## GSI Fast-Pulsed Synchrotron Magnet Project: 

## MNW Report GSI 8 version 2

## Martin Wilson 7 Apr 02

**[Analysis of the AC Loss Measurements at University of
Twente]{.underline}**

##### Introduction

Using a calorimetric technique, the ac losses of 5 different cables have
been measured in sinusoidally varying fields at the University of Twente
by Andries den Ouden and colleagues. In addition they have measured the
magnetization of individual wires at several different ramp rates. These
measurements are described and analysed in Report UT-GSI 01-01
\'Calorimetric loss measurement of NbTi Rutherford cables for fast
pulsed magnets\' by A. den Ouden, WAJ Wessel and HJG Krooshoop. Their
analysis, which uses only the calorimetric results for whole cables,
fits the measured losses by adjusting the wire transverse resistivity
ρ~t~ and the interstrand resistances R~c~ and R~a~. In order to get a
reasonable fit, it is necessary to postulate different values of ρ~t~ in
different cables.

In this report, I adopt a different approach:

i)  use values of ρ~t~ and J~c~ calculated from the Twente measurements
    of magnetization for single wires.

ii) use values of R~c~ and R~a~ calculated from the measurements of Arup
    Ghosh and summarised in my Report 7-1.

iii) calculate losses using the formulae presented our MT-17 paper
     \'Design studies on superconducting cosθ magnets for a fast pulsed
     synchrotron\'.

iv) compare calculated loss per cycle with that measured.

v)  use the ratio between measured and calculated to predict losses and
    temperature rise in the synchrotron magnets.

This revised version 2 incorporates three changes

a.  the transverse resistivity of the wire ρ~t~ has been factored by the
    resistivity ratios, because the rate dependent magnetization was
    measured on non-heat treated wires.

b.  reversible magnetization has been included in the derivation of J~c~
    from the magnetization curves.

c.  hysteresis loss in the dipole is computed at 5 radial points for
    each azimuthal point, as described in Report (9).

To save you reading it all again, changes are shown in red

**2. Transverse resistivity from magnetization of single wires**

Appendices 1- 2 and 9 reproduce the Twente measurements on single wires
of magnetization at different ramp rates (there is one other data sheet
for wire sample 1 which I have not used). This magnetization comprises
two components:

a\) magnetization of the individual filaments given by
$M_{f} = \lambda_{f}\frac{2}{3\pi}J_{c}d$

b\) eddy currents coupling the filaments
$M_{e} = \frac{2}{\mu_{o}}\overset{˙}{B}\tau$ where
$t = \frac{\mu_{o}}{2\rho_{t}}\left\{ \frac{p_{w}}{2\pi} \right\}^{2}$

Thus, by plotting the measured *M = M~f~ + M~e\ ~* against *B\`* at a
defined field, we can get *J~c~* from the intercept and *ρ~ t~* from the
gradient. Appendix 1 makes this plot for sample 3, which has a 13mm
twist pitch and was exposed to the widest range of *B\`*. Linear
interpolation was used to obtain the magnetization at a series of exact
field points 0.1T, 0.2T etc, in all four quadrants of the hysteresis
loop. At each field, the magnetization is calculated as the mean between
the half-heights of the hysteresis loop at positive and negative fields.
Fig 1 shows this magnetization versus *dB/dt* at higher fields; to get
the plots on the same scale, I have subtracted the intercepts, ie *M* at
zero *B\`.*

![](media/image1.wmf){width="5.1618055555555555in"
height="3.079861111111111in"}

Fig 1: Dependence of magnetization on ramp rate for wire sample 3

Fig 2 shows the gradient of M versus B\` as a function of field. It may
be seen that this gradient is roughly independent of field - as it
should be. Magnetoresistance might be expected to cause a reduction of
gradient with increasing field, but Fig 2 shows that, if anything, there
is a slight increase.

![](media/image2.wmf){width="5.135416666666667in" height="2.9125in"}

Fig 2: Gradient versus field

From the mean gradient, Appendix 1 calculates the transverse resistivity
as *ρ~t~* = 9.2 x 10 ^-10^ Ωm. However, as pointed out by Juris
Kaugerts, the wire samples were not heat treated whereas the cables
were. According to Arup Ghosh (email 12 Mar 02) 'the virgin wire prior
to cabling had a low RRR of 48, after the cure cycle the RRR increases
to 200-210'. Until we get direct measurements on heat treated wires, the
best we can do is to scale *ρ~t~* by the resistance ratios, to get a an
estimated *ρ~t~* = 2.15 x 10 ^--10^. With such a low value, one would
expect to see some magnetoresistance at higher fields, but I without
direct measurements, I propose to stay with this (low field) estimate.

Wires which are more tightly twisted show correspondingly less
dependence of magnetization on dB/dt. Appendix 9 repeats the above
analysis for Sample 2, which has a 6mm twist pitch. Here the eddy
current component of magnetization is a tiny fraction of the total and
drift in the magnetometer reading makes a strong impact on the
calculated gradient. If we take a single value of gradient near the
maximum field (where the effects of drift are smallest because it is
nearest to the sweep reversal point) we find *ρ~t~* = 5.5 x 10 ^-10^ Ωm,
in fair agreement with sample 3. However, the gradient calculated at
intermediate fields is up to 4 x bigger, implying a very low value for
*ρ~t~*. In my judgement, although the quality of these measurements is
as good as any I have seen, the results for tightly twisted wires are
not a reliable predictor of *ρ~t~* and I have chosen to ignore them,
using only Sample 3. Sample 1, with a 4mm twist pitch, has even smaller
eddy current magnetization and I have not even attempted to extract
*ρ~t~* from this data.

**3. Current density from magnetization of single wires**

To get a good fit of J~c~ versus field, it is desirable to measure
magnetization over the widest possible range of field. Sample 4 was the
only one to be measured up to 4T and for this reason I have used it to
estimate J~c~. As pointed out by Arup Ghosh however, magnetization of
the filaments contains a (usually neglected) reversible component in
addition to the critical state term given in section 2(a).

![](media/image3.wmf){width="4.631944444444445in"
height="2.682638888888889in"}

Fig 3: Flux screening and trapping regions of the magnetization loop.

The magnetization loop is measured in four quadrants as sketched in Fig
3, but for analysis I plot the four regions in the all positive
quadrant. It is useful to define flux screening and flux trapping
regions as shown in Fig 3. Magnetization currents flowing in the bulk
material (critical state) depend on the direction of change of field and
the resulting magnetization is therefore antisymmetrical about the axes
as shown, with the two trapping regions being similar to each other and
the two screening regions likewise. Reversible magnetization, coming
from surface currents is different however and depends only on the
external field direction; it is negative when the external field is
positive and vice versa. For data analysis, one must first use these
symmetry properties to determine the magnetometer offset, shown by the
purple line in Fig 3

In processing the results I have used the reversible magnetization
measured by Arup for an early SSC wire (listed in the spreadsheet
IGC521D RevA.xls) and scaled it to the matrix ratio of 2.25:1 of the
wire measured here.

As laid out in Appendix 2, the calculation has 4 stages.

a)  compensate the slope of the hysteresis loop, caused by the
    magnetometer offset: I do this by applying a correction which is
    linear with field and which makes the difference between the
    screening and trapping curves at ± 3.5T equal to twice the
    reversible magnetization scaled from Arup's spreadsheet 'IGC521D
    RevA.xls'.

b)  subtract the eddy current component to obtain the persistent current
    magnetization *M~h~* at zero *B\`*: I use the mean eddy current term
    (gradient) measured for sample 3, scaled for 6mm instead of 13mm
    twist pitch.

c)  calculate four curves of steady state *M~h~* versus *B*,
    corresponding to positive or negative fields, and positive or
    negative magnetization; these curves fall into two pairs: screening
    and trapping

d)  fit the screening and trapping curves with a modified Kim Anderson
    equation, ie.

> $$
> []{.mark}$M_{f} = \lambda_{f}\frac{2}{3\pi}J_{c}d = \lambda_{f}\frac{2}{3\pi}d\left\{ \frac{J_{o}B_{o}}{B + B_{o}} \right\} + A_{o} + A_{1}$
>
> $$(Note: because of the enormous volume of data, I did the fitting by
> eye)

e)  calculate J~c~ from the mean of the two fitted curves (which is
    equivalent to subtracting the reversible magnetization component)

f)  calculate the reversible magnetization as half the difference
    between the two fitted curves

Appendix 2-2 sets out stages a) - c) on the sheet \'corr M vs B\' and
stages d) -- f) on the sheet \'fitted M and J~c~\'. Fig 4 summarizes the
results and Fig 5 shows a local area to emphasize the difference between
the shielding and trapping branches. Note that magnetization is defined
per unit volume of wire, ie including the copper.

![](media/image4.wmf){width="5.688194444444444in"
height="4.122222222222222in"}

Fig 4: Magnetization curves, corrected for magnetometer calibration and
coupling (plotted in the all positive quadrant).

![](media/image5.wmf){width="5.498611111111111in"
height="4.038194444444445in"}

Fig 5: magnified region of Fig 3

The modified Kim Anderson parameters used to fit J~c~ in

[]{.mark}$J_{c} = \left\{ \frac{J_{o}B_{o}}{B + B_{o}} \right\} + A_{o} + A_{1}$

are listed in Table 1.

Table 1: Parameters used for fitted curves in Figs 4 & 5.

  ------------ ---------------- ---------------- ------------------------
  branch       trapping         screening        mean (ie Jc without
                                                 reversible M)

  J~0~ =       3.00E+10         4.70E+10         3.85E+10

  B~0~ =       0.15             0.11             1.30E-01

  A~0~ =       4.20E+09         4.50E+09         4.35E+09

  A~1~ =       -7.50E+08        -4.20E+08        -5.85E+08
  ------------ ---------------- ---------------- ------------------------

Values listed in the right hand column are used to calculate hysteresis
loss in the magnet.

Finally we calculate the reversible magnetization as half the difference
between the screening and trapping curves. It is plotted in Fig 6,
together with the data supplied by Arup for IGC521D. The data coincide
at 3.5T, merely because this was the data point used to set up the
magnetometer offset. Elsewhere however they disagree strongly, with the
data supplied by Arup looking much more credible. I don't know the
reason for the discrepancy, perhaps my assumption of a magnetometer
offset which varies linearly with field was incorrect (Andries please
comment). In practical; terms however, the effect on calculation of
hysteresis loss will be negligible.

![](media/image6.wmf){width="5.448611111111111in"
height="3.973611111111111in"}

Fig 6: Reversible magnetization inferred from this data compared with

**4. Coupling losses in the cables**

Appendix 3 sets out the calculation of total loss for each of the five
cables tested and Appendix 4 is a Mathcad sheet providing more detail on
the calculation plus an independent check on the first cable in App 3.
Table 2 lists the common parameters of all cables tested.

Table 2: Common parameters of all cables tested

+-----------------------------------+-------+---------------+-----------+
| > cable half width                | c     | > 4.87E-03    | m         |
+-----------------------------------+-------+---------------+-----------+
| > cable half thickness            | b     | > 5.83E-04    | m         |
+-----------------------------------+-------+---------------+-----------+
| > cable twist pitch               | p     | > 7.40E-02    | m         |
+-----------------------------------+-------+---------------+-----------+
| > cable slope                     | cosθ  | > 0.967       |           |
+-----------------------------------+-------+---------------+-----------+
| > number of strands               | N     | > 30          |           |
+-----------------------------------+-------+---------------+-----------+
| > permeability free space         | μ~ο~  | > 1.26E-06    | henry/m   |
+-----------------------------------+-------+---------------+-----------+
| > wire filling factor             | λ~w~  | > 0.872       |           |
+-----------------------------------+-------+---------------+-----------+
| > filament filling factor         | λ~f~  | > 0.308       |           |
+-----------------------------------+-------+---------------+-----------+
| > wire transverse resistivity     | ρ~t~  | > 2.15E-10    | Ω.m       |
+-----------------------------------+-------+---------------+-----------+
| > filament diameter               | d~f~  | > 6.00E-06    | m         |
+-----------------------------------+-------+---------------+-----------+
| > Kim Anderson Jc fit             | J~o~  | 3.85E+10      | A/m\^2    |
+-----------------------------------+-------+---------------+-----------+
| > Kim Anderson Jc fit             | B~o~  | 1.30E-01      | Tesla     |
+-----------------------------------+-------+---------------+-----------+
| > Kim Anderson Jc fit             | A~0~  | 4.35E+09      | A/m\^2    |
+-----------------------------------+-------+---------------+-----------+
| > Kim Anderson Jc fit             | A~1~  | -5.85E+08     | A/m\^2    |
+-----------------------------------+-------+---------------+-----------+
| > field amplitude                 | B~a~  | 0.3           | Tesla     |
+-----------------------------------+-------+---------------+-----------+
| > frequency                       | f     | 0.25          | Hz        |
+-----------------------------------+-------+---------------+-----------+
| > angular frequency               | ω     | 1.571         | rad/sec   |
+-----------------------------------+-------+---------------+-----------+

Table 3 lists those parameters which change with cable type and the
calculated loss per cycle (in Joules.m^-3^) at the maximum measured
frequency of 0.25 Hz, for each of the five loss components. The only
losses which can be observed experimentally are the last three
highlighted rows, ie hysteresis and total rate dependent (eddy current)
loss with field transverse and parallel.

Also listed in Table 3 are the factors *ωτ*  which are important because
they determine the importance of self shielding by the eddy currents. At
high frequencies, the losses per cycle actually decrease because the
induced eddy current shield the interior or the wire or cable form the
external field. For example, the full expression for eddy current loss
in a single wire is:

![](media/image7.wmf)

The last part of this expression is plotted in Fig 7, where it may be
seen that the initial linear rise is followed by as peak and then a
decrease. The linear approximation, used in Appendix 3, is shown by the
dashed line (it may be obtained from the constant B\` formula by
substituting a root mean square B\`). Thus we see that a condition for
negligible self shielding and consequent validity of the linear
approximation is *ωτ* \< 0.3. The values listed in Table 3 show this
condition to be fulfilled comfortably in all cases.

Table 3: Loss per cycle (J.m^-3^) for each of the tested cables

  ---------------- ------------------ ---------- ---------- ---------- ---------- ----------
  cable                               001SSNHT   001SSHT    002TiNHT   002TiHT    002NCNHT

  crossover        R~c~               5.30E-02   5.30E-02   7.80E-04   7.80E-04   5.91E-05
  resistance                                                                      

  adjacent         R~a~               4.80E-05   2.74E-04   9.40E-05   7.60E-05   3.00E-05
  resistance                                                                      

  wire twist pitch p~w~               6.00E-03   6.00E-03   4.00E-03   4.00E-03   4.00E-03

  transverse       Q~tc~              37.5       37.5       2549.2     2549.2     33616.5
  crossover                                                                       
  loss/cycle                                                                      

  transverse       ω∗τ~tc~            2.83E-05   2.83E-05   1.92E-03   1.92E-03   2.53E-02
  crossover time                                                                  
  const                                                                           

  transverse       Q~ta~              884.6      155.0      451.7      558.7      1415.3
  adjacent                                                                        
  loss/cycle                                                                      

  transverse       ω∗τ~ta~            4.41E-04   7.72E-05   2.25E-04   2.78E-04   7.05E-04
  adjacent time                                                                   
  const                                                                           

  parallel         Q~pa~              8.9        1.6        4.6        5.6        14.3
  adjacent                                                                        
  loss/cycle                                                                      

  parallel         ω∗τ~pa~            8.47E-05   1.48E-05   4.32E-05   5.35E-05   1.35E-04
  adjacent time                                                                   
  const                                                                           

  filament         Q~f~               1646.8     1646.8     731.9      731.9      731.9
  coupling loss                                                                   

  filament         ω∗τ~f~             4.20E-03   4.20E-03   1.86E-03   1.86E-03   1.86E-03
  coupling time                                                                   
  const                                                                           

  filament         Q~h~               9930.1     9930.1     9930.1     9930.1     9930.1
  hysteresis                                                                      

  total rate       Q~tc~+Q~ta~+Q~f~   2569       1839       3733       3840       35764
  dependent                                                                       
  transverse                                                                      

  total rate       Q~pa~+Q~f~         1656       1648       736        738        746
  dependent                                                                       
  parallel                                                                        
  ---------------- ------------------ ---------- ---------- ---------- ---------- ----------

![](media/image8.wmf){width="4.21875in" height="2.8333333333333335in"}

Fig 7: General dependence of loss per cycle on frequency

Finally, Table 4 lists the experimental measurements of hysteresis loss
and eddy current loss at 0.25 Hz (as calculated from the mean gradient
\* 0.25Hz), and compares them with the calculation.

Table 4: Calculated loss compared with measurement at 0.25 Hz

  ----------------------------- ------------- --------- ---------- --------- ----------
  Twente results file           GSILOS3.xls                                  

  Twente sheet title            001SSNHT      001SSHT   002TiNHT   002TiHT   002NCNHT

  Hysteresis transverse J.m^-3^ 11960         11398     12256      11526     12092

  Hysteresis transverse:        1.20          1.15      1.23       1.16      1.22
  exp/theo                                                                   

  Hysteresis parallel J.m^-3^   11707         11491     11243      11817     11606

  Hysteresis parallel exp/theo  1.18          1.16      1.13       1.19      1.17

  Rate dep\'t loss transverse   3990          2082      1370       6420      34557
  J.m^-3^                                                                    

  Rate dep\'t loss transverse   1.55          1.13      0.37       1.67      0.97
  exp/theo                                                                   

  Rate dependent loss parallel  1005          1105      415        1905      280
  J.m^-3^                                                                    

  Rate dep\'t loss parallel     0.61          0.67      0.56       2.58      0.38
  exp/theo                                                                   
  ----------------------------- ------------- --------- ---------- --------- ----------

It may be seen that the hysteresis calculation is consistently low by
about 20%, which is surprising, given that the Jc data is taken from a
magnetization loop - so that errors in filament diameter, shape etc
would automatically be taken into account. Perhaps the reason is the
rather small field amplitude of the cable loss measurement.

The lower value of ρ~t~ has brought the transverse coupling loss for the
uncored and stainless cables into fair agreement with the theory, but
the measured parallel loss is now much lower than calculated. Indeed the
parallel loss is even lower than the calculated filament coupling loss
in the individual wires -- which is difficult to believe, regardless of
what is happening in the cable. The only answer has to be a direct
measurement of filament coupling in strands extracted from the cable.

The titanium core measurements are all over the place, but I don\'t
think it\'s worth spending time on them, given that the core may be
punched through.

**5. Loss in the magnet**

Taking our preferred cable of stainless core non heat--treated with a
4mm twist pitch, I now use the above \'cooking\' factors of
experimental/theoretical values to predict loss and temperature rise in
a magnet subject to the latest cycle, as defined in Table 5.

Table 5: Magnet cycle for loss calculation

  -------------------------------------- ---------- ------------ ----------
  extraction aperture field              B~ap~      4.0          Tesla

  ramp ratio B~inj~ / B~ext~             f          0.1          

  ramp time                              T~r~       3.6          sec

  ramp rate                              B\`        1.0          T/s

  injection time                         T~i~       0            sec

  extraction time                        T~e~       5            sec
  -------------------------------------- ---------- ------------ ----------

Appendix 5 details the calculation and the results are summarised in
Table 6

**Table 6: Summary of magnet losses**

  ---------------------------------------------------- ------------------
  ramping total loss / metre                           5.3 Watt

  ramping total loss / magnet                          13.8 Watt

  time averaged loss / magnet                          8.2 Watt

  transverse crossover loss / total                    0.38%

  transverse adjacent loss / total                     11.5%

  parallel loss / total                                0.13%

  filament coupling loss / total                       30.4%

  hysteresis loss / total                              57.5%
  ---------------------------------------------------- ------------------

It may be seen that, compared to the losses of cable A6 in our MT-17
paper, the coupling components have increased in importance, although
hysteresis remains the most important single component.

**6. Temperature rise**

Using the peak power of 4.46E+03 W.m^-3^ listed in Appendix 5 at angle
73°, we now calculate the highest temperature rise in the cable.
Appendix 6 calculates the temperature rise across the cable and, for the
case of no cooling slots, across the Kapton insulation. Appendix 7 looks
at heat transfer to the helium coolant. Appendix 8 looks at the case
where there is a gas gap between the cable and Kapton insulation, here
we take the worst case of conduction across the helium, because
convection will not occur across gaps near the upper pole where the
heated surface is uppermost . Results are summarised in Table 6

Table 6: Calculated temperature rises

  -------------------------------------------------------- --------------
  across cable                                             0.017K

  heat transfer with a 26% cooling slot                    0.100K

  total with cooling slot                                  0.117K

  across gas film between cable and Kapton                 1.020K

  across Kapton                                            0.145K

  heat transfer without cooling slot                       0.026K

  total without cooling slot, assuming gas film between    1.191K
  cable and Kapton                                         

  total without cooling slot, assuming good thermal        0.171K
  contact between cable and Kapton                         
  -------------------------------------------------------- --------------

**7. Concluding remarks**

i)  Calculated values of hysteresis loss are about 20% lower than
    measured. The discrepancy, though not serious, is surprisingly in
    view of the fact that all parameters were derived from magnetization
    (and hence loss) measurements on single wires.

ii) Calculated values of cable coupling loss in transverse field are in
    agreement with experiment for the uncored cable and a factor \~ 1.5
    less than experiment for the cored cable.

iii) Calculated values of cable coupling loss in parallel field are a
     factor 1.5 -- 3 greater than experiment for all cables (except the
     Ti core which has punch through).

iv) The fact that the calculated filament coupling loss is greater than
    the measured total loss in parallel field seems to indicate that the
    estimated lower ρ~t~ is in fact too low and that we cannot explain
    the discrepancies solely in terms of ρ~t~.

v)  Although the discrepancies are intellectually displeasing,
    calculations of loss in the magnet using an experimental/theoretical
    \'cooking\' factor show that they do not threaten the project, given
    the new lower ramp rate.

vi) Temperature rises with cooling slots in the Kapton are negligible

vii) Temperature rises if the Kapton is thermally bonded to the cable
     are small

viii) Temperature rises with a gap between Kapton and cable are too
      large

**8. Recommendations**

i)  In all future loss measurements, without warming up, a VI
    measurement should be made on the same cable sample. In this way we
    could hope to get a better estimate of *R~a~* for the calculation.

ii) Magnetization should be measured at different ramp rates on isolated
    single strands extracted from cable 001SSNHT after heat treatment to
    get an accurate ρ~t~ for the wires.

iii) The direct measurement of *R~c~* should be repeated for 001SSNHT,
     to give us some idea of the scatter.

iv) Transport current *J~c~* should be measured for sample 4 (wire code
    91-0-80122A-05) to compare with the *J~c~* derived here from
    magnetization.

**List of Appendices**

Appendix 1: Transverse resistivity of sample 3

Appendix 2: Current density fit for wire sample 4

Appendix 3: Calculated ac loss in cables compared with Twente
measurements

Appendix 4: Check on loss formulae and calculation for first cable in
App 3

Appendix 5: Calculation of losses in dipole with cycle 5 and
001SSNHTAppendix 5

Appendix 6: Temperature Rise across Rutherford Cable

Appendix 7: Temperature Drops across gas films

Appendix 8: Force flow cooling with 5 bar Helium

Appendix 9: Transverse resistivity for Sample 2