NBSIR 85-3027

Reference NBS

PUBLICATIONS

DEVELOPMENT OF STANDARDS FOR SUPERCONDUCTORS

Interim Report

January 1982— December 1983

L.F. Goodrich, J.V. Minervini, A. F. Clark, F.R. Fickett, J.W. Ekin, E.S. Pittman

Electromagnetic Technology Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

JANUARY 1985

-QC

100

, U56

35-3027

1935

Work performed under Department of Energy contract DE-AI01-76PR06010

NBSIR 85-3027

national bureau

OF STANDARDS

(Library

DEVELOPMENT OF STANDARDS FOR SUPERCONDUCTORS

Interim Report

January 1982—December 1983

L.F. Goodrich, J.V. Minervini, A.F. Clark, F.R. Fickett, J.W. Ekin, E.S. Pittman

Electromagnetic Technology Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

JANUARY 1985

Work performed under Department of Energy contract DE-AI01-76PR06010

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

1. INTRODUCTION 1

1.1 References 3

2. EVALUATION OF PRESENT STATUS 4

2.1 Summary of Major Accomplishments 1977-1984 5

2.2 Possible Future Research Topics 8

3. PREPARATION AND DISSEMINATION OF STANDARDS 10

3.1 ASTM B01.08 Subcommittee Minutes 11

a. Minutes from Knoxville, Tennessee (November 1982) .... 11

b. Minutes from San Francisco, California (April 1984) ... 12

3.2 Summary of Laboratory Visits 16

3.3 Summary of Standard Reference Material Development 18

3.4 Summary of Study to Assess Development of

Stability Standards 20

3.5 Summary of Critical Field Assessment 24

3.6 References 27

4. EXPERIMENTAL PROGRAM 28

4.1 Effect of Geometry on Current Transfer in

Critical Current Measurements 29

4.2 Lap-Joint Resistance of Multifilamentary

Superconductors 54

4.3 AC Losses 86

4.4 References 134

APPENDIX A - PUBLICATIONS AND PRESENTATIONS 136

APPENDIX B - EFFECT OF TWIST PITCH ON SHORT-SAMPLE V-I

CHARACTERISTICS OF MULT I FILAMENTARY

SUPERCONDUCTORS 139

APPENDIX C - THE EFFECT OF FIELD ORIENTATION ON CURRENT

TRANSFER IN MULTIFILAMENTARY SUPERCONDUCTORS 149

APPENDIX D - CRITICAL CURRENT MEASUREMENTS ON A NbTi

SUPERCONDUCTING WIRE STANDARD REFERENCE

MATERIAL 153

APPENDIX E - ANNOUNCEMENT OF SRM 1457 161

APPENDIX F - CERTIFICATE OF SRM 1457 162

iii

DEVELOPMENT OF STANDARDS FOR SUPERCONDUCTORS

Interim Report

January 1982 —December 1983

L. F. Goodrich, J. V. Minervini, A. F. Clark, F. R. Fickett,

J. W. Ekin, E. S. Pittman

Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado 80303

A cooperative program with the Department of Energy, the

National Bureau of Standards, and private industry is in progress to

develop standard measurement practices for use in large scale

applications of superconductivity. The goal is the adoption of

voluntary standards for the critical parameters and other

characterizations of practical superconductors. Progress for the

period January 1982 through December 1983 is reported. The major

effort was the procurement, selection, and certification of the first

superconducting wire for critical current measurements as a Standard

Reference Material (SRM 1457). Other work reported here includes:

effect of geometry on current transfer; lap-joint resistance; and ac losses

.

Key words: critical current; critical parameters; losses;

measurement methods; standards; superconductor.

1. INTRODUCTION

The development of standard measurement practices is essential to the success of any developing technology. In order to help ensure success in the

field of large scale applications of superconductivity the Department of

Energy and the National Bureau of Standards have undertaken a program to

establish a uniform terminology and reliable measurement techniques for the

many new aspects of superconductivity that are essential to good design. This

report is the fourth in a series summarizing the progress in this program,

which is jointly supported by the National Bureau of Standards and the Department of Energy through the Office of Fusion Energy and the Division of High

Energy Physics.

The first three reports [1.0-1, 1.0-2, 1.0-3] summarized the progress in

the first several years. The first report included: a large effort on the

standardization of terminology, a preliminary assessment of measurement

capabilities in the United States, the formation of an ASTM Subcommittee on Superconductors, some preliminary transient loss measurements, a comparison of

1
critical temperature measurements on practical superconducting materials, and

extensive research on the many factors that affect critical current measurements. The second report concentrated on the development of a standard method

for measuring critical current and included reports on contracts to the four

U.S. wire manufacturers for research on various aspects of this measurement.

The second report also included a survey of the state of the art of critical

current measurement (including a round robin and an assessment of the criteria

used), the final publication of the definition of terms, and the development

of a draft standard for the determination of critical current for superconductors with a critical current less than 600 amperes. The major effort described

in the third report was the development of a standard test method for critical

current, the necessary back-up research, and the coordination of the adoption

of the test method and standard terminology through the subcommittee level in

ASTM. The two ASTM standards on superconductors [1.0-4, 1.0-5] were approved

in December 1982.

This fourth report covers the period January 1982 through December 1983.

It includes an evaluation of the present status (section 2), a summary of the

preparation and dissemination of standards (section 3), three previously

unpublished papers reporting work performed on the experimental program, and

appendices including: papers published during this reporting period, a publication list, and copies of the announcement and certificate for SRM 1457.

2
1.1 References

[1.0-1] F. R. Fickett and A. F. Clark, "Development of Standards for Superconductors, Annual Report FY 79," NBSIR 80-1629, National Bureau of

Standards, Boulder, Colorado (Dec. 1979).

[1.0-2] F. R. Fickett, L. F. Goodrich, and A. F. Clark, "Development of

Standards for Superconductors: Annual Report FY 80," NBSIR 80-1642

National Bureau of Standards, Boulder, Colorado (Dec. 1980).

[1.0-3] A. F. Clark, L. F. Goodrich, F. R. Fickett, and J. V. Minervini,

"Development of Standards for Superconductors, Interim Report, Oct.

1980 to Jan. 1982," NBSIR 82-1678, National Bureau of Standards,

Boulder, Colorado (July 1982).

[1.0-4] Standard Definitions of Terms Relating to Superconductors, Annual

Book of ASTM Standards, ASTM B713-82, Part 2.03, pp. 591-594,

American Society for Testing and Materials, Philadelphia,

Pennsylvania (1983).

[1.0-5] Standard Test Method for D-C Critical Current of Composite Supercon

ductors. Annual Book of ASTM Standards, ASTM B7 14-82, Part 2.03, pp

595-598, American Society for Testing and Materials, Philadelphia,

Pennsylvania (1983).

3
2. EVALUATION OF PRESENT STATUS

It is important in an interactive research program such as this to

continually assess the results with respect to the direction of future efforts.

In this section we will summarize the major accomplishments and outline the

possible future research topics. Due to the timing of this report, the

accomplishments through 1984 are listed, and the list of possible future

research topics is all encompassing to stimulate future efforts.

4
2.1 Summary of Major Accomplishments 1977-1984

I. Superconductivity

* Survey, workshops, and continuous interactions performed with manufacturers and users on measurement practices and problems for superconducting parameters. Determined the priorities and direction of the

research and standards effort.

* Formed ASTM Subcommittee on Superconductors B01.08 as forum for

impartial discussion of measurement problems. Involved many as 30

active participants representing all U.S. and one European wire

manufacturer and all DoE and DoD laboratories.

* Standard definition of terms for superconductivity adopted by ASTM

after extensive review, four open literature papers containing proposed

definitions, and the ASTM voting procedure.

* Published more than 30 publications presenting research results in the

open literature to the superconductivity community.

II. Critical Current

A. Small conductors (<600 A)

* Round robin test organized to assess the initial state-of-the-art

measurement techniques. Critical evaluation done of test methods

and needs.

* Extensive research assessment completed on 34 measurement parameters that affect the accurate determination of critical current.

Results published in open literature.

5
* The first user, producers, and third party consensus ASTM standard

test method for the measurement of critical current was adopted.

* First standard reference material (SRM) for the determination of

critical current completely characterized and available for

purchase

.

* Current transfer identified as major parameter affecting critical

current measurements and joint behavior. Experimental characterization and theoretical model completed.

B. Large conductors (>600 A)

* Parameters evaluated to identify those that affect large conductors. New or scaled up problems of major concern in research

now.

* Survey and interactions performed with the National Laboratories

on present methods and problem areas.

III. AC Losses

* Performed comparison of calorimetric, lock-in amplifier, and

digital oscilloscope techniques. Literature reported factors of

10 difference reduced to 2 or 3.

* Developed digital multiplication technique for use with ac current and

ac fields.

* Developed theoretical model for both ac current and ac field.

IV. Critical Field

* Assessment and critical evaluation done of test methods and needs.

6
V. Stability

* Assessment and critical evaluation done of test methods and needs.

* Parameters in I versus T studied on SRM.

c

7
2.2 Possible Future Research Topics

I. Critical Current

A. Large Conductor (600-6,000 A)

* Completion of parameter assessment on monolithic conductors.

* Large conductor SRM (ten times the small conductor critical

current)

.

* Cable conductor measurement parameters.

* Set up of a critical current measurement service.

* Small lab test method.

B. Continuous Testing

* Moving current and voltage contacts and mechanical logistics.

* Standard test method.

C. Electromechanical Properties

* Bending and biaxial strain effects on critical current measurement.

* Standard test method for strain dependence of critical current.

* Strain effects on present SRM.

II, AC Losses

* Standard method development.

* Development of theoretical model of AC loss.

* ASTM coordination.

* Assessment of SRM needs.

8
III. Stability

* Standard method development.

* Disturbance spectra determination, sources and characteristics.

* Joint resistance measurements.

* ASTM coordination.

IV. Critical temperature, critical field

* Prove extrapolation techniques.

* Provide experimental guidelines.

V. Nonsuperconducting Magnet Materials

* Residual resistance ratio method in presence of superconductors.

* Very low level magnetic susceptibility.

* Internally supported conductors, effect on J

c measurements.

VI. Interactions

* Workshops with users and producers.

* Laboratory visits.

* ASTM coordination.

An essential component to the development of a research program such as this

is input from producers, users, and researchers working with practical superconductors. Feedback about the relative importance of the topics listed above

or other areas that need research is solicited and always welcome.

9
3. PREPARATION AND DISSEMINATION OF STANDARDS

The final goal of this program is the adoption of voluntary standards for

practical superconductors. The first two standards, definition of terms and

small conductor critical current, were adopted in December 1982. The next

steps were to start research on a large conductor critical current test and to

develop a Standard Reference Material (SRM) for the measurement of critical

current. The SRM is now available through the NBS Office of Standard Reference

Materials as "SRM-1457 Superconducting Critical Current - NbTi Wire" (See

Appendices E and F) . The minutes of two ASTM meetings held during this

reporting period are given in section 3.1. The major problem areas where

research is needed were identified at those meetings. Further discussions and

observations were made at conferences and laboratory visits. A list of visits

is given in section 3.2. A summary of the selection and certification of

SRM 1457 is given in section 3.3. Also included as sections 3.4 and 3.5 are summaries of the evaluation of the measurement state of the art and an assessment of measurement needs of two possible research areas: stability and

critical field.

10
3.1 ASTM B01.08 Subcommittee Minutes

by A. F. Clark

3.1 a. Minutes from Knoxville, Tennessee (November 1982)

ASTM B01.08 Subcommittee on Superconductors

November 29, 1982 7:30 pm

Andrew Jackson Room, Knoxville Hyatt Regency

A meeting of the ASTM B01.08 Subcommittee on Superconductors was held in

conjunction with the 1982 Applied Superconductivity Conference on November 29,

1982 in Knoxville, Tennessee. The purpose of the meeting was primarily to assess the problem areas for a standard for critical current measurements on high-current (up to 20 kA) superconductors.

The chairman of the subcommittee, A. F. Clark, opened the meeting with a brief discussion of the progress in the DoE-NBS program for the development of

standards for superconductors. Two draft standards, definitions of terms, and

small conductor critical current, are presently out for ASTM society ballot.

Research is underway on large conductor measurements and ac losses, and assess

ments are in progress on critical field and stability. Except for a short

discussion on continuous testing, the meeting was devoted to presentations and

discussions on the problems of large conductor critical current testing.

The discussion was initiated by informal presentations of methods of

measurement now in use in their laboratories by John Miller of Oak Ridge

National Laboratory, A1 Mclnturff of Fermilab, and Bill Sampson of Brookhaven

National Laboratory. Others contributing their experience were Karl Best of

Vacuumschmelze, L. Intichar of Siemens, and Yuki Iwasa of Francis Bitter

National Magnet Laboratory. John Miller listed several problem areas that

proved to be common among the rest and included expense, sample mounting and

orientation, current transfer resistance, criteria, self field, and cable vs. strand measurements. A1 Mclnturff pointed out some of the problems that arise

if production measurements are needed, and also the role of the power supply

as a noise source in the voltage signals. Bill Sampson described an apparatus

that used the distinct advantages of a dipole magnet and methods of data

processing. A discussion ensued addressing such things as scaling experiments

11
common test facilities, methods for small labs, the inclusion of Nb^Sn conductors, and other methods. The question of whether a standard was even needed,

and if so, what form it should take was raised by the chairman.

In summary, the subcommittee felt that a standard was definitely needed.

The decision had the full support of the wire manufacturers and national labs

that were represented. It was felt that it could start as a set of guidelines,

but that would only be a step on the way to an ASTM standard. Major problem

areas for research at NBS were identified as a) the expense of the test in

money, time, and liquid helium, b) sample mounting techniques to contain the

large stresses, c) power supply noise and alternate current sources, d)

voltage signal processing including noise pickup, and e) current transfer. It was also felt that Nb^Sn and all other developmental conductors should be

included, and that one central test facility could probably not be responsive

to all the needs.

In a final discussion, the need and desirability of a method for continuous testing of a length of wire or superconductor critical current was addressed. It was felt that this could be done, but that an assessment should

first be made that would include a survey of potential users.

All in all, it was a fruitful exchange of information, a lively discussion,

and generated some helpful guidance for future work.

3.1 b. Minutes from San Francisco, California (April 1984)

ASTM B01.08 Subcommittee on Superconductors

April 5, 1984 1:00 pm

Ramada Hotel, San Francisco, California

Chairman, A. F. Clark, called the meeting to order at 1:00 p.m. and

presented the agenda, indicating that we would first evaluate the presently

adopted standards, discuss the recent development of a standard reference

material by NBS, hear an NBS research report on present progress and

12
measurement techniques, and finally, discuss future tasks. In attendance were

seven members plus three guests.

The two present standards, B713-82 Definitions and B714-82 Critical

Current, were brought up for discussion as to their applicability and usefulness. There were no comments on the Definitions. It was felt that people

used them without question. On the standard measurement technique, several

comments indicated a need for publicity and the need for soliciting feedback

on its usefulness. Suggestions to accomplish this were: a short article in

Cryogenics or a direct solicitation to wire manufacturers including both

Europe and Japan. As far as the measurement technique, the value of including

the bending strain was indicated, as well as a more varied range of electric

field criteria. The question of the quench limit in NbTi and Nb^Sn were

compared as a justification for the need of a wider ranging electric field.

Other than these criticisms it was felt that the standard measurement technique was useful, but needed to be applied more often. Also that a standard

measurement needed to be extended to higher critical currents and we must, for

that, add such things as aspect ratio, self field, and current transfer

considerations

.

Loren Goodrich presented the present status of the standard reference

material being developed by NBS. It will be available in less than a month at

a price of $219 for 2.2 meters of well-characterized, round, NbTi multifilamentary conductor. Questions were raised as to whether it was too short, the answer being no, and, are all the data available for the user. Loren indicated

that a complete report is available and will probably be sent out with each

SRM. The SRM will be publicized in the ASTM Standardization News and through

a general mailing to the ICMC and Applied Superconductivity mailing lists. It was felt this should be adequate exposure.

Loren Goodrich and Alan Clark of NBS then gave a research report on the

development of large conductor critical current tests and ac losses. Briefly,

NBS is exploring aspect ratio, self field, and current transfer effects in

conductor measurements up to 3,000 amperes, and using three or four ac loss

measurement techniques to compare measurements on similar conductors. In the

former a test technique should be available soon, and in the latter there are

13
extraordinary difficulties in getting measurements to agree. The committee

felt that current transfer and large conductors is a major problem and that

the extension to 6,000 amperes or more should come fairly quickly. The upper

limit for large conductors should probably be on the order of 10-12,000 amperes.

Whether an SRM should be created was discussed and before this were to take

place, the committee felt that a survey of needs should be performed by NBS,

which could also include a round robin testing of large conductors by those

who can make such measurements. The only laboratories with this measurement

capability are Oak Ridge, General Dynamics, Lawrence Livermore, and the M.I.T.

Magnet Laboratory. The General Dynamics representative indicated they were

setting up large conductor measurements because the wire manufacturers were

showing differences of 40% on the same wire. The possibility of measuring

forced flow conductors was raised, but the committee felt that because the

currents were too high and the flow characteristics were too variable, the

development of this kind of measurement was probably premature and that it is more important to go into complex geometries (i.e., cables, etc.), higher

currents, and pursue the fundamental understanding of these measurements

before the development of a test technique. As for ac losses, the committee

felt that this measurement problem should be ranked second only behind large

conductor critical current. The factors of 10 are still present in many

measurements and form a big unknown for design parameters.

Finally, the areas of stability, critical field, and other measurements

were raised. For stability the variability of magnet design is still wide

ranging and still there, but the committee felt that it would be worth a try

to develop a standard measure of this difficult parameter. This is principally because some magnet designs are getting used more and more and if some intrinsic properties could be identified for stability, this would be an immense help in magnet design. A principal point of research would be the

influence of the disturbance spectrum on these measurements. As to critical

field measurements, it was felt that H _ was not a commercial parameter, but

c l

that the J^-H curve is a very definite need. Thus, the cookbook extrapolation

for may be adequate, but this would have to be proved by further research.

Of much more importance is the absolute field measurement. Presently, Hall

probes, search coils and NMR probes are used to try and develop the precision

14
to 8 T that is needed in magnet design (i.e., 1 part in 10 5

) . Most people use

some sort of local in-house calibration and ancient techniques in these

measurements. Another real need for the residual resistance ratio (RRR) was

raised when these measurements are in the presence of a superconductor. The

measurements of RRR are okay when they are done on the bulk copper, but there

is a strong need to define a RRR in the presence of a superconductor and

develop a technique for its measurement.

On Friday, April 6th, the subcommittee met with the full B1 Committee on Conductors, about 35 people, to give a report on the subcommittee meeting and

on the progress of various tasks. In general, the main committee gave full

support in the difficult task in the development of standards in a new field

and offered their full support wherever possible, in particular, with properties of the copper stabilizer.

15
3.2 Summary of Laboratory Visits

Name

A. F. Clark

and F. R. Fickett

A. F. Clark and

L. F. Goodrich

A. F. Clark*

F. R. Fickett*

A. F. Clark*

A. F. Clark*

* Travel funded by

Date

3/1-3/82

4/4-8/82

Location

Lawrence Livermore

National Laboratory;

Univ. of California

at Berkeley; Stanford

Linear Accelerator

Center

Westinghouse (Pittsburgh) ; Oak Ridge

National Laboratory

Richmond, Virginia

5/8-22/82 Kobe, Japan

9/27-10/4/82

9/29-10/3/82

Japanese National

Railway Test Center;

University of Kyushu;

Tohoku University and

Research Institute for

Iron, Steel, and Other

Metals; National

Research Institute of

Metals (NRIM); Electrotechnical Laboratory

(ETL) ; High Energy

Physics Laboratory (KEK)

Furukawa Electric Company, Central Research

Laboratory

National Magnet Laboratory, Boston, MA

Stanford University

and Stanford Linear

Accelerator Center

11/29-12/3/82 Knoxville, Tennessee

NBS

Subject

Discuss contract

and proposed

research

Discuss measurement programs and

large conductor

critical current

measurements

Attend ASTM B1

Committee meeting

Attend and chair

a session at the

joint ICEC9-ICMC

Obtain the most

recent developments

of materials and

measurements in

Japan

Obtain critical

field measurements

Discuss and

observe superconductor measurements

Attend Applied

Superconductivity

Conference; chair

ASTM meeting;

attend Cryogenics

Editorial Board

meeting

16
Name Date

L. F. Goodrich

A. F. Clark*

L. F. Goodrich

A. F. Clark*

A. F. Clark*

* Travel funded

Location Subj ect

11/29-12/3/82 Knoxville, Tennessee Attend and present

a paper at the

Applied Superconductivity Conference and attend

ASTM meeting

5/15-19/83 Massachusetts Institute of Technology;

Magnetic Corporation

of America (Boston,

Massachusetts)

Discuss superconductor measurements

8/15-19/83 Colorado Springs,

Colorado

Attend and present

paper at ICMC

8/21-23/83 Copper Mountain,

Colorado

Cochair workshop

on Problems in

Superconductivity

8/28-9/16/83 Paris, France;

Grenoble, France;

Attend HR and MT8

conferences and

present paper

Karlsruhe, FRG;

Oxford, England

Southampton, England

Perform superconductor measurements and research

by NBS

17
3.3 Summary of Standard Reference Material Development

by L. F. Goodrich

This is a summary of the work and the report [3.3-1] that led to the

first superconducting wire for a critical current measurement Standard Reference Material (SRM 1457). The report reviews the selection and certification

of the SRM. The SRM is intended to provide a means for checking the performance of measurement systems used in the commerce and technology of superconductors. SRM 1457 should prove valuable in determining the overall accuracy of a critical current measurement system that is dependent on numerous variables

and effects that can make this seemingly easy measurement very difficult. SRM

1457 may be purchased for $219 from the Office of Standard Reference Materials,

Room B311, Chemistry Building, National Bureau of Standards, Washington, DC

20234.

An effort was made to keep the use of this SRM as unrestricted as possible.

The precautions listed on the certificate, together with the American Society

for Testing and Materials (ASTM) Standard Test Method (B714-82), are sufficient

for any valid user measurement technique. Some deviations in testing technique

from the method on which the certification was based were accommodated by in

creasing the total uncertainty in the certified critical current. The deviations that are allowed, and the ones that are not allowed, are identified in

the precautions sections of the certificate.

The critical current was measured using a 2 cm voltage tap separation on

a coil specimen holder with a 3.18 cm diameter (specimen bend diameter of about

3.23 cm) and a two turn per centimeter pitch length. The general technique

used to measure the critical current was the ASTM standard test method B7 14-82

[3.3-2]; however the variables were held to much tighter limits in order to

get a lower total uncertainty including the sample inhomogeneity. The acquisition and analysis of the raw data were carefully developed to measure the criti

cal current with accuracy and precision. The limits to systematic and random

errors of the principal variables that effect the critical current (current,

electric field, magnetic field, temperature and strain) were estimated. Other

concomitant variables and effects considered in the development and certification of this SRM were: voltage filtering and response time; current ramp

18
rate; liquid helium hydrostatic head and stratification; inductive and thermoelectric voltage; magnetic field of the specimen coil; and winding tension.

Preliminary screening measurements were performed on each of five candidate SRM samples. Two of these conductors displayed short- or long-range

inhomogeneity that made them seem unfit for use as an SRM. One of the remaining conductors was eliminated becaused the length delivered was considered too

short. One of the two candidates left had the lowest copper-to-superconductor

ratio, so it was eliminated in favor of the other. Further measurements and

tests were made on the conductor chosen to be the SRM.

The conductor designated as SRM 1457 was wound onto 500 distribution

spools, each with approximately 2.2 m of wire. Nine of these spools were

selected at nearly equal distances along the whole length of wire, including

the spools at each end. Critical current measurements on the sample spools were obtained for 36 combinations of three factors affecting critical current:

magnetic field, temperature, and electric field.

Although there were no obvious trends along the length of wire, substantial variation in critical current was evident, especially at lower magnetic

fields. These variations were associated with material variability, or inhomogeneity, of the wire, and were incorporated into a statistical model

that was derived from an empirical equation for the dependence of critical

current on temperature and electric field at any given magnetic field.

SRM 1457 has been certified at magnetic fields of 2, 4, 6, and 8 T for

temperatures from 3.90 to 4.24 K and electric field criteria from 0.05 to

0.2 yV/cm. Because material variability could not be ignored, the uncertainty

in the certified values of critical current (excluding systematic errors) is a statistical tolerance interval. The resulting tolerance limits allow for

inhomo gene it}7 by estimating limits for the critical current of individual

spools, rather than limits on the average critical current of all spools.

The statistical tolerance limits and estimated systematic error have been

combined to give a total uncertainty on the certified values. The final

estimated uncertainties are no greater than 2.57% of the reported critical

current at any of the four magnetic fields.

19
3.4 Summary of Study to Assess Development of Stability Standards

by J. W. Ekin

A study of the various types of superconductor stability has been made

along with a survey of experimental methods currently used to measure stability. The potential for developing stability measurement standards was assessed.

This is a summary of the main conclusions of that study. A more detailed

report is under preparation.

STABILITY TYPES

A major conclusion of the study is that there are only two fundamentally

different types of stability: flux-jump stabilization and cryogenic stabilization. There are many names for cryogenic stabilization corresponding to

different assumptions used in solving the power-balance equation that is the

heart of this type of analysis. They are all solutions to the same general

equation under different simplifying conditions. This is shown in detail for

the so-called Stekly criteria, the Cold-End Recovery criterion, the Cold-End

Minimum-Propagation-Zone criterion, the Critical-Current Margin criterion, and

the Spherical and Elliptical Minimum-Propagating-Zone criteria.

STABILITY STANDARDS

From the standpoint of standards, two questions need to be answered

affirmatively before a stability standard can be considered: 1) Is there a meaningful measure of stability; and 2) Is a stability measurement likely to

be repeated often in the utilization of superconductors (i.e., does anyone

care?)

.

For flux-jump stabilization, the answer to the first question is yes but

the answer to the second question in most cases is no. There usually is not

much need to test for flux jumps — they are taken care of through proper

sizing of superconductor filaments.

For cryostabilization, on the other hand, both questions can be answered

affirmatively, as discussed in the following two sections.

20
A PROPOSED PRACTICAL MEASURE OF CRYOSTABILITY

For each of the cryostability criteria the study shows that the expression for obtaining cryostability contains the relative current density,

j(E J/J ) , as a parameter. Cryostability can always be obtained by operating

c at a low enough j . This current stability limit can be measured for any

system. Thus, although there is no consensus on how stability should be

measured, at least one meaningful measure of cryostability exists: the

current stability limit. Also, it is an engineering parameter which is needed

in most large magnet designs. For many large magnet designs it is as important as, if not more important, than the superconductor critical current.

Because its measurement is not easy to do correctly, there probably is an eventual need for a standard procedure to measure it.

MEASUREMENT OF CURRENT STABILITY LIMIT

Two basic experimental methods are possible for determining such a current stability limit prior to the construction of the final winding.

Basically they go hand in hand. The first method is to determine the parameters needed if cryostability has been achieved, particularly the cooling rate

which is quite variable. It depends strongly on the surface condition,

cooling channel geometry, and orientation of each conductor relative to

vertical. The second method is to construct a section of winding which

simulates as closely as possible the projected cooling conditions of the final

winding structure. The current stability limit is then measured as the

steady-state conductor current when thermal runaway is initiated by a heater

embedded in the winding. The heater drives a portion of the experimental

winding into the normal state.

The measurement is critically dependent on simulating the cooling conditions to be actually encountered in the final magnet, and therefore any

standard measurement procedure would have to be adaptable to individual magnet

cooling conditions. As such it is a measurement standard that probably would

have to be specified or performed by the magnet manufacturer, as opposed to

the conductor manufacturer.

One exception to this would be superconductors with complicated surface

geometries, such as cable conductors, internally cooled conductors, or conductors with augmented surfaces (such as cut surfaces or sintered coatings). It

such superconductors it would be useful to have a measurement procedure for

21
determining the surface cooling rate for an individual conductor. Such a measurement could be performed by the conductor manufacturer and might well be

useful in a calculation of the current stability limit, or at least as a figure of merit in conductor selection.

SUMMARY

To summarize, stability measurements are still in the evolutionary stage,

but at least there appear to be methods for measurement of the current stability limit. The current stability limit is, furthermore, a critical parameter

that is variable enough that there is a need for its measurement as a common engineering parameter, at least in cryostable magnet design. It is a measurement performed at present primarily by magnet designers, but surface cooling

rate data (under pool-boiling conditions) for certain conductors could become

a common critical parameter supplied by the conductor manufacturers.

A key question for advanced stability considerations is the quantification of the internal disturbance spectrum. As long as this question is not

solved, potentially powerful measurements for achieving limited cryostability

remain unfortunately of only limited significance. In addition to studying

trends, such as scaling of disturbance magnitude with magnet size and field

level, future work should possibly be concentrated more strongly on disturbance spectra, particularly information on detailed stress distributions and

microscopic energy release events in common magnet structures.

CONCLUSIONS OF THE STUDY

1) There is no need for a flux-jump stability standard.

- reliability has been achieved by making the filament and strand size

small

- not of engineering concern

2) There is a strong engineering need for a cryostability standard.

- first need to answer some basic questions about measurement specification (heater geometry, cooling geometry, heat pulse duration, etc.)

- magnet designs also need some settling time

3) It is suggested that a first practical engineering parameter for

determining cryostability is the current stability limit. Proposed

symbol "I .

22
4 ) I is already measured in some form or another for most cryostable magnet

designs because of its dependence on:

- conductor surface condition

- internal conductor configuration (thermal conductivity, heat capacity)

- cooling channel geometry

- conductor orientation

- disturbance spectrum

5) Measurement of I is performed at present primarily by magnet designers,

not conductor manufacturers.

6) There are two methods for determining I a priori:

s a) Calculate from measured quantities, such as surface cooling rate,

using formulas given in papers.

b) Measure directly using a small coil with heaters.

7) Surface cooling rate data could become a common parameter supplied by

conductor manufacturers for certain superconductors.

- conductors with augmented surfaces (fins, sintered coatings)

- cabled conductors

- internally cooled conductors (cooling defined by conductor design, not

magnet design)

- augmented internal heat capacity

8) Key question for improved performance — WHAT IS THE DISTURBANCE

SPECTRUM?

23
3.5 Summary of Critical Field Assessment

by F. R. Fickett

This is a summary of a paper entitled "Standards for Measurement of the

Critical Fields of Superconductors," which has been prepared by F. R. Fickett

and is to be published in the March-April 1985 issue of the NBS Journal of

Research.

In the paper, the origins, definitions, and measurement of the various

critical magnetic fields associated with superconductors are reviewed. The

potential need for a consensus standard for the measurement of these fields is

evaluated. Measurement techniques, as practiced both in industry and in the

national laboratories, and extrapolation techniques commonly used to determine

the upper critical fields of the newer materials are presented. Sources of error in the experimental determination of critical fields are assessed for

the various common techniques. A comprehensive bibliography of the modern

literature on critical field measurement and interpretation is included in the

publication. Here we briefly summarize the conclusions reached in that paper.

The purpose of the investigation was to evaluate the desirability and

feasibility of creating a standard for measurement of the critical field of

practical superconducting materials. The type of standard under consideration

is that typically produced by organizations such as ASTM to assist in commerce. Two standards already exist in the field of superconductivity, one for general

definitions and the other for the measurement of critical currents below

600 A. Such standards are created by consensus among all interested parties

and must be able to be used by industrial laboratories in their day-to-day

operation. A further consideration is that there should be a demonstrated

need for the standard, at least in the foreseeable future.

A standard of this type can take several forms. It may be any of the

following: a list of definitions; a manual outlining accepted measuring and

reporting methods; a detailed method of measurement in which apparatus,

technique, and report format are specified; or an artifact or standard reference material (SRM). Whatever the form, it is essential that the standard be

backed up with adequate research to document the need for each requirement of

the standard. This is not a trivial problem, and it is often neglected in the

rush to create a standard to solve a particular problem. Our feelings regarding the need for and structure of a H^

standard are given below. In summary.

24
it seems that the time is not yet ripe for a full-fledged standard, but there

is some justification for creating a list of standard definitions and, perhaps,

an "operation manual" or similar document for the determination of upper

critical field. A standard reference material approach might also prove

useful but would be quite expensive.

Standards of the sort discussed here are usually created in response to a need expressed by the community. In the case of critical field, there has

been a limited expression of need. The commercial materials now in use are,

in general, adequately characterized by their critical current versus field

characteristic. Critical field information is of most use to that group of

researchers who are trying to construct better practical materials for high

field applications by modification of the crystallographic or electronic

structure of various existing materials. This group should agree among

themselves on the requirements for an acceptable measurement of the critical

field, but that is not adequate reason for creating a standard. It is entirely

conceivable that very high field materials may become feasible in the future,

and the need could become great for a critical field standard for commercial

versions of those superconductors. We do feel that a few definitions related

to the critical field measurement should be added to the general definitions

standard, mostlv the various modifications of H „ now in common use. Furthered

more, it is possible that certain groups, such as DoE, might want to specify a critical field measurement method and data analysis technique for a particular

material. This could be done, but it would require that some of the research

mentioned below be performed first if the document were to have very wide

application.

Our investigation indicates that the creation of a detailed single

measurement standard for critical field is probably impossible given the

current state of knowledge regarding the factors that influence H and the

inhomogeneous nature of the superconducting portion of the practical conduct r composite. However, if an attempt were to be made, there are a few items that

should be considered. The only measurement technique that is likely to be

widely used in industry is the resistive method applied at 4.2 K. A clever

application of an inductive technique might also be possible, but none has

appeared to date. Similarly, the possibility of routine direct measuremer.c : ^ remote because of the expense of high field magnets. Thus, ext rape 1 at i or. techniques would have to be used and most likely those would be erh:

25
current versus field extrapolations with the measurements made at 4.2 K.

Extensive research on the pinning force phenomenon would be necessary.

Significant advances in understanding the effect of crystallographic and

metallurgical variations on the critical field would be required. All these

requirements could be mitigated somewhat by the use of a standard reference

material, but research on measurement methods and their related errors would

still be needed. A standard method should use a relatively high current

density, probably in excess of 100 A/mm2

, so as to avoid some potential

problems

.

The SRM approach to the standardization of critical field measurements is

probably the most appealing for the present circumstances. Unfortunately, it

is also a very expensive solution. The idea is to make a series of very well

characterized materials that could then be distributed for the calibration of

apparatus. Such materials could also be used to evaluate the various extrapolation techniques. The characterization would require very careful work,

expertise in several measurement and analysis techniques, access to high field

magnets, and a consensus as to the proper choices for the important parameters.

However, considerable progress is now being made in understanding the interactions between the metallurgy and the superconducting properties of these

materials which may well result in an advanced (practical) superconductor with

well-documented homogeneity and internal structure in the near future. This

conductor, if it can be made in significant quantities, would be an ideal

candidate for a critical field SRM.

26
3.6 References

[3.3-1] L. F. Goodrich, D. F. Vecchia, E. S. Pittman, J. W. Ekin, and A. F.

Clark, "Critical Current Measurements on an NbTi Superconducting Wire

Standard Reference Material," NBS Special Publication 260-91,

National Bureau of Standards, Boulder, Colorado (1984).

[3.3-2] Standard Test Method for D-C Critical Current of Composite Superconductors, Annual Book of ASTM Standards, ASTM B7 14-82, Part 2.03, pp.

595-598, American Society for Testing and Materials, Philadelphia,

Pennsylvania (1983).

27
4 . EXPERIMENTAL PROGRAM

This section contains three unpublished papers on work performed in the

experimental program. The first, section 4.1, describes how the critical

current measurement geometry will affect the observed current transfer characteristic. Current transfer voltages are ever present, although at different

magnitudes. These voltages can interfere with the determination of critical

current even for small NbTi conductors and it will be more of a problem for

large conductors. In the second paper, section 4.2, the measurement ana modelling of lap-joint resistance between multifilamentary superconductors is

presented. The results can be used to estimate the lap-joint resistance. An

unexpected result was the dominance of the joint interface resistance. The

third paper, section 4.3, is on the principles of ac loss theory and the

methods by which the losses can be experimentally measured. The general focus

is to develop an understanding of the loss mechanisms within an individual

superconducting filament.

28
4.1 Effect of Geometry on Current Transfer in Critical Current Measurements

by L. F. Goodrich

The experimental results and discussion of a program to evaluate current

transfer in multifilamentary superconductors are presented with comparisons to

existing theories. Specific areas of current transfer covered are: magnitude

and current dependence of the current-transfer electric field as a function of

distance away from the current contact in both parallel and perpendicular

magnetic fields; dependence on current contact length; variation in a region

of changing magnetic field angle or magnitude; symmetry of current transfer;

response to multiple parallel and perpendicular magnetic field regions.

INTRODUCTION

The voltage that results from current transfer in multifilamentary

superconductors can interfere with the determination of critical current. The

experimental results, comparisons to existing theories, and discussion presented here summarize the information gathered on the source, magnitude, and

shape of the transfer characteristic for various sample measurement configurations .Current transfer results from a number of sources, namely; joints,

inhomogeneities (change in local superconducting properties), strain, self

field, and changes in the magnitude and angle of the magnetic field. For each

of these, the current distribution among the superconducting filaments is

locally disturbed from equilibrium. This non-equilibrium distribution of

current causes spatial, flux-flow voltage gradients among the filaments and

resistive voltages from the normal metal. These spatial voltage gradients

drive some of the current through the interfilament material (normal metal) in

the process of attaining a new equilibrium distribution. The result is a redistribution of the current among the superconducting filaments over a length of wire that causes a voltage drop along the wire. This voltage drop

is referred to as the current-transfer voltage, V , and the corresponding

electric field, E^. The current-transfer voltage will be a function of both

current and position along the wire due to each source. The physical extent

of current transfer depends on the magnitude of the redistribution (largest near a joint), the resistivity of the interfilament material (the resistivity

29
of bronze is much greater than that of copper), the diameter of the wire, and

the sharpness of the superconductor’s flux-flow resistivity transition.

Current transfer exists in commercial multifilamentary superconductors.

The most direct evidence of this phenomenon is the electric field, E, as a function of current, I, and distance from the current contact, x. A plot of

this is given in figure 4.1-1. The lower current portion of the curves change

significantly with distance from the current contact. This indicates that the

lower current portion is not intrinsic to the sample and the position dependence indicates that its source is the current contact (joint). The joint is

the predominant source of current transfer exhibited here because of the

magnitude of the current redistribution and the measurement conditions, which

minimized the effect of the other sources. Some of the other sources of

current transfer are discussed below.

The theory of current transfer in multifilamentary superconductors has

been the subject of several papers [4. 1-1-3]. These theories deal only with

transfer adjacent to a current contact, which is where the effect is largest.

Each of these theories assumes that the intrinsic resistivity of the superconducting filaments, p

g

, is given by

P s = P c (f>" (4.1-1)

c

where is the critical current density, P c is the corresponding resistivity

criterion (in general these would be reference values, but for convenience let

them be the critical values) , J is the current density and n is a constant for

the superconductor (Ref. [4.1-3] uses m-1 instead of n) . The value of n is

determined from the shape of the flux-flow resistivity transition near the

critical current, and this equation is a reasonable representation of the

shape [4.1-4]. Typical values of n range from 10 to 60. Notice that this

assumption implies that the superconductor has a finite resistivity at any

non-zero J, thus current transfer will take place among the superconducting

filaments at every current. One result of these theories is an approximate

expression (for large n and within geometric factors) for the current transfer

resistivity, p^. The expression in Ref. [4.1-2] is

30
V/

cm

I , A

Figure 4.1-1. The electric field as a function of current for various distances from the current contact, x, at 4 T.

31
p = (0. l/n)p (d/x) (4.1-2) 2

, t m

and Ref. [4.1-3]’s expression is

p = (0. 25/ (n+2)) p (d/x)

t m

(4.1-3)

where n is defined by eq (4.1-1), p is the transverse resistivity of the m wire’s filament region, d is the diameter of the wire's filament region, and x

is the distance from the current contact. If the only variable in these

expressions is x, then the current transfer portion (low current) of the

voltage-current (V-I) curve should be linear. Another result is the introduction of a current transfer length, x^, in Ref. [4.1-2] which is a specific

value of x in eqs (4.1-2) and (4.1-3) where p is equal to a chosen value of

resistivity, p*. Usually p* is of the order of the critical current criterion.

Thus, x is a measure of the distance from the current contact at which

t -12

P

t < p*. For example, with p* = 10 fl*cm, x would be approximately 100 d

for a Nb^Sn superconductor and 8 d for a NbTi superconductor.

All of the current transfer data presented here were taken on a commercial multifilamentary superconductor based on niobium-tin (Nb^Sn) . A photograph of the cross section is given in figure 4.1-2. The wire has a diameter

of; 0.70 mm. It has an outer copper jacket separated by a tantalum diffusion

barrier from the core of bronze, Nb, and Nb^Sn. The diameter, d, of the

filament region enclosed by the barrier, is 0.40 mm. The wire was originally

not twisted and unreacted, thus it could be either twisted or left untwisted

and formed to shape prior to its reaction. Critical current data on this wire

are given in table 4.1-1 for various critical current criteria.

Table 4.1-1. Sample critical current data.

I , A for various criteria

c V- T n E = 1 yV/cm E = 1 nV/cm p c c c -12

= 10 ft’cm

4 15.5 ± 1 275 181 232

8 12.4 ± 2 124 74 97

32
r »*v>

v.y.v, ;.V 4

J.*'. ».» V‘ /*'v.*y

- ^

tip *' • 'U rZ::~ ••*•>. *&•>,

• v * V/** *•*•• A . V«V #V/# «*»

; ='1: - ••••'• •',:•• V-.v

!• *‘'i' -'»*»• #

»* •'• . v.v •••* V

yjn^jv.* 'V V:

>.&" !\:V; tt* v.;.

• *•*»< »«. 1

Sk *V.v

v

I iHfl# A*

*

'i*ae>v

Figure 4.1-2. A cross-sectional photograph of the Nb^Sn multif ilamentary

superconductor sample.

33
The rest of the paper is divided into five major sections presenting the

results from particular measurement configurations representing the sample

shape: coil, long straight, hairpin, short straight, and "nested" geometries.

Each of these sections details different aspects of current transfer as exhibited by the geometry. The first four geometries are commonly used in

critical current measurements and are illustrated on figure 4.1-3. The

"nested" geometry is a unique one used to study current transfer symmetry and

current transfer response to multiple parallel and perpendicular field regions.

COIL GEOMETRY

The coil geometry is the best geometry for studying current transfer in a relatively uniform applied magnetic field with long current contacts and a long active sample length (length between the current contacts) . In this

geometry the magnitude and current dependence of E were measured as a function of current contact length, i , and of distance, x, from the current

cc contact with the magnetic field essentially perpendicular to the wire axis.

Both twisted and untwisted multifilamentary Nb^Sn samples were measured to

determine if twisting had any effect on current transfer. A comparison of E

(I, x, i, ) is made below between data from this geometry and the theories.

The effect of current contact length on current transfer was determined

by making a set of measurements of the voltage between adjacent taps on the

coil sample and then cutting the sample in the current contact region and

repeating the measurements. This allowed a systematic study of the effect of

the length of the current contact. The value used for x (the distance from

the current contact) of a pair of voltage taps was obtained by an iterative

process that takes into account the electric field as a function of x. The

taps were relatively close together so x was not a strong function of the

iterative process, especially at large x.

The current dependence of E^ is more complicated than was implicit in the

theories. Figure 4.1-1 shows that this dependence is not linear as predicted,

but can be approximated by a power law,

E

t a I P ^ , (4.1-4)

34
Critical Current Measurement

Hairpin Long Straight

Figure 4.1-3. The four common critical current measurement configurations.

35
where p is nearly independent of distance from the current contact (except

very near the current contact) and may be a function of sample as well. This

empirical approximation works well for all of the data, but the power does

tend to increase slightly with current. The value of p was determined from

-12 . data (30 to 60% of I at 10 fi’cm) on a twisted sample with three different

values of (0.29, 1.27, and 5.15 cm) in a magnetic field of 4 T. The

values of p as a function of x are plotted on figure 4.1-4. For the lowest x

(x = 0.18 cm, x/d = 4.2) p is approximately 1.6 and it increases with distance

very quickly to an asymptote of 2.5 ± 1 for x greater than 1 cm. For the

larger values of x/d there was more uncertainty in p because of the low E^,

thus the larger range. The value of p had little dependence on either i or

cc the magnetic field, and data on an untwisted sample gave similar results.

Extending the theory of Ref. [4.1-2] can give insight into the empirical

power law dependence of E^ on current and on position [4.1-5]. The theory in

Ref. [4.1-2] starts by modeling the conductor as two regions, a cylindrical

rod and a concentric cylindrical shell. What results from considering current

transfer between these two regions is eq (4.1-2), which has an implicit linear

current dependence of E^. Extending this to a continuum of concentric cylindrical shells, however, does not give the same linear result. If the transfer

between two shells is linear, then the transfer among three shells will be

linear for the lower currents, until J in the second shell approaches J

c for

that shell, then there will start to be superposed another linear dependence.

Thus, together there will be two linear segments with different slopes.

Extending to a continuum of concentric cylindrical shells should give a smooth

curve with p

> 1. The transfer voltage at small x may be dominated by the

current flowing in the outer copper jacket (essentially two conducting shells)

rather than the current transfer taking place through the interfilament copper

matrix (approaching a continuum of conducting shells) . This could be why the

transfer characteristic has a slope close to one for small x and an almost

constant (asymptotic) slope for large x.

There are cases where a linear transfer characteristic (p = 1) would be

expected. These cases have a single or dominant transfer barrier between

parallel conductive paths. An example of a single transfer barrier would be a monofilament or a single layer superconductor in a normal metal matrix. An

36
d £O*

37

Figure

A.

1-4.

The

value

ofp

determined

ona

twisted

sample

with

three

different

values

of£

(0.29

cm:

square,

1.27

cm:

triangle

5.15

cm:

circle)

ina

magnetic

field

of4

T.
example of a dominant transfer barrier would be a monolithic, multif ilamentary

Cu-NbTi composite soldered over its entire length to a copper shunt. In this case the solder interface could dominate the transfer due to the relative

resistance of the interface compared to the copper. If the superconductor is

long enough, all of the current will transfer into the superconductor; however,

the voltage drop across the solder may dominate the nonlinear transfer voltage

of the copper matrix. Multiple linear segments in the transfer characteristic

may be caused by a filament distribution that has rings or regions of filament

that are separated by a relatively thick ring of normal metal, a high resistivity diffusion barrier, or the intrastrand material of a cabled conductor.

Data taken on the magnitude of E as a function of x and £^ c is shown in

figure 4.1-5 for a 338 cm long coil sample of twisted Nb„Sn. Plotted are -12 J values of E at 140 A (60% of 1^ at 10 Q*cm) in a magnetic field of 4 T, as

a function of x for various £ . Similar data were obtained at 8 T. The

cc points on the plot at the largest x were taken on voltage taps positioned

around the center of the coil sample. E^ was also measured on the other side

of the sample center at two values of x, 3.2 and 14.6 cm. These data indicate

that the current transfer is symmetric; the transfer in has the same profile

as the transfer out. This symmetry will be described further in the discussion of the other sample geometries.

E

t (x) has a strong dependence on at low x (see fig. 4.1-5). This is

most likely due to variations in the local critical current. The percent

difference between the measured values of I (x) of the twisted coil and I at

c c the center of the coil (264 A) is also given on figure 4.1-5. The largest

change in I occurs in the region that is not monotonic. The variations in I c c were caused in part by the defects (inhomogeneities, filament breakage, and

nonuniformity) introduced by the twisting and also by the slight change in

magnetic field along the coil. The latter effect could account for a 2.5%

change in 1^. The defects introduced by the twisting are thought to account

for the rest of the variations in I , and these give the irregular E (x) . The

c t dominant source of current transfer is still the joint, except at the very low

E or large x, where the variations in I start to dominate,

t c

The other two sources of current transfer, bending strain and self field,

are probably not significant, although if either of these two sources were

38
V

/cm

Figure 4.1-5. E as a function of x and i at 140 A (60% of I at 10

t cc c fl*cm) and 4 T. The percentage difference of the critical

currents are indicated under each position.

39
present, they would cause E to be less dependent on x. The bending strain

effect should be negligible, because the sample was reacted in shape. The

self field source, just like bending strain, only occurs near 1^, so it should

be negligible at this current.

A comparison of data taken on a twisted sample (same as fig. 4.1-5) and

on an untwisted sample (85 cm long coil sample) is shown on figure 4.1-6. As

mentioned above, the dependences of on I and &cc are very similar for the

twisted and untwisted samples. Furthermore, as can be seen on figure 4.1-6,

the magnitude of E^ as a function of x is only slightly different for the two

samples at low x. This difference is probably due to a systematic difference

in the current contacts or of the position x = 0, for the two runs. The

points at the largest x, for the untwisted sample, are those from voltage taps across the center of the coil; this is why they are slightly higher than the

twisted data. Thus, the effect of twisting is insignificant, except as mentioned above for the very low E^ where the inhomogeneities introduced by

twisting cause E^ to be higher.

A comparison of the experimental p

fc (x/d) with that predicted for this

sample by the current transfer theories is shown on figure 4.1-7. Plotted are

the experimental values of extrapolated to I

c (using eq (4.1-4)) in a magnetic field of 4 T, as a function of x/d for various l . The theories are

cc

in reasonable agreement with each other (within geometric factors) , and they

are close to the experimental r p at the shortest l . This indicates that

t cc their initial conditions are closest to this current contact length.

HAIRPIN GEOMETRY

The hairpin geometry is an example where the sample axis changes along

its length from parallel, to perpendicular, to parallel with the relatively

uniform applied magnetic field. This creates a change in the local critical

current of the sample along its length that results in current transfer. In

general the transition from parallel to perpendicular can be gradual in the case of the round bottom hairpin (constant radius semicircular) or more abrupt

as in the flat bottom hairpin (two small radius of curvature sections and a flat section) . Measurements made on samples in both these geometries are described below.

40
V/

cm

x , cm

Figure 4.1-6. E as a function of x and

untwisted samples.

I at cc

140 A and 4 for twisted and

41
cm

x / d

Figure 4.1-7. p as a function of x/d extrapolated to I at 4 T for three

different values of JU, compared with eqs (4.1-2) and (4.1-3).

42
The current transfer profile, E

t (x), was measured on the parallel field

sections of a hairpin sample and compared with that in a perpendicular (coil

geometry) field of the same magnitude. The profiles are shown on figure 4.1-8

for a current of 140 A, 4 T field, and the same £ ,1.3 cm. The data for the

cc perpendicular field were adjusted from data with £ = 1.27 cm to Z^

c = 1*3 cm

(the same as the hairpin), but this adjustment was only 3% at the lowest x and even less for larger x. Values of E for the two geometries at the lowest x are very close and E in the parallel field drops much faster with x, especially for the larger x. Also, when E = 1 nV/cm, x is ^4 cm in the parallel

field and about ten times this value, 40 cm, in perpendicular field. The

difference between current transfer in these two orientations is due to the

dependence of on the critical current. It is not clear if these data

indicate whether or not the assumption in eq (4.1-1) is valid; however, these

data do indicate that, if at all, it would take a very long length in parallel

field before the current distribution would be uniform. In any case, however,

it is advantageous to keep the joint and associated current transfer away from

the critical current measurement region. The electric field profiles of each

end of the sample in the parallel field were symmetric, implying that the

transfer into the conductor occurs in the same way as the transfer out. A

similar parallel-field current transfer profile was measured on the parallelfield sections of the "nested coil" geometry described below. Also, the

results of the measurements on current transfer in the low-field region of the

long geometry were consistent with these data, which suggest an equivalence of

the current transfer in low field with that in parallel field.

Additional current transfer takes place on the bottom of the hairpin,

since the distribution has to be more uniform than in the parallel field

section. As the angle between the applied magnetic field and the sample axis

changes, I changes; thus, p changes and the rest of the transfer occurs,

c s The distributed nature of this cause for current transfer makes the current

transfer profile more complex. The current transfer profile, E^(x), for the

round bottom hairpin geometry is given at 140 A and 4 T in figure 4.1-9. The

horizontal line sections of the histogram-like curve are the measured average

electric field between the voltage taps. The dashed line is an estimate of

the shape of the actual curve. The current transfer electric field increases

from the parallel-field value of less than 1 nV/cm to a peak of several

43
V/cm

x , cm

Figure 4.1-8. E as a function of x at 140 A, 4 T, and l = 1 . 3 cm for

cc perpendicular and parallel magnetic fields.

44
V

/cm

Figure 4.1-9. E

t as a function of x at 140 A and 4 T for the hairpin geometry

(parallel and perpendicular magnetic field regions).

45
hundred nV/cm at the center of the round bottom, the essentially perpendicularfield region. For the flat bottom hairpin geometries the current transfer

electric field increases more rapidly from the parallel-field value to peak

value of several hundred nV/cm just around the corner in the perpendicular

field. It decreased slightly at the center of the flat bottom. In both

hairpin geometries the current transfer electric fields were symmetric about

the center of the perpendicular-field region just as in the parallel-field

region. This implies that the current starts to transfer to the outer filaments on the other side of center, symmetric with the inward transfer. The

results of measurements on current transfer in the field-gradient region of

the long geometry showed a similar symmetry about the center of the field.

LONG GEOMETRY

In the long geometry, figure 4.1-3, the sample traverses several magnetic

field regions; from low, to gradient, to relatively uniform perpendicular, to

gradient, and then to low again. This creates a change in the local critical

current of the sample along its length similar to that in the hairpin geometry

and thus a current transfer phenomenon that is similar to that of the hairpin

geometry. The measurements made on the sample in the low-field regions of the

long geometry were consistent with that observed in the parallel field regions

of the hairpin geometry.

Additional current transfer takes place in the gradient and uniform

magnetic field regions, since the distribution has to be more uniform than in

the low field regions. The current transfer profile, E^Cx), for the long

geometry is given in figure 4.1-10 at 140 A and 4 T. The horizontal line

sections of the histogram-like curve are the measured average electric field

between the voltage taps and the vertical line sections just connect the

adjacent horizontal lines. Some percentages of the central magnetic field are given on the linear x axis which correspond to the vertical lines of the

histogram curve. The dashed curve is an estimate of the shape of the actual

curve. E

t rises sharply between 50% and 75% of the central magnetic field, it

peaks between 95% and 99%, and decreases slightly in the central region.

Measurements were made with the current in the other direction, and the values

of the voltage at the same current magnitude were the same to within 1% ±5 nV.

46
V/

cm

2

1

. 1

*

4J Uj

.01

001

in LO in

r-» in

05

05

05

05

05

in

05

0 8 10 12 14

X , cm

Figure 4.1-10. E as a function of x at 140 A and 4 T for the long geometry,

with the percentage of the central magnetic field also indicated along the x-axis.

47
These data, as with the hairpin data, imply that the current starts to transfer

to the outer filaments on the other side of center, symmetric with the transfer

in.

SHORT GEOMETRY

The short geometry, figure 4.1-3, can be the worst case as far as current

transfer is concerned since the overall sample length, current contact length,

and separation of current contact and voltage taps are limited by the bore of

the magnet. It is, however, the easiest geometry and thus it is often used.

Generally in this simple geometry eq (4.1-2) or (4.1-3) can be used to estimate the current transfer resistivity. However, in cases where the current

injection into the sample is not symmetric, the current transfer voltage can

be larger than calculated by these equations. It can be negative and can even

be significant for NbTi superconductors [4.1-6]. This was the subject of

extensive study in Ref. [4.1-6], which will only be summarized here.

Data taken on short samples of commercial multifilamentary superconductors have uncovered anomalous V-I characteristics. A voltage was detected

at currents well below the sharp upturn in the V-I characteristic near I c< It was apparently due to current transfer but larger in magnitude than would be

expected from previous current-transfer analysis [4.1-2, 4.1-3]. Further data

indicated that the voltage was strongly dependent on the voltage tap location.

In fact, the voltage measured below I in the current direction between some

taps was negative. In all cases, as I was approached, the V-I characteristic

returned to "normal." Experimental data on a Cu-NbTi superconductor is given

in figure 4.1-11. This effect can be much larger in Nb^Sn samples. It was demonstrated that this effect was due to a combination of current injection,

current transfer, and superconducting filament twist pitch. This study also

demonstrated that if the current injection were made more symmetric, either by

making the current contact longer than a twist length or having a symmetric

current contact, then the current transfer could be estimated by eq (4.1-2) or (4.1-3). Due to the large difference in the resistivity of the copper jacket

and the bronze core, the current injection into most Nb^Sn samples is symmetric. However, most Cu-NbTi superconductors do not have this large difference, so current contacts as long as a twist length may be required.

48
Voltage

,

/xV

Figure 4.1-11. Experimental voltage versus current on a Cu-NbTi superconductor

with nonsymmetric current injection.

49
NESTED GEOMETRY

A logical extension of the hairpin and long-geometry results is to ask

what would happen if there were multiple parallel and perpendicular field

sections? Would the current transfer in and out among the filaments on every

perpendicular field section? If it did, this would have a serious implication

for superconducting magnet design in cases where the windings are such that

the magnitude or angle of the magnetic field cycles many times. It would

affect the persistence, refrigeration load, and the stability of the system.

The nested geometry was designed to answer these questions. It was the

subject of a recent paper [4.1-7] and the main point will only be summarized

here

.

The specially shaped sample for this study was formed, prior to its reaction, in a rectangular pancake coil, nested geometry, as illustrated in figure

4.1-12. Measurements were made with the magnetic field in the plane of the

coil, pointed down in the figure. This configuration has a number of sections

that are essentially parallel or perpendicular to the magnetic field. The

ends of the sample are in parallel field. These sections are sequentially

numbered for reference on figure 4.1-12 so that all of the odd-numbered

sections are in parallel field and the even-numbered in perpendicular field.

Evidence of the response of the current to multiple parallel and perpendicular field sections is also given on figure 4.1-12. The measured voltage

and tap separation of each adjacent pair of taps is indicated for a current of

75 A in a field of 8 T (same sample as in table 4.1-1). These voltages

changed sign and had about the same magnitude when the current direction was

reversed. This indicated a symmetry of current transfer with the direction of

electron flow. Furthermore, there is a symmetry of current transfer voltage

about the center of the configuration, section 6.

The symmetry of current-transfer voltage can be explained with equipotential lines as indicated by the data. When the current reaches the first

perpendicular-field section, 2, it then has to transfer into the inner filaments, since the reduced critical current forces the distribution of current

among the filaments in this region to be more uniform than that in the parallel

50
Section 2

Figure 4.1-12. Nested sample geometry with voltage in nV (tap separation in

cm) for each adjacent pair of taps at a current of 75 A, with

a magnetic field of 8 T in the plane of the coil.

51
field region, 1. Now consider the symmetric section, 10, here the voltage

profile indicates that the current, in this the last perpendicular field

section, starts to transfer to the outer filaments in the same way that it

transferred to the inner ones in section 2. This may seem surprising at

first, but can be explained using a time-reversal argument. If the presence

of the current contact creates equipotential lines in the superconductor for

current entering, it will have the same equipotential lines for current

leaving the superconductor. Thus, these data indicate that once the current

distribution is uniform among the filaments, it will not change much (except

due to inhomogeneities in the superconducting properties) until the last part

of the last perpendicular-field region.

CONCLUSIONS

Transfer characteristic measurements on samples with short current

contacts (10 wire diameters) had good agreement with the existing current

transfer theories. The transfer voltage for samples with long current con- tacts (100 wire diameters) was smaller by as much as a factor of 10 in the

region adjacent to the current contact (within 10 wire diameters). This would

reduce the heating and increase the stability of the critical current sample.

The transfer voltage far away from the current contact (more than 100 wire

diameters) was relatively independent of current contact length.

The largest discrepancy between the measurements reported here and the

theories is the shape of the transfer characteristic. The transfer voltage

can depend on current to a power as high as 2.5. There are cases where the

relationship can be linear (a small effect can always be approximated as linear), but the continuum filament distribution should give a curve.

Another conclusion of this study is that the full-magnitude, perpendicular magnetic field is necessary to complete the transfer. The length in

lower or parallel magnetic field will not complete the transfer no matter how

long it is. Associated with this, the current transfer into a conductor is

symmetric with the transfer out of a conductor. The current will transfer to

the outer filaments of a conductor in the last full-magnitude, perpendicular

magnetic field before a lower or parallel field region is reached. A possible

52
application of this effect would be to put the two ends of the wire through a seperate high field coil to force the current into a more uniform distribution.

Current transfer that results from nonsymmetric current injection can

cause very strange voltage-current curves, often with peaks or negative

voltages that are larger than theory would predict. This is very common in

low voltage measurements on copper stabilized, multifilamentary NbTi superconductors. It has been explained by a combination of current transfer and

filament twist. The magnitude of this effect can be reduced by making the

current injection more symmetric.

53
4.2 Lap-Joint Resistance of Multif ilamentary Superconductors

by L. F. Goodrich and E. S. Pittman

INTRODUCTION

Lap joints between superconductors are important for most magnet designs,

and studying them will also help understand the role of the current contact in

critical current measurements. Lap joints were made by overlapping two

lengths of a superconducting wire and joining them with Pb-Sn solder. Lap

joints of various lengths were measured on commercial multifilamentary niobiumtitanium (NbTi) and niobium-tin (Nb^Sn) samples. The voltage-current characteristics of the joint were measured at 4 K as a function of joint length, L_. , magnetic field, H, and orientation of the magnetic field with respect to the

j oint

.

This work is an extension to the study reported in Ref [4.2-1], Here,

measurements were made on Nb^Sn samples and on longer NbTi joints. The

components of the lap-joint resistance (solder and boundary; surface copper;

and transfer) were identified and separated. The emphasis here is on the

shape and magnitude of the current transfer component. A model for the

lap-joint resistance is compared to the experimental data.

EXPERIMENT

The lap-joint resistance data presented here were taken on two commercial

multifilamentary superconductors. These two were superconductors based on NbTi and Nb^Sn. The NbTi sample had cross-sectional dimensions of 0.53 x 0.68

mm, with 180 filaments and a copper to non copper ratio of 1.8. The NbTi

filaments had a twist length of 1.25 cm per twist and the wire had a critical

current of about 103 A at 8 T, 4 K and 0.1 yV/cm. The Nb^Sn sample was 0.70 mm in diameter, with 2869 filaments and a copper to non copper ratio of

1.7. The Nb^Sn sample was originally not twisted and unreacted, thus it could

be twisted and formed to shape prior to its reaction. The samples measured

here were twisted with a twist length of about 1.3 cm per twist and reacted on

a coil mandrel. The Nb^Sn wire had a critical current of about 93 A at 8 T,

4 K and 0.1 yV/cm.

54
Digital measurements were made of the voltage, V, across the lap joint as

a function of current, I. The lap-joint resistance, R, as a function of

current was defined as V/I (not as the local slope of the V-I curve). At each

magnetic field, several current sweeps were made to reduce the time between

the zero-current readings (thermoelectric voltage check points) and thus

reduce the change in thermoelectric voltage. Repeat measurements were made at

the lower currents because of the increased error in those measurements due to

the lower voltage levels. The estimated limits to accuracy were the larger of

50 nV or 1% for the voltage and 50 mA or 0.25% for the current. The estimated

limits to precision were the larger of 10 nV or 0.2% for the voltage and 10 mA

or 0.05% for the current.

The magnetic field and current dependence of the lap-joint resistance was

used to separate the three components of this resistance. Figure 4.2-1 shows

an illustration of the lap joint with the components indicated. The value of

the resistance extrapolated to zero current was defined as R0. RO has two

separable components, a part that does not have magnetoresistance and a part

that is separated by assuming a certain magnetoresistance. The first part of

R0 is the combination of solder and boundary (solder/copper) resistance which

will be defined as Rl. The second part of R0 is the resistance of the surface

layer of copper, defined as R2, that was separated from R0 using the estimated

magnetoresistance. The third component, defined as R3, of the lap-joint

resistance is the current dependent part or transfer resistance above R0.

This component may also have a contribution from the boundary resistance

between the copper and the NbTi filaments. The current dependence of the

transfer resistance scales with magnetic field when the current is normalized

by dividing by the critical current, 1^. Data from a wide range of magnetic

fields were used to test this normalization.

Magnetoresistance measurements were not made on the copper or the solder

used in this study. The copper values were estimated using the transverse

resistivity Kohler plot [4.2-2] and the measured resistivity ratio (70.5).

Longitudinal magnetoresistance does not follow a universal Kohler-like plot.

It saturates with magnetic field, and the difference between transverse and

longitudinal magnetoresistance is less for lower resistivity ratio copper

[4.2-3]. A further discussion of this is given in the NbTi result section.

55
Figure 4.2-1. Illustration of the lap joint with the components of resistance

indicated

.

56
The magnetoresistance of the solder was obtained from Ref. [4.2-4]. The

magnetoresistance of the solder was ignored because the resistivity of the

solder was expected to change by about 7% at 9 T, compared to an expected 157%

change for the copper. Furthermore, it was not known a priori how much of R1

is solder resistance and how much is boundary resistance.

Resistivity measurements were made on both wire samples and a number of

solders. The low temperature measurements were made just above the transition

temperature. The copper resistivity was determined from measurements on the

composite superconductor and on the superconductor after the copper was chemically removed. The low temperature resistivity of two Sn-50Pb solders

were measured, one with a resin flux core (0.36 yft*cm) and the other solid

(0.58 yft'cm). The cross-sectional area of the cored solder was determined by

density measurements. The resistivity ratio, room temperature to low temperature, was 35.9 for the cored and 26.2 for the solid, which indicates that

impurity variation is the cause of this difference. The resistivity from Ref.

[4.2-4] for Sn-50Pb was 0.59 yft cm. A sample of the solder used to make the

lap joints was not available, so the more conservative value of resistivity

was used, 0.58 yft cm. The measured low temperature resistivity of Sn-37Pb

solder y?as 0.64 yft cm.

RESULTS FOR NbTi JOINTS

The NbTi joints were made by lapping the wider sides of two pretinned

wires and soldering with Pb-Sn solder. An effort was made to keep the joint

aligned and as thin as possible, using a minimum amount of solder. Measurements were made with a coil sample geometry in a simple solenoidal magnet and

with a straight sample geometry in a radial access magnet.

Six lap joints were measured on the coil geometry. The joints were

clamped and soldered in a pair of concentric semicylinders of fiberglassepoxy. All of the coil geometry joints were soldered with Sn-50 Pb alloy.

Two samples were made; each with three joints. Adjacent joints were separated

by about 10 cm (190 times the wire thickness). Each voltage tap was more than

3.8 cm (70 times the wire thickness) from the edge of the lap joint.

57
The joint was oriented so that the magnetic field was essentially perpendicular to the wire axis (coil pitch approximately 5 degrees) and parallel to

the wider face of the wire (in the curved plane of the joint interface). A

fiberglass-epoxy shim was used to support one wire on each joint as it made

the transition to the coil form. A small amount of varnish was used to hold

the wire and shim in place. The range of overlap lengths was from 2.4 to

29.6 mm (4.5 to 56 times the wire thickness). Joint areas were from 1.6 to

20 mm2

.

Two lap joints were measured on the straight geometry in a radial access magnet, where the effect of joint interface orientation was studied (keeping

the magnetic field perpendicular to the wire axis). The f iberglassepoxy

sample holder had a step machined in the bottom of a groove to support the

wire in the joint region. The sample was also varnished in the groove to

reduce motion as the Lorentz force was rotated from into the bottom of the

groove to one side of the groove. One of the joints was soldered with Sn-50 Pb

alloy (3.8 mm lap) and the other with Sn-37 Pb eutectic alloy (4.6 mm lap).

Each voltage tap was more than 1.2 cm (20 times the wire thickness) from the

edge of the lap joint.

Typical lap-joint resistance data as a function of normalized current is

shown in figure 4.2-2. The normalizing 1^ was about 690 A at 1 T and 63 A at

9 T. This normalization produces a relatively uniform family of curves over

this wide range of magnetic fields and critical currents. The increase in R0

(the resistance extrapolated to zero current) with magnetic field is attributed

to the magnetoresistance of the surface copper layer (R2) . The difference

between the joint resistance extrapolated to 1^ and R0 (the transfer resistance,

R3) also has a magnetoresistance due to the interfilament copper. There is a slight heating effect evident in these data at the highest current for the

lowest magnetic field, where the resistance is higher than expected. This

effect was more pronounced in the shorter joints, but it should only affect

the determination of R3. Figure 4.2-3 is a plot of the current dependent

joint resistance, normalized by R3, versus normalized current at 7 T for all

eight NbTi joints. The shape is very similar for all of the joints tested.

58
G

£

G• r-H O

i/i

c

Figure 4.2-2. Typical NbTi lap-joint resistance as a function of normalized

current for various magnetic fields.

59
(R—

R0)/R3

1.2

1 -

0.8

0.6

0.4

0.2

0

O = 2.4 mm

A = 3.8 mm

+ = 4.6 mm

x = 5.2 mm

o = 7.4 mm

v = 11.2 mm

H = 19.7 mm

x = 29.6 mm

x 4 !

X

X

£

X

S'X

$

£

ff

* ff * £ & * £ S' x

f * * 9 x

x

X

X

H

-0.2

0 0.1 0.2 0.3 0.4 0.5

i/i.

0.6 0.7 0.8 0.9

Figure 4.2-3. Current dependent joint resistance, normalized by the

resistance at I , versus normalized current at 7 T for all

c eight NbTi joints.

60
A simplified model of the current dependent resistance of a rectangular

wire lap joint is given in Appendix 4.2-A. The result is

R = -I— » (4.2-1)

wL I c

where p is the resistivity of the filament region, t is the thickness and w is

the width of the filament region of each wire. This gives a resistance that

increases linearly with I (and a voltage that goes as I

2

) . The measured R

increases with current faster than linear (see fig. 4.2-2), approximately as the current raised to the power of 1.52 ± 20% for magnetic field of 4 T and

above, for all eight joints. The nonlinear resistivity of the superconducting

filaments and their distribution in the cross section are the most likely

cause of this current dependence of the joint resistance.

The R0 values were determined for each magnetic field, for each lap

joint. The lower current resistance data (up to 33% of 1^) were fitted with a least-squares routine to the empirical current dependence

R(I) = R0 + cl

1,5

. (4.2-2)

The determined K0 values were not a strong function of the value of the power,

as long as the fit was restricted to the lower currents. This restriction was

somewhat arbitrary, but it was a compromise between forcing the fit or having

a small number of points. These R0 values for each joint were then fit with a least-squares routine to the magnetic field dependence

R0 (H) = R1 + R2 F (H) , (4.2-3)

where

F (H) =

r (H)

r (H=0)

(4.2-4)

61
a unitless parameter that has the magnetoresistance dependence expected for

the surface copper (1+Ar/r). Notice that F=1 at H=0.

An intrinsic parameter, p£, for the solder and boundary resistance can be

obtained for each lap joint by taking the product of R1 and the lap-joint

area. This quantity, p£, is the product of the resistivity and length in the

direction of current transfer from one wire to the other. Table 4.2-1 gives

the R1 and p£ values for each lap joint, and figure 4.2-4 is a plot of p£ versus L_. . The average value of p£ is 3.5 nfi'cm2 (±40%). The variation in

this parameter with lap-joint length seemed to be random and is probably due

to variation in the thickness of the solder layer and boundary resistance.

The 3.8 and 4.6 mm lap joints were measured in the straight geometry. The

4.6 mm lap joint was the only one soldered with eutectic SnPb solder (the

others were soldered with Sn-50Pb) , but its p£ wasn't significantly different

from that of the others.

The thickness of the solder layer was measured for the five longest

joints in order to estimate the solder contribution to Rl. Each joint was ground and polished perpendicular to the curved plane of the joint interface.

Photographs were taken along the length of the joint at two levels, approximately 1/3 and 2/3 of the way into the joint width. The average solder

thickness was estimated from measurements taken on these photographs. The

average solder thickness for each of the three longest joints was about 3 ym,

Table 4.2-1. Rl and R2 components of NbTi lap-joint resistance.

L.

J (mm)

Rl

(nft)

p£

(nfl’ cm)

R2

(nft)

P

(nfi* cm)

2.4 251.5 4.2 55.2 44

3.8 113.8 3.0 39.1 49

4.6 119.1 3.8 31.8 49

5.2 119.3 4.3 33.5 58

7.4 59.5 3.1 23.8 59

11.2 63.2 4.9 14.2 53

19.7 18.5 2.5 8.9 58

29.6 10.5 2.2 5.3 52

62
p

l

,

nfi

cm

w

0 3 6 9 12 15 18 21 24 27 30

L: , mm

Figure 4.2-4. Product of resistivity and length in the direction of current

transfer, pi for the solder and boundary resistance versus L

63
which would give a p£ of about 0.2 nft*cm 2 for the solder alone. The average

was the reciprocal of the sum of the reciprocal thicknesses, which is appropriate for parallel path resistance. This implies that as much as 90% of R1 is

due to boundary resistance and the variation in p £ is due to variation in

boundary conditions. The wires in the other two joints photographed were

tipped. This was indicated by the very different average solder thicknesses:

2 and 34 ym for the 5.4 mm joint, and 5 and 12 ym for the 7.4 mm joint. So

for thin solder layers, the solder contribution is a small part of Rl.

The boundary resistance between the solder and copper was the main

component of p£ (3.5 yfi’cm 2 ). Not much attention was given to wire surface

preparation. The insulation was chemically stripped and the wire surface

cleaned with alcohol. This could account for the relatively large variation

in p£. Similar results have been reported for the boundary resistance in

aluminum stabilized superconductors [4.2-5] where p£ was 3 to 8 yft cm

2

.

The surface copper resistivity can be computed from R2 for each joint.

It is not obvious what area, width, and thickness are appropriate for this

resistance extrapolated to zero current. The filament region is not rectangular and has an all copper region in the center. Using the width of the first row of the filament region (0.4 mm) and twice the average distance from the

copper layer to the first row of filaments (60 ym) gives the values listed in

table 4.2-1 and plotted on figure 4.2-5. The value expected from the resistivity ratio measurement (25 nft'cm) is smaller than this value, 53 nft cm. A

number of possible sources that would make the copper resistivity appear

larger follow:

1. Thickness variation and voids in the solder layer and variation in

boundary resistance will cause a nonuniform current density that

will effectively decrease the cross-sectional area of the copper

layer.

2. Joint misalignment will cause the current to flow a longer distance

through the copper.

64
,

nfi

cm

a

Figure 4.2-5. Surface copper resistivity versus L.,

65
3. The fact that the filaments do not form a solid layer will cause the

current to focus on each filament.

4. The current will not flow exactly perpendicular to the joint interface, which would make the effective length longer.

These effects are expected to be sufficient to explain the discrepancy. Other

effects, like a solder alloyed copper layer or filament boundary resistance,

would not be included in R2 due to the magnetoresistance separation method.

The R3 values were determined for each magnetic field using a leastsquares fit routine to the emperical current dependence of the joint resistance,

R(I) - RO = R3(I/I ) K

, (4.2-5)

c

where R3 and the exponent, K, were the fitting variables. A linear fit was made of the logarithm of this equation and the fit was restricted to I/l

c values from 0.3 to 0.9. The values of K determined by this fit are shown versus magnetic field for all of the NbTi lap joints in figure 4.2-6. K

tended to be smaller for larger magnetic fields and for shorter joints.

In order to find out what information can be obtained from the R3 values,

consider the model (Appendix 4.2-A) for the current dependent resistance at

the critical current.

R3

pt

wL

One way to view this expression is that the average distance that the current

penetrates into the filament region of each wire is t/2 when the current is at

I . For a wire with twisted filaments this model is still valid if the

c lap-joint length is much smaller than one twist length. For the lap-joint

length longer than one twist length, this distance will become half of the

transposition thickness. The effective wire thickness can be calculated from

the R3 data after an assumption of the value of the effective transverse

resistivity. If the transverse resistivity of the filaments is zero, then the

66
M0H,T

Figure 4.2-6. The exponent, K, of the current dependence versus magnetic

fields for all of the NhTi lap joints.

67
effective transverse resistivity of the filament region would be about one half that of the copper (approximately 1 to 1 copper to superconductor by

volume in the filament region). With this assumption, the effective width of

the wire can be calculated taking into account the copper layer on the sides

and in the center of the conductor. The values for the effective wire thickness are given in table 4.2-2 for all magnetic fields, and on figure 4.2-7 at

a magnetic field of 7 T for all eight joints. In table 4.2-2 t ^is almost

constant with magnetic field for each joint. A magnetoresistance factor, F, was included in the resistivity of the copper. The only significant departure

from this occurred for the shorter joints, at the lower magnetic field (higher

current), where the heating was creating a significant temperature rise. The

plot of the effective thickness, t^^, illustrates that the expected long

joint limit (greater than a twist length) approached the transposition thick- ness (0.15 mm), the smallest thickness region in which all of the filaments

occupy somewhere along the conductor. The 3.8 and 4.6 mm joints were measured

embedded with varnish in a fiberglass epoxy groove, so the heating at low

field (high current) was significantly more than for the coil geometry. The

short joint values of t however, were almost twice the thickness of the

filament region (0.44 mm). This is most likely due to the effective resistivity

assumption. For the shorter joints, the transverse current density must be

large enough that the effective resistivity is closer to that of copper.

Table 4.2-2. Effective wire thickness (in units of mm) for each

NbTi lap-joint at various magnetic fields.

L.

J (mm) 1 2 3

Magnetic Field

4 5

(T)

6 7 8 9

2.4 1.48 1.12 1.03 0.93 0.87 0.83 0.81 0.78 0.74

3.8 2.83 1.24 1.04 0.89

4.6 1.64 0.97 0.89 0.83

5.2 0.94 0.86 0.86 0.84 0.83 0.82 0.82 0.81 0.76

7.4 0.70 0.64 0.64 0.64 0.63 0.62 0.62 0.62 0.60

11.2 0.39 0.36 0.37 0.37 0.37 0.37 0.36 0.37 0.35

19.7 0.25 0.23 0.24 0.24 0.24 0.24 0.24 0.25 0.25

29.6 0.24 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.20

68
eff

»

S

a

Figure 4.2-7. Effective wire thickness at 7 T for eight joints. The filament

twist length is about 12.7 cm. The expected limits for the

short and long joints are marked with dashed lines.

69
Other explanations such as heating would show a stronger magnetic field

dependence

.

The two straight geometry samples were measured as a function of the

orientation of the magnetic field with the joint interface, keeping the

magnetic field transverse to the wire axis. The interest was in determining

the orientation effect on the magnetoresistance dependence of the joint

resistance. The orientation angles are approximate, within a few degrees.

The expected symmetry of the results indicated that the orientation was off by

1° to 2°. The zero angle data were taken with the magnetic field in the plane

of the joint interface so that the transferring current is in essentially

transverse field. The 90° data have the magnetic field perpendicular to the

joint interface. The lap-joint resistance was measured as a function of

current and magnetic field at 0° and at 90°. The value of R0 was determined

for each set of 0° and 90° data. The 0° R0 was separated into R1 and R2 using

the assumed magnetoresistance dependence. The R1 was subtracted from R0 for

the 90° data. Dividing this difference by R2 gives the F appropriate for

longitudinal field. Table 4.2-3 lists the 90° F determined for each joint,

along with 0° F for comparison. Figure 4.2-8 is a plot of AR/R versus magnetic field for the two orientations. The longitudinal magnetoresistance does

not increase as fast with magnetic field as in the transverse case.

The lap-joint resistance of one of the joints (0.38 cm) was measured as a function of current at 7 T for various magnetic field orientations. The

current for each set of data (each orientation) was normalized by measured

critical current for that orientation. The change in I c between 0° and 90° was about 5.2% [4.2-6]. As above, F as a function of angle was determined

Table 4.2-3. Longitudinal magnetoresistance determined using the R0

values for 0° and 90° orientations on two NbTi lap joints.

Magnetic field Transverse Longitudinal F

T F L. = 3.8 mm L. = 4.6 mm

3 3

1 1.093 1.054 1.058

3 1.382 1.169 1.180

5 1.738 1.308 1.334

7 2.137 1.472 1.482

70
AR/R

1 1 a

1 1 1 1 (

i >

i \

• =L< = T:

>ngit

•ans

idin

ver

al

3e

0.8 1 10

Magnetic Field , T

Figure 4.2-8. Magnetoresistance, AR/R, versus magnetic field for two lapjoint orientations, 0° transverse and 90°longitudinal

.

71
from R2 (the surface copper component). It was also similarly determined

using the transfer resistance component, R3. These values are given in

table 4.2-4. The values of F determined from these two resistance components

are in good agreement. The difference at 90°, longitudinal, is probably due

to differences in the current path in the two regions of the lap joint.

RESULT OF Nb

3 Sn JOINTS

The Nb^Sn joints were made on the coil geometry with a wire that had been

wound and reacted on a mandrel with the same diameter as the coil test fixture. The joints were made by cutting the wire in the middle and rotating the

two coiled wires so that they overlapped. This overlapped region was then

soldered to make the lap joint. The other end of each wire was soldered to

each current contact, which was more than 160 cm from the lap joint. Voltage

taps were placed 10 cm from each end of the lap joint, whose length was about

250 times the diameter of the filament region, 0.4 mm. With this configuration, the magnetic field was oriented perpendicular to the joint interface.

The joint was made this way, keeping both wires on the mandrel surface, so that the wire would not be damaged by an unsupported Lorentz force or by

bending it in order to lap with a different orientation. The joint orientation with the magnetic field was not important, however, because the current

will flow in many directions in the copper jacket. This is due to the relatively lower resistivity of the copper jacket (20 nfl’cm) compared to that of

Table 4.2-4. Magnetoresistance as a function of

R0 and R3 from 7 T data on the 3.8

angle determined using mm NbTi lap-joint.

Angle (°) F from R0 F from R3

0 2.160 2.137

20 2.016 2.109

45 1.707 1.846

70 1.497 1.669

90 1.457 1.619

110 1.532 1.664

135 1.738 1.902

-20 2.037 2.081

72
the interfilament bronze (about 3.4 yfi cm). The current will flow around in

the copper in order to enter the bronze through the whole perimeter. This

fact combined with the extra complications of a round wire lap joint made it

unreasonable to determine R1 and R2 for the Nb^Sn joints. Thus, this part of

the study concentrated on R3.

Two Nb^Sn lap joints were measured one at a time in the coil geometry.

These joints were 13 and 51 mm long. The lap-joint resistance data as a function of normalized current for various magnetic fields for these two

joints are shown on figures 4.2-9 and 4.2-10. The normalizing I was about

560 A at 1 T and 93 A at 8 T. This normalization produces a relatively

uniform family of curves. The empirical current dependence of resistance is

the normalized current raised to the power of 1.6. This is a little higher

than for the case of the NbTi joints, but the geometry is different as well.

The R0 values were determined for each magnetic field and for each joint,

assuming this current dependence for the lower currents (up to 33% fo I )

.

The R0 values were then subtracted from the total joint resistance to leave

the current dependent part. The current dependent part was then fit from 30%

to 90% of 1^ to give R3 and the exponent. Those values are given in table

4.2-5 and plotted in figure 4.2-11. The high exponent at 1 T for the shorter

joint may be due to heating. The average exponent was 1.62 ± 9% (excluding

the 1 T value for the shorter joint).

Table 4.2-5. Effective resistivity of the Nb^Sn filament region

determined from lap-joint data compared to the direct

measured value of resistivity.

Magnetic field

(T)

L. =

3 P ef f

(nft cm)

13 mm

^measured

P ef f

L. =

3 P ef f

(nft cm)

51 mm

Pmeasured

P ef f

1 570 6.0 410 8.3

2 565 6.0 496 6.9

4 583 5.8 532 6.4

6 604 5.6 559 6.1

8 598 5.7 565 6.0

73
a

a

•*—

75

70 -

65

60

55

50

45

40

T" —r — 1 T— T '

0 = 2.0 T

A = 4.0 T + = 6.0 T

x = 8.0 T

-K

* *

+

A A

X

*

+

X A

+

X

*

&

A

x ±

v + A * * a

v AO

X ^ O * . A O

+ A ^ O

°

A & * n O°

A oo

o

°°8|8o§°

go o

A

4 ogo°°°

88

§00 °

J L

o o

o

J I I L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

i/ic

Figure 4.2-9. Lap-joint resistance versus normalized current for the 51 mm

long Nb^Sn joint at various magnetic fields.

74
I/I,

Figure 4.2-10. Lap-joint resistance versus normalized current for the 13 mm

long Nb^Sn joint at various magnetic fields.

75
2.2 T T

• = 13 mm

= 51 mm

w

1.6

0 2 4 6 8 10

m0h,t

Figure 4.2-11. The exponent of the current dependence versus magnetic field

for the two Nb^Sn joints.

76
In order to model the transfer resistance, R3, the resistivity of the

interfilament bronze has to be estimated. The Nb^Sn reaction was not complete

(30 hours at 730°C), so there might have been as much as 5 wt.% Sn (2.74 at.%)

remaining out of the starting 13.5 wt.% bronze alloy. This alloy would have a

4 K resistivity of about 7.3 yft'cm [4.2-7]. The low temperature (nonsuperconducting) resistivity of the noncopper part of the wire (bronze, Nb^Sn, Nb, and

Ta) was measured to be 3.4 yft'cm. A bronze with that resistivity would be

1.24 atomic percent Sn (2.29 wt.%). The resistivity is directly proportional

to impurity concentration. For an unreacted sample, the noncopper low temperature (nonsuperconducting) resistivity was 1.1 yfl'cm, so it is clear that the

Ta and Nb were affecting the measurement. The 3.4 yfTcm value will be used

here as an estimated value.

The model for R3 (Appendix 4.2-B) is greatly simplified in the case of

Nb^Sn where significant current transfer is known to take place past the end

of the joint. Nevertheless, relating the results to a model is necessary in

order to extend these results. One way to view R3 is

R3

eff

2ttL

9

where p is the effective transverse resistivity. The values of p -- and

err eff

the ratio of the estimated p and p are listed in table 4.2-5. This ratio eff

is very high, close to 6, which means that R3 is much lower than expected.

Also, notice that the values for the two joints are very close to each other, even though the lap-joint lengths were different by a factor of 3.9. The

longer joint shows some magnetoresistance that may be due to current sharing

in the copper jacket away from the joint, which was not included in R0. A

number of mechanisms that would lower R3 below the model value follow:

1. The transfer is not restricted to the length of the joint. This was known to be true, but an unexpected result was that R3 scaled rather

well with the length of the joint for these two joints.

2. The effective transverse resistivity will be smaller because approximately 20% of the non-copper volume (area) is superconducting.

Superconducting paths between filaments are not expected.

77
3. The resistivity of the bronze could be somewhat lower than the

measured resistivity of the noncopper material, but the value used

was equivalent to a bronze with a Sn content that was half of what

was expected.

4. The Sn content of the bronze will be spatially dependent, so the

resistivity of the bronze will be lower in the center of the wire

than in the ring just inside the Ta barrier.

All of these mechanisms are likely, but the factor of 6 may be beyond these

explanations

.

DISCUSSION

An earlier report on lap-joint resistance [4.2-1] was based on measurements using a flat bottom hairpin geometry. There was an overlooked effect

due to this choice of geometry. The joint was oriented so that the transferring current was in the direction of the applied magnetic field and the

resistance separation was based on transverse magnetoresistance of the copper.

This caused the p£ to be overestimated, 4.5 nfi'cm2 (here 3.3 nSl’cm2 ), and the

p of the copper to be underestimated as 30.8 nfi'cm (here 53 nfl’cm) . Considering this, these two studies are in good agreement.

CONCLUSION

The model and separation scheme seemed to work well for the NbTi lap

joints. The added elements of twist length, transverse current density, and

variation in solder thickness explain the magnitude differences. The boundary

resistance can be as much as 90% of the joint interface resistance. The

nonlinear resistivity of the superconducting filaments and their distribution

in the cross section are believed to be the cause of the faster than linear

dependence of the joint resistance on normalized current. With these additions the model can be used to estimate the lap-joint resistance of NbTi

wires

.

78
The model predicted a lap-joint resistance for the Nb^Sn wires that was

about six times higher than the measured value. Part of this discrepancy can

be explained with the known limited assumption of the model, but not all of

it. The current dependence of the lap-joint resistance was a little stronger

than the case of NbTi. This could be due to the round, as opposed to rectangular, wire cross section.

79
APPENDIX 4.2-A

CURRENT TRANSFER RESISTANCE FOR A RECTANGULAR CONDUCTOR

Some simplifying assumptions are made in order to model the current

transfer resistance of a lap joint. These assumptions are:

1. All of the current transfer takes place within the length of the

joint, L.

2. The transverse resistivity of the filaments is independent of

current and thus the resistivity of the filament regions can be

modeled as a variable that is independent of current and position.

3. The filaments have a uniform critical current density, J , and a

c continuous distribution throughout the filament region of the

conductor.

With these assumptions, the voltage drop across one filament region of a conductor can be derived and from the functional form of the transfer resistance of a lap joint. The diagram in figure 4.2-A1 defines the geometric

variables of one of the conductors in a lap joint. A current, I q

, enters this

conductor through the joint area (shaded). The current, I, that is transferred across the conductor is a function of z (the distance into the filament

region) . This is so because the transverse current will change to longitudinal

current up to a current density of J

c until the transverse current reaches

zero. The equations for this are:

I = I - zwJ , o c

I =

I I o

(1

where x is I /I . o c The differential voltage drop across an area with thickness

80
< w >

Figure 4.2-A1. Rectangular-conductor lap-joint diagram.

81
dz is

dV = dz.

wL

The total voltage drop across this wire is

V = Is!

wL (1 - tx) dz.

where y is the value of z where the transverse current is zero, which in this

geometry is tx. Evaluating the integral gives

v

i

2 o 2wL I

and

R = —2wL

The resistance will increase linearly with current and that the effective

transverse distance at the critical current is t/2 (half the thickness of this

region). The total resistance of a lap joint will involve two wires, and the

factor of 2 will be removed:

R(I) T

82
APPENDIX 4.2-B

CURRENT TRANSFER RESISTANCE FOR A ROUND CONDUCTOR

The same assumptions used for the rectangular conductor, plus one more,

will be used here. The additional assumption applies only to a round Nb^Sn

conductor with a copper jacket:

1. The current enters the filament region with a uniform current

density along the cylindrical surface area of the joint.

This assumption is due to the relatively low resistivity copper jacket compared to the high resistivity bronze interfilament material. See figure

4.2-A2 for the geometric variables. The equations are analogous to the rec- tangular conductor's:

I tt (a 2 - r

2

) J

c

)

I I c

)

dV

I p dr

2TrrL

9

V

~P

2ttL a

r I dr '

where y

= a/ ( 1-x) ,

e = ^l (1 (1-x»-

The second term of this expression is well behaved. It goes to -1 and R goes

to zero as x goes to zero. The second term also goes to zero as x goes to 1. R increases faster than linear with current and R at I becomes

c

83
Figure 4 2-A2. Round-conductor lap-joint diagram.

84
R

P 4ttL*

Two wires are involved for a lap joint and the resistance at critical current

will be

p

— P R

2 ttL

'

85
4.3 AC Losses

by J. V. Minervini

INTRODUCTION

This writeup is intended to document the principles of the ac loss theory

that underlie the experimental apparatus and test conditions to be presented.

It is not intended to be a rigorous treatise, but it is designed to give a relatively simple understanding of hysteresis losses in rectangular (one

dimensional) and round (two dimensional) superconductors and the methods by

which the losses can be experimentally measured.

The focus of this work is to develop an understanding of the loss mechanisms within an individual superconducting filament for two main conditions:

1) dc transport current in an ac external field,

2) ac transport current in a dc external field.

Only hysteresis loss in the filaments will be considered and not coupling or eddy-current losses. The loss will be computed from solution of Maxwell's

magneto-quasistatic equations for the field and current distributions within

the filament for the two conditions of external field and transport current

listed above.

A principal consideration is the form of the field dependence of the

critical current density, J^. In all cases it will be assumed that the BeanLondon model applies, i.e., that J

c is constant and independent of the local

field H. Although the Kim-Anderson model could be used, in which is

inversely proportional to H, the experimental evidence indicates that the

Bean-London model is appropriate for most of the operating conditions of

interest

.

The loss expressions will be developed first for the simple slab model to

establish the method and indicate preliminary results. Then the two-dimensional expressions will be developed. However, the exact mathematical formulation for computation of the flux penetration profiles will only be cited since

86
a complete treatment is given in a previous paper by Minervini [4.3-3].

BASIS OF LOSS MODEL

First we must establish the type of material under consideration. Most

present and projected engineering applications use a multifilamentary composite conductor, either in monolithic, cabled or braided form. The materials

are primarily NbTi and Nb^Sn. The loss expressions will be applicable to both

types of superconductor and most other nonideal type II superconductors. However, the expressions are probably not suitable for "in situ" type conductors.

There are three well known loss mechanisms in multifilamentary composite

superconductors. These are:

1) hysteresis loss in the superconducting filaments,

2) coupling loss due to circulating currents flowing axially through the

filaments and returning through the matrix,

3) eddy current loss in the normal matrix material.

In addition there is a surface loss which may be significant at high frequencies in superconducting compounds such as Nb^Sn which have a reaction layer on

the surface of the filaments. The only loss mechanism considered here is the

hysteresis loss in the filaments. This is the only significant contributor to

the total loss in the composite if the frequency of the field or current

change is less than the relaxation time, x, for coupling currents induced in

the matrix given by

v

o °i ,v 2

> T 2 2tt

where = 4 tt x 10 H/m

aj_ = conductivity normal to the filament axis, A/(V*m)

L^_ = twist pitch of the filaments, m.

87
Under these conditions the eddy current flow is dominated by the composite resistance, and the reaction field of the eddy currents is ignored. No

shielding takes place and all the superconducting filaments are subjected to

the same changing field and thus to the same hysteresis loss. The total loss

for the wire is just the hysteresis loss for one filament multiplied by all

the filaments in the composite. Since the loss is hysteretic and therefore

frequency independent, the loss expressions are good for any time dependence

of the field or current change.

ONE-DIMENSIONAL FLUX PENETRATION: NO TRANSPORT CURRENT

A one-dimensional slab of nonideal type II superconductor is often used

as a simple model to illustrate the fundamentals of the hysteresis loss

calculation. Consider the slab of thickness d in figure 4.3-1 to be of

infinite extent in the y and z directions. If a uniform external field, H^,

oriented parallel to the sides of the slab in the y direction is increased

from 0 to a value above H^, flux will begin to penetrate the material. For

our purposes assume H

^ is very small so that we can ignore the Meissner

state. Just how small is small will be determined later. As the flux penetrates from both surfaces bulk screening currents are induced to flow near the

surfaces to screen the interior region of the conductor. If we assume the

material to be in the critical state as in the Bean-London model then the

screening currents flow just at the critical current density, J

c> and are independent of the local field variation.

The field and current distribution will be given by solution of the

Maxwell equation V x H = J with the boundary condition that H = H at the

e surfaces. The solution has the form

H = H + J (x - d/2) , x > 0.

c (4.3-1)

e

The field penetrates to a point X given by

x = d/2

P (4.3-2)

88
The slab is fully penetrated when = H^, the full-penetration field, where

H = J d/2. (A. 3-3)

P c

The field and current distributions that correspond to flux penetration

for the Bean-London model are shown in figures A. 3-la through A.3-lc for an increasing external field and figures A.3-ld through A.3-lg for a decreasing

external field. The hysteresis inherent in the superconductor is illustrated

by the trapped flux remaining in the slab when the external field is reduced

to zero (fig. A.3-le). As long as << H^, the effects of the Meissner

state can be ignored.

The magnetization of the slab is given by

, +d/2

M = -j [ / H dx] - H , (A.3-A)

d

-d/2

and the magnetization curve for a complete cycle to the upper critical field

limit of is shown in figure A. 3-2. A more complete treatment of the

computation of the magnetic moment as a function of the applied field is given

in Appendix A.3-A. The energy loss per cycle per unit volume is easily

computed from the area under the magnetization curve and given by

W

h

= / [ § M dH] dV . (A. 3-5)

volume cycle

Carrying out the integration we get for the loss per unit volume per cycle,

W,

, , H , \T -

3 y

0

H

P H~

Hm — H

p

(A . 3-6a)

P

and

r-

- I “o

H

p

2

[ 3 (ir>- 21 H„,± Hp’ <4 - 3- 6b)

p

where H is the maximum external field for the loop, m v

89
Figure A.

3

Field Distribution

(a)

(b)

(c)

Current Distribution

1 u

k *

+J

c n

L -J

1. Field and current distribution in a type II superconducting

slab of thickness d with external field applied parallel to the

slab surfaces.

90
-M

Figure 4.3-2. Magnetization curve for a type II superconducting slab.

91
We can normalize field values to the full penetration field H

the full penetration field loss given by

P

and losses to

W

V

0 2 U

2 3 y 0

H

p

to get the reduced results:

W, H

h _ , m. 3

Wn " ' 0 p

H

-s < 1 H -

and

W, H H

^=[3(/)-2 Op ] f>l.

p

and, if H /H >> 1, m p

H — s 3 (—

)

Wn

V H' 0 p

These results were originally derived by London,

ONE-DIMENSIONAL EFFECT OF TRANSPORT CURRENT

(4. 3-7a)

(4. 3-7b)

(4. 3-7c)

Consider the one-dimensional slab to be in a uniform external field and

carrying a transport current 1^ in the z direction. Let I

t be a fraction of

the critical current I such that 0 < i < 1, where i = I /I . The current and

c t c field distributions for the slab are shown in figure 4.3-3. The current flows

everywhere in the cross section at the critical current density ±J

c which has

been assumed to be a constant as per the Bean-London model. The net current

flowing through the slab will always be given by i as can be seen by summing

the positive and negative current-carrying areas and dividing by the total

area. If the external field, H^, is cycled by an amount ±H^ about the

92
Field Distribution

H

H =H+ H

Current Distribution

. J

(a)

m

(b)

(c)

(d)

H = H - H

H + H

0 m

e t

(e)

(f)

H

e+ H

t <g)

H - H +H e o m

00

- X

+J

- X

-J -

h" d—I

/

1 i r

i % -j

shielding transport shielding

currents current currents

Figure 4.3-3. Bean model of field and current distribution in a slab carrying

a transport current, I /I = 0.5.

93
value Hq the current and field distributions will be as shown in figures

4.3-3a through 4.3-3h.

Since one can picture the central region of the slab as being occupied by

the transport current, the field required to fully penetrate to this region,

H (i), will be less than the field required to penetrate to the center of the

slab for zero transport current. That is, H (i) < H (0), where H (0) = J d/2.

P P P c The full penetration field now depends upon the fraction of transport current

and is given by

H (i) —E_— = c i H (0)

U

P

i). (4.3-8)

The magnetization and the hysteresis loss can be computed as before from

eqs (4.3-4) and (4.3-5), respectively. The results are given below in normalized form for the energy loss per unit volume per cycle:

V£ , H, ,3

W

0

- <H <0)> • Hm

H (i) P

rr < 1 (4. 3-9a)

w

h

(i) H H H wn

- (1 - «

3+ - (!moT>] (1 + i2)

- FTiy 5 1 (4 - 3- %)

0 P P P

where now W^/V is given by

W-0 2 2 '

3 *0 H

p

<0> (4.3-10)

If i = 0, eqs (4.3-9a) and (4.3-9b) reduce to eqs (4.3-7a) and (4.3-7b),

respectively. It is interesting to note that if is less than H^(i) the

hysteresis loss is totally unaffected by the transport current. This is so because the flux front does not penetrate deep enough into the slab to reach

the central transport current carrying region during any portion of the cycle.

94
One of the most significant features of this model is that the loss

during a cyclic field change is partly due to flux motion in the portion of

the conductor carrying screening currents and partly due to flux motion in the

portion of the conductor carrying the transport current. The two components

of the loss are additive and can be computed separately. The results are given below. For partial penetration the total loss is identical to that

given in eq (4.3-9a).

H H

~ in m m / . , \ • For 7

;—rrr <1 or ——tttt < (1 - 1 ) H

p

(i) - y°> -

W

h

(1) W

s (1) Wt (1)

. Hm ,3

W

0

W

0 w

0 ' H

p

(0)

Total Loss (4.3-lla)

W

s (i) H

m 3

~W

= W Shielding Loss (4. 3— lib)

W

t (i) —rr = 0 Transport Loss

0

(4.3-llc)

H H m m

For 1 or

h

' (o

') — ^ “ i) > the full penetration loss is

P P^

given by:

W (i) W (i) W (i) H H (i)

~W^

= + ~W

Q

(1 " + 3 ^H

p

(0)^ “ H^(O) ^ (1 + 1 ) Total Loss (4.3-1 Id)

W (i) o H H (i) — = (1 - i) +3 [ (jJ—COT") ” H (0) ^ “ i ) Shielding Loss (4.3-1 le)

W.(i) \ H (i) . ~wl = 6[(H (0)

) “ ( H (0)

)] 1 U p p

Transport Loss (4.3-llf)

95
The normalized hysteresis loss per cycle per unit volume versus maximum

external field change is shown in figure 4.3-4 with the transport current as a parameter. The shielding and transport loss components are shown as a function of the transport current in figures 4.3-5 and 4.3-6, respectively, and

the total loss in figure 4.3-7. The maximum field change is the parameter.

It is clear that the transport current does not affect the loss unless the

field change is large enough to penetrate into the transport current region,

which for the slab occurs when H /H (0) > (1 - i). m p

The two loss components can be measured separately because the shielding

loss component is provided by the external field source power supply and the

transport current loss is provided by the transport current power supply. For

a summary of the loss expressions see Appendix 4.3-A.

AC TRANSPORT CURRENT IN A DC FIELD: SLAB MODEL

Consider the case of a slab carrying an alternating transport current

while in a dc background field. For simplicity, let us assume there are two

adjacent slabs, side by side, carrying the same transport current in series

opposition, as illustrated in figures 4.3-8a and 4.3-8b. The net field

surrounding the slabs will be zero while the field between the slabs will be

the sum of the self field of each slab carrying a transport current i, the

fraction of critical current. The field and current distributions are shown

in figures 4.3-8a and 4.3-8b, respectively. The field has penetrated to x^

which is just i d. Figures 8c and 8d show a sequence of field profiles for

decreasing current and increasing current, respectively. Again we have

assumed that the slabs are in a large uniform dc bias field parallel to the

surfaces and also that the Bean-London model applies.

The magnetic moment as a function of the transport current can be computed

as before by application of eq (4.3-4), giving the results for increasing

current

,

H (0) P 12 i i 2 4

i

Z t-i + 2 (f-) z m i + cf-n i m m

(4. 3-12a)

96
w

h

(l)

Figure 4.3-4. Total hysteresis loss in a slab as a function of the naxi-u:

field change for different values of transport current.

97
SHIELDING

LOSS

2.0

TRANSPORT CURRENT, i

Figure 4.3-5. The shielding loss in a slab as a function of the transport

current with maximum field change as a parameter.

98
TRANSPORT

LOSS,

4.0

3.6

3.2

2.8

2.0

1.6

1.2

0.8

0.4

0

TRANSPORT CURRENT, i

Figure 4.3-6. The transport loss in a slab as a function of transport current

with maximum field change as a parameter.

99
TOTAL

LOSS

TRANSPORT CURRENT,

i

Figure 4.3-7. Total hysteresis loss in a slab as a function of transport

current with maximum field change as a parameter. These data

are the same as that in Figure 4.3-4.

100
J

A

H H

Figure 4.3-8. a) Field and b) current distributions in a slab carrying an alternating current in a constant background field; c) and d) are sequences of field profiles for decreasing and increasing

current, respectively.

101
and for decreasing current.

M

H

i ± . : (0) 2

1m

[1 + 2 (i-) - (i-) 2

] m m

(4. 3-12b)

where i is the maximum value of i < 1, and H (0) is as previously defined, m — p

Integration of the magnetization around a cyclic change of the transport

current yields for the normalized hysteresis loss per cycle per unit volume

"h . — . 3 = 4 i w

o (4.3-13)

where W^/V is given in eq (4.3-10).

This result is similar to that for a slab carrying zero transport current

in a changing external field (eq (4.3-7a)). Note that i^ cannot exceed 1. For a current change i that gives an equivalent field change H < J d/2, then m m — c the loss due to a changing current is half the loss due to a changing external

field.

Some interesting results can be obtained if we note that the total flux

in the filament is just the integral of the field distribution across the

filament

,

d

<J> = vIq£ / H dx ’

where & is the length of the sample and the induced voltage is

Vv = -ii

dt*

102
Assuming a sinusoidal time dependence of the current with frequency to,

i = i sin (tot) , (4.3-14) m

we get the terminal voltage as a function of time for increasing current.

•rp = -cos (tot) [1 + sin (tot) ] (4.3-15a)

V

0

and for decreasing current.

= -cos (tot)[l - sin (tot)] (4. 3-15b)

where Vn = H (0) d£ i U U p m

to. The voltage and current as a function of time

are shown plotted in figure 4.3-9, and voltage versus current in figure 4.3-10.

These theoretical curves are useful to compare with measured waveforms.

TWO-DIMENSIONAL FLUX PENETRATION IN A CYLINDRICAL FILAMENT

The loss calculations of the previous sections have all been computed

based upon the one-dimensional slab model. Until recently these solutions

were all that was available due to the complexity of solving the two-dimensional field equations. A mathematical technique has been developed to

determine the two-dimensional contours of flux penetration into cylindrical

shaped superconducting filaments exposed to a transverse ac external field

[4.3-1, 4.3-2, 4.3-3]. These solutions have been found for both circular and

elliptic shaped cylinders.

The problem is to compute the region of flux penetration in a cylindrical, type-II superconducting filament. The model is based upon the following assumptions: (a) the constant bias field is uniform, transverse to the

filament and much greater than H (0), the field required to fully penetrate

the filament at zero transport current, and (b) the filament is in the

critical state and the Bean-London model holds.

103
Figure 4.3-9. Terminal voltage, V, if a slab carrying an alternating current,

i, in a constant background field as a function of time.

104
V

Vo

Figure 4.3-10. Terminal voltage of a slab as a function of the alternating

transport current.

105
Consider the filament to be in a transverse field, H The induced

e current distribution creates a uniform magnetic field in an interior region

that is equal in magnitude and antiparallel to the external field so that the

interior region is completely shielded. The net magnetic field distribution

is just the superposition of the external field and the reaction field generated by the induced current distribution.

As the external field is decreased from the fully penetrated condition

(fig. 4.3-lla) the flux exits from the surface, reversing the current distribution in the region where the flux has decreased. The boundary that separates

the shielded region from the flux change region penetrates deeper into the

filament as the external field is further decreased (figs. 4.3-llb and 4.3-llc).

Once the flux front reaches the center of the filament it is considered to be

fully penetrated (fig. 4.3-lle). At this point the filament can no longer

shield itself. Thus, if the field is increased beyond this point, the current

distribution will remain constant as the flux continues to penetrate uniformly. After a complete cycle the current distribution will return to that

pictured in figure 4.3-lla. Reversal of the field change before full penetration will begin a second flux front (fig. 4.3-lld).

The sequence of flux fronts shown in figure 4.3-11 is only schematic.

The actual shapes of the flux front profiles have been computed by the mathematics of complex variables and an iterative numerical technique [4.3-1,

4.3-2, 4.3-3]. Figure 4.3-12 shows the computed flux front profiles for a circle (fig. 4.3-12a), an ellipse of aspect ratio 0.5 (fig. 4.3-12b), and an ellipse of aspect ratio 0.5 (fig. 4.3-12c). The numbers on the curves refer

to the ratio of the external field change to the full penetration field of the

circular filament at zero transport current. This value of the full penetration field has been computed to be

J d

H (0)

c (4.3-16)

P 7T

where d is the diameter of the filament

106
Figure 4.3-11. a) Initial condition at H 0 + AH . b) Field reduced to H,_ +

U e U

AH el- AH., c) Lower limit of field change H - AH . d) Field

U e increased to H n - AH + AH_. e) End of half cycle with H +

U e 4 . U AH . Note: currents bounded by contour C shield AH/2,

e

107
1.0

Figure A. 3-12. a) Limits of transverse flux penetration into cylindrical

filaments of circular cross section for different values of

the external field change AH^/H^CO).

108
- 1.

0

0.

0

1.

0

Figure A. 3-12 b) Limits of transverse flux penetration into cylindrical

filaments of elliptical cross section for different values of

the external field change AH /H (0) along the minor axis.

e p

109
1.0

k=0.5

Figure 4.3-12 c) Limits of transverse flux penetration into cylindrical

filaments of elliptical cross section for different values of

the external field change AH /H (0) along the major axis.

e p

110
The curves of flux penetration illustrated in figure 4.3-12 were generated

by assuming their shape to be in the form of an even polynomial, i.e..

, 2 4 6 8

y

= f (x) = a

Q + a^x + a^x + ag x + a

g x , (4.3-17)

where the coefficients a^ were determined by iteration to satisfy the boundary

conditions of the governing equations for each value of external field.

The analytic form of the flux penetration curves allows the calculation

of the filament magnetization for a discrete number of field points. Since at

a given value of the external field the current distribution in the filament

is known, the magnetization at a point due to this current distribution is

given by [4.3-4]

m = j(r x j) (4.3-18)

and the average magnetization is given by

M = / (r x j) dV (4.3-19)

The current density J is just and the average value of the ydirected magnetization component is zero by symmetry. Thus, eq (4.3-19)

reduces to

, H a a / 1- (x/a) s M = 3727 [/ dz / dx / (y J ) dy]

0 0 f(x)

C (4.3-20)

which in nondimensional form becomes

M

H

p

(0)

1 y l— x

* z

2 [/ dx' / y ’ dy '

]

0 f(x’)

(4.3-21)

111
where all primed dimensions have been normalized to the filament radius a_ and

the form of f(x) is given by eq (4.3-17). Evaluation of eq (4.3-21) for the curves shown in figure 4.3-12a gives values for the magnetization which can be

closely approximated by the following expression [4.3-5]

m o AH . AH . AH

M = \—( fLj) ^ _ o

( 2 + 0 ( ——~) 1 H (0) 3^H (0)

; ZVH (0)

; Z n (0)

;J * P P P P

AH

< Hp(°) - 1 (4.3-22)

The expression for M in eq (4.3-22) gives the magnetization for an initial increase of AH^ from zero. The complete hysteresis curve can be con- structed by noting that once the filament is fully penetrated, any change in

external field must induce a shielding current change of magnitude 2J

c to reverse the current distribution in the shielding current region (fig. 4.3-11).

The complete hysteresis curve is shown in figure 4.3-13.

TWO DIMENSIONAL LOSS: NO TRANSPORT CURRENT

The hysteresis loss per cycle per unit volume can be computed by integration of the magnetization over a complete cycle. This must be done for two

conditions: partial penetration and full penetration. The partial penetration

loss W _ results from substitution of eq (4.3-22) for the magnetization into

pO

eq (4.3-5) to give

W - AH 0 AH

P.9 = 0 ( - ) ^ WOp^H (Or n (0)

p

) (4.3-23)

where Wn /V = (4/3) H (0) is the loss per unit volume for AH = H (0)

.

0 Op e p

To compute the full penetration loss we note that the current

distribution cannot change once the full penetration field H^(0) has been

exceeded. Thus the field penetrates uniformly. The total loss over one cycle

is the sum of the partial penetration loss plus the loss that occurs when AH^

exceeds H^(0) and is given by

112
Figure A. 3-13. Magnetization loop for circular superconducting filament.

113
(4.3-24)

W

fnO Wn

(H (0))

f P° = e + - r r w v v

L 0 volume cycle

§ ( J • E) dt dV]

The electric field E is found from the Maxwell equation.

d <H

c

> (4.3-25)

and given by

E

z -*0 fi 2 (4.3-26)

Carrying out the integration in eq (4.3-24) gives for the loss per cycle.

W£ . AH AH

= r 2 ( *-) - n — , W„ 1 V H (Or J

’ H (0) -

0 P P

> 1 . (4.3-27)

TWO DIMENSIONAL LOSS: WITH TRANSPORT CURRENT

The transport current destroys the symmetry of the problem due to the

addition of the self field of the filament. In the previous case the net

current in the filament was zero whereas now the current must sum to the

transport current. The following condition must always be satisfied:

A - A_

1 (4.3-28)

where A^ = area of ±J , and i = (I /I ) is the ratio of transport to critical

± c t c current. In addition, the field required to fully penetrate the filament must

become a function of the transport current such that H (i) < H (0) where H (i)

p — p p

is a function yet to be determined.

Assume that the filament carries a normalized transport current i, 0 < i

< 1 , in a uniform external field Hp >> H^(i). The problem is to compute the

114
current distribution as the external field is cycled between H

Q

± AH^ while

the transport current remains constant for two cases: (1) AH < H (i) and (2) r e — p

AH > H (i). The exact initial current distribution is not known. In fact,

e - p we cannot assume any one specific initial state since the current pattern is

determined by the most recent history of current and field change. The

crucial element in this reasoning process is that only the most recent history

of change is important.

After several cycles of external field change with constant transport

current the initial current distribution will be wiped out or, more accurately,

rearranged. The process is guided by three requirements:

1) flux can enter or leave only from the filament surface,

2) current can only flow at ±J , and

c 3) the condition of eq (4.3-28) must always be satisfied.

This reasoning will result in a time-dependent steady-state current distribution as shown in figure 4.3-14 with the transport current redistributed into a central core of the filament surrounded by positive and negative shielding

currents near the outer surface. The exact number of cycles required to

completely rearrange the current distribution is not known but must be a function of i and AH^. If there are many cycles of external field change the

details of the transition stage will have an insignificant effect on the loss.

This type of current distribution in a circular filament is analogous to

that in a slab under similar conditions as indicated in figure 4.3-3. The

shielding currents will alternate in sign in a region near the surface while

screening the transport current in the core while AH < H (i) . Thus, the

e p

definition of the full-penetration field as a function of transport current

Hp(i) follows in that it is just the magnitude of the external field change,

AH^, that, if exceeded, will no longer change the current distribution in the

filament. The curves that define the boundary between shielding currents and

transport current are also known. They are the same curves that define the

field penetration profiles for the case when i = 0. This is so because the

current distributions defined by these profiles totally shield the region

within them.

115
Figure 4.3-14. Current distribution in a circular filament showing the

shielding current region and the transport current region.

116
A curve of H (i) versus i was computed from the previously determined

P shielding profiles (fig. 4.3-12a). At each value of applied field the shielding currents surround an area which is proportional to the transport current

i. The integration to determine this area is trivial since the curves that

define this boundary are simple polynomials. The fraction of transport

current is then

1 7(a

o m

+ a m

2 3 + a m

4 5 + a m

6 7 + a m8 9

(4.3-29)

where the polynomial coefficients a^ and the x-axis intercept, x^, were previously computed for several values of applied field. The full penetration

field versus current is shown in fig. 4.3-15. At zero transport current H^(0)

is equal to J^d/Tr while at i = 1 H (1) goes to zero since the entire filament

is filled with transport current and thus cannot carry any shielding currents.

To compute the loss for the case of partial penetration with transport

current W is quite simple. The current distribution and the full penetration field are given in figures 4.3-14 and 4.3-15, respectively. The transport current contributes nothing to the loss since it is completely shielded

from the external field change throughout the cycle. Thus, the loss is

identical to that for the case of partial penetration without transport

current as long as the condition AH < H (i) holds, i.e.,

e - p

W AH AH

ujr

= 2(H (0)

) “ ( H (0)^ Op * (4.3-30)

p

Full penetration occurs when AH^ exceeds H^(i). At this point

penetrates the filament uninhibited by shielding currents and the

dissipation for this cycle is given by

the flux

energy

W (H (i))

_Rt PV

[ / § (J*E) dt dv]

volume cycle

(4.3-31)

117
Figure 4.3-15. Full penetration field as a function of the transport current.

118
The electric field E is determined from eq (4.3-26), but the computation

is more complicated than the zero current case because the position of zero electric field no longer lies on a line of symmetry along the x axis. It now

coincides with the boundary separating regions of positive and negative

current. This boundary is a function of the transport current and its shape

is given by the appropriate curve in figure 4.3-12a for AH = H (i) . Thus E

e p is given by

E = — I AH |y 4 for -1 < x < -x and x < x < 1 (4.3-32a)

z u e l m m

and

E = -y n | AH I 4[y - f(x)] for -x < x < x , (4.3-32b)

z U e Z m m

where x and y are normalized to the filament radius a = d/2, and f(x) is the

function defining the transport current region boundary (eq (4.3-17)). See

figure 4.3-16.

The effect can be summed up as follows. The magnetic flux penetrates

uniformly into the filament from both sides inducing a positive electric field

in regions of positive current and a negative electric field in regions of

negative current. No flux crosses the surface separating the two regions and

thus E = 0 along this boundary. The power dissipation is given by J^'E and can be separated into two contributions: one from the shielding currents and

the other from the transport current. The external field source supplies the

energy dissipated by the shielding currents and the current source provides

the energy dissipated by the transport current.

Substituting eq (4.3-32) into eq (4.3-31) and carrying out the integration over a complete cycle gives the loss per unit volume for AH > H (i)

:

e p

W H (i) H (i) AH H (i)

-*r - “'iMoi1 - [ /roy] } + 2 {[ rw' - 'iuot11 Y(i) • (4 - 3 ' 33>

U p p p p

119
FLUX

MOTION

Figure 4.3-16. Full penetration flux motion for a) increasing external field,

and b) decreasing external field.

120
The non-dimensional function y(i) is shown in figure 4.3-17. The twodimensional loss expression in eq (4.3-33) is very similar to the one-dimensional loss expression, eq (4.3-9b). The first term on the right side of each

expression is the portion of the loss that occurs when the conductor is

partially penetrated. The second term on the right side of each expression is

the portion of the loss that occurs when the conductor is fully penetrated.

The factor y(i) for the two-dimensional case is directly analogous to the

(1+i 2

) term for the slab loss. However, it cannot be expressed analytically

but must be computed numerically for each of the flux penetration curves of

figure 4.3-12a.

Just as in the slab model, a portion of the loss occurs in the transport

current region and a portion in the shielding current region during the part

of the cycle when AH exceeds H (i). Thus, y(i) can be written as y(i) =

e p

a(i) + B(i), where a(i) represents the fraction of the loss that occurs in the

shielding current region and B(i) represents the fraction of the loss in the

transport current region. The functions a and B are shown along with y as functions of current in figure 4.3-17. They can be computed from the

following expressions:

a(i) (4 [/ dx -xm

A-x2 x A-x2 / y dy] +2 [/

m

dx / (y-f(x)) dy]

0 0 f(x)

x

- 2 [/

m

0

-f(x)

dx /

-A—

x

2

(y - f (x)) dy] } (4 . 3-34a)

and

„ Xm -f(x)

B(i) = t- i2[f dx f (y - f(x)) dy]}. (4.3-34b)

0 f(x)

The shapes of these curves are not unexpected since at zero transport

current the loss is all shielding loss, a(0)=l, and no transport loss, B(0)=0.

121
Figure 4.3-17. Functions a, 3, and y as functions of transport current.

122
As i is increased a(i) must decrease and 3(i) must increase since a greater

area of the filament is filled with transport current, leaving less area for

shielding currents until the filament is totally filled with transport current

at i=l. The two-dimensional factors a(i) and 8(i) are analogous to the slab

2 2 factors ( 1— i ) and i , respectively.

The total loss can be separated into two components and summarized as follows :

Ww

ft

0

w w

+ — w

o wo

9

where, for AH < H (i),

e - p

W AH _ AH . — = 2 ( —

)

3 - ( —

)

4

WnOpH (0) H (0)

p

t

0

0

and for AH > H (i)

e p

W H (i) — = {2 (—2

W

q

H (0)

H (i) AH H (i)

( h

£W) } + 2{(

-h (o)^ " <hV } a(l -) ’ P P P

(4 . 3-35a)

(4. 3-35b)

(4 . 3-35c)

(4. 3-35d)

W AH H (i)

W7

= 2{(H (0)'

) “ ( H

2 Up W)} * (4 . 3-35e)

p

The shielding, transport, and total losses are shown as functions of the

transport current with external field change as a parameter in figures 4.3-18,

4.3-19, and 4.3-20, respectively. The two-dimensional loss expressions are summarized in Appendix 4.3-B.

123
SHIELDING

LOSS

4.0

Figure 4.3-18. Shielding loss versus transport current for different values

of external field change.

124
Figure 4.3-19. Transport loss versus transport current for different values

of external field change.

125
TOTAL

LOSS,

Figure A. 3-20. Total loss versus transport current for different values of

external field change.

126
APPENDIX 4.3-A

SUMMARY OF ONE-DIMENSIONAL LOSS FORMULA

All losses are given as a normalized loss per unit volume per cycle.

CONDITIONS: Changing external field with constant transport current.

Partial Penetration Without Transport Current

H

... — < i Hp(0) - ’ i=0

!e0 '3

W

0

- ( H (0)

J

Full Penetration Without Transport Current

Hm

Hp(0)

> 1, i=0

W

W

f pO

0

' 3(FToT>-

2 5 p

Partial Penetration With Transport Current

Hm

Hp(0) i 1 , 0 < i < 1

W

pt =

, Hm s 3

W

Q " ( H

p

(0)

)

127
Full Penetration With Transport Current

Hm y°>

> 1, 0 < i < 1

W. H (1) H H (1)

f P.t = (_E_—

)

3 + 3 r c—n*—) _ ( P_ —)

i n + i

2

) W- (0)

; UH (0)' 'H (0)' Op J U ; p p

/Wq is the total loss. It can be further broken down into a shielding

current loss, W^/Wq and a transport current loss, W^/Wq.

where

W, W W

_f£L = _s

+ _t w w w

0 0 0

W H (i) H H (i)

W7

=

^H

P (0)^ + 3 ^H (0)

^ "

^H

P

(0)^ ^ “ l2 ') Op p p

and

W H H (i)

w7

= 6 [(fw) ” (/(oy)] 3-2 Op p

if H >> H (0) m p

W. . H

_££L ~ 3(—!B—) q + ±

2

) Wn

" (0)

; u ' 0 p

Note (1): These losses are for a constant transport current, i=I /I and for

t c an alternating external field, H^, given by

H = H ± H (t)

e 0 m

128
where is the uniform background field,

H

0 ~ H

P (0)

J d

H (0) = -f- P

2

w

0 _

2 „ V 3 y 0

H

p

Note (2): The partial penetration field as a function of transport current,

H (i)

,

is given by

P

H (i)

H (0) ^ ^

P

CONDITIONS: Constant External Field With Alternating Transport Current

Loss expression 0 < i < 1 — m —

"h *

. 3

77

- = 4 1 w

o

max

1

m

"

I

W

0 2 V~ '

3

M 0

H

p

2(0)

J d

V°>

= -§-

Additional Formulas

for i = i sin(wt)

m

129
increasing current

magnetization A_

H

p( °) = \

i

2 [-1 + 2 (f-) + (^)Z

] l m 1 l

i x2. m m

V

i

terminal voltage — = -cos(wt) [1 + sin(wt)]

0

decreasing current

magnetization

H

p

(°)

1 . 2

TT X

2 m

[1 + 2(i-) - (^)

2

] m m

terminal voltage

V,

V,

-cos(a)t) [1 - sin(oot)]

where : V = y H (0) d i w i

2 U U p m

d = slab thickness

l = slab length

w = frequency

/

130
APPENDIX 4.3-B

SUMMARY OF TWO-DIMENSIONAL LOSS FORMULA

All losses are given as a normalized loss per unit volume per cycle.

Partial Penetration Without Transport Current

AH

0

H (0) - l ’ P

i=0

AH AH . 2 ( —) ^ - ( —) 4

^H (0)

; ^H (0)

; P P

Full Penetration Without Transport Current

AH

0

H (0)

> l ’ P

i=0

W, n AH

f P° = 2 ( —

W

Q

Hp (0)

) - 1

Partial Penetration With Transport Current

AH

1, 0 < i < 1

w-21 = W

0

AH

2 ( Hp(°)

6

) 3

AH

( L-)

4

( 0 )

; P

131
Full Penetration With Transport Current

AH

H

p

(i)

fr > 1, 0<i< 1

W H (i) H (i) , AH H (i)

W

= ^H (O)^ ”

^H (0)^ ^ + 2[(H_

(0T

) ”

^H (0)

)]

0 p p p p

/Wq is the total loss. It can be further broken down into a shielding

current loss, W /W„ and a transport current loss, W /W_.

s U t U

!f£t V!i w

o

" w

o wo

where

W

£ W,

H

p

(1)

3

H

p

(1)

4

AH

e

^ 2( H ( 0 )

) ( H ( 0

)^ ^ + 2 ^ ( h ( 0 ) p p p

H (i)

^ "

^H (0)

)]

P

and

W AH H (i)

wT

= 2 ^h (0)^

" (/(oy Up ) ^ p

If AH >> H ( 0 ) e p

W. „ AH

fpt _ „ , e AH

W,

= 2(— [a(i) + 6(i)] = 2(—7^-) y(i) Hp(0> H (0)

P

Note ( 1 ): These losses are for a constant transport current, i = I /I and

t c for an alternating external field, H

e> given by

H = H + AH

e 0 e

132
where is the uniform background field.

H

o >> H

p

(0)

J d

yo) - -tw

o = I

u o V<°>

Note (2) : The partial penetration field as a function of transport current

H (i) , is given in figure 4.3-15.

P

Note (3): y(i) = a(i) + 3(i)

The coefficients a(i), B(i), and y(i) are given in figure 4.3-17.

The losses W /W_, W /W_, and W. /Wn are given in figures 4.3-18,

s U t U rpt U

4.3-19, and 4.3-20, respectively.

133
4.4 References

[4.1-1] M. N. Wilson, Rutherford Lab. Report SMR/l, Chilton, Oxfordshire,

England, 1977 (unpublished).

[4.1-2] J. W. Ekin, J. Appl. Phys . 49, 3406 (1978).

[4.1-3] L. Dresner, Cryogenics 18, 285 (1978).

[4.1-4] L. F. Goodrich and F. R. Fickett, Cryogenics 22, 225 (1982).

[4.1-5] J. W. Ekin, private communication.

[4.1-6] L. F. Goodrich, J. W. Ekin, and F. R. Fickett, Advances in Cryogenic

Engineering - Materials 28, 571 (1982).

[4.1-7] L. F. Goodrich, IEEE Trans. Magn. MAG- 19, to be published.

[4.1-8] A. F. Clark, L. F. Goodrich, F. R. Fickett, and J. V. Minervini,

NBSIR 82-1678, National Bureau of Standards, Boulder, Colorado

(July 1982).

[4.2-1] L. F. Goodrich and J. W. Ekin, "Lap Joint Resistance and Intrinsic

Critical Current Measurements on a NbTi Superconducting Wire," IEEE

Trans. Magn. MAG-17, 69 (1981).

[4.2-2] F. R. Fickett, "Electrical Properties," Chapter 5 in: Materials at

Low Temperatures, edited by R. P. Reed and A. F. Clark (American

Society for Metals, Ohio, 1983).

[4.2-3] F. R. Fickett, private communication.

[4.2-4] M. Kobayashi, M. Mruzek, and W. Craddock, "Electrical and Mechanical

Properties of Lead/Tin Solders and Splices at 4.2 K," presented at

the International Cryogenic Materials Conference, May 11-14, 1983,

Kobe, Japan (unpublished).

[4.2-5] D. Yu, Y. M. Eyssa, and P. Zollikei, "Electrical Boundary Resistance

in an Aluminum-Stabilized Superconductor," Adv. Cryog. Eng. - Materials, 28, 701 (1981).

[4.2-6] L. F. Goodrich and F. R. Fickett, "Critical Current Measurements: A

Compendium of Experimental Results," Cryogenics 22^ , 225 (1982).

[4.2-7] F. R. Fickett, "Electric and Magnetic Properties of CuSn and CuNi

Alloys at 4 K," Cryogenics 22, 135 (1982).

[4.3-1] R. A. Beth, J. Appl. Phys. ^38 (12), 4689 (1967).

[4.3-2] J. V. Minervini, Adv. Cryo. Eng. 28, 587 (1982).

134
[4.3-3] J. V. Minervini, "Analysis of loss mechanisms in superconducting

windings for rotating electric generators," Ph.D. Thesis, Department

of Mechanical Engineering, Massachusetts Inst, of Technology,

Cambridge, MA (1981).

[4.3-4] J. D. Jackson, "Classical Electrodynamics," John Wiley and Sons,

New York, 146 (1967).

[4.3-5] C. Y. Pang, P. G. McLaren, and A. M. Campbell, in: Proceedings of the

Eighth International Cryogenic Engineering Conference, IPC Science

and Technology Press, Guildford, Surrey, England (1980).

135
APPENDIX A

PUBLICATIONS AND PRESENTATIONS

1. A. F. Clark and J. W. Ekin, "Defining Critical Current," IEEE Trans. Mag.

MAG-13 , 38-40 (1977).

2. R. L. Powell and A. F. Clark, "Definitions of Terms for Practical Superconductors, 1. Fundamental States and Flux Phenomena," Cryogenics 17 , 697-701 (1977).

3. R. L. Powell and A. F. Clark, "Definitions of Terms for Practical Superconductors, 2. Critical Parameters," Cryogenics _1_8, 137-141 (1978).

4. J. W. Ekin, "Current Transfer in Multifilamentary Superconductors.

I. Theory," J. Appl. Phys. 49, 3406-3409 (1978).

5. J. W. Ekin, A. F. Clark, and J. C. Ho, "Current Transfer in Multifilamentary Superconductors. II. Experimental Results," J. Appl. Phys. 49 , 3410-3412 (1978).

6. A. F. Clark, J. W. Ekin, R. Radebaugh, and D. T. Read, "The Development

of Standards for Practical Superconductors," IEEE Trans. Mag. MAG-15,

224-227 (1979).

7. F. R. Fickett and A. F. Clark, "Development of Standards for Superconductors, Annual Report FY 79," NBSIR 80-1629, National Bureau of Standards,

Boulder, Colorado (Dec. 1979).

8. F. R. Fickett and A. F. Clark, "Standards for Superconductors," Prof.

1979 Mech. and Mag. Energy Storage Contractors' Review Meeting, Conf-790854,

U.S. Dept, of Energy, Washington, DC (Dec. 1979).

9. D. T. Read, J. W. Ekin, R. L. Powell, and A. F. Clark, "Definitions of

Terms for Practical Superconductors: 3. Fabrication, Stabilization, and

Transient Losses," Cryogenics J_9, 327-332 (1979).

10. F. R. Fickett and A. F. Clark, "Development of Standards for Superconductors," Prof. 8th Int'l. Cryo. Eng. Conf . (ICEC8) , IPC Science and

Technology Press (1980) pp. 494-498.

11. F. R. Fickett and L. F. Goodrich, "NBD Superconductor Standardization

Program," Prof. 1980 Superconducting MHD Magnet Design Conference A. M.

Dawson, ed., (Report published by Bitter National Magnet Laboratory, MIT,

Oct. 1981) p. 87-89.

12. F. R. Fickett, S. B. Kaplan, R. L. Powell, R. Radebaugh, and A. F. Clark,

"Definitions of Terms for Practical Superconductors: 4. Josephson

Phenomena," Cryogenics 20_, 319-325 (1980).

13. G. Fujii, J. W. Ekin, R. Radebaugh, and A. F. Clark, "Effect of Thermal

Contraction of Sample Holder Material on Critical Current," Adv . Cryog . Eng . Vol. 26 , Plenum Press, New York (1980) p. 589-598.

136
14. G. Fujii, "Present Practices in Japan for the Measurement and Definition

of Various Superconducting Parameters," Tech. Report ISSP Univ. of Tokyo,

Ser. A, No. 1063 (July 1980).

15. G. Fujii, J. W. Ekin, R. Radebaugh, and A. F. Clark, "Effect of Thermal

Contraction of Sample-Holder Material on Critical Current," Tech. Report

ISSP Univ. of Tokyo, Ser. A, No. 1074 (Aug. 1980).

16. A. F. Clark, "Development of Standards for Practical Superconductors,"

Proc . Superconductivity Technical Exchange Meeting, IAPG, PIC-ELE-SC

209/1 (1980), p. 103 (presentation only).

17. L. F. Goodrich and J. W. Ekin, "Lap Joint Resistance and Intrinsic

Critical Current Measurements on a NbTi Superconducting Wire," Proc. 1980

Applied Superconductivity Conference, IEEE Trans. Mag. MAG- 1

7

, 69-72

(1981).

18. R. Radebaugh, G. Fujii, D. T. Read, and A. F. Clark, "A Standards Program

for ac Losses in Superconductors," Proc. IVth Int'l. Congress on Refrigeration (1980), Comm. A 1/2-10 (1980)., pp. 1-4.

19. G. Fujii, "Present Practices in Japan for the Measurement and Definition

of Various Superconducting Parameters," Cryogenics 2J_, 21-38 (1981).

20. H. R. Segal, Z. J. J. Stekley, and T. A. DeWinter, "Development of

Critical Current Measurement Standards," Prof. 1980 Applied Superconductivity Conference, IEEE Trans. Mag. MAG- 17 , 73-76 (1981).

21. G. Fujii, M. A. Ranney, and A. F. Clark, "Thermal Expansion of Nb^Sn and

V Ga Multifilamentary Superconducting Cables, Fiberglass-Epoxy and

Cotton-Phenolic Composite Materials," Jap. J. Appl. Phys. 20_, L267-L270

(1981) .

22. L. F. Goodrich, J. W. Ekin, and F. R. Fickett, "Effect of Twist Pitch on Short-Sample V-I Characteristics of Multifilamentary Superconductors,"

Adv. Cryog. Eng. - Materials, Vol. 28, Plenum Press, New York (1981),

p. 571-580.

23. A. F. Clark, G. Fujii, and M. A. Ranney, "The Thermal Expansion of

Several Materials for Superconducting Magnets," IEEE Trans. Mag. MAG-17,

2316-2319 (1981).

24. F. R. Fickett, L. F. Goodrich, and A. F. Clark, "Development of Standards

for Superconductors: Annual Report FY 80," NBSIR 80-1642, National

Bureau of Standards, Boulder, Colorado (Dec. 1980).

25. L. F. Goodrich and F. R. Fickett, "Critical Current Measurements: A

Compendium of Experimental Results," Cryogenics 22_, 225-241 (1982).

26. A. F. Clark, L. F. Goodrich, F. R. Fickett, and J. V. Minervini, "Development of Standards for Superconductors, Interim Report Oct. 80 to Jan. 82,"

NBSIR 82-1678, National Bureau of Standards, Boulder, Colorado (July 1982).

137
27. Standard Definitions of Terms Relating to Superconductors, Annual Book of

ASTM Standards, ASTM B713-82, Part 2.03, pp. 591-594, American Society

for Testing and Materials, Philadelphia, PA (1983).

28. Standard Test Method for D-C Critical Current of Composite Superconductors,

Annual Book of ASTM Standards, ASTM B7 14-82, Part 2.03, pp. 595-98,

American Society for Testing and Materials, Philadelphia, PA (1983).

29. L. F. Goodrich, "The Effect of Field Orientation on Current Transfer in

Multifilamentary Superconductors," IEEE Trans. Mag. MAG-19, 244-247

(1983).

30. L. F. Goodrich, E. S. Pittman, and A. F. Clark, "Critical Current Measurements on Standard Reference Material," presented at the Int’l. Cryog.

Materials Conf., August 15-19, 1983, Colorado Springs, Colorado.

31. A. F. Clark, L. F. Goodrich, and F. R. Fickett, "Experience in Standardizing Superconductor Measurements," J. Physique 45 (Cl) , 379-382 (1984).

32. L. F. Goodrich, D. F. Vecchia, E. S. Pittman, J. W. Ekin, and A. F. Clark, "Critical Current Measurements on an NbTi Superconducting Wire

Standard Reference Material," NBS Special Publication 260-91 (1984).

138
APPENDIX B

EFFECT OF TWIST PITCH

ON SHORT-SAMPLE V-I CHARACTERISTICS

OF MULTIFILAMENTARY SUPERCONDUCTORS*

L. F. Goodrich, J. W. Ekin, and F. R. Fickett

Electromagnetic Technology Division , National Bureau of Standards Boulder Colorado

INTRODUCTION

Precise determination of the critical current of practical

superconductors requires measurement of the voltage-current (V-I)

characteristic of the conductor at various magnetic fields. The

measurement usually requires the detection of quite small voltages

since very sensitive critical current criteria are necessary for

the design of practical devices. Furthermore, most laboratories

have only relatively small-bore solenoidal magnets, leading to the common use of very short sample lengths for routine critical

current measurements. This situation may lead to some difficulties, as we show here.

Data taken on short samples of commercial multifilamentary

superconductors have uncovered anomalous V-I characteristics. A

voltage was detected at currents well below the sharp upturn in the

V-I characteristic near I

c

. It was apparently due to current

transfer, but larger in magnitude than would be expected from

previous current-transfer analyses. 1 Further data indicated that

the voltage was strongly dependent on the voltage tap location. In fact, the voltage measured below I in the current direction

between some taps was negative. In all cases, as I

c was approached, the V-I characteristic returned to "normal." Extensive

experiments have shown that there are two extreme anomalous shapes

of the V-I curves. These are illustrated in Fig. 1. Depending on

the test geometry, the magnitude of many of these anomalous voltages can be on the order of commonly used critical-current criteria

and may significantly affect the determination of I £

.

*Partially funded by the Department of Energy.

571

Adv. Cryog . Eng . - Materials . Vol. 28 . Plenum Press, New York (1982), 571-580.

139
572 L. F. Goodrich, J. W. Ekin, and F. R. Fickett

Fig. 1 . Experimental data at 8 T

showing anomalous V-I characteristic of short NbTi

sample ( £ab

= 1 * 5 cm» 2,5

filament twists between current contact centers). The

horizontal dimensions of the

inset are approximately to

scale.

In this paper, the experimental investigation of the anomalous

behavior and a phenomenological model developed to account for the

observations are presented. Several techniques are discussed that

minimize the effect and, thus, allow precise critical current

determination in short samples.

APPARATUS AND SAMPLE PREPARATION

The apparatus used in this experiment was typical for a short,

straight sample critical current measurement. It consisted of the

following: a Dewar with a 9-T superconducting solenoid (3.8-cm

bore), a cryostat with 600-A vapor-cooled leads, a series-regulated

600-A battery current supply, an analog nanovoltmeter, and an X-Y

recorder. The magnetic field measurements were made to a precision

of 0.1X with a calibration accuracy of 0.2X. The voltage and

current measurements had an accuracy of 2Z and 0.4% and a precision

of IX and 0.2X, respectively. Typical noise voltages were ±5 nV.

Thermal voltages were checked at zero current and usually did not

vary by more than ±10 nV. The sample holder was made of NEMA G-ll

epoxy-fiberglass. Superconducting bus bar current dontacts were

set flush with one surface of the G-ll (see the Inset in Fig. 1)

such that the Lorentz force on the sample could be supported by the

G-ll, either directly or by a thin layer of varnish. Small slots were routed into the G-ll for voltage taps on the underside of the

wire where needed.

The measurements were made principally on two samples: a twisted multifilamentary NbTi (twist pitch 1.27 cm, Cu:NbTi of

1.8:1, RRR of the copper ~70, 0.53 x 0.68 mm, 180 filaments) and an

140
Effect of Twin Pitch on Short*Sample V-l Characteriitict 573

untwisted multifilamentary Nb^Sn (0.70-mm diameter, 2869 filaments). The Nb.Sn wire had an outer copper jacket separated by a tantalum diffusion barrier from the core of bronze, niobium, and

NbjSn.

Sample preparation was typical for short sample testing except

for the two following techniques. A technique was developed to

spot-solder a pair of voltage taps directly across the wire from

each other to allow measurement of transverse voltages. The

alignment of these taps was checked by measuring the room temperature resistivity and the voltage polarity of the pair and thus

deducing the approximate misalignment. The worst case misalignment

was -0.2 mm, but more usually Z 0.1 mm. The other special technique was selective etching of the copper jacket from the NbjSn

wire. This was accomplished using an enamel insulating paint as a mask and a nitric acid etch. Small copper islands were left on the

sample for ease in soldering voltage taps and current contacts.

EXPERIMENT

To investigate the anomalous voltage seen on the NbTi critical

current sample as described in the Introduction, tests were made

with two pairs of voltage taps spaced 0.5 and 1.5 cm apart. The

V-I characteristics are 6hown in Fig. 1. Here the voltage definition V . V

fi - Vb was used. These curves were reproducible and

reversible to within 12.

It was observed that both V . and V . changed sign when the

current was reversed, but not when the field direction was re- versed. There were slight differences (-102) in the magnitudes of

these voltages, especially close to I , as the direction of the

current and the field were changed. These are attributed to the

Hall effect and are discussed below, but they do not significantly

affect the unusual shape of the V-I characteristics.

During the development of the phenomenological model, experiments were made using several unique sample configurations of both

the twisted NbTi and the untwisted NbjSn. These data and the

voltage tap and current lead arrangements are presented in the

appropriate places in our discussion of the model.

PHENOMENOLOGICAL MODEL AND SUPPORTING DATA

The unusually shaped V-I characteristics may be understood in

terms of the interaction between current transfer and the twist

pitch of the superconductor. Filaments nearest the current contacts carry current near their critical current density and exhibit

a flux-flow resistivity. Conversely, filaments on the opposite

side of the superconductor from the current contacts carry very

little current because the resistive matrix separates them from the

141
574 L. F. Goodrich, J. W. Ekin, and F. R. Fickctt

point of current injection. Therefore, it is possible for a voltage tap to be sampling either a resistive or nonreslstive group

of filaments and, thus, the voltage between taps may be quite

different, depending on the relationship of tap spacing to twist

pitch. Also, significant transverse voltages should be observed

across the wire.

Results of transverse voltage measurements made on the NbTl

sample are shown in Fig. 2. Notice that these curves are similar

in shape and size to those in Fig. 1 except near Ic

. As Ic is

approached, the transverse voltage tends to go to zero as the

current distribution among the filaments becomes more uniform. The

distance between current injection points (approximately center to

center) for the data shown in Fig. 2 was 2.6 times the twist

pitch. Thus, the group of filaments nearest to the current bus bar

on one end of the sample are not the same as the group nearest to the bus bar at the other end. Current must therefore transfer

between the two groups of filaments by flowing through the resistive matrix material of the wire. This generates the large trans- verse voltages. The unusual V-I characteristics shown in Fig. 1 are simply the sum of the usual flux-flow V-I characteristic and

Fig. 2. Experimental data at

8 T on transverse voltages of short NbTi

sample (£ = 1.25 cm, 2.6 filament twists

between current contact

centers)

.

Fig. 3. Experimental data at

9 T on a selectively

etched Nb,Sn conductor

(£

ae

55 1.25 cm, no fila

ment twists) . Note the

bus bar locations.'

142
Effect of Twin Pitch on Short>Sample V-I Characteriitici 575

the transverse voltages caused by current transfer through the

resistive matrix from one group of filaments to another.

Understanding of the effect Is simplified considerably If the

conductor used to obtain the V-I characteristics in Figs. 1 and 2 is untwisted. In this arrangement one current bus bar was placed

on top, the other on the bottom to provide a half-integral number

of twi s t s between the current injection points. Data were obtained

on the untwisted Nb^Sn conductor in this geometry and are shown in

Fig. 3. These data can be explained by the model shown in Fig. 4.

Note that in multifilamentary conductors the equipotential lines at low currents are much more closely aligned with the conductor axis

than in a conductor with an isotropic resistivity. The voltage

between taps e and f starts from aero at I « I

c (Fig. 4A), rises

in magnitude as I increases (Fig. 4B), and decreases toward zero as all the filaments become resistive near I

£ (Fig. 4C). Similarly,

the voltage between taps a and f starts from zero at I « I c (Fig. 4A) , rises in magnitude to a negative peak at an Intermediate

value of I (Fig. 4B), and then becomes positive as the entire conductor becomes resistive at I = I (Fig. 4C). The voltage between

taps b and e rises from a low value at I « I

c to an intermediate

high at an intermediate value of 1, back to a low value as the

A. I<< I,

Fig. 4.

Equipotentiol Lines A\ c

| e _ 1 n i n i ^\y i "^\\ i

y ' \\w- n \\n- M \ o 1 t u t)

0 f

c. I-Ic

Model of equipotential lines in a superconductor with an odd halfintegral number of filament

twists between the current contacts: (A) current much less

than I c

; (B) intermediate

currents less than I : and

c (C) current near I c

.

143
576 L. F. Goodrich, J. W. Ekin, and F. R. Fickett

resistivity becomes more isotropic at I 3 I

£

, and finally to a high

value as I exceeds I£

. Note that the curves in Fig. 3 have about

the same shape as those in Fig. 1.

In Fig. 3 it is easy to separate the anomalous V-I characteristics (V^, ^

a f)

* nto an essentially intrinsic characteristic

(V^j, V ) and an anomalous current-transfer characteristic (transverse voltages V-). These separations ’ Vbe -, V

bf

" V

ef and

V

£ V + Vg £, were demonstrated experimentally with agreement of

about 1IT ±10 nV. So it is possible to have V-I characteristics

with these shapes (and everything in between) depending on voltage

tap location. Remember that this discussion and the data shown in

Figs. 1, 2, and 3 correspond to a sample with a half-integral

number of twist lengths between the points of current injection.

When there is an integral number of twist lengths between the

points of current injection, the current transfer pattern is

altered. Data obtained on the NbTi conductor with a current

contact spacing of about two twist lengths are shown in Fig. 5. In

this geometry, the V-I characteristic of the resistive filament.

Fig. 5. Experimental data at

8 T on a NbTi conductor

(£ae

3 1.25 cm, 2 filament twists between current contact centers).

Fig. 6. Experimental data at

9 T on a selectively

etched Nb

3 Sn conductor

(£ =1.25 cm, no fila

ae ment twists).

144
Effect of Twi»t Pitch on Short-Sample V-I Characteriitici 577

V

#e , is the sum of the V-I characteristics of nonresistive filaments, V.

f (essentially intrinsic characteristic), and the transverse voltages, V^, V

g £ (anomalous current-transfer characteristic). The equation V

fle - - V * was demonstrated experimentally with an agreement or about 1.5X ±15 nV. Data obtained on

the NbjSn conductor in this geometry are shown in Fig. 6. Note

that these curves are very similar in shape to the curves in Fig. 5 and can also be separated into an intrinsic V-I characteristic and

an anomalous current-transfer characteristic, V.

£ Vge + V

g

x -

V

b» with experimental agreement of about IX ±10 nv. The model for

this configuration is shown in Fig. 7. The current is injected and

extracted from the same group of filaments. The voltage between

taps b and f is indicative of this group of filaments and rises

much more rapidly than the voltage between taps a and e. Current

does not transfer through the matrix to the far group of filaments

sampled by taps a and e until I approaches I

c

« In fact, the

voltage taps on the top of the conductor remain at about the same potential until I reaches I

£

.

TECHNIQUES TO MINIMIZE THE ANOMALOUS CURRENT TRANSFER VOLTAGE

Note from the above discussion that these unusually shaped V-I

characteristics result from nonsymmetric current injection. If the

Voitoqe Tops

/ \

\ — -

~\ y f~

- 1 u

t u

a u

1 -

A I << i c /)

Current Bus Bor

0ncnen —

\ / m "~rrr:

,1, 1 f r1 * u 111 V | - i

1 b a i 1 I 1

B. I < I c

Equipotentiol Lines

/ o c , n p

1 \ r'"T'*r r- \\ V

\ \W \

Fig. 7.

f ^

i

,

1

, I j

ll/s:

1 i b a f I

“

'

7 ^

J / ///*;

c. I»I r

Model of equipotential lines in a superconductor with an integral

number of filament twist6 between

current contacts: (A) current

much less than I c

; (B) intermediate current less than I c

; and

(C) current near I c.

145
578 L. F. Goodrich, J. W. Ekin, and F. R. Fickett

current is uniformly Introduced around the circumference of the

superconductor, the current transfer voltage should be independent

of twist pitch considerations and reduced in magnitude.

There are at least two possible methods of making the current

injection symmetric in short sample testing. One method is to make

the solder joint at the current contact longer, at least one twist

length, so that the current is Introduced into all of the outer

filaments. In the second method, current injection is made more uniform by having a symmetric current contact. Both of these

techniques have been shown experimentally to reduce the anomalous

voltages greatly, thus permitting a more accurate determination of

the critical current.

The long current contact method was tested on the twisted NbTi

wire. First the voltages were measured with the current contacts

covering more than one twist length, then the wire was cut to

shorten the current contact, and the same voltage taps were measured again. The current-transfer voltages were about a factor of

10 lower for the long current contacts and of a magnitude consistent with symmetric current-transfer analysis. 1

Measurements on the Nb,Sn wire with the copper jacket intact

illustrate the second method. The copper jacket gives a more uniform injection of the current into the superconducting filament

region because of the relatively high resistivity of the bronze in

that region. The results are shown in Fig. 8. The magnitude of

the current transfer voltage is reduced over that of in Fig. 6,

again consistent with symmetric current-transfer analysis. Also,

the voltage drop along the top filaments is about the same as that

along the bottom filaments, which indicates a removal of the twist

pitch dependence.

A COMMENT ON GALVANOMAGNETIC EFFECTS

Observations of strange voltages in a current-carrying conductor at low temperatures in a high magnetic field are often

explained by arguments involving one or more of the many classical

galvanomagnetic effects. It is our contention that, although

several of these effects are present in our data, none of them cause the unusual shape of the V-I curves. Most of the effects are seen only in metals where the product of the cyclotron frequency,

u), and the electron collision time, x, is quite large, usually

~100. Even in the copper stabilizer on our wires, wt Z 2.4 at 4 K

and 10 T. Furthermore, large effects are most common to single

crystals and our metals are highly polycrystalline.

Two galvanomagnetic effects do appear in our data: the Hall

effect and transverse magnetoresistance. Of these, only the former

146
Effect of Twi»t Pitch on Short-Sample V-l Characterise! 579

Current , A

Fig. 8. Experimental data at 9 T on NbjSn conductor with copper

jacket giving uniform current injection (*ae

s 1.25 cm, no filament twists).

can be seen directly in I-V curves made at fixed field. The Hall

voltages appear most strongly on the probe pairs transverse to the

current. They reverse with both field and current and are developed almost entirely in the normal metal components of the wire.

In the wires measured here, at currents well below I , we noted

Hall voltages of ~10 nV in the NbTi at 8 T (on top of a ~200 nV

current transfer voltage) and ~100 nV in the NbjSn at 9 T (on top

of a ~15 pV transfer voltage). These values appear to be consistent with the known Hall coefficients for the matrix materials 2 and

the other parameters, but even an order of magnitude calculation

requires a much more detailed model than we have space for.

Suffice it to say that the Hall effect, although observable,

represents only a small contribution to the measured transverse

voltages.

Magnetoresistance is an even effect, it does not depend on current and does not reverse with field, thus it does not affect

the shape of the V-I curves. It shows up most strongly in measurements on voltage taps along the sample as an increase in the

resistance of the relatively pure copper stabilizer as the field is

increased. For our wires this effect causes the zero field resistivity of the stabilizer at 4 K to increase by about a factor of 3 in going to 9 T.

147
580 L. F. Goodrich, J. W. Ekin, mod F. R. Fickett

CONCLUSIONS

Anomalous voltages may be observed In short sample critical

current tests on multlf llamentary superconductors. These voltages

may In some Instances even be negative, but In any case they can

Interfere with the correct determination of I

c (even for NbTi)

especially when using sensitive (but realistic) electric field or resistivity criteria.

All of the observed behavior can be adequately explained by

use of a model that considers the combined effects of the twist

pitch of the filaments and the details of current injection and

transfer within the sample.

The anomalous voltages can be reduced In short sample testing

by Insuring that the current Is injected symmetrically Into the

conductor by long (at least one twist pitch) current contacts or by

symmetric current contacts. The current-transfer voltages then

will be Independent of twist pitch and reduced In magnitude, making

the usual methods of treating current-transfer voltages applicable. 1

ACKNOWLEDGMENTS

The authors have benefited greatly from numerous discussions

with A. F. Clark. E. Pittman has been most helpful in the care and

feeding of the instrumentation. Ms. V. Grove prepared the manuscript—many times. Our thanks to them all.

REFERENCES

1. J. W. Ekln, J. Appl. Phys. 49:3406 (1978).

2. C. M. Hurd, "The Hall Effect In Metals and Alloys," Plenum

Press, New York (1972).

148
V

IEEE TRANSACT1 0VS ON MAGNETICS. VOL. MAG-19 NO. 2. MAY 19*3

7 hi E-nr: if f:h: ofientaticn ck : c-aEsi teasstep is kdiii"z_i-w.zi“.iJT s

N ~ - r ea _ cf Standards

rad: 5-1303

Ads tract

Experimental data ar.d disc.ssicn are presented ::

the c.rrer.t distrib.ticn aicr.g the length cf a supercc-.c-ctir.g were when subjected tc multiple parallel

and cerrerocc_lar magnetic fields. The exper inencal

cata -ere tax.ee cr a rectangular partake ceil with the •relied magnetic field ir the plane of the coil.

Tre se cata indicate that significant current transfer

ccc-rs in the first and last perpendicular sagnetic

field secticr.s arc little transfer occurs between

fese owe secticns. Tne implication for superconducting nagr.et design will alsc be discussed.

Introduction

•as.lts cf recent measurement s

- on current

transfer in hairpin and long straight critical current

geometries shewed curious effects in multiple parallel

arc perpercic_lar field secticns on the current

transfer in aultif ilamer.rary superconductors . These

two critical c.rrer.t geometries are illustrated cn

Fig. 1. In the hairpin geometry, the sample axis

changes along its length free parallel, to perperdic.lar, to a return parallel with respect tc the relatively uniform applied magnetic field. In the

leng geometry, the sample traverses several r egretic

field regicns; from lew, to gradient, to relatively

uniform perperdic_lar , to gradient, and then to low

again. All of the measurements reported and discussed

here were made on the same material. A crcss-sect local view cf this racerial is given in Fig. 2. 1c

is a commercial multi filament ary Vo -. In superconductor

with a diameter cf 1." mm, 2559 filaner.es, and a ccpper cc non-copper racic cf 1.7. Its critital

current is about 75 A at 5 T, 4 K using an electric

field critericr. of 1 nV cm.

Hairpin Long Straight

Figure 1. Schematic representation cf the hairpin and

long straight critical current measurement geometries.

The following experimental observations on the symmetry cf current transfer were nade cn both the

ccntatts were mace at the surface of the wire and rhe ccpper jacket ensured that the current entering the cere of the conductor was nearly radially symmetric.

The length of the sample in the parallel cr low field

regicn was long enough that the current transfer

density was essentially completed before the wire

*'«brk supported in pare by the Department cf Energy,

Tivision cf High Energy Physics, and the If dice cf

Manuscript received Sov ember 30, 1952

Figure 2. Cress section ct „r.e s _p erccnc.cc mg sanc.e with an cuter ccpper jacket separated by a tar.cal_r

diffusion barrier iron the tore cf brcnce. , arc

enters the perpendicular, high-field region. This was

contact, x. E - close cc the t.rrent contact -as a few

with x, and that there is a finite c.rrer.t oar.sif , Jf, below which the superccncucting filanar.t resistivity is zero, then the current distribution among one filaments must have essentially reachec the distribution that the conductor would rave in a paralle. cr lew field if Che current contact were an infinite

distance away. This postulate only states fat the effect would be observed ar.d would nave ab-c.t f e sane

discettce away. The electric field profile :: e;v er.t cf the sample in parallel cr low fielt -as s mmetnt,

implying chat the transfer into ire cond.tt :r t; t.rs in the sane way as the transfer cut. Ir. the a: e-:e cf a perper.ditular-f ield region, no additional

fer 'in would take place at f is curia*:. *:-a at. additional transfer dees tax.e place w-e* fa

enters the perpendicular , high-fielc regt:* reta.se

the lower critical c.rrenc density free; f e c.rre-t

its t r Tr. non among Che filamer.es tc ca e re _rt: *m This transfer results in Et

’ s as rig.- a; 1M * cm These transfer electric fielcs were symetri:

the tenter cf the magnetic field j.st a a ir : . - r parallel field region. Trts incite; fat :

• starts tc transfer t: the e.ter filaments :t f e :f tr side of center, symmetric with the transfer tr..

A logical extension of these res. Its !• t: what would rapper, if there were n.ltt:.. tar... . * . perpend ieular field settlors' tr-. - . : : r

*

: transfer in and c.t aacng the filane-:; :* perpendicular field seeder.' If t: ; - have a serious iaplicaticr. s for s.ter - - • ~ »

'

design ir. cases w-ere the wt* cf.gs art s •»

' magmt.ee cr angle of the nag-.et . > - times. It would affect the persist e* e. -< lead, ar.d the stadlif cf the s .r T n r- - below was deslgr.ee to ar;.er f es*.

00 IS- 9*6* 53 W00-C;«S01 00 X 19*3 IEEE
Experimental

245

The special shape sample for this study was

formed prior to its reaction, in a rectangular pancake

coil as illustrated in Fig. 3. The bends in the sample have a radius of curvature of about 2 mm. The

results did not show any significant change in the superconducting properties due to the bends, except,

perhaps, for some minor filament inhomogeneities and

breakage. Measurements were made with the magnetic

field in the plane of the coil, pointed down in the figure. The critical current at 1 uV/em was measured

for the seven pairs of adjacent voltage taps in

perpendicular field -sections, and the resulting

critical current data had a range of only 22 indicating a relatively uniform sample. This configuration

has a number of sections that are essentially (within

10°) parallel or perpendicular to the magnetic field.

The ends of the sample are in parallel field. These

sections are sequentially numbered for reference on

Fig. 3 so that all of the odd-numbered sections are in

parallel field and the even-numbered in perpendicular

field. The current contacts at each end of the wire

were 0.5 cm long. The first and the last sections, 1 and 11, were 6.6 cm and 6.3 cm long respectively. As mentioned above, with a parallel field section this

long, the current transfer due to the parallel field

critical current density is essentially complete

before the first perpendicular field section. Notice

also the fact that the magnetic field at the ends of the Bample is not exactly parallel with the sample

(because of alignment and field profile) would only

cause the effect in the first perpendicular field

section to be smaller than if the ends were in a perfectly parallel field.

The data were taken by recording the voltage as a function of current as the current was ramped up and

then down. The voltage was measured to an accuracy of

22 ±2 nV and the current to an accuracy of 0.52 ±0.2 A. The current ramp rate was relatively slow (a few

minutes from zero to I c ), but there still were significant voltages induced in the voltage leads to the sample because of this ramp. These induced voltages

were accounted for by using the recorded voltages, up and down, and the known current ramp rate. The

resulting voltage-current curves were consistent with

voltage-current points obtained by holding the current

fixed (actually the current was still drifting, but at

a rate at least 20 times less). Thus the final

voltage-current curves were independent of ramp rate

in the range tested and the observations are not the result of any inductive effect.

Evidence of the response of the current distribution to multiple parallel and perpendicular field

sections is also given on Fig. 3. The measured

voltage and tap separation of each adjacent pair of

taps is indicated for a current of 75 A in a field of

6 T. The critical current (75 A) at 1 nV/cm was

determined from a measurement of another specimen, so the small additional voltage on the inner configuration taps could be due to a difference in the critical

current, the continuing transfer, or filament inhomogeneities. All of the voltages are positive. They

changed sign and had the same magnitude (within the accuracy of the measurement) when the current direction was reversed. This indicates a symmetry of current transfer (in and out) with the direction of current flow. Furthermore there is a symmetry of current transfer voltage about the center of the configuration, section 6. Notice the similarity of the voltages seen on the following pairs of sections:

1 and 11; 2 and 10; 3 and 9; 4 and 6; 5 and 7.

Section 2

Figure 3. Rectangular pancake coil sample with voltage

in nV (tap separation in cm) for each adjacent pair of taps at a current of 75 A with a field of 8 T in the plane of the coil.

Discussion

The above symmetry of current transfer voltage is somewhat surprising on first consideration. Notice that when the current reaches the first (for now assume electron current flow and section 1 is connected to the negative contact) perpendicular field section, 2, the current then will have to transfer into the inner filaments, since the reduced critical current density forces

the current distribution among the filaments to be more uniform than that in the parallel field region, 1. The expected voltage profile due to this current transfer

is observed along this perpendicular section, 2. Now

consider the symmetric section, 10, here the voltage

profile indicates that the current 6tarts to transfer

to the outer filaments in the same way that the current transferred to the inner ones in section 2. In fact,

almost all of this transfer takes place in section 10,

relatively little takes place around the corner in sec- tion 11, just as happened in section 1. Why does the current start to transfer out in section 10? If the current wants to transfer out before it reaches a parallel field section, why doesn't it transfer out at the end of sections 2, 4, 6, or 6? These results suggest that the presence of the current contact must be

affecting the current distribution for a long length.

The effect of the current contact can be illustrated with equipotential lines as indicated by the

150
246

data. A schematic diagram of the equipotential lines

is given on Fig. 4. Using a time-reversal argument,

these lines will be the same for current entering and leaving the superconductor. The outer copper jacket

is not shown on this figure. The equipotential

aurfaces in the copper jacket should extend radially

outward from the surface of the core, except near the current contact. The parallel and perpendicular field

regions are treated here separately. The effect that

the presence of the perpendicular region has on the

parallel region is discussed in the appendix. The

appendix also has a list of rules for qualitative

construction of equipotential lines in superconductors

and another example. The equipotential lines in

Fig. 4 do take into account the interaction of these

two regions; however, in the following discussion this

interaction was omitted for clarity.

Consider first the easier case to imagine, that

of the current entering the superconductor. The

copper jacket on the superconductor will ensure that

the current entering the core of the conductor at the current contact will be essentially radially symmetric. 2 The current is injected into the outer Superconducting filaments at the joint and it starts to flow along these filaments. However, the current

density is above the critical current density (except

at the very low currents), and thus, a flux-flow

voltage drop will occur along these filaments which

will cause some of the current to transfer through the resistive interfilament material (bronze) into the next inner layer of superconducting filaments. This process continues until the current density in the outer filaments is essentially J*, where J* is the current density below which the superconducting

filament resistivity is zero. I c in parallel field is about four times I c in perpendicular field, so it is expected that J will not be uniform throughout the conductor in parallel field, no matter the length, for any current less than the I c measured in perpendicular

field. There will be a low J in the inner filaments,

thus all the inner filaments will be at about the same potential and there will be very little voltage drop

along them. As the current enters the perpendicular

field region, J c drops and the outer filaments will

again have a flux-flow voltage drop which causes more

transfer. If the current is close to I c , then the equipotential surface will eventually become flat and

perpendicular to the axis of the conductor. Some transfer may occur due to inhomogeneities in the superconducting properties, but other than this, the

equipotential surface will stay flat through the parallel and perpendicular regions until the last

perpendicular region. The current will start to transfer out in the last perpendicular field region.

This transfer out is a little harder to imagine

because the presence of the current contact effects

the equipotential surfaces in the multifilamentary

superconductor a relatively long distance away, more

than 6 cm or 150 times the diameter of the core. The

extent of this long range effect has been observed

before in this sample and in Cu-NbTi samples where

anomalous voltages were reported. 2 The current starts

to transfer out of the center filaments because these

filaments are at a higher potential than the adjacent

filaments. This transfer out will cause the outer filaments to have a flux-flow voltage drop, the inner

filaments will be at about the same potential and

there will be very little voltage drop along the inner

filaments. Once the current has entered the parallel

field region the flux-flow voltage drop along the outer filaments will become very low and very little

transfer out will occur. As the current gets closer

to the contact, the transfer out will increase and more and more flux-flow voltage will appear along the wire and ultimately the current will leave the conductor at the current contact.

Figure 4. Schematic diagram of the equipotential

lines in the core of the superconductor with the magnetic field pointing down the figure.

These data also indicate that the ultimate

current distribution of a multifilamentary superconductor is not uniquely defined by the local conditions. Where the electric field is zero, the current density in a superconducting filament can be any value

below J c

. 3 The value that it will have is determined

by the boundary conditions (current contact) and the external magnetic field in regions where the electric

field is non-zero. The voltage drops along the configuration's inner parallel field sections (3, 5, 7, and 9) indicate there is very little transfer

occurring. The voltage drops along the configuration's

inner perpendicular field sections (4, 6, and 8) could

result in part from slight inhomogeneit ie9 and fila- ment breakage at the bends. So the transfer that occurs here because of the field orientation change la at least eight times smaller than that of the first

and last perpendicular field sections (2 and 10; , if this transfer occurs at all. This suggests that the interfilament material, the bronze, is effectively

isolating the superconducting filaments from each

other. t This does not rule out the possibility that a transfer out would occur if the low or parallel field

section were much longer. Conceivably, this could occur by means of selective transfer (toward the lower

free energy state of the superconductor) at filament

inhomogeneities and discontinuities. If th-s were to occur, the transfer back in would take place at the next perpendicular field section.

t If the conductor were a monofilament, then the current distribution in the conf lgurat lc n

' s inner

parallel field sections would not be as uniform as that for the configuration's inner perpendlc..

sections because the free energy of the »jpt: .d . r is lower for the current flowing in the outer area.

151
Conclusion

247

The current will complete the transfer to a more

uniform distribution In the first perpendicular

magnetic field section. After the current has a uniform distribution, it will not transfer among the filaments everywhere the magnetic field changes. The

current will start the transfer to the outer filaments

during the last perpendicular magnetic field section.

The implication of this for superconducting magnet

design is that once the current has transferred to a uniform distribution among the filaments, it may stay

in this distribution through low or parallel field

sections until the last perpendicular field section.

Thus, putting each end of the magnet wire through an external, high-field region would force the current

into a uniform distribution that would remain for the whole length within the magnet.

Acknowledgments

The author wishes to thank F. R. Fickett and A.

F. Clark for our numerous discussions; J. U. Ekin and

J. R. Clem for their review; and V. L. Grove for

preparing this manuscript.

References

1. L. F. Goodrich and F. R. Fickett, "Critical

current measurements: a compendium of experimental results," Cryogenics 22 , 225-241, 1982.

2. L. F. Goodrich, J. W. Ekin, and F. R. Fickett,

"Effect of twist pitch on short-sample V-I characteristics of multifilament ary superconductors,"

Adv. Cryo. Eng. 28, 571-580, 1982.

3. J. R. Clem, private communication.

Appendix

The presence of the perpendicular region has an

effect on the equipotential lines in the parallel

field region. First consider some rules for qualitative construction of equipotential lines in superconductors:

1)

Equipotential lines can not cross a superconducting filament where J in that filament is less

than J*.

2)

The potential of a super conduct ing filament where

J in that filament is less than J*. is determined

at the first or last place where J “ .

3)

If J in a superconducting filament is everywhere

less than j£, then the potential of that filament

is determined by the potential of the adjacent

material

.

Figure Al. Schematic diagram of the equipotential

lines in the core of the superconductor with a uniform

external magnetic field.

voltage for each equipotential surface will scale

approximately with the current. The equipotential

lines on each end will look like those on Fig. 4 from

the current contact to the 0.3 line (the lines on

Fig. 4 marked 0.00 to 0.2 will not be there). Notice

that if the current is below I c and the distance

between the current contacts is long enough, then a region such as that between the two 0.0 potential

lines will exist and the potential everywhere in this

region will be the same. As the current approaches

I*, the space between the 0.0 lines (on Fig. Al) gets

smaller (both along and across the wire) and ultimately

meet at 1*.

4) The current will flow toward a lower potential even if it is in a superconducting filament that

is below its J*.

5 ) The electric field in the superconducting fila- ments at any point along the wire will be larger

for the outer filaments than for the inner fila- ments (except when zero or near inhomogeneities).

First consider the case where the whole superconductor is in a uniform magnetic field, either zero, parallel, or perpendicular, and the current is below I*

(the critical current equivalent to J* in all of the filaments). A schematic diagram of the equipotential

lines for this case is given on Fig. Al . The equipotential surfaces, shown in cross section as lines, are rotationally symmetric about the wire axis. The

Now consider the case represented on Fig. 4, where

there are two regions of different l c s (parallel and perpendicular field regions in this specific case) with

the second region having a lower Ic* The other end of the configuration will be an image of this end. The presence of the perpendicular field region will cause equipotential lines such as the 0.1 and^0.2 lines (in

Fig. 4) when the current is above the I c of the^perpendicular field. When the current is below the I c of the perpendicular field, a line such as the 0.2 line will

be the last line before the image lines at the other

end of the configuration. Notice that the center fila- ments in the parallel field region have a different

potential because of the presence of the perpendicular

field. This results in a radial electric field and,

thus, some current transfer even^though the filaments

in this region are below their Jc152
APPENDIX D

CRITICAL CURRENT MEASUREMENTS ON A NbTi SUPERCONDUCTING WIRE

STANDARD REFERENCE MATERIAL*

L. F. Goodrich, D. F. Vecchia, E. S. Pittman, and

A. F. Clark

Electromagnetic Technology and Statistical

Engineering Divisions

National Bureau of Standards

Boulder, Colorado

INTRODUCTION

The National Bureau of Standards is producing a standard

reference material (SRM) for the measurement of the superconducting critical current (I ). This SRM and the recently adopted

American Society for Testing and Materials (ASTM) standard test

method (B714-82) will aid both the commerce and technology of

superconductors through the promotion of more uniform measurements. The SRM will serve as an artifact for interlaboratory

comparison to further advance the consensus and evolution of the new test method. The general use and philosophy of an SRM are given in Ref. 1.

To perform well as an SRM, the conductor chosen should be as homogeneous as possible. Conductors were purchased from the

inventories of each of the United States wire manufacturers. Each

conductor was selected by the manufacturer as a good candidate for an SRM. Preliminary screening measurements were performed on each

conductor to determine the short- and long-range spatial variations

in the critical current. The choice of the SRM was based primarily on these data.

The conductor selected for the critical current SRM was despooled onto 500 distribution spools, each with ^2.2 m of wire.

Work supported in part by the NBS Office of Standard Reference

Materials and the Department of Energy.

953

Adv. Crvog . Eng . - Materials , Vol . 30 , Plenum Press, New York (1984), 953-960.

153
954 L. F. Goodrich et al.

Measurements on a sample of the spools were used to determine the

likely outcomes If the spools for distribution were measured. The

statistical analysis presented Is for data taken at 4.07 R. A

complete description of the measurement procedure , temperature

dependence, and statistical analysis will be reported elsewhere. 2 This analysis will be based on the user making one measurement, on one spool, at one of the given magnetic fields (2, 4, 6, and 8 T)

and at any temperature from 3.90 to 4.24 R.

EXPERIMENT

The only significant change from the common instrumentation 3 used to measure I , was the addition of a digital processing

oscilloscope. This allowed the automation of the data analysis

and higher precision in the processing of the voltage-current

(V-I) curves than using an analog X-Y recorder alone. A number of

acquisition and analysis variables were identified and their

effect on the precision and accuracy of the critical current

measurement determined. The acquisition variables considered

were: current ramp rate, digital sampling rate, and voltage

filtering and amplification. The analysis variables and corrections considered were: corrections for the inductive voltage and

changes in thermoelectric voltage, and the number of points in the curve fitting. A correction for the magnetic field profile was made to I measured on adjacent voltage taps. The hydrostatic

head and stratification of the liquid helium bath were also

measured and the correct liquid helium vapor pressure used for the

temperature determination.

The principal sample geometry used was a helical coil. There

are many tradeoffs for each geometry (e.g., long straight, short

straight, hairpin, etc.), but the coil seemed the best for this

type of testing. With the coil, a number of segments of a given

specimen could be measured to test the short range variations.

Also the problems of negative voltage, 4 magnetic field angle and

field uniformity are less in this geometry. The only major problem

is the effect of bending strain which is not well known and is

hard to measure 2 for NbTi superconductors; however, an upper limit can be put on the effect using uniaxial strain data. 3 The plastic

flow of the copper matrix, 3 which would reduce the bending strain

on the NbTi filaments, would make the estimated correction based

on uniaxial strain data an upper limit. With a coll diameter of

3.2 cm and a sample of 0.05 cm diameter, the peak bending strain

would be approximately 1.6Z, assuming the neutral axis does not

shift. A uniaxial strain of 1.6Z would decrease I by about 8Z at

a magnetic field of 8 T. The integral of this effect over the cross section results in a 2Z decrease in the overall value of I of the wire assuming the twist pitch is longer than the current

transfer length, 3 which is a reasonable assumption for these NbTi

superconductors. Because the bending strain correction is not

154
Critical Current Measurements on a NbTi Superconducting Wire SRM 955

exactly known, the correction of the I data to the unbent state

was not made, but it should be less than 22 at 8 T and even less

at the lower magnetic fields.

The calibration of the instruments used to measure voltage,

current, magnetic field, pressure, and length was performed. 2 The

estimated systematic errors (inaccuracies) and random errors

(precision limits) are given in Table 1. The random errors are included in the observed variation in the critical current and for

this reason were separated from the systematic errors. The error

in each of the critical current variables is expressed in terms of

the resultant percentage error in the critical current at each of

the magnetic fields. These errors were estimated using the known

dependence of the critical current on each of the variables. The

periodic and random deviation (PARD) of the current and magnetic

field are not included because the sample current source was a battery power supply and the magnet was used in persistent mode.

The sum and the root-mean-square of the estimated errors are given

at the bottom of the table for each magnetic field.

PRELIMINARY SCREENING

The selection, from the five candidate conductors for the

SRM, was based on the best balance of properties for use as an SRM. The chosen conductor may not be the best for any other

Table 1. Systematic and Random Errors Expressed in Percent Error

in I c at Magnetic Fields of 2, 4, 6, and 8 T •

Variable

Systematic

2 T AT

error

6 T

a)

8 T

Random

2 T 4 T error6 T

(2)

8 T

Current .25 .25 .25 .25 .05 .05 .05 .05

Electric field .05 .05 .05 .07 .05 .05 .05 .07

Magnetic field .10 .12 .21 .45 .04 .05 .05 .09

Temperature .02 .02 .03 .04 .06 .08 .10 .17

Magnetic field

profile

.02 .03 .05 .10

Magnetic field

angle

.10 .10 .10 .10

Tensile strain .15 .15 .15 .15 .07 .07 .07 .07

IA .69 .72 .84 1.16 .27 .30 .32 .45

.33 .34 .38 .56 .12 .14 .15 .22

155
956 L. F. Goodrich et al.

application. A brief preliminary srreening was designed to test

two properties, the long-range (spool-to-spool) and the short-range

(tap-to-tap) homogeneity of I . Two other key properties were an adequate length and a usable copper-to-superconductor ratio. These

four properties were sufficient to pick the conductor to be the SRM.

The homogeneity of I for each candidate was determined by

measuring two specimens, one 5 and one 50 meters from the end of

the shipping spool of each. Five pairs of voltage taps were

placed on each specimen. The length of the specimen measured by

each pair of voltage taps was about 2 cm and the centers of the

adjacent pairs were separated by about 10 cm. The results of this

limited preliminary screening pointed out problems with two of the

candidates. Although the problem might not be present throughout,

both candidates with a possible problem were eliminated.

In no case were the above mentioned problems identifiable with

the choice of a given or measured physical parameter or combination

of parameters. So the particular parameters for each sample will

not be identified. Only the values will be listed here, nonrespectively, for completeness. The wire diameters were: 0.40, 0.51,

0.51, 0.51, and 0.64 mm. The number of superconducting filaments

were: 54, 54, 60, 126, and 180. The approximate superconducting

filament diameters were: 23, 26, 34, 42, and 50 ym. The copperto-superconductor ratios were: 1.4, 1.6, 1.8, 1.9, and 2.0 to 1.

The results of the tests on short and long range homogeneity

for all five conductors are summarized in Figs. 1 and 2. Figure 1 is a plot of the percentage difference between I of the two speci- mens for each sample at various magnetic fields. Notice that the

differences for each sample are about the same at all magnetic

fields measured. Sample 4 had a large shift in I . Another

specimen of this sample, 150 m into the spool, was £hen measured.

The I of this specimen had decreased another 1.1Z from the value

of the 50 m specimen. This may have been due only to an end

effect, but this trend did not look good, so sample 4 was eliminated. Figure 2 is a plot of the percentage difference between the

measured I on the center voltage tap and that of the other four

taps for both the 5 and 50 m specimen of each sample. The expected

tap profile due to the magnetic field profile has been corrected.

Notice that sample 5 seems to have substantially larger variations

and, for this reason, was eliminated.

The other two key properties were then used to select the one conductor to be the SRM. One of the remaining candidates was eliminated because the length delivered was considered to be too

short (less than 400 m) and an adjacent spool could not be obtained.

One of the two candidates left had the lowest copper-to-superconductor ratio (1.4:1). It was eliminated even though the low ratio

did not seem to adversely affect the I c determination, but it

156
Critical Current Measurements on a NbTi Superconducting Wire SRM 957

could affect the usage of the SRM throughout the many kinds of

testing anticipated for an SRM. This left one conductor for

further testing.

PROVISIONAL DATA AND STATISTICAL ANALYSIS

Critical current measurements were performed on the chosen

SRM. Specimen spools were selected by a systematic sampling

procedure to assure that spools were selected along the whole

length of wire. No trend or cycle related to distance along the

wire was found.

The critical current data were obtained by placing three

pairs of voltage taps on each specimen. The length of wire

measured by each tap was about 2 cm and the centers of adjacent

pairs were separated by about 20 cm. At magnetic fields of 2, 4,

6, and 8 T, two repeat determinations of I on each tap were

recorded, producing a total of six measurements on each specimen

at given (controlled) temperature. Data have been collected from

nine spools equally spaced along the length of wire. The physical

theory on the shape of the V-I curve and dependence on the weak

Sample number

Fig. 1. The percentage change

in I between the 5 and §0 m specimens of

each sample, long

range homogeneity.

Fig. 2. Short range homogeneity

of the 5 and 50 m specimens of each sample

(percentage differences

relative to center tap).

157
958 L. F. Goodrich et al.

link suggests that critical current measurements may be nearly

summable on a logarithmic scale. Therefore, the data were first

transformed to estimate the quantities of interest, and then

results on the adjusted scale were converted to the original units

of measurement. In the discussion to follow, the estimates of

critical current are, in fact, geometric means of the original

data because they are derived from simple averages of data on the

logarithmic scale.

Critical current measurements at a given field strength can

be represented by the statistical model given by.

ijk

* D

1 * T

1J eijk

!

i - 1, ... ,9;

j - 1, 2, 3; k - 1, 2,

where Y .. is the logarithm of the measured critical current for

the k-trr repeat determination at the j-th tap on the i-th sample

spool. Each measurement is thought of as the sum of four components: the mean critical current, v; a long-range inhomogeneity,

D^, corresponding to distance along the wire; short-range inhomogeneity, T , corresponding to the particular tap location on a given spool, and a random measurement error, e... . The longand short-range inhomogeneity terms are considered to be random

deviations from the mean critical current for any given spool and

tap location. Both the D 's and T 's are assumed to follow

Gaussian probability distributions with mean zero and variances

Vp and ouu o_ ,y respectively. icapct Wi.VCi.jr • The measurement errors, e assumed to have variance o

2

. ijk* are

An important question for certification is whether all the

available spools for this SRM can be treated as identical. The

components of variance associated with the long- and short-range

material variability of the wire provide a quantitative measure of

inhomogeneity in the critical current SRM, so estimates of each

type of variation were obtained. The estimated standard deviations, expressed as a percentage error in the critical current at

each of the magnetic fields, are given in Table 2. The table

illustrates that long-range lnhomogeneity Is more evident at 2 T

and decreases with Increasing magnetic field.

Because the critical current measurements revealed substantial long- and short-range variation, the uncertainty statement

for the reported critical current at each magnetic field is a statistical tolerance interval. This statistical procedure allows

for the observed variation in critical current by estimating

limits for the critical current of individual spools, rather than

limits on the overall average critical current of all spools.

158
Critical Current Measurements on a NbTi Superconducting Wire SRM 959

Table 2. Estimated Longin Percent Error

and Short-Range Inhomogeneity Expressed

in I at 4.07 K and 0.2 yV/cm.

c

Magnetic

Field

Spool-to-Spool

Standard Dev. (o^)

Taps within Spool

Standard Dev. (o^)

Total , * V>

2 0.41 0.33 0.53

4 0.29 0.26 0.39

6 0.19 0.25 0.31

8 0.18 0.25 0.30

A tolerance Interval for the distribution of critical currents

for a length of wire of about 2 cm has the form, exp [Y ± K S_ ]

,

where Y is the sample mean of all measurements on the logarithmic

scale, S y is the estimated standard deviation of a single measurement, ana K is usually taken from tables such as in Weissberg and

Beatty. 7 The value of K * K(N, f, P, y) depends on:

(1) N: the effective number of observations for Y;

(2) f: the degrees of freedom associated with S^;

(3) P: the proportion of critical current measurements to be

covered; and

(4) y: the probability level associated with the tolerance

interval.

The estimated superconducting critical currents for an electric field criterion 0.2 yV/cm for a wire length of 2 cm are given

in Table 3. The uncertainty of the reported value, ignoring systematic errors, is the statistical tolerance interval constructed

such that it should cover 99Z of critical current determinations

with probability 0.95. The tolerance limits in Table 3 are expressed in terms of the resultant percentage error at each magnetic field and apply to a single measurement on any given 6pool

for a length of wire of about 2 cm. Uncertainties for wire lengths

greater than 2 cm are expected to be less than those in Table 3. The experimental systematic error and the tolerance limit are summed to give the total uncertainty and are all expressed as percent error in I c> CONCLUSIONS

A carefully controlled acquisition and analysis system was developed to measure I . Two of the five candidate conductors displayed short- or long-range inhomogeneity that made them seem unfit

for use as an SRM. The conductor chosen for the SRM had substantial

long- and short-range variation in I c

; therefore the uncertainty

159
960 L. F. Goodrich et al.

Table 3. Provisional Results at 4.07 K and 0.2 yV/cm.

Magnetic

Field

(T)

Critical

Current

(A)

Total

Uncertainty

(2)

Tolerance

Limits

(2)

Systematic

Error

(2)

2 301.34 2.40 2.07 0.33

4 193.87 1.86 1.52 0.34

6 130.65 1.53 1.15 0.38

8 75.58 1.68 1.12 0.56

statement vas calculated using statistical tolerance limits. The

resulting total uncertainty was within the objective of the study.

ACKNOWLEDGMENTS

The help and support of the Superconductors and Magnetic

Materials Group, especially F. R. Fickett and J. W. Ekin, are gratefully acknowledged. Suggestions on the content and presentation of this paper from W. S. Liggett, Jr. of the Statistical

Engineering Division are appreciated.

REFERENCES

1. J. D. Cali, T. W. Mears, R. E. Michaelis, W. P. Reed, R. W. Seward, C. L. Stanley, H. T. Yolken, and H. H. Ku, "The

Role of Standard Reference Materials in Measurement

Systems," NBS Monograph 148, National Bureau of Standards,

Washington, D.C. (1975) 54 pp.

2. L. F. Goodrich, D. F. Vecchia, E. S. Pittman, A. F. Clark,

and J. W. Ekin, to be published.

3. L. F. Goodrich and F. R. Fickett, Critical current measurements: A compendium of experimental results. Cryogenics

22:225-41 (1982).

4. L. F. Goodrich, J. W. Ekin, and F. R. Fickett, Effect of twist

pitch on short-sample V-I characteristics of multifilamentary superconductors, Adv . Cryog . Eng . 28:571-80 (1982).

5. J. W. Ekin, Mechanical and properties and strain effects in

superconductors, in: "Superconductor Materials Science:

Metallurgy, Fabrication, and Applications," S. Foner and

B. B. Schwartz, eds. (Plenum Press, New York, 1981) p. 455.

6. J. W. Ekin, private communication.

7. A. Weissberg and G. H. Beatty, Tables of tolerance - Limit

factors for normal distributions. Technometrics 2:483-500

(1960).

160
NBS

Standard c

Reference Jr

Materials P

-p r"

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

APPENDIX E

Standard Reference Material 1457

Superconducting Critical Current

NbTi Wire

Winter 1984

The NBS Office of Standard Reference Materials announces the availability of the first superconducting wire for critical current measurements as a Standard Reference Material (SRM). It is intended to provide a means for testing the perfor- mance of measurement systems used in the development of superconductors. This SRM consists of approximately 2.2 m

of a multifilamentary niobium-titanium, copper stabilized, superconducting wire wound in a single layer onto a spool

with a core diameter of 8.7 cm. SRM 1457 should prove valuable in determining the overall accuracy of a critical current measurement system that is dependent on numerous variables and effects that can make this seemingly easy measurement very difficult. The critical current for SRM 1457 has been certified at magnetic fields of 2, 4, 6, and 8 T, for temperatures from

3.90 to 4.24 K, and electric field criteria from 0.05 to 0.2 jxV/cm.

An effort was made to keep the use of this SRM as unrestricted as possible. The precautions listed on the certificate,

together with the American Society for Testing and Materials (ASTM) Standard Test Method (B7 14-82). are sufficient for a valid user measurement technique. Some deviations in testing technique, from the method on which the certifica- tion was based, were accommodated by increasing the total uncertainty of the certified critical current. The deviations

that are allowed, and the ones that are not allowed, are identified in the precautions sections of the certificate SRM 1457 may be purchased from the Office of Standard Reference Materials, Room B3I1, Chemistry Building

National Bureau of Standards, Washington, DC 20234, at a price of S219.

0384

161
1). S. Depanmem of Commerce APPENDIX F

Malcolm ^fialdrige ^uram of j^fandarrfs

(Eertiftcate

Reference Material 1457

Superconducting Critical Current - NbTi Wire

This Standard Reference Material (SRM) is intended for use in checking the performance of measurement systems used

in superconductor technology. This SRM consists of 2.2 m of a multifilamentary niobium titanium, copper stabilized superconducting wire wound in a single layer onto a spool with a core diameter of 8.7 cm. Critical current (I c) for the SRM is certified over a range of magnetic fields, temperatures, and electric field criteria. Measurement Technique: Adherence to the precautions given in this certificate (see section Precautions), together with

the American Society for Testing and Materials (ASTM) Standard Test Method B7 14-82 [ 1], is necessary and sufficient for a valid certification. Measurements for certifying SRM 1457 were obtained on a coil of diameter 3.18 cm with a

voltage tap separation of 2 cm. The critical current is defined as the average of the values measured with increasing and

decreasing current.

Certified Critical Currents: The certified critical currents in amperes at 4.2 K for an electric field criterion of 0.2 jiV/cm

are given in the table below. At respective magnetic fields, critical currents of this SRM can be calculated for temperatures from 3.90 to 4.24 K and electric field criteria from 0.05 to 0.2 #iV/cm using:

Ic (T, E) = Ic (4.2, 0.2) • jexp[A(4.2 - T) + B(4.2 - T)

2

] )

• [E/0.2]C

National oTSiandards

Ernest Ambler. Director Rational

Standard

where I c (4.2, 0.2) and the coefficients A, B, and C are given in the table. Critical currents for SRM 1457 were derived

from an empirical equation for the dependence of critical current on temperature (T) and electric field criterion (E):

£n(Ic )= £n(Ir )+A(TrT)+B(TrT)

2 fC UnfE/Er).

In this equation, £n(Ic ) is the natural logarithm of the critical current and I r is the critical current at the reference temperature, Tr , and reference electric field criterion, Er. The experimental data at each magnetic field were fitted by a maximum likelihood procedure using a statistical model [2] that combines the empirical expression above with terms

that allow for material variability (inhomogeneity) among the spools of wire.

Certified Value of Critical Current (I c) at 4.2 K and 0.2 pV/ctn and

Coefficients for Temperature and Electric Field Extrapolation.

Magnetic Critical Total Coefficients for Extrapolation

Field Current Uncertainty A B C

(T) (A) (%)

2.000 293.30* ±2.57 0.218625 -0.04755 0.0172089

4.000 187.38 ±2.01 0.266361 -0.04682 0.0176600

6.000 124.72 ±1.71 0.369479 -0.10488 0.0194218

8.000 69.72 ±1.97 0.649242 -0.27906 0.024831

1

•Extra digits are provided for accurate extrapolation.

Statistical design and data analysis were provided by D.F. Vecchia of the Statistical Engineering Division. Measurements for certification of SRM 1457 were coordinated by L.F. Goodrich. The measurements leading to the development

and certification of SRM 1457 were performed by L.F. Goodrich, E.S. Pittman, and A.F. Clark of the Electromagnetic

Technology Division.

The technical support aspects involved in the preparation, certification, and issuance of this Standard Reference

Material were coordinated through the Office of Standard Reference Materials by R.K. Kirby.

Gaithersburg. MD 20899

June 19, 1984 162

Stanley D. Rasberry, Chief

Office of Standard Reference Materials
Interpretation of Uncertainty: The uncertainty of a certified critical current at each magnetic field is the sum of an

estimated systematic error and statistical tolerance limits computed from the experimental data. The total uncertainty is expressed as percent error in I c and does not change for extrapolated critical currents over the allowable range of

temperatures and electric field criteria. The statistical tolerance limits were constructed so that they should include 99 percent of the critical current measure- ments with probability 0.95. The resulting tolerance interval (and total uncertainty) is valid fora single measurement on

any given spool that is made as directed on a coil of diameter 3.18 cm with a voltage tap separation of 2 cm.

Precautions:

1) This SRM should be carefully handled and stored to protect it against physical damage such as: excessive bending,

scraping, and other deformation. Any excessive physical damage will invalidate the certification.

2) On each spool, the twisted wire ends and an additional 2 cm on each end of the spool core should be discarded.

These sections of the wire are not certified.

3) This certification is invalidated if this SRM is mechanically cycled by demounting and remounting on a specimen

holder. Mechanical cycling can concentrate handling stress, which would lower I c in the stressed regions of the conductor.

4) This certificate is based on a slow cooling of the specimen mounted on aG-1 1 tube (circumferential fiber direction) by

gas heat exchange with a liquid nitrogen precooled Dewar [2). For a valid certification, the specimen must be

measured on a suitable specimen holder [1]. The effect of a rapid cooling by immersion into liquid nitrogen or liquid

helium can change I c owing to dynamic differential thermal contraction. For rapid (immersive) cooling, 0.25% must

be added to the total uncertainty even if a suitable specimen holder is used. It is conceivable that a user could

demonstrate that the particular system and technique employed does not have a cumulative thermal or mechanical

cycling effect. In this case, the specimen still has utility, but this SRM cannot be certified beyond one thermal

cycle.

5) This certification is only valid for a zero-to-l c ramp time in the range of 30 to 300 seconds for all magnetic fields [2] Also, the certification was based on the assumption that voltage filtering and instrumentation response times

contribute negligible error to the measured value of I c . A nonnegligible effect can be removed by averaging the I c values measured with increasing and decreasing current at a constant ramp rate. A nonnegligible effect must be

removed for a valid certification.

6) A chemical wire stripping compound should be used to remove the insulation from this SRM. A phenol/methylene

chloride wire stripping compound was found to adequately remove this insulation.

7) If the specimen temperature exceeds 250 °C, the certification is invalidated. The current and voltage contacts should

be soldered carefully to avoid overheating and physical damage. If a specimen enters the normal state (quenches)

while carrying a high current density, it could melt within a few seconds. An adequate quench protection circuit may

be necessary [1], A typical current shutdown time of 10 ms is adequate.

8) For a voltage tap separation of more than 2 cm, the uncertainty in I c should be less. For a voltage tap separation of

less than 2 cm, the uncertainty in I c may be more. This certification is only valid for a voltage tap separation greater

than or equal to 2 cm.

9) If this SRM is measured with a bend diameter other than 3.23 cm (coil of diameter 3.18 cm), the results may be

different. Uniaxial strain data was used to determined the expected upper limits to the bending strain effect For

bending diameters from infinity (straight) to 1.6 cm, the certified critical current values can be used only if the following amount is added to the total uncertainty:

G •

| 1 -(3.23/d)

2 |,

where d is the bend diameter in centimeters and G=1.10, 1.20, 1.36, and 1.70% at 2, 4, 6, and 8 T, respectively

Noncertified Values at Other Criteria: Critical current measurements were made on the sample spools at electric field criteria 0.02 and 0.5 jiV/cm. However, SRM 1457 is not certified to the extended range of electnc field because

measurements at the additional criteria did not conform to the required measurement procedure A comparison of observed critical currents to values extrapolated from the certifying equation is given in reference 2 for information on l\ Most of the measurements were within the range computed using tolerance limits that are only valid between 0 05 and

0.2 jjV/cm.

References:

1. Standard Test Method for D-C Critical Current of Composite Superconductors, Annual Book of AS IM Standards

ASTM B7 14-82, Part 2.03, pp. 595-98, American Society for Testing and Materials. Philadelphia, F’a (1983

2. Goodrich, L.F., Vecchia, D.F., Pittman, E.S., Ekin, J.W., and Clark, A F., Critical Current Measurement^ on an NbTi Superconducting Wire Standard Reference Material, NBS Special Publication 260-91 f I9S4>

163
U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

SHEET (See instructions) NBSIR 85-3027 January 1985

4. TITLE AND SUBTITLE

DEVELOPMENT OF STANDARDS FOR SUPERCONDUCTORS

Interim Report

January 1982 —December 1983

5. AUTHOR(S) l Goodrich, J. V. Minervini, A. F. Clark, F. R. Fickett, J. W. Ekin,

E. S. Pittman

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Department of Energy

Washington, D.C. 20545

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gn i fi cant

bi bliography or literature survey, mention it here) A cooperative program with the Department of Energy, the National Bureau of Standards,

and private industry is in progress to develop standard measurement practices for use

in large scale applications of superconductivity. The goal is the adoption of voluntary standards for the critical parameters and other characterizations of practical

superconductors. Progress for the period January 1982 through December 1983 is re- ported. The major effort was the procurement, selection, and certification of the

first superconducting wire for critical current measurements as a Standard Reference

Material ( SRM 1457). Other work reported here includes: effect of geometry on current transfer; lap-joint resistance; and ac losses.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

critical current; critical parameters; losses; measurement methods; standards;

superconductor

13. AVAILABILITY

{£\ Uni imited

| | For Official Distribution. Do Not Release to NTIS

n Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

(jQ^Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

168

15. Price

U.S. GOVERNMENT PRINTING OFFICE:1985— 579-977 / 25313 USCOMM-DC 6043-P80