f Fermilab Fermi National Accelerator Laboratory

P.O. Box 500 • Batavia, Illinois • 60510

TD-01-015

March 19, 2001

## Peak Field and Lorentz Force Calculations for the

## Return End of the First High-Field Dipole Model-HFDA01 

S. Yadav

# *Fermi National Accelerator Laboratory*

# *MS 316, PO Box 500*, *Batavia, IL, USA*

***Abstract**---*In this note we present a brief 3-D analysis of the
peak fields and Lorentz forces for the return end of the first
high-field dipole model (HFDA01). The understanding gained from this
study would be useful in optimizing the magnet end design for the next
dipole models. Also we envision a combined use of ROXIE and BEND to
perform both the magnetic and mechanical analysis of the magnet ends at
the same time.

1.  **Introduction**

The 2-D cross-section design for the first high-field dipole model
(HFDA01) is reported in \[[^1]\]. This cross-section was optimized using
program ROXIE \[[^2]\]. The chosen cross-section produces a central
field of 12.02 T for an excitation current of 19.6 kA. The iron yoke in
\[1\] was simulated by an area starting at 60mm radius from the center
of coil and with a constant magnetic permeability of 1000 (please see
Appendix in the end). Fig. 1 and 2 show the 2-D cross-section of the
final design along with the numbering system for the blocks and the
conductors as adopted in program ROXIE. Fig. 3 presents the magnetic
field distribution in the conductors for an excitation current of 19.6
kA. The 2‑D peak fields for each of the conductor blocks are summarized
below in Table 1.

Table 1: 2-D peak fields for each of the conductor blocks.

+-----------------------+-----------------------+-----------------------+
| Block No.             | Conductor No.         | Peak Field (in T)     |
|                       |                       |                       |
|                       |                       | for I = 19.6 kA       |
+-----------------------+-----------------------+-----------------------+
| 1                     | 5                     | 6.095                 |
+-----------------------+-----------------------+-----------------------+
| 2                     | 11                    | 8.470                 |
+-----------------------+-----------------------+-----------------------+
| 3                     | 13                    | 9.150                 |
+-----------------------+-----------------------+-----------------------+
| 4                     | 14                    | 11.811                |
+-----------------------+-----------------------+-----------------------+
| 5                     | 20                    | 12.083                |
+-----------------------+-----------------------+-----------------------+
| 6                     | 24                    | 12.568                |
+-----------------------+-----------------------+-----------------------+

![](media/image1.png){width="4.46875in" height="4.28125in"}

Figure 1: 2-D XY cross-section (with bare cable) of the dipole model
showing the block numbers.

![](media/image2.png){width="4.4375in" height="4.09375in"}

Figure 2: 2-D XY cross-section (with bare cable) of the dipole model
showing the conductor numbers.

![](media/image3.png){width="3.917361111111111in"
height="3.4854166666666666in"}![](media/image4.png){width="1.9909722222222221in"
height="5.013888888888889in"}

Figure 3: Magnetic field contours for the 2-D cross-section, I = 19.6
kA.

The coil ends were designed using program BEND \[[^3], [^4]\] to
minimize internal stresses in the cable. Some of the design and
fabrication related issues for the coil ends are reported in \[[^5]\].
Fig. 4 shows the YZ cross section of the return end along with the block
numbers that correspond to the bock numbers for the 2-D cross-section
shown in Fig. 1. The same YZ cross-section is shown in Fig. 5 with the
z-axis dimensioned in mm.

In general, some of the design objectives for the coil end optimization
are (from \[[^6]\]):

1.  Minimum integrated multipole content in the coil ends.

2.  Low enhancement of the field in the coil ends.

3.  Minimum internal stresses in the cables.

4.  Short length of the ends.

5.  Homogeneous Lorentz force distribution in the innermost turn of the
    inner layer and in particular minimization of the variation in
    pressure on the narrow face of the conductor in order to avoid slip
    between the turns.

![](media/image5.png){width="4.520833333333333in" height="1.78125in"}

Figure 4: Return end YZ cross section (with insulated cable) of the
dipole model with the block numbers shown.

![](media/image6.png){width="4.979166666666667in"
height="1.6354166666666667in"}

Figure 5: Return end YZ cross-section (with insulated cable) of the
dipole model. The z-axis is dimensioned in mm.

In this note, we present the results of a 3-D magnetic analysis for the
coil return end. Such an understanding is useful for optimizing the coil
end design for the future dipole models. In particular, we have focused
our attention on the peak fields and Lorentz force distribution in the
ends. For the first dipole model, the coil ends were not optimized for
minimum multipole content in the ends. The longitudinal distribution of
the dipole field and multipole components in the ends is reported in
\[[^7]\], which also presents the results of a 3D magnetic analysis
performed to optimize the length of the iron yoke.

2.  **Peak Field Calculations**

The peak field calculations were performed using the computer program
ROXIE. These calculations are based on the Biot-Savart law. For the 2-D
cross-section, the iron yoke is regarded as a magnetic mirror of
circular shape of inner diameter 120 mm and with a magnetic permeability
of 1000. For the computer model, the iron yoke starts at z=0 and extends
for 1 m inwards away from the end. However, note that the imaging method
used in ROXIE does not allow to take into account the contribution of
the iron yoke for the peak fields or Lorentz force calculations.[^8]
This would imply that the field values computed in the proximity of the
edge of the iron yoke (z = 0 mm) are not accurate. However, since the
actual peak fields normally occur close to the coil termination and it
has been shown previously \[[^9]\] that the distance from the edge of
the iron yoke to the region with peak field is sufficiently large to
ensure that the contribution of the iron yoke can be neglected,
therefore the present analysis is sufficient. The transport current in
the cable is modeled by defining a current filament at the position of
each strand (i.e., 2 x 14 filaments). The grading of current density in
the conductor due to the keystoning is accounted for in the program
ROXIE. The coil ends were modeled using the standard features available
in program ROXIE. For this initial investigation, the dekeystoning of
the cable at the ends was not accounted for.

Table 2 lists the maximum field in the end for each conductor block
along with the *z* location of the point at which the peak field occurs.
The corresponding linear field load line coefficients for the maximum
field in each block of the return end as a function of the current are
reported in Table 3. For comparative purposes, similar values for the
straight section (obtained from Table 1) are also reported in Table 3.
It is observed that the peak field in all the return end conductor
blocks is higher than in the magnet body. Future optimization efforts
would try to reduce the peak fields in the magnet ends.

Table 2: Peak fields in the end for I=19.6 kA.

+-----------------+-----------------+-----------------+-----------------+
| Block No.       | Conductor No.   | Peak Field (in  | z (mm) location |
|                 |                 | T)              | for peak field  |
|                 |                 |                 |                 |
|                 |                 | for I = 19.6 kA |                 |
+-----------------+-----------------+-----------------+-----------------+
| 1               | 5               | 6.346           | 27.2            |
+-----------------+-----------------+-----------------+-----------------+
| 2               | 11              | 8.818           | 74.37           |
+-----------------+-----------------+-----------------+-----------------+
| 3               | 13              | 9.445           | 38.78           |
+-----------------+-----------------+-----------------+-----------------+
| 4               | 14              | 12.053          | 28.02           |
+-----------------+-----------------+-----------------+-----------------+
| 5               | 20              | 12.339          | 18.87           |
+-----------------+-----------------+-----------------+-----------------+
| 6               | 24              | 12.831          | 24.93           |
+-----------------+-----------------+-----------------+-----------------+

Table 3: Comparison of field load line coefficients in the straight
section and the return end.

  ----------------- ----------------- ----------------- -----------------
  Block No.         Conductor No.     B~max~/I (T/kA)   B~max~/I (T/kA)
                                      for straight      for return end
                                      section           

  1                 5                 0.311             0.324

  2                 11                0.432             0.450

  3                 13                0.467             0.482

  4                 14                0.603             0.615

  5                 20                0.616             0.630

  6                 24                0.641             0.655
  ----------------- ----------------- ----------------- -----------------

Figure 6 shows the variation of the magnetic field along the z
coordinate for each of the return end conductor blocks (for conductors
with the peak field), while Figure 7 shows the enhancement in the
magnetic field at the ends as compared to the magnetic field in the
straight section for each conductor block. Figures 8 and 9 show the
contour plots of magnetic field distribution for the outer and inner
layers respectively.

![](media/image7.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 6: Magnetic peak fields in the end for each conductor block
(OR-outer layer return end & IR-inner layer return end) for excitation
current I = 19.6 kA.

![](media/image8.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 7: Normalized peak field (peak fields at the end normalized with
the corresponding 2-D peak fields provided in Table 1) distribution for
the return end.

![](media/image9.png){width="3.88125in" height="3.9229166666666666in"}
![](media/image10.png){width="1.8784722222222223in"
height="4.090277777777778in"}

![](media/image11.png){width="3.8986111111111112in"
height="2.826388888888889in"}
![](media/image12.png){width="2.058333333333333in"
height="4.168055555555555in"}

Figure 8: Magnetic field distribution for the return end- outer layer.

![](media/image13.png){width="3.81875in" height="3.6069444444444443in"}
![](media/image10.png){width="1.8784722222222223in"
height="4.090277777777778in"}

![](media/image14.png){width="3.88125in" height="2.8229166666666665in"}
![](media/image12.png){width="2.058333333333333in"
height="4.168055555555555in"}

Figure 9: Magnetic field distribution for the return end-inner layer.

3.  **Lorentz force distribution in the ends**

Figure 10 shows a contour plot of the Lorentz forces per unit length for
the 2-D cross-section for an excitation current of 19.6 kA. As expected
these contours are similar to the magnetic field contours shown in Fig.
3. The resulting *x* and *y* direction Lorentz forces for each conductor
block are summarized in Table 4.

![](media/image15.png){width="4.289583333333334in"
height="3.8868055555555556in"}![](media/image16.png){width="1.5895833333333333in"
height="4.951388888888889in"}

Figure 10: Lorentz force contours for the 2-D cross-section, I = 19.6
kA. Note that as expected these are similar to the magnetic field
contours shown in Fig. 3.

Table 4: Lorentz force per unit length for the 2-D cross-section.

  ----------------------- ----------------------- -----------------------
  Block No.               F~x~ (kN/m)             F~y~ (kN/m)

  1                       242.6                   -202.4

  2                       368.9                   -364.9

  3                       243.4                   -163.4

  4                       484.7                   -48.4

  5                       875.5                   -170.3

  6                       596.5                   -112.9

  Sum of all              2811.6                  -1062.4
  ----------------------- ----------------------- -----------------------

Table 5 summarizes the Lorentz force per quadrant in the *x*, *y*, and
*z* directions for each conductor block of the return end. The total
resulting longitudinal force per quadrant of the return end is 75.1 kN.
Note that this information is useful for the design of the end plates.

Table 5: Lorentz force per quadrant for each conductor block of the
return end.

  ----------------- ----------------- ----------------- -----------------
  Block No.         F~x~ (kN)         F~y~ (kN)         F~z~ (kN)

  1                 42.6              -30.0             10.8

  2                 52.4              -43.6             21.2

  3                 22.0              -12.9             9.0

  4                 73.1              -5.5              6.0

  5                 106.0             -18.1             19.9

  6                 50.0              -8.2              8.2

  Sum of all        346.1             -118.3            75.1
  ----------------- ----------------- ----------------- -----------------

Another important parameter in the design of the ends is the variation
of Lorentz forces in the longitudinal direction of each block. The force
in the z-direction for each individual conductor for the return end (RE)
is shown in Fig. 11 for a current, I = 19.6 kA. Note that the Lorentz
forces scale quadratically with current and thus forces at any other
excitation current could be computed. Also note that the contribution of
the iron yoke to the magnetic field in the end region is not taken into
account.

![](media/image17.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 11: Longitudinal force in all conductors, I = 19.6 kA

Fig. 12 shows the variation in the azimuthal Lorentz force per unit
length for the return end. As expected the azimuthal force increases as
one moves away from the end towards the magnet straight section and
approaches the value for the 2-D cross-section near *z*=0.

![](media/image18.png){width="5.996527777777778in"
height="5.350694444444445in"}

Figure 12: Azimuthal Lorentz force per unit length (N/mm) at I =19.6 kA
for the return end. Note that for the straight section (z\<0), the
azimuthal Lorentz force approaches the value for the 2-D cross section
(-2336 kN/m as given in Table 2 of Ref. \[[^10]\]).

Figures 13 and 14 show the pressure distribution due to the Lorentz
forces for the outer and inner layers respectively. As mentioned
previously, we need to strive to achieve homogeneous Lorentz force
distribution in the innermost turn of the inner layer and in particular
to minimize the variation in pressure on the narrow face of the
conductor in order to avoid slip between the turns.

![](media/image19.png){width="4.010416666666667in"
height="4.116666666666666in"}
![](media/image20.png){width="1.7409722222222221in"
height="4.022222222222222in"}

![](media/image21.png){width="3.7215277777777778in"
height="2.859722222222222in"}
![](media/image22.png){width="1.7083333333333333in"
height="3.8819444444444446in"}

Figure 13: Pressure due to Lorentz forces on surfaces for the return
end-outer layer.

![](media/image23.png){width="3.7506944444444446in"
height="3.4381944444444446in"}
![](media/image20.png){width="1.7409722222222221in"
height="4.022222222222222in"}

![](media/image24.png){width="3.2111111111111112in"
height="2.348611111111111in"}
![](media/image22.png){width="1.7083333333333333in"
height="3.8881944444444443in"}

Figure 14: Pressure due to Lorentz forces on surfaces for the return
end-inner layer.

4.  **Summary**

The peak field and Lorentz force distribution in the return end of the
first dipole model, HFDA01 has been investigated. This study would form
the basis for optimization of the ends for the next dipole models.

**APPENDIX**

For all the calculations reported here, a constant magnetic permeability
of 1000 was used for the iron yoke for a current of 19.6 kA. It was
later realized that iron saturates at a much lower current and a
magnetic permeability value between 4 and 5 is a more real
representative value for the iron yoke at higher currents. This would be
incorporated in our future studies. However, note that since the effect
of the iron yoke for the end fields and forces is very small (since the
ends have a non-magnetic steel yoke surrounding them), the effect of
using a higher magnetic permeability for the iron is very small for the
end fields and force computations.

For the 2-D fields, using an iron permeability of 1000 results in higher
peak fields (Table 1) by a factor of 1.1 to 1.2 than using an iron
permeability of 5, with the maximum increase for the block number 1.
Similarly, the X- direction Lorentz forces (Table 2) for the 2-D cross
section are higher by a factor of 1.12 to 2.22, with the maximum
increase again for the block number 1. Note that there is not a
significant increase in the Y-direction Lorentz forces (\<1.02) due to
the use of a higher iron permeability value.

**References**

[^1]: \[?\] V. Kashikhin, "First high field magnet cross-section
    design," Fermilab TD-99-027, (1999).

[^2]: \[?\] S. Russenchuck, "A Computer Program for the Design of
    Superconducting Accelerator Magnets," CERN AT/95-39, LHC Note 354,
    Geneva, Switzerland, 26 Sept. 1995.

[^3]: \[?\] J.M. Cook, "Strain energy minimization in SSC magnet
    winding," IEEE Transaction on Magnetics, Vol. 27, pp. 1976-1980,
    (1991).

[^4]: \[?\] S. Yadav, "Design Parameters for the Coil Ends for the First
    High Field Dipole Model-HFDA01," Fermilab Technical Note TD-01-056,
    (2001).

[^5]: \[?\] S. Yadav, D.R. Chichili & I. Terechkine, "Coil end parts
    design and fabrication issues for the high field dipole at
    Fermilab," IEEE Transactions on Applied Superconductivity, Vol. 11
    (1), pp. 2284-2287, (2001).

[^6]: \[?\] S. Russenschuck, "Coil end optimization for the LHC main
    dipole," IEEE Transactions on Magnetics, Vol. 31 (3), pp. 1960-1963,
    (1995).

[^7]: \[?\] G. Ambrosio, N. Andreev, A.V. Zlobin, et. al., "Magnetic
    design of the Fermilab 11T Nb3Sn Short Dipole Model," IEEE
    Transactions on Applied Superconductivity, Vol. 10 (1), pp. 322-325,
    (2000).

[^8]: A volume magnetization integral equation method is foreseen to be
    used in future in order to estimate the influence of the iron on the
    peak fields.

[^9]: \[?\] G. Sabbi, "Magnetic field analysis of HGQ coil ends,"
    Fernilab TD-97-040, (1997).

[^10]: \[?\] D. R. Chichili & G. Ambrosio, "Mechanical and sensitivity
    analysis of 43.5 mm bore Nb~3~Sn dipole model," Fermilab TD-99-035,
    (1999).