Dynamic behavior of the S2C2 magnetic circuit
FFAG13 September 2013
Wiel Kleeven
The New IBA Single Room Proton Therapy Solution: ProteusONE
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Synchrocyclotron with superconducting coil:
S2C2
- 2 -
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Patient treatment room
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S2C2 overview
General system layout and parameters
A separate oral contribution on the field mapping of the S2C2 will be given by Vincent
Nuttens (TU4PB01)
Several contributions can be found on the
ECPM2012-website
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Overview
Some items to be adressed
4.
5.
2.
3.
6.
7.
8.
9.
1.
Goal of the calculations
Different ways to model the dynamic properties of the magnet
What about the self-inductance of a non-linear magnet
Magnet load line and the critical surface of the super-conductor
Transient solver: eddy current losses and AC losses
A comparison with measurements
Study of full ramp-up/ramp-down cycles
Temperature dependence of material properties
A multi-physics approach and a qualitative quench model
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Efforts to learn more on the superconducting magnet
Coil and cryostat designed and manufactured by the Italian company ASG
For the coming years, the proteus®one and as part of that, the S2C2, will be the number®one workhorse for IBA
Succes of this project is essential for the future of IBA
A broad understanding is needed to continuously improve and develop this new system
The S2C2 is the first superconducting cyclotron made by IBA.
The superconducting coil was for a large part designed by ASG but of course by taking into account the iron design made by IBA/AIMA. This was an interactive process
For us many things have to be learned, regarding the special features of this machine.
Some items under study now, or to be studied soon are:
1.
2.
Fast warm up of the coil for maintenance
Cold swap of cryocoolers for maintenance
The present study on the dynamics of the magnet must be seen as a learning-process and any feedback of this workshop is very welcome
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Different models for the S2C2 magnetic circuit
3.
4.
1.
2.
5.
Opera2D/Opera3D static solver
Opera2D transient solver
Opera2D transient solver coupled to an external circuit
Semi-analytical solution of a lumped-element circuit model
Multi-physics solution of a lumped element circuit with temperaturedependent properties
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Magnetic circuit-modeling
OPERA3D full model with many details
Long and tedious optimization process
Yoke iron strongly saturated
Influence of external iron systems on the internal magnetic field
Stray-field => shielding of rotco and cryocoolers
pole gap < => extraction system optimization
Influence of yoke penetrations
Median plane errors
Magnetic forces
ITERATIVE PROCESS WITH STRONG
INTERACTION TO BEAM SIMULATIONS
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The static Opera2D model
What information can we obtain
1.
2.
3.
The magnet load line with respect to the superconductor critical surface
Magnetic field distribution on the coil
Maximum field on the coil vs main coil current
Compare with critical currents at different temperature
The static self-inductance of the magnet
From stored energy
From flux-linking
The dynamic self-inductance of the magnet
Essential for non-linear systems like S2C2
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What do we get from Opera2D static solver
Load line relative to critical surface maximum coil field Magnet load line and critical currents (from ASG)
7
6
5
4
3
2
1
0
0
S2C2 Field in the center and maximum field on the coil
B_tot_center
B_iron_center
B_max_coil
100 200 windings/coil=3145
300 400 500
PSU current (Amps)
600 700 800 maximum coil field during ramp up
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The static self-inductance of the magnet
1.
The static self-inductance of the magnet
From the stored energy: 𝑈 =
1
2
L 𝐼 2
From flux-linking:
Flux for a single wire in the coil:
Relation with vector potential:
Total flux over coil: 𝜙 = 2𝜋
0 𝑟 𝑟𝐴 𝜃
=
0 𝑟 𝑟𝐵 𝑟𝐵 𝑧 𝑧 𝑟 𝑑𝑟 𝑟 𝑑𝑟 𝜙 𝑡𝑜𝑡
=
2𝜋𝑁
𝐴 𝐴(𝑐𝑜𝑖𝑙) 𝑟𝐴 𝜃 𝑑𝑟𝑑𝑧
Self of one coil from flux-linking: 𝐿 =
2𝜋𝑁
𝐴𝐼 𝐴(𝑐𝑜𝑖𝑙) 𝑟𝐴 𝜃 𝑑𝑟𝑑𝜃
2 nd method allows to find difference between upper and lower coil
Can be calculated directly in Opera2D
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Self-inductance from stored energy
Calculated with Opera2D static solver
S2C2 Stored energy and self-inductance
16
14
12
10
8
6
4
2
0
0 100 stored energy static self windings/coil=3145
200 300 400 500
PSU current (Amps)
600 700
400
100
50
800
00
350
300
250
200
150
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Static self from flux-linking
Asymmetry may induce a quench? => probably not;
D
V=0.3 mV is too small
Small vertical symmetry in the model
Introduces a voltage difference between upper and lower coil during ramp
10
8
6
Asymmetry in self-inductance of upper and lower coil
400
L(upper coil)-L(lower coil) self windings/coil=3145
320
240
0.3 mV
160 4
2
0
0
80
100 200 300 400 500
PSU current (Amps)
600 700 800
0
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What do we get from the 2D transient solver?
Eddy currents and related losses
Current density profiles Losses
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
Losses in former, cryostat walls and yoke iron during a ramp
50 ramp-rate=2.7 Amps/min
100
Magnet current (Amps) former iron yoke cryo-walls
150 200
Apply a constant ramp rate of
2.7 Amps/min to the coils
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Eddy current losses during ramp up and quench
1.
2.
During ramp-up
Eddy current losses in the former (max about 1.5 W) are important because they contribute to the heat-balance
Losses in iron and cryostat walls are (of course) negligible
During a quench
When current decay curve is known, losses in former, iron and cryostat walls can be calculated with OPERA2D transient solver
In the former: up to 15 kWatt
In the iron: up to 8 kWatt
The yoke losses help to protect the coil
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Opera2d transient solver coupled to external circuit
PSU drive programmed as in real live
Cyclotron `impedance ´ is calculated in real time by the transient solver
Circuit currents are calculated in real time by the Opera2D-circuit solver
Allows to study full dynamic behaviour of the magnetic circuit during ramp up
Quench study is of qualitative value only and has not been done in Opera2D
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The full ramp-up/ramp-down cycle
Default PSU-ramping for the S2C2
Used in the OPERA2D external circuit simulations
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A full ramp-up and ramp-down cycle
Coil current compared to dump current coil
Dump (x10)
It is seen that for a given
PSU current the magnetic field in the cyclotron is different for ramp-up as compared to ramp-down
This is due to the fact the dump-current changes sign when ramping down
Higher coil currents in down ramp
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Tierod-forces during ramp-up and ramp-down
Seems to be in agreement with previous slide
0
-10
-20
-30
-40
-50
-60
200
Fx
Fy total
Horizontal forces on cold mass (040613)
300 400 500
PSU-current (Amps)
600
10
700
0
30
20
60
50
40
Larger forces during down ramp
However:
Current split between dump and coil can not explain completely the difference in forces iron hysteresis also seems to play an important role
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AC losses during ramp-up
From Martin Wilson course on superconducting magnets
Hysteresis losses (W/m 3 )
𝑃 𝑓
= 𝜆 𝑠𝑢𝑝
2
3𝜋
𝐽 𝑐
(𝐵)𝑑 𝑓 𝑑𝐵 𝑑𝑡
Coupling losses (W/m 3 )
𝑃 𝑒
= 𝜆 𝑤𝑖𝑟𝑒
( 𝑑𝐵 𝑑𝑡
) 2 𝜌 𝑡 𝑝
(
2𝜋
) 2
Tool developed in Opera2D-Transient solver that integrates above expressions in coil area
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J c
(B) d f l sup l wire r t p dB/dt
=> critical current density
=> filament diameter
=> fraction of NbTi material
=> fraction of wire in channel
=> resitivity across wire
=> pitch of the wire
=> B-time derivative in coil
Critical surface => Bottura formula
Needed for AC losses calculation
Bottura formula
𝐽 𝑐
=
𝐶
0
𝐵 𝑏 𝛼 (1 − 𝑏) 𝛽 (1 − 𝑡 𝑛 ) 𝛾
𝑇 𝑡 = (reduced temperature)
𝑇 𝑐0
𝐵 𝑏 =
𝐵 𝑐2
(𝑇)
(reduced field) critical field at zero current
𝐵 𝑐2
𝑇 = 𝐵 𝑐20
1 − 𝑡 𝑛 a,b,g
,C
0
=> fitting coefficients
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Critical surface => Bottura formula (2)
7
6
5
4
3
2
1
0
0 2
Critical surface at T=4 K (Bottura-formula)
Normalized to unity at 5
Tesla/4.2 K
Spencer
Somerkoski
Green
Morgan
Hudson
4 6
Magnetic field (Tesla)
8 10 12
16
14
12
10
8
6
4
2
0
0
Critical field as function of temperatue at zero current
(Bottura)
Spencer
Somerkoski
Green
Morgan
Hudson
1 2 3 4 5 6
Temperature (K)
7 8 9 10
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Critical surface => S2C2 wire
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
0
Critical surface of S2C2 wire (3500 Amps @ 5Tesla/4.2 K)
2 4 6 8
Magnetic field (Tesla)
10
T=3 K
T=4K
T=5 K
T=6 K
T=7 K
12 14
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AC losses obtained with OPER2D transient solver
Initial results => maybe can be improved
Hysteresis losses somewhat larger than eddy current losses
Coupling losses very small
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A lumped element model of the circuit
Turns out to give very good predictions 𝑑𝐼 𝑐 𝑑𝑡
=
𝑅 𝑑
𝐿
𝐼 𝑝 𝑐
− 𝐼 𝑐
1 +
1
𝑁 2
+
𝑅 𝑑
𝑅 𝑓
𝐿 𝑐
𝑁 𝑑𝐼 𝑑𝑡
2
(1 +
− 𝑅
𝑅 𝑐
𝑅 𝑑
)
𝑅 𝑑
𝑅 𝑓 𝑐
𝐼 𝑐
SOLVED IN EXCEL
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Primary circuit
PSU
Coil self-inductance
Coil resistance (only with quench)
Dump resistor
Secondary circuit
Former self-inductance
Former resistance
Perfect mutual coupling (k=1)
Ideal transformer
Compare both models with experiment
Voltage on the terminals of the coils during ramp-up
Blue: measured
Black:OPER2D transient-circuit model
Red: analytical lumped element model
• Perfect match with
OPERA2D
• Not a good match with lumped element model
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The concept of dynamic self-inductance
Important for non-linear magnets
Definition of self-inductance: 𝜙 = 𝐿𝐼
Faraday’s law: 𝑑
Combine: 𝑉 = 𝑑𝑡
𝐿𝐼 = 𝐿 𝑑𝐼 𝑑𝑡 𝑑𝜙
𝑉 =
+ 𝐼 𝑑𝐿 𝑑𝑡 𝑑𝑡 𝑑𝐼
= 𝐿 𝑑𝑡
+ 𝐼 𝑑𝐿 𝑑𝐼 𝑑𝐼 𝑑𝑡
= 𝐿 𝑑𝐼 𝑑𝑡
(1 +
𝐼
𝐿 𝑑𝐿
) 𝑑𝐼
For a non-linear system the dynamic self must be used in lumped element circuit simulations
𝐿 𝑑𝑦𝑛
𝐼 𝑑𝐿
= 1 +
𝐿 𝑑𝐼
𝐿 𝑠𝑡𝑎𝑡
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S2C2 self-inductance
A large difference between static and dynamic self
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Compare both models with experiment
Voltage on the terminals of the coils during ramp-up
Blue: measured
Black:OPER2D transient-circuit model
Red: analytical lumped element model with static self
Green: analytical lumped element with dynamic self
An almost perfect match is obtained
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Compare both circuit-models
Resistive losses in the former during ramp-up
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Blue:OPER2D transient-circuit model
Red: analytical lumped element model
Very good agreement between both models
- 29 -
Further applications of lumped element model
Introduce a kind of « multiphysics »
Since this simple model works so well: can we push it a little bit further?
Resistors in model become temperature-dependent
Introduce additional equations for temperature change
𝑑𝑇 𝑑𝑡
𝑅(𝑇)𝐼
2
= 𝑚𝐶 𝑣
(𝑇)
= ℎ𝑒𝑎𝑡 𝑠𝑜𝑢𝑟𝑐𝑒 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 𝑐ℎ𝑎𝑛𝑔𝑒
R(T) => resistance => 𝑅 𝑇 = 𝜌 𝑇 𝐿
𝐴 r
(T) => resistivity
C v
(T) => specific heat
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Specific heat of copper and aluminium
Very accurate fitting is possible
𝐶 𝑝
= 𝑎 + 𝑏𝜃 + 𝑑𝜃 2
+ ⋯ + 𝑖𝜃 8 𝜃 = log(𝑇)
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Electrical resistivity of copper and aluminium
Same kind of fitting is possible
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A qualitative model for quench behavior
Based on (« multi-physics ») lumped element model
Five different zones with four different temperatures in the cold mass
1.
2.
Upper coil superconducting zone (T
0
)
Upper coil resistive zone heated by resistive loss (T
1
) expanding due to longitudinal and transverse quench propagation
3.
4.
5.
Resistive former heated by eddy current losses (T
2
)
Lower coil superconducting zone (T
0
)
Lower coil resistive zone heated by resistive loss (T
3
) expanding due to longitudinal and transverse quench propagation
Start quench in upper coil
Lower coil will quench when former temperature above critical temperature
ADIABATIC APPROXIMATION => no heat exchange between zones
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Model for quench propagation
From Wilson course
Introduce the fraction f=f l
*f t
1.
f l of the coil that has become resistive
=> Longitudinal propagation (fast
10 m/sec): 𝑑𝑙 𝑑𝑡
= 𝑣 𝑙𝑜𝑛𝑔
⇒ 𝑣 𝑙𝑜𝑛𝑔
𝐽
= 𝛾𝐶 𝑣
𝐿
0 𝜃
0 𝜃 𝑡
− 𝜃
0
2.
f t
=>Transverse propagation (slow
20 cm/sec): 𝑑𝑟 𝑑𝑡
= 𝑣 𝑡𝑟𝑎𝑛𝑠
⇒ 𝑣 𝑡𝑟𝑎𝑛𝑠
= 𝛼𝑣 𝑙𝑜𝑛𝑔
J => current density
G => mass density
C v q
0 q t
=> specific heat
=> base temperature
=> contact temperature
L
0
=> Lorentz number
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Maximum temperature in the coil
Occurs at position where the quench started
Resistive loss per m 3 equals increase of enthalpy per m 3
𝐽 2 𝑡 𝜌 𝑇 𝑑𝑡 = 𝛾𝐶 𝑣
𝑇 𝑑𝑇 𝑑𝑇 𝑚𝑎𝑥 𝑑𝑡
𝐽 2 𝜌
= 𝛾𝐶 𝑣
Where J is current density and g is mass density
Allows to calculate T max also from a measured decay curve
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Solution of quench module in Excel
Several differential equations are integrated in parallel
1.
2.
3.
4.
6.
7.
8.
5.
1 equation for the circuit current (slide 24)
3 equations for the average temperatures in resistive zone of both coils and in the coil former (slide 30)
1 equation for the maximum temperature in the coil (slide 35)
2 equations for the longitudinal and transverse quench propagation in the upper coil (slide 34)
2 equations for the longitudinal and transverse quench propagation in the lower coil (slide 34)
Dynamic self is fitted as function of coil current
Material properties are fitted as function of temperature
All circuit properties (currents,voltages,resistances,losses) are obtained
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Current decay and quench propagation
Main coil current decay and quench propagation
700
600
500
400
300
200
100
0
0 20
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Icoil upper coil fraction lower coil fraction
40 60
Time after quench (sec)
- 37 -
80
0.20
0.15
0.10
0.05
0.00
0.35
0.30
0.25
After 50 seconds main coil current already reduced with a factor
10
At that time, about 25% of both coils have become resistive
Cold mass temperatures during the quench
Lower coil quenches about 0.1 seconds later
T max
T coil
170 K
120 K
T form
40 K
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Ohmic losses during the quench
Iron losses may be obtained from Opera2D transient solver
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Voltages during the quench
Large internal voltages in resistive zones may occur
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Conclusions
Many things have to be learned; this is only a start on one aspect
For learning we have to start doing
For example study of the quench problem will force us to learn:
1.
2.
3.
4.
5.
More about material properties
More about heat transport in the cold mass
More about mechanical/thermal stress in the coldmass
Multi-physics approach
….
A precise quench study needs to be done with 3D finite element codes
Quench model in Opera3D?
Comsol ?
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