Dynamic behaviour of the S2C2 magnetic circuit

Dynamic behavior of the S2C2 magnetic circuit

FFAG13 September 2013

Wiel Kleeven

The New IBA Single Room Proton Therapy Solution: ProteusONE



High quality PBS cancer treatment: compact and affordable

Synchrocyclotron with superconducting coil:

S2C2

- 2 -

New Compact Gantry for pencil beam scanning

Patient treatment room

Protect, Enhance and Save Lives

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

Protect, Enhance and Save Lives - 3 -

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

- 4 Protect, Enhance and Save Lives

Efforts to learn more on the superconducting magnet

Coil and cryostat designed and manufactured by the Italian company ASG

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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


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


Magnetic circuit-modeling

OPERA3D full model with many details





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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

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 pole gap < => extraction system optimization

Influence of yoke penetrations

Median plane errors

Magnetic forces

ITERATIVE PROCESS WITH STRONG

INTERACTION TO BEAM SIMULATIONS


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


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


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


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


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


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


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

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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


Opera2d transient solver coupled to external circuit

PSU drive programmed as in real live

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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


The full ramp-up/ramp-down cycle

Default PSU-ramping for the S2C2

Used in the OPERA2D external circuit simulations


A full ramp-up and ramp-down cycle

Coil current compared to dump current coil

Dump (x10)

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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


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

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Larger forces during down ramp

However:

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

Current split between dump and coil can not explain completely the difference in forces iron hysteresis also seems to play an important role


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


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


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


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


AC losses obtained with OPER2D transient solver

Initial results => maybe can be improved

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

Hysteresis losses somewhat larger than eddy current losses

Coupling losses very small


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

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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


The concept of dynamic self-inductance

Important for non-linear magnets

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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


S2C2 self-inductance

A large difference between static and dynamic self


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


Compare both circuit-models

Resistive losses in the former during ramp-up


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


Specific heat of copper and aluminium

Very accurate fitting is possible

𝐶 𝑝

= 𝑎 + 𝑏𝜃 + 𝑑𝜃 2

+ ⋯ + 𝑖𝜃 8 𝜃 = log(𝑇)


Electrical resistivity of copper and aluminium

Same kind of fitting is possible


A qualitative model for quench behavior

Based on (« multi-physics ») lumped element model





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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


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


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


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


Current decay and quench propagation

Main coil current decay and quench propagation

700

600

500

400

300

200

100

0

0 20


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


Ohmic losses during the quench

Iron losses may be obtained from Opera2D transient solver


Voltages during the quench

Large internal voltages in resistive zones may occur


Conclusions

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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 ?


Dynamic behaviour of the S2C2 magnetic circuit

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