dynamic behavior of the s2c2 magnetic circuit ffag13 september 2013 wiel kleeven

Dynamic behavior of the S2C2 magnetic circuitFFAG13 September 2013Wiel Kleeven

Protect, Enhance and Save Lives - 2 -

The New IBA Single Room Proton Therapy Solution: ProteusONE

Synchrocyclotron with superconducting coil: S2C2

New Compact Gantry for pencil beam scanning

Patient treatment room

High quality PBS cancer treatment: compact and affordable

30.4 m

12.8 m


S2C2 overviewGeneral 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


1. Goal of the calculations

2. Different ways to model the dynamic properties of the magnet

3. What about the self-inductance of a non-linear magnet

4. Magnet load line and the critical surface of the super-conductor

5. Transient solver: eddy current losses and AC losses

6. A comparison with measurements

7. Study of full ramp-up/ramp-down cycles

8. Temperature dependence of material properties

9. A multi-physics approach and a qualitative quench model

OverviewSome items to be adressed


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. Fast warm up of the coil for maintenance

2. 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

Efforts to learn more on the superconducting magnetCoil and cryostat designed and manufactured by the Italian company ASG


1. Opera2D/Opera3D static solver

2. Opera2D transient solver

3. Opera2D transient solver coupled to an external circuit

4. Semi-analytical solution of a lumped-element circuit model

5. Multi-physics solution of a lumped element circuit with temperature-dependent properties

Different models for the S2C2 magnetic circuit


Magnetic circuit-modelingOPERA3D 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


1. 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

2. The static self-inductance of the magnet From stored energy From flux-linking

3. The dynamic self-inductance of the magnet Essential for non-linear systems like S2C2

The static Opera2D modelWhat information can we obtain


What do we get from Opera2D static solverLoad line relative to critical surface

0

1

2

3

4

5

6

7

0 100 200 300 400 500 600 700 800

Mag

netic

fiel

d (T

esla

)

PSU current (Amps)

S2C2 Field in the center and maximum field on the coil

B_tot_centerB_iron_centerB_max_coil

windings/coil=3145

maximum coil field Magnet load line and critical currents (from ASG)

maximum coil field during

ramp up


1. The static self-inductance of the magnet From the stored energy: L From flux-linking:

Flux for a single wire in the coil: Relation with vector potential: Total flux over coil: Self of one coil from flux-linking:

2nd method allows to find difference between upper and lower coil Can be calculated directly in Opera2D

The static self-inductance of the magnet


Self-inductance from stored energyCalculated with Opera2D static solver

00

50

100

150

200

250

300

350

400

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600 700 800

Self

(Hen

ri)

stor

ed e

nerg

y (M

J)

PSU current (Amps)

S2C2 Stored energy and self-inductance

stored energy

static self

windings/coil=3145


Static self from flux-linkingAsymmetry may induce a quench? => probably not; DV=0.3 mV is too small

0

80

160

240

320

400

0

2

4

6

8

10

0 100 200 300 400 500 600 700 800

L tot(H

enri)

L up-L

low

(mH

)PSU current (Amps)

Asymmetry in self-inductance of upper and lower coil

L(upper coil)-L(lower coil)

self

windings/coil=3145

Small vertical symmetry in the model

Introduces a voltage difference between upper and

lower coil during ramp

0.3 mV


What do we get from the 2D transient solver?Eddy currents and related losses

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 50 100 150 200

Loss

es (W

att)

Magnet current (Amps)

Losses in former, cryostat walls and yoke iron during a ramp

former

iron yoke

cryo-walls

ramp-rate=2.7 Amps/min

Current density profiles Losses

Apply a constant ramp rate of 2.7 Amps/min to the coils


1. 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

2. 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

Eddy current lossesduring ramp up and quench


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


The full ramp-up/ramp-down cycleDefault PSU-ramping for the S2C2

Used in the OPERA2D external circuit simulations


A full ramp-up and ramp-down cycleCoil current compared to dump current

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

up

downcoil

Dump (x10)


Tierod-forces during ramp-up and ramp-downSeems to be in agreement with previous slide

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

0

10

20

30

40

50

60

-60

-50

-40

-30

-20

-10

0

200 300 400 500 600 700

tota

l (kN

)

Fx,F

y (k

N)

PSU-current (Amps)

Horizontal forces on cold mass (040613)

Fx

Fy

total


Hysteresis losses (W/m3)

Coupling losses (W/m3)

Tool developed in Opera2D-Transient solver that integrates above expressions in coil area

AC losses during ramp-upFrom Martin Wilson course on superconducting magnets

𝑃 𝑓=𝜆𝑠𝑢𝑝23𝜋

𝐽 𝑐(𝐵)𝑑𝑓𝑑𝐵𝑑𝑡

𝑃𝑒=𝜆𝑤𝑖𝑟𝑒

( 𝑑𝐵𝑑𝑡

)2

𝜌𝑡( 𝑝2𝜋

)2

Jc(B) => critical current densitydf => filament diameterlsup => fraction of NbTi materiallwire => fraction of wire in channelrt => resitivity across wirep => pitch of the wiredB/dt => B-time derivative in coil


Critical surface => Bottura formulaNeeded for AC losses calculation

Bottura formula

(reduced temperature)

(reduced field)

critical field at zero current

, ,a b g,C0 => fitting coefficients


Critical surface => Bottura formula (2)

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12

Criti

cal c

urre

nt (n

orm

alize

d)

Magnetic field (Tesla)

Critical surface at T=4 K (Bottura-formula)

SpencerSomerkoskiGreenMorganHudson

Normalized to unity at 5 Tesla/4.2 K

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8 9 10

criti

cal fi

eld

(Tes

la)

Temperature (K)

Critical field as function of temperatue at zero current (Bottura)

SpencerSomerkoskiGreenMorganHudson


Critical surface => S2C2 wire

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 2 4 6 8 10 12 14

Criti

cal c

urre

nt (n

orm

alize

d)

Magnetic field (Tesla)

Critical surface of S2C2 wire (3500 Amps @ 5Tesla/4.2 K)

T=3 KT=4KT=5 KT=6 KT=7 K


AC losses obtained with OPER2D transient solverInitial results => maybe can be improved

Hysteresis losses somewhat larger than eddy current losses

Coupling losses very small


A lumped element model of the circuitTurns out to give very good predictions

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

𝑑 𝐼 𝑐𝑑𝑡

=𝑅𝑑 ( 𝐼𝑝− 𝐼 𝑐)+

𝑅𝑑

𝑅𝑓

𝐿𝑐𝑑𝐼𝑑𝑡

𝑁 2 −𝑅𝑐 𝐼𝑐

𝐿𝑐 [1+ 1

𝑁 2 (1+𝑅𝑐

𝑅𝑑

)𝑅𝑑

𝑅 𝑓 ]SOLVED IN EXCEL


Compare both models with experimentVoltage 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


Definition of self-inductance: Faraday’s law: Combine:

For a non-linear system the dynamic self must be used in lumped element circuit simulations

The concept of dynamic self-inductanceImportant for non-linear magnets


S2C2 self-inductanceA large difference between static and dynamic self


Compare both models with experimentVoltage 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-modelsResistive losses in the former during ramp-up

Blue:OPER2D transient-circuit model

Red: analytical lumped element model

Very good agreement

between both models


Resistors in model become temperature-dependent Introduce additional equations for temperature change

R(T) => resistance => r(T) => resistivity Cv(T) => specific heat

Further applications of lumped element modelIntroduce a kind of « multiphysics »

Since this simple model works so well: can we push it a little bit further?


Specific heat of copper and aluminiumVery accurate fitting is possible


Electrical resistivity of copper and aluminiumSame kind of fitting is possible


Five different zones with four different temperatures in the cold mass1. Upper coil superconducting zone (T0)

2. Upper coil resistive zone heated by resistive loss (T1) expanding due to longitudinal and transverse quench propagation

3. Resistive former heated by eddy current losses (T2)

4. Lower coil superconducting zone (T0)

5. Lower coil resistive zone heated by resistive loss (T3) 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

A qualitative model for quench behaviorBased on (« multi-physics ») lumped element model


Introduce the fraction f=fl*ft of the coil that has become resistive

1. fl => Longitudinal propagation (fast 10 m/sec):

2. ft =>Transverse propagation (slow 20 cm/sec):

Model for quench propagationFrom Wilson course

𝑑𝑙𝑑𝑡

=𝑣 𝑙𝑜𝑛𝑔⇒𝑣 𝑙𝑜𝑛𝑔=𝐽

𝛾 𝐶𝑣 √ 𝐿0 𝜃0𝜃𝑡−𝜃0

𝑑𝑟𝑑𝑡

=𝑣𝑡𝑟𝑎𝑛𝑠⇒𝑣𝑡𝑟𝑎𝑛𝑠=𝛼𝑣 𝑙𝑜𝑛𝑔

J => current densityG => mass densityCv => specific heatq0 => base temperatureqt => contact temperatureL0 => Lorentz number


Resistive loss per m3 equals increase of enthalpy per m3

Where J is current density and g is mass density

Allows to calculate Tmax also from a measured decay curve

Maximum temperature in the coilOccurs at position where the quench started

𝐽 2 (𝑡 )𝜌 (𝑇 )𝑑𝑡=𝛾𝐶𝑣 (𝑇 )𝑑𝑇

𝑑𝑇𝑚𝑎𝑥

𝑑𝑡= 𝐽 2𝜌𝛾 𝐶𝑣


1. 1 equation for the circuit current (slide 24)

2. 3 equations for the average temperatures in resistive zone of both coils and in the coil former (slide 30)

3. 1 equation for the maximum temperature in the coil (slide 35)

4. 2 equations for the longitudinal and transverse quench propagation in the upper coil (slide 34)

5. 2 equations for the longitudinal and transverse quench propagation in the lower coil (slide 34)

6. Dynamic self is fitted as function of coil current

7. Material properties are fitted as function of temperature

8. All circuit properties (currents,voltages,resistances,losses) are obtained

Solution of quench module in ExcelSeveral differential equations are integrated in parallel


Current decay and quench propagation

After 50 seconds main coil current already reduced with a factor 10

At that time, about 25% of both coils have become resistive

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0

100

200

300

400

500

600

700

0 20 40 60 80

frac

tion

of c

oil t

hat i

s qu

ench

ed (-

)

Mai

n co

il cu

rren

t (Am

ps)

Time after quench (sec)

Main coil current decay and quench propagation

Icoil

upper coil fraction

lower coil fraction


Cold mass temperatures during the quenchLower coil quenches about 0.1 seconds later

Tmax 170 K Tcoil 120 K Tform 40 K


Ohmic losses during the quenchIron losses may be obtained from Opera2D transient solver


Voltages during the quenchLarge internal voltages in resistive zones may occur


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. More about material properties

2. More about heat transport in the cold mass

3. More about mechanical/thermal stress in the coldmass

4. Multi-physics approach

5. …. A precise quench study needs to be done with 3D finite element codes

Quench model in Opera3D? Comsol ?

Conclusions

dynamic behavior of the s2c2 magnetic circuit ffag13 september 2013 wiel kleeven

Documents

s2c2 magnetic circuit

superconducting magnet

s2c2 magnetic circuit

superconducting coil

adressed slide

external circuit

coil maximum field

lumpedelement circuit