6.3 Total resistance measurements of the original sample holder

6

6.1 Introduction

Since 1998, critical current measurements at the Short Sample Test Facility (SSTF) of the TD have been performed using a sample holder set-up that had been designed based on the so-called ITER barrel [16]. Such barrel was used with a set-up that employed pressure contacts to transfer the current between the copper rings of the barrel and the copper lugs. It was designed a few years ago for testing strands developed for the International Thermonuclear Experimental Reactor (ITER), which featured a Jc (12 T) ~ 700 A/mm2, and carried only a few hundreds Amps in the magnetic field range of interest.

1736

1436

1180

961

773 612 400

800 1200 1600 2000

9 10 11 12 13 14 15 16

Magnetic Field [T]

Critical Current [A]

Tc = 18 K Bc20 = 26 T

Cu% ~

Jc_strand(12 T) ~ 3000 A/mm²

Fig. 6.1 Current and field ranges needed for high Jc strands

However, it is not uncommon for nowadays Nb3Sn wires to have Jc (12 T) ~ 2000 A/mm2, and state-of-the-art superconductors show Jc (12 T)’s as high as 3000 A/mm2. In the case of 1 mm diameter strands, this results in currents between 600 A and 1800 A at magnetic fields from 15 T down to 10 T.

The first step to be undertaken was that of measuring the current limits of the existing set-up. This could be done also thanks to the recent upgrade of the sample power supply from 1000 A to 1800 A.

The main question, to which this chapter tries to answer, is whether the present contact (used to connect the currents leads to the sample holder) introduces a resistance low enough to measure critical currents ( larger than 1000 A ) without heating and/or quenching the sample. Transport current measurements were performed using the original probe and the upgraded power supply system to experimentally evaluate the current limits of the set-up.

To estimate the maximum contact resistance that would allow carrying 2000A, the measured total set-up resistance was correlated to the maximum transport current obtained with the available cooling. An extrapolation was made at higher currents.

An analytical evaluation of the sample holder resistances was also performed and various models were used to predict the values of different kinds of contact resistance. These models were validated with measurements and used to understand where to improve the set-up.

Fig.6.2 2 x 6680A AGILENT 895 A/5 VDC PS in parallel (PS LIMIT 1800 A)

6.2 High current measurements with the original probe

The upgraded power supply system generates a maximum current of 1800 A.

We performed a high critical current measurement with the old probe to check its current limit. For such experiment we used NbTi wire, because it is very stable also at low field where we reach high currents. The following plot shows the results for two samples.

Fig.6.3 Old probe performance at high current.

According to this plot one can see that the current limit of the old probe is about 1400A.

6.3 Total resistance measurements of the original sample holder

6.3.1 Introduction

Three kinds of contact resistances in series give the total contact resistance of the set-up (resistance between the bottom ends of the two currents leads). These are the resistance of the soldered parts between the copper rings and the SC wire, the contact resistance between the copper lugs and the copper ring, and between the brass nut and the inner current lead, and the copper resistance of all the other parts. To improve the contact a thin layer of indium is used between the copper lugs and the copper rings, and between the bottom copper lug and the brass nut.

0 500 1000 1500 2000

0 1 2 3 4 5 6

B field [T]

Ic or Quench Current [A]

NbTi - 2 turns NbTi - 4 turns

0 500 1000 1500 2000

0 1 2 3 4 5 6

B field [T]

Ic or Quench Current [A]

NbTi - 2 turns NbTi - 4 turns

Fig. 6.4Voltage taps.

Fig. 6.5: Voltage taps

The original sample holder set-up is typically instrumented with three voltage taps located as shown in Fig 6.5.

The total sample holder resistance is evaluated as a function of magnetic field to take into account the effect of magnetic induction. Such resistance is then plotted over the whole field range against the maximum transport current obtained in each measurement with the cooling provided in the Variable Temperature Insert (VTI). The maximum contact resistance that would allow carrying 2000 A is extrapolated using the maximum equal-power-generation curve.

1

2

Current

lead Current

Superconducting wire lead

V34

V16 V25

6.3.2 Error analysis

Statistical errors affecting the total resistance measurement are due to two distinguishable sources: one related to the instrumentation precision and small disturbances in the cooling, and the other related to variations in sample mounting.

Statistical errors due to the first source are evaluated, for each mounted sample, by repeating the measurement several times. Because we use previously available data (that do not have repeated measurements) the contribution of this source of error was not evaluated.

Errors introduced by mounting the sample are evaluated by repeating the resistance measurement on 34 samples independently prepared. The same mounting procedures are followed for all the samples (the torque applied to the nut is measured). The average resistance value (Rt) and its standard deviation (σst) are calculated. σst is then used as an estimate of the statistical error.

The major sources of systematic errors are: instruments calibration and the operator.

As the instruments are periodically calibrated and submitted to maintenance, the accuracy valued provided by the manufacturing company is used as be an estimate of the deviation from the operating value.

In order to evaluate the systematic error due to the operator in mounting the sample, two samples were prepared by two different people and measured by the same one. The average value of the resistance was calculated for both samples and compared. The difference between the two average values was smaller by two orders of magnitude than the standard deviation due to the statistical errors. For this reason this error is considered negligible.

To evaluate the systematic error introduced by the operator in analysing the data, the same plot was analysed four times by two different people, and the difference between the average values were compared. This difference was two orders of magnitude smaller than the standard deviation due to the statistical error. The sensitivity to the operator performing the analysis is thus negligible.

6.3.3 Total resistance

The total resistance was obtained as the slope of the voltage between the voltages taps located on the current leads (see Fig 6.6) as a function of the sample current.

The next Figure shows an example of the plot from which the resistance was calculated as the slope of the linear part. The upper non linear part is due to the transition of the superconductor to normal.

To evaluate how the contact resistance changes with the applied field, the average resistance was calculated from measurements at seven different fields. The

0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04

0 20 40 60 80 100 120

Current, A

Voltage, V

Fig. 6.7Plot from which the total sample holder resistance was calculated as the slope of the linear part.

Fig. 6.6 Voltage tapes location for the total resistance measurement.

plot that was obtained is shown in the following Figure (the spread is given by the root mean square of the distribution). The result is a linear trend as expected.

The estimated contact resistance is the sum of three different terms: current leads copper resistance, Indium resistance (Indium is placed between the contact surfaces to improve the contact) and the solder resistance (between the superconductor and the sample holder). All these resistances are proportional to the electrical resistivity of the material (depending on applied field) by a form factor. The Copper resistance is inversely proportional to the RRR, whereas the Indium and solder resistivity are constant for magnetic fields larger than 0.1 T. Therefore a linear trend is obtained by summing up the three resistances (two constant and one linear).

This is confirmed by the RRR as a function of the applied field, as found experimentally and shown in Fig. 6.9. Its inverse, also shown in Fig. 6.9, is linear.

The variation of the Rc with B between 6 and 15 T is small, 1 micro Ohm, and is due to the magnetic resistivity.

y = 0.0677x + 3.0727 R² = 0.9914

0 1 2 3 4 5 6

4 6 8 10 12 14 16

Field, T

Rc, micro Ohm

Fig. 6.8 Total sample holder resistance as function of magnetic field.

Fig. 6.9RRR and 1/RRR as function of magnetic field.

6.3.4 Extrapolation of maximum current capability

The total sample holder resistance and the maximum current at seven different fields were determined for 71 samples (34 samples were used to evaluate the total resistance and 37 were added later), subdividing the samples in two groups: the samples that quenched prematurely (premature quenches) at some field, and the samples that had or had started a regular transition (SC transition).

0 50 100 150 200 250 300

0 5 10 15

Field (T)

RRR

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0 5 10 15

Field (T)

1/RRR

Premature Quench

SC Transition

Fig. 6.10

In the latter group the maximum current is certainly determined by the properties of the superconductor, while in the former group the premature quench could also be due to mechanical instability or heating. The total sample holder resistance has an effect only on premature quenches due to heating. A plot encompassing all data is shown in the following Figure.

V34 (zeroed)

-0.000005 -0.000003 -0.000001 0.000001 0.000003 0.000005 0.000007 0.000009

0 200 400 600 800 1000

V25 (zeroed)

-0.000005 0 0.000005 0.00001 0.000015 0.00002

0 50 100 150 200

0 20 40 60 80 100 120 140 160

0 200 400 600 800 1000

Maximum Current,A

Total Resistance, micro Ohm

SCTransition Flat Quench

0 1 2 3 4 5 6 7 8

0 2 00 4 00 6 00 80 0 1 0 00

M a x im um Curre nt, A Total Resistance, micro Ohm

T ra n sitio n F la t Q u e n ch

Fig.6.11Total sample holder resistance as a function of maximum current in the probe.

1

The data points associated to premature quench that are located in the same area as the data points associated to regular transitions cannot have quenched due to heating. As they indeed feature similar resistance values, it means that such resistances allow most samples to have a regular transition without excessive heating at least up to the currents shown in the plot.

However, the data points associated to premature quenches that are in the upper part of the plot are more representative as due to heating because the total resistance of these samples is much larger than average.

The data points that allow transition while presenting a large resistance are the most interesting to use to infer information on what are the maximum allowed contact resistances to reach specific maximum transport currents with a given set-up. Data point “1” in Figure 6.11 was that associated to the maximum resistance in the group of samples showing regular transition. Using this experimental information, it can be stated that the heat generated by such resistance allows SC transition. The following plot shows the resistance as a function of transport current for the constant heat generation associated to this case.

Using this method, it was found that the maximum resistance that would allow carrying 2000 A is 1µΩ, which is about ¼ of that of the existing Fnal holder.

0 20 40 60 80 100

0 500 1000 1500 2000

Maximum Current [A]

Contact Resistance

[µΩµΩµΩµΩ]

R

= P/I

Fig.6.12Resistance extrapolation

6.4 Individual resistance measurements of the original sample holder

6.4.1 Error analysis

The sources of statistical and systematic errors affecting the contact resistance measurements are the same as those affecting the total resistance measurements.

In this case to evaluate the statistical errors for each mounted sample, we repeated the measurement four times. The average resistance value (Ri) and its standard deviation (σsi) are calculated. The relative error is then estimated.

Statistical errors introduced by mounting the sample are evaluated by repeating the measurement on 5 samples independently prepared. The standard deviation of the weighted mean is used as statistical error.

Systematic errors are negligible.

6.4.2 Copper RRR

The copper resistance at 4.2K was obtained as the slope of the voltage between two voltage taps located on copper lug 1 (see Fig. 6.4) at a distance of 1 inch from each other below the zero helium level, as a function of the sample current. The measurement was repeated four times at 0 T on the same sample and the average resistance is R=1.06⋅10⁻⁷ ±5.6⋅10⁻⁹Ω, where the error is the standard deviation of the distribution.

Then, knowing the geometry, we found a resistivity at 4.2 K Ω

⋅

= ⁻¹⁰

2 .

4 _K 1.481 10

ρCu that corresponds to a RRR of 113.

6.4.3 Contact resistances

Fig. 6.13 shows the position of the voltage taps used for the measurements.

Those used for R1 and Rs are placed as shown also during regular tests in order to calculate the total resistance and that of the sample, whereas the others were placed as shown only for the specific task of measuring the individual contact resistance measurements.

Fig. 6.13Voltage taps position for the contact resistance measurements.

Single contact resistances have been obtained from voltages measured between the taps indicated in Figure 6.13 as R2, R3, and R4. The measurements are performed with a magnetic field of 12 T and the results are illustrated in the following Table.

Table 6.1 Measured contact resistances

To evaluate copper contributions we use the measured copper residual resistivity ρ_Cu4.2K =1.481⋅10⁻¹⁰Ω⋅m. For the brass we have not explicitly found its low temperature resistivity, but have made estimates. The company catalogue indicated that the brass alloy would be either 270 or 360, which include 60-65%

copper, 35% zinc, small amounts of lead and a trace of iron. These alloys are commonly called "yellow brass" or "free cutting brass". Also for other copper alloys like copper-nickel alloys and even 2% beryllium copper the resistivity tends to be fairly constant with temperature. It is the equivalent of a RRR approaching 1.0 for a very impure copper. So we estimate the room-temperature value for resistivity to be roughly the value at 4 K (within 30%). For these brass alloys, the resistivity is

WHEIGHTED MEAN ERROR

R2 1.38193E-07 2.46919E-10

R3 3.21835E-07 4.88165E-10

R4 2.81456E-06 1.41257E-09

K cm

In_4.2 =0.031122⋅10⁻⁶Ω⋅

ρ , and the thickness of the foils used between the contact surfaces is about 0.1 mm.

The results are illustrated in the following Table (see also Appendices 1-2).

Table 6.2 Resistances due to the various materials (calculated)

COPPER BRASS INDIUM TOTAL

R2 1.455⋅10⁻⁹ 1.353⋅10⁻¹⁰ 1.59⋅10⁻⁹ R3 1.286⋅10⁻¹⁰ 1.378⋅10⁻⁷ 1.353⋅10⁻¹⁰ 1.388⋅10⁻⁷

10 10

982 .

5 ⋅ ⁻ 1.45⋅10⁻¹⁰

R4 1.978⋅10⁻⁸ 1.768⋅10⁻⁶ 6.703⋅10⁻⁶

The main contribution comes from the brass nut while the Indium contribution is negligible. The resulting contact resistances are:

Table 6.3 Summary

6.4.4 Effect of Indium heating

Indium has a low resistivity and is extremely malleable and ductile. It stays soft and workable down to cryogenic temperatures. It is used to increase the effective contact surface by filling the space between two contact areas. The experiment of heating the In to decrease the contact resistance was attempted.

In has a low melting point of 430K. The thin indium foils were warmed up to better conform to the surfaces, and then the nut was turned again to maintain the

2.2.8811EE--0066 3 3..2222EE--0077 1.1.3388EE--0077

11..0033EE--0066 1.1.7777EE--0077

11..9988EE--0066

1 1..8833EE--0077 1

1..3388EE--0077 2

2..8800EE--1100 7

7..2277EE--1100

11..3366EE--0077 11..3355EE--1100

11..4455EE--0099

ρρ ρρ

ρρ !!

"

"## "" "" $$ %% %% &&'' !!(())**

!! ""

pressure. Experiments have been performed on five samples, and the measurements repeated four times for each sample.

We expected lower R2 and R3 when using melted indium, but as seen in the following Table, Indium heating does not affect the contact resistance appreciably.

Table 6.4

WHEIGHTED MEAN ERROR

No In heating In heating No In heating In heating R2 1.38193E-07 2.21918E-07 2.46919E-10 2.37E-09 R3 3.21835E-07 4.91225E-07 4.88165E-10 1.61701E-09

Fig. 6.14 Contact resistances measurements.

The last plot reports all the contact resistance measurements. The main contribution to the total resistance is due to the brass nut.

6.4.5 Splice resistance

The total current It (2000A) is transferred from the copper barrel ring to the wire through a resistive layer of 60Sn-Pb solder. To evaluate the splice resistance we used two voltage taps near the ends of the welded wire. By repeating the measurements four times we obtained a mean resistance of

Ω

⋅

± ⁻

−8 1.48 10 9

4310 .

3 (the error is the standard deviation of distribution).

0.0E+00 2.0E-06 4.0E-06 6.0E-06 8.0E-06 1.0E-05 1.2E-05 1.4E-05 1.6E-05

0 1 2 3 4 5

Heated In (5 samples) Non-heated In (7 samples)

Fig. 6.15 Voltage taps location for splice resistance measurements.

6.5 Resistance calculation

6.5.1 Introduction

The purpose of this study was to understand which parameters affect the value of the different kinds of resistance. A simple model was made for the R2 contact resistance and for the resistance of the splices. When possible, calculations were validated with experimental data.

During critical current measurements the short sample is immersed in liquid helium and we perform the measurement at various magnetic fields. When not otherwise stated, material properties at 4.2 K and 0 T field were used in the following.

#$%&

##&' $(&'&(&)( *')+

'&++*'& ) ,%- )(%*.

/'++)*

6.5.2 Splice resistance

The simplified model that was used to evaluate the total resistance of the junction is illustrated in the following Figure.

Fig. 6.16 Simplified model used for total resistance junction evaluation.

Two turns of superconducting wire are soldered around the copper ring, but we consider a homogeneous layer of superconductor (1 mm thick, like the wire diameter) and one of solder (thickness of 0.3 mm).

As the angle of the wire spiral is about 3°, the strand can be considered as being nearly perpendicular to the sample axis. In addition, the turns of the wire are sufficiently close to each other to be considered as a continuous.

K m

Cu_,4.2 =1.48⋅10⁻¹⁰Ω

ρ

K m

Solder_,4.2 =2⋅10⁻⁹Ω

ρ (It is constant for field above 1.8T)

Cu =31⋅10⁻3m φ

m t_Cu =1.05⋅10⁻³

R = Copper resistance per unit length =

m t_Cu

Cu K

Cu = ⋅ Ω

⋅

⋅

−4 2

. 4

, 0.01448 10 φ

π ρ

r = contact resistance across the solder per unit length in x direction

= m

t

Solder Solder K

Solder = ⋅ Ω

⋅

⋅ ₋₉

2 . 4

, 0.0316 10

φ π ρ

Fig. 6.17Superconducting wire mounted on the barrel and soldered around the copper rings.

If V (x) is the copper voltage relative to the superconductor, which is taken to be V=0, and I (x) the current flowing in the copper, one may easily show that:

⋅

=

⋅

=

R x x I dV

dx x r dI x V

) ) ( (

) ) (

(

φ

φ

+

%'&

Combining and solving these two equations (the boundary conditions are I (0)=0; I (L)= It it is found:

L L

x x

t e e

e r e

R I x

V _α^α ₋⁻_α^α

−

⋅ +

⋅

⋅

= ( )^1/2 )

(

where:

α= (R/r) ^½ = 214 m^-1 L = 2 10^-3m

α L = 0.428

For It = 2000A,

e V e

e r e

R I L

V _{t L}^L _L^L = ⋅ ⋅

−

⋅ +

⋅

⋅

= ( )^1/2 ₋⁻ 33.51 10⁻⁶ )

( _α^α _α^α

R junction = V (L) = 0.01675 10^-6 Ω It

Now we can find the total power dissipated by the junction as W (L):

W = It² (Rr)^1/2 e^αL+ e^-αL = 0.067 W e^αL- e^-αL

We can also write:

R junction = (Rr)^1/2 e^αL+ e^-αL = (Rr)^1/2 f (α L) e^αL- e^-αL

where:

(R r) ^1/2 = 0.006764 µΩ (total resistance of the junction except for a geometrical factor)

f (α L) = e^αL+ e^-αL (geometrical factor)

To reduce the junction resistance we can make use of the geometrical factor that is plotted as a function of αL in the following Figure:

Fig. 6.18 Geometrical factor as function of αL.

From this plot we see that by increasing L we can reduce the junction resistance up to αL ≈ 2, but that beyond this point the junction resistance remains at a constant value of (R r) ^1/2 = 0.006764 µΩ.

In our case α L = 0.428, so we can reduce the junction resistance by increasing L up to L = 2/α = 0.0047 m, that means 4 soldered wire turns instead of 2.

6.5.3 Contact resistance

The contact resistance consists of the constriction resistance and of the surface contaminant resistance.

The constriction resistance between two solid bodies in contact originates from the existence of surface asperities. If nominally plane bodies are placed on top of each other the whole covered area is called apparent contact area A No surface of a solid

, , ( ( - -

! !

! !

, , ( (

( (

.. //

$ $ Ω* Ω *

+ + $ $ * *

&

&

0 0

Table 6.5

body is perfectly plane and if the contact members were infinitely hard, the load (the force that presses the contact members together, P) could not bring them to touch each other in more than three points. Since actual materials are deformable, the points become enlarged to small areas and simultaneously new contact points may set in.

The sum of all these areas (spots) is the load bearing area, Ab. Ab may be of a much smaller order of magnitude than the apparent contact area. The current flow is constricted through these small spots, causing an increase of resistance compared with

the case of fully apparent contact surface. The surface Ab is usually partly covered by insulating tarnish film and then only a fraction of Ab has metallic (conducting) or quasi-metallic contact (when it is covered with a thin film that is penetrable by electrons by means of tunnel effect). This increase of resistance is the surface contaminant resistance.

In absence of contamination film on the surface, the true contact area is also the same as the load bearing area. The true contact area is expected to increase with the application of a load. It is apparent that the contact resistance is inversely proportional to the true contact area.

The scheme of the pressure contact is:

Fig. 6.19 Pressure contact scheme

+

"

+

"1 & ' / % '/ & &

The brass nut is tightened by hand with a torque (T) of 10 Nm.

If we make the assumption that the friction coefficient is about 0.15 for a standard nut, we can use the following rule to estimate the force (F) transmitted to the junction by the nut.

T = 0.2FD [page 395, Juvenal]

with:

T= 10 Nm (torque)

D = 7.5 10^-3 m (thread nominal diameter) F = 6667 N

then the differential thermal contraction of the system would produce an additional tightening force.

The thermal contraction of the Ti alloy mandrel is 0.15% from 293 to 4 K, whereas the copper thermal contraction is 0.32 % between the same two temperatures.

Ti Ti Cu

Cu⋅L − ⋅L

=ε ε

δ

mm L

L

L_Cu = _Ti = ₀ =34.925 mm 05 .

=0 δ

N E

A P L

i i

i

=364

⋅

= δ

We represent the asperities by humps shaped as spherical segments. As contact is made these humps are hit first, and they may deform plastically and strain harden, while the underlying metal may deform mainly elastically. The contacts are flat, which means that the waves on their surface have small amplitude. The estimated average height of the elevations is 5 ⋅ 10^-8m. Combining this with the assumption that the maximum slope is one over twenty, and considering the elevations as spherical segments, we find that the average diameter, d, is 4 ⋅10^-6m and the radius of curvature, r, is 4⋅ 10^-5m.

The humps cannot lie closer than one per area of d². Consequently, if the

2

107

707 .

2 m

N A

P p F

a

a = + = ⋅ ,

the average load per hump is P_i = p_a⋅d² =4.331⋅10⁻⁴⋅N

Fig. 6.20 Contact point.

By considering the case of purely elastic deformation for a ball against a semi- infinite plane body of the same material with a Poisson ratio of 0.3, as it is approximately realised in iron, nickel and copper, one obtains as single spot diameter

m E r

a =1.11³ Pⁱ ⋅ =5.807⋅10⁻⁷⋅

The average pressure p in the load bearing area will be:

2 8 0

2 2 3

10 816 . 4 5

. 1

m N a

dx x a a

P p

a i

⋅

⋅

=

⋅

−

⋅ ⋅

⋅

= π

795 .

=17 pa

p

Under the circumstance considered the true load bearing area would be less than:

2

1 ⋅ = ⋅ 5⋅

= ⁻

'

Because all of the a-spots lie at distance from each other which is large compared to the radius (a<<d), we can make the assumption of long constriction for every single a-spot (the constriction lines of flow from different a-spot don’t deflect each other). Their conductances add up. Thus:

=

i

ai

R

4 1 1

ρ

and

Ω

⋅

⋅

=

= 2.877 10⁻⁶ 4A_b

R ρ

is the total constriction resistance between the copper rings and the copper lugs. For more details see Appendix 3.

6.5.4 Summary

Table 6.6 Summary

The Comparison between the analytical values and the experimental data shows that the splice resistance model reproduces quite well the data.

The fact that the contact resistance calculated value is larger than the measured one by a factor of 20 is consistent with the effect of In to dramatically increase (in this case by a factor of ~20) the effective contact area.

To reduce the splice resistance we could increase the number of turns up to 4.

For the contact resistance we should increase the actual contact area. There are constraints in increasing the pressure because it is important to minimize the transfer of torsional strain to the sample. It is also evident that since the brass nut produces a

(( ,,,,µΩµΩ + +

&

& &&

&

& // 22 &&

µΩµΩ µΩµΩ

!! ΩΩ ,, ΩΩ

&

& &&

3

3// 11 '/'/ +

+

&

& &&

4444 &&

&

&

%

%''&&&'&'

((55 (( +

+

%

%''&&&&''

((

large fraction of the resistance, a sample holder design that does not need such a nut would be preferable.

6.6 Summary

A low resistance (of about 1 µΩ) sample holder is needed to perform transport current measurements up to 2 kA at low fields in addition to regular critical current tests.