Practical magnet design
1. Iron saturates, when
B  2.14T
•Fields higher than 2.2 T with super conducting magnets
•high I
•
B   0 H  M 
Free currents
(coils)
magnetization 0 M  2.14T
2. For a given field we need a certain current (Ampereturns)
•Can be approximated
•Final values from computer codes (FEMM,
POISSON, TOSCA, etc.
Current in the coils
   B  0 M
h
B  0 H
e.g. B = 2 T, h = 20 cm, NI = 320000 At
  B
NI   H  dl  h
0
Coil parameters
•NI = (NI)needed, but N = ?, I = ?
•i.e. which conductor?
Dissipated power (heat) P = I2R
•Must be removed from the system
•Air cooling
•Indirect cooling
•Direct water cooling
•Hollow conductor
Relatively ineffective
Coil cooling
•The dissipated electric power in converted into heat (water
temperature).
•The temperature rise rise between input and output is
 
T  0.06 I 2 R / 4.2q
where
T = temperature rise [K]
I = electric current [A]
R = coil resistance [W]
q = water flow [l/min]
Usually
Tin  20 oC
Tout  100 oC
Often the requirement is
T  20 oC
The resistivity of copper
Cu  1.72 108 Wm (cold )
 1.95 108 Wm (T  50C )
R l

A
So: The smaller the T, the larger q is needed
For a large q a large pressure difference (pressure gradient) is
needed
Semi empirically

P  l 8qa
where
P = pressure difference [bar]
q = water flow [l/min]
a = flow cross section [mm2]
l = flow length [m]

1.334 1.79
Note!
q  I2
 P  I 3.58
So, the desired pressure increases very rapidly with
electric current
Usually P  7  8 bar
 15 bar 
If the desired pressure becomes too high the cooling circuit
can be divided into several parallel circuits
(pressure decreases as 1/n2)
I2 R
T
q
T
q
q
n
q
n
q
n
n parallel cooling circuits
(electrically in series)
q
n
q
n
•Division into parallel cooling circuits is done e.g. with a
double pancake structure
•“wind” starting from the outer radius into the inner
radius, come back along the next layer
•These double pancakes are connected in series
electrically and in parallel
Note: The conductivity of the cooling water
must be low 
ion exchanged water,
closed cooling circuit, heat exchangers
Hollow conductor
R
f
1
R 2   R 2
4
 2  2
ACu  d  f  4 R  R 
4
4 

d 
2
Cu
d

2

4
2
f 2  R 2  4 R 2
380 mm
Example: K130 cyclotron
< 400 mm
parallel
20 layers (axially)
per coil
10 turns (double conductor)
Desired NI = 400.000 At
In one conductor pair 400.000 At/(2x200 t) = 1000 A
In one conductor: 500 A
Conductor pair is connected in parallel also
electrically
Length of a double pancake
l  2  N  2 r
 2 10  2 1.435m
 180m
Conductor parameters
d = 17 mm
= 7 mm
R = 1 mm
A = 250 mm2
Conductor resistance (double pancake/ one conductor)
Rdpc  180m 1.95 10-8 Wm/250 10-6 m 2
 14 mW
for one conductor
Dissipated power:
Pdpc  I 2 R  500A  14mW  3.5 kW
2
P2 coils  2  2 10  3.5 kW  140 kW
For a hot coil
Pnominal = 130 kW
Cooling
We allow T = 20 oC
Hence the cooling water flow:
 
0.06 I 2 R 0.06  3.5 103
q

 2.5 l/min
T  4.2
20  4.2
Required pressure
drop
for one conductor/dpc



P  180   8  2.5    7 2 

4


 6.3 bar
1.334 1.79




bar
Both coils consist of 10 double pancakes (dpc)
Each dpc has two conductors in parallel (both
electrically and for cooling)
40 pieces of 180 m long cnductors
Total flow = 40 x 2.5 l/min = 100 l/min
So: the cooling system must be capable of producing about 6.5
bar pressure difference at a 100 l/min flow rate in order to keep
the temperature increase below 20 oC.
Note: the cooling system has to cool other devices as well!
The conductor was of type
What would happen, if it was replaced with a conductor
where Acu and Awater are the same as well as the conductor
outer dimensions?


The flow rate in the cooling hole would be doubled
and the cross section would be doubled
Change in the pressure drop

P  qa 1.334



1.79

P2
2q  2a 

1.79
P1
q  a 1.334

1.334 1.79



 22

1.334 1.79
 0.66
Would be beneficial to use one hole with the same
total cross section
But: winding would be difficult
r
Alternatively we could use a square conductor with the same
cross section
•Number of turns would increase and the conductor
length as well
•Higher P
•Also the resistance of one dpc would increase
More flow
Higher pressure
Let us calculate this as an example:
Conductor dimensions
2 17 f 2  7
=
24 / f 9.9
In one coil 400.000 At/2 = 200.000 At
Keep the current density unaltered
 I = 2 x 500 A = 1000 A
 N = 200
Full double pancakes  N = (2 x 7 ) x 15 = 210
 I = 952 A
i.e. 7 double pancakes (15 turns)
The conductor is insulated 


diameter 26 mm
r= 390 mm
r  1.240  0.390 / 2m
 Ldpc  30  2   1.435m  270m


 Rdpc  270 1.95 108 / 2  250 106 W
 10.53 mW
Power dissipation for a double pancake
Pdpc  10.53 103  9522 W  9.54kW
Total power
P  2  7  9.54kW  134kW
Water flow rate
0.06  9.54 103
qdpc 
l / min  6.81 l/ min
20  4.2
Pressure drop



P  270   8  6.81  9.92 

4


1.334 1.79




bar  10.8 bar
Starts to on the limit!
In both cases

Ptot  135 kW and
Power supply: 1000 A/140 V
I  1000 A
Power supply Current/Voltage
These have also some limitations
Current must be taken into account when dimensioning the
transport cables from the PS to the magnet
The dimensioning (f) depends naturally upon the heat transfer
properties of the surroundings
Typically power dissipation (heat) can be < 10 W/m
e.g. Acu = 400 mm2 and I = 425 A 
P/L = 7.8 W/m
Note:

ACu  I 2
For I = 1000 A, Acu = 1600 mm2 (7.8 W/m)
4 cm
4 cm
In practice the feeding cable is divided into several thinner
cables
More cooling surface/unit power loss
More flexible (mounting)
For example: K130 cyclotron
Inominal = 1000 A, Imax = 1170 A
4 x 150 mm2 cable
Maximum current 320 A/cable (depends on the
assembly – cooling circumstances)
Note also:
For high currents the transfer losses (Voltage) may be
significant in a long transfer cable
The Power Supply sees both the load and the transfer line
Important note: choose the hollow conductor dimensions from the
manufacturer’s stock list – otherwise you may be surprised by the
price!
RECTANGULAR WITH A ROUND HOLE
Dimensions in mm
NO.
OD.
ID.
x/y
R
Kg/m
8329
8330
8356
8357
8290
8327
8300
8314
8370
8342
8115
6861
8364
8521
6833
8280
6864
8260
8331
6879
8160
8328
8103
8287
8399
4x3
4.3x4
4.67x4.19
5.08x4.77
5.8x3.7
6x4
6x5
6x5.5
6.1x4
7x5
7x6
7x6.3
7x8.5
7.2x4.2
7.5x5
7.5/7x6
7.5x6
7.5x7
7.5x8
7.6x7.2
7.8x6.25
8x6
8x6
8x6
8x6.7
2
2.5
2.41
2.64
2.5
2.5
3
3.5
3.2
3
4
3.4
4
2.5
3
3.3
3.5
5.2
4
3.5
2.5
3.2
3.2
4
4.5
1/0.5
0.9/0.75
1.13/0.89
1.22/1.065
1.65/0.6
1.75/0.75
1.5/1
1.25/1
1.45/0.4
2/1
1.5/1
1.8/1.45
1.5/2.25
2.35/0.85
2.25/1
1.98/1.35
2/1.25
1.15/0.9
1.75/2
2.05/1.85
2.65/1.87
2.4/1.4
2.4/1.4
2/1
1.75/1.1
0.8
0.8
0.76
0.76
0.8
0.8
1
0.8
0.6
0.6
1
1
1
1
0.5
1
1
1/0.25
1
1
0.5
0.8
1
1
1
0.070
0.100
0.130
0.163
0.143
0.170
0.200
0.200
0.143
0.250
0.256
0.304
0.412
0.218
0.270
0.310
0.308
0.275
0.420
0.395
0.389
0.350
0.350
0.309
0.329
8270
8279
8242
8513
6863
6862
6857
8269
8448
8195
8518
8192
6146
8363
8348
8362
6145
6161
8391
6812
8258
8372
8133
8355
8236
8166
8392
8436
8413
8259
8x6.9
8x7
8x7
8x7.4
8.2x7.2
8.2x7.2
8.5x5.5
9x7.5
9x8.5
9.07x8
9.3x8
9.4x8.15
9.5x7.1
9.5/9.1x8
9.5/9.1x8
9.5x8/7.4
10x7.1
10x7.8
10x9
10.5x8.2
10.6x7
10.7x8
10.7x8
10.77x9.95
11x7
11x7
11/10.4x7
11x8
11x8.8/8.6
11x9.5
3.5
3.5
4
4.6
3.8
5
2.7
5
5
4.77
6
5.05
3.5
3.2
5
5
3.9
4.5
4
4.8
4
4.25
5.8
5.82
3
5
5
4
5
5
2.25/1.7
2.25/1.75
2/1.5
1.7/1.4
2.2/1.7
1.6/1.1
2.9/1.4
2/1.25
2/1.75
2.15/1.61
1.65/1
2.17/1.55
3/1.8
3.15/2.4
2.25/1.5
2.25/1.5
3.05/1.6
2.75/1.65
3/2.5
2.85/1.7
3.3/1.5
3.225/1.875
2.45/1.1
2.475/2.065
4/2
3/1
3/1
3.5/2
3/1.9
3/2.25
1
1
1
1
1
1
1
1
1
0.79
1
0.8
1
1
1
1
1
1
1
1
1
1
1
1.25
1
1
1
2
1
1
0.400
0.410
0.381
0.370
0.417
0.343
0.360
0.420
0.500
0.484
0.400
0.500
0.512
0.593
0.490
0.484
0.520
0.540
0.685
0.590
0.543
0.631
0.520
0.708
0.620
0.510
0.497
0.640
0.677
0.750
6144
8442
8444
8266
8298
8507
8527
11x9.8
11x10
11.3x7.8
11.5x9.5
11.8x10.3
11.9x8.58
12x9
6.5
5
4.82
6.7
6
3.96
5
2.25/1.65
3./2.5
3.24/1.49
2.4/1.4
2.9/2.15
3.97/2.31
3.5/2
1
1
1.25
1
1.2
0.8
0.5
0.650
0.800
0.613
0.654
0.820
0.798
0.788
8144
6893
8212
6801
8177
8274
8511
6751
8289
8307
8116
8253
8495
6824
8378
8528
8412
8461
8500
6180
8233
6800
6132
12x9.5
12x10
12x11
12.3x11.8
12.6x10
12.7x6.35
12.7x10.66
12.7x12.2
13x11
13.2x9
13.5x8.25
13.5x11.5
13.5x12.5
13.8x13.1
14x7.6
14x8
14x11.9/11.5
14x12.5
14x12.6
14x12.6
14.3x11.4
14.4x12.3
14.9x11.9
6.5
5
7
7.4
5
3.81
6.6
5.6
6
5
5.1
6.3
9.8
7.9
4.5
5
5
5.86
6
8
6
6.5
6.5
2.75/1.5
3.5/2.5
2.5/2
2.45/2.2
3.8/2.5
4.445/1.27
3.05/2.03
3.55/3.3
3.5/2.5
4.1/2
4.2/1.58
3.6/2.6
3.7/2.7
2.95/2.6
4.75/1.55
4.5/1.5
4.5/3.45
4.07/3.32
4/3.3
3/2.3
4.15/2.7
3.95/2.9
4.2/2.7
1.5
1
1
1
1
1
1.25
0.79
1
1
1
1
1
1
1
0.5
1
1
1
1
1.5
1
1
0.700
0.890
0.830
0.900
0.940
0.611
0.892
1.160
1.018
0.880
0.810
1.100
0.827
1.160
0.801
0.824
1.292
1.320
1.317
1.120
1.200
1.270
1.300
8482
8305
8142
8219
8180
6802
8427
6140
8322
8497
6896
6822
8351
8483
8179
6181
8451
8455
6160
8181
8278
6855
8394
6890
8460
8190
8429
8170
8196
8221
6832
15x8
15/14x8
15x8
15x9.5
15x10
15x11.8
15/14.2x11.8
15x12.5
15x13
15x13.7
15.2x12.75
15.4x10
16x10
16x14
16x14
16x14.5
16/15.2x15
16.2x10
16.3x10.8
16.5x8.9
16.5x12.5
16.5x12.5
16.5x12.5
16.75x8.6
17x14.5
17x15
17x16
17x16
17.4x10.7
18x8
18x10
5
4.5
4.5
6.5
5
6.5
6.5
7
6.1
8
8
4
5
6.5
7
10
9
4.77
6.6
5.2
7.35
8.2
8.2
5.5
10
6.4
6
9
6.1
4
6
5/1.5
5.25/1.75
5.25/1.75
4.25/1.5
5/2.5
4.25/2.65
4.25/2.65
4/2.75
4.45/3.45
3.5/2.85
3.6/2.38
5.7/3
5.5/2.5
4.75/3.75
4.5/3.5
3/2.25
3.5/3
5.715/2.615
4.85/2.1
5.65/1.85
4.58/2.58
4.15/2.15
4.15/2.15
5.63/1.55
3.5/2.25
5.3/4.3
5.5/5
4/3.5
5.65/4.6
7/2
6/2
1
1
1
1
1
1
1.75
1.5
1
2
1.5
2
1.5
1.5
1
1
1
1
1.5
1
1
0.5
0.5
1
2
1.5
1.5
1.5
1.5
0.5
0.890
0.910
0.920
0.970
1.160
1.270
1.256
1.300
1.460
1.380
1.250
1.250
1.220
1.690
1.640
1.360
1.543
1.260
1.240
1.120
1.460
1.360
1.370
1.080
1.494
1.960
2.160
1.840
1.385
1.160
1.340
8400
6888
8336
8428
8439
8232
8326
6806
8169
8508
8122
8453
8167
8517
8498
8776
8506
8430
8119
6825
8271
8473
8465
8146
8189
8478
6859
8148
8117
8403
8335
8315
8359
18x11.2
18x12
18x12.65
18/17.2x14
18x14
18x15
18x15
18.54x10.46
18.9x11.4
19x12.5
19x13
19x16
19.05x7.92
19.05x15.88
19.05x15.88
19.05x15.88
19.3/18.3x15.5
20x10
20x12
20x12
20x14
20x15
20.1/19.1x19.6
20.2x15
20.4x13.4
20.5x18
20.5x18
20.83x15.75
20.9x15.65
20.9x20.5
21x12
21x16.5
21x18
6.6
8
9.5
10
10
8
8.5
4.96
7
9
8
7
4.75
8
10.80
12.7
10.5
6.5
8
9
6
8
11
10
8
7
8.5
6.35
7
9.5
5
7.7
9
5.7/2.3
5/2
4.25/1.58
4/2
4/2
5/3.5
4.75/3.25
6.79/2.75
5.95/2.2
5/1.75
5.5/2.5
6/4.5
7.15/1.59
5.525/3.94
4.13/2.54
3.18/1.59
4.4/2.5
6.75/1.75
6/2
5.5/1.5
7/4
6/3.5
4.55/4.3
5.1/2.5
5.7/2.7
6.75/5.5
6/4.75
7.24/4.7
6.95/4.33
5.7/5.5
8/3.5
6.65/4.4
6/4.5
1.5
1
1
1
1
2
2
1.52
1.5
1.5
1
1
1.59
2
4.75
4.75
2
2
1
1
2
1
2.5
1.5
1
1
1
2
1.5
1.5
1
1
1.5
1.480
1.480
1.390
1.520
1.534
1.930
1.880
1.540
1.560
1.540
1.750
2.400
1.170
2.220
1.700
1.400
1.845
1.460
1.690
1.560
2.220
2.230
2.580
1.990
1.860
2.950
2.780
2.620
2.570
3.180
2.070
2.670
2.790
8199
8388
8410
8524
8352
8421
6895
8382
6811
8476
8164
8284
8462
8198
8109
8262
8210
8367
8312
8218
6803
8126
8440
8191
6882
8320
8457
8514
21x18.2
21.21x12.67
21.5x18.5
21.5x18.5
22x12.24
22x18.9/17.7
22.2x18.3
22.22x15.88
22.5x17.8
22.7/19
22.86x16.51
23x12
23x14
23x15
23x15
23x17.8
23x20
23x21.9
23.01x18.87
23.5x13.98
23.5x16
24x11
25x15
25x17
25x20
25x21
25x21
25x23
7.45
9.52
8
8.5
6.35
8
11
7.92
13
8
11.13
7.5
10
6.55
8
11.8
8
6.5
12.3
10
9
7.5
10
6.5
11.5
9.1
14.5
15.5
6.77/5.37
5.845/1.575
6.75/5.25
6.75/5
7.825/2.945
7/5.45
5.6/3.65
7.15/3.98
4.75/2.4
7.35/5.5
5.86/2.69
7.75/2.25
6.5/2
8.22/4.22
7.5/3.5
5.85/3
7.5/6
8.25/7.7
5.36/3.29
6.75/1.99
7.25/3.5
8.25/1.75
7.5/2.5
9.25/5.25
6.75/4.25
7.95/5.95
5.25/3.25
4.75/3.75
1.5
1.6
1
1
1.52
2
0.8
1.7
2
2
1.2
2
1.5
1.5
1.5
1.5
1.5
1.5
1.2
1
2.5
1
1
1.5
1
2
3
2
3.000
1.747
3.099
3.04
2.110
3.110
2.790
2.692
2.320
3.380
2.490
2.040
2.159
2.780
2.600
2.740
3.650
4.190
2.810
2.230
2.730
1.960
2.643
3.470
3.520
4.080
3.150
3.420
8184
8343
8398
8274
8152
8425
8104
8431
8162
8161
8332
8321
8243
6874
6876
6868
8466
6837
8263
8432
6839
8499
8443
8520
6816
8433
8145
8296
25x24
25.4x14.35
25.4x14.35
25.4x19.05
25.5x18.3
26x18
26.5x23.7
27x22.5
27.84x23.42
27.84x23.42
28.1x20.1
29x19.5/19
30x22
30x22
30x23
30.3x23.7
31.75x27.94
32x22.5
32.5x29
33x17
33x24
33.78x30.63
34x27
36x16
37x23
37x24
37x28
38x35
12
6.35
7.92
12.7
9
6
13.5
13.5
8.64
11.3
6.1
9.8
13
14
8
12
19.05
10.5
11
10.5
6.5
12.7
19
10
8
8
22
12.7
6.5/6
9.525/4
8.74/3.215
6.35/3.18
8.25/4.65
10/6
6.5/5.1
6.75/4.5
9.6/7.4
8.27/6.06
11/7
9.6/4.73
8.5/4.5
8/4
11/7.5
9.15/5.85
6.35/4.445
10.75/6
10.75/9
11.25/3.25
13.25/8.75
10.54/8.965
7.5/4
13/3
14.5/7.5
14.5/8
7.5/3
12.65/11.15
1
1.6
1.6
1.6
1.5
1.5
1
2.5
1.58
1.58
2
2
1
5
2
1.5
4.76
1.5
1
3
1.5
2.38
4
1
1.5
1.5
1.5
3.5
4.340
2.960
2.798
3.170
3.590
3.920
4.330
4.103
5.280
4.920
4.760
4.290
4.700
4.330
5.680
5.400
5.200
5.650
7.570
4.175
6.770
8.050
5.540
4.440
7.100
7.470
5.840
10.66
8404
6804
6871
8505
8201
8107
8268
8333
8267
8240
8120
8316
8295
8108
8308
8515
8250
8375
8137
8297
8434
8226
8114
8360
8168
8358
8273
8265
8101
8422
8275
8123
39x31
39.37x22.86
40x12
40.5x22.2
41x23
42x28
42.2x30.7
43.1x14.6
43.2x24.7
43.5x33
45x14
47.1x41
47.7x38
48x15
48x19
48x24
48x38
50x10
50x35
50.8x22.2
52x21
52.2x24.5
53.8/53.2x39.5
55x22
60.3x25.4
60.32x25.4
65x47
75x32
82x35
82x35
85x35
95x40
9
8.89
7.5
9.5
8
8
8
6.8
8
15
10
13.5
12.7
8
11
14
8
5
12
12.7
9
11.11
15
9.8
15.9
12.7
10
8
12
15
8
15
15/11
15.24/6.99
16.25/2.25
15.5/6.35
16.5/7.5
17/10
17.1/11.35
18.15/3.9
17.6/8.35
14.25/9
17.5/2
16.8/13.75
17.5/12.65
20/3.5
18.5/4
17/5
20/15
22.5/2.5
19/11.5
19.05/4.76
21.5/6
20.54/6.69
12.25/19.4
22.6/6.1
22.2/4.75
23.81/6.35
27.5/18.5
33.5/12
35/11.5
33.5/10
38.5/13.5
40/12.5
2
2.54
1
3
2.5
2
2.5
2
2.5
3
2
2
3.5
5
1.5
2
2
1.5
5
3.5
1.5
2.38
2
2.5
2
3.17
4
2
2
2
5
3
10.21
7.400
3.890
7.340
7.940
10.00
11.08
5.270
9.040
11.20
4.190
15.95
14.97
5.810
7.290
8.890
15.82
4.280
14.46
8.860
9.190
10.52
17.28
10.10
11.89
12.49
26.50
20.97
24.60
24.00
26.00
32.30
SQUARE WITH A ROUND HOLE
Dimensions in mm
NO.
OD
ID
x=y
R
Kg/m
MARGIN Eur/kg
6844
8529
8350
8301
8153
8206
8150
6867
8215
6831
8416
6881
8339
8235
8283
6860
6835
8408
8204
8405
8417
8149
8424
6873
8385
6858
8325
6869
8156
8125
8132
4
4
4.19
4.3
4.5
4.62
4.76
5
5
5
5.6
5.6
5.79
6x6/5.6
6
6
6
6x6/5.6
6
6.15
6.35
6.35
6.40
6.4
6.48
6.5
6.5
6.5
6.5
6.5
6.7
2.5
2.0
2.4
2.34
2.5
2.39
3.18
2
2.5
3
3.05
3.6
3.18
3
3
3.5
4
4
4.5
2.54
3.15
4.75
3
4.4
3.18
3
3.1
3.5
4
4.5
4
0.75
1
0.895
0.98
1
1.11
0.79
1.5
1.25
1
1.275
1
1.305
1.5
1.5
1.25
1
1
0.75
1.805
1.6
0.8
1.7
1
1.65
1.75
1.7
1.5
1.25
1
1.35
0.5
0.5
0.8
0.76
1
0.76
1
1
1
0.7
1
1
1
1
1
0.8
1
1
1
0.635
1
1.27
1
1
1
1
0.8
1
0.5
1
1
0.098
0.113
0.110
0.120
0.136
0.150
0.124
0.179
0.172
0.157
0.210
0.183
0.220
0.250
0.250
0.230
0.200
0.197
0.173
0.290
0.280
0.190
0.300
0.230
0.297
0.300
0.310
0.280
0.260
0.220
0.280
11,80
10,40
10,00