A11102 Sbllb?

NATL INST OF STANDARDS & TECH R.I.O.

!

REFf All 102561 167

mbs TRANSIENT LOSSES

publications SUPERCONDUCTORS

Final Report

IN

R. B. Goldfarb, Editor

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

June 1986

- QC

ion

• U56

86-3053

1986

RESEARCH

INFORMATION CENTER

NBSIR 86-3053

TRANSIENT LOSSES IN

SUPERCONDUCTORS

Electromagnetic Technology Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

June 1986

Prepared for: Air Force Office of Scientific Research

Bolling Air Force Base

Washington, DC 20332

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

Final Report

R. B. Goldfarb, Editor

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TRANSIENT LOSSES IN SUPERCONDUCTORS

Final Report

R. B. Goldfarb, Editor

Electromagnetic Technology Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Under steady-state conditions, there are no losses in superconducting

wires. However, when subjected to alternating or transient magnetic fields or transport currents, losses in type-II superconductors can become significant.

This report deals with hysteresis losses at 4 K measured by magnetization and

complex magnetic susceptibility. The theoretical and experimental relationships between ac susceptibility and magnetization as functions of dc field were examined in terms of the critical-state model as developed by Carr and

Clem. Minor-loop hysteresis loss was shown to be obtainable by direct measurement of loop area, from the imaginary component of ac susceptibility, and from

the reversible susceptibility plus the critical current density or fullpenetration field. Hysteresis and transport losses measured simultaneously in

a Nb-Ti superconducting coil were found to agree substantially with the

predictions of Minervini’s two-dimensional model. Hysteresis loss measurements

in a series of fine-filament Nb^Sn superconductors showed that the effective

filament diameter is a function of interfilament separation and local area

ratio of matrix material to Nb. A review of internal fields in superconductors showed the importance of demagnetization factors and internal fields

for the correct analysis of magnetic data. A theoretical method of calibrating

ac susceptometers for cylindrical specimens, which is based on a mutualinductance calculation, was developed.

Key words: ac losses; hysteresis; magnetization; magnetometers; niobium-tin;

niobium-titanium; superconductors; susceptibility; susceptometers; transient

losses

iii

TRANSIENT LOSSES IN SUPERCONDUCTORS

Final Report

EXECUTIVE SUMMARY

Under steady-state conditions, there are no losses in superconducting

wires. However, when type-II superconductors are subjected to alternating or transient magnetic fields or transport currents, losses can become significant.

Consequently, they are an important design consideration in many practical

superconductor applications. AC losses may be categorized into hysteresis

losses, eddy-current and coupling losses, and self-field losses.

This report deals mostly with hysteresis losses at 4 K. The experimental

methods developed primarily involve the measurement of (1) magnetization as a function of field, and (2) complex magnetic susceptibility as functions of dc

field and ac-field amplitude and frequency. Most measurements were made on variations of commercial multifilamentary Nb-Ti conductors with no transport

current in longitudinal fields. However, some results are reported for Nb-Ti

with transport current and Nb^Sn conductors, both in transverse fields. In

general, the experimental relationship between complex ac susceptibility and

magnetization as a function of dc magnetic field was examined in terms of the

critical-state model as developed by Carr and Clem. Good agreement between

experiment and theory was found.

While the dc magnetic methods measure only the hysteresis component of

total loss, hysteresis is usually the largest component of loss below 1 kHz.

Additionally, magnetic hysteresis measurements give information on the remanent

magnetization due to trapped flux in superconducting wires. This information

is useful in selecting conductors for specific applications. For these

reasons, it is likely that magnetic methods will emerge as a standard measurement technique for quantifying ac losses.

In addition to magnetic techniques, two other ways to measure transient

losses in superconductors were evaluated. These methods were an electrical

wattmeter and a calorimeter. These two other methods were found to be

experimentally rather difficult compared to the magnetic methods. The

electrical method requires the measurement of very small changes in phase

angle between nonsinusoidal voltage and current. The calorimetric method

requires the measurement of very small incremental helium flows above the

normal static boil-off. More development would be required, compared to the

magnetic methods, for the rapid acquisition of useful materials information.

Program Obj ectives

The following were program objectives and their related outputs:

1. Correlate hysteretic losses in a given material for different sample

configurations

Paper I: Hysteretic losses were studied in Nb-Ti superconductors with

different filament diameters, numbers of filaments, and ratios of matrix

v
material to superconductor. Hysteresis decreased with decreasing filament

diameter and was independent of conductor cross-sectional shape. The loss per

volume of superconductor, expected to be independent of matrix-tosuperconductor ratio, sometimes showed significant unexplained dependence.

2. Study the effect of matrix material on the losses for a given filamentary

superconductor. Study coupling losses as a function of filament density and

filament-matrix bonding.

Paper V: The effect of local area ratio of matrix material to superconductor

was studied in fine-filament Nb^Sn conductors. The hysteresis losses depended

on the interfilament separation and area ratio.

3. Extend existing theoretical model for transient transport current and

transient external field.

Paper IV: The two-dimensional model for shielding-hysteresis and transportcurrent losses was experimentally verified for the first time.

Results

A summary of significant results follows:

1. Hysteresis loss was shown to be accurately predicted within a few percent,

using the London-Bean critical-state model, from the full-penetration field as

a function of maximum applied field. It was shown that the full-penetration

field could be deduced directly from the hysteresis loop, without reference to

the critical current density. (See paper I.)

2. Minor-loop hysteresis loss was shown to be obtainable in three ways: by

direct measurement of loop area, from the imaginary component of ac susceptibility, and from the reversible susceptibility plus the critical current

density. The loss expressions derived by Carr were shown to be the same as those derived by Clem in the limit where the reversible susceptibility is zero. This was the first correlation of the different experimental magnetic

methods. The relationships between complex magnetic susceptibility and

permeability were discussed. (See paper II.)

3. A theoretical expression for the reversible susceptibility plus experimental measurements of full-penetration field were found to predict the measured

real and imaginary components of ac susceptibility as functions of applied

field using Clem's model. This was the first test of the applicability of the

model to practical superconductors. The full-penetration field was found to

be inversely proportional to applied longitudinal field, thus following a Kim-type relationship. The shape of the hysteresis loop was used to deduce

the superconductor penetration depth and approximate the lower critical field.

(See paper III.)

4. Hysteresis and transport losses measured simultaneously in a Nb-Ti superconducting coil were found to substantially agree with the predictions of

Minervini's cylindrical model. This was the first experimental verification

of the two-dimensional theory. (See paper IV.)

vi
r

> . Systematic hysteresis loss measurements in a series of fine-filament Nb„Sn

superconductors showed that the effective filament diameter is a function of

interfilament separation and local area ratio of matrix material to Nb. The

effective filament diameter was obtained by applying the critical-state model

to measured hysteresis loops. There would be little filament interaction due

to bridging for Nb^Sn interfilament separation greater than 2 ym. This was

the first study of filament coupling in Nb^Sn. The study is expected to have

an impact on the design of practical wires. (See paper V.)

6. A review of internal fields in superconductors was prepared. The importance of demagnetization factors and internal fields for the correct analysis

of magnetic data was discussed. The review will be a useful guide for the

calibration of magnetometers used in measuring hysteresis losses in superconductors. (See paper VI.)

7. A new theoretical method of calibrating ac susceptometers for cylindrical

specimens that is based on a mutual-inductance calculation was developed. The

results were in excellent agreement with calibrations using standard reference

materials. (See paper VII.)

8. A multifunction circuit was designed to protect instruments connected or coupled to a superconducting magnet against large voltage surges in the event

of a quench. (See paper VIII.)

9. A review was prepared of the methods used for characterizing alloys,

especially superconductors, with regard to matrix-to-superconductor volume

ratio, filament radius, density, and weight and atomic percents. This review

will be useful for some users and manufacturers of multif ilamentary wires.

(See paper IX.)

This project required the development of instrumentation for magnetic

measurements and computer programs for data acquisition and processing. It

resulted in the publication of five papers, the preparation of four other manuscripts (three of which will be submitted for publication), and participation

in a workshop on problems in superconductivity [Cryogenics 24, 378 (1984)].

vii
t
CONTENTS

Abstract iii

Executive Summary v

I. Hysteretic Losses in Nb-Ti Superconductors

R. B. Goldfarb and A. F. Clark 1

II. Magnetic Hysteresis and Complex Susceptibility as Measures

of AC Losses in a Multif ilamentary NbTi Superconductor

R. B. Goldfarb and A. F. Clark 5

III. AC Losses in Nb-Ti Measured by Magnetization and Complex

Susceptibility

R. B. Goldfarb and A. F. Clark 9

IV. Losses in a Nb-Ti Superconductor as Functions of AC Field

Amplitude and DC Transport Current

M. Dragomirecky , J. V. Minervini, J. W. Ekin,

R. B. Goldfarb, and A. F. Clark 17

V. Hysteresis Losses in Fine-Filament Internal-Tin

Superconductors

R. B. Goldfarb and J. W. Ekin 23

VI. Internal Fields in Magnetic Materials and Superconductors

R. B. Goldfarb

' 35

VII. Calibration of AC Susceptometer for Cylindrical Specimens

R. B. Goldfarb and J. V. Minervini 43

VIII. Quench Circuit for Electronic Instruments Used with

Superconducting Magnets

R. G. Benson, R. B. Goldfarb, and E. S. Pittman 47

IX. Characterization of Alloys

R. B. Goldfarb 53

X. Units for Magnetic Properties

R. B. Goldfarb and F. R. Fickett 57

Hysteretic losses in Nb-Ti superconductors

R. B. Goldfarb and A. F. Clark

Electromagnetic Technology Division, National Bureau ofStandards. Boulder, Colorado 80303

When subjected to transient magnetic fields, superconductors exhibit losses. At low frequencies, most of the dissipation is hysteretic. Magnetization was measured in an axial field for eight

multifilamentary Nb-Ti superconducting wires with different filament sizes and different ratios of copper to superconductor. The full-penetration field Hp was estimated from the high-field ends of

the hysteresis loops. The estimate ofHp provides a method to assess the critical current density Jc

. There was good agreement between measured losses and those predicted from Hp and the peak

applied field.

INTRODUCTION

An ideal type I superconductor is perfectly diamagnetic

below its critical temperature Tc and critical field Hc

. This is a consequence of the Meissner effect, B=n0(H + M) = 0, X = M/H=- 1, (1)

in SI units, where B is the magnetic flux density, H is the magnetic field strength, Mis the magnetization, and \ is the

susceptibility. The diamagnetism arises from circulating

persistent surface currents. In type II superconductors the pure Meissner state ap- pears only below Hcl . Between HcX and Hcl (lower and upper critical fields) the “mixed” state occurs in which surface currents coexist with magnetic vortices, usually containing one flux quantum each, formed by local persistent currents. These vortices are analogous to paramagnetic moments. 1 The effective magnetization (magnetic moment per unit vol- ume), owing to surface and vortex currents, can be measured

in the same way as the magnetization of magnetic materials. According to the “critical state” model, as the applied

field increases above Hcl , the magnetic flux and a current

density J penetrate toward the center of the superconductor.

The magnitude of / is Jc , the critical current density, taken

to be independent of field. The field at which complete penetration occurs is termed the full-penetration field Hp

. For a

cylindrical filament with a field applied axially, Hp =Jc r, (2) where r is the filament radius. In “hard” type II superconductors, such as Nb-Ti, pinning sites impede the motion of flux vortices, which gives a

higher value of Jc

. Consequently, when the applied field is

TABLE I. Physical properties of multifilamentary Nb-Ti superconductors.

cycled, hysteresis occurs. Flux pinning is the largest source

of bulk hysteresis, and hysteresis is the largest component of

losses for ac fields below about 1 kHz. It is important to minimize these losses in non-dc superconducting-magnet

applications in fusion, accelerators, and pulsed power

sources. To minimize hysteresis, almost all commercial NbTi wires are composed of many fine superconductor fila- ments, 1 to 25 nm in radius, within a thermally stabilizing

Cu matrix. The weak diamagnetic contribution of the Cu to the effective magnetization is negligible.

Simple magnetic measurements can be useful to fabri- cators and users of superconductors. In this paper, measured

hysteresis losses are compared to those predicted from the- ory for a number of superconductors.

EXPERIMENT

A vibrating-sample magnetometer (VSM) was used to obtain dc hysteresis loops at about 4.3 K. The magnetic field was supplied by a superconducting magnet. Field accuracy was estimated to be within ± 4 kA/m (50 Oe), and magnetization accuracy to be within ±5%.

Each sample consisted of about 5-14 1.5-cm-long seg- ments of insulated multifilamentary Nb-Ti wire, bundled to- gether. For such lengths the filaments may be treated as uncoupled. The field and all measurements were along the longitudinal axis. There was no transport current applied to the wires. Magnetization was calculated per volume of supercon- ductor filaments. Demagnetization factor effects were negligible; the internal field was taken to be the same as the exterSample number

Dimensions without

insulation (mm) Number offilaments

Filament

radius ( /im) Cu/Nb-Ti volumeratio

1 if = 0.51 180 12.2 1.46 2 0.57X0.84 132 13.0 4.98

3 d = 0.68 2046 4.3 2.08 4 d = 0.64 180 15.0 1.50

5 0.52x0.70 180 16.4 1.30 6 0.64x0.89 168 19.9 , 1.59 7 d= 1.01 114 19.5 4.84

8 0.63X0.88 240 19.3 091

3809 J. Appl. Phys. 57 (1 ), 1 5 April 1 985 3809

1
Externa] Field (kOe)

o>c

ooa2

c

50

<a 2

<ue

00

(0 2

-20 -10 0 10 20

Sample

<

-*i l«~ 2Hp

Pinal

1 ^branch

Initial p

branch

i iJL I

External Field (kOe)

-20 -10 0 10 20

-1 o 1 External Field (MA/m)

2-2 -1 o 1 2 External Field (MA/m)

External Field (kOe)

-2-10 1 2 External Field (MA/m)

FIG. 1. dc hysteresis loops for eight multifilamentary Nb-Ti superconductors. Note that the vertical scales are not all the same.

nal applied field. Before each set of measurements was made,

the sample was warmed above Tc (about 10.2 K for Nb-Ti)

and then cooled in zero field. MEASUREMENT OF HYSTERESIS

Table I summarizes the physical properties of the su- perconductor wires studied. Figure 1 shows the hysteresis

loops for the samples for a peak applied field H0 of 1.6 MA/m (20 kOe). Occasional spurious nonsmooth behavior

is accounted for by slight deviations in temperature during measurement. The area enclosed by each loop is equal to the

loss W,

W= ti 0 j)HdMs*n0 j)MdH, (3)

3810 J Appl. Phys., Vol. 57, No. 1 , 15 April 1985

where H ranges from + H0 to — H0

. The full-penetration field Hp can be estimated from the

hysteresis loop. As the loop is traversed, the filaments go from full penetration in one direction to full penetration in the other direction. Therefore, at a high-field end of the loop,

Hpis approximately one-half the field required to reverse the magnetization. This reversal field 2Hp

is determined by a peak in d

2 M/dH 2 and is indicated on the loop for sample 1 in Fig. 1. For the case when the peak applied field H0

is greater than Hp

, Carr derived an expression for the loss, assuming

Jc to be independent of magnetic field.

2 In SI units, W = (A^0H0Hp

/'i){\ - Hp /1H0) . (4) Clearly H0

is much larger than Hp for the loops in Fig. 1.

R. B. Goldfarb and A. F. Clark 3810

2
TABLE II Hysteresis loss W for a peak field H0 of 1.6 MA/m (20 kOe|.

Sample

number

Loss from loop

area (kJ/m3

) (kA/m)

",

(Oe)

Loss from

H„ (kJ/m3

)

1 114 45 550 120

2 116 45 550 120

3 50 20 250 50 4 152 50 650 130

5 158 65 800 170 6 141 60 750 160

7 177 65 800 170

8 224 80 1000 210

Losses calculated numerically using Eq. (3) and computed

using Eq. (4) are compared in Table II. Considering the ap- proximate nature of the Hp determination, the agreement is good.Based on Eq. (2), Jc can be estimated from Hp

. This

could be useful for new superconductor materials where it is difficult to measure^ directly in wire samples. In this paper, Jc is the critical current density for the filaments, not the

entire wire. For H0 >Hp

, Eqs. (4) and (2) give Jc in terms of measured loop area W-. Jc =lW/\nJlj. (5) COMPARISON OF HYSTERESIS LOOPS

Although different wires from different manufacturers

are listed in the tables, some comparisons may be made.

Equations (4) and (2) indicate the well-known fact that hysteresis decreases with decreasing filament radius. This is il- lustrated in a comparison of the losses in samples 1 and 4. The wires are virtually identical except for filament radius. An extreme case is sample 3, with a large number of very fine filaments. A comparison of samples 4 and 5 shows that con- ductor geometry (round vs. rectangular cross section) has no

significant effect on the loss. Since magnetization is plotted per unit volume of super- conductor (Nb-Ti) rather than per unit wire volume, the cop- per-to-superconductor volume ratio is not expected to have a

large effect on the magnetization values. A comparison of samples 1 and 2 shows little difference. However, samples 6 and 8, with similar filament radii, show significant differ- ences in measured losses.

3811 J. Appl. Phys., Vol. 57, No. 1, 15 April 1985

The shapes of the loops are somewhat intriguing. The

loops for samples 1, 2, 3, and 4 are generally trapezoidal and

negatively sloped. Those for samples 5, 6, 7, and 8 are generally hexagonal and more hysteretic. The difference in types is highlighted if the initial branch of each loop is compared to the final branch (see Fig. 1, sample 1). For the hexagonal set, there is an attempt, upon increasing the field from zero, to retrace the peak of the initial branch. These curves most

resemble the ideal reversible magnetization curve that is ex- pected from the critical state model.

CONCLUSION

Measured hysteresis losses for eight multifilamentary

Nb-Ti superconductors were found to be in good agreement

with predicted values based on the maximum applied field H0 and the full-penetration field Hp

. The latter, a function of

critical current density Jc and filament radius r, was estimat- ed from the hysteresis loop. This estimate of Hp can be a convenient method of assessing the critical current density.

ACKNOWLEDGMENT

This work was sponsored by the Air Force Office of

Scientific Research.

1 W. J. Carr, Jr., IEEE Trans. Magn. MAG-21, 355 ( 1985).

2 W. J. Carr, Jr., ACLoss and Macroscopic Theory ofSuperconductors (Gor- don and Breach, New York, 1983), p. 70.

R. B Goldlarb and A. F Clark 3811

3

332 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 2, MARCH 1985

MAGNETIC HYSTERESIS AND COMPLEX SUSCEPTIBILITY AS MEASURES OF

AC LOSSES IN A MULTI FILAMENTARY NbTi SUPERCONDUCTOR

R. B. Goldfarb and A. F. Clark Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado 80303

Abstract

Magnetization and ac susceptibility of a standard

NbTi superconductor were measured as <. function of longitudinal dc magnetic field. The ac-f ield-amplitude

and frequency dependences of the complex susceptibility

are examined. The magnetization is related to the susceptibility by means of a theoretical derivation

based on the field dependence of the critical current

density. Hysteresis losses, obtained directly from dc hysteresis loops and derived theoretically from ac susceptibility and critical current density, were in reasonable agreement.

Introduction

When subjected to alternating or transient magnetic

fields, superconducting wires exhibit losses. These

may be categorized into hysteresis losses, eddy-current

losses, coupling losses, and self-field losses. Bulk hysteresis losses are frequency independent and arise mostly from flux pinning in the superconducting filaments. There may be, in addition, hysteretic losses

due to pinning at the filament surface. At high

frequencies, eddy-current losses occur within the matrix and on the wire surface (skin effect). Coupling

losses in the matrix, important over about 1 kHz, are caused by the transfer of coupling currents between

filaments. If there is a transport current, matrix

eddy-current losses arise from the self field. There are a few general methods of measuring ac losses in superconductors. The most direct is calorimetric, in which a superconducting coil is exposed to transient fields and the dissipated heat is measured

by the boil-off of liquid helium. Another method is electrical, by measuring the phase angle between voltage and current. Wattmeters work on this principle.

The final methods are magnetic. Magnetization

measurements are possible because magnetic moments are associated with circulating persistent surface currents

and, in type-II superconductors in the mixed state,

flux vortices. There are basically two types of magnetic methods. In the first, magnetization (mag- netic moment per unit volume) is measured directly

using, for example, a vibrating-sample magnetometer.

Though lock-in detection at the frequency of vibration

is used, this is a dc measurement, since the time required to cycle the field is on the order of 30 minutes. Measurements may be made while slowly ramping

or stepping the field. The latter gives more accurate

results if there is any imbalance in the pick-up coils.

In the second type of magnetic measurement, a sinusoidal ac field is applied (sometimes superimposed

upon a dc field) and the derivative of the magnetization is detected by induction in a pickup coil. If the pickup coil is uncompensated, the induced voltage is proportional to the permeability u. If the coil is compensated (i.e., zero voltage when no sample is present) , the induced voltage is proportional to the susceptibility x* In SI units,

Contribution of the National Bureau of Standards, not subject to copyright.

’-ip" * Uq

1 + x) * Pq(1 +x'-1x")

(la)

C o

+ X (lb)

u 0 x". (lc)

where single primes indicate the real (dispersive,

inductive) components, the double primes indicate the imaginary (absorptive, resistive, loss) components, and

Up is the permeability of free space. The induced voltages may be integrated with respect to field (or with respect to time, since field is swept as a function of time) to yield the frequency-dependent

magnetization

1 The area enclosed by the dc or low-frequency ac hysteresis loop of magnetization vs. field is a direct measure of the hysteresis loss. Alternately, the imaginary component of susceptibility (or permeability)

at low frequencies is an indirect measure of the hysteresis loss. This paper will discuss dc hysteresis

loops and complex susceptibility. The commonly used method of integrated voltages will not be discussed.

The conductor carries no transport current.

Experimental Methods

DC hysteresis loops at about 4.1 K were obtained

using a vibrating-sample magnetometer (VSM) . Field values up to 1.6 MA/m (20 kOe) were determined from the current in a multifilamentary Nb-Ti superconducting

magnet. No correction was made for trapped flux; zero- field-value errors are thus estimated to be ±4 kA/m

(30 Oe). AC susceptibility was measured at 4 K with a lock-in amplifier and a compensated susceptometer of the type described in Ref. 2. If the induced waveform is nonsinusoidal , the lock-in amplifier will detect only

the fundamental-frequency component. Losses that appear as harmonics are not detected. Therefore, this method is most suitable for relatively small ac field amplitudes. For very small ac field amplitudes, x" is known to be about zero, and this fact is used to adjust

the phase 4> on the lock-in amplifier:

X

1 = | x I cos<t> , (2a)

x" = | X | s in <f> . (2b)

The phase setting was kept constant for all fields, but was readjusted for each frequency. The background

signal in the absence of a sample was subtracted from

all measurements. The susceptometer was positioned in the bore of the same superconducting magnet as used with the VSM. Both apparatus were computer controlled. Magnetization and susceptibility accuracies are esti- mated to be within 5%. The sample consisted of an insulated, Nb-Ti wire,

cut into 14 straight segments 1.5 cm long, and bundled

together. At low frequencies and for short samples (or short twist pitch), the filaments may be taken to be uncoupled. The field and all measurements were along

the longitudinal axis. Before each set of measurements

was made, the sample was warmed to above its critical

temperature and then cooled in zero field. Demagnetization-factor corrections were negligible; the Internal Manuscript received September 10, 1984. field was taken to be the same as the external applied

00 1 8-9464/85/0300-03 32$0 1.00© 1985 IEEE

5
333

field. Magnetization and susceptibility were calculated per volume of superconductor filaments. Mass and volume measurements were made on a length of the wire, before and after dissolving the Cu matrix in HNO^, to experimentally determine the filament radius and the volume and density of NbTi. The characteristics of the wire are given in Table I. 3

AC Field Dependence of Complex Susceptibility

The ac susceptibility of the sample was measured for

a few field amplitudes and frequencies as the dc bias field was stepped. Figures la-e show a comparison of the results. Only three branches are shown for each

case to avoid redundancy. The shapes of the x' and x" curves are dependent upon both field amplitude and frequency. The frequency dependence is small, which is evidence that hysteresis is the major loss mechanism.

For the smaller field amplitudes there are large differences between ascending and descending branches,

and the branches are not symmetrical for positive and negative external dc fields (Figs. la-c). These

effects may be attributed to relatively large filamentsurface losses when Che ac field is small. Bussifere and Clem 4 discuss the applicable case when losses are associated with the bending of vortices trapped at a high angle to the surface, and the field is much less

than the lower critical field H .. Short samples facilitate this effect. C

External Field (kOe)

-20 -10 o 10 20

External Field (MA/m)

GOo

a

a>o

V)D

cn

CD£

3

"o>

toMO

aIDO

to3

CO

3

"o>

Table I. Characteristics of NBS SRM 1457

Number of filaments: 180 Filament radius: 12.20 um Copper-to-NbTi volume ratio: 1.456 Density of NbTi alloy: 6.004 g/cm 3

* NbTi-to-insulated-wire mass ratio: 0.3025

External Field (kOe)

-20 -10 0 10 20

External Field (MA/m)

Figure 1. Imaginary (x") and real ( x

* ) components of ac susceptibility as a function of applied

longitudinal dc field at 4 K. AC field amplitudes and frequencies are shown. Erratum: Page 333, first column, next-to-last sentence: "field" should be "ac field"

6
)J4

External Field (MA/m)

Figure 2. Magnetization M vs. dc field H at 4.1 K obtained with a VSM. The initial branch for zero trapped flux and four minor loops (a-d) are shown.

Theoretical Derivation of Complex Susceptibility

The following theoretical description of complex

susceptibility is based on the development by Clem for

the case of bulk-pinning hysteretic losses in type-II

superconductors. 5 The model applies when the ac field amplitude is small and the field equations can be linearized. It will be shown that the dc hysteresis

loop and the field dependence of the critical current

density are predictive of the ac susceptibility.

The critical current as a function of transverse

field 6 for the sample was converted to critical current

density J using the total superconductor crosssectional area (not including matrix material). A good

fit was obtained to the Kim equation,

J (H) - J (0) / ( 1+H/H.) ,, (3)

C C U for fields H up to 4.8 MA/m (60 kOe) . The adjustable

parameters J (0) and H„ were found to be 9.2 GA/m 2 and

1.1 MA/m (14

c k0e), respectively. While the longitudinal-field J is larger than the transverse-field

J , they are, of course, identical at zero field.

c The full-penetration longitudinal field H is

a function of H owing to the field dependence^of J^:

H (H) - J r, (4)

P c where r is the filament radius. It is useful to define

a parameter x as

x(H) - h/H - h/J r, (5)

P

c where h is the peak amplitude of the ac field that is superimposed upon H. As shown by Clem and using Eqs. 1, the complex

susceptibility may be expressed in terms of an Idealized differential susceptibility for type-II

superconductors X rev: (Uxrev

) *l

(x) " 1 * (6) anrev>s 2 (*>. (7)

where

X - dM /dH rev rev

( 8 )

where M is the reversible magnetization, and

rev °

gj(x) - x-5x2 /16, 0 < x < 1, (9a)

g (x) - [ (-l+x-5x 2

/ 16) 0 + (-4/3x+2-7x/ 4+1 3x 2

/ 24) sin

+ (-l/2+x/2-x 2 /6)sin28 (-x/ I2+x s /24) sin36

+ (-x 2

/ 192)sin46 ] /ir + 1, x > 1, (9b)

where 8(x) - 2sin *(x S , 8(1) - it, 8(®) - 0, and

g^(x) - (4x-2x 2

) /3it , 0 < x < 1, (10a)

g

2 (x) - (4/x-2/x 2

) /3ir, x > 1. (10b)

Using Eqs. 6 and 7, x‘ and x" will be calculated for

the sample. The major hysteresis loop for the sample

is shown in Fig. 2. The lower critical field H . is

C 1

External Field (kOe)

-20 -10 0 10 20

o.oe

0.03

000

-0.03

-0 06

-2-10 1 2 External Field (MA/m)

Figure 3. Imaginary (x") and real ( x

' ) components of ac susceptibility calculated for an ac field amplitude

of 56.2 kA/m (706 Oe) using the theoretical model, the data in Fig. 2, and the field dependence of (Eq. 3)

CC£

2

*>a

v

6

3

•>

0.5

0.0

-0.5

_i n

7

Volume

Susceptibility

(cgs)
Table II. Comparison of hysteretic losses obtained by dc and ac methods

for a field amplitude of 56.2 kA/m (706 Oe).

335

Center

(kA/m) field

(kOe) Loss from minorloop area (kJ/m 3

) Loss from x"

Eq. 11 (kJ/m 3

) Loss from Eqs.12, 13 (kJ/m 3

) 0 0 0.6 0.7 0.9

400 5.0 1.3 2.0 2. 1 800 10.0 1.4 1.8 2.6 1540 19.3 1.3 1.6 2.6

about 0.12 MA/m (1.5 kOe) . The tangent to the curve

at any H is the differential susceptibility Xj

c

- As an approximation, Xj will be used for X rev

- This is somewhat reasonable for the initial branch of the hysteresis loop, but rather unreasonable for the later irreversible branches. However, theoretical x and experimental x dc are both about zero for fieldl e much larger than and therefore do not greatly affect the computation.

A program was written to compute, for each value of H and Xj • the predicted J (H) , H (H) , x(H) , g (x)

g 2 (x). X'(x), and x"(x)> given r

P and h. For h - 56.2 kA/m (706 Oe) , x' and x" as computed are plotted as a function of H in Fig. 3. As in Fig. 1, only the first three branches are shown. No adjustable parameters were used. A comparison with actual measured x' and x" in Figs. ld,e shows good agreement in both shape and magnitude. Ideally, the computed curves in Fig. 3 would be symmetrical with respect to the external dc field and show no difference between the initial segment and the return portion, as expected from critical state theory. 5 Deviations from this are because x, was used instead of x dc rev Hysteresis Loss

In this section, the minor loops in Fig. 2 are examined. The loss corresponding to each is calculated

in three ways: minor-loop area, from x"> and from J . The energy loss per unit volume per cycle is 5

W = nu

0h

2 x". (11)

Substituting for x" using Eqs. 7, 10b, and 5 for full field penetration (x > 1),

w = (l+X rev )(4u 0hJ

c r/3)(l-Jc r/2h). (12)

Except for the factor (1+x ), the same equation is derived, in cgs emu, by CarrY 7 But for H >> ^

X =0. For partial field penetration (x < 1), usfng Eqs. 7, 10a, and 5, Eq. 11 may be expressed as

W = (1+x )(4u nh

3 /3J r) ( 1 — h / 2 J r) , (13) rev U c c again similar to Carr's expression for this case. Figure 2 shows four minor dc hysteresis loops for

the sample determined experimentally with the VSM. The peak amplitude for each minor loop is 56.2 kA/m (706 Oe) . The area enclosed by each loop was calculated numerically by

B-U0 fHdH: u 0 |MdH. (14)

This hysteresis loss is compared in Table II to the losses derived from Eq. 11, using x" measured at 10 Hz

in 56.2 kA/m (706 Oe) peak field (Fig. Id). Another comparison can be made with losses obtained using Eqs.

12 and 13, with J (at the center field) from Eq. 3. For the first minor loop, x < 1. Once again, x is approximated by x dc from the initial branch of reV

the hysteresis loop. The agreement is reasonable,

indicating that hysteretic losses may be measured directly with a hysteresis loop, or estimated by measurement of the imaginary component of low-frequency ac susceptibility, or from a reasonable estimate of J (H)

c

Possible inaccuracies owing to the following should

be considered: (1) J (H) was measured in transverse

rather than longitudinal field; (2) losses are not entirely hysteretic; (3) induced voltage in suscep- tometer pickup coil contains higher-order harmonics;

(4) applied ac field amplitude was not small enough to allow linearized field equations. Only the first of these is believed to be of significance, affecting the data in the last column of Table II.

Conclusion

The technique used to measure hysteretic losses might depend on one's application. If a field is stepped slowly, a dc hysteresis loop is obtained.

If an alternating field is swept and the induced voltage integrated, an ac hysteresis loop may be obtained. For the general case, a dc bias field may be superimposed upon the stepped or alternating field. If the induced voltage is not integrated, but separated

into real and imaginary parts, the complex susceptibility is obtained. Often, an ac susceptibility measurement is a convenient method. In this paper, it was shown that the complex susceptibility in an ac field could be reasonably predicted from the dc hysteresis loop, the critical current density, and the filament radius. Hysteresis losses obtained directly

(Eq. 14) were compared to those obtained indirectly

using experimental measurements of x" (Eq- H) and average values of J and x. (Eqs. 12 and 13).

c ac Acknowledgments

The authors are grateful to J. R. Clem for critical

comments on an earlier draft of the manuscript. This work was sponsored by the Air Force Office of Scientific

Research

References and Footnotes

1. W. A. Fietz, "Electronic Integration Technique for Measuring Magnetization of Hysteretic Superconducting Materials," Rev. Sci. Instrum. 36, pp. 1621-26

(1965).

2. R. B. Goldfarb and J. V. Minervini, "Calibration of AC Susceptometer for Cylindrical Specimens," Rev.

Sci. Instrum. 55, pp. 761-64 (1984).

3. NbTi wire, Standard Reference Material 1457 for critical current measurements, available from Office

of Standard Reference Materials, National Bureau of Standards, Gaithersburg, MD 20599. The data on

this wire in Table I are not certified by NBS.

4. J. F. Bussi&re and J. R. Clem, "Effect of trapped

magnetic flux on ac losses of Nb.Sn," IEEE Trans. Magn. MAG-15, pp. 264-67 (1979).

5. J. R. Clem, "AC Losses in Type-II Superconductors,"

Ames Lab. Tech. Rept. IS-M 280, 1979.

6. Critical current was measured at 4 K by L. F. Goodrich using a transport current technique with a 0.2 pV/cm criterion. The self-field contribution

was not included. The demagnetization field, which would correct values of applied field, is about 0.6%

at 1.6 MA/m (20 kOe) and 0.2% at 4.8 MA/m (60 kOe)

7. W. J. Carr, Jr., AC Loss and Macroscopic Theory

of Superconductors, New York: Gordon and Breach,

1983, p. 70.

8
International Cryogenic Materials Conference - ICMC

Cambridge, Massachusetts - August 1985

Paper DZ-8

To be published in:

"Advances in Cryogenic Engineering (Materials)"

Vol. 32, R. P. Reed and A. F. Clark, eds.

Plenum Press, New York (1986)

AC LOSSES IN Nb-Ti MEASURED BY MAGNETIZATION AND COMPLEX SUSCEPTIBILITY

R. B. Goldfarb and A. F. Clark

Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado

ABSTRACT

DC magnetization and complex ac susceptibility were measured at 4 K

as functions of longitudinal dc field for a multif ilamentary Nb-Ti superconductor with no transport current. Minor hysteresis loops were obtained

in the dc measurements. The full-penetration field, H , a function of

applied field, H, was deduced directly for each minor lBop. The values for

H were fit to the Kim-type equation, H (H) = H (0) / ( 1+H/H^) , where H (0)

afid H, are constants. The minor hyster!sis-loof> areas gave losses thlt were In excellent agreement with Carr's theoretical critical-state

equation, W = (4u_H,,H / 3 ) ( 1—H /2Hn ), where FL is the maximum applied field

for each loop.

° ° P P ° °

An expression was obtained for the ideal reversible differential

susceptibility: X gv

=

4>q / (H“H , ) , where <J>q is the flux quantum, H

^ is the lower critical field, and A c is the penetration depth. H and X C for the sample were deduced from the shape of the major hysteresis loop.

Clem's theoretical expressions for the real (x') and imaginary (x")

components of ac susceptibility are functions of x , H , and ac field

amplitude, h. The predicted susceptibilities based r §n thlse expressions

were in good agreement with measured curves of x' and x" as functions of h

and H. The measured x' and x" were independent of frequency up to 1 kHz,

as expected when bulk hysteresis is the primary loss mechanism.

INTRODUCTION

In an earlier work 1 we discussed the relationship between dc magnetization and ac susceptibility in a type-II superconductor. That work described

how magnetic measurements provide information on hysteresis losses. In this paper, as in Ref. 1, we examine magnetization and susceptibility for

longitudinal fields and no transport current. Here, however, we obtain the

full-penetration fields, H , directly from the hysteresis loops rather than

estimate H from measurements of critical current density, J . Also, we derive an Expression for the reversible susceptibility, x

C

> rather than

using the experimental susceptibility, x^ » as an approximtYion. A superconducting wire whose low-field magnetization approached a reversible curve was selected for study. 2 Several minor hysteresis loops were obtained in

addition to the major loop. The susceptibility curves were more nearly

reversible and virtually independent of frequency, as expected from theory.

9
Table 1. Characteristics of Nb-Ti Wire

Cross section bare wire: 0.63 * 0.88 mm

Twist length: 1.7 cm Cu/Nb-Ti volume ratio: 0.91

Number of filaments: 240 » Filament radius: 19.3 pm Density of Nb-Ti alloy: 6.20 g/cm 3

EXPERIMENT

The magnetization measurements were made with a vibrating-sample

magnetometer (VSM) at 4 K. Magnetization was computed as magnetic moment

per unit volume of Nb-Ti. The volume of the Cu matrix was not included.

It is useful to compare the advantages of the VSM method vis-A-vis the

popular integration method of Fietz. 3 The VSM method (1) is useful for

small samples, (2) is a dc measurement, not sensitive to coupling losses,

(3) does not require precise pick-up coil balance when the applied field is

stepped, and (4) is not subject to integrator drift. The integration

method (1) measures frequency dependences, (2) detects flux jumps, and

(3) is easily adaptable to measurements with transport current.

The experimental methods are more fully described in Ref. 1. The

sample in this study is different and its length is 3.0 rather than 1.5 cm. The purpose of a longer sample was to avoid significant end effects. The

characteristics of the wire are given in Table 1. This is the same wire as sample 8 in Ref. 2.

DC HYSTERESIS LOOP

Full-Penetration Field

The major and minor hysteresis loops are shown in Fig. 1. The fullpenetration field, H , a function of applied field, H, may be estimated

directly for each miSor loop. 2 As the loops are traversed, the filaments

go from full penetration in one direction to full penetration in the other

direction. Therefore, at a high-field end of a loop, H is approximately

one-half the field required to reverse the magnetizatioS. The reversal

field, 2H , for the major loop is shown in Fig. 1.

The well-known Kim model for critical current density is

J

c

(H) = J^O/U+H/iy , (1)

where J (0) and H, are constants. In the critical-state model, for a field

applied C axially,

H -Jr, (2)

P c where r is the filament radius. We did a linear least-squares fit of each

loop's Hp to the expression

H (H) = H (O/Cl+H/H^ , (3)

where H (0) is a constant equal to J (0)r. The fit was excellent. We obtained*

3^(0) = 140 kA/m (1.76 kOe) and = 1.33 MA/m (16.7 kOe)

10
External

Field

(kOe)

(gtuo/nuia) uoryaznauS^Wm o

o m cj up

lO oi o I I

<2

2

Ct,

13c

in

a)xw

(ui/v5[) uorjazTiauStefl

I <D

V-< *H

0) tw

CL3 X

OD <U H CL

I CL C CO 2 rH ^ CO

»-* C

CO *H

•U X

C P

<D hi B *H

cO 00

rH C

*H o

f-H 05 P > 6 X

CO QJ H> H Hi o o

M-t rH

CL

05CL 05 O *H O

rH C

O

05 *H

•H H

05 CO 0) N U

0)

05

•U

<uc ^ 00 C COe

H

O CJ C Cl

Xl H

c o

co Hiu

H 3 O X

C

c0 O 2 u

00

•H

U-*

11
Table 2. Comparison of Losses Obtained

from Eq. 4 and Loop Area.

H H Eq. 4 Area

(kA/m) P (kJ/m3

)

1591 63 165 159

1194 74 143 136

796 87 110 105

398 108 62 58

Hysteresis Loss

For the case when the peak applied field, H^, is greater than H , Carr

derived an expression for the loss, assuming J to be spatially conltant

and field independent in the London-Bean approximation. 5 In SI units,

» < 4wy3 ,<i-y2v (4)

Clearly Hq is much larger than H for the loops in Fig. 1 lated by numerically integratingP the loops are compared in Table

losses obtained from Eq. 4. They agree to within 6%.

Losses calcu2 to

REVERSIBLE SUSCEPTIBILITY y rev An exact equation for the ideal, reversible magnetization-versus-f ield curve of a type-II superconductor, H > H , is not obtainable in analytic

form. The usual procedure is to divide tne field into three regions:

immediately above H (H^ 1 < H) , the middle region (H^, < H < H_ 0 ), and

immediately below H

£ (H

C

! H

cl , c2

, and make judicious approximations.

One tractable case is < H < H when the separation between flux

vortices is larger than the coherence Sength, £, but much smaller than the

penetration depth, X. Thus the Ginzburg-Landau parameter < = X/£ is much

greater than 1. For Nb-Ti at 4 K, 6 X s 394 nm and £

= 5 nm, and this case

is applicable. To first order, the magnetization as a function of magnetic

field strength M(H) is 7

M = (4>q/8itvJqX 2

) {ln[4Try

0 (H-Hcl

)X 2

/<j> 0

] + a} - H

cl , (5)

where is the magnetic flux quantum = h/2e = 2.068 x 10 -15 Wb (T*m 2

) and

a is a constant on the order of unity. The functional dependence of M(H) can be seen by dropping the a term and using the relationship 8

Hd

= (4> 0

/ 4ir 'J o x2 ^ In (A/C) . ( 6 )

Thus

M =

-(<f> 0

/8TTU 0

X

2

) ln[$

0

/4THi 0 (H-H

cl )£ 2

] . (7)

The volume magnetic susceptibility x

® dM/dH can be obtained from

either Eq. 5 or 7 and is simply

X = <j> 0

/8irp 0 (H-H

cl

)X 2

. (8)

We take this x to be equal to the ideal reversible susceptibility x the range H < H < H For H < H , X r y

= -l. X represents ifle susceptibility under conditions of 2 hermolynamic equ-lfYbrium.

12
Our goal is to obtain numerical values for x for our wire. The

differential slope of the third-quadrant branch of

e?he major magnetization

curve (Fig. 1), from 0.2 to 1.6 MA/m (2.5 to 20 kOe) , was fit to Eq. 8 using linear least squares. While this portion of the magnetization curve

itself is not reversible, we expect that its slope will simulate x • For

the adjustable parameters H

^ and X, we obtain 92 kA/m (1 kOe) and

r 5K nm, respectively. These values are not too unreasonable, though this fit risks

overestimating H with values closer to H (0). Substituting into Eq. 8 we get x for £nis wire for H < H < H ^ (SI units) : ° rev cl c2

X - 10772/ (H-91751) . (9) rev This equation will be used in the next section to calculate x' and x"* An alternate approach would be to use textbook values of H and X in

Eq. 8. 9 This would give slightly different values of x' ana x" for

H i H For larger fields, the differences are negligible.

c 1

MEASURED AND IDEAL x' AND x"

The real and imaginary components of susceptibility, x' and x"» were

measured as functions of frequency (10, 100 and 1000 Hz), sinusoidal ac field amplitude, h [11 and 56 kA/m (140 and 700 Oe)

, and dc bias field, H

[0 - 1.6 MA/m (0 - 20 kOe) ] . The susceptibilities were independent of frequency to within 3%. Since x" is 3 measure of the losses and hysteresis is

known to be frequency independent, bulk hysteresis is probably the primary

loss mechanism. Frequency-dependent eddy-current and coupling losses are likely insignificant for this sample.

Figures 2 and 3 show x' and x" measured at 10 Hz. The initial and

decreasing-f ield branches are plotted. Some irreversibility may be seen.

In Fig. 3, h is on the order of H . As shown by Clem, 10 x' and x" as functions of H may be predicted f?om h, H , and x • For our cases,

h < H , the equations are: 1 ’ 10 13 r£V

P

X' - (1+X Hh/H -5h 2 /16H 2 )-l , (10) rev p p X" = (l+Xrev )(4h/H

p -2h 2 /H

p

2 )/3Tr . (11)

Using the expressions for H and x (Eqs. 3 and 9), we computed the

theoretical curves of x' and x" fSr this rwYre for ac field amplitudes of 11 and 56 kA/m (140 and 700 Oe) . They are shown in Figs. 4 and 5. They compare quite favorably with the actual curves in Figs. 2 and 3. For H < H ., x' = -1 and x" = 0. There are discontinuities at H as expected. In tne actual curves these are rounded owing to flux pinning.

CONCLUSION

DC magnetization curves and measurements of complex susceptibility

provide information on hysteretic losses in multifilamentary superconductors

such as Nb-Ti. These measurements may be performed on small samples.

Interpretation of the data yields information on full-penetration field,

critical current density, lower critical field, and penetration depth. The

frequency independence of susceptibility suggests bulk hysteresis as the

primary loss mechanism.

ACKNOWLEDGMENT

This work was sponsored by the Air Force Office of Scientific Research.

13
External Field (kOe)

-20 -10 0 10 20

External Field (MA/m)

0.06

0.03

0.00

-0.03

-0.06

GO

oxi o

Q,

0)o

w

3

cn

o>

Fig. 2. Complex ac susceptibility as a function of dc bias field.

The ac field amplitude is 11 kA/m (140 Oe) at 10 Hz.

External Field (kOe)

-20 -10 0 10 20

External Field (MA/m)

0.06

0.03

0.00

-0.03

-0.06

W

OX) o

Q.

<Dom

3

m

o>

Fig. 3. Complex ac susceptibility as a function of dc bias field.

The ac field amplitude is 56 kA/m (700 Oe) at 10 Hz.

14
-20 20

External Field (kOe)

-10 0 10

0.06

0.03

- 0.00

- -0.03

-0.06

w

OjO o

a.

<uow3

in

cu£

O

>

External Field (MA/m)

Fig. 4. Theoretical complex susceptibility vs. dc bias field for an ac field amplitude of 11 kA/m (140 Oe)

External Field (kOe)

-20 -10 0 10 20

External Field (MA/m)

Fig. 5. Theoretical complex susceptibility vs. dc bias field for

an ac field amplitude of 56 kA/m (700 Oe)

15
REFERENCES

1. R.

2. R.

3. W.

4. Y.

5. W.

6. M.

7. A.

8. A.

9. Cf

10. J.

B. Goldfarb and A. F. Clark, Magnetic hysteresis and complex

susceptibility as measures of ac losses in a multifilamentary NbTi

superconductor, IEEE Trans. Magn. MAG-21: 332 (1985).

B. Goldfarb and A. F. Clark, Hysteretic losses in Nb-Ti superconductors , Appl . Phys. 57 : 3809 ( 1985)

A. Fietz, Electronic integration technique for measuring magnetiza

tion of hysteretic superconducting materials, Rev. Sci. Instrum. 36:1621 (1965).

B. Kim, C. F. Hempstead, and A. R. Strnad, Magnetization and

critical supercurrents, Phys. Rev. 129:528 (1963).

J. Carr, Jr., "AC Loss and Macroscopic Theory of Superconductors,"

Gordon and Breach, New York (1983), p. 70.

R. Beasley and T. P. Orlando, cited by R. J. Donnelly, in: "Physic

Vade Mecum," H. L. Anderson, ed., American Institute of Physics,

New York (1981), ch. 7, p. 121.

L. Fetter and P. C. Hohenberg, in: "Superconductivity," R. D. Parks, ed., Marcel Dekker, New York (1969), ch. 14, p. 838.

L. Fetter and P. C. Hohenberg, in: "Superconductivity," R. D. Parks, ed., Marcel Dekker, New York (1969), ch. 14, p. 843.

. H s 8 kA/m (100 Oe) , in: A. K. Ghosh and W. B. Sampson, Magneti

zation and critical currents of NbTi wires with fine filaments,

paper DZ-7, this conference.

R. Clem, "AC Losses in Type-II Superconductors," Ames Lab. Tech.

Rept. IS-M 280, Iowa State University, Ames (1979), pp. 23-24.

16
Eleventh International Cryogenic Engineering Conference - ICEC 11 West Berlin - April 1986

Paper 09-4

LOSSES IN A Nb-Ti SUPERCONDUCTOR AS FUNCTIONS OF AC FIELD AMPLITUDE

AND DC TRANSPORT CURRENT

M. Dragorairecky , J.V. Minervini, J.W. Ekin, R.B. Goldfarb, and A.F. Clark

National Bureau of Standards, Boulder, Colorado 80303, USA

Hysteretic shielding losses and transport losses were measured

in a multifilamentary Nb-Ti superconducting coil as functions of

transverse ac field amplitude and dc transport current. The

conductor was biased with a dc field. There was significant

agreement with the predictions of Minervini 's two-dimensional

theoretical model.

INTRODUCTION

AC losses for the case of current-carrying conductors in transient applied magnetic

fields are of importance in practical applications. The theory for the one-dimensional slab model is well known and a few experimental studies have been made.

Predominant among these is the work of Ogaswara, et al. [1-4]. For multifilamentary conductors, they emphasize the dependence of loss on the rate of change of the

field. Their field amplitudes were rather large, on the order of 200 kA/m (2.5

kOe) . Ciazynski also examined the dependence of hysteresis and transport losses on

the sweep rate of the applied field. The analysis was in terms of an equivalent

electrical network [5], Bruzzone and Kwasnitza discussed the dependence of the

hysteresis loss on the ratio of ac field amplitude to fullrpenetration field [6].

References [1-6] are primarily concerned with coupling losses in multifilamentary

conductors. De Reuver et al. [7] recently reported measurements of losses in

conductors carrying dc transport current in a transverse ac field. They analyzed

their data in terms of the slab model and found qualitative agreement.

The first complete two-dimensional analysis for a cylindrical conductor with a transport current in a transverse ac field was presented by Minervini in 1981 [8].

The theory is based on the London-Bean approximation where the critical current

density, J , is independent of field, H. Ignoring eddy-current and coupling losses

(for low-frequency ac fields) , the total ac loss can be separated into two components. (1) Shielding hysteresis loss occurs in the outer volume of the superconductor filament not occupied by transport current. (2) Transport loss occurs

owing to flux motion through the central portion of the superconductor filament

when the shielding region is saturated with shielding currents. The shielding

hysteresis loss, W , and transport loss, W , as functions of transport current

density, J, are shown in Figs. 1 and 2 [8]. The total loss would be equal to the sum of W

g and VJ . Curves are shown for different ac field amplitudes, h. The

energies and W^_ are normalized to W^, J is normalized to J , and h is normalized

to H^(O), ‘where, for filaments of diameter d,

H (0) = d J / TT (1) p c is the full-penetration field for zero transport current (J = 0) , and

W

Q = 4 P 0 H

p

(0) 7 3 (2 >

is the volume energy density when h = H (0) and J = 0 [9]. H (J) is a monotonically decreasing function of J. In thisppaper we report measu?ements of shielding

hysteresis loss and transport loss that were made in an effort to test the two- dimensional model. The results are in good agreement with the model. More experimental data are needed to adequately determine the detailed shape of the curves.

17
EXPERIMENT AND RESULTS

A multifilamentary Nb-Ti conductor was noninductively wound on a coil form. Inner

and outer balanced pick-up coils were wound coaxially. The sample was immersed in

liquid helium. It was exposed to a 3-MA/m (40~k0e) dc bias field. The purpose of

the bias field was primarily to operate on thefc relatively flat portion of the J

- versus-H curve, where, as assumed in the London-Bean approximation, J is not a strong function of H [10]. A secondary purpose of the bias field was to reduce J to a value more accessible in the laboratory and to operate under conditions

similar to those in practical applications.

The wire selected for study had a diameter of 0.635 mm, 180 filaments 30 ym in diameter, and a copper-to-superconductor volume ratio of 1.50. This wire was the same as sample 4 in Ref. [11]. The sample was 18 m long, wound in two layers of

72 and 74 turns. The critical current while in the bias field was 310 A, giving a filament J of 2.44 x 10 g A/m 2 at 3 MA/m (40 kOe) . H (0) was 2.33 * 10

1* A/m

(290 Oe) and W

Q was 910 J/m 3

. P

Data were acquired with a computer-controlled, four-channel digital recording

oscilloscope. The voltage waveform of the pick-up coil was integrated with respect

to time to give magnetization, which was plotted as a function of the ac-field

waveform. The resulting hysteresis loop was numerically integrated to get the

shielding hysteresis loss. The energy for the loss was provided by the ac-field

power supply.

The magnetization axis was calibrated by noting that the maximum magnetization

value for ac fields greater than H (0) for zero transport current is 2H (0) / 3 [9]. An alternate calibration would be P to use the initial susceptibility inP zero bias

field with an ac field amplitude less than the lower critical field, H However

* C. 1 the external susceptibility when the field is applied along the axis or the sample

coil depends on the effective demagnetization factor, a function of how tightly the

coil is wound [12]. This could not be determined accurately.

Transport loss was numerically computed as the integral, over two complete cycles,

of the product of sample voltage and transport current. The energy for the transport loss was provided by the transport-current power supply.

The frequency of the ac field was 1 Hz, far below the frequency for which coupling

currents or eddy currents become important. Unlike in earlier studies [1-7], the

frequency of the ac field was not a variable. The experimental set-up is shown in

Fig. 3. Particular care was taken to avoid ground loops and thermal voltages.

Very low-noise (< 3 yV) preamplifiers were used. Any dc offsets of the amplifiers were subtracted from the measured waveforms. The results are shown in Figs. 4 and 5

DISCUSSION

Generally speaking, this was a difficult experiment. The agreement of the experimental results (Figs. 4 and 5) with the theory (Figs. 1 and 2) is fairly good. The

need for more data to fill-in the curves is noted.

For the case J = 0 and h > H (0) , the theoretical expression for W /W. [8]

. P s u reduces to *

W /wn

= 2h/H (0) - 1 . (3)

s 0 p Thus, the consistency of the values for h/H (0) and W

g

/ W

Q

(-1 = 0) in Fig- 4 can be

evaluated with reference to Fig. 1. For exRmple , the

S curve for h/H (0) = 1 should

intercept the W^/W^ axis at unity. P

The theoretical curves for the one-dimensional slab model are only qualitatively

similar to those for the two-dimensional cylindrical model. However, when the data

18
were normalized to the values of H (0) and W. for the slab model and compared to the theoretical slab-model curves, P the quantitative agreement was poor. For shielding loss, the measured values were greater by a factor of about 2. For transport loss, the measured curves had a different curvature. Furthermore, the

total-loss curves (W + W ) for our data are in excellent quantitative agreement

with the theoretical total-loss curves for the two-dimensional model [8] but in

poor agreement with the theoretical curves for the slab model [1-4,7],

Though our data do not include large ac field amplitudes, we consider this case for

the purpose of comparison to the high-field work of others. When h/H (0) >> 1, W /W. = 2h/H (0). In this limit, from Eqs. 2 and 3,

P

s 0 p W = 8 y. H (0) h / 3 , (4)

s Op

in agreement with Carr's derivation [13].

CONCLUSION

The two-dimensional cylindrical model seems to be better than the one-dimensional

slab model for analyzing shielding hysteresis losses and transport losses in

multifilamentary conductors with round filaments.

ACKNOWLEDGEMENT

This work was sponsored by the Air Force Office of Scientific Research.

REFERENCES

1 Ogaswara, T. , Takahashi, Y. , Kanbara, K. , Kubota, Y., Yasohama, K. , and

Yasukochi, K. , 'Alternating field losses in superconducting wires carrying dc

transport current. Part 1: single core conductors'. Cryogenics, vol. 19 (1979), pp. 736-740.

2 Ogaswara, T., Takahashi, Y., Kanbara, K. , Kubota, Y., Yasohama, K. , and

Yasukochi, K. , 'Transient field losses in multifilamentary composite conductors

carrying dc transport currents'. Cryogenics , vol. 20 (1980), pp. 216-222.

3 Ogaswara, T. , Takahashi, Y. , Kanbara, K. , Kubota, Y. , Yasohama, K. , and

Yasukochi, K. , 'Alternating field losses in superconducting wires carrying dc

transport current. Part 2: multifilamentary composite conductors'.

Cryogenics , vol. 21 (1981), pp. 97-101.

4 Ogaswara, T., Itoh, M. , Kubota, Y. , Kanbara, K. , Takahashi, Y., Yasohama, K.

and Yasukochi, K. , 'Transient field losses in multifilamentary composite

conductors carrying transport currents'. IEEE Trans. Magn., vol. MAG-17

(1981), pp. 967-970.

5 Ciazynski, D., 'Effect of a transport current on the losses of a superconducting composite under fast changing magnetic field'. IEEE Trans.

Magn. , vol. MAG-21 (1985), pp. 169-172.

6 Bruzzone, P. and Kwasnitza, K. , 'Scaling law for filamentary superconductor

hysteresis losses in superimposed dc and ac magnetic fields'. J. Physique

Colloq. Cl, vol. 45 (1984), pp. 483-487.

7 De Reuver, J.L., Beunk, H.D., Roovers, A.J.M., and Van der Klundert, L.J.M.,

'AC field losses in a dc current-carrying conductor containing one layer of

NbTi filaments'. Inti. Conf, Magnet Tech. , MT-9, Marinucci, C. and Weymuth, P.,

eds., Swiss Inst. Nucl. Res., Zurich (1985), pp. 501-504.

19
8 Minervini, J.V., 'Two-dimensional analysis of ac loss in superconductors

carrying transport current' . Advances in Cryogenic Engineering (Materials) , Vol. 28, Reed, R.P. and Clark, A.F., eds.. Plenum, New York (1982), pp, 587-599.

9 Pang, C.Y., McLaren, P.G., and Campbell, A.M. , 'Losses in superconducting

cylinders in transverse fields'. Inti. Cryo. Engr. Conf

, Vol. 8, Rizzuto, C.,

ed. , IPC Sci. and Tech. Press, Surrey, England (1980), pp. 739-743.

10 Carr, Jr., W.J., AC Loss and Macroscopic Theory of Superconductors , Gordon and

Breach, New York (1983), p.55.

11 Goldfarb R.B. and Clark, A.F., 'Hysteretic losses in Nb-Ti superconductors'.

J , Appl . Phys , vol. 57 ( 1985), pp. 3809-3811

12 Sumivoshi, F., Irie, F., and Yoshida, K. , 'The effect of demagnetization on the

eddy-current loss in a single-layered multif ilamentarv superconducting coil'.

J . Appl . Phys

, vol. 51 (1980), pp . 3807-381 1

13 Carr, Jr. , W.J. , AC_ Loss and Macroscopic Theory of Superconductors , Gordon and

Breach, New York (1983), p.67.

o5

(V®

U)>> X

O)s2

0> s.V)o

&

N

reE

Fig.

Normalized Current Density, J / J c

1 Theoretical shielding hysteresis

loss for different ac fields [8].

Normalized Current Density, J/J c

Fig. 2 Theoretical transport loss for

different ac fields [8].

20
Shielding

Hysteresis

Loss,

W

s

/W

Fig. 3 Experiment for measuring shielding hysteresis loss and transport loss.

The coils are immersed in liquid helium.

Current Density, J/J

c

Fig. 4 Experimental shielding hysteresis

loss for different ac fields.

Current Density, J/J c Fig. 5 Experimental transport loss for

different ac fields.

21

HYSTERESIS LOSSES IN FINE-FILAMENT INTERNAL-TIN SUPERCONDUCTORS

R. B. Goldfarb and J._ W. Ekin

Electromagnetic Technolqgy Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

Hysteresis losses were measured on a series of fine-filament Nb^Sn

superconductors made by the internal-tin process. Hysteresis was measured as

a function of filament diameter and interfilament separation using a vibratingsample magnetometer in transverse field. Losses were greater than expected

from the critical-state model that expresses loss as a function of filament

diameter. Micrographs of the reacted wire cross sections showed some interfilament bridging for all wires. This gave rise to effective filament diameters

that were greater than actual diameters. The critical interfilament separation,

above which the losses would be expected to follow the critical-state model,

was determined.

INTRODUCTION

Fine-filament superconductors are of interest in ac-loss studies because,

according to the critical-state model, hysteresis loss is proportional to

filament diameter. Thus, one would expect that wires composed of fine filaments would exhibit small losses. In this study we used a vibrating-sample

magnetometer in transverse magnetic field to measure the dc hysteresis loss

for a series of Nb^Sn superconductors made by the internal-tin process. There

was no transport current. Filament diameter and edge-to-edge interfilament

separation were systematically varied to determine whether filament coupling

caused larger-than-expected hysteresis losses. Such magnified losses may be

23
expressed in terms of an effective filament diameter that is greater than the

actual filament diameter. The effective diameter is deduced from the hysteresis loss using the critical-state model.

There has been some previous work on hysteresis losses in fine-filament

superconductors. Dubots et al. compared measured and calculated hysteresis in

Nb-Ti conductors with filament diameters ranging from 10 to 0.1 ym [1], They

found that, for diameters smaller than 1 ym, effective filament diameters

remained about 1 ym. Information on interfilament separation was not given.

Carr and Wagner attributed the asymmetrical hysteresis curve they observed in

Nb-Ti with 1 . 6-ym filament diameter to relatively large surface currents [2],

Ghosh and Sampson studied fine-filament Nb-Ti with filament diameters in the

range 1-5 ym [3], They note that, for diameters less than 3 ym, the magnetization (and therefore hysteresis loss) is larger than expected from the criticalstate model. They attribute this to degradation of the magnitudes of critical

current owing to filament necking or damage.

In this paper we show that certain Nb^Sn internal-tin conductors with

filament diameters smaller than 5 ym have effective filament diameters greater

than expected from the critical-state model. The ratio of effective to actual

Nb^Sn filament diameters appears to be a linear function of interfilament

separation. The slope of the line is a function of the local area ratio of

matrix material to Nb for each wire. The greater effective filament diameters

appear to be a consequence of bridging between filaments.

SAMPLE PREPARATION AND MEASUREMENT

The multifilamentary Nb^Sn wires used in this study were composed of

subelements containing 150 filaments each. The array of subelements was surrounded by a tantalum diffusion barrier. Unreacted Nb^Sn wires were wrapped

24
on the threads of size 10-24 stainless steel screws. The thread pitch was

such that the windings were not in contact with each other. The screws had

been previously oxidized to prevent the wires from sticking to the screws.

The internal-tin-process wires were reacted in vacuum, with cold ends out of

the furnace, according to the following schedule: 1 day at 340°C, 4 days at

580°C, and 4 days at 700°C. This schedule is adequate to fully react the

wires for the filament diameters in this study [4], The residual tin content

of each wire after reaction was estimated by the manufacturer to be 8 wt.%.

After reaction, the wire coils were unscrewed from the stainless screws

and the long ends cut off. Each coil was about 1.3 cm in height. The wire

volumes were measured using Archimedes’ principle. The wire diameter was the

same for each sample, 0.681 mm. Thus, the wire length for each coil was

obtained

The average Nb-filament diameters before reaction and number of filaments

in each sample were provided by the manufacturer. The reacted filament diameters and total volume of Nb^Sn in each coil were calculated based on a presumed increase of 38% in filament cross-sectional area after reaction [5],

Local area ratios of matrix to Nb were determined by the manufacturer from the

arrangement of the Nb rods in the billets before drawing: the Nb may be

thought of as enclosed by adjacent hexagons and the cross-sectional areas of

Nb and matrix material measured. The average center-to-center filament

distance, s, may be determined from the local area ratio, L, and Nb filament

diameter, d, by the equation

L = (2/3/tt) (s/d) 2 - 1 . (1)

The edge-to-edge Nb separation is simply s - d.

Table 1 gives relevant parameters for the six samples. Note that Nb

diameter plus Nb separation equals Nb^Sn diameter plus Nb^Sn separation.

25
Both sums are equivalent to the center-to-center filament distance (not

shown) . The matrix-to-Nb local area ratios of 1.5 and 1.8 correspond to area

ratios for reacted Nb^Sn of 1.0 and 1.2, respectively.

Magnetic moment was measured at 4.2 ± 0.2 K with a vibrating-sample

magnetometer in the axis of the coil (i.e., field transverse to the wire if we

ignore thread pitch of about 5.5°). Maximum applied field was 1.6 MA/m

(20 kOe) . There was no transport current and the coils were open-circuited.

Magnetization was computed as magnetic moment per unit volume of Nb^Sn (i.e.,

not including the volume of the matrix material) . Magnetization was plotted

versus external (applied) magnetic field. A typical magnetic hysteresis loop

for one of the samples (sample 18) is shown in Fig. 1.

RESULTS

Hysteresis loss was computed by numerically integrating the loop area,

ignoring the initial branch. According to the critical-state model, for

transverse field geometry [6]:

W = 8 J d cc y„. Hn / 3tt (SI units) (2)

c err U U

Thus, an effective filament diameter, d^^., may be defined that is a function

of the hysteresis loss, W, for a maximum applied field, H^. Since we did not

have data for the critical current density, J^, at 1.6 MA/m (20 kOe) , we

deduced these J based on the critical current of each wire measured at 6.4

c MA/m (80 kOe) together with the total Nb^Sn cross-sectional area and the

measured value of the full-penetration field, H , at 1.6 MA/m (20 kOe)J , P c H^, and filament diameter, d, are related according to the critical-state

model [ 6]

H = d J / tt (SI units) (3)

p c H may be obtained as one-half the field required to reverse the magnetization

26
[7]. From the hysteresis loops, at 1.6 MA/m (20 kOe) is roughly 75 kA/m

(940 Oe) for all the loops. This suggests a factor of 10 increase in

compared to measured J

c at 6.4 MA/m (80 kOe) . Applying this factor of 10 to

each conductor gives the values of shown in Table 2. We then compute the

effective filament diameters of the wires, and additionally, the ratio of the

effective to actual Nb^Sn filament diameters.

Optical and scanning-electron microscopy revealed that some of the

filaments in each wire were not entirely isolated but were connected by

"bridges". An example of this is shown in Fig. 2. Such filaments had typically lost their original hexagonal spacing and aligned themselves in rows in

the process of wire drawing [8],

DISCUSSION

As shown in Fig. 3, the relationship between actual and effective Nb^Sn

diameters is not monotonic. However, some insight is obtained by plotting the

ratio of effective to actual Nb^Sn diameters versus Nb^Sn interfilament

separation (see Fig. 4). It appears that effective filament diameter is a

function of both actual filament diameter and interfilament separation. Thus,

loss per volume of Nb^Sn superconductor is a function of both parameters.

Further refinement is possible by distinguishing among those conductors

with the same matrix-to-Nb local area ratios. These are identified in Figs. 3

and 4 for the two ratios. We see in Fig. 3 that there is less of an increase

in effective filament diameter for the conductors with the larger local area

ratio. However, we note that, in Fig. 4, the abscissa (Nb^Sn interfilament

separation) is not independent of local area ratio (see Eq. 1). The apparently

linear dependence of the ef f ective-to-actual diameter ratio on inter f i lament

separation has no physical significance.

27
A critical interfilament separation can be defined. It corresponds to

the separation above which the hysteresis losses would depend on filament

diameter as expected from the critical-state model. We determined the critical

separation for reacted Nb^Sn, as shown in Fig. 4, by linearly extrapolating

the data to a unity value of ef f ective-to-actual diameter ratio. The critical

separation appears to be a function of the local area ratio. Critical values

for the separation between reacted Nb^Sn filaments were about 1.6 ym for an

area ratio of 1.5 and about 2.0 ym for a ratio of 1.8. The corresponding

critical values for the separation between unreacted Nb filaments were 2.4 and

1.5 ym, respectively.

CONCLUSION

Certain fine-filament superconductors made by the internal-tin process

have effective filament diameters that are greater than actual filament

diameters. The differences between effective and actual diameters increase as

actual diameters decrease, the effect being more pronounced in those wires

with the smaller local area ratio of matrix material to Nb . When the ratio of

effective to actual diameters is plotted as a function of edge-to-edge interfilament separation, a more refined picture emerges. The diameter ratio

increases linearly with decreasing interfilament separation for each local

area ratio. The larger-than-expected effective filament diameters are likely

due to interfilament bridging in these wires.

For the practical purpose of conductor design, a critical interfilament

separation can be determined. It corresponds to the spacing above which the

hysteresis loss would depend on filament diameter as expected from the criticalstate model. The critical Nb^Sn interfilament separation for the conductors

used in this study was about 1.6 and 2.0 ym for matrix-to-Nb area ratios of

28
1.5 and 1.8, respectively. The critical separation is offered simply as a

practical design parameter related to the critical-state model and for guidance

in the manufacture of fine-filament internal-tin superconductors such as

these .

ACKNOWLEDGMENTS

M. Suenaga made the high-field critical current measurements and generously shared samples of the unreacted wires. A. K. Ghosh astutely pointed out

the importance of local area ratios. G. M. Ozeryansky provided valuable

information on the physical characteristics of the wires. E. S. Pittman made

the optical and scanning-electron micrographs. The wires were manufactured by

Intermagnetics General Corporation [9]. The experimental work was sponsored

by the Air Force Office of Scientific Research. Most of the results of this

study were presented at the Fourth U.S. -Japan Workshop on High Field Superconducting Materials for Fusion, 1986. This workshop and the microanalysis were

sponsored by the U.S. Department of Energy, Office of Fusion Energy.

REFERENCES

1. P. Dubots, A. Fevrier, J. C. Renard, J. P. Tavergnier, J. Goyer, and

Hoang Gia Ky , NbTi wires with ultra-fine filaments for 50-60 Hz use:

Influence of the filament diameter upon losses, IEEE Trans. Magn. MAG-21,

177 (1985).

2. W. J. Carr, Jr. and G. R. Wagner, Hysteresis in a fine filament NbTi

composite, in: "Advances in Cryogenic Engineering (Materials)," Vol. 30,

A. F. Clark and R. P. Reed, eds., Plenum, New York (1984) p. 923.

3. A. K. Ghosh and W. B. Sampson, Magnetization and critical currents of NbTi

wires with fine filaments, in: "Advances in Cryogenic Engineering

(Materials)," Vol. 32, R. P. Reed and A. F. Clark, eds., Plenum, New York

(1986) p. 809.

4. A. K. Ghosh, private communication.

5. G. M. Ozeryansky, private communication.

29
6. W. J. Carr, Jr., "AC Loss and Macroscopic Theory of Superconductors,"

Gordon and Breach, NY (1983) p. 67.

7. R. B. Goldfarb and A. F. Clark, Hysteretic losses in Nb-Ti superconductors, J. Appl. Phys. 57, 3809 (1985).

8. Evidence for interfilament bridging in these wires was first presented by

A. K. Ghosh and W. B. Sampson in: "Magnetization studies of multifilamentary Nb^Sn wires" at the Fourth U.S. -Japan Workshop on High Field

Superconducting Materials for Fusion, 1986 (unpublished).

9. Certain commercial materials are identified to adequately specify the

experimental study. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards, nor does it imply

that the material is the best available for the purpose.

TABLES

Table 1. Parameters for six Nb^Sn multifilamentary superconductors.

Sample

Nb

Diam.

(urn)

Nb^Sn

Diam.

(ym)

Local

Area

Ratio

Nb

Sep

(ym)

Nb

3

Sn

Sep

(ym)

No

Fil. Length

(cm)

Vol

Nb„Sn

(mm3

)

3 4.2 4.9 1.5 2.1 1.4 2850 17.8 9.71

6 4.0 4.7 1.8 2.4 1.7 2850 17.3 8.55

9 . 3.0 3.5 1.5 1.5 1.0 5550 17.1 9.24

12 2.8 3.3 1.8 1.7 1.2 5550 17.0 8.01

15 2.3 2.7 1.5 1.2 0.8 9150 17.0 8.89

18 2.2 2.6 1.8 1.3 0.9 9150 16.8 8.04

Table 2. Hysteresis loss and effective filament diameters for Nb^Sn.

Sample

Estimated

J

(GA/m2

) W

(MJ/m3

)

Eff. Nb Sn

Fil. Diam.

(ym)

Eff . /Actual

Diameter

Ratio

3 49.4 0.870 10.46 2.12

6 50.6 0.714 8.37 1.78

9 50.6 1.369 16.05 4.55

12 54.3 0.885 9.67 2.94

15 53.8 1.419 15.66 5.80

18 51.9 0.805 9.20 3.56

30
250

-20 20

250

External Field (kOe)

-10 o 10

Figure 1. Typical hysteresis loop for a multif ilamentary Nb Sn

the internal-tin process (sample 18). Magnetization

as magnetic moment per unit volume of Nb^Sn.

- 125

- 0

- -125

J -250

2

wire made by was computed

31

Magnetization

(emu/cm

3)
i

Figure 2. Typical scanning-electron micrograph showing some interfilament

bridging (sample 18, center subelement).

32
Effective

NbgSn

Filament

Diameter

(yu,m)

2 3 4 5

Actual NbgSn Filament Diameter (yum)

Figure 3. Effective filament diameter as a function of actual Nb^Sn filament

diameter for two local area ratios of matrix to Nb.

33
Effective/Actual

NbgSn

Diameter

Ratio

NbgSn Interfilament Separation (/rm)

Figure 4. Ratio of effective to actual Nb^Sn filament diameter as a function

of Nb^Sn interfilament separation for two local area ratios of

matrix to Nb.

34 /
INTERNAL FIELDS IN MAGNETIC MATERIALS AND SUPERCONDUCTORS

R. B. Goldfarb^

Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

This paper reviews some of the concepts needed for the correct analysis

of magnetization data, both for magnetic materials and superconductors.

Demagnetization factors, initial susceptibilities, and hysteresis losses are

discussed

INTRODUCTION

Magnetization measurements have acquired an important function in the

measurement of ac losses in superconductors. Many of the traditional analysis

methods used for magnetic materials apply to superconductors with the assumption of perfect diamagnetism. Some refinement is necessary for type-II and

multifilamentary superconductors

DEMAGNETIZATION FACTORS

The magnetization curve characteristic of a magnetic material is obtained

by plotting magnetization, M, versus internal field, H . H is obtained

by correcting the external applied field, H , by the demagnetization field,

^demag

= w^ere ® is the geometric demagnetization factor. In SI units.

H. = H - DM . int ext ( 1 )

For a sphere, D = 1/3. For a long cylinder, with H applied axially, D = 0;

for applied transversely, D = 1/2. corresponds to the field H in

35
the equation B = p (H+M) . H. is sometimes referred to as the Maxwell field.

0 xnt

Here we ignore any ac effects such as eddy currents.

INITIAL SUSCEPTIBILITIES

Often the initial slope of the magnetization curve is used for

calibration of magnetometers. We define the internal susceptibility, x

. t

> as

dM/dH. and the external susceptibility, x > as dM/dH . Thus, from Eq . 1, int y ext ext

int

= Xext / (1 - D Xext

) ' (2)

ext

= X int / (1 + D w • (3)

Let us consider the case of a soft ferromagnet with a very large X^nt

•

In a small applied field H , we would measure M such that, from Eq. 3, 0 Xt

X = dM/dH = 1/D.

ext ext

We can treat a simple type-I superconductor as a perfectly diamagnetic

material: X- = ”1. In an applied field H , from Eq. 3,

ITT t GXt

X _ = dM/dH = 1/ (D-l) . ext ext

(4)

Thus, for a sphere, x = -3/2. For a long cylinder axially, x = -1;

0Xt GXt

transversely, x = “2. J ext

An interesting case is that of an infinitely thin sheet of finite

dimensions. For H normal to the surface, D = 1. From Eq. 4, x , and

ext ext

thus M, will approach infinity. While this seems unphysical at first, we note

that M is calculated as magnetic moment per unit volume, V. M gets infinitely

large only as V approaches zero. Furthermore, for H >0, flux immediately

GXl

begins to penetrate the sheet in this "intermediate" state; full penetration

occurs at H = H , the critical field,

c For these cases, we plot M versus H in Fig. 1 [1]. Note that M versus

GX t H_^

t would be characteristic of the material, independent of geometry. The

plot of M versus H. would be the same as that for the axial cylinder (D = 0)

int

36
The same initial X^nt

would be obtained for a type-II superconductor.

The lower critical field, H is signaled when the initial M-versus-H.

cl int curve deviates from linearity, indicating flux penetration and the "mixed"

state. (The intermediate state is also possible in type-II superconductors.)

HYSTERESIS LOSS

We now consider hysteresis loss in superconductors. Theoretical loss

expressions are usually given in terms of the applied field, H . While a

hysteresis loop plotted as M versus H is generally shaped differently from

one plotted as M versus H , the areas enclosed by the loops are the same.

This can be seen by integrating H_^ as a function of M and referring to Eq.

1, as we now show.

Consider a hysteresis loop plotted as M versus H_^ . Let the end points

of the loop be denoted as 1 and 2. The two points are joined by curves A and

B. Using the usual convention of integrating H as a function of M, the area

of the loop enclosed by curves A and B is

W = /i H

A

dM - /i H

B

dM

int xnt

(5)

= / 1 H

A

dM - D /, MdM - / 1 H

B

dM + D A M dM

ext ext 1

= / 1 H

A

dM - /i H

B dM . ext ext ( 6 )

Thus, the loop areas are the same for both M versus H. and M versus H

xnt ext

MULTIFILAMENTARY SUPERCONDUCTORS

The fields in the somewhat special case of a multifilamentary superconductor have been analyzed by Carr [2-4], In the equations, M now refers to

the magnetic moment per unit volume of superconductor, not including matrix

material. Often, however, M is expressed as magnetic moment per total

37
volume and the volume fraction of superconductor is carried in the equations.

For hysteresis calculations, the local field, H , applied to each filament

is used rather than H applied to the composite conductor. H., is someext rr r loc

times referred to as the Lorentz or cavity field:

H, EH. + DM

loc mt (7)

For H applied axially to the conductor (and filaments), H_ = H = H.

ext rr j loc ext int

For H applied transversely, H. = H. + M/2. For the special

ext loc mt r

case when H is changed slowly and there is no transport current, Eq. 1

applies: rr H. = H - M/2. Thus, H- = H . Therefore, we can continue to int ext loc ext use in the hysteresis loss calculations for this special case. Similarly,

X-, , equal to dM/dH for multif ilamentary conductors, will equal x • For

JL o c _Loc ext

the general transverse case, however, shielding effects prevent us from using

and H interchangeably.

We summarize the equations for x. and x when there is no shielding mt ext

and the volume fraction of superconductor, A, is carried in the equations

[2,4], We have, for H applied axially,

X int

’ Xext

= - X •

The expressions for H applied transversely are

X. _ mt = -2A/(1+A) ,

X = -2A . ext

Equations 9 and 10 are related by Eqs . 2 and 3.

( 8 )

(9)

( 10 )

EFFECTIVE DEMAGNETIZATION FACTOR FOR COILS

Sumiyoshi et al. derived an effective demagnetization factor, D f°r a

single-layer coil of multifilamentary superconductor parallel to the coil axis

[5], That is, is applied transversely to the conductor.

D __ = [1 - cos (irr/d) ] / [cosh(irr/d) - cos(iTr/d)] (11)

eft

38
Here, r is the radius of the filament bundle and 2d is the center-to-center

separation of the bundles. D

^ takes into account the interaction between

the turns of the coil. We assume that there- is no current in the conductor.

In the limiting case where d >> r, D ^

= 1/2, as for isolated transverse

cylinders. In the other limit, d = r, D ^

= 0.1588. Thus from Eq. 4, Xgxt can range from -2 to -1.189.

The effect of coil-turn interaction has also been discussed by Carr et

al. [6], Zenkevitch and Romanyuk [7], and Campbell [8].

CONCLUSION

The correct analysis of magnetization data requires that one account for

demagnetization factors and internal fields. This applies to magnetic materials

as well as superconductors.

ACKNOWLEDGMENT

This work was sponsored by the Air Force Office of Scientific Research.

REFERENCES

1. E. A. Lynton, "Superconductivity," Methuen, London (1962) p. 25.

2. W. J. Carr, Jr., Electromagnetic theory for filamentary superconductors,

Phys. Rev. B 11, 1547 (1975).

3. W. J. Carr, Jr., "AC Loss and Macroscopic Theory of Superconductors,"

Gordon and Breach (1983) pp. 97-101.

4. W. J. Carr, Jr. and G. R. Wagner, Hysteresis in a fine filament NbTi

composite, in: "Advances in Cryogenic Engineering (Materials)," Vol. 30,

A. F. Clark and R. P. Reed, eds. (1984) pp. 923-30.

5. F. Sumiyoshi, F. Irie, and K. Yoshida, The effect of demagnetization on the eddy-current loss in a single-layered multif ilamentarv superconducting

coil, J. Appl, Phys. 51, 3807 (1980).

6. W. J. Carr, Jr., B Clawson, and W. Vogen, AC losses in superconducting

solenoids, IEEE Trans. Magn. MAG-14, 617 (1978).

39
7. V. B. Zenkevitch and A. S. Romanyuk, Effect of magnetic properties of a composite superconductor on the losses in a variable magnetic field.

Part 3: Collective interaction of turns, Cryogenics 20, 79 (1980).

8. A. M. Campbell, A general treatment of losses in multifilamentary

superconductors, Cryogenics 22, 3 (1982).

40
Magnetization

Applied Field, Hext

Figure 1 Magnetization versus external applied

superconductor of various geometries,

ties are given by the initial slopes,

flux penetration for the intermediate

field for an ideal type-I

The external susceptibiliThe broken lines indicate

states. After Lvnton fl].

41

Calibration of ac susceptometer for cylindrical specimens

R. B. Goldfarb and J. V. Minervini

Electromagnetic Technology Division. National Bureau ofStandards, Boulder, Colorado 80303

(Received 1 1 November 1983; accepted for publication 30 January 1984)

The absolute magnetic susceptibility of cylindrical specimens is obtained with an ac susceptometer whose calibration is based on a calculation of mutual inductance. An axially

magnetized cylinder is modeled as a solenoid of the same size. The mutual inductance between

such a solenoid and a pickup coil of arbitrary dimensions is computed. The susceptibility is then a function of the mutual inductance, the cylinder length, the magnitude and frequency of the ac magnetizing field, and the voltage induced on the pickup coil. Demagnetization factor and eddycurrent effects are considered, an example is given, and pickup coil compensation is discussed. Other calibration methods are also presented.

PACS numbers: 07.55. + x

INTRODUCTION

A useful method of measuring the magnetic susceptibility of nonferromagnetic materials employs an ac field source and a pickup coil. A typical circuit is shown in Fig. 1 . Such pickup

coils usually require calibration with standards of similar

size to that of the specimens under study, except for the case of thin pickup coils with ellipsoidal samples, whose dipolar

fields allow a closed-form mathematical solution. 1 A method of calibration of an ac susceptometer for relatively long cylindrical specimens of arbitrary dimensions,

coaxial with a pickup coil, is based on a calculation of mutual

inductance. The mutual inductance between two coaxial

coils (inner and outer) may be numerically computed. Since

the flux density distribution of a uniformly, longitudinally

magnetized cylinder of uniform composition is the same as

that of a solenoid of the same size, a cylindrical specimen

may be modeled as an inner coil. The calculation of mutual

inductance leads to the absolute magnetic volume susceptibility of the cylinder, when an ac field is applied, in terms of

the emf induced on the outer pickup coil.

I. CALIBRATION BY CALCULATION OF MUTUAL

INDUCTANCE

A. Equivalent solenoid

In general, the flux density B may be thought of as due

to Amperian currents of volume density V x M and surface

density Mxn, where M is the magnetization and ft is the

unit vector normal to the surface. The flux density B external

to a cylinder of uniform axial magnetization (VxM = 0) is equivalent to that of a model solenoid of the same size with ampere-turn density NI //, where M has units A/m, Nis the number of turns, / is the current, and / is the length:

M = NI/l. (1) The near field cannot be expressed in closed form, but may

be computed by superposition of B due to elemental uni- formly magnetized disks or elemental circular current loops

for the cylinder or equivalent solenoid, respectively. The

near field of a current loop is well known. 2,3

The total magnetic flux (flux linkage) <P sensed by the pickup coil may be calculated from either B or the vector

potential A:

<t> = JjB*</a = (jLws, (2)

where da is the incremental area, ds is the incremental contour, and the integrations are over the entire pickup coil. The

mutual inductance per solenoid turn is L* = L. 2/N=<P/NI= —i \-ds, (3) NI J

where L , 2 is the mutual inductance between the solenoid and

the pickup coil. L , 2 is proportional to N;L* is independent of Abut depends on the geometries of the solenoid (i.e., specimen) and the pickup coil, and the number of turns on the pickup coil. A general mutual inductance calculation is detailed in Sec. I B. The emf induced in the pickup coil due to the solenoid is

v = — d<P /dt — — L l2 dl /dt. (4)

Fig. 1. One method [see number (6) in Table II) of measuring at su .. rpnt'i

ity.

761 Rev. Scl. Instrum. 55 (5), May 1984 761

43
Converting current to magnetization using Eq. 1, u= -L*ldM/dt. ( 5 ) To this point, the ac field source has not been part of the

derivation. In fact, it has been and will continue to be as- sumed that the emf induced in the pickup coil by the field source is nulled (see Sec. II B). The ac field is assumed to be

uniform. Substituting M = where \ is the volume susceptibility (dimensionless), H = H0 cos 2nft is the applied

ac magnetic field, and rearranging Eq. (5), the final result is obtained:

X =

L*l2nfHr,

( 6)

where/is in hertz and^ is in SI units. To convert to cgs units, the value of\ is divided by 4n. The assumption of uniform M

and H is a good one for relatively long cylinders characterized by small demagnetiztion factors (see Sec. I C).

B. Mutual inductance of coaxial solenoids

The mutual inductance [L n in Eq. (3)] may be calculat- ed using computer techniques for the general case of coaxial, but not necessarily concentric, thick solenoids. Figure 2 shows the geometry and defines some of the variables. The

mutual inductance is calculated by integrating A, due to a current density J in coil 1, around the turns of coil 2. The

vector potential A has only a <t> component. At the point

[r,4>j] in cylindrical coordinates, 4

A#(r, z) = r da a

f dd cos# ln[£ + [£

2 + r2 + a2 — 2ar cos 9 ) 1/2

] If

1 4tt J„,, Jo = rdaa2 Cdd(— irsm20

- .W^A(r>z)i 4ir Ja „ Jo V(a 2 + r

2 — 2ar cos d)[£ 2 + r

2 + a

2 — 2ar cos 9 ) 1/2 /

fl \v ( 7)

where £ = z — A, g, = z — l /2, £2 = 2 + / /2, and the cylin- drical coordinates of each current element are a, 9, A. J is assumed constant and is simply

* — • / (a 1

— a 1

) From the expression for L l2 in Eq. (3), and using Eqs. (7) and

( 8 ) L„ =

4vl(a {2 -au) where / is canceled.

A(r, zyds. ( 9 )

For the purpose at hand,

a i 2 ~ a n = d /2, where d is the specimen diameter. All that remains is to compute the contour integral of A. For purposes of computation, coil 2 is divided into nXm circular rings, usually not coincident with the actual windings (see Fig. 2). The radius of each ring is

r< = a2l + (i - {)(a 22 - a 2l )/n, i = 1,2 n, (10) and its position is Zj=zd -{l2 + (j-{)l2 /m, j = 1,2, (11) The double integral A [given in Eq. (7)] is evaluated for each

ring located at (r, , zf using a standard library routine. Then

the contour integral of A in Eq. (9) is simply

&A(r,z)'ds= ]T £ 2irri A(ri ,z]

), (12) J mn i - i j= i where ds is rd</>. The factor N2/mn is introduced to scale L , 2 to reflect the actual number of turns in coil 2. If the coil displacement zd

is zero (the usual case when the specimen

cylinder is centered in the pickup coil), the index j can run from 1 to m/2 and the result for L 12 doubled, thereby taking

advantage of the axial symmetry. The number oinXm dis- crete segments should be chosen to optimize the tradeoff between accuracy and computation time. Other procedures may be used to calculate L X1 . Thincoil approximations are clearly suitable for the model sole- noid and are often adequate for the pickup coil.

5

C. Demagnetization factor considerations

When susceptibilities are large or when cylinders are not too long, the measured x should be corrected for demagnetization fields *in . =*/(! -DX ), (13) where ^,nl is the “internal” susceptibility, corrected for de- magnetization, and D is the demagnetization factor (0 < D

< 1 in SI). Exact values of D are obtainable only for ellip- soids. Average values for uniformly magnetized cylinders

have been computed by Brown6 7 and Crabtree8 and are giv- en in Table I. These “magnetometric” values are greater than those “ballistic” values reported by Bozorth and Chapin9 for highly permeable rods. The latter were experimentally determined and based on the magnetization at the middle of a rod rather than an average over the entire volume. Note that the approximation of uniform longitudinal magFig. 2. Geometry for mutual inductance calculations.

762 Rev. Scl. Instrum., Vol. 55, No. 5, May 1984 ac susceptometer 762

44
netization is realistic only in the cases of materials with small Mscepubilnies, or relatively long cylinders characterized by

small demagnetization factors. Therefore, calculations with

cylinders are always approximate to some degree.

D. Eddy current considerations

The eddy currents induced in a conducting material exposed to an alternating magnetic field are a function of the

material's susceptibility, resistivity, and geometry. These

currents give rise to an internal field which differs from the applied field in magnitude and phase. For accurate susceptibility measurements, these deviations should be minimized.

Zijlstra

10 examines the case of an infinitely long cylin- der subject to an ac axial field. The deviations m field magnitude and phase are a function of the reduced radius

a0 = d A25. (141 where d is the cylinder diameter and 5 is the skin depth, S = (p/irfp\ U2

, (15) where p is the material resistivity, /is the frequency, and /a is the permeability equal to p0

\ 1 — \ • where p0

is the permeability of free space. The analysis shows that magnitude errors m x are less than 3% when co <0.9 and that errors in both magnitude

an'' phase are negligible when a0 < 0.2. The restriction on a0 may be met by reducing d and /, since p and u are materials

properties.

II. EXPERIMENT

A. Example

The dependence of L * upon a cylinder’s length and di- ameter was computed for a specific pickup coil for illustrative purposes. A plot is shown in Fig. 3. The normalization of

the variables applies to this particular coil only.

The room temperature susceptibility of a sample of AlSI-type 316 stainless steel was measured using this coil.

T *BLE I Longitudinal demagnetization factors D for cylinders as a function of the ratio of length to diameter I/d [after Crabtree Ref. 5 ].

l/d D I/d D !/d D

0.0 1.000 2.0 0.181 4.0 0.0978

0.1 0.7% 2.1 0.174 4.1 0.0956

0.2 0.680 2.2 0.167 4.2 0.0935

0.3 0.594 2.3 0.161 4.3 0.0914

0.4 0.528 2.4 0.155 4.4 0.0895

0.5 0.474 2.5 0.149 4.5 0.0876

0.6 0.430 2.6 0.144 4.6 0.0858

0.7 0.393 0.140 4.7 0.0841

0.8 0.361 2.8 0.135 4.8 0.0824

0.9 0.334 2.9 0.131 4.9 0.0808

1.0 0.311 3.0 0.127 5.0 0.0793

1.1 0.291 3.1 0.123 5.5 0.0723

1.2 0.273 3.2 0.120 6.0 0.0666

1.3 0.257 3.3 0.116 6.5 0.0616

1.4 0.242 3.4 0.113 7.0 0.0573

1.5 0.230 3.5 0.110 75 0.0536

1.6 0.218 3.6 0.107 8.0 0.0503

1.7 0.207 3.7 0.105 8.5 0.0473

1.8 0.198 3.8 0.102 9.0 0.0447

1.9 0.189 3.9 0.100 10.0 0.0403

The result was verified with a SQUID susceptometer. A cyl- inder of length 10.2 cm and diameter 1.2 cm was prepared.

L * was computed to be 14 pH for the combination of cy fin- der and pickup coil. An ac field of 570-A/m rms at 100 Hz

was applied, x was found to be 2.9 X 10 _J! SI . This approxi- mates the smallest values of susceptibility which can be ac- curately measured with the present apparatus. A small piece of steel was obtained from the same rod stock for SQUID

measurement. Using a density of 7.958 g/cnr. % was

3.0 X 10 -3

, m good agreement with the mutual inductance

method. Taking a resistivity oilSpfl cm," and applying the eddy-current criterion of Sec. I D. a0

is found to be 0.195. which is very acceptable.

B. Coil compensation

As mentioned in Sec. II A, the background emf of the pickup coil with no sample present needs to be subtracted from the measured emf with the sample in place. This may be done by the various methods shown in Table II. Methods i4

(5 1, and 6' seem to be satisfactory: method 6 . illustrated m

Fig. 1, is the least expensive and most straightforward.

III. OTHER CALIBRATION METHODS

A. Change in self-inductance

A variation of the method described in Sec. I A is that of

relating the change m self-inductance of a pickup coil, after a specimen is inserted, to the magnetic susceptibility of the specimen. This technique may be used with a commercial

inductance meter equipped with a convenient zero offset. Such devices may be thought of as applying a sinusoidal current to a test coil and measuring the voltage across it. The

following equations apply:

dl

dt’ where L, is the self-inductance. I = I0 cos 277ft, 17

(18)

— L, '16.

FlC. 3. Dependence of L * upon c> ..r.der length and c ameter The . > -

coil parameters are: a- = 119 mm. J-- = 13 3 — 3" —

.V, = 6042. z, = 0.

7S3 Rev, ScL Inetrum., Val, 55, No. 5, May 1984 ac susceptometer 7M

trratum: Page 763, second column , line 3 and Fig. 3: "L*" should re

45
Table II. Various methods of pickup coil compensation.

Method of coil compensation Remarks

( 1

)

( 2) (3)

(4)

(5)

( 6)

Subtract dc output of lock-in amplifier for empty pickup coil Null dc output of lock-in amplifier using internal offset Subtract ac signal of pickup coil using digital recording oscilloscope Subtract rms level of ac signal of pickup coil using digital oscilloscope Subtract an ac waveform equal to that of empty pickup coil, using a

differential amplifier, before input to lock-in amplifier

Series matched, counterwound “bucking” coil to null sample pickup coil

Simple; poor resolution because Ao/v is small Simple; offset works only over limited range Any phase shift caused by eddy currents gives errors

Digital processing and noise filtering needed Need stable phase-lock phase shifter with variable gain; eddy-current phase shifts give errors Both coils must be exposed to same ac field

From Eq. (6), the sample contribution to the voltage is

”nn, =L*lxlirfHma . (19) The field//TM is proportional to the current in the coil Hms = clms , (20) where c is a constant of dimension m

- '. The total voltage is

"r.rms + "rms = 2vfIrms {Ls + CL * ly) =2ir//rms (Lf +4L). (21) The volume susceptibility is obtained from the increase in inductance AL when the specimen is inserted in the coil

This method, relying on the coil for both field and pickup, is

less sensitive than the method in Sec. I A.

B. Calibration with standards

It is often desirable to calibrate susceptometers experimentally, which requires the use of standards. The resulting calibration is strictly valid only for specimens of the same

size and shape as the standard used.

1. Materials with known susceptibility

The NBS Office of Standard Reference Materials sells four susceptibility standards: Al, Pt, Pd, and MnF2 , with y

ranging from 2xl0 -5

to 6x 10 -3(SI) at room temperature.

13 They are available in various forms.

2. Soft ferromagnets with “infinite” internal susceptibility

and known demagnetization factor An example is a spherical sample of Gd near its Curie

temperature. Some provision for cooling is needed to experimentally obtain the maximum pickup voltage.

The susceptibility y > s generally proportional to the pickup coil voltage v, and inversely proportional to the sample volume V, the ac field H, and its frequency /: y = av/ VfH. (23) The empirical calibration constant a is obtained by measuring the pickup coil voltage v0 for a sphere of volume V0 and

demagnetization factor 1/3 in a field H0 of frequency /0

. Assuming an infinite ^,nt , Eq. (13) yields

1 -Dy0 = 0, (24) whence the measured susceptibility y0 = 3. Thus, a = 3F0 /0//0

/i;0

. (25)

764 Rev. Sel. Inetrum., Vol. 55, No. 5, May 1984

3.

Superconducting materials with known

demagnetization factor An example is a spherical sample of Nb below its criti- cal temperature. The ability to cool to cryogenic temperatures is required. The perfect diamagnetism (^ml = — 1) of superconductors arises from the Meissner effect. From Eq.

(13), Xo = Dy0 -\, (26) and for the sphere, y0 = — 3/2, or 3/2 but 1 80° out of phase.

Equation (23) applies with a = 3F0 /0tf(/2t,0

. (27)

Similar calibrations may be done, of course, with cylin- drical standards using the approximate demagnetization

factors in Table I. Methods of preparing spheres are de- scribed in Ref. 14. ACKNOWLEDGMENTS

F. R. Fickett, S. M. Goodnick, C. A. Hamilton, and D.

A. Krueger commented critically and valuably on the manuscript. The work was partially funded by the Air Force Office of Scientific Research.

'See, for example, the result given in W. R Abel, A. C. Anderson, and J. C. Wheatley, Rev. Set. Instrum. 35, 444 1 1 964).

2 J D. Jackson, Classical Electrodynamics { Wiley, New York, 1962), pp. 142 and 143. 'W R. Smythe, Sialic and Dynamic Electricity, 3rd ed. (McGraw-Hill, New York, 1968), pp. 290 and 291. “The derivation of the vector potential for a thick solenoid may be found in G. V Brown, L Flax, E. C. Itean, and J C. Laurence, NASA Tech Report No. TR R-170, December 1963, available at NTIS, Springfield, Virginia, N64-15331. 'Formulas and tables for the mutual inductance of coaxial, single-layer

coils may be found in F. W. Grover, Inductance Calculations (Van Nostrand, New York, 1946), Chap. 15; E. B. Rosa and F. W. Grover, Bull. Bur Stand. 8, 1 (1912).

RW. F Brown, Jr., Am. J. Phys. 28, 542 (1960).

7 W F Brown, Jr., Magnetostatic Principles in Ferromagnetism (NorthHolland, Amsterdam, 1962), pp. 187-192. The magnetometnc demagnetization factors may be obtained from the re//-inductance of the model sole- noid.

H G. W. Crabtree, Phys. Rev. B 16, 1 1 17 (1977).

9 R. M. Bozorth and D. M. Chapin J. Appl. Phys. 13, 320(1942).

I0H. Zijlstra, Experimental Methods in Magnetism (North-Holland, Amsterdam, 1967), Vol. 2, pp. 72-79. "H. M. Ledbetter, Met. Sci. 14, 595 (1980).

I2 F. R. Fickett, in Materials at Low Temperatures, edited by R. P. Reed and

A. F. Clark (American Society for Metals, Metals Park, Ohio, 1983), Chap. 5, p. 197.

I3 NBS Spec. Publ. 260 (Office of Standard Reference Materials, National Bureau ofStandards, Washington, D. C., 1981).

I4 J. N. Paranto and C. E. Patton, Rev. Sci. Instrum. 52, 262 (1981).

ac susceptometer 764

46
QIIF.NCH circuit for electronic instruments used with superconducting magnets

R. G. Benson, R. B. Goldfarb, and E. S. Pittman

Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

A multifunction circuit is described that protects, in the event of a

quench, instruments connected or coupled to a superconducting magnet.

INTRODUCTION

Most laboratory superconducting magnets are suitably protected against

damage if they revert to the normal state (quench). Often it is desirable to

provide protection for instruments connected or coupled to such a magnet.

Such protection is provided by a circuit that performs three functions: (1)

sends an interrupt command to a computer via an interface card, (2) protects

an analog-to-digital (A/D) card that monitors the magnet voltage from large

voltages generated during a quench, and (3) shorts the output of a digital-toanalog (D/A) programming-voltage card and thus nulls the output of the magnet

current supply. A diode bank that protects a magnet power supply is also

described

QUENCH DETECTION CIRCUIT

A superconducting magnet in our laboratory is powered by a constantcurrent supply. This power supply, programmed with the dc voltage output of a

D/A card, is bipolar and referenced to ground. The D/A card is controlled by

a microcomputer. The magnet voltage, L dl/dt, is sensed by an A/D card which

47
in turn is connected to the computer. By monitoring this voltage, the

computer is able to determine when the magnet has finished charging or

discharging

In the event of a quench, we wish to: (1) inform the computer that a

quench has occurred by enabling the interrupt card, (2) protect the A/D card

against possible high magnet voltages, and (3) bring the power supply output

to zero by nulling the programming voltage of the current supply.

We designed and constructed a circuit to perform these tasks. The input

is connected across the magnet. The circuit’s input impedance is greater than

500 kfi. A comparator-threshold adjustment sets the voltage criterion for a

quench (e.g., 3.5 V). A time-constant adjustment prevents the quench detector

from triggering on random, transient voltage spikes. This sets a time delay

of, typically, 10 to 250 ms for actual quench detection. When a quench is

detected, an LED turns on. Once the magnet voltage has decreased to below the

quench-detection threshold, pressing the RESET button rearms the circuit.

The circuit schematic is shown in Fig. 1. There are three outputs:

INTERRUPT, A/D OUT, and D/A OUT. (1) When a quench is detected, a logical "1"

is sent to the interrupt card. (2) A/D OUT normally sends the magnet voltage

to the A/D card. In the event of a quench, a relay breaks this connection.

Additional protection is provided by a series fuse. Finally, two Zener

diodes, D3, in parallel with the output prevent the magnitude of the output

voltage from exceeding 12 V. (3) D/A OUT is normally an open switch in

parallel with the output of the D/A card. When a quench occurs. Relay A

closes, forcing the programming voltage of the magnet current supply to zero.

(The D/A card is internally current limited.)

The circuit is divided into several stages: input protect, absolute

value, voltage comparator, latch logic, and output stages.

48
Input protect . Large voltage spikes dissipate power in resistor Rl. The

input voltage to the rest of the circuit is limited to about 10 V by the two

Zener diodes Dl.

Absolute value . The absolute value stage uses two operational amplifiers, A1 and A2, and has a positive output with unity gain. For positive

input voltages, diode D2 is reverse biased, A2 is effectively removed from

the circuit, and A1 is a unity-gain amplifier. For negative voltages, D2 is

forward biased, the positive input to A1 is grounded, and A1 acts as an

inverter

Voltage comparator . The comparator produces a low output of zero volts

for inputs greater than the reference voltage and a high output of 4 V otherwise. The feedback resistor provides some hysteresis in the comparator. The

QUENCH-THRESHOLD adjustment R2 sets the comparator reference voltage V ^

The TIME-CONSTANT adjustment R3 is part of a simple RC filter.

Latch logic . When the circuit is turned on, the 10-pF capacitor Cl holds

the CLEAR input to ground long enough to allow the flip-flop to start in the

cleared state (Q low and Q high). After about 10 ms, the CLEAR input goes

high. It can be set low with the RESET button. When a quench is detected,

the output of the voltage comparator goes low and SET becomes active, causing

Q to go high and Q low.

Output stages . When Q is high, a logical "1" is sent to the interrupt

card. Also, transistor T1 is turned on, which lights the LED and closes Relav

A. This in turn shorts D/A OUT, which zeros the programming voltage to the

power supply. When Q is low, transistor T2 is turned off. This opens Relay

B, which protects the A/D card.

49
POWER SUPPLY PROTECTION

Power supplies designed for use with superconducting magnets are often

built with quench protection. A quench is accompanied by large voltages

(typically a few hundred volts) and currents of polarity opposite that of the

charge current. Most power supplies are not designed to sink reverse current.

In the case of a unipolar dc supply, adequate protection may be provided by a

reverse-biased, heat-sinked power diode across the output terminals, in

parallel with the magnet leads.

In the case of a bipolar supply, however, the forward voltage drop of the

diode would be too small for negative-current operation; magnet charging

voltages are often as great as 3 V. We have constructed an array of heatsinked IN 3111 diodes to protect such bipolar supplies. The circuit is shown

in Fig. 2. It is connected across the output of the power supply, in parallel

with the magnet leads. The diode array does not conduct for typical magnet

charging voltages. If quench voltages are large in either polarity, the array

conducts, thereby protecting the power supply. The six diodes in series in

either direction begin to conduct at 4 V and conduct well at 5 V.

ACKNOWLEDGEMENT

This work was sponsored by the Air Force Office of Scientific Research.

50
input

absolute

voltage

latch

output

Z)o

Do

< Q

o o

O z

51

Figure

1.

Quench

protection

circuit
Figure 2. Diode array for power supply protection.

52
CHARACTERIZATION OF ALLOYS

R. B. Goldfarb

Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado 80303

I. PHYSICAL CHARACTERISTICS OF MULT I FILAMENTARY Nb-Ti SUPERCONDUCTORS

We have found that, except for the number of Nb-Ti filaments, superconductor wire manufacturers often do not reliably report the physical characteristics of their wires. For accurate magnetization and loss calculations,

however, we must know the copper-to-superconductor volume ratio and the

filament radius in addition to measured magnetic moment. It is also useful to

know the density of the Nb-Ti alloy. This note describes how we determine

these parameters.

A 1.000-meter length of wire is stripped of its insulation. We measure

its mass, m . . Using Archimedes' principle, we determine its volume, V . wire wire

The wire is then soaked in a 50% solution of nitric acid and water to remove

the copper, leaving only the Nb-Ti filaments. The filaments are rinsed and

thoroughly dried. We then measure their mass, m^^. We do not directly

determine the volume of the filaments, because of the danger of trapping

air bubbles upon immersion in water. Instead, we calculate the mass of

3 copper, nu = m . - mri1 , and use the known density of pure Cu (8.93 g/cm ) Cu wire fil

to calculate the volume of copper, V„ . Then, V,. . .. = V . - V_ . Cu fil wire Cu

The copper-to-superconductor volume ratio is then V /V^^. The density

n

of the Nb-Ti alloy is m^^/V^ The filament radius is (V^^/ttNL) , where N

is the number of filaments given by the manufacturer and L is the length

(1.000 m).

53
II. CONVERSION OF ATOMIC AND WEIGHT PERCENTS

Let the atomic masses of the elements be denoted by A , the atomic

x percents be denoted by a^, and the weight percents be denoted by w.. By

definition, Za. = 1 and Zw. = 1. 1 x To convert atomic to weight percent:

w. = a.A./Za.A. . (1) i i i 3 3 The average atomic mass of the alloy is A:

A = Za . A. . (2)

3 3 Thus

w.xxx= a. A. /A . (3)

To convert weight to atomic percent:

a. = (w. /A. ) /Z (w. /A.

)

. (4) i ii 3 3 As pointed out by H. M. Ledbetter (private communication), substituting Eq. 4

into Eq. 2 gives

A = [Z(w /A )]

_1

. (5)

Thus

a. = (w./A.) A . (6)

X XX

III. DENSITIES OF SUBSTITUTIONAL BINARY ALLOYS

Often we wish to deduce the density of a binary alloy by interpolation

from the densities of the pure elements. There are two approaches. The first

is to assume that both elements have the same atomic volumes. We then

interpolate by substituting the atomic mass of one for the atomic mass of the

other. The second approach is to assume the same atomic masses. We then

interpolate by substituting the atomic volumes. Both methods are presented

below, applied to the Nb-Ti system. In this example, where the atomic masses

of Nb and Ti are known to be very different, the first approach is better.

54
Symbols

Subscripts indicate element 1 or 2

p = density

m = mass per atom

v = volume per atom

n = number of atoms per cm

3

Same Atomic Volumes

q = number of atoms per g

a = atomic percent element 1 w = weight percent element 1

There is the same number of atoms per cm

3 for elements 1 and 2 (n^ = n^) .

The atomic mass of one is substituted for the atomic mass of the other.

(7)

( 8 )

A "standard" alloy

p = nm

P alloy

= n[am

l + (1_a)m2

]

= ap

1 + ( 1-a) p

For example, p = 8.57 g/cm3 and p^ = 4.51 g/cm 3

,

composition is 46.5 wt.% Ti = 63.0 at . % Ti. Thus,

P NbTi

= 6,012 §/ cm3 (9)

This compares favorably with a measured density of 6.004 g/cm3

.

Same Atomic Masses

There is the same number of atoms per g for elements 1 and 2 (q^ = q 9

)

The atomic volume of one is substituted for the atomic volume of the other.

( 10 ) -1P = qv

P alloy ’ q[uV

l + < 1 -“) v2

1

= u/p

1 + (

1 —u) / P 2

P alloy

' P

1 P2 /[(PI +u(P2

-P

1 )]

( 11 )

( 12 )

For the example of Nb-Ti of "standard" composition, we would obtain an inac-

.3 P NbTi

= g/cm3 curate estimate: . The first method, assuming equal

atomic volumes for the elements, is more suitable.

If the densities of certain alloy compositions are known, further refinement is possible by interpolating between them instead of the densities of the

pure elements.

55

UNITS FOR MAGNETIC PROPERTIES

Quantity Symbol Gaussian & cgs emu*2 Conversion

factor, C

b SI & rationalized mks ‘

Magnetic flux density, magnetic induction B gauss (G) d 10“ 4 tesla (T), Wb/m2

Magnetic flux <t> maxwell (Mx), G-crrr 1(T 8 weber (Wb), volt second (V-s)

Magnetic potential difference, magnetomotive force U, F gilbert (Gb) 10/477 ampere (A)

Magnetic field strength, magnetizing force H oersted (Oe).

f Gb/cm 103 /477 A/m

(Volume) magnetization s M emu/cm3 h 10 3 A/m

(Volume) magnetization X-rrM G 103 /477 A/m

Magnetic polarization,

intensity of magnetization

J. I emu/cm 3 477 X 10~ 4 T, Wb/m2 '

(Mass) magnetization cr. M emu/g

1477X 10- 7 A-mVkg

Wb-m/kg

Magnetic moment m emu, erg/G 10- 3 A-m2

, joule per tesla (J/T)

Magnetic dipole moment j emu, erg/G 477 X 10- 10 W'b-m '

(Volume) susceptibility X. K dimensionless, emu/cm3 477

(477)

2 X 10' 7 dimensionless henry per meter (H/m), W’b/(A-m)

(Mass) susceptibility Xp. Kp cmVg, emu/g

477 X 10 -3

(477)

2 X 10- 10 m

3 /kg

Hm: /kg

(Molar) susceptibility XnioU Kmol cm3 /mol, emu/mol

477X 10- 6

(477)

2 X 10- 13 m

3 /mol

H-mVmol

Permeability F dimensionless 477 X 10“ H/m, Wb/(A-m)

Relative permeability 1 M'r not defined dimensionless

(Volume) energy density, energy product

w erg/cm3 10“' J/m 3

Demagnetization factor D. N dimensionless 1/477 dimensionless

a. Gaussian units and cgs emu are the same for magnetic properties. The defining relation is B =H +4n\1 .

b. Multiply a number in Gaussian units by C to convert it to SI (e.g., 1 G X 10~ 4 T/G = 10

4 T).

c. SI (Syst'eme International d'Unites) has been adopted by the National Bureau of Standards. Where two conversion factor-, .ire given, the upper one is recognized under, or consistent with, SI and is based on the definition B = no(H - M i. where

p.o = 47rX 10“ H/m. The lower one is not recognized under SI and is based on the definition B = \x<JI ~J where the umh'l

/ is often used in place of J

d. 1 gauss= 105 gamma (y).

e. Both oersted and gauss are expressed as cm" 1/2 -g

1/2

-s _l

in terms of base units

f. A/m was often expressed as "ampere-turn per meter" when used for magnetic field strength

g. Magnetic moment per unit volume.

h. The designation "emu" is not a unit.

/. Recognized under SI, even though based on the definition B = p.oH +J

See footnote c

j. jx r = p./po = 1+X' aH >n SI. \x, is equal to Gaussian p..

k. B H and have SI units J/m3

; M H and B-H /Av have Gaussian units erg/cm'

R B Goldfarb and F. R. Fickett, U.S. Department of Commerce, National Bureau of Standards. Boulder. Col. rad 1 > M NBS Special Publication 696 For sale by the Superintendent of Documents, U.S Government Printing Office W,i\h ngt •

I 11 ;

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National Bureau of Standards

6b OFFICE SYMBOL

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Bolling Air Force Base

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11 TITLE (Include Security Classification)

TRANSIENT LOSSES IN SUPERCONDUCTORS (Unclassified)

12 PERSONAL AUTHOR(S)

Goldfarb, R. B. (Editor)

13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year, Month, Day) 15 PAGE COUNT

Final Report FROM 10/82 TO 9/85 1986 June 72

16 SUPPLEMENTARY NOTATION

17 COSATI CODES

FIELD GROUP SUB-GROUP 20 03

20 12

18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number) ac losses; hysteresis; magnetization; magnetometers;

niobium-tin; niobium-titanium; superconductors;

susceptibility; susceptometers ; transient losses

19 ABSTRACT Under steady-state conditions, there are no losses in superconducting

wires. However, when subjected to alternating or transient magnetic fields or transport currents, losses in type-II superconductors can become significant.

This report deals with hysteresis losses at 4 K measured by magnetization and

complex magnetic susceptibility. The theoretical and experimental relationships between ac susceptibility and magnetization as functions of dc field were examined in terms of the critical-state model as developed by Carr and

Clem. Minor-loop hysteresis loss was shown to be obtainable by direct measurement of loop area, from the imaginary component of ac susceptibility, and from

the reversible susceptibility plus tne critical current density or fullpenetration field. Hysteresis and transport losses measured simultaneously in

a Nb-Ti superconducting coil were found to agree substantially with the

predictions of Minervini’s two-dimensional model. Hysteresis loss measurements

in a series

filament diamet

ratio of matrix material to Nb. A review of internal fields in superconductors showed the importance of demagnetization factors and internal fields

for the correct analysis of magnetic data. A theoretical method of calibrating

ac susceptometers for cylindrical specimens, which is based on a mutualinductance calculation, was developed.

of fine-filament Nb-,Sn superconductors snowed that the effective

iter is a function J of interf ilament separation and local

20 DISTRIBUTION /AVAILABILITY OF ABSTRACT S UNCLASSIFIED/UNLIMITED SAME AS RPT OTIC USERS

21 ABSTRACT SECURITY CLASSIFICATION Unc lassi f iecJ 22a NAME OF RESPONSIBLE INDIVIDUAL Harold Weinstock

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BIBLIOGRAPHIC DATA

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1. PUBLICATION OR

REPORT NO.

NBSIR 86-3053

2. Performing Organ. Report No. 3. Publication Date

June 1986

4 . TITLE AND SUBTITLE

Transient Losses in Superconductors

5. AUTHOR(S)

R. B. Goldfarb, Editor

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final Report

1 Oct 1982 - 30 Sep 1985

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Air Force Office of Scientific Research

Bolling Air Force Base

Washington, DC 20332

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 in formation . If document includes a significant

bi bl iography or literature survey, mention it here)

Under steady-state conditions, there are no losses in superconducting

wires. However, when subjected to alternating or transient magnetic fields or transport currents, losses in type-II superconductors can become significant.

This report deals with hysteresis losses at 4 K measured by magnetization and

complex magnetic susceptibility. The theoretical and experimental relationships between ac susceptibility and magnetization as functions of dc field were examined in terms of the critical-state model as developed by Carr and

Clem. Minor-loop hysteresis loss was shown to be obtainable by direct measureand from

.simultaneously in

a NbyTi superconducting coil were found to agree substantially with the

predictions of Minervini's two-dimensional model. Hysteresis loss measurements

in a series of fine-filament NboSn superconductors snowed that the effective

filament diameter is a function~of interfilament separation and local area

ratio of matrix material to Nb. A review of internal fields in superconductors showed the importance of demagnetization factors and internal fields

for the correct analysis of magnetic data. A theoretical method of calibrating

ac susceptometers for cylindrical specimens, which is based on a mutualinductance calculation, was developed.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon

ac losses; hysteresis; magnetization; magnetometers; niobium-tin; niobium-titanium;

superconductors; susceptibility; susceptometers; transient losses.

13. AVAILABILITY

Uni i mi ted

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r]] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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PRINTED PAGES

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15. Price

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U. S. GOVERNMENT PRINTING OFFICE 1986—672-376 -P, (if , N >.