PFC/RR-83-21
DOE/ET-51013-92
MFTFB+T MAGNETOSTATIC AND CIRCUIT
FAULT INDUCED FORCES
R. D. Pillsbury, Jr., R. J. Thome
W. R. Mann, and W. G. Langton
Massachusetts Institute of Technology
Plasma Fusion Center
Cambridge, MA 02139
August 1983
Foreword
This work was performed for the Fusion Engineering
Design Center, Oak Ridge, Tennessee in support of their
activi-:y for the Mirror Fusion Test Facility at the
Lawrence Livermore National Laboratory. The effort was
carried out by the Plasma Fusion Center Magnet Technology
Division under contract no. DOE-AC02-78ET-51013.
TABLE OF CONTENTS
Page No.
0.0
INTRODUCTION. . . . . . . . . . .
1
1.0
SUMMARY . . . . . . . . . . . . .
3
2.0
COIL AND CIRCUIT MODELS . . . . .
22
3.0
MAGNETOSTATIC LOAD ANALYSES . . .
30
4.0
TRANSIENT LOAD ANALYSES . . . . .
41
5.0
CONCLUSIONS . . . . . . . . . . .
64
0.0 INTRODUCTION
Magnet systems consisting of multiple, inductively coupled coils can
experience substantial changes in their usual operating load patterns in
the event of a fault condition such as a short circuit followed by a rapid
system discharge. The methods for determining the load changes are
straightforward, but time consuming and the number of cases requiring
consideratlrn grows rapidly as the number of coils and circuits in a system
increases.
As magnet systems become larger and more complex, the electromagnetic
load redistribution or other effects such as excessive temperature rise can
become so severe that a thorough fault condition examination is essential.
The Magnet Technology Division at the Plasma Fusion Center has been
developing codes to improve the efficiency and turn around time for model
construction and fault condition analysis. In this report the codes were
applied to a system of 24 coils in nine circuits which constitute MVTFB+T,
a joule mirror fusion system under consideration by LLJL. The process
begins by system field/force model construction using sufficient detail to
analyze parameters selected as indicative of the severity of a fault
condition (e.g. - total load in a particular direction on a coil or load per
unit length at a point in a coil), but not so detailed that turn around
time is unattractive. If a severe condition is found it is assumed that
the design croup will investigate the condition with a more detailed model.
An estimation of "severity" of a fault condition requires a well
defined set of "usual' operating conditions for comparison. We define the
latter as the extreme maximum or minimum values of the critical parameters
for any combination of the circuits in a fully "on" or "off" condition. In
itself, this magnetostatic analysis is nontrivial in a complex system. For
example, for the nine circuits in MFTFB+T, there are 512 (2 to the ninth
power) usual operating conditions. It should be emphasized that these
represent feasible conditions since it is conceivable that any combination
of circuits :ould be fully charged during shakedown tests or system
operation. This is indicative of the need for an efficient procedure and
codes for fault analysis since the number of credible faults is often many
t imes the number of usual operating conditions in a multiple coil/multiple
In this report, the fault conditions which have been selected arise
from assuming that all circuits are charged to their operating level, a
short circuiu occurs across one coil and a discharge of the entire system
follows. Each case requires a transient circuit analysis required by em
load determination as a function of time because, in general, the extreme
loads do not necessarily occur at the time a discharge current in a circuit
passes throuch its maximum value.
NOW&
In general, the four C-shaped magnets at each machine end (I4'2, 1I11,
WT2, and WTI) experience the worst case loading under static current
conditions. The central cell solenoids (WS6, WS5, WS4, and WS3) experience
their peak loads during discharge transients with a neighboring coil
short-circuited. The coils at the machine center (WS2, WS1,
WCC1, WTDFS2,
etc.) have peak axial loads that occur primarily in the static condition.
However, the peak radial loads occur during the discharge transients.
The currient scenarios chosen for the analyses were by no means
exhaustive from a fault condition standpoint. An entire class of faults
which requires consideration involves system behavior during discharge
execution with a section of one of the superconducting coils in the
resistive state. Even though the coils are designed to be stable under
usual conditions, that is, complete LHe coolant immersion, a resistive
region is possible in the event of a fault in the form of a low helium
level. The tEimperature rise in a resistive region which is not immersed is
strongly dependent on the coupled discharge character of the circuits and
the level of operating current and dump voltage selected.
Substantial progress has been made in the ability to turn around
certain classes of fault conditions as evidenced by the cases treated in
this report. is indicated above, however, other classes of severe
conditions ex. st and development of codes for rapid turnaround of these
cases should continue.
-2-
I
I
1. 0 SLMVY
This report presents the results of work performed to date on the
estimation of the Lorentz forces acting on the various coils in the MFTFB+T
mirror machine. The analyses include: (1) static loading under normal
operating conditions; (2) transient loading during a normal discharge; and
(3) transient loading for one coil short circuits with all coils
discharging. For the purposes of this memo, normal operation is defined as
any possible combination of circuits fully charged to their operating
current level or at zero current. This definition would then include any
possible scenario for the entire system or partial system at full operating
current.
Figure 1.1 shows both an isometric and side view of the coils in the
machine. Txwenty four coils are shown and consist of two large central
solenoids, iwo high field choke coils, and twelve identical solenoids in
the central cell. Each of the machine ends has four coils: two transition
coils and a yin-yang pair. Only the coil centerlines are shown.Superimposed are the coordinate system, the coil numbering system and coil
nomenclature adopted for this effort. The coils are numbered consecutively
from the coil with the largest positive axial (z) coordinate (coil 1) to
the largest negative coordinate (coil 24). Table 1.1 lists the coil number,
name, their circuit number, and full operating current. The circuit
numbering is arbitrary, but collects the 24 coils into nine circuits
corresponding to the expected power supply configuration for this machine.
Note that each of the four yin-yang coils are in an independent circuit
whereas other similar coils are in series in the same circuit. Results may
change considerably if different current levels or circuit configurations
are postulated.
The models of the coils and circuits are described in section 2.0. The
results froff the static analyses are given in section 3.0. The transient
analyses are presented in section 4.0. The remainder of this section
summarizes the results.
Table 1.2 summarizes the peak values of the net axial force which can
act on each coil out of the possible normal operating conditions. Both the
largest positive and largest negative loads are given along with the
current scenario that produces those forces. The current scenarios are
gii'en as a nine digit number. Each digit represents the current multiplier
in each circiit. For example, the current scenario 110011001 implies that
circuits 1,2,5,6, and 9 are at full current, while circuits 3,4,7, and 8
are at zero :urrent. The forces given represent the total axial load acting
on the coil.
It can be seen that, except for the two end coils, the worst case
do rot occur when all coils are at full operating current.
Similarly, none of the cases of the single circuit on - all others off
generates maimun or minimum forces. In general, the worst case loading
occurs when as many coils as possible to one side of the coil under
II:adings
-3-
High Field
Choke Coils
FR (y =0)
Transition
-(X z
y
Pair
FR(x
y
=o)
x
Cc4 D
N
I-
w~ w
CY
VM
04
C-1
1
N
V'OWNWOVM
r- VN p-T
19
0N N
Figure 1.1
-T
C
:D
-.
~.
NT-O0ONO~fle
.
MFTF+T Coil Centerlines, Numbers and Nomenclature
-4-
z
rable 1.1 Coil/Circuit Numbering and Nomenclature
coil
#
coil
name
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
WM2
WM1
WT2
WT1
WS6
WS5
WS4
WS3
WS2
WS1
WCC1
WTDFS2
ETDFS2
ECC1
ESI
ES2
ES3
ES4
ESS
ES6
ETI
ET2
1
2
3
4
5
5
5
5
5
5
6
7
7
6
5
5
5
5
5
5
4
3
4403
3872
5206
2322
1823
1823
1823
1823
1823
1823
107190
8000
8000
107190
1823
1823
1823
1823
1823
1823
2328
5206
3
[Ml1
8
3872
24
EM2
9
4403
Circuit
#
-5-
Operating
Current (A)
Table 1.2
coi I
#
1
3
4
5
6
7
6
9
16
11
12
13
14
15
16
17
18
19
20
21
23
n4A
coil
name
WM2
WM1
WT2
WT1
WS 6
WSS
WS4
WS3
WS2
WS1
WCC1
WTDF32
ETDF32
ECC1
ES1
ES2
ES3
ES4
ES5
ES6
ETI
ET2
EMI
E- Ic
Peak Axial Loads Under Normal
Operating Conditions
Maximum
Fz (MN)
0
4.75
4.96
1.04
0
0
Current
Scenario*
Minimum
Fz (M)
Current
Scenario*
110000000
-5.27
-5.02
-1.51
-0.48
011111111
-4.44
000011111
000011111
111110000
111110000
111110000
-2.17
-1.63
-1.84
-2.80
-4.82
111111000
-1.68
111000000
111100000
0
0.43
1.41
3.91
0.21
-162.
0
102.
1.68
4.82
2.66
1.84
1.63
2.17
4.44
0.48
1.51
5.02
110000100
110001100
5.27
111111111
110011100
110011100
110011100
110011100
110011100
110011100
110111100
111111100
111111110
+ each diqit -epresents a circuit on (1)
111111111
661111111
000111111
000011111
000011111
000011111
000011111
000001111
000000111
0
-0.21
001111011
-3.91
001110011
001110011
001110011
-1.41
-0.43
0
0
0
-1.04
-4.%
-4.75
0
001100011
001000011
000000011
or off (0)
e.g. 116011100 means circuits 1,2,5,6,7 are on and circuits 3,4,8,9 are off
consideratiOn have zero current and as many as possible on the other side
have full cirrent. Due to the connection symmetry of most of the circuits
the worst case load does not correspond to all coils on one side on and all
coils on the other off.
Force; that are listed as 0 imply there is no scenario that gives a
force of that sign or direction on that coil. For example, WS6 (coil #5)
has a zero maximum (positive) force. This means that there is no possible
combination of circuits completely on or-off that produces a net force on
WS6 that is away from the machine center. However, there are several
combination; that produce a net force on WS6 toward the machine center. The
worst case .oad is -4.44 MN and occurs when circuits 1,2,3 and 4 are off
and circuits 5-9 are on (i.e. a current scenario of 000011111).
Figure 1.2 shows the same information in bar chart form. The worst
case maximu (positive) and minimum (negative) net axial force on each coil
is shown. For scaling purposes the forces on the large solenoids have been
clipped and the peak value written adjacent to the bars. There is
anti-symmetry about the machine center, as is expected.
Table 1.3 summarizes the peak radial loading per unit length acting
on each coil for normal operating conditions. In a solenoid, the forces per
unit length are indeed radial. For C-shaped coils, these forces are local
running loads (N/m) which contribute to the lobe-opening forces. For the
purposes of this memo these forces are denoted as radial for all coils.
There are tuo values presented. The first
is in the X=0 plane; the second
in the Y=0 plane. The two radial force directions are illustrated in Figure
1.1 for a typical C-shaped magnet and for a typical solenoid.
Note that
the force in the X=G plane is y-directed and the force in the y=0 plane is
x-directed.
In a purely solenoidal system, these two forces would be
equal. The lack of equality is a function of the non-solenoidal fields
produced by the C-shaped magnets in the system. The radial forces are
pu.It~ie fur dll cases and are usually extreme for all circuits on.
However, som of the coils (e.g. WT2) experience the worst case radial
loading when an adjacent coil is off and unable to hold the axial flux from
the system away from the winding in question.
Figure 1.3 shows the information from this table in bar chart form.
The sclid lines represent the radial forces (per unit length) in the X-0
plane; and the dashed lines the Y=0 plane. Only the C-shaped coils at the
machine ends experience a significant loading difference in the two planes.
The end solenoids do experience a slight difference in force, but the
difference it; too small to see at this scale and the solid lines hide the
dashed ones. A more detailed discussion and presentation of some of the
individual current scenarios for usual operating conditions can be found in
section 3.0. The cases considered in detail were all coils on, and
individual circuits off with all others on.
-7-
11
X
a4vi
U.0
-D
c)
-
C-C--D I I I I I I..ILI f I I I
M4
C
I
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I
101.8
CO
z
IZI
.LLLLI
N
IL
.7
-
I
m
.
I
.
I
.
.
i
I
.
-iOt.9
.
i
I
24232221 20 19181716151413
i I
9I
211.10
I
COIL P
Figure 1.2
Peak Axial Loads - Static
-8-
8I
7I
8
' I '
5
4
I
3
I
2
1
Table 1.3
coil
#
2
3
4
5
6
8
9
10
11
12
13
14
15
16
coi l
name
Peak Radial Loads per Unit Length Under
Normal Operating Conditions
X=6 Plane
Fr (M/m)
WM2
WM1
4.95
6.03
WT2
WT1
WS6
WS5
WS4
0.80
WS2
WS1
WCC L
WTDFS2
ETDFS2
ES I
ES2
0.87
0.67
0.75
0.82
0.90
1.01
1.16
34.0
28.1
28.1
34.0
1.16
1.01
F'
0.90
18
L4
19
ESS
0.83
0.75
0.65
0.66
1.34
3.99
5.41
20
21
22
23
24
ET1
ET2
E 1I
E12
Current
Scenar 1o*
Y-0 Plane
Fr (MN/m)
Current
Scenario*
111111111
5.41
111111111
111111111
10111111
111111111
3.99
1.34
0.66
0.65
0.75
1111111
111111
1111111
116111111
11111111
111111111
111611111
111611111
111111111
111111111
111111111
111110111
111111111
111110111
111110111
111111111
111110111
111111111
111111111
111111111
111111111
116111111
111111111
111111111
111111111
111111111
* each di9 it represents a circuit on (1)
0.83
0.90
1.01
1.16
34.0
28.1
28.1
34.0
1.16
1.01
0.90
0.82
0.75
0.67
0.87
0.80
6.03
4.95
111111111
111111111
111116111
111111111
111116111
111116111
111111111
111116111
111111111
111111111
111111111
111611111
111611111
111111111
111111161
111111111
111111111
or off (0)
e.g. 110011166 means circuits 1,2,5,6,7 are on and circuits 3,4,8,9 are off
-9-
w wu W WWW UWWLWJI*Jt
i.
CC-
*
V3
C4
-01M G'-ww~q
V V OC C C"
0a .f-oU4
ca
Lo
X = 0 Plane
Y = 0 Plane
0-
(v)
in
i
E
z
CLL
C),
0
,;T
N42
22
01
1
7I
1
41
CO
121I 10 9
Figure 1.3
8
7
e
5
4
3
2
Peak Radial Loads Per Unit Length - Static
-10-
1
The next set of analyses investigated the coil loading experienced
during discharge transients. The lumped circuit parameter model used is
discussed in section 2.0. The scenarios considered were: (1) all coils on
and then discharged simultaneously, and (2) all coils on then a single coil
short-circuited with all others discharging. The extreme loads on different
coils can occur at different times during the transient, and, therefore,
cannot be used to represent a subsystem loading - e.g. peak axial loads on
coils 1 and 2 (WM2 and WM1) cannot be summed to get a worst case loading on
the set. Such an assumption would be conservative, but possibly too much so.
Table 1.4 presents the worst case axial forces acting on each coil
for the normal discharge case. Some coils actually experience a load
reversal during discharge, because all coils do not discharge at identical
rates. Figure 1.4 shows the same information in bar chart form. Table 1.5
and Figure 1.5 present the worst case radial forces per unit length in the
X-0 plane (solid lines) and Y-0 plane (dashed lines) for the normal
discharge c'ase.
Similarly, Tables 1.6 and 1.7 and Figures 1.6 and 1.7 present the
worst case loads on each coil for all the single coil short-circuit
transients, Each coil from 1 to 12 (M2 to WTFDS2) in turn was assumed to
have a short across the entire coil with a short resistance of 1 micro-Ohm.
The other coils discharge through their dump resistors. The case numbers
correspond to the coil number that contains the short during the worst
transient v'or the coil. Case 0 is the normal discharge of all coils. The
notation 0+ implies a small load (order of kN). WT2 experiences a small
load reversal in the radial force per unit length in the X-O plane. Since
only half the coils were shorted individually because of the system
symmetry, the worst case forces on the coils are tabulated for coils 1-12
only. The loads on all the coils are presented in the figures.
A mort detailed discussion of the transient analyses is given in
Also presented are the loadings for the individual current
scenarios.
Snetiui 4.A.
Finally, Tables 1.8 and 1.9 list the worst case loadings for each coil
from all the usual and fault current scenarios investigated. If the load is
from a static scenario, the scenario is given as the nine digit number
describlng which circuits are charged and which are not. If the load arises
during a ccil short-circuit transient, the coil being shorted is given.
Only the first 12 coils are listed. The peak forces on the other coils will
have the expected syimmetry or anti-symmetry. These worst case loads are
sho'wn in Figures 1.8 and 1.9. All coils are shown.
-11-
11
1
Table 1.4
Peak Axial Loads Under Normal
Discharge Conditions
coil
#
coil
name
Maximum
Fz (MN)
Minimum
Fz (MN)
1
0
0
3.44
0.56
0
0
0
0
0.15
0+
0
0
-5.27
-0.28
5
6
WM2
WMIl
WT2
WT1
WS6
W35
2
3
4
7
WS4
8
9
WS3
WS2
10
11
WS1
13
14
WCC 1
WTDFS.?
ETDFS2
ECCI
15
ESI
16
17
18
19
20
ES2
ES3
ES4
ESS
ES6
ETI
12
21
22
23
24
90.3
1.47
4.78
2.75
1.74
1.46
1.84
0
0
-3.89
-1.84
-1.46
-1.74
-2.75
-4.78
-1.47
-90.3
0
0
0+
-0.15
3.89
0
0
0
0
ET2
EMIl
0
0
-0.56
-3.44
0.28
EM2
5.27
0
0
0+ denotes a small force on the order of kN.
-12-
X
£4~~
~
Z~
-4
I
I
I
I
I
I
I
I
I
-
I-
'O6I-S w4V.
44
N~
M
I
I
N
.4(
ho
~
QI.9.fV
io.b
*
0)
fC
I
I
9I-
I I
I
*I
II
II
II
I
I
I
z
LL
Si
24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9
Figure 1.4
II
8
7
0
i
5
Peak Axial Forces - Normal Discharge
-13-
I
4
*
I
3
2
1
Table 1.5 Peak Radial Loads per Unit Length Under
Normal Discharge Conditions
coil
#
1
C-
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2- I
22
23
24
coil
name
WM2
WMIl
WT2
WT1
WS6
WSS
WS4
WS3
WS2
WS1
WCCI1
WTDFS2
ETDFS2
ECC 1
ESI
ES2
ES3
ES4
ES5
ES6
ETI
ET2
EM1
EM2
X=0 Plane
Fr (MN. m)
Y=0 Plane
Fr (MN/m)
5.41
3.99
1.34
0.66
0.65
0.75
0.83
0.90
1.01
4.95
6.03
0.47
0.87
0.64
0.75
0.82
0.90
1.01
1.18
34.0
28.1
28.1
34.0
1.18
1.01
0.90
0.83
0.75
0.65
0.66
1.34
3.99
5.41
1.18
34.0
28.1
28.1
34.0
1.18
1.01
0.90
0.82
0.75
0.64
0.37
0.47
6.03
4.95
-14-
x
N
iu
w
It
w
O
wwwtWW
64
n
C4CIv
-j
Inj
X
Ln-
=
0 Plane
-Y =O0Plane
0
UI)
C4
E
, ,n
z
LL
Ca
In
v-4
24
2-322
Figure 1.5
2120
19 18
17
18 15 14 19
21 L O
8
7
0
Peak Radial Forces Per Unit Length
-
-15-
5
3
2
1
Normal Discharge
Table 1.6
Coil
#
Coil
Name
1
WM2
W41
WT2
WTI
2
3
4
5
6
1436
7
W34
WS5
8
9
10
W:.31
11
WCC1
12
wTrDSF2
Table 1.7
Coil
#
Coil
Name
Peak Axial Loads On Coils Due to Single
Coil Short-Circuits
Maximum
Fz(MN)
Case
#
Minimum
0
0.56
0+
-0.04
-3.99
4
0.56
0.75
5
0.89
1.12
7
8
9
1.85
0
0.90
-5.27
-1.39
-1.12
1
2
0
1.09
3.60
6
10
X=0 Plane
Maximum
Fr (M/m)
Case
#
ll
WM2
4.95
6.03
3
WT2
WTI
0.51/-.03 A
0.87
WS"
WS!;
1.15
0
5
1.53
1.78
6
7
2.05
2.46
6
9
6
7
8
9
10
11
12
WS4
IA&
WS2
WSJ,
wCc1
WTEISF2
5
-2.29
-1.86
6
8
-2.10
8
-3.06
-4.83
-1.47
-90.3
10
12
3.23
49.0
30.4
Y-0 Plane
Minimum
Fr (t./m)
5.41
0
0
4.13
1.34
o.66
1.15
1.53
1.78
2.05
2.46
3.23
49.0
30.4
3/2
10
11
12
0+ denotes a simall force on the order of kN.
A
12
3
4
4
9
12
Peak Radial Loads Per Unit Length On Coils
Due to Single Coil Short-Circuits
1
2
4
5
Case
Fz(MN)
a slight foroie reversal
-16-
Case
*
0
2
0
0
5
6
7
8
9
10
11
12
N
x
C
(*VC4 cIIIII.0 IN~~ff
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24 23 22 21 20 19 I
17 16 15 14 13 12 1I LO 9 8
7
6
5 4 3
2
1
COIL #
Figure 1.6 Peak Axial Forces Due To A Single Coil Short Circuit
-17-
x
w
w
cj CflwflCI~Oi-
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3:3I0CflI-4
3.-
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3121
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10
Figure 1.7
432
9
a
7
6
5
4
3 f2
Peak Radial Forces Per Unit Length Due To A
Single Coil Short
-18-
,11
Peak Axial Loads Under All Conditions
Table 1.8
coil
coil
Maximum
#
name
Fz (MN)
1
2
Wi2
Wi 1
3
W72
0
4.75
4.96
1.04
0+
0.56
0.75
0.89
1.41
3.91
0.21
0.90
4
5
6
7
8
9
10
11
12
W26
WS4
WE3
WS2
W1
WCCI
Wi DFS2
Table 1.9
Current
Scenario*
11000
11100600
liltn m
short 4
short 5
short 6
short 7
111110000
11111600
11111100
short 10
name
X=6 Plane
Fr (MN/m)
Current
Scenario*
1
2
3
4
5
6
7
8
9
10
11
12
WM2
WHi
4.95
6.03
W12
W1
W6
0.80/-.03 A
111111111
111111111
161111111
111111111
short 5
WS2
14,1
W( Cl
WTDFS2
111111111
011111111
001111111
000111111
000011111
short 6
short 8
short 8
0011111
short 10
00001111
600000111
Peak Radial Loads Per Unit Length
All Conditions
coii
W$3
-5.27
-5.02
-1.51
-0.48
-4.44
-2.29
-1.86
-2.10
-2.60
-4.83
-1.68
#
WS4
Current
Scenario*
-102.
coil
WS5
Minimum
Fz (MN)
0.87
1.15
1.53
1.78
2.05
2.46
3.23
49.0
30.4
short 6
short 7
short 8
short 9
short 10
short 11
short 12
Y-0 Plane
Fr (Mf/m)
5.41
4.13
1.34
0.66
1.15
1.53
1.78
2.05
2.46
3.23
49.0
30.4
Current
Scenario*
111111111
short 2
111111111
111111111
short 5
short
short
short
short
short
short
short
6
7
8
9
10
11
12
or off (0)
e.g. 11011100 means circuits 1,2,5,6,7 are on and circuits 3,4,8,9 are off
* each dig it represents a circuit on (1)
0+ denotesl a small force of the order of kN.
^ force rtversal for short 2.
-19-
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COIL M1
Figure 1.8
Peak Axial Loads For All Current Scenarios Considered
-20-
x
OLL0
JZ;
"goA VM"
W~
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Y = 0 Plane
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Figure 1.9
1211 10 9 8 7 8
COIL #
5
4
3
2
1
Peak Radial Loads Per Unit Length For All Current
Scenarios
-21-
2.0 COIL
MD CIR UIT MODELS
The individLal coils in the mirror machine were modeled with lumped
current filaments made up of piecewise straight segments or sticks. The
number of filamer ts used to represent the coil cross-section varied from
coil to coil. Fil aments were used in the solenoids for convenience. Checks
on the accuracy cf both inductance and force were made using uniform
current density s olenoid models.
Figure 2.1 shows the typical coil geometries for a C-shaped and
solenoid magnet. Also indicated are the necessary inputs to generate a
stick model of the coil. Table 2.1 lists the inputs and other critical
parameters for each coil in the machine. In addition to the geometry, the
table also lists the overall current density, the ampere-turns, the full
operating current , the number of turns, the size of the dump resistor, and
the coil self-ind uctance.
The model usd for both the inductance and force calculations is shown
in Figure 2.2. Th! coils WM2, WM1, EM2, and EMI were modeled(aith 6
filaments (2 by 3). The other four C-shaped magnets were modeled with 4
filaments (2 by 2). The central cell solenoids were modeled with 2
filaments (radially). The choke coils each had 4 filaments (2 by 2). Each
layer of the larg' solenoids was modeled with 2 filaments (axially). Each
filament was brok'n up into 40 sticks for both solenoids and C-shaped
magnets.
The effect o the number of filaments on the calculation is shown in
Figure 2.3. The a:Aal field profile is shown for both the model described
above as well as or a single filament model for each coil. Note that the
peak field under ,he choke coils decreased from 14.7 to 12.7 T. The primary
effect of a finer model is to determine more accurately the peak fields
under the choke a8id C-shaped coils. Since the model was to be used for
calculating induc lances and fields at the coils, the finer model was used.
The fine mod4l was used to construct three force influence coefficient
matrices. Each eliment of these 24 by 24 matrices represented the force on
a coil (or force 1 er unit length at a point) due to another coil. The terms
were per unit curd ent in all coils. Therefore, given a current scenario,
the necessary mul iplication yields the axial force. That is:
24
F(
)=IVk)
Z
f(k,j)*I(j),
J=1
where I(k) is the current in the kth coil. f(k,j) is the force on coil k
per unit current n coil k due to a unit current in coil j. Similar
expressions hold for the radial forces per unit length.
-22-
Magnet
Model of a C-Shaped
x
ROLL
y
E. G. to get f rom
x
R2
-z
z
to
x
0=900
) z
d%
x
PITCH
z
E.G.to get from
x
R1
z
to
x
'=180 0
44w
z
INPUTS C-Shaped:
R,R
2 ,dR,dR2 ,
I
PITCH
R
ROLL
Figure 2.l.a C-Shaped Magnet Model
-23-
Solenoid Input
dz
r
dr
r
zc
VC
___ ___
__i______
INPUTS: zc, rc,dz,dr,
Figure 2.1.b
Solenoid Magnet
-24-
Model
cnr
m
nc
in
ink
0
U"
r_ a
C 0
qr
A
0
0
0
N
-
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-201
-25'L - -- 201 - - +- - -- 10- - +- - - 0- -25
-1-s
5
-5
15
Z
Figure 2. 2
Side View of Multiple Filament Model of MFTFB+T
-26-
25
111,11.1
Islet
aim mess,
seem,
15111 ems am imesmi
'ii.
v
tea,,
I
Sing e Filament Peak
Mult iple Filament Peak,
-4
4
N
ca - -
4
in
-
-30
25[
-20
-3-5
-10
-5
0
5
Single
Filament
10
15
Z(m)
Figure 2.3
Axial Field Profile for Two Models
-27-
Multiple
Filament
20
a,
The ax al force calculations are on a coil basis. That is, the J x B
body force d nsities are integrated over the coil volume to obtain a total
force. The r dial forces, on the other hand, are a local force calculated
on a per uni, running length basis. The integrated radial force is, of
course, zero The radial forces per unit length were calculated at two
points in ea h coil. These points are in the X-0 and Y-0 planes and
correspond t local y-directed and x-directed forces, respectively. In a
purely solen idal system, these forces (per unit length) would be equal
because of t e axial symmetry. The presence of the C-shaped coils perturbs
the solenoid 1 nature of the field and produces nonuniform loadings on the
"-shaped coi s and on the closest solenoids.
The sta ic load analyses discussed in section 3.0 used these matrices
along with a static current scenario vector to determine the forces on the
coils.
An inductance matrix was calculated using the fine model discussed
above. This natrix was used in the transient load calculations along with
the lumped cilcuit model discussed below. The transient currents were
calculated using an eigenvalue/eigenvector expansion for each transient
current case. These expansions were used to determine the current versus
time behavior of each coil. These current versus time behavior then
multiplies thl force influence matrices to get the force versus time. The
maximum and minimum forces for each scenario were calculated throughout the
transient to determine the extreme loads.
The cir uit model for the machine is shown in Figure 2.4. As can be
seen there ar, nine independent circuits. The four coils that form the two
yin-yang pairs at either end of the machine are on individual circuits. The
two transitio (C-shaped) coils at either end are connected across the
machine to th mirror image coil at the other end. The ten solenoids are on
one circuit. "he two choke coils are on a single circuit as are the two
large solenoi s. Also shown are the circuit parameters assumed for the
transient ana yses. The only transients considered for this effort were
full coil sho ts with all other coils discharging through their individual
dump resistor. . Under this assumption the circuits become 24 independent
L-R circuits coupled only through the mutuals inductances from coil to
coil.
-28-
MFTB+T
£IP
CIRCUITS
EM2
WM2
0.085
=4403A
L = 11.I H
Mi
EM
L=ll.IH
TL5 f
0.0444
ETI
i'z
~
I
0.085
10 =3872A
ET2
L = 0.86H
WT I11*=2328
0851
L=2.6 H
W2
1 1s WT2
1
A
.0_
.
-02
ES6
o~ozES5
o.02 *
ES4
90.02
ES3
L =16.4 H
10=8000 A
ETDFS2
o.125
.125T
WTDFS2 H
4.65 x
ES2
0.02
o.o2a
ES I
0.02 9
WSI
0.02)
WS2
0.029
WS3
0.029
WS4
02 9
WS5
1T2
9
L= 5.6 x
10- 4 H
'o = 107'90
4
Lu36HECCI
L = 3.6
H
lo==1823
WS6
Figure 2.4
0-4
MFTFB+T Circuits and Circuit Parameters
-29-
4
-
3.0 MAGNETOSTA IC LOAD ANiALYSES
The first Iset of analyses performed investigated the worst case
loading on the :oils under various static normal operating conditions. A
normal operat in scenario is defined as any combination of one to nine
circuits at full current and the remaining circuits at zero current. The
data obtained fOom this analysis are summarized in Tables 1.2 and 1.3 and
Figures 1.2 and! 1.3.
This secti yn presents the loading for a few of the individual current
scenarios. The cenarios given are: (1) all circuits at full operating
current; and (2 certain circuits at full current with all others at zero
current. These Ire not necessarily the worst case scenarios.
In the fol owing set of figures, the axial force on each coil is
presented in on figure. For scaling purposes, the bars representing the
two large solen ids are clipped and the peak values written. The two radial
forces per unit length are presented in one figure. The radial force in the
X-0 plane i sh wn as a solid line and the force in the Y-0 plane as a
dashed line. For coils with little or no difference in the two forces, the
dashed line is hidden by the solid one. Shown at the top of each figure is a
side view of the machine with the coils numbered. The coils that have zero
current are denoted with an asterisk (*).
-30-
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4.0 TRAMSIET LOAD ALYSES
The transient load analyses consisted of a sequence of runs in which
all coils start from a fully charged condition; each coil from I to 12 (1411
to WTDFS2) was assumed to have a short across the entire coil while the
other coils are assumed to discharge through their individual dump
resistors. The two choke coils are resistive and have an internal
resistance rather than an external dump resistor.
Figure 2.4 shows the nine independent circuits in the machine. For a
discharge scenario, the nine circuits become 24 independent L-R circuits
coupled through the coil to coil mutual inductances. The matrix set of
first order ordinary differential equations to be solved for the unknown
current versus time is given by:
[LJ(I(t))
+
[RJCI(t)) -
(0)
where CU is the system inductance matrix. CRJ is a diagonal matrix of coil
dump resistors. The initial condition for these runs was that each coil is
at full operating current. For a coil short, the dump resistor value in [RJ
is replaced by a small number (0 is unacceptable from a numeric standpoint).
If the terms in the ELI and ER] matrices are constant, the set can be
solved usini3 an eigenvalue/eigenvector approach. The current at any time
can then be found from the eigenexpansion. Figure 4.1 shows the set of
currents vesus time for the normal discharge case. Only the currents in
1-12 are shown since by symmetry the currents in 13-24 are identical. It
can be seen that the current in coil 10 (WS1) initially increases. Because
the force on a coil is a product of a row of the influence coefficient
matrix and the current vector and then multiplied by the coil current, it
is not obvious from the current plot when the peak forces occur. In order
to find the peak forces during each scenario, the eigenexpansion was used
to calculate currents at various times. The forces were calculated at each
of these times and the maximum (positive) and minimum (negative) values
found.
The forces versus time are shown in Figure 4.2 through 4.4 for the
normal discharge case. Only the forces on coils 1-12 are shown. Figure 4.2
shows the peak axial force acting on each coil. It can be seen that certain
coils actually experience a force reversal during the discharge (e.g. coil
#10, WS6). Figure 4.3 shows the radial forces per unit length acting on the
coils in the X=0 plane and Figure 4.4 the radial forces per unit length in
the Y=0 plane (dashed lines).
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Figure 4. I Coil Currents vs Time for Normal Discharge
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Figure 4.2
Coil Axial Forces vs Time for Normal Discharge
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Figure 4.3
Coil Radial Force Per Unit Length (X = 0)
Normal Discharge
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Figure 4.4
Coil Radial Force Per Unit Length (Y = 0 )
Normal Discharge
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One of the more interesting cases was a short circuit across coil 2
(W14). Them! is a slight radial force reversal in coil #3 (14T2) in the X-0
plane. The current versus time for the coils is shown in Figure 4.5. The
net axial load versus time for coils 1-12 is shown in Figure 4.6. The
radial force per unit length in the X-0 plane for each of the coils is
shown as a Punction of time in Figure 4.7. There is indeed a small reversal
for this radial force in coil 3. Figure 4.8 shows the radial force per
unit length in the Y-0 plane for coils 1-12.
The remaining figures in this section are the peak loads during
discharge fOr single coil shorts. The figures have a side view of the
machine with coil numbers superimposed at the top. The coil with the short
is denoted by an asterisk (*). The case number in the plot title
corresponds to the coil with the short. Both the worst case axial forces
and radial forces per unit length are shown. The radial force per unit
length is the X-0 plane is denoted by a solid line; the the force per
unit length in the Y-0 plane by a dashed line. The worst case loadings
across all transient runs are summarized in Tables 1.7 and 1.8 and in
Figures 1.7 and 1.8.
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Coil Currents vs Time for Coil #2 Short-Circuited,
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Coil Axial Forces vs Time for Coil #2 Short-Circuited;
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Figure 4.7 Coil Radial Forces Per Unit Length (X = 0 Plane)
for Coil #2 Short-Circuited; All Others Discharging
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Coil Radial Forces Per Unit Length (Y = 0 Plane)
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5.0 CONCLUSIOI5
A large number of possible normal operating and fault conditions for
the MFTFB+T mirror machine have been investigated to estimate the worst
case coil loadings. The results from these analyses are summarized in
section 1.0. More detailed results are presented in sections 2.0 through
4.0.
In general, the four C-shaped magnets at each machine end (W112, WMi,
WT2,and WT1) experience the worst case loading under static current
conditions. The central cell solenoids (WS6, WS5, WS4, and WS3) experience
peak loads during discharge transients with a neighboring coil shortcircuited. The coils at the machine center (WS2, WSI, WCC1, WTDFS2, etc.)
have peak axial loads that occur primarily in the static condition.
However, the peak radial loads occur during the discharge transients.
The axial loading investigated was the net force on a coil and tells
nothing of the distribution of the load. The radial loading was considered
at two points within the coils as an indicator of when the worst case load
probably occuro. The load conditions are predicated on a certain
coil/circuit configuration. If this configuration changes or if a coil
operating current is altered, the new static and transient load scenarios
should be reconsidered.
The current scenarios chosen for the analyses were by no means
exhaustive from a fault condition standpoint. An entire class of faults
which requires consideration involves system behavior during discharge
execution with a section of one of the superconducting coils in the
resistive state. Even though the coils are designed to be stable under
usual conditions, that is complete LHe coolant immersion, a resistive
region is possible in the event of a fault in the form of a low helium
level. The temperature rise in a resistive region which is not immersed is
strongly dependent on the coupled discharge character of the circuits and
the level of operating current and dump voltage selected.
-64-
PFC BASE MAILING LIST
Argonne National lahoratory, TIS, Reports Section
Associazione EURATOM - CNEN Fusione, Italy, The Librarian
CRPP, Switzerland, Troqon, Prof. F.
Central Research Institute for Physics, Hungary, Preprint Library
Chinese Academy of Sciences, China, The Library
Eindhoven University of Technology, Netherlands, Schram, Prof. D. C
The Flinders University of S.A., Australia, Jones, Prof. I.R.
General Atomic Co., Overskei, Dr. D.
International Atomic Energy Agency. Austria,
Israel Atomic Energy Commission, Soreq Nucl. Res. Ctr., Israel
JET, England, Gondhaltkar, Dr. A.
Kernforschungsanlage Julich, FRG, Zentralbibliothek
Kyushu University, Japari, Library
Max-Planck-Inslitut fur Plasma Physik, FRG, Main Library
Nagoya University, Institute of Plasma Physics, Japan
Physical Research Laboratory, India. Sen, Dr. Abhijit
Rensselaer Polytechnic bistitute, Plasma Dynamics Iab.
South African Atomic Eiergy Board, S. Africa, Hayzen, Dr. A.
UKAEA, Culham Laboratory, United Kingdom, Librarian
Universite de Montreal, Lab. de Physique des Plasmas, Ca..ada
University of Innsbruck, Inst. of Theoretical Physics, Austria
University of Saskatchewan. Plasma Physics Lab., Canada
University of Sydney. Wlls Plasma Physics Dept., Australia
INTERNAL MAILINGS
MIT libraries
Industrial Liaison Office
G. Bekefi, A. Bers, D. Cohn, B. Coppi, R.C. Davidson,
T. Dupree, S. Foner, J. Freidberg, M.O. Hoenig, M. Kazimi,
L. Lidsky, E. Marmar, J. McCune, J. Meyer, D.B. Montgomery,
J. Moses, D. Pappas, R.IL Parker, N.T. Pierce, P. Politzer,
M. Porkolab, R. Post, II Praddaude, D. Rose, J.C. Rose,
R.M. Rose, B.B. Schwartz, R. Temkin, P. Wolff,
T-F. Yang
I