using a new

variable s. This odd function must satisfy u(0) = 0 and u(1) = a, and its derivative, )()(' svsu = ,

vanishes at s =1, i.e., 0)1( =v . For instance, two such functions are

⎟⎠⎞

⎜⎝⎛ −= 3

2

1

2

3)( ssasu and )()1(

2

3)(' 2 svsasu =−= (3.15)

⎟⎠⎞

⎜⎝⎛ +−= 53

8

15

4

5

8

15)( sssasu and )()1(

2

3)(' 22 svsasu =−= (3.16)

To transform Eqs. (3.13) and (3.14) into discrete forms, variable s is discretized into 2N

equidistant points, 1/)2/1( −−= Nisi , with Ni 2...,2,1= . With the variable substitution we

have:

∑∑∫∫==−−

===N

i

i

N

i

i

a

a

wNsvdssvdy2

1

2

1

1

1

/)()( (3.17)

where Nsvw ii /)(= . Equations (3.13) and (3.14) then become

)()()(

)( 2

10

)(

tJQtByBJ

tJ

JE j

N

j

ijiiz

Bn

c

i

i

i

c&& ∑

=

=−∇+⎟⎟⎠

⎞⎜⎜⎝

⎛μϕ

J (3.18)

and

64

i

N

i

i wJtI ∑=

=2

1

)( (3.19)

with )( ii vyy = , ),()( tyJtJ ii = , ),()( tyBtB ii&& = , and jjiij wyyQQ ),(= .

Returning to the problem of the divergence of ijQ when ji = , with the variable

substitution )(syy = as shown above, 0)( ≠sv for 10 << s . Thus, the difference ji yy − is

replaced by Nvi π2/ when ji = . The complete definition of the kernel matrix is therefore given

by:

ji

i

ij yyw

Q −= ln2π

, for ji ≠ (3.20)

ππ 2ln

2ii

ij

wwQ = , for ji = (3.21)

Using the inverse matrix ijQ 1− of ijQ and integrating Eqt. (3.18) over the width of the

conductor, we get:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ+=Δ+ ∑∑

=

−

=

)()(

)()()(

)(2

1

12

10

tByBJ

tJ

JEQw

ttIttI jjz

Bn

c

j

j

j

c

N

j

ij

N

i

i&ϕ

μJ

(3.22)

Similarly in 2D calculations, the scalar potential ϕz∇ is calculated from Eqt. (3.22). The current

density is therefore calculated by using Eqt. (3.18). The total loss, transport loss and the

magnetic field distribution can be calculated in a similar way as Eqts. (3.8-3.11) in 2D

calculations. Brandt’s calculation for YBCO conductor can be performed fast and accurately

even by a personal computer. It takes only 10 to 20 minutes to calculate AC loss at each point

(each value of current and/or magnetic field). This model therefore has potential for 2D

investigation of AC loss of a HTS coil. In 2D, a coil is equivalent to a stack of tapes with each

tape carrying the same current. AC loss and electrodynamics in the tapes is more complicated in

this case due to the electromagnetic interactions between tapes. As discussed earlier, because of

the high aspect ratio in YBCO tape, the FEM calculations face a problem of a large number of

65

meshes in this case. This problem can be overcome with the Brandt model since each tape can

be treated as a thin strip with a small error.

3.3.2. Eddy current loss

The eddy current power loss per unit length in the stabilizer at time t is calculated using

∫=CS

eee dxdytyxJtyxEtQ ),,(),,()( (3.23)

where SC is the cross-sectional area of the Cu stabilizer. The electric field ),,( tyxEe and eddy

current density ),,( tyxJ e obey Ohm’s law

),,(),,( tyxJtyxE eCe ρ= (3.24)

where Cρ is the Cu resistivity.

Since Sda >> and Cda >> , both the superconductor and stabilizer layers are considered as thin

films and therefore one can ignore the x–dependence of the eddy current density and electric

field along the thickness of the stabilizer. Using Eqt. (3.17), the integral in Eqt. (3.23) becomes

the discrete form:

Fig. 3.3 An imagined loop on the stabilizer formed by the negative edge and a line at yi position(not to scale)

66

Ci

N

i C

i

e dwtE

tQ ∑=

=2

1

2 )()(

ρ (3.25)

where )( iei yEE = and assuming that the sample is long enough to ignore end effects. Faraday’s

Law is applied for a unit length of a loop formed between the negative the edge of the stabilizer

at y = -a and a line at position yi as shown in Fig. 3.3

dt

tEE i

i

)(0

φ∂−=− (3.26)

In this equation, magnetic flux )(tiφ is defined as

∑∫ ∑=− =

===i

j

jjj

y

a

i

j

ji wtHwtyHdytyHti

10

100 )(),(),()( μμμφ (3.27)

Thus, Eqt. (3.26) becomes

∑=∂

∂+=

i

j

jji wtHt

EE1

00 )(μ (3.28)

Using charge conservation, the total eddy current along the z-axis in the stabilizer must be zero.

Therefore,

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+=== ∑∑∑∑====

i

j

jj

N

i C

CiCi

N

i C

iCi

N

i

i wtHt

Edw

dwE

dwj1

00

2

1

2

1

2

1

)(0 μρρ

(3.29)

or

∑∑∑=== ∂

∂−=

N

i

i

i

j

jj

N

i

i wwtHt

wE2

11

2

100 )(μ (3.30)

For each time step, the magnetic field distribution is calculated using Eqts. (3.8-3.9). Using these

results for Eqt. (3.30), 0E is determined and then substituted into Eqt. (3.28) to calculate iE . The

eddy current loss in the stabilizer is then determined by Eqt. (3.25).

67

If the Ag layer is not very thin (for instance, the Ag layer thickness is greater than 10 percent of

the Cu stabilizer thickness), the eddy current loss generated in this layer must also be considered.

In this case, the total eddy current losses in both Cu and Ag layers are calculated using an

effective thickness, deff, which replaces dC in Eq. (29):

effi

N

i C

i

e dwtE

tQ ∑=

=2

1

2 )()(

ρ (3.31)

The effective thickness effd is calculated by

A

CAAC

eff

ddd

ρρρ +

= (3.32)

where Aρ and Ad are resistivity and thickness of Ag layer, respectively.

3.3.3. Ferromagnetic loss

Recent publications show that the ferromagnetic loss can be significant, particularly when only a

transport current is present [54-58]. The dependence of ferromagnetic loss on the amplitude of

the applied magnetic field can be measured from magnetization hysteresis of an HTS tape in a

SQUID magnetometer at T > Tc [56-58]. This dependence can be fitted by an appropriate

mathematic relation. Combining this relation and the magnetic field distribution inside the

substrate calculated by Eqt. (3.8-3.9), the ferromagnetic loss therefore can be estimated. The

results of ferromagnetic loss calculations will be presented in chapter 5 to compare with

experimental data.

In the case of either transport current or magnetic field only, the magnetic field amplitude at each

point inside the substrate is found at the time when the peak transport current or applied

magnetic field occurs. Otherwise, the magnetic field at any point is compared for all the time

steps in a half cycle to determine the magnetic field amplitude at that point.

68

3.4. Effect of the initial condition

The current density and magnetic field distribution inside the conductor are symmetric or

asymmetric over the x and y axes in parallel or perpendicular applied field. Therefore the

calculations only need to be performed in a half of the tape cross-section. However, when the

field directs at some angle other than 90° or 0°, the calculation must be performed in the whole

cross-section. In addition, to save the computing time, AC loss calculations also need to be

performed in a half of cycle. In Fig. 3.4, the electric field in the conductor was calculated for five

consecutive periods using power law property for Bi-2223 sample B1 with specifications shown

in Table 5.1 in chapter 5. It can be seen that the magnitude of the electric field in first half-period

is smaller than the other periods. From the second half cycle and after, the electric field varies

periodically as expected. This observation suggests that there is the possibility of a significant

error if AC loss is calculated only from the first half-period. To demonstrate the necessity of

using more than one half of a cycle for accurate calculation of AC loss, transport and total AC

losses were calculated from four consecutive half-periods at applied field Ba = 15.6 mT. The

results are depicted in Fig. 3.5. For comparison, the data on measured total AC loss (filled

symbols) and transport losses (open symbols) are also included in Fig. 3.5. As seen from the

figure, AC losses calculated from only the first half-period are much smaller than losses

Fig. 3.4. Calculated electric field along the conductor with perpendicular field Ba = 15.6 mT and transport current It = 75A

69

calculated from other halves, while the results obtained from the second, third and forth half-

periods are equal and agree well with experimentally measured results. Thus, for accurate

calculation of AC losses and the least computing time, the second half-period should be used.

The variation between the first half-period and the other halves is due to effects related to the

initial conditions assumed in the calculation. In the calculations, the current density in the entire

sample are assumed to be zero at t = 0. As seen in the Fig. 3.4, however, the current density over

the cross-section of the sample at ωt = 3π is not zero. Thus, the “initial condition” of the nth half-

period (n ≠ 1) is different from that of the first one. That difference of current distribution results

in an error in AC loss calculation if the first half-period is used.

Fig. 3.5. Calculated transport losses and total losses obtained from the first, second, third andforth half-periods for sample B1. For comparison, experimental results of transport losses andtotal losses are also plotted. Ba = 15.6 mT.

70

(a) ωt = π/2

(b) ωt = π

Fig.3.6. Current density profile captured at (a) ωt = π/2 and (b) ωt = π in 4 mm x 0.25 mm tapewhen it carries a transport current of 80 A

3.5. Electrodynamics and AC loss in HTS tapes with rectangular cross-section

3.5.1 Electrodynamics

In this section, the Brandt model is used to model the time evolution of the current distribution in

a rectangular HTS tape when it carries an AC transport current and/or is placed in an external

AC magnetic field. The simulated tapes are assumed to have the same cross-sectional area of 2a

x 2b = 1 mm2 (a is the half width while b is half-thickness of the tape), critical current of 100 A

(or Jc = 100 A/mm2) and n = 20. The electrodynamics in a square tape (2a x 2b = 1 mm x 1 mm,

α = b/a = 1) and a thin rectangular tape (2a x 2b = 4 mm x 0.25 mm, aspect factor α = b/a =

1/16) will be presented and compared in this section. The transport current and magnetic field are

assumed to have the same phase: )sin()( tItI ω= and )sin()( tBtB ω= . Based on the Critical

State Model, the penetration field on a rectangular conductor is given by:

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++=

20 1

1lnarctan2

αα

απμ c

p

bJB (3.33)

Thus, the penetration fields for the simulated square and thin tape are 90.5 mT and 37.7mT,

respectively.

71

The current distribution at ωt = π/2 and ωt = π when the 4 mm x 0.25 mm tape carries an AC

transport current of magnitude I0 = 80 A = 0.8Ic is shown in Fig.3.6 . In general, the current

distribution in this case is nearly similar as in the case for elliptical tape described in context of

the Critical State Model in section 2.2.3. When the transport current changes between its positive

and negative peaks, the cross-section of the tape is separated into 3 regions, the outer and the

middle region carrying current with opposite sign and a free-current core (green color) in the

center of the tape (Fig. 3.6(b)). The oval shape of the free-current core is almost unchanged in

time.

Fig. 3.7 (a-d) shows the electrodynamics in the 4mm x 0.25mm tape when a perpendicular field

of 30 mT is applied. As discussed in chapter 2, the HTS conductor responds to applied magnetic

field by distributing the shielding current in such a way that the interior of the conductor is best

shielded from external magnetic field. When ωt = π/2 or magnetic field reaches its positive peak,

(a) ωt = π/2

(b) ωt = 2π/3

(c) ωt = 5π/6

(d) ωt = π

Fig. 3.7. Current profile in 4 mm x 0.25 mm tape (a, b, c, d) and in square wire (e, f) forperpendicular applied magnetic field of 30 mT

1 2 3 4 5

(e) ωt = π/2 (f) ωt = π

72

the conductor is separated into 3 regions with positive current (red color), free-current core

(green) and negative current (dark color). Those regions correspond to the regions with current

density of +Jc, 0, -Jc in the context of the Critical State Model as discussed in chapter 2. The

existence of the small free-current core means that, in perpendicular field of 30mT, the tape has

not reached its full penetration state as expected. The cross-like shape of the free-current core is

the result of the high aspect ratio of the rectangular conductor. When the magnetic field

decreases from the peak value, the current density changes its sign starting from the edge and the

surface of the sample, to form 5 regions with negative, positive, zero, negative and positive

current density corresponding to regions 1, 2, 3, 4, 5 as seen in Fig. 3.7 (b-d). During the

descending half cycle of magnetic field, regions 2 and 4 gradually shrink and disappear as the

field reaches its negative peak. The current density is highest near the edge and near the surface

of the tape.

Unlike the Critical State Model, which limits the critical current density to Jc, the local current

density near the edge of the tape in in Fig. 3.7 (a) increases to 130 A/mm2, 130% of Jc of the

sample. In such regions with J > Jc, flux-flow loss will be generated. Therefore, most of AC loss

is generated near the edge or the surface of the tape.

Fig. 3.7 (e-f) depicts the current distribution in a square wire in perpendicular field Ba = 30 mT.

With aspect ratio α = 1, the shape of the free current-core in the square wire has an oval shape

when ωt = π/2. The larger free-current core in square wire for the same applied magnetic field

suggests that the penetration field of square wire is higher than that of the thin tape. This agrees

with values of the full penetration full for these conductors estimated from Eqt. (3.33)

When the applied magnetic field 30mT is applied in angle θ = 60°, the current profile in the tape

is shown in Fig. 3.8. In this case, there is no free-current region; the tape is in full penetration.

When the field decreases from its positive peak to negative peak, the current also changes its

sign, from the edge and the surface of the tape inwards. There are only two regions with current

density of opposite sign formed in this case. Fig. 3.8 shows how the boundary between these

regions moves when the magnetic field decreases from positive peak to zero in a quarter-period.

73

Now consider the electrodynamics in the conductor when it carries an AC transport current and

is placed in magnetic field. Fig. 3.9 (a-b) depicts the current profile in the tape for I = 80A =

0.8Ic and B = 5mT at times ωt = π/2 and ωt = π. The current profiles in Fig. 3.9 (a-b) are

qualitatively similar to those for transport current shown in Fig. 3.6 except the current free core

is shifted to the left. In this case the current density at both edges of the tapes has the same sign

when ωt = π/2. This is a characteristic of the current distribution with transport current. The

current profile observed in Fig. 3.9 (a-b) therefore is the distribution of the high current –low

field regime that was discussed when analytical model for calculating total ac loss of a thin strip

was presented in chapter 2.

The current distribution in the tape when carrying a transport current of 50 A in a 30 mT

magnetic field is shown in Fig.3.9 (c-f). In this case there is no free-current core and the tape is

fully penetrated. The current distribution at both edges of the tape seems to have opposite sign

and changes in time in a opposite way. This is a characteristic for current distribution created by

(a): ωt = 3π/6

(b): ωt = 4π/6

(c): ωt = 5π/6

(d): ωt = π

Fig. 3.8. Current profile in 4 mm x 0.25 mm when magnetic field of 30 mT is applied at angle θ= 60°

74

applied magnetic field. Thus, the current distribution in this case describes the electrodynamics

of the low current –high field regime.

These results of the current distribution for the case that both transport and magnetic field are

applied illustrate why the total AC loss in a thin strip needs to be calculated analytically by two

equations, one for the low current-high field regime and one for the high current –low field

regime (section 2.25).

3.5.2 AC loss in rectangular tapes

In order to estimate the dependence of the shape of cross-section of the HTS tape on AC loss,

calculations were performed for three rectangular tapes with the same cross-section and area of 1

mm2. These simulated tapes have dimensions of 1mm x 1mm (aspect ratio α = 1), 2.5mm x

0.4mm (α = 1/6.25) and 4mm x 0.25mm (α = 1/16). The critical current density, Jc=100A/mm2,

is also assumed for all three tapes.

(a) ωt = π/2, I = 80 A, B = 5 mT

(a) ωt = π, I = 80 A, B = 5 mT

(c) ωt = π/2, I = 50 A, B = 30 mT (d) ωt = 2π/3, I = 50 A, B = 30 mT

(e) ωt = 5π/6, I = 50 A, B = 30 mT (f) ωt = π , I = 50 A, B = 30 mT

Fig. 3.9. Current profile in the tape in the high current – low field regime with B0 = 5 mT, I0 =80 A (a, b) and the low current – high field regime with B0 = 30 mT, I0 = 50 A (c, d, e, f)

75

Figure 3.10 depicts the simulated self-field AC loss in the HTS rectangular tapes in comparison

with the results obtained from Norris analytical model for a thin strip and an elliptical tape (Eqts.

2.47 and 2.35). The self-field loss for rectangular tape decreases as the aspect ratio of the tape

decreases. The loss in the square conductor is the highest and close to the loss in an elliptical

wire. The slope of the calculated loss curves is slightly changed at current I0 = 30A. Below 30A,

the slope is about 4, which is the same as the slope of Norris loss curve for a thin strip. When I >

30A, the slope is about 3, the same as the slope of Norris loss curve for an elliptical conductor.

This behavior may be related to the power law I-V characteristics used in the numerical

calculation and is also observed in the experimental results for Bi-2223 tape presented in chapter

5.

0.00000001

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

10 100

Transport Curre nt (A)

Tra

ns

po

rt l

os

s (

J/m

/cy

cle

)

1 m m x 1 m m

2.5 m m x 0.4 m m

4 m m x 0.25 m m

Norris's Strip

Norris's Eliipse

Fig. 3.10. Transport loss in rectangular tapes with different aspect ratios as a function of currentamplitude

76

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1 10 100

Magnetization (mT)

Mag

neti

zati

on

lo

ss (

J/m

/cycle

)

Elliptical tape

4 mm x 0.25 mm

2.5 mm x 0.4 mm

1 mm x 1 mm

Fig. 3.11. Numerically calculated magnetization loss of rectangular tapes in perpendicularapplied field. Analytical results for elliptical tape of thickness 0.25 mm and width 4 mm is alsoplotted

Contrary to the transport AC loss case, for a perpendicular magnetic applied field, the higher

aspect ratio tape generates more AC loss as seen in Fig. 3.11, which shows the magnetization

loss calculated numerically for three rectangular tapes with different cross-sectional dimensions

and the analytical results for magnetization loss in an elliptical tape of thickness 0.25 mm and

width 4 mm. For the field higher than 20 mT, the losses in elliptical and rectangular tapes with

the same thickness and width are almost equal. In the lower field region, the elliptical tape

generates higher loss than the rectangular tape.

The dependence of magnetization loss in the tape with dimensions of 4mm x 0.25mm on the

magnetic field orientation is shown in Fig. 3.12. The magnetic field is applied at 4 angles: 0

(parallel), 30°, 60°, and 90° (perpendicular). The slope of the loss curve is similar for all field

angles. For the whole range of applied field, magnetization loss reduces slowly when the field

angle changes from 90° to 60° and then drops faster when the field angle changes from 60° to 0°.

77

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1 10 100

magnetic field (mT)

Ma

gn

eti

za

tio

n lo

ss

(J

/m/c

yc

le)

0

30

60

90

θ (o)

Fig. 3.12. Magnetization loss in the 4 mm x 0.25 mm tape with applied magnetic field atdifferent angles

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

0.1 1 10 100

Peprpendicular field component (mT)

ma

gn

eti

za

tio

n lo

ss

(J

/m/c

yc

le)

30

60

90

θ (o)

Fig. 3.13. Magnetization loss as a function of the perpendicular field component for differentfield angles (tape 4 mm x 0.25 mm)

78

0

5

10

15

20

25

30

35

0 20 40 60 80 100

Magnetic field (mT)

4 mm x 0.25 mm

2.5 mm x 0.4 mm

Fig. 3.14. The loss ratio Q (θ = 90°)/Q(θ = 0°) for two tapes: 2.5mm x 0.4mm (α = 1/6.25) and4mm x 0.25mm (α = 1/16)

Q (θ

= 9

0°)

/Q(θ

= 0

°)

In fact, for such a thin tape, magnetization loss is mainly generated by the perpendicular

component of magnetic field. For more clarity, magnetization loss as a function of perpendicular

field component Bsin(θ) is plotted in Fig. 3.13. The loss curves for θ = 90° and θ = 60° are

superimposed together while the curve for θ = 30° is a little higher. This means that, for field

angle larger than 60°, contribution of the parallel field to magnetization is small. This result

agrees with experimental data reported in [86-87] for Bi-2223 tape. For HTS tapes with very

high aspect ratio, such as YBCO conductor, the contribution of parallel field is much smaller and

can be ignored for any field angle.

In the context of the Critical State Model, it was shown in section 2.2.4 that the ratio between

AC losses in perpendicular and by parallel magnetic field for an elliptical tape is 1/α, Q(θ =

90°)/Q(θ = 0°) = 1/α, if the field is high enough (Fig. 2.17). A similar behavior is also observed

for numerically calculated magnetization loss in rectangular tapes as described in Fig. 3.14. The

loss ratio was plotted for two rectangular tapes with dimensions: 2.5mm x 0.4mm (α = 1/6.25)

and 4mm x 0.25mm (α = 1/16). The loss ratio is less than 10% different from 1/α for magnetic

field higher than 5 mT.

79

0.00001

0.0001

0.001

0.01

10 100 1000

Tra nsport curre nt (A)

AC

lo

ss (

J/m

/cycle

)

Transport loss

Magnetiz ation loss

90o

60o

30o

0o

Fig. 3.15. Dependence of transport and magnetization loss in 4mm x 0.25mm tape on field angleand B = 30 mT

The orientation of applied magnetic field not only affects magnetization loss but also strongly

impacts the transport loss component when an HTS tape carries an AC transport current in an

applied magnetic field. Fig.3.15 depicts the magnetization loss and transport loss components

generated in the 4mm x 0.25mm tape when the transport current changes from 10A to 100A and

applied field of 30mT is applied with field angles 90°, 60°, 30° and 0°. Summing the transport

loss and magnetization loss results in the total AC loss. Similar to magnetization loss, the

transport loss component also decreases rather slowly when θ decreases from 900 to 600 and

drops faster when θ decreases from 600 to 00.

Contrary to high aspect ratio tapes, the effect of the field orientation on AC losses is quite small

for a square wire. Fig 3.16 shows transport loss and magnetization loss when a square wire

carries a transport current in B0 = 20 and 40mT with two orientations, θ = 450 and θ =900. The

magnetization loss is independent of θ while the transport current loss at θ = 450 is slightly

smaller than that of θ = 900. Theoretically, for a round wire, the loss characteristics will be the

same for all field orientations. The small anisotropy in the cross-sectional shape of square wire

results in a small effect of field angle on ac loss in this kind of conductor.

80

0.000001

0.00001

0.0001

0.001

0.01

10 100

Transport current (A)

AC

lo

ss

(J

/m/c

yc

le)

Series1 Series2

Series6 Series5

Series7 Series4

Series3 Series8

45o, 20 mT

90o, 20 mT

45o, 40 mT

90o, 40 mT

45o, 20 mT

90o, 20 mT

45o, 40 mT

90o, 40 mT

Magnetization Transport

Fig. 3.16 Transport and magnetization loss in square wire for several field orientations

3.6. Electrodynamics and AC loss in YBCO tapes

The numerical calculations presented in section 3.3 are employed to study the general AC loss

characteristics and electrodynamics in the HTS and stabilizer layers of a YBCO coated

conductor. The calculation was performed for a YBCOconductor which is used for AC loss

measurements presented in chapter 5. The simulated conductors consist of a thin YBCO layer

(1μm) and a 75μm thick Cu stabilizer. The critical current and n-value at 77 K are 186 A and 22,

respectively. The results are presented for three specific situations: (A) magnetic field applied

perpendicular to the tape without transport current, (B) transport current is passed through the

tape without a background magnetic field, and (C) both a perpendicular magnetic field and a

transport current are applied.

3.6.1 Conductor in perpendicular magnetic field

81

Fig. 3.17. One full cycle of magnetic field, illustrating the seven times at which snapshots ofcurrent and magnetic field profiles are taken

Consider the case of an AC magnetic field, B(t), applied perpendicular to the wide surface of the

conductor. To determine the current and magnetic field distributions versus time, consider seven

moments in time during a half-cycle such that B(t) decreases from B0 = 20 mT to –B0 = -20 mT.

The seven “snapshots” in time are illustrated in Fig. 3.17.

Figs. 3.18(a) and 3.18(b) show the current distribution J(y) and magnetic field B(y)

corresponding to these seven times as calculated from both the numerical model (dashed lines)

and the analytical Eqts. 2.61 and 2.62 (solid lines) given by Brandt [31]. The analytical results

are plotted only in the positive half of the conductor for clarity in the figure. The numerically

calculated J(y) and B(y) with a power law (n = 22) are in general agreement with those calculated

by Brandt using the Critical State Model. As a result of Critical State Model, the current density

calculated by the analytical model is equal to Jc in the penetrated region, slightly smaller than

that calculated numerically with power-law characteristics. Fig. 3.18(c) shows the distribution of

the eddy current density Je(y) calculated numerically over the stabilizer width.

82

Fig. 3.18. (a) Current density J(y), (b) magnetic field B(y) and (c) eddy current density Je(y) areplotted for the seven times indicated in Fig. 3.17. In (a) and (b), analytical results (solid lines)are plotted for the positive half of the conductor and numerical result (dashed lines) are plottedeverywhere. (c) shows only numerical results.

83

Fig. 3.19. Numerical results of Qm and Qe for frequency of 51 Hz (dotted lines with opensymbols) and 151 Hz (dotted lines with filled symbols). Corresponding analytical results (solidlines) are also plotted.

The eddy current density is asymmetric with respect to the z-axis because of the symmetry of the

magnetic field distribution as shown in Fig. 3.18(b). For current continuity, the eddy currents

must form closed loops. In this case, the eddy current loops are formed between two regions near

the edges of the stabilizer. The eddy currents run along one edge, cross the stabilizer and then

run along the other edge in the opposite direction. Therefore, when B(t) changes from B0 to –B0,

the eddy current density is higher near the edges of the tape and becomes zero at a certain

distance from the center of the stabilizer. The zero-eddy current portions near the tape center

correspond to the non-varying portions of B(y) in Fig. 3.18(b).

The magnetization loss in the HTS layer, Qm, and eddy current loss, Qe, calculated numerically

for applied magnetic field ranging from 1 to 100 mT at frequencies 51 and 151 Hz, are plotted in

Fig. 3.19. Open symbols correspond to the results at 51 Hz and filled symbols are results at 151

Hz. The loss component Qm is frequency independent as expected. The eddy current loss

component in the Cu layer, Qe, increases by a factor of two to four as the frequency changes

from 51 to 151 Hz. Overall, Qm is the dominant AC loss component. The eddy current loss

84

changes rapidly with increasing field and frequency and needs to be considered in the high field

region. Analytical results for Qm calculated from Eqt. (2.64) and for Qe calculated from Eqt.

(2.66) are also plotted in Fig. 3.19 for comparison. Good agreement between numerical and

analytical results is observed for magnetization loss. There is a small variation between the

calculations of eddy current loss at low magnetic field. This suggests that n-value affects Qe

more significantly than Qm.

3.6.2 Conductor with a transport current

Now consider the situation where there is only transport current I(t) in the conductor and no

applied magnetic field. The current distribution J(y) and magnetic field B(y) for I0 = 167.4 A (0.9

Ic) are plotted in Figs. 3.20(a) and 3.20(b) for the same seven times shown in Fig. 3.17. Again,

good agreement between numerical (dashed lines) and analytical (solid lines) results is observed.

B(y) calculated analytically using Eqt. 2.51 and 2.52 penetrates the conductor slightly further

than the computational result. Unlike the previous case, the eddy current density Je(y) shown in

Fig. 3.20(c) is symmetric with respect to the z-axis as a result of the asymmetric magnetic field

distribution shown in Fig. 3.20 (b). In this case, the separate eddy current loops are formed

between either edge of the stabilizer and its center. Therefore, Je(y) near the center of the sample

is a non-zero constant, corresponding to the “frozen-like” portions of B(y) observed in figure

3.20(b).

The AC losses for this case are shown in Fig. 3.21. The transport loss in HTS layer, Qt, and the

eddy current loss, Qe, calculated analytically from Eqts. (2.46) and (2.55) are also plotted. There

is good agreement between the numerical and analytical results for transport in HTS layer, Qt

while some differences are observed for Qe, especially for the low and high current regions. The

difference between numerical and analytical calculations of Qe is explained by two factors. The

first is the difference in the assumed n-value; the second is from an assumption in Müller’s

calculation [67] that the eddy current electric field in the center of stabilizer is zero. In fact, as

seen in Fig. 3.20(c), Ee is a non-zero constant near the tape center.

85

Fig. 3.20. (a) Current density J(y), (b) magnetic field B(y) and (c) eddy current density Je(y) areplotted for the seven times indicated in Fig. 3.17. In (a) and (b), analytical results (solid lines)and numerical results (dashed lines) are shown. (c) shows only numerical results.

86

10 10010

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

AC

lo

ss (

J/m

/cycle

)

Transport current (A)

Qt (Numerical)

Qe (Numerical)

Qt (Analytical)

Qe (Analytical)

Fig. 3.21. Numerical results (open symbols) of transport loss Qt, and eddy current loss Qe at 51Hz. Corresponding analytical results for Qt and Qe (filled symbols) are also plotted.

3.6.3. Current carrying conductor in a perpendicular applied magnetic field

Lastly, consider the case when the transport current and magnetic field are simultaneously

applied to a YBCO coated conductor. The total AC loss generated in the HTS layer, Qs,

calculated numerically (open symbols) and analytically (solid lines) using Schönborg’s Eqts 2.68

and 2.69 are plotted in Fig. 2.22. The numerical results reproduce the analytical results well. A

slight variation between the two calculations in high-field and high-current regions may be

explained by the difference in n-value used in each calculation. In the numerical model, n = 22,

while in the analytical calculation, n is assumed to be infinite (Critical State Model).

87

Fig. 3.22. The total loss Qs in the superconducting layer is plotted for I/Ic = 0.1, 0.3, 0.5, 0.7 and0.9. Numerical (open symbols) and analytical results (solid lines) are seen.

3.6.4. AC loss and the critical current distribution

With the suitable variable substitutions, electrodynamics in a thin strip can be numerically

calculated accurately. Good agreement between analytical results and numerical results was

observed in this case. A thin HTS tapes with elliptical cross-section is possibly treated as a thin

strip with non-uniform critical current density. Assume a thin conductor with elliptical cross-

section as shown in Fig. 3.23. In this case, the sheet critical current density at position y, Jc(y),

will be proportional to the thickness of the conductor at that position. Thus,

2

022

21

42)( ⎟

⎠⎞

⎜⎝⎛−=−=

a

yJya

a

IyJ c

cc ππ

(3.34)

with Jc0 is the uniform sheet critical current density, Jc0 = Ic/2a. Hence, the elliptical critical

current density is independent of the thickness b of the tape.

88

a

b

-b y-a y

x

a

b

-b y-a y

x

Fig. 3.23. Elliptical cross-section of a HTS tape in xy plane

Jc(y)/Jc0

0

0.5

1

1.5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y/a

Elliptical

k = 1.5

k = 0.5

Fig. 3.24. Normalized critical current distribution Jc(y)/ Jc0 as a function of y/a for three different Jc(y) distributions

To understand more about the effect on Jc(y) distribution of AC loss, convex and concave Jc(y)

distributions are also considered. Assume that an YBCO tape is placed in xyz-coordinates as

shown in Fig. 3.2. Since Jc(y) should be symmetric over its center (z-axis), a simple second order

polynomial bayyJ c += 2)( is used to describe Jc(y) distribution. A parameter k with k = Jc(a)/

Jc(0) is introduced to determined the shape of Jc(y). Distribution Jc(y) is concave for k > 1, and

convex for k < 1. The uniform Jc(y) distribution corresponds to k = 1. Because integrating Jc(y)

over the tape width must yield the critical current Ic, the Jc(y) therefore can be give by:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

+−

−+

=2

0 2

)1(2

2

3)(

a

y

k

k

kJyJ cc (3.35)

Fig. 3.24 illustrates the normalized critical current distribution Jc(y)/ Jc0 for three different cases,

k = 1.5 (concave distribution), k = 0.5 (convex distribution) and elliptical distribution (Eqt. 3.34).

89

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

10 100 1000

Transport current (A)

Tra

ns

po

rt lo

ss

(J

/m/c

yc

le)

k =1.5

k = 1

k = 0.5

Elliptical distribution

Strip (Norris model)

Elllipse (Norris model)

Fig. 3.25. Numerical results of the self-field loss as a function of transport current for differentJc(y) distributions. Analytical results obtained from Norris model for elliptical and strip tapes arealso plotted.

AC loss characteristics are analyzed for 1 cm wide tape for three Jc(y) distributions plotted in

Fig. 3.24. Similar to previous simulations, the Ic = 186 A and n = 22 are assumed. Fig. 3.25

depicts the self-field loss calculated numerically for a thin strip with different Jc(y) distributions.

Analytical results obtained from Norris model for elliptical and thin strip conductors are also

plotted for comparison. As seen in the figure, effect of the Jc(y) distribution on self-field loss is

significant. As expected, the numerical results for the uniform Jc(y) distribution (k =1) is nearly

identical to the analytical results obtained from the Norris model for a thin strip. The sample with

convex Jc(y) distribution generates considerably higher self-field loss than the sample with

concave Jc(y). The numerical results for a thin strip with elliptical Jc(y) agree well with Norris’s

results for an elliptical cross-sectional tape. Thus, AC loss numerical calculation for a thin HTS

with elliptical cross-section can be performed in 1D but with elliptical Jc(y) distribution. This

finding is important since it can help to reduce the computing time and error for calculation of

AC loss in an elliptical tape with low aspect ratio, α = b/a << 1.

Fig. 3.26 illustrates the magnetization loss in the HTS strip for those Jc(y) distributions. In the

low field region, the sample with convex Jc(y) distribution generate higher AC loss than the

90

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100

Magnetic field (mT)

Mag

neti

zati

on

lo

ss (

J/m

/cycle

)

k = 1.5

k = 0.5

Elliptical distribution

Fig. 3.26. Numerical results of the magnetization loss as a function of magnetic field fordifferent Jc(y) distributions

sample with concave Jc(y) distribution, similar to the self-field loss case. A contrary effect is

observed in the high field region. The sample with convex Jc(y) distribution generate slightly

lower AC loss than the sample with concave Jc(y) distribution.

3.7. Chapter summary

Numerical calculation using Brandt’s model is successfully demonstrated to study

electrodynamics and to calculate all the individual loss components generated in HTS tapes with

various cross section of shapes. With an appropriate variable substitution, this model provides

accurate results for AC loss calculations in YBCO coated conductors. Good agreements between

analytical results and numerical results were observed in this case. The critical current density

distribution strongly affects on AC loss characteristics of HTS tapes. In order to compare with

experimental results of actual HTS tapes, other important practical properties of HTS conductors,

such as field dependence of Jc and n-value should be taken in to consideration as presented in

chapter 5.

91

CHAPTER 4

AC LOSS MEASUREMENTS

4.1 Introduction

In general, the AC losses in HTS conductors can be measured by either electromagnetic (EM)

[49, 54-57, 88-103] or calorimetric (CM) [55, 104-114] methods. In the EM method,

magnetization losses and transport current losses are measured separately and then added to

obtain the total loss of the conductor. The EM method, however, is only applicable when the

applied magnetic field and the transport current are in phase. A small phase difference between

the applied field and the transport current may result in considerable error. The CM method,

however, can be used to measure the AC loss with any phase difference between the magnetic

field and transport current. This method, however, is less sensitive and more time consuming

than the EM method. Some advantages and disadvantages of these methods are summarized in

Table 4.1.

A main part of this dissertation is to measure the total AC loss in HTS tapes by both EM and CM

methods. Especially, a novel total AC loss measurement for variable temperatures from 30 K to

100 K has been successfully built.

92

Table 4.1. Comparison between electromagnetic and calorimetric methods

Electromagnetic method Calorimetric method

Advantages

• Higher sensitivity, rapid measurement

• Easy to measure in a wide range of

frequency, transport current and

magnetic field

• Can measure transport loss and

magnetization loss separately

Disadvantages:

• Transport current and magnetic field

must be in phase

• Accurate phase control and proper

compensation are needed

Advantages

• Applicable for any phase difference

between transport current and magnetic

field

Disadvantages:

• Lower sensitivity, relatively time

consuming

• Only provides data for the total AC loss

• Temperature rise on the sample may

affect the sample properties and cause

error

• Must be able to measure small

temperature rise, in mK scale

4.2. Measurement principle of the electromagnetic method

4.2.1. Transport current loss measurement

Transport current loss in HTS tapes is caused by the self-field created by the transport current

within the conductor. Self-field therefore must be taken into account when measuring transport

AC loss.

In general, electromagnetic loss per unit length, per cycle, is calculated by integrating Poynting’s

vector on any closed surface S surrounding the sample in a period T:

93

∫∫ ×−=S

T

dSdtl

W HE0

1 (4.1)

where l is the length of the sample located inside surface S. E and H are local electric field and

magnetic field on surface S.

Consider an HTS tape with width a placed in xy-coordinates as shown in Fig. 4.1. An AC

transport current I(t) is applied along the y axis. The voltage leads are soldered to the tape and

expanded to a distance r from the tape center. The distances between the soldering taps are l.

Because of the existence of a free-current core in the HTS conductor, the electric field along its

axis (y-axis) must be zero. Thus, consider to a loop formed by the voltage leads and the center of

the tape (the red loop in Fig. 4.1). The relation between the electric field at a distance r and the

voltage picked up on the voltage loop becomes:

l

trVtrE

),(),( = (4.2)

r

x

Iy

V(r)

S

a

l

r

x

Iy

V(r)

S

a

l

Fig. 4.1. Sketch of a HTS conductor with a voltage loop to measure transport AC loss

HTS conductor

94

Imagine a cylindrical surface S of radius r that has the same axis as the conductor. If the

conductor is a homogeneous round wire, the electric field will be uniform in surface S as a

consequence of the symmetry. The magnetic field on the surface is also uniform and given by:

r

tItrH

π2

)(),( = (4.3)

Thus, using Eqt. (4.1-4.3), the transport loss per unit length per cycle is derived from integrating

Poynting’s vector on the surface S :

)(1

),()(1

rIVlf

dttrVtIl

WT

i∫ == (4.4)

Where Vi(r) is the root-mean-square (RMS) value of the in-phase component of V(r) measured

by the voltage leads. This is a well-known formula to evaluate the transport loss for a normal

conductor. From Eqt. (4.4), the loss is determined only from the in-phase component of voltage

picked up on the voltage leads. Experimentally, that component can be measured using of lock-in

amplifier with the reference signal in phase with the transport current.

For a round wire, Eqts. (4.2- 4.3) hold for any distance r > a. The voltage lead therefore can be

placed in the surface of the conductor to reduce the noise from the inductive (out-of-phase)

component.

For a tape conductor, the assumption that E(r) and H(r) are uniform on S surface holds if r » a.

In principle, r must be infinite to obtain the true transport, but this is not practical. It was proven

by both experiments and calculations that r must be at least three times the half-width of the tape

to obtain a results of 95% or higher of the true loss [88-90]. Moreover, if a large loop is used, the

inductive signal and the noise from the surrounding environment may include more error.

Therefore, r = 3a is the optimum half-width of the voltage loop for transport loss measurements

of a HTS tape

Physically, for a homogeneous circular wire, the current distribution is always symmetric about

its axis. Thus, when the current distribution changes inside the wire, no magnetic field around the

conductor is in quadrature with the transport current. Therefore, the in-phase component of V(r)

95

is independent of ar ≥ . This is not the case for a non-circular wire. Therefore the in-phase

component of V(r) varies with ar ≥ .

4.2.2 Magnetization loss measurement

Magnetization loss in a HTS conductor placed in an AC applied field, B(t), per unit length and

per cycle is calculated conventionally from the area of the magnetization loop by the following

formula [17, 45, 62]:

( )dt

t

tMtBQ

T

am ∂∂

= ∫ )( (4.5)

where M(t) is magnetization per unit length of the conductor which is created by the screening

and eddy currents in the sample. Measurements of magnetization M(t) can be performed using a

conventional pick-up coil.

This study focuses on AC loss in HTS tapes, and in particular YBCO coated conductors. With

high respect ratio tapes, the effect of the parallel field component is very small and

magnetization losses therefore depend mainly on the component of magnetic field that is normal

to the wide surface of the tape conductors [95]. Therefore, this dissertation focuses only on

magnetization loss of an HTS tape exposed to a perpendicular magnetic field. An in-plane

y

x

B

Pick-up coilHTS tape

a W y

x

B

Pick-up coilHTS tape

a W

Fig. 4.2. A thin tape HTS of width 2a and an in-plane pick-up coil of width 2W

96

rectangular pick-up coil is a convenient configuration in this circumstance. Assume that a HTS

tape with the half-width a and an in-plane pick-up coil with a half-width W are placed in a xyz-

coordinate system as seen in Fig 4.2. The sample is assumed to be long enough so that end

effects can be ignored. With the in-plane pick-up coil, the time derivatives of magnetization M(t)

on the tape can be determined from the induced voltage in pick-up coil [90-92]:

P

P

NL

tVWC

t

tM

0

)()(

μπ=

∂∂

(4.6)

In the above equation, C is a calibration factor, N, W and LP are respectively the number of turns,

half-width and length of the pick-up coil, and Vp(t) is the voltage of the pick-up coil . The

calibration factor C accounts for conversion of the measured induced voltage on the pick-up coil

to the magnetic moment of the sample. Using Eqt. (4.5), the magnetization loss was calculated

using the following formula [90-92]:

fNL

VHWCQ

P

iPa

m

,π= (4.7)

In Eqt. (4.7), Ha is the RMS value of applied magnetic field and f is the frequency. Vp,,i is the loss

voltage component on the pick-up coil that is in phase with the applied magnetic field and can be

measured by a lock-in amplifier.

Factor C has been estimated experimentally [91, 93] and numerically [92, 102] for some specific

pick-up coils. Similar to the transport current case, when W > 3a, C ~ 1 with an error of few

percents [90, 92]; i.e., the magnetization loss measured by a large in-plane pick-up coil can be

determined by:

fNL

VHWQ

P

Pam π= (4.8)

Hence, in magnetization loss measurements, only the in-phase component is needed from the

pick-up coil to determine the magnetization loss. Thus, to reduce the error, a cancellation coil is

used to eliminate the inductive voltage component of the pick up coil, which might be few orders

higher than the in-phase component. The cancellation coil must be designed to pick-up the same

97

inductive signal as the pick-up coil but it must be placed far from the sample to ensure that it

picks up only the inductive voltage caused by the applied magnetic field. For a wide sample,

therefore, there is sometimes insufficient room for both the pick up coil and compensation coil

with widths about three times the sample width. In addition, the pick-up coil should not be too

large in order to increase the signal-to-noise ratio. Therefore, in [41], the dependence of factor C

on the width of the pick up coil is investigated in detail by two methods: numerical calculation

based on uniform magnetization approximation [100] and analytical calculation based on field

distribution in thin strip conductors using Brandt’s method. The simulations were performed for

YBCO tape with half-width a = 5 mm and Ic = 160 A. For this tape, the characteristic field

4.620 == aIB cc πμ mT.

Figure 4.3 plots the dependence of C on applied field simulated by the Brandt method is shown

for pick-up coils with different widths. C is a constant in the high field and low field region and

the transition between these region occurs at magnetic field around Hc = 6.4 mT. When the half-

width of the pick-up coil W = 15 mm (or normalized width W/a = 3), C is almost constant and

equal to 0.96 for whole range of applied field with an error of ±2 %. If W = 10 mm (or

Fig. 4.3. Dependence of C of pick-up coils with different widths on the applied magnetic field,simulated by Brandt’s equations for a thin strip conductor

Magnetic field (mT)

Fac

tor

C

98

normalized width W/a = 2), C = 0.91 for the whole range of applied field with an error of ± 4 %.

If W = 6 mm (or W/a = 1.2), C varies from 0.55 to 0.85 when the applied field changes from 0.1

mT to 100 mT. Thus, magnetization loss measurements using such a small pick-up coil must

consider to the dependence of the calibration factor C on the applied magnetic field if they are

performed over a wide range of field.

Figure 4.4 shows dependence of C calculated from Brandt’s method on the pick-up width for

different values of applied magnetic field. To compare, C calculated from a uniform

magnetization approximation (C calculated by this method is independent of the applied field) is

also plotted. C is consistent for both methods. The dependence of C on the pick-up coil width

estimated by the uniform magnetization approximation is nearly identical to that calculated from

Brandt’s thin strip model for applied field 5 mT.

Fig. 4.4. Calibration factor C calculated by Brandt equations for different values of appliedmagnetic field. Dependence of C on the pick-up coil width calculated by uniform magnetizationapproximation (the solid line) is also plotted.

Normalized width, W/a

Fac

tor

C

99

In some cases, instead of putting the cancellation coil far from the sample, a double pick-up coil

with both “pick up” coil and “cancellation” coil in a same plane as shown in Fig. 4.5 can be used.

This configuration consists of two pick-up coils, the inner coil with width W1 and N1 turns and

the outer coil with width W2 and N2 turns. If (W1N1) = (W2N2), the inductive part cancels when the

two coils are connected in anti-series. Thus, in principle, a double pick-up coil formed by those

two coils connected in anti-series will have zero inductive component. Assume C1 and C2 are the

calibration factors of the inner and outer coils, respectively. Then, AC loss can be measured by

the double pick-up coil using:

fL

VH

WCNWCN

WCWCQ

P

Pam

112221

2211

2 −=

π (4.9)

4.2.3 Total AC loss measurement

When transport and magnetic fields are simultaneously applied, the transport loss generated by

transport current and the magnetization loss generated by the applied field are still measured by

the voltage leads and pick-up coil as presented in the previous section [91]. The voltage lead

configuration, however, must be considered to avoid coupling between the external magnetic

field and the self-field, ensuring that the pick-up coil only measures the “magnetization” signal

y

x

B

HTS tape

a W1 W2

Inner coil

Outer coil

y

x

B

HTS tape

a W1 W2

Inner coil

Outer coil

Fig. 4.5. An HTS thin tape of width 2a and a double pick-up coil

100

and the voltage loop only measures the “transport” signal. The self-field created transport current

will go around the tape, thus it does not affect the pick-up coil signal. If the voltage loop of the

transport measurement is arranged as in Fig. 4.1, however, it will pick up some signal from the

external magnetic field. Therefore, a symmetric “figure 8” shaped voltage loop, as shown in Fig.

4.6, has been used in this situation [91].

With the “figure 8” shape, two parts of the voltage loop have opposite orientations to the external

magnetic field to minimize the effect of the applied field. If the areas of two parts are exactly

equal, the correct transport loss is measured. A spiral voltage loop is another solution for this

problem [94-95, 105]. More detailed discussion about the error from the voltage loop

arrangement will be given in section 4.5.

4.3. Experimental setup

4.3.1 Electrical setup

The electrical set-up for the measurement is shown in Fig. 4.7. The experiment is automatically

controlled by a computer with Labview through a GPIB bus system. Two function generators are

used with reference time-bases locked together through a 10 MHz synchronization signals. With

this set-up the phase difference between the function generator outputs can be set at any value

I

B

Voltage loop

Pick-up coil

HTS Tape

I

B

Voltage loop

Pick-up coil

HTS Tape

Fig. 4.6. Pick-up coil arrangement and a figure “8” shaped voltage loop for total ac lossmeasurement

101

between 0° to 360°. The phase-locked signals from these function generators are used to

modulate two AE Techron power amplifiers. Each amplifier has power of 12 kW, enough to

provide high AC current and magnetic field. One power amplifier feeds current for the dipole

magnet and the other provides transport current for the sample. In principle, this setup works

over a wide range of frequencies, from DC to 10 kHz.

To compensate for magnet inductance, a high power capacitor system is connected in series to

the magnet. To achieve high magnetic field, the measuring frequency must be near the resonance

frequency of the capacitor-magnet system. To measure at different frequencies, the capacitor

system must be adjusted by changing the connecting combinations between the capacitor

components.

The current feeding the magnet is measured by a current transducer and current going through

the sample is measured by a non-inductive shunt or another current transducer. Since the

transport circuit load is very small, a current transformer is used for amplification. With the

current setup, a current as 600 A at 500 Hz can be achieved. A transformer plays important role

in isolating the measuring transport circuit and the power transformer. Without the transformer,

it is very difficult to measure the transport loss due to the significant contribution of the common

mode noise, even with the lock-in amplifier having a common mode rejection ratio (CMRR)

higher than 100 dB.

The signal picked up from the current transducer is also used for the reference signals to all three

lock-in amplifiers. Lock-in amplifier 3 (see Fig. 4.7) monitors the phase difference between the

transport current and applied magnetic field to ensure that the current and magnetic are in phase

when AC loss data are acquired. After the transport current and magnetic field are set, there will

be some phase shift between them due to several reasons, such as the transformer response time,

and the coupling between transport current and magnetic field circuit. Lock-in amplifier 3 is

therefore needed to measure that phase shift and feedback to the Labview program to adjust the

phase difference between two function generators so that exactly in-phase transport current and

applied magnetic field are achieved.

102

AC Supply

Transformer

Lock –in Amplifier 1 (Magnetization Voltage)

Lock –in Amplifier 2 (Transport Voltage)

Lock –in Amplifier 3 ( Δφ )

Digital Voltmeter 1

Digital Voltmeter 2

AC Supply

Function Generator

Function Generator

Δφ

Compensation coils

Pick-up coil

Cancel coil

Non-inductive shunt

TransducerDipole Magnet

Sample

LABVIEW

Ref

eren

ce s

igna

l

C

Fig. 4.7. Electrical setup for total AC loss measurement

103

Transport loss voltage is measured between voltage taps by a figure-8-shaped voltage lead and

using lock-in amplifier 2 (see Fig. 4.7). A compensation coil is connected in series with the

voltage lead to minimize the induced voltage component.

For magnetization losses, a pick-up coil is connected in series but in opposite direction to a

cancellation coil for reducing the inductive voltage component due to applied magnetic field.

Another compensation coil placed outside the magnet is also used to minimize the “left-over”

inductive component before using lock-in amplifier 1 (see Fig. 4.7) to measure the loss (in-

phase) voltage.

4.3.2 Magnet

Two specially designed double helix magnets have been built for AC loss measurements. Both

have an identical winding concept; the small magnet (7.6 cm diameter bore) is used for 77 K

measurements and the large one (16.5 cm diameter bore) is used for the variable temperature

measurements. Figure 4.8 (a) shows a photograph of the large magnet used to produce the

background transverse AC field. The double-helix dipole magnets have two sets of tilted coils,

each consisting of three coil layers. The coils are wound around concentric glass fiber reinforced

plastic (GFRP) tubes with litz copper wire. The magnet is suspended in a cryostat so that it can

be cooled with liquid nitrogen during operation. There is a gap between layers to allow liquid

nitrogen flow for efficient cooling. The magnet conductor sits in helical grooves machined in

GFRP tubes and is directly exposed to liquid nitrogen without any epoxy or insulation.

Figure 4.8(b) illustrates the concept of the magnetic field orientation in a double-helix dipole

magnet. As shown in the figure, the axial components of the magnetic field of the concentrically

wound helical coil layers which are tilted at opposite angles will be cancelled, while the dipole

content of those coils are added to produce a total magnetic field perpendicular to the solenoid

axis. The advantage of this kind of magnet in AC loss measurement is that the HTS tape sample

can be placed along the axis of the dipole magnet and can be rotated easily to investigate the AC

losses for different magnetic field orientations. In addition, this dipole can produce highly

uniform magnetic field in a long sample space [111-112].

104

The small magnet produces a highly uniform magnetic field (within 1% homogeneity) over a

region of 13 cm in length with field constant of 2.31 mT/A [110]. With a maximum operatiing

current of 100 A, the magnet can create a maximum magnetic field of 230 mT. The large magnet

produces a highly uniform magnetic field over a longer length (20 cm) but with lower field

constant (1.487 mT/A) [112]. The inductance is 8.1 mH for the small magnet and 49.5 mH for

the large one. Inductance in the magnet is unavoidable and it limits the magnet operation in an

AC environment, especially for high frequency. The inductance, however, can be compensated

by a capacitor to reduce the output load for the power amplifier. Near the resonance frequencies,

high magnetic field is therefore still achieved.

Fig. 4.8. Picture of the large double-helix magnet and magnetic field configuration of twohelically wound coils at opposite tilt angle

105

4.4 Variable temperature measurement by the electromagnetic method

4.4.1 Measurement design

Superconducting properties of materials strongly depend on temperature. In addition, HTS

devices usually work in a wide range of temperatures from 20 K –77 K. Information about the

dependence of AC loss on temperature is therefore crucial, not only for optimizing the operation

temperature of HTS devices, but also to understand the dependence of AC loss and other basic

properties of HTS conductors on temperature.

Most AC loss measurements are performed at liquid nitrogen temperature, 77K, or in a narrow

temperature range between 65 K to 77 K by nitrogen pumping. Only a few magnetization loss

measurements in variable temperatures from 30 K to 77 K using the calorimetric method have

been reported in the literature [113-114]. Variable temperature total AC loss measurement using

the electromagnetic method is desired. It has some challenges, however, as listed here:

• Using the EM method, metal surrounding the sample must be avoided to reduce the eddy

current noise that is in phase with the loss signal. Therefore cooling the sample and

maintaining uniform temperature along the sample during the measurements are

significant obstacles

• End cooling is insufficient. A heatsink is therefore needed to stabilize and protect the

sample from over-heating. Especially at low temperature, the sample will carry very high

current and has high n-value and the cooling power is more limited.

• Heat generated from current leads must be considered

• The magnet bore must be sufficient to accommodate the sample holder and part of the

cold head while also to reduce the temperature gradient. Thus, the inductance of the

magnet must be considered.

As a part of this dissertation, a novel total AC loss measurement system using the EM method in

variable temperature has been successfully built. The identical electrical set-up as for the 77K

measurements was used. At temperature controller and DC power supplies are added to monitor

106

and control the sample temperature. Fig. 4.9 is a sketch of the magnet with the sample holder for

this measurement. The double helix magnet is immersed in liquid nitrogen to produce a high

transverse field. The sample holder is accommodated in a GFRB tube sealed at the bottom. The

top of the tube is attached to a rotatable stainless steel CF flange and the flange-tube joint sealed

with Stycast. A feed-through collar with four KF40 ports rests on the top of the tube flange. The

bottom of the collar seals the steel flange that supports the GFRP tube. The top of the collar is

sealed with the cold head flange via a CF joint. The collar, the GFRP tube, and the cold head

flange assembly together form the vacuum tight measurement chamber. The collar provides

access to the current leads, instrumentation wiring, and vacuum ports. The rotatable flange

allows for the rotation of the sample to change its relative orientation with respect to the

magnetic field.

Magnet

Liquid nitrogen

Cold head

Sample holder

GFRP sample chamber

B

Current leads

CF Flange

Magnet

Liquid nitrogen

Cold head

Sample holder

GFRP sample chamber

B

Current leads

CF Flange

Fig. 4.9. Sketch of the cryostat with the sample holder and magnet for the variable temperaturemeasurement

107

Two current leads are anchored to the cold head. A thin sapphire plated is placed between the

cold-head and current leads for electrical insulation. Thus, the current leads are also used to cool

the sample.

Fig. 4.10. shows a photo and sketch of the sample holder with a HTS sample soldered between

the current leads. There are two heaters wound around current leads (one on each lead). The

sample temperature was monitored by 5 to 10 thermocouples and several temperature sensors

mounted along its length. The heaters were controlled to ensure that the temperatures at both

ends of the sample are equal, and thus to achieve uniform sample temperature. The heaters were

also used as compensation heat sources. When the transport current and/or magnetic field are

turned on, AC loss in the sample and heat generated in the current leads causes a temperature rise

in the sample. The temperature controller will respond fast to that temperature rise by reducing

the heating power of the heaters. To protect the sample and to improve the temperature

Heaters

Samples

Sapphire plate

Current leads

Heaters

Samples

Sapphire plate

Current leads

Heaters

Samples

Sapphire plate

Current leads

Fig. 4.10. Photo and sketch of the sample holders

108

uniformity, a sapphire plate was placed beneath the sample to serve as a heat sink/stabilizer for

the sample. The sapphire plate has a 1 cm thickness and contacts the current leads at both ends.

Helium pressure is an important factor for controlling the sample temperature. As the helium gas

pressure decreases, the thermal exchange between the sample and the surroundings is reduced.

As a result, the sample can be cooled to lower temperature but becomes less stable. Figure 4.11

plots the sample temperature profile recorded by seven thermocouples mounted along its length

when the helium gas pressure is about 1.5 bar. At that helium pressure, good temperature

uniformity (variation of ±0.5K along the conductor) was obtained for any temperature setting

between 30 K and 100 K. The sample temperature stabilizes quickly in five minutes for each

temperature setting. Thermocouple T5 was separated from the tape surface and therefore shows

higher temperature than other thermocouples at low temperature.

30

40

50

60

70

80

90

0 500 1000 1500 2000 2500 3000 3500

Time (s)

Te

mp

era

ture

(K

)

T1

T2

T3

T4

T5

T6

T7

T1 T2 T3 T4 T5 T6 T7

30

40

50

60

70

80

90

0 500 1000 1500 2000 2500 3000 3500

Time (s)

Te

mp

era

ture

(K

)

T1

T2

T3

T4

T5

T6

T7

T1 T2 T3 T4 T5 T6 T7T1 T2 T3 T4 T5 T6 T7

Fig. 4.11. Sample temperature profile measured by seven thermocouples T1 to T7 mountedalong the sample.

109

4.4.2 Measurement procedure

At low temperature, the critical current of HTS conductor is very high. Measuring AC loss with

high current must be performed properly, especially in a limited cooling condition. Heat

generated in the sample and current leads causes a temperature rise and thus reduces the critical

current. As a consequence, a quench might occur and the sample may be damaged.

Firstly, the temperature is set at a desired value. When the sample temperature is stable, the

magnetic field is turned on. The sample temperature increases due to AC loss generated but it

stabilizes quickly to the set temperature because of the compensation from heaters. The transport

current is then passed through the sample and AC loss data is acquired in a duration of 15

seconds. After that, the current is turned off and the sample restabilizes at the set temperature,

only then can experiment continue at the next value of transport current.

The magnetic field is turned on continuously at a constant amplitude during the measurement of

an AC loss curve. Magnetic field and transport current should not be turned on simultaneously so

as to avoid too much heat generated by AC loss. The heaters might not respond sufficiently fast

to such a large amount of AC loss and a quench may result.

Figure 4.12 shows results of a cooling test at 45 K. The sample is a 4 mm wide YBCO coated

conductor with a 15 μm thick surround copper stabilizer. The successful test for this tape ensures

safe measurement for other samples with thicker stabilizer. At 45 K, in a 50 mT DC field, the

sample has Ic = 333 A and n = 23. Temperatures shown in Fig. 4.11 are recorded from

thermocouples T1 and T7 mounted at the ends of the sample and from a Cernox RTD sensor

mounted in the middle of the tape.

Firstly, sample is stabilized at 45 K. When a 50 mT AC magnetic field is applied at 51 Hz, the

tape temperature increases due to the magnetization loss, but then quickly returns to 45 K

because of a compensation from heater (i.e. the heater power is reduced). A 305 A transport

current (~0.95Ic) is then applied, resulting in a rapid temperature increase to 45.6 K, followed by

a steady increase at 1.2 K/minute. The cooling power is insufficient and the temperature

continues to rise. The sample transport current is turned off and the tape temperature returns to

110

45 K. A transport current of 280 A is then applied and a similar temperature evolution results but

at lower rate, 0.8 K/minute.

At 240 A, the tape is stable at 45.5K even when the heaters are turned off. In this case, the total

AC loss balances the cooling power. Since the total AC loss measurements are performed in 15

seconds after the current is set, the rates of increasing temperature are sufficiently slow to

perform AC loss measurements with acceptable error. Figure 4.13 plots the temperature versus

time on the tape when measuring AC loss in an AC field of 50 mT and with a 272 A transport

current. While acquiring the AC loss data, the temperature rise on the tape is only about 0.6 K.

42

43

44

45

46

47

48

0 100 200 300 400

Times (s)

Te

mp

era

ture

(K

)

T1

T7

RTD

B = 50 mT, ONB = 230, ON

I = 305 A, ON

I = 280 A, ON

Current off Current off

Fig. 4.12. Temperature along the tape when it carries AC transport currents in an ACperpendicular applied field of 50 mT, f = 51 Hz

I = 240 A, ON

111

4.5. Errors in the electromagnetic method

AC loss measurements are quite simple in principle but difficult in practice since they require the

measurement of a small signal, sometimes as small as 50 nV, in a “noisy” AC environment.

Therefore, sources of errors must be considered carefully to obtain accurate results. Sources of

errors include phase error, common mode error, error associated with geometry of the voltage

loop and error from the empty coil effect.

4.5.1. Phase error

The non-inductive resistor used in the measurement setup has very good phase accuracy (less

than 0.1°). It was limited to 300A, however, and cannot be used for the magnetic circuit because

of the high voltage in this circuit. Therefore, a fast response current transducer is convenient to

use for a reference signal, since the signal from the current transducer is isolated from the circuit,

and is therefore safe for lock-in amplifiers. The phase error in the current transducer is somewhat

higher than the non-inductive resistor, about 0.3°.

40.00

42.00

44.00

46.00

48.00

50.00

0.00 5.00 10.00 15.00 20.00 25.00

Times (s)

Tem

pera

ture

(K

)

T1 T4 T7Current on

Current offLoss measurement

40.00

42.00

44.00

46.00

48.00

50.00

0.00 5.00 10.00 15.00 20.00 25.00

Times (s)

Tem

pera

ture

(K

)

T1 T4 T7Current on

Current offLoss measurement

Fig. 4.13. Temperature versus times for three locations along the tape during AC lossmeasurements at I0 = 272 A, B0 = 50 mT and f = 51 Hz

112

Assume that the actual inductive and resistive signals from the voltage leads or pick-up coil are

Ui and Ur, the phase error of the reference signal of the lock-in amplifier is σ, and the recorded

resistive voltage on the lock-in amplifier is Vr. Since σ is small, the relative error is given by:

r

i

r

rr

U

U

U

UV δsin≈

− (4.10)

Without compensation, the ratio Ui/Ur may reach 100s and the measurement can deliver errors of

hundreds of percents. With proper compensation, however, the inductive component is almost

eliminated and the ratio Ui/Ur can go down to 1 to achieve error of 0.5%. So Ui/Ur can be in the

range between 1 to 3 to obtain error less than 1.5%.

Another important source of error comes from the compensation coil. The compensation coil

signal must be purely inductive in order to not add any resistive signal to the measured signals

from the sample. The undesired resistive signal can be created by eddy current on conductors

near the compensation circuit. The compensation circuit consists of two small coupled coils. One

coil was formed by winding a few turns of the current carrying cable around a small plastic tube

(~3 cm diameter). The compensation coil is made of small copper wire (0.05 mm diameter) and

also wound around that tube. The compensation coil is connected to a potentiometer. By

changing the potentiometer, the compensation signal was adjusted to obtain the best cancellation

for the inductive component of the measured signal. The compensation coils must be placed at a

distance from transformer or magnet or any conductor to avoid an in-phase signal.

4.5.2. Error from common mode signal

The common mode noise can be identified by changing the polarity of the voltage leads or pick-

up coil for the lock-in amplifiers input. This error usually affects transport measurements only.

Assume the voltages at the taps where the voltage leads are soldered to the sample are V1 and V2.

The loss voltage is determined from V12 = V1 – V2. The common mode noise is proportional to V1

+ V2. If there is some difference between AC loss obtained by measuring V12 and V21, common

mode noise must be suspected. To avoid common mode noise, the following steps need to be

considered:

113

• The lock-in amplifier should have a good common mode rejection ratio (CMRR), about

90-100dB

• A transformer is needed to isolate the power amplifier and transport current circuit.

• The sample must be grounded

4.5.3. Error from the voltage loop geometry

Finally, when transport loss is measured in the presence of an applied AC magnetic field, there

may be an error caused by the non-symmetry of the voltage loop. If the two loops in a “Fig 8”

shaped voltage loop are not symmetric, it will pick-up an additional signal from the sample

magnetization. That “addition” is proportional to the magnetization loss and can be estimated by

measuring the loss voltage on the loop in an AC field with zero transport current [60]. If that

voltage is considerable, it must be subtracted from the transport loss data.

4.5.4. Error from the empty coil effect

The pick-up coil may have a resistive signal, even when the sample is not mounted. That in-

phase signal comes from the magnetic field created by eddy currents in surrounding conducting

materials, such as the current leads, cryostat, and magnet wires. The signal is proportional to the

applied magnetic field and can be estimated by measuring an empty pick-up coil at several

values of magnetic field. If the empty coil effect is considerable, it must be subtracted from the

measured magnetization loss data. In 77 K measurements, the effect is estimated to be less than 4

%. For the variable temperature set-up, however, the empty coil effect may be as high as 10%.

The actual magnetization loss is very small at low temperature, making the empty coil effect

relatively important. In addition, the large current leads in this set-up increase the empty coil

effect.

4.6. Calorimetric method

The electromagnetic method measures the total AC loss with high sensitivity but is limited to in-

phase transport current and magnetic field only. Many applications of HTS tapes (for example,

HTS transformer, HTS motor, three phase-power transmission cables, etc.) require conditions

114

where the magnetic field and transport current are not in phase. Therefore, understanding the

effect of relative phase difference on total AC loss is necessary. The calorimetric method (CM)

measures the total AC loss for any phase difference between current and magnetic field and thus

is suitable for this situation. In addition, the CM method is also used to compare with the results

obtained from the EM method to confirm the measurement validity.

4.6.1 Measurement procedure

Fig. 4.14 illustrates the sample arrangement for the calorimetric method. The sample is covered

by polystyrene foam blocks to thermally insulate the sample from boiling liquid nitrogen. If a

DC transport current higher than critical current is passed through the sample for a duration of

time, heat is generated in the sample causing the sample temperature to increase. The heat

generated by the DC current can be measured accurately by the four-probe technique. If the

temperature rise on the sample is measured, a calibration curve that describes the relation

between the generated heat and sample temperature increase can be obtained. When an AC

transport current and/or magnetic field are applied for the same time duration, AC losses also

cause the sample temperature to rise. From that temperature increase and the calibration curve,

Fig. 4.14. Arrangement of sample holder for the calorimetric method

115

the total AC loss is estimated.

The temperature increase on the sample is measured by a type-E differential thermocouple with

one tip placed on the sample and a reference tip placed in liquid nitrogen as seen in Fig 4.14

[105-106]. For a wide sample (1 cm wide tape, for instance), however, the sample dimensions

are much larger than the dimensions of the thermocouple tip. Therefore, the temperature

measured by the thermocouple represents only a local temperature rather than the sample

temperature. Moreover, the smallness in size and heat capacity of the thermal couple, compared

to the dimensions of the sample, may cause temperature rise measurements to be unstable. In

addition, the mK scale of the temperature rise will create a microvolt scaled voltage between the

terminals of the thermocouple. Therefore, inductive noise from the AC current or magnetic field

can pose another problem with the use of thermocouples. Cernox temperature sensors have

proven to improve temperature rise measurements by overcoming the above problems [45].

With a resistance on the order of 103 Ω and sensitivity of 15 Ω/K at temperatures around 77 K,

Cernox sensors with model CX-1080 manufactured by Lakeshore are preferable.

Fig. 4.15. Temperature along the tapes during AC loss measurements at I0 = 272 A, B0 = 50and f = 51 Hz

116

Figure 4.15 compares the temperature rise on the sample measured by differential thermocouple

and Cernox sensor when an AC transport current of 122.2 A at frequency 301 Hz passes through

a YBCO sample (1 cm wide, Ic = 160 A, 75 μm thick Ni-5%W substrate) in for 40 seconds. As

seen in the figure, when the transport current is applied, the thermocouple signal oscillates. This

did not happen when the same thermocouple was used to measure temperature rise on Bi-2223

tape samples [110], suggesting that the oscillation on the differential thermocouple is mainly

caused by the ferromagnetic substrate used with YBCO coated conductors. The ferromagnetic

substrate enhances the self-field (or applied magnetic field) and induces a significant voltage in

the thermocouple loop, although the loop area was minimized as much as possible. There is no

oscillation, however, in the temperature rise of the YBCO sample measured by the Cernox

sensor. This means effect of the induced voltage is negligible in this case.

1

10

100

1000

0 10 20 30 40 50

Time (s)

Te

mp

era

ture

ris

e (

mK

)

193.4 A

172.6 A

152.3 A

131.9A

91 A

111.3 A

70.6 A

40.4 A

40.2 A

20.2A

Fig. 4.16. Temperature rise on the sample when it carries an AC transport current with amplitudevarying from 20 A to 193 A in an AC perpendicular magnetic field B0 = 10.4 mT, f = 51 Hz

117

In addition, with a larger size and higher heat capacity, results measured by Cernox sensor are

also confirmed to be more stable as seen in Fig. 4.16. In that figure, the temperature rise on a

YBCO sample as a function of AC transport current in an AC perpendicular magnetic field B0 =

10.4 mT for a duration of 40 s is measured by a Cernox sensor. Nearly identical temperature rise

curves are obtained when the same current of 40 A is applied on the sample repeatedly. The

Cernox sensor stably measured temperature rise as small as 10 mK.

4.6.2. Error and limitation of the calorimetric method

Inductive noise and instability cause some errors in temperature rise measurements by using

differential thermocouple. This error can be limited by replacing thermocouple with Cernox

sensor.

In the calorimetric method, the temperature rise should not be too high to avoid significant

changes of the superconducting properties of the sample. In addition, an overheated sample may

create large nitrogen bubbles around the sample set-up. When those bubbles rise off the sample,

Fig. 4.17. Calibration curve before and after compensating for the effect of heat generated in thecurrent leads

Before Compensation

118

the sample temperature will decrease significantly, causing instability in the measurement.

Therefore the temperature rise should be less than 1500 mK for acceptable error. The thickness

of the insulation blocks and the duration of the magnetic field and current pulses must be

optimized in order to produce appropriate temperature rise on the sample for a certain range of

AC loss. This means that several calibration curves are required to cover a wide range of

transport current, magnetic field and frequency.

Finally, as reported in [45], for a sample with high critical current, another considerable issue is

the effect of heat generated in the current leads and soldering junctions. Due to design limitations

of the sample holder, the current leads may not be large enough to carry a high current with

negligible heat produced. To estimate effect of the current leads, the temperature rise is measured

on the sample when it carries a DC current well below its critical current. Since the heat

generated from the current leads is purely resistive, it will be proportional to the square of the

RMS value of the transport current. From that fact, the temperature rise on the sample due to

background heat from the current leads is estimated for any AC transport current and it can be

subtracted from the measured data for better accuracy. Figure 4.17 depicts a calibration curve

obtained from the above YBCO sample when some DC currents higher than Ic are applied for a

duration of 40 seconds. The open and the filled symbols are the calibrating data before and after

subtracting the effect of current lead, respectively. The calibrating data after taking account of

the current lead effect is fitted well by a linear relation and the fitting line goes through the origin

of the coordinates as expected.

119

CHAPTER 5

AC LOSS CHARACTERISTICS IN (Bi,Pb)2Sr2Ca2Cu3O10

AND YBa2Cu3O7-δ TAPES

In this chapter, the experimental and calculated results of AC losses of Bi-2223 and YBCO tapes

are presented. Transport loss, magnetization loss and total loss in those samples are investigated

thoroughly at 77 K to understand the AC loss behavior as well as the contribution of individual

AC loss components generated from different mechanisms. Finally, dependence of critical

current Ic and AC losses in Bi-2223 and YBCO tapes on temperature is discussed. The time

dependence of the transport current and perpendicular magnetic field are I(t) = I0sin(ωt) and B(t)

= B0sin(ωt-Δφ), respectively. Frequency f = 51 Hz and phase difference Δφ = 0 are assumed

throughout this chapter, except when stated otherwise.

5.1. Sample specification

AC losses were measured in three Bi-2223 tapes and four YBCO tapes. The specifications of

those tapes are summarized in table 5.1 and table 5.2. All Bi-2223 tapes are non-twisted tapes

with pure silver matrix and sheath. In table 5.1, the dimensions 2at x 2bt and 2as x 2bs refer to the

dimensions of the tapes and their superconducting core, respectively. Samples B1 and B2 were

cut from the same batch. They have nearly identical specifications with small variation of Ic and

n-value.

120

Table 5.1. Specifications of the Bi-2223 samples

Sample 2at x 2bt

(mm x mm) 2as x 2bs

(mm x mm) n Ic (A) Number of filaments

B1

B2

B3

4 x 0.21

4 x 0.21

2 x 0.21

3.6 x 0.14

3.6 x 0.14

19

18

18

119

117

65

55

55

37

Table 5.2. Specifications of the YBCO samples

Y1 Y2 Y3 Y4

Ic

n-value

Tape width

YBCO layer width

YBCO layer thickness

Substrate material

Substrate thickness

Stabilizer thickness

Stabilizer type

186

22

10 mm

10 mm

1 μm

Ni-5%W

75 μm

75 μm

One side

160

21

10 mm

10 mm

1 μm

Ni-5%W

75μm

75 μm

One side

81

36

4 mm

3.9 mm

0.8 μm

Hastelloy

50 μm

15 μm,

Surrounding

83

36

4 mm

3.9 mm

0.8 μm

Hastelloy

50 μm

15 μm,

Surrounding

121

In table 5.2, samples Y1 and Y2 are YBCO coated conductors fabricated by RABiTSTM

technique on a Ni-5%W substrate, while samples Y3 and Y4 are manufactured by the IBAD

technique with Hastelloy substrates. Samples Y1 and Y2 are stabilized on one side with a 75 μm

thick copper layer. Samples Y3 and Y4 are from the same batch and are surrounded by a 15 μm

thick copper stabilizer.

5.2. Field dependence of Jc and n-value

Critical current density and n-value of HTS conductors decrease in an applied magnetic field.

Based on experimental data of low temperature superconductors, Kim proposed a model to

describe the field dependence Jc(B) in a DC perpendicular magnetic field B [115]:

)(

||1

)(),(

TB

B

TJTBJ

c

coc

+= (5.1)

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150

DC perpendicular magnetic field (mT)

I c(B

)/I c

(0)

Y1 Y3

Fig. 5.1. Normalized critical current Ic(B)/Ic(0) of samples Y1 and Y3 as a function of perpendicular DC magnetic field.

122

where B is the magnitude of the flux density, Bc(T) is a temperature dependent constant and

Jco(T) is the zero field critical current density at temperature T. In many cases, this model does

not fit well with experimental data [116-117]. Recently, a more accurate model based on the

granular nature of HTS tapes was proposed [117-118]. In this model, the current density is

described as a combination of two parallel current paths: a network of the weak-link currents (the

currents flowing between the HTS grains) and the strong-link currents (the intragranular current,

or superconducting currents flowing inside grains). The field dependence Jc(B) is written as

follows:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=)(

)( )(exp).,0(

)(1

),0()(

T

s

csT

w

cwc

TB

BTI

TB

B

TIBI

α

β (5.2)

In Eqt. (5.2), the first term is the contribution from the weakly linked current path and the second

term is the contribution from the strongly linked current path. Icw(0,T), Ics(0,T), Bw(0,T), Bs(0,T)

)(Tβ , )(Tα are fitting parameters determined from experimental data. Fig. 5.1 depicts the

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150

DC perpendicular magnetic field (mT)

n(B

)/n

(0)

Y1 Y3

Fig. 5.2. Normalized n-value, n(B)/n(0), of sample Y1 and Y3 as a function of the perpendicularDC magnetic field

123

normalized critical current I(B)/I(0) for samples Y1 and Y3 in perpendicular DC field. As seen in

this figure, Ic of sample Y3 decreases in DC field more quickly than that of sample Y1. The

same situation happens for n-value. As shown in Fig. 5.2, the n-value of sample Y3 drops more

quickly than that of sample Y1 when a perpendicular DC field is applied.

In some cases, in order to have accurate relations Ic(B) and n(B) used in AC loss numerical

calculations, the field dependence of Jc and n-value can also be fitted by polynomials, if neither

Eqts. (5.1) or Eqt. (5.2) are sufficient [82-83]. In this case, the wider fitting range requires the

higher order of the fitting polynomial.

In this dissertation, for better computing accuracy, the Jc(B) and n(B) of the sample Y1 are fitted

by polynomials:

( ) 1.00 + 0.84B +67.60B - 310.88B + 357.68B-)0()( 234cc IBI = (5.3)

( )1 + 1.45B - 31.02B + 398.37B - 1177.49B)0()( 234nBn = (5.4)

Ic(B) and n(B) of sample Y3 are fitted by functions:

⎟⎠⎞

⎜⎝⎛

+++−=

093.0/1

1

81

70)05.0/)008.0exp((

81

1181)(

BBBI c (5.5)

⎟⎠⎞

⎜⎝⎛

++−=

25.0/1

1

119

109)018.0/exp(

36

1136)(

BBBn (5.6)

and Ic(B) and n(B) of sample B1 are fitted by the functions:

( )( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

++−=

9.12.1

058.0/1

1

119

42)12.0/(exp

119

77119)(

BBBI c (5.7)

( )( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

++−=

88.18.1

2.0/1

1

19

11)023.0/(exp

19

819)(

BBBn (5.8)

The fitting curves of sample Y1 and Y3 are also plotted in Figs. 5.1 and 5.2.

124

Flux pinning in parallel magnetic field is much better than in perpendicular field. Therefore, the

degradation of Ic and n-value are quite small in parallel applied field up to 100 mT, as seen in

Fig. 5.3. In the numerical calculation, the dependence of Ic and n-value in parallel field is

negligible.

5.3. Comparison between calorimetric and electromagnetic methods

In general, good agreement between results obtained from the calorimetric and electromagnetic

methods have been achieved and reported; the details of the comparison between these two

experimental methods can be found in references [101, 104-105,110]. Fig.5.4 shows the self-

field power loss in sample B1 measured by both calorimetric and electromagnetic methods at

frequencies of 91 Hz and 111 Hz. As seen in the figure, good agreement between the results

obtained by both methods was observed

Due to low sensitivity and resolution, the calorimetric method was only used for peak transport

currents, I0, greater than 60 A, corresponding to I0/IC > 0.5. In general, the variation between two

methods is less than 10% for all the compared currents.

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

Paralell magnetic field (mT)

I c(B

)/I c

(0)

an

d n

(B)/

n(0

)

Ic(B)/Ic(0)_Y1 n(B)/n(0)_Y1

Ic(B)/Ic(0)_Y2 n(B)/n(0)_Y2

Fig. 5.3. Normalized critical current and n-value of samples Y1 and Y3 as a function of parallelDC magnetic field

125

0.00001

0.0001

0.001

0.01

0.1

1 10 100

Magnetic field (mT)

Ma

gn

eti

za

tio

n l

os

s

(J/m

/cy

cle

)

electromagnetic method

calorimetric method

Numerical calculation

Fig. 5.5. Magnetization loss in sample B1 measured by the electromagnetic and calorimetricmethods at 51 Hz

Fig. 5.4. Self-field power loss in sample Y2 measured by the electromagnetic and calorimetric methods at frequencies 91 Hz and 111 Hz

126

The magnetization loss at 51 Hz in sample B1 is shown in Fig. 5.5. The open symbols are the

results obtained from the CM method and the filled symbols are the data obtained from the EM

method. The numerical results calculated for that rectangular sample are also plotted. Again,

good agreement between experimental and numerical results is observed for magnetization loss.

In high magnetic fields, the temperature rise on the sample for this calorimetric set-up is up to

2000 mK. Such a high temperature rise on the sample decreases the critical current of the

sample. As a consequence, the results obtained from the calorimetric method are affected and

slightly smaller than those obtained from the EM method.

Finally, Fig. 5.6 compares the total AC loss in YBCO sample Y2 obtained by the EM and CM

methods. The measurements were performed at 51 Hz for three different values of perpendicular

applied field, 10.4, 30.8 and 51.6 mT. The transport current changes from around 10 A to 200 A.

As seen in the figure, quite good agreement between the two experimental methods was

observed for total AC loss measurements.

Fig. 5.6. The total AC loss in sample Y2 measured by the electromagnetic (open symbols) andcalorimetric (filled symbols) methods at 51 Hz.

127

In summary, the agreement between experimental results measured by the CM and EM methods

for either individual AC loss components or for the total AC loss in both Bi-2223 and YBCO

samples confirms the validity of the measurements. Both these methods will be used to study AC

loss characteristics of HTS conductors as presented in the following sections.

5.4. Self-field AC loss in HTS tapes

5.4.1 AC loss components in the self-field loss

Figure 5.7 plots the self-field AC losses for Bi-2223 samples B1, B2 and B3 as the normalized

loss versus the normalized current. The self-field losses per cycle are normalized to πμ /20 cI to

make it dimensionless and become a function of the normalized current i = I0/Ic, as suggested by

Eqt. 2.36. As seen in the figure, the measured self-field losses of all three Bi-2223 samples agree

well with the analytical results predicted by Norris’ equation for an elliptical tape. This is not

always true and there are several publications reporting variations between measured results and

Norris’ model [76, 119-120]. This difference may be caused by the shape of the sample cross-

section [74, 76] or by the critical current distribution over the tape cross-section [119-120].

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

0.01 0.1 1 10

Normalized current i = I0/Ic

No

rmali

zed

self

-fie

ld l

oss q

(i)

B1

B2

B3

Norris's ellipse

Norris's strip

Fig. 5.7. Self-field loss in Bi-2223 samples B1, B2 and B3 measured by the EM method at 51Hz. Analytical results obtained from Norris’ model for a thin strip and an elliptical tape are alsoplotted.

128

Norris’ model describes the hysteresis loss in the tape when I0/Ic < 1. When I 0 > Ic, the transport

loss increases rapidly due to the contribution of the resistive and flux-flow losses. To verify the

contribution of the loss components in the total self-field loss, three voltage loops with different

arrangements, as shown in Fig. 5.8, were used. Voltage loop V1 is to measure the total self-field

loss. The contactless voltage loop V2 starts from the center of the tape and is also extended to the

same distance as the loop V1. This loop is to measure the hysteresis loss component. Finally, the

loop V3 soldered at the edge of the tape and pulled back to the center of the tape is to estimate

the resistive loss component.

The loss components in sample B2 obtained from those three voltage loops are plotted in Fig.

5.9. When I 0 < Ic, AC losses obtained from V1 and V2 almost coincide while the loss measured

by the loop V3 is significantly smaller. This confirms that the main component for that current

region is hysteresis loss. The slope of the hysteresis loss obtained from V2 changes suddenly and

saturates at I0 = 150 A. This observation suggests that the current at which the conductor is fully

penetrated is 150 A, quite higher than the transport critical current Ic = 119 A of this sample.

Above the full penetration current, the resistive loss measured by loop V3 is the main component

and nearly same as the total self-field loss.

I

V2

V1

V3

I

V2

V1

V3

Fig. 5.8. Voltage loop configurations to measure the total self-field loss (V1), hysteresis loss(V2) and resistive, flux-flow losses (V3)

129

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1 10 100 1000

Transport current (A)

Se

lf-f

ield

lo

ss

(J

/m/c

yc

le)

V1 :Total transport loss

V2: Hysteresis loss

V3: Resistive loss

Fig. 5.9. The self-field loss components in sample B2 measured from the three voltage loopsshown in figure 5.8

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

0.01 0.1 1 10

Normalized current i = I0/Ic

No

rma

iliz

ed

se

lf-f

ield

lo

ss

q(i

) Y1

Y3

Norris's ellipse

Norris's strip

Fig. 5.10. The self-field loss in YBCO samples Y1, Y3 measured by the EM method at 51 Hz.Analytical results obtained from Norris’ model for a thin strip and an elliptical tape are alsoplotted.

130

Figure 5.10 plots the normalized self-field loss in YBCO samples Y1 and Y3. The self-field

losses calculated from Norris’ model for elliptical and thin strip tapes are also plotted for

comparison. In the high current region, the losses in these YBCO coated conductors seem to

follow the Norris’ model for a thin strip. For low current, however, the slope of the loss curves of

these samples decreases. When i = I0/Ic < 0.5, the self-field loss in sample Y1 is higher than the

loss calculated by Norris’ model for an elliptical tape. The main loss component in this sample

for the low current region is the ferromagnetic loss generated in the Ni-W substrate by the self-

field that will be elucidated in section 5.4.2

5.4.2 Self-field AC loss in a DC magnetic field

The self-field AC loss in sample Y1 in perpendicular DC magnetic field up to 90 mT is shown in

Fig.5.11. In general, the DC magnetic field decreases the critical current of the conductors. Thus,

the self-field AC loss is expected to be increased with the presence of the applied DC magnetic

field. As seen in Fig. 5.11, the self-field loss is increased in the high current region where the

hysteresis and flux-flow losses are dominant. However, when I0 < 100 A, the self-field loss

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

10 100 1000

Current amplitudes (A)

Tra

nsp

ort

lo

ss (

J/m

/cycle

)

0

30 mT

50 mT

70 mT

90 mT

Fig. 5.11. The self-field AC loss in sample Y1 when the perpendicular DC applied magnetic fieldincreases from 0 to 90 mT.

131

decreases with increasing magnetic field. In this current region, the ferromagnetic loss is the

main component. The partial magnetization of the substrate caused by DC perpendicular

magnetic field reduced its ferromagnetic loss, thus reducing the self-field loss measured in this

sample.

The effects of parallel DC field on the self-field loss in sample Y1 are seen in Fig. 5.12. In

parallel DC field from 0 to 50 mT, the critical current change is less than 10% as seen in Fig. 5.3.

Therefore, in such parallel DC field, the change of hysteresis loss and flux-flow losses dissipated

in the HTS layer would be small. As a consequence, the transport losses in the high current

region, where these losses are the main contributions, are almost unchanged in parallel field.

However, in the low transport current region, the self-field loss is reduced dramatically because

of the decrease of the ferromagnetic loss. For more clarity, the self-field loss as a function of DC

magnetic field is plotted in Fig. 5.13 for different current amplitudes. Effect of DC parallel field

on the ferromagnetic loss in the substrate is much stronger than that of perpendicular field. The

ferromagnetic loss decreases quickly when the parallel DC field increases from 0 to 10 mT and is

nearly zero at 20 mT. When the parallel DC field is higher than 20 mT, the self-field loss is

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

10 100 1000

Transport current (A)

Tra

nsp

ort

lo

ss (

J/m

/cycle

)

0 mT

2 mT

4 mT

6 mT

8 mT

10 mT

15 mT

20 mT

30 mT

50 mT

Strip

Fig. 5.12. Self-field AC loss in sample Y1 when parallel DC applied magnetic field increasesfrom 0 to 50 mT.

132

mainly the hysteresis loss in the YBCO layer and it agrees well with Norris’ model for a thin

strip as expected (Fig. 5.12).

With a high aspect ratio, the diamagnetizing factor of the substrate in parallel field is N// = 0.015,

about 64 times smaller than the diamagnetizing factor in perpendicular field, N⊥ = 0.985. The

substrate is therefore saturated much more easily in parallel field than in perpendicular field.

Thus, the effect of parallel DC field on the ferromagnetic loss in the substrate is much stronger

than that of the perpendicular magnetic field.

The ferromagnetic loss in the substrate of sample Y1 was estimated from the magnetization loop

measured by SQUID magnetometer at a temperature higher than the critical temperature Tc of

YBCO to exclude superconductivity of the HTS layer [57-58]. The dependence of the

ferromagnetic loss in the substrate of sample Y1 on the amplitude of parallel magnetic field is

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0 10 20 30 40 50 60

Parallel DC field (mT)

Tra

ns

po

rt l

os

s (

J/m

/cy

cle

)

15.1 A

30.1 A

45.2 A

60.4 A

75.3 A

90.6 A

105.7 A

121.1 A

136.2 A

151.4 A

166.7 A

181.7 A

196.2 A

Fig. 5.13. Self-field loss in sample Y1 as a function of the parallel DC magnetic field at different AC transport current amplitudes.

133

shown in Fig. 5.14 [58]. Based on classical theory for magnetization, the experimental data in

Fig. 5.14 is fitted with a function [121]:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2.12.1

)28.0(

1)28.0coth(2.0)(

BBBQ f (5.9)

where B is the DC magnetic field in mT and Qf in mJ/m/cycle. The fitting curve is plotted in

Fig.5.14. It is worth noting that the ferromagnetic loss is frequency independent and nearly

temperature independent in the range of 50 K to 100 K [58]. From Eqt. 5.9, the ferromagnetic

loss in the substrate can be estimated by the way described in section 3.3.3.

The calculated results for the self-field loss, Qs, generated in the YBCO layer and the

ferromagnetic loss, Qf, generated in the substrate are shown in Fig. 5.15. The eddy current loss

generated in the stabilizer at this frequency is small with respect to the hysteresis and

ferromagnetic losses, and thus it can be ignored. At low transport currents, the ferromagnetic

loss, Qf, dominates. When the transport current is above 100 A, the ferromagnetic loss begins to

saturate. Thus, Qs becomes more significant in the high transport current region. With the

significant contribution of ferromagnetic loss, the measured transport loss is quite higher than the

transport loss calculated by Norris’s model for the HTS layer. In Fig. 5.15, the numerical and

Fig. 5.14. Experimental data for dependence of ferromagnetic loss in the substrate on amplitudeof parallel DC field [58], the fitting curve from Eqt. 5.9 is also plotted.

Fitting curve

134

experimental results of the total transport loss Qtt that is the sum of the ferromagnetic loss in the

substrate and the self-field loss, in the YBCO layer are also plotted. When the ferromagnetic loss

is taken into consideration, good agreement between numerical and experimental results of self-

field AC loss dissipated in sample Y1 is observed, especially at high current. The numerical

calculations can be improved by improving the fitting function for the relation between

ferromagnetic loss and DC magnetic field.

The situation is different for sample Y2 with Hastelloy substrate. When either perpendicular or

parallel DC field is applied, the self-field loss in sample Y2 increases in the entire range of

transport current, as a consequence of the decrease of Ic (Figs. 5.16 and 5.17). In DC parallel

field up to 50mT, the decrease of Ic is small. The loss increase is therefore observed clearly only

in the high current region. To have better understanding of the effect of DC applied magnetic

field on the self-field loss in this sample, the self-field loss was replotted on a normalized scale

(Figs. 5.18 and 5.19). On the normalized scale, all the loss curves in both perpendicular field and

parallel field are superposed together. This confirmed the contribution of ferromagnetic loss in

10 10010

-7

10-6

10-5

10-4

10-3

10-2

10-1

Substrate:

Qf (Numerical)

HTS layer:

Qs (Numerical)

Qs (Norris)

AC

loss (

J/m

/cyc

le)

Transport current (A)

Total self-field loss:

Qtt (Numerical)

Qtt (Experimental)

Fig. 5.15. The self-field loss in sample Y1 measured by the EM method at 51 Hz andnumerically calculated AC loss components: the self-field loss in the YBCO layer and the ferromagnetic loss in the substrate

135

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1 10 100

Transport current (A)

Tra

ns

po

rt l

os

s (

J/m

/cy

cle

)

B =0 mT

B = 15 mT

B = 30 mT

B = 40 mT

B = 50 mT

Fig. 5.17. The self-field AC loss in sample Y3 when parallel DC applied magnetic fieldincreases from 0 to 50 mT.

the substrate of this sample is negligible and the increase of self-field loss in sample Y2 due to

DC applied magnetic field is mainly caused by degradation of the critical current.

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1 10 100

Transport current ( A)

Tra

nsp

ort

lo

ss Q

(J/m

/cycle

) B= 0 mT

B= 20 mT

B = 30 mT

B = 50 mT

B = 70 mT

Fig. 5.16. The self-field AC loss in sample Y3 when perpendicular DC applied magnetic fieldincreases from 0 to 70 mT.

136

0.1 110

-5

10-4

10-3

10-2

10-1

100

101

No

rma

lize

d s

elf-f

ield

lo

ss q

(i)

Normalized current i = I0/I

c

B = 0 mT

B = 20 mT

B = 30 mT

B = 50 mT

B = 70 mT

Norris's strip

Norris's ellipse

Fig. 5.18. Normalized self-field AC loss in sample Y1 when perpendicular DC applied magneticfield increases from 0 to 70 mT.

0.1 110

-5

10-4

10-3

10-2

10-1

100

101

Norm

aliz

ed s

elf-f

ield

loss q

(i)

Normalized current i = I/Ic

B = 0 mT

B = 15 mT

B = 30 mT

B = 40 mT

B = 50 mT

Norris's strip

Norris's ellipse

Fig. 5.19. Normalized self-field AC loss in sample Y1 when parallel DC applied magnetic fieldincreases from 0 to 50 mT.

137

The superposition of the loss curves in Figs. 5.18 suggests that the change of n-value from 36 to

15 in perpendicular DC field does not change the AC loss behavior, critical current plays a much

more important role in determining the ac loss in a HTS tape. Without a ferromagnetic substrate,

Bi-2223 tapes have the same self-field loss behavior in DC magnetic field as sample Y3 [122-

124].

Although there is no contribution from ferromagnetic loss, in the low current region, the

experimental data for the self-field loss in sample Y2 is still higher than the self-field loss

calculated from Norris’ model for a thin strip. As presented in the next section, the contribution

of eddy current at 51 Hz is also very small. Therefore, the reason for the difference between

theoretical and measured results is possibly caused by non-uniform critical current distribution

along the width of the tape and further study is needed. As presented in section 3.6.4, AC losses

in a HTS tape strongly depend on the critical current distribution in the tapes.

5.4.3 Frequency dependence of self-field AC loss

AC loss components generated by different mechanisms have different responses to frequency f.

In the unit W/m, the resistive loss is independent of frequency f, the hysteresis loss is

proportional to f and the eddy current/coupling loss is proportional to f2. Therefore, the total

power self-field loss can be written as a second order polynomial:

)()()( 2 IQfIQfTQP rnes ++= (5.10)

where Qe(I), Qn(I) and Qr(I) are the current dependent parameters that present for the eddy

current loss, hysteresis loss, and resistive loss, respectively. Thus, the loss per cycle is written as:

f

QQfIQ

f

PQ r

ne

s

s ++== )( (5.11)

Figure 5.20 plots the self-field AC loss in sample B2 for frequencies ranging from 51 Hz to 2500

Hz. The self-field loss per cycle increases with increasing frequency for currents smaller than 60

A because of the contribution of eddy current loss. However, the eddy current loss is quite small.

The self-field loss per cycle is almost unchanged up to a frequency of 350 Hz. At I0 = 30 A, the

loss per cycle increases by 100% when the frequency increases from 50 Hz to 2500 Hz. Based on

138

Eqt. 5.11, the resistive loss contribution at this current is negligible. Therefore, the eddy current

loss is only about 2% of the total self-field loss when I0 = 30 A and f = 55 Hz.

For the intermediate current values, the main loss component is the hysteresis loss. Therefore,

the self-field loss in this current region is nearly independent of frequency. For high current,

where the transport current is close to or higher than Ic, the resistive loss becomes significant.

The self-field loss therefore decreases as the frequency increases.

Figure 5.21 plots the self-field loss in sample Y3 for several frequencies. For currents smaller

than Ic, the self-field loss per cycle is nearly constant even when the frequency of the current

changes nearly five-fold, from 51Hz to 250 Hz. This suggests that, similar to sample B1, the

eddy current contribution in this sample is very small at those frequencies.

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

10 100 1000

Transport current (A)

Se

lf-f

ield

lo

ss

(J

/m/c

yc

le)

51 Hz

101 Hz

201 Hz

350 Hz

500 Hz

750 Hz

1000 Hz

1300 Hz

1900 Hz

2500 Hz

Fig. 5.20. Self-field AC loss in sample B2 at different frequencies from 51 Hz to 2500 Hz.

139

5.5. Magnetization loss at 77 K

Magnetization loss in sample B1, measured by the calorimetric method, for different orientations

of the applied magnetic field, is shown in Fig. 5.22. The numerically calculated data for a

rectangular tape with specifications similar to sample B1 are also plotted for comparison. In

general, calculated results describe well the AC loss behavior of the tape. Numerically calculated

losses are a little higher than the measured magnetization loss. With a low aspect ratio, b/a ≈

1/25, the magnetization loss strongly depends on the orientation of the magnetic field and mainly

depends on the perpendicular components of the magnetic field as discussed in chapter 3. In

parallel field, the loss is small, therefore the calorimetric method only can measure for the

applied field higher than 30 mT and the results obtained from this field orientation are less stable

due to the lower signal-to-noise ratio. To measure at lower magnetic fields and to have better

sensitivity for measurement in a parallel field, the thermal insulation in the calorimetric method

must be improved. Hence, to measure the comprehensive AC loss by the calorimetric method,

several experimental setups and one calibration curve for each setup must be prepared.

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

1 10 100

Transport current (A)

Se

lf-f

ield

lo

ss

(J

/m/c

yc

le)

51 Hz

150 Hz

250 Hz

Fig. 5.21. Self-field AC loss in sample Y3 at several frequencies.

140

Consider the magnetization AC loss in YBCO conductors, the Brandt equation for estimating the

magnetization loss in a thin strip (Eqt. 2.64) can be rewritten as

⎟⎟⎠

⎞⎜⎜⎝

⎛=

c

cm

B

Bf

IQ 0

20

πμ

(5.12)

The normalized magnetization loss ( )bf is a function of the normalized field cBBb /0= :

( ) )tanh())ln(cosh(2 bbbbf −= (5.13)

The characteristic field, πμ aIB cc 20= , of samples Y1 and Y3 are Bc(Y1) = 7.44 mT, Bc(Y3) =

8.03 mT. The magnetization losses of samples Y1 and Y3 in AC perpendicular magnetic field at

51 Hz are plotted on a normalized scale in Fig. 5.23. The magnetization loss of sample Y3 agrees

well with the analytical results obtained from Brandt’s equation. In the high current region, the

experimental data are lower than the theoretical results. This can be explained by the decrease in

the critical current caused by applied magnetic field that is neglected in the analytical

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1 10 100

Magnetic field (mT)

Ma

gn

eti

za

tio

n lo

ss

(J

/m/c

yc

le)

90 degree

60 degree

30 degree

0 degree

Numerical results

Fig. 5.22. Magnetization loss in sample B1 measured by the CM for different orientation ofapplied magnetic field. Numerical results are also plotted.

141

calculations. The numerical calculation presented at the end of this section therefore will give a

better estimation of magnetization loss.

For sample Y3, the measured magnetization loss is smaller than the theoretical results for the

entire range of magnetic field. The strong ferromagnetism in the Ni-W substrate of this sample

possibly shields the HTS layer from the external magnetic field, thus reducing the magnetization

loss. This explanation is favored by both experimental and simulated results reported recently in

several publications [125-128]. This shielding effect needs to be further studied since it would be

a possible way to reduce AC loss in HTS conductors.

In the case of self-field, the shielding effect of the ferromagnetic substrate, however, might be

small. As discussed in section 5.4.2, the substrate of sample Y1 does not affect self-field loss

generated in the YBCO layer of this sample, though it adds significant ferromagnetic loss to the

total self-field loss in this sample.

In an applied magnetic field, the Ni-5%W substrate of sample Y1 also contributes to the

ferromagnetic loss. Fig. 5.24 plots the numerically calculated results of the different loss

components in sample Y1 when a perpendicular magnetic field at 51 Hz is applied. The

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0 5 10 15 20

b = B0/Bc

No

rma

iliz

ed

ma

gn

eti

za

tio

n l

os

s f

(b)

Y1

Y3

Brandt's equation

Fig. 5.23. Normalized magnetization loss in YBCO samples Y1 and Y3. The normalized lossfunction given by Brandt’s equation is also plotted.

142

measured magnetization loss is also plotted. As seen in the figure, the ferromagnetic loss, Qf, is

comparable to the hysteresis loss in the YBCO layer, Qs, only in the very low field region; it

saturates at applied field higher than 10 mT. The eddy current loss, Qe, at 51 Hz is small in

comparison to Qs. The eddy current loss per cycle, however, is proportional to frequency.

Therefore, at high frequency, the eddy current loss contribution must be considered. In general,

hysteresis loss in the YBCO layer Qs is a dominant component. The shielding effect of the

ferromagnetic substrate is not taken into consideration in the numerical calculations, the

calculated Qs is therefore overestimated. Consequently, the calculated total magnetization loss

Qm is higher than the measured data. The numerical calculation becomes significantly more

complicated and finite element calculations may be needed if the shielding effect of

ferromagnetic substrate is considered.

The magnetization loss in the IBAD sample Y3 as a function of magnetic field is shown in Fig.

5.25 for several frequencies up to 550 Hz. As presented in section 5.4.2 when the sell-field loss

in DC magnetic field is studied, the ferromagnetic loss in the substrate is very small for this

sample. The measured magnetization loss is therefore the sum of the hysteresis loss in the YBCO

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100

Magnetic field (mT)

AC

lo

ss

(J

/m/c

yc

le)

Qs (Calculated)

Qe (Calculated)

Qf (Calculated)

Qm (Calculated)

Qm (measured)

Fig. 5.24. Magnetization loss in sample Y1 measured by EM method at 51 Hz. The numericallycalculated results for the magnetization AC loss components: hysteresis loss in YBCO layer,ferromagnetic loss in the substrate and the eddy current loss are also plotted

143

layer and eddy current loss in the copper layer. The dependence of measured magnetization loss

per cycle on frequency in this sample is small and the difference between the loss curves for

frequencies from 51 Hz to 550 Hz is difficult to be observed in logarithmic scale. Thus, the

contribution of eddy current loss is small compared to the hysteresis loss in the YBCO layer.

To quantitatively estimate the eddy current loss, the measured magnetization losses at 51 Hz and

200 Hz are plotted in linear scale as seen in Fig. 5.26. The total magnetization losses, Qm,

calculated numerically for those frequencies, are also plotted for comparison. When the eddy

current loss in the stabilizer and the field dependence of Jc(B) and n(B) are taken into account,

the numerical results provide a better description of experimental magnetization loss than the

analytical data shown in fig. 5.23.

The effect of eddy current at these frequencies is observable when the magnetic field is higher

than 50mT where the loss curves separate. The individual eddy current loss component at 200Hz

is also plotted in the figure. At 200Hz, eddy current loss contributions to the total magnetization

loss are about 4% at 50 mT and 10% at 100 mT.

0.00001

0.0001

0.001

0.01

0.1

1 10 100 1000

Magnetic field (mT)

Ma

gn

eti

za

tio

n lo

ss

(J

/m/c

yc

le)

55 Hz

80 Hz

150 Hz

200 Hz

250 Hz

350 Hz

450 Hz

550 Hz

Fig. 5.25. Magnetization loss in sample Y3 measured by EM method at different frequencies,from 51 Hz to 550 Hz

144

The magnetization loss in a wide range of frequency has not been studied extensively because of

the experimental limitation of high frequency AC fields. With YBCO tapes striated into many

small filaments to reduce the hysteresis loss, the effect of frequency on magnetization loss is

significant, even at low frequencies near the power line frequency [129-130]. The coupling

between filaments through the metal layers generates some coupling loss in the metals. In these

conductors, the frequency dependence of AC loss needs to be considered.

5.6. Total AC loss at 77 K

In this section, the AC loss characteristics in Bi-2223 and YBCO tapes when they carry an AC

transport current in an AC applied magnetic field are presented. In section 5.6.1 and 5.6.2, the

transport current and applied magnetic field are in phase. The dependence of total AC loss on the

phase difference will be discussed in section 5.6.3.

5.6.1. Total AC loss in a Bi-2223 tape

Figure 5.27 illustrates the experimental and numerical results of transport loss when sample B1

carries an AC current and is exposed in a perpendicular AC magnetic field. As discussed in

0

0.01

0.02

0.03

0.04

0 50 100 150

Magnetic field (mT)

Mag

neti

zati

on

lo

ss (

J/m

/cycle

)

55 Hz_measured

55 Hz_calculated

200 Hz_measured

200 Hz_calculated

Calculated eddy current

loss at 200 Hz

Fig.5.26. Magnetization loss in sample Y3 measured by EM method at 51 Hz and 200 Hz. Theeddy current loss in the stabilizer at 200 Hz is also plotted.

145

section 3.6.4, a thin elliptical tape can be treated as a thin strip with an elliptical critical current

distribution. In Fig. 5.27, the numerical results obtained from the 2D calculation for a

rectangular tape with uniform current distribution (filled triangle symbols) and from the 1D

calculation for a thin strip with elliptical Jc distribution (solid lines) are also plotted for

comparison. Both these numerical approximations result in good agreement with experimental

results for the transport loss.

The sum of the transport loss and the magnetization loss gives the total AC loss generated in that

sample. The experimental and numerical results of the total AC loss are shown in Fig. 5.28. In

general, the calculated results obtained from both numerical approximations describe well the

measured data. The numerical data obtained for elliptical tape, however, seem to be a little

smaller than the measured results. This suggests that the magnetization loss component

calculated by this approximation is somewhat smaller than the measured magnetization loss.

The numerical calculations with an actual cross-section and actual critical current distribution of

the tape would provide more accurate results. The tape cross-section can be obtained using a

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

10 100 1000

Transport current (A)

Tra

ns

po

rt A

C lo

ss

(J

/m/c

yc

le)

10.4 mT

15.6 mT

31.5 mT

52.5 mT

Filled friangles: Numerical for rectangular tape

Solid lines: Numerical for thin elliptical tape

Fig. 5.27. Transport AC loss component as a function of transport current at different values ofperpendicular applied field. The numerical results obtained from a rectangular uniform tape anda thin strip tape with elliptical Jc distribution are also depicted

146

microscope. The critical current distribution, however, is more difficult to determine. The sheet

critical current density along the width of a tape, however, can be estimated experimentally by a

“magnetic knife” [131-132] or by using micro hall sensor [132-133].

5.6.2. Total AC loss in YBCO tapes

The transport loss in sample Y3 as a function of applied magnetic field is plotted in Fig. 5.29 for

several values of transport current. Again, with Jc(B) and n(B), the numerically calculated data

describe well the behavior of the measured transport loss. For a given transport current, the

transport AC loss is increased with increasing applied magnetic field. The effect of applied

magnetic field on the transport loss is more significant for lower transport currents. The

contribution of flux flow loss is observed in the high current, high field region. For example,

when I0/Ic = 0.9, the transport loss increases rapidly and the loss curve changes slope as magnetic

field is higher than 30 mT.

Fig. 5.30 shows the experimental and calculated magnetization loss components in sample Y3 as

a function of magnetic field. The effect of transport current on the magnetization loss is stronger

1.0E-04

1.0E-03

1.0E-02

1.0E-01

10 30 50 70 90 110 130 150

Transport current (A)

To

tal A

C lo

ss

(J

/m/c

yc

le)

10.4 mT 15.6 mT 31.5 mT 52.5 mT

Solid triangles: Numerical for rectangular tape

Solid lines: Numerical for thin elliptical tape

Fig. 5.28. The total AC loss as a function of transport current at different values of perpendicularapplied field. The numerical results obtained from a rectangular uniform tape and a thin striptape with elliptical Jc distribution are also depicted.

147

for the lower magnetic fields. When B0 < 30 mT, the magnetization loss increases with the

transport current. Inversely, the magnetization loss component decreases slightly with transport

current as the magnetic field is higher than 30 mT.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1 10 100

Magnetic field (mT)

Mag

neti

zati

on

lo

ss (

J/m

/cycle

) 0.1 Ic 0.5 Ic 0.9 Ic

Solid lines: numerical

Symbols: Experimental

Solid lines: numerical

Symbols: Experimental

Fig. 5.30. The numerical and experimental results of the magnetization AC loss components insample Y3 as a function of perpendicular AC field at different transport current values.

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1 10 100

Magnetic field (mT)

Tra

nsp

ort

lo

ss (

J/m

/cycle

)

0.1 Ic 0.3 Ic 0.5 Ic

0.7 Ic 0.9 Ic 1.1 Ic

Solid lines: numerical

Symbols: Experimental

Fig. 5.29. The numerical and experimental results of the transport AC loss components in sampleY3 as a function of perpendicular AC field at different transport current values.

148

Fig. 5.31 plots the numerical and experimental total AC loss in sample Y3 at transport I/Ic = 0.1,

0.5, and 0.9. In general, the numerical data reproduce well the experimental results. In the low

magnetic field region, the calculated results are a little higher than the experimental data. The

smaller actual width and/or the non-uniform critical current distribution in the sample may be the

reason for this small variation.

The comparison between the numerically calculated and experimental results of the transport

loss component and the total AC loss in YBCO sample Y1 are illustrated in Fig. 5.32 and Fig.

5.33, respectively. Good agreement between measured and calculated results for transport loss,

Qt, is seen in Fig. 5.32. The experimental data (the symbols) for the total AC loss, however, was

found to be quite a bit smaller than numerical results (the solid lines). The differences between

the experimental and numerical results for Qtt mainly result from the error in the calculations of

the magnetization loss component, similar to what was discussed in section 5.5. Since the

ferromagnetic shielding of the Ni-5%W substrate in RABiTSTM sample Y1 is not taken into

consideration in the numerical calculations, the calculated data for the magnetization loss

component are somewhat higher than the measured results.

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1 10 100

Magnetic field (mT)

To

tal A

C lo

ss

(J

/m/c

yc

le)

0.1 Ic

0.5 Ic

0.9 Ic

Solid lines: numerical

Symbols: Experimental

Fig. 5.31. The numerical and experimental results of the total AC loss in sample Y3 as a function of perpendicular AC field at different transport current levels.

149

Fig. 5.33. The numerical and experimental results of the total AC loss in sample Y1 as a functionof perpendicular AC field at different transport current levels.

Increasing current

Fig. 5.32. The numerical and experimental results of the transport AC loss components in sampleY1 as a function of perpendicular AC field at different transport current levels.

Increasing current

150

5.6.3. Dependence of phase difference on AC losses

In many power applications, the transport current in the HTS tapes and the magnetic field to

which it is exposed are out of phase. Therefore, understanding the effect of the phase difference,

Δϕ, on AC losses is necessary for the design of efficient superconducting electric machines.

Since the electromagnetic method is applicable only when the transport current and magnetic

field are in phase, the calorimetric method was employed to evaluate the effect of Δϕ on total AC

loss in HTS tapes.

The measured and numerically calculated total AC loss in sample B1 as a function of transport

current at perpendicular applied magnetic field Ba = 31.5 mT at various values of Δϕ are shown

in Figs. 3.34 and 3.35. The loss behavior with respect to Δϕ is similar for the experimental and

numerical results. This can be seen more clearly in Fig. 5.36, which presents the normalized total

loss, Qt(Δφ )/Qt(Δφ = 0°) as a function of Δϕ. The open symbols are the experimental results and

the solid lines are the numerically calculated results. The good agreement between the numerical

and experimental results is seen. To estimate the effect of Δϕ on AC loss, a loss ratio R was

defined as R(%) = 100(Qmin/Qmax), where Qmax, Qmin are maximum and minimum values of AC

loss determined at the phase difference Δφmax and Δφmin, respectively.

At transport current It =15 A (It ~ 0.13 Ic), the effect of Δϕ on the total AC loss is insignificant.

The minimum loss Qmin obtained at Δφmin = 75° is about 95% of the maximum loss Qmax obtained

at Δϕmax = 150°. At It = 112.5 A (It ~ 0.95 Ic), the minimum loss was obtained at Δφmin = 110°

while the maximum loss was obtained at around Δϕmax = 20°. For this transport current, the

effect of Δϕ on total AC loss becomes significant and R = 60%. As seen in Fig. 5.36, the values

of Δφmax and Δφmin shifted slightly with changing transport current. Physically, )(tI , )(tB , and

their time derivatives )(tI& , )(tB& affect the current distribution inside a HTS tape. )(tI , )(tB ,

and )(tI& , )(tB& , however, are 90° different in phase. Therefore, the competition between )(tI ,

)(tB and their time derivatives )(tI& , )(tB& impacts the current distribution that determines Δφmax

and Δφmin.

151

Fig. 5.35. The numerical results of the total AC loss in sample B1 as a function of the transportcurrent and the phase different Δϕ, B0 = 31.5 mT and f = 51 Hz.

Fig. 5.34. The experimental results of the total AC loss in sample B1 as a function of thetransport current and the phase difference Δϕ, B0 = 31.5 mT and f = 51 Hz.

152

The interplay between the transport current and the magnetization losses affect the total AC loss.

At low values of transport current, the magnetization loss is dominant, while at high values of

transport current, the transport loss dominates. Therefore, it is interesting to study the effects of

Δϕ on the magnetization and transport loss separately. This is difficult experimentally, however,

since calorimetry measures the total loss only, while electromagnetic measurements are not

usable for nonzero Δϕ. Thus the numerical calculations are particularly valuable. The calculated

magnetization loss at Ba = 31.5 mT with varying transport current is shown in Fig. 5.37. The

effect of Δϕ on the magnetization loss strongly depends on transport current, and the effect is

more significant for higher transport currents. At the highest transport current considered, It =

115 A (It = 96.6% Ic), the loss ratio for magnetization loss Rm is only about 50%. The minimum

magnetization losses are obtained at Δφmin in the range from 60° to 90°, depending on the

transport current amplitude. In some HTS devices, such as HTS transformers, where

magnetization losses are dominant and the phase difference in the operating regime is around

90°, this is advantageous for minimizing losses. Comparing Fig. 5.36 and Fig. 5.37, the curves

Fig. 5.36. Numerical (the lines) and experimental (the symbols) data of normalized total lossQt(Δφ )/Qt(Δφ = 0°) in sample B1 as a function of the phase different Δϕ, Ba = 31.5 mT and f =51 Hz.

153

corresponding to It = 15 A have nearly identical shape, which is consistent with the expectation

that, at low current, the magnetization losses dominate.

Calculations of the effect of Δϕ on transport loss, Qtr, for Ba = 31.5 mT are shown in Fig. 5.38. In

the figure, the normalized transport loss, Qtr(Δφ )/Qtr(Δφ = 0°), is plotted as function of the phase

difference Δφ. At the highest transport current, It = 115 A (It = 96.6% Ic), the effect of Δϕ is the

least significant but the transport loss ratio Rt remains small at ~50%. The strongest effect of Δϕ

on transport loss is observed at It = 55 A (It = 46.2% Ic), where Rt ~25 %. The maximum

transport loss is obtained around Δφmax = 30° while the minimum transport loss is observed at

Δφmax ~ 120° -130°. When the transport current is below 55 A, the effect is nearly independent of

transport current with Rt ≈ 25%. For transport current greater than 55 A, Rt increases as the

current increases.

Fig. 5.37. Numerical magnetization loss in sample B1 as a function of the phase different Δϕ, B= 31.5 mT and f = 51 Hz.

154

5.7. Temperature dependence of AC loss characteristics

5.7.1. Temperature dependence of Ic and n-value

Since AC loss and critical current have a tight relationship, the transport critical current needs to

be measured before measuring AC loss. Fig. 5.39 depicts the I-V curves of sample Y4 for several

temperatures from 45 K to 85 K. The critical current and the n-value were determined using a

μV/cm criterion.

The n-value reduces from 37 to 25 when temperature decreases from 85K to 45K. This can be

seen as the decrease in the slope of the I-V curves as the temperature decreases. This behavior

was also observed in Bi-2223 sample B2 and is consistent with results reported where else [34-

35]. There is no accepted explanation for this phenomenon but it might be related to the

temperature dependence of flux pinning properties in HTS conductors. Temperature dependence

of Ic for samples B2 and Y4 is shown in Fig. 5.40. The critical current increases significantly

with decreasing temperature. The temperature dependence of Ic in HTS conductors follows the

Fig. 5.38. Numerical transport loss in sample B1 as a function of the phase different Δϕ, B= 31.5mT and f = 51 Hz.

155

power law: Ic ~ (1-T/Tc)n with n = 1-3.5 [135-136]. As seen in Fig. 5.40, Ic(T) is fitted well by

function Ic(T) = 790(1-1/Tc)1.2 for sample Y4 and by function Ic(T) = 1050(1-T/Tc)

1.8 for sample

B2.

0

100

200

300

400

500

30 40 50 60 70 80 90 100

Temperature (K)

I c (

A)

Y3 B2

Fig. 5.40. The temperature dependent critical current of sample B2 and sample Y4.

-0.000001

0

0.000001

0.000002

0.000003

0.000004

0.000005

0 100 200 300 400

DC current (A)

Ele

ctr

ic f

ield

(V

/m) 85 K

77 K

65 K

55 K

45 K

Fig. 5.39. The DC I-V curves measured in sample Y4 at different temperature.

156

5.7.2. Self-field AC loss at variable temperatures

The transport loss in samples B2 and Y4 at several temperatures are shown in Figs. 5.41 and

5.42. For a given transport current, the self-field loss is smaller at lower temperature as a result

of the increase in critical current. At low I0/Ic, the signal is small and thus it may be affected by

the vibration of the sample holder caused by the cryo-cooler. As a consequence, the data is less

stable than the 77 K measurement when I0/Ic < 0.2.

Figures 5.43 and 5.44 show the normalized self-field losses q(i) = Q/(μ0Ic2/π) in samples B2 and

Y4 versus the normalized transport current i = I0/Ic. As seen in the figures, the loss curves at all

the measured temperatures are superposed together. This confirms that, in these samples, the

main contribution in the self-field losses is the hysteresis loss from superconducting material and

it changes with temperature mainly due to the change of the critical current. For sample B2, the

measured data agree well with calculated results obtained from Norris’ model for an elliptical

tape. This agreement is expected since this sample should have similar self-field loss behavior as

sample B1 since they are both cut from the same batch. For the sample Y4, the self-field loss

behavior is also nearly identical to that of sample Y3 measured by the 77 K measurement setup.

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

10 100 1000

Transport current (A)

Tra

nsp

ort

lo

ss (

J/m

/cycle

)

77 K

65 K

55 K

45 K

Fig. 5..41. Temperature dependent self-field loss in sample B2 as a function of the transportcurrent.

157

Thus, it can be concluded that the variable temperature setup delivers reliable results for AC loss

measurements.

Since the loss curves at different temperatures are superposed in the normalized scale, the self-

field loss at any temperature therefore can be estimated from the critical current at that

temperature and the self-field AC loss at 77 K.

2

)(

)77()77,(),( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

TI

KIKiQTiQ

c

c (5.14)

This may not be a case for RABiTSTM samples with Ni-W substrates because of the significant

contribution of ferromagnetic loss in the self-field loss in this type of conductor. The

ferromagnetic loss depends on the substrate properties. The ferromagnetic loss and the

superconducting hysteresis loss would have different temperature dependences. It therefore

would be interesting to study temperature dependence of AC loss in a YBCO tape fabricated by

RABiTSTM method. Some initial data of the self-field loss in RABiTSTM YBCO samples at

variable temperature was reported in [137]

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

10 100 1000

Transport current (A)

Tra

nsp

ort

lo

ss (

J/m

/cycle

)

45 K

55 K

65 K

77 K

Fig. 5.42. Temperature dependent self-field loss in sample Y4 as a function of the transportcurrent.

158

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0.01 0.1 1 10

Normalized transport current i = I/Ic

No

rma

lize

d t

ran

sp

ort

lo

ss

q(i

)

45 K

65 K

55 K

77 K

Norris_strip

Norris_ellipse

Fig. 5.44. The normalized self-field loss in sample Y4 as a function of the transport currentmeasured at several temperatures.

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.01 0.1 1

Normalized transport current i = I/Ic

No

rma

lize

d t

ran

sp

ort

lo

ss

q(i

)

77 K

65 K

55 K

45 K

Norris's ellipse

Fig. 5.43. The normalized self-field loss in sample B2 as a function of the transport currentmeasured at several temperatures.

159

5.7.3. Magnetization at variable temperature

Temperature dependent magnetization loss in sample Y4 is plotted in Fig. 5.45. The

magnetization loss was measured in the temperature range from 35 K to 95 K for different values

of perpendicular magnetic field at 51Hz. For a given magnetic field, the dependence of

magnetization loss on temperature is significant, especially for low magnetic fields. For each loss

curve, the magnetization loss is maximum at a certain temperature, Tmax, and drops more quickly

for T > Tmax than for T < Tmax. When applied magnetic field increases from 5 mT to 60 mT, the

corresponding Tmax(B) decreases from 86 K to 59 K. Information on temperature dependence of

magnetization loss is very important for optimizing the design and operation temperatures of

HTS devices. From the energy efficiency standpoint, a HTS device in which the HTS conductor

operates in the magnetic field B should not operate at temperature Tmax(B) of that conductor.

When the temperature is higher than the critical temperature of YBCO, Tc = 90K, the

magnetization is almost independent of the temperature for all magnetic fields. When T > 90K,

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

30 40 50 60 70 80 90 100

Temperature (K)

AC

lo

ss (

J/m

/cycle

) 60 mT 50 mT 40 mT 30 mT

20 mT 15 mT 10 mT 5 mT

Fig. 5.45. Magnetization loss in sample Y4 as a function of the temperature and appliedmagnetic field.

160

the superconducting layer becomes a normal conductor and the hysteresis loss in HTS layer

becomes very small. Thus, the main component of magnetization loss at T > 90 K is the eddy

current loss generated by the normal conductors. It is worth noting that at T > Tc, the eddy

current is caused by the uniform external field along the width of the sample since there is no

shielding effect caused by the superconducting layer at that temperature. This is not the case for

low temperatures when the sample is in superconducting state. In superconducting state, the

center part of the sample is shielded and the eddy current loss is only generated near the edge of

the tape. Thus, the eddy current loss at low temperatures will be considerably small.

Mathematically, as discussed in section 3.6 and section 5.5, both analytical and numerical

models have shown that eddy current loss strongly depends on b = B0/Bc. At low temperatures,

Bc is higher thus B/Bc will be smaller. As a consequence, eddy current loss is very small.

From the data shown in Fig. 5.45, the magnetization losses at several temperatures are plotted in

Fig. 5.46. As presented in section 5.5, the experimental data of the magnetization loss in sample

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1 10 100

Magnetic field (mT)

mag

neti

zati

on

lo

ss (

J/m

/cycle

)

45 K

55 K

65 K 77 K

85 K

CalculatedMeasured

Fig. 5.46. Magnetization loss in sample Y4 as a function of applied magnetic field. Thecorresponding analytical results obtain from Brandt equation are also plotted.

161

Y3 follows quite well the Brandt equation for magnetization loss of a thin strip. Thus, the

magnetization loss calculated analytically by the Brandt equation from Ic(T) is also used in this

case to compare with the experimental data. As seen in Fig. 5.46, the theoretical results agree

with experimental magnetization loss for 85 K and 77 K. At lower temperature, the theoretical

results are somewhat smaller than the measured data, especially for low fields.

To explain for the magnetization loss behavior in variable temperatures, Eqt.2.36 is recalled

here:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

c

mB

BgB

aQ 02

00

24

μπ

(5.15)

where function g(b) is given by Eqt. 2.65. At a given applied field B0, the magnetization loss is

maximum when g(b) is maximum. Function g(b) reaches its peaks at b = B/Bc ≈2.45. Therefore,

for each magnetization loss curve in Fig. 5.45, the applied magnetic field is kept constant. The

characteristic field Bc, however, changes with temperature as a result of the change of critical

0

0.001

0.002

0.003

0.004

20 30 40 50 60 70 80 90 100

Temperature (K)

Mag

neti

zati

on

lo

ss (

J/m

/cycle

)

B0 = 20 mT

Tmax = 78.5 K

Fig. 5.47. Temperature dependent magnetization loss in sample Y4 when magnetic field B0 =20 mT is applied.

162

current. Thus, for each curve, the condition B0/Bc ≈2.45 is satisfied at temperature Tmax of that

curve. For example, the magnetization curve corresponding to applied magnetic field B0 = 20 mT

(as seen in Fig. 5.47) has Tmax = 78.5 K. Thus, Bc(78.5 K) = 20/2.45 = 8.16 (mT). Since

πμ aIB cc 20= , a new critical current at Tmax, called the magnetization critical current, Icm(Tmax)

(to distinguish it from the conventional Ic determined from transport measurement) can be

obtained. Magnetization Icm(Tmax) is calculated from external field B0 only, not from any value of

magnetization loss.

The magnetization and transport critical currents are plotted in Fig. 5.48. At temperature T > 77

K, the magnetization critical current Icm and the transport critical current Ic are nearly equal.

When the temperature is further decreased, Icm is somewhat higher than Ic. In fact, the transport

critical current is determined based on an electric field criterion that may not physically describe

the nature of the electrodynamics in the conductor. A similar relation between the transport

critical current measured by the “four-probe” technique and the magnetization critical current

determined from magnetization curve measured by SQUID magnetometer was reported in [138].

0

50

100

150

200

250

300

350

30 40 50 60 70 80 90

Temperature (K)

I c (

A)

transport

Magnetization

Fig. 5.48. Temperature dependence of the transport and magnetization critical current of sampleY4.

163

The magnetization loss calculated by Brandt’s equation from magnetization critical current Icm is

compared to the experimental data as shown in Fig. 5.49. Good agreements between calculated

and measured results are seen in this case. Hence, for magnetization loss, an effective (or

magnetization) critical current rather than the conventional transport current may be needed for

Brandt’s equation to have better magnetization loss estimation for a thin strip.

5.7.4. Total AC loss in variable temperatures

The total AC is the sum of the transport loss and magnetization loss. The transport loss and

magnetization loss at 45 K are plotted in Figs. 5.50 and 5.51 for three different values of applied

field, 10 mT, 30 mT and 50 mT. At such a low temperature, the dependence of Ic and n-value on

magnetic field can be ignored for applied fields up to 50mT. The numerically calculated results

with the field independent Ic and n-value are also plotted. In general, the transport loss and

magnetization loss are in the same order of magnitude. At 10 mT, the transport loss plays a more

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1 10 100

Magnetic field (mT)

mag

neti

zati

on

lo

ss (

J/m

/cycle

)

60 K

65 K

77 K

85 K

Measured Calculated

Fig. 5.49. Temperature dependence of the transport and magnetization critical current of sampleY4.

164

important role as I > 100 A. At a magnetic field of 50 mT, the magnetization loss is generally

higher than the transport loss for the entire range of measured transport current.

0.00001

0.0001

0.001

0.01

0.1

1

10 100 1000

Transport current (A)

Tra

ns

po

rt l

os

s (

J/m

/cy

cle

) 10 mT

30 mT

50 mT

MeasuredCalculated

Fig.5.50. The experimental and numerical transport loss component in sample Y4 at 45 K.

0.00001

0.0001

0.001

0.01

0.1

1

10 100 1000

Transport current (A)

Tra

ns

po

rt lo

ss

(J

/m/c

yc

le) 10 mT

30 mT

50 mT

Calculated Measured

Fig. 5.51. The experimental and numerical magnetization loss component in sample Y4 at 45 K.

165

The total AC loss at for several temperatures from 45K to 77K is plotted in Fig. 5.52 (B0 = 10

mT) and Fig. 5.53 (B0 = 50 mT). When B0 = 10 mT, the transport loss plays a more important

role than the magnetization loss. Consequently, the loss curves have transport-like shape and the

total loss increases quickly as the transport current increases. The total loss at 77 K is highest due

to the lowest Ic at this temperature.

In Fig. 5.53, at magnetic field B0 = 50 mT, the magnetization loss is more significant than the

transport loss component. The loss curves have “flatter” slope with increasing current. In this

case, the lowest total loss is observed at 77 K, but only in a narrow range of current. When the

current is higher than 70 A, the loss increases dramatically due to the flux-flow loss contribution.

In general, the total AC loss at 65 K is highest in this case. This is consistent with what was

observed in Fig. 5.45. At 50 mT applied magnetic field, the Tmax of the loss curve is around 65 K.

As discussed in chapter 1, the cooling penalty is larger at lower operating temperatures. To have

an efficient HTS device, the operating temperature and design need to be optimized by balancing

the cooling penalty, AC loss and Ic(T).

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250 300 350

Current (A)

To

tal lo

ss

(J

/m/c

yc

le)

45 K

55 K

65 K

77 K

B = 10 mT

Fig.5.52. Experimental total AC loss in sample Y4 at temperature several temperatures from 45K to 77 K, B0 = 10 mT, f = 51 Hz.

166

0.00001

0.0001

0.001

0.01

0.1

1

10 100 1000

Current (A)

To

tal lo

ss

(W

/Am

) 50 mT 30 mT 10 mT

Fig. 5.54. Total AC loss in sample Y4 measured at 45 K for several of magnetic fields.

B = 50 mT

0

0.02

0.04

0.06

0.08

0 50 100 150 200 250 300

Current (A)

To

tal lo

ss

(J

/m/c

yc

le)

45 K

55 K

65 K

77 K

Fig.5.53. Experimental total AC loss in sample Y4 at several temperatures from 45 K to 77 K, B0

= 50 mT, f = 51 Hz.

167

For application purposes, the total AC loss at 45 K as a function of transport current is plotted in

unit W/A-m as seen in Fig. 5.54. Assuming that the cooling penalty factor for 45 K is 20, the AC

loss must be smaller than 1 mW/A-m to be comparable with copper to achieve energy savings, as

presented in chapter 1. As seen in Fig. 5.54, the total AC loss at 10 mT meets that limitation.

Thus, for HTS applications with small AC magnetic field, such as cables or fault current limiters,

samples Y3 and Y4 are a potential conductor. However, higher magnetic field applications, for

instance, transformer, gradient coil, or generator, the AC loss may be a limiting consideration.

The further development of low loss conductors for those applications is required. Some

concepts for low loss HTS conductors will be presented in the next chapter.

168

CHAPTER 6

SUMMARY AND FUTURE WORK

6.1. Primary findings

Table 6.1. Measurement capabilities for total AC loss

Parameter Small magnet Large magnet

Typical sample length 10 cm 25 cm

Transport current 0 – 1000A at 100 Hz

400 A at 3 kHz 0 – 600A, 100 Hz

Background magnetic field 0 – 200 mT at 100 Hz

0 - 30 mT at 1 kHz 0 – 120 mT at 50 Hz

Temperature range 77 K 30 – 100 K

Phase relation between transport current and magnetic field

Any In phase

Background magnetic field orientation

Any Any

169

As a main part of this dissertation, an advanced AC measurement loss facility has been

successfully built, tested and validated. The facility consists of two measurement setups, a

universal setup to measure total AC loss at 77 K and an experimental setup to measure total AC

loss with variable temperatures, ranging from 30 K to 100 K. Both measurement setups utilize

specially designed double-helix magnets that were proven to be convenient for AC loss

measurement. In the 77 K measurements, both electromagnetic and calorimetric methods were

used to have more thorough understanding of the AC behavior in HTS conductors. The

agreement between the results obtained from these experimental methods validated the

measurements. The variable temperature AC loss measurement uses the electromagnetic method

to study the total AC loss in variable temperatures ranging from 30 K to 100 K. The sample

holder design was optimized to overcome the cooling difficulties and to maintain the temperature

uniformity along the sample with error ± 0.5 K during the measurement. The measurement

capabilities are summarized in the table 6.1.

The AC loss and electrodynamics in HTS conductors were studied by numerical calculations

based on Brandt’s model. In the numerical calculations, the superconducting electrical behavior

is assumed to follow the power law model and the field dependence of Jc and n-value are fitted

from the experimental data. Numerical calculations were employed to study electrodynamics in

rectangular tapes for several values of aspect ratio. The current distribution was found to be

highest near the edge or surface of the tape causing the highest AC loss power density there. The

magnetization AC loss in high aspect ratio samples strongly depends on the orientation of the

magnetic field. For a relatively high applied magnetic field (20 mT or higher for the investigated

tapes), the ratio between magnetization losses in parallel and perpendicular magnetic field are

approximately equal to the aspect ratio of the tape. This finding was observed in both numerical

calculation for rectangular tapes and analytical results obtained for elliptical tapes.

YBCO tapes can be treated as thin strips due to their very small aspect ratio. Using variable

substitutions, the numerical calculations for a thin strip yield precise results. In this case, the

superconducting hysteresis loss in the YBCO layer, eddy current loss in the stabilizer and the

ferromagnetic loss in the substrates can be estimated. Good agreement between numerical and

analytical results for AC loss components as well as current and magnetic field profiles was

observed.

170

The numerical results showed that AC loss in a HTS tape is strongly affected by the Jc

distribution. The simulated self-field loss in a HTS strip tape with convex Jc distribution is

higher than that of a tape with concave Jc distribution. This observation is also true for

magnetization loss in the low magnetic field region. In the high field region, a reverse effect,

however, was seen. Therefore, the slope of the magnetization loss curve for a HTS strip tape

with convex Jc distribution is smaller than that of a tape with concave Jc distribution. The

numerical self-field loss in a thin strip with elliptical Jc distribution is equal to the analytical

results obtained from the Norris’ model for an elliptical tape. This finding suggests that a thin

HTS tape with an elliptical cross section also can be treated as a thin strip with elliptical Jc

distribution.

Using the experimental systems and analytical and numerical calculations, the electromagnetic

characterization, AC loss characteristics, and electrodynamics in Bi-2223 and YBCO tapes have

been studied. The following conclusions result:

1. With different substrates, the critical current and the n-value of IBAD YBCO tapes

decrease with perpendicular DC field relatively fast as compared to RABiTSTM samples.

2. The measured self-field losses in all the Bi-2223 tapes agree well with Norris’ model

while the self-field losses in the YBCO tapes seems to follow Norris’ model for a thin

strip in the high current region (near critical current). The main contribution of self-field

loss in region I < Ic for Bi-2223 tapes and IBAD YBCO tapes is the hysteresis loss

generated in the superconducting materials.

3. The sudden change of the slope of the self-field hysteresis loss in a Bi-2223 tape

measured by a contactless voltage loop suggests that the actual full penetration current in

this tape is much higher than the critical current measured by the four-probe method.

4. For the RABiTSTM samples, the ferromagnetic loss plays a significant role in the total

self-field loss for transport bellow than 100 A. At the currents higher than 100 A, the

ferromagnetic loss is saturated and becomes smaller than the superconducting hysteresis

loss. Furthermore, this ferromagnetic loss is decreased in the presence of a DC applied

field, especially in parallel DC field. With small demagnetizing factor in the parallel

171

direction, the substrate is almost saturated in only 20 mT DC parallel field. Thus, the

ferromagnetic loss in the substrate reduces quickly when parallel field increases from 0 to

10 mT and completely disappears at 20 mT field. When ferromagnetic loss is taken into

account, the numerical results of the total self-field loss in a RABiTSTM sample agree

quite well with the experimental data.

5. The experimental data for magnetization loss in Bi-2223 and IBAD YBCO tapes agree

well with the numerical results. The measured results in a wide range of frequencies for

both self-field loss and magnetization loss show that the contribution of the eddy current

loss is smaller compared to the hysteresis loss. The simulated eddy current loss interprets

well for the difference between the total magnetization loss (per cycle) measured for the

IBAD sample at 51 Hz and 200 Hz.

6. The measured magnetization loss in the RABiTSTM sample is smaller than both the

analytical and numerical results. This variation is possibly caused by the shielding effect

of the sample Ni-5%W substrate that is ignored in the calculations. The electromagnetic

interaction between the superconducting layer and Ni-W substrate in this kind of sample

requires further study.

7. The agreement between the numerical and experimental results was also achieved for

both transport and magnetization loss components in the total AC loss of Bi-2223 and

IBAD tapes. For applied field higher than 10 mT, the magnetization loss dominates and

the transport loss is comparable with magnetization loss only when the current is close to

Ic. Thus, in many HTS devices, the magnetization loss would be the dominant

component.

8. When the temperature decreases from 85 to 45 K, Ic in IBAD YBCO tape and Bi-2223

tape increase by 6-8 times while the n-value is slightly reduced. The change of self-field

loss with variable temperatures was mainly due to the change in Ic. As a result, all the

self-field loss curves at different temperatures for each sample are superposed together.

The loss at any temperature can be estimated from the self-field AC loss measured at 77

K and the temperature dependence of critical current.

172

9. The magnetization loss strongly depends on temperature. At a given magnetic field B0,

there is a temperature Tmax at which the magnetization loss is maximum. The

magnetization critical current at Tmax can be determined from the applied magnetic field

B0. It was found that the magnetization critical current is higher than the conventional

transport critical current when the temperature is above 77 K. The measured

magnetization at several temperatures agrees well theoretical results obtained from the

magnetization Ic. An effective (or magnetization) critical current may be needed for

Brandt’s equation to give a better estimation of the magnetization loss in YBCO tapes.

6.2. AC loss reducing solutions

Based on Brandt’s model, for a given sheet critical current density, the magnetization in a YBCO

tape is proportional to the square of its width. Hence, AC loss can be reduced by decreasing the

tape width. The YBCO tapes therefore can be striated into a multifilamentary tape to reduce the

magnetization loss. However, striation also decreases Ic and thus increases the transport loss. In

addition, if the filaments are not electrically insulated, the coupling between them may cause

coupling loss. The loss reduction by this idea needs to be further studied and improved.

Increasing the sheet critical current density will lower the transport AC loss. Increasing the

thickness of YBCO layer, however, is not an efficient method to increase the critical current

density as discussed in chapter 1. A multi-layer YBCO tape might be a solution for this

challenge. Furthermore, because the superconducting materials are diamagnetic, the inner YBCO

layers in a multilayer tape can be shielded from the external magnetic field by the outer

superconducting layers. Thus, the magnetization loss can be reduced. For example, the AC loss

is expected to be improved in a double layer YBCO tape with a ferromagnetic substrate

sandwiched by YBCO layers deposited on its both sides. In this tape, the substrate is shielding by

YBCO layer thus the contribution of ferromagnetic loss may be reduced significantly.

Additionally, the ferromagnetic substrate and one YBCO layer may help to shield partially the

other YBCO layer from external magnetic field, thus decreasing the magnetization loss in YBCO

layers. However, fabrication of this kind of sample is left for future research.

For HTS devices, AC loss will be decreased if the perpendicular magnetic field is minimized. In

addition, the operating temperature and the design of HTS devices must be optimized to avoid

173

operating at the temperature Tmax, at which the magnetization loss in HTS tape is maximum.

Finally, for applications in which HTS tapes are working with the presence of DC field (MRI

gradient coil, for instance), the parallel DC magnetic field can help to eliminate the contribution

of ferromagnetic loss.

6.3. Future work

The interaction between the HTS layer and ferromagnetic substrate is an interesting problem and

needs to be further studied. It would be interesting to study the electromagnetic properties and

AC loss characteristics of a YBCO tape that is covered on both sides with strong ferromagnetic

materials. The numerical calculations for modeling the interaction between the HTS layer and

ferromagnetic layers can be useful tools for designing low loss conductors. However, the

problem will be complicated in this case and can only be solved out by the finite element

method.

In practical application, HTS tapes are usually working in the form of coils (transformer, magnet.

motor,…), matrix (fault current limiter) or cables. It is therefore necessary to study AC loss in a

stack of HTS tapes or a cluster of in-plane tapes. In this case, AC loss characteristics in a HTS

tape are affected by the self-field and /or the shielding effect of its neighboring HTS tapes. For a

stack of HTS tapes, the different configurations may have different AC loss properties. For

example, with a stack of two HTS tapes, the substrate-in configuration may have smaller self-

field loss than that in the substrate-out arrangement because with the substrate-in arrangement,

the substrates are shielded by HTS layers. In addition, the distance and electrical conduction

between the tapes in a stack will also be important parameters impacting AC losses. Without the

electrical insulation between the tapes, the coupling between the tapes may occur and generate

coupling loss. The FEM calculations to study the electrodynamics and AC loss in a stack of HTS

tape remains a challenge due to the extensive CPU and memory. As presented in this

dissertation, with the variable substitutions, the numerical calculation based on Brandt’s model

yields good estimation for AC losses in YBCO tapes. This numerical calculation can be extended

to study AC in a stack of YBCO tapes.

The temperature dependence of AC loss in RABiTSTM YBCO tapes with Ni-W substrates is also

an interesting problem for future research. The different temperature dependence of the

174

ferromagnetic loss in the substrate and the superconducting hysteresis loss in the YBCO layer

may give interesting information on AC loss characteristic for this kind of sample in variable

temperatures.

The experimental setup can be adapted to measure AC loss in HTS coils. The information of AC

coils will help to have more precise evaluation of the AC loss in HTS devices such as a

transformer or generator. With a high power amplifier and compensating capacitors, the system

is suitable for testing HTS coils with considerable inductance.

Finally, it is interesting to make use of the variable temperature experimental setup to study AC

quench stability in HTS tapes. This will be a novel and useful study for practical power

applications. The AC losses (both transport and magnetization losses) will strongly affect the

stability of HTS devices and limit their operation. The information on AC quench and stability

will be very important for HTS devices.

175

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186

BIOGRAPHICAL SKETCH

Doan Ngoc Nguyen was born on April 18, 1979, in Hai Duong City, Vietnam. He completed his

Bachelor of Science degree in Physics (Honors Program) in 2001 at Vietnam National

University, Hanoi, Vietnam. In the same year, he began his Ph.D. degree at the Department of

Physics, Florida State University. He joined Dr. Schwartz lab in Summer 2002. He has ten

published papers since 2002 that are listed bellow:

[1] P. V. P. S. S. Sastry; D. N. Nguyen; P. Usak and J. Schwartz, “Verification of a thermal interpretation of BSCCO-2223/Ag current-voltage hysteresis,” Supercond. Sci. and

Technol., 17 (3), 314-319 (2004)

[2] D. N. Nguyen, P. V. P. S. S. Sastry, G. M. Zhang, D. C. Knoll and J. Schwartz, “AC loss measurement with a phase difference between current and applied magnetic field,” IEEE

Trans. Appl. Supercond., 15 (2), 2831-2834 (2005)

[3] D. N. Nguyen, P. V. P. S. S. Sastry, G. M. Zhang, D. C. Knoll and J. Schwartz

“Experimental and numerical studies of the effect of phase difference between transport current and perpendicular applied magnetic field on total ac loss in Ag-sheathed (Bi, Pb)2Sr2,” J. Appl. Phys., 98 (073902), 1-6 (2005)

[4] G. M. Zhang, D. N. Nguyen, A. Mbaruku, P. V. P. S. S. Sastry and J. Schwartz, “Critical current and AC loss of Bi2Sr2Ca2Cu3O10/Ag tapes subjected to tensile stress,” IEEE

Trans. Appl. Supercond., 15 (2), 2835-2838 (2005)

[5] D. N. Nguyen, P. V. P. S.S. Sastry, D. C. Knoll and J. Schwartz, “Electromagnetic and calorimetric measurements of AC loss on YBCO coated conductor with Ni-aloy substrate,” Supercond. Sci. and Technol. 19, 1010-1017, (2006)

[6] D. N. Nguyen, P. V. P .S. S. Sastry, G. M. Zhang, D. C. Knoll and J. Schwartz, “Relationship Between Critical Current Density and Self-Field Losses of Ag-Sheathed (Bi,Pb)2Sr2Ca2Cu3Ox Superconducting Tapes,” Advances in Cryogenic Engineering

Materials 52, 869 (2006)

[7] D. N. Nguyen, P. V. P .S. S. Sastry and J. Schwartz, “Waveform of Loss Voltage in Ag-Sheathed Bi2223 Superconducting Tape Carrying AC Transport Current,” Advances in

Cryogenic Engineering Materials 52, 696 (2006)

[8] D. N. Nguyen, P. V. P. S. S. Sastry and J. Schwartz, “Numerical calculations of the total AC loss of Cu-stabilized YBCO coated conductors with a ferromagnetic substrate,” J.

Appl. Phys. 101, 053905 (2007)

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[9] G. M Zhang, D.C Knoll, D. N. Nguyen, P. V. P. S. S. Sastry, U. Trociewitz, X. Wang, and J. Schwartz, “Temperature Dependence of Critical Currents and AC Losses of (BiPb)2Sr2Ca2Cu3Ox and YBa2Cu3Ox Tapes,” Supercond. Sci. Technol. 20, 516-521 (2007)

[10] P. V. P. S. S Sastry, D. N. Nguyen, G. M. Zhang, D. C. Knoll, U. Trociewitz and J. Schwartz, “Variable temperature total AC Loss and stability characterization facility,” IEEE Trans. Appl. Supercond. (in press)