A Three-Stage Cryogenic Pulse Magnet Program for BNL Targetry Experiment

Robert J. Weggel; BNL, bldg. 901A; Upton, NY 11973-5000

631-344-2428 (fax = 3248); <weggel@bnl.gov>

**General Description of Magnet System**

![](media/image1.png){width="3.1930555555555555in"
height="2.7006944444444443in"}

Fig. 1. Cross section of Targetry magnet system: cryostat, magnet
windings, target (mercury jet), and proton beam. The bore is 15 cm. The
winding pack is of 100 cm length, 20 cm I.D., and 80 cm O.D. Channels
for coolant separate the three subcoils, each of \~10 cm radial depth.

This report documents the conceptual design of a cryogenic pulse magnet
to generate up to 14.5 teslas in a room temperature bore of 15 cm. The
winding I.D., O.D. and length are 20 cm, 80 cm and 100 cm, respectively;
the conductor is 3.6 metric tons (1,800 turns) of high-purity copper.
Cooling of the magnet prior to each pulse is by helium gas or liquid
nitrogen forced through axial channels in the magnet; recooling of the
helium gas is by either liquid nitrogen or liquid hydrogen in a heat
exchanger inherited from the SSC. To hasten cooldown of the magnet, it
includes coolant channels not only at its inner and outer radius, but
also at two intermediate radii.

At the outer of these radii (30 cm) is an additional current lead, to
allow operation with the outermost coil omitted from the electrical
circuit. The remaining magnet, of 60 cm O.D. and 1,200 turns, is
appropriate for liquid-nitrogen operation; the full magnet has too high
a resistance to reach full current unless cooled to less than \~34 K.
The option to energize just the 1,200-turn magnet enables a staged
approach to the full-field mode of operation. One can postpone
fabrication of the outermost coil until needed by the last of the three
stages that we propose.

Three Proposed Stages: 5 T, 10 T & 14.5 T

![](media/image2.wmf){width="3.24375in" height="3.3090277777777777in"}

Fig. 2. Field vs. time for cryogenic pulse magnet for BNL Targetry
Experiment upon completion of each of three stages culminating in
14.5-tesla operation. Stage 1: 1,200 turns, 0.54 MVA; Stage 2: 1,200
turns, 2.16 MVA; Stage 3: 1,800 turns, 2.16 MVA.

Figure 2 graphs the field versus time for the three proposed modes of
operation. Stage 1 employs the 1,200-turn magnet cooled to 87 K or less
by liquid nitrogen at barely above atmospheric pressure (77-78 K). The
power supply is one of many at BNL that is rated at 3.6 kiloamperes and
150 volts (540 kVA). At this voltage, the magnet takes about seven
seconds to reach its full field of 5 T, which one holds for up to a
second. Discharge at full negative voltage, --150 V, takes an additional
three seconds.

Stage 2 employs the same 1,200-turn magnet, but energized by four BNL
power supplies in series/parallel: 7.2 kA x 300 V = 2.16 MVA. The pulse
is about \\10% shorter than for Stage 1. Simultaneously doubling both
the current and the voltage calls for a magnet of the same resistance at
peak current as Stage 1. However, because of the fourfold increase in
peak power, the magnet heats up nearly four times as much during each
pulse; therefore, it must start from a lower temperature---no more than
75 K. To reach this temperature, one can employ evaporative cooling at
reduced pressure to subcool the nitrogen to within a few degrees of its
64 K freezing point. BNL has roughing pumps ample to accomplish this.
Cryogenic investments will be in hardware and engineering associated
with the SSC heat exchanger and cryogen delivery systems. Investments in
the power supply will be hardware and software to ensure that its four
0.54 MVA modules share the load equally.

Stage 3 employs the same 2.16 MVA power supply as Stage 2, but now
energizing all 1,800 turns. In order for the same 300 volts to suffice
to drive full current through the much greater length of conductor, one
must reduce its electrical resistivity by cooling it to \~30 K with
helium gas whose temperature may be as low as 22 K. The heat sink is
liquid hydrogen trucked to BNL from an industrial liquefaction plant.
With 1½ times as many turns as Stage 2, Stage 3 generates 14.5 T instead
of 10 T. The pulse is much longer than for the previous cases: \~15
seconds to full field, a flat top of about a second, and eight seconds
to return to zero, when driven down at --300 V.

Table I tabulates parameters of Stages 1, 2 and 3. The last two rows
give the temperature rise and cumulative heating for a typical pulse, in
which, after a one second flat top, the power supply ("two-quadrant," or
"bipolar") discharges the coil with negative voltage of the same
magnitude used to charge the magnet.

# Table I: Pulse Magnet Systems for E951 Targetry Experiment

+-----------------------------------+--------+--------+--------+--------+
| #                                 | Units  | Case 1 | Case 2 | Case 3 |
+-----------------------------------+--------+--------+--------+--------+
|                                   |        |        |        |        |
+-----------------------------------+--------+--------+--------+--------+
| Peak on-axis field                | T      | 5.0    | 10.0   | 14.5   |
+-----------------------------------+--------+--------+--------+--------+
| No. of 0.54 MVA power supplies    | \- -   | 1      | 4      | 4      |
+-----------------------------------+--------+--------+--------+--------+
| Mode of ganging supplies          | \- -   | None   | 2 x 2  | 2 x 2  |
+-----------------------------------+--------+--------+--------+--------+
| Initial temperature               | K      | 84     | 74     | 30     |
+-----------------------------------+--------+--------+--------+--------+
| Number of turns utilized          | \- -   | 1200   | 1200   | 1800   |
+-----------------------------------+--------+--------+--------+--------+
| Charge time                       | sec    | 7.2    | 6.2    | 15.2   |
+-----------------------------------+--------+--------+--------+--------+
| Temperature rise at end of pulse  | K      | 6      | 21     | 48     |
+-----------------------------------+--------+--------+--------+--------+
| Cumulative heating at end of      | MJ     | 2.4    | 7.9    | 13.5   |
| pulse                             |        |        |        |        |
+-----------------------------------+--------+--------+--------+--------+

The remainder of this report supports the choice of magnet parameters of
Table I. Each choice is to some extent subjective, involving judgment
calls on the relative importance of costs and benefits. Of interest is
the rate of change of each cost and benefit with a change in each magnet
parameter. The following nine sections look at perturbations from a
base-case design for each of the three stages of operation, examining
the effects of just one or two parameters at a time.

**How Long Should the Magnet Be?**

An easy parameter to fix is that of magnet length, because the Targetry
Experiment has a requirement for field homogeneity, which depends very
strongly on magnet length and much less strongly on anything else. The
field homogeneity is to limit the magnetohydrodynamic forces on the
mercury jet target traversing the field; the field should be uniform to
10\\% over a length of 60 cm. Figure 3 reveals that the magnet should be
about 100 cm long.

![](media/image3.wmf){width="3.047222222222222in"
height="3.0722222222222224in"}

Fig. 3. On-axis field profile of magnets of 20 cm I.D., 80 cm O.D., and
either 80 cm or 100 cm length. The longer magnet achieves the desired
profile very well over the target, from --30 cm to +30 cm; the shorter
magnet does not.

The "ideal field" in figure 3 is that which would retain quite well most
of the pions captured in the 60 cm long target region, while
transitioning to its much lower value in downstream components of a
neutrino factory or muon collider. The field ramps downward by a factor
of sixteen over a length of three meters from the end of the target
region, while the magnet bore increases by a factor of four. This field
profile would allow one to test downstream components, such a
radio-frequency cavity, in the intense shower of particles emanating
from the target. Unfortunately, given the present budgetary climate, the
Targetry Experiment cannot afford to build these downstream components,
nor the more elaborate, expensive, and power-consumptive magnet in which
to test them. Therefore, it is irrelevant that beyond the target region
neither of the above field profiles matches very well the ideal field.

**How much does electrical conductivity improve at low temperature?**

![](media/image4.wmf){width="3.042361111111111in" height="2.825in"}

Fig. 4. Electrical resistivity , heat capacity c~p~, and ratio /c~p~
between room temperature and 30 K, for copper with a residual
resistivity of 0.05 cm below \~20 K.

The motivation to cryogenically cool any magnet, pulsed or not, is to
improve the electrical conductivity of its conductor. Figure 4 shows
that the resistivity of high-purity copper declines greatly from room
temperature down to 30 K. The incentive to operate at cryogenic
temperatures is great indeed. Cooling to 80 K (with liquid nitrogen at
atmospheric pressure) improves the electrical conductivity by a factor
of nearly seven. Cooling to 66 K (with liquid nitrogen subcooled to
nearly its freezing point of 64 K) gives a ratio of about ten. Cooling
to 30 K (with liquid hydrogen as the heat sink, for example) can achieve
a ratio of about 30.

![](media/image5.wmf){width="3.0430555555555556in"
height="3.1555555555555554in"}

Fig. 5. Fig. Low-temperature electrical resistivity , heat capacity
c~p~, and ratio /c~p~ for copper with a residual resistivity of 0.05 cm
below 20 K.

There is very little motivation to cool the conductor still further; one
has entered a regime of diminishing returns. Figure 5 illustrates two of
the reasons for this. One reason is that electrical resistivity improves
rather little. This is true even for copper that is exceedingly pure,
unless it is so completely annealed as to be too weak for a very high
field magnet. The other reason is that, if the copper is uncooled, with
only its heat capacity to limit its temperature rise, it will heat up
very rapidly, because its heat capacity plummets, approximately as T^3^,
below \~30 K. The heating rate, proportional to /c~p~, is three times
worse at 20 K than at 30 K.

**How Beneficial is Cryogenic Operation?**

![](media/image6.wmf){width="2.879861111111111in"
height="3.564583333333333in"}

Fig. 6. Field vs. initial temperature of cryogenic pulse magnets of 100
cm length, 20 cm I.D. and either 60 cm or 80 cm O.D.

With the maximum magnet power, 2.16 MVA, deliverable at the targetry
site, a Stage 3 targetry magnet could generate only about 4 T if
operated at room temperature. Figure 6 shows that even precooling to the
magnet to 80 K fails to deliver sufficient field intensity to duplicate
the 14-15 T desirable for a neutrino factory of muon collider. To
achieve this field, one needs to precool the targetry magnet to about 30
K.

The field-versus-temperature predictions of figure 6 (and all the other
relevant figures in this report) acknowledge that electrical resistivity
depends not only on temperature, as shown in figure 5, but also on
magnetic field. This magnetoresistance can significantly degrade
electrical conductivity at very low temperatures.

**How detrimental is magnetoresistance?**

![](media/image7.wmf){width="3.2993055555555557in" height="3.26875in"}

Fig. 7. Electrical resistivity versus temperature from 20 K to 100 K and
magnetic fields from zero to 20 T.

Magnetoresistance can be severe at low temperatures and high fields. It
becomes ever more significant at low temperature. Figure 7 graphs, as a
function of temperature and magnetic field, the electrical resistivity
of copper with a residual resistivity of 0.05 cm. Note that at 20 K the
magnetoresistance at 20 T nearly triples the resistivity at zero field.
Even at fields that are more modest and temperatures that are less
frigid, magnetoresistance is non-negligible. At 10 T and 100 K, the
effect is about 13%.

**What does a typical pulse look like?**

![](media/image8.wmf){width="3.04375in" height="2.9902777777777776in"}

Fig.9. Current, magnet resistance and power supply voltage of Stage 3
pulse magnet for Targetry Experiment.

Figure 9 graphs the time dependence of some of the more important
parameters of the highest field (Stage 3) pulse magnet for the Targetry
Experiment. Note that the resistance increases by a factor of about 3½
during the pulse. The slight decline in resistance during the last
several seconds of the pulse is because the magnetoresistance is
decreasing faster with the decreasing field than the zero-field
resistance is increasing from the still-increasing temperature.

Note also that the peak resistive voltage of the magnet, when pulsed
from this temperature, is about 240 V---i.e., about 80\\% of the full
voltage available from the power supply. To obtain the flat top to the
field requires the power supply voltage instantly to drop to 240 V from
300 V, because the inductive back voltage abruptly has dropped to zero.
Then, for the duration of the flat top (shown here as one second), the
voltage must increase several percent to track the increase in
resistance as the windings heat up. Full negative voltage drives the
current to zero.

**How should one discharge the magnet?**

![](media/image9.wmf){width="3.0430555555555556in"
height="2.9902777777777776in"}

Fig.10. Peak temperature rise and cumulative heating in cryogenic pulse
magnets of 20 cm I.D., 80 cm O.D. and 100 cm length. Left-hand curves:
peak temperature rise; right-hand curves: cumulative heating.

Figure 10 graphs, as a function of time, the temperature rise and
cumulative heating at the end of each field pulse of the Stage 3 magnet
when discharged in any of four ways. One way is that of a typical pulse,
in which one drives the current down with reverse voltage of the same
magnitude used to charge the coil. Driving the current down, from a flat
top of one second, limits the peak temperature rise to 48 K and the
heating to 13 MJ. Another way is with the current coasting down,
dissipating in the magnet all of the magnetic energy that it stored.
This would occur if the power supply tripped out at the end of the flat
top, and therefore did not deliver the reverse voltage expected from it.
The windings must absorb all, instead of just a part of, the magnetic
energy stored in the coil. The additional 6 MJ leads to a total
temperature rise of nearly 60 K. This is still a very comfortable value
from the standpoint of thermal stresses, but the magnet will take longer
to recool.

The other two modes of discharge employ an external resistor. A 100 mΩ
resistor, implying a peak discharge voltage of --720 V, gives an energy
dissipation and temperature rise roughly equivalent to that when
discharged through the power supply at a constant voltage of --300. With
a 1 Ω resistor (--7.2 kV peak discharge voltage), the energy dissipation
and temperature rise are about 9 MJ and 38 K, respectively. At such a
high discharge voltage, the external resistor absorbs nearly all the
magnetic energy stored in the magnet.

**How big in diameter should the magnet be?**

![](media/image10.wmf){width="3.2430555555555554in"
height="3.2881944444444446in"}

Fig. 11. Central field vs. outer diameter of cryogenic pulse magnets of
20 cm inner diameter and 100 cm length.

Figure 11 reveals that to generate the very most field with a fixed
power supply calls for a targetry magnet with an outer diameter of about
one meter, irrespective of the power level or initial temperature. Note,
however, that the maximum of each curve is very broad. Well before one
has reached a diameter as large as a meter, the increase in field
becomes too gradual to justify the rapid increase in conductor mass and
cost. Already at an outer diameter of 80 cm, the conductor mass has
risen to 3.6 metric tons. Increasing the outer diameter to one meter
would require 60\\% more conductor, for less than a 3\\% increase in
field.

![](media/image11.wmf){width="3.248611111111111in"
height="3.2743055555555554in"}

Fig. 12. Cumulative heating vs. outer diameter of cryogenic pulse magnet
of 20 cm inner diameter and 100 cm length.

An additional reason for limiting the outer diameter of the targetry
magnet is to limit the heat that one must remove after each pulse.
Figure 12 shows that the cumulative heating increases rapidly with
diameter. For example, doubling the diameter approximately triples the
cumulative heating, tripling the recool time in any system that is
limited by the mass flow of coolant. This consideration, like the one
concerning conductor mass and cost, favors magnets no larger than 80 cm
in diameter.

**How warm does the conductor get?**

![](media/image12.wmf){width="3.046527777777778in" height="3.04375in"}

Fig. 13. Temperature at end of pulse of Stage 3 magnet energized to
7,200 A, 14.5 T (with a half-second flat top) from an initial uniform
temperature of 30 K.

Figure 13 plots temperature contours for the Stage 3 magnet pulsed to
full field (with a flat top of ½ sec.) from a uniform initial
temperature of 30 K. The resulting temperature is highest, 76 K, at the
inner radius of the magnet midplane (r = 10 cm, z = 0), where
magnetoresistance adds most to the resistivity, and therefore the
heating rate. Near the kernel, where the field and magnetoresistance are
zero, the maximum temperature is only 62 K.

![](media/image13.wmf){width="3.046527777777778in" height="3.04375in"}

Fig. 14. Temperature distribution in Stage 3 magnet if energized from an
illustrative non-uniform temperature: 34 K at radii of 15 cm, 25 cm and
35 cm, falling parabolically to 22 K at radii of 10 cm, 20 cm, 30 cm and
40 cm.

For comparison, figure 14 plots temperature contours for the Stage 3
magnet pulsed to full field (again with a flat top of ½ sec.) from an
initial temperature with the same average as before, 30 K, but with a
distribution suggestive of that in a magnet pulsed to full field and
then recooled too briefly for temperatures to equilibrate. The peak
temperature, coincidentally, is the same, 76 K, as when energized from a
uniform temperature, but occurs at a different radius, \~14 cm, where
the initial temperature was high. At r = 10 cm the peak temperature is
less than 75 K, despite the magnetoresistance being greatest there,
because the initial temperature was only 22 K.

**How do field and heating depend on initial temperature?**

![](media/image14.wmf){width="3.0444444444444443in"
height="3.1416666666666666in"}

Fig. 15. Field and cumulative heating in Stage 3 magnets of 100 cm
length, 20 cm I.D., 80 cm O.D. and 1,800 turns, pulsed from an initial
temperature of 20 K to 50 K.

If one pulses a cryogenic magnet from too high a temperature, it will
fail to reach its design current and, hence, its design field. Figure 15
shows that for the Stage 3 magnet, the critical temperature is 35 K. A
magnet pulsed from 40 K, for example, can generate only 14 T, instead of
14.5 T. Pulsing the magnet from any temperature lower than 35 K will not
increase its field, because one runs out of current. However, one now
has spare voltage with which to shape the pulse. For example, a magnet
pulsed from 33.7 K not only can reach full field, but also can hold it
for the desired one second. A further benefit is the reduction in
dissipation, thereby improving the rep rate of the magnet. Pulsing from
25 K instead of 30 K decreases by 10\\% the heat that one must remove.
Relative to a magnet pulsed from 33.7 K, the improvement is 23\\%.

![](media/image15.wmf){width="3.048611111111111in"
height="3.234027777777778in"}

Fig. 16. Central field and cumulative heating of Stage 2 magnet of 100
cm length, 20 cm I.D., 60 cm O.D. and 1,200 turns, pulsed from an
initial temperature of 60 K to 90 K. Comparative magnet is of same
dimensions but has 5\\% more turns.

Figure 16 plots data similar to that of figure 15, but now for the Stage
2 magnet, whose intended coolant is subcooled nitrogen. 75.5 K is the
maximum temperature at which the magnet will generate full field with a
flat top of one second. Cooled to 80 K (as with nitrogen that is not
subcooled) limits the field to 9.6 T instead of 10 T. Cooled to 66 K,
the minimum likely with subcooled nitrogen, reduces the cumulative
dissipation to 5.6 MJ, an improvement of 37\\% relative to 75.5 K
operation. The comparison magnet, with 5\\% more turns, will generate
5\\% more field at any temperature below 69 K, but at the price of
significantly more heating. For example, at 66 K its cumulative heating
is 7.4 MJ, one third more than for the magnet with 1,200 turns.

![](media/image16.wmf){width="3.04375in" height="3.2256944444444446in"}

Fig. 17. Central field and cumulative heating for Stage 1 magnet and
comparative magnets with 5\\% and 10\\% more turns, each pulsed from an
initial temperature of 65 K to 90 K.

Figure 17 presents data as in the previous two graphs, but now for
variants on the Stage 1 magnet, whose intended coolant is liquid
nitrogen at atmospheric pressure. A magnet with 1,200 turns generates
full field, complete with a flat top of one second, at all temperatures
below 87.4 K. This temperature is so far above that of liquid nitrogen
that cooldown should be rapid, if governed by heat transfer to the
coolant. However, if cooldown is limited by mass flow through the
coolant channels, then operation from a lower temperature may be
superior. Pulsed from 80 K, the magnet dissipates 37\\% less heat---only
1.94 instead of 3.08 MJ. One may even wish to subcool the nitrogen.
Cooled to 66 K, the magnet dissipates only 1.13 MJ, 42\\% less than at
80 K. The two comparison magnets, with 5\\% and 10\\% more turns,
respectively, generate correspondingly more field, but with
substantially narrower temperature ranges of operation and higher
dissipations at any given temperature. These considerations, plus the
convenience of making a single magnet suffice for both Stages 1 and 2,
make the 1,200 turn magnet the compelling choice.