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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 New Compact Gantry for pencil beam scanning Patient treatment room Protect, Enhance and Save Lives -2- 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 1. 2. 3. 4. 5. 6. 7. 8. 9. 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 Protect, Enhance and Save Lives -4- 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. 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 Protect, Enhance and Save Lives -5- Different models for the S2C2 magnetic circuit 1. 2. 3. 4. 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 Protect, Enhance and Save Lives -6- 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 Protect, Enhance and Save Lives -7- The static Opera2D model What information can we obtain The magnet load line with respect to the superconductor critical surface 1. 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 2. From stored energy From flux-linking The dynamic self-inductance of the magnet 3. Essential for non-linear systems like S2C2 Protect, Enhance and Save Lives -8- 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) S2C2 Field in the center and maximum field on the coil 7 Magnetic field (Tesla) maximum coil field during ramp up B_tot_center 6 B_iron_center B_max_coil 5 4 3 2 windings/coil=3145 1 0 0 100 200 300 400 500 PSU current (Amps) Protect, Enhance and Save Lives 600 700 800 -9- The static self-inductance of the magnet The static self-inductance of the magnet 1. From the stored energy: From flux-linking: 1 𝑈 = 2L𝐼 2 Flux for a single wire in the coil: 𝜙 = 2𝜋 Relation with vector potential: 𝑟𝐴𝜃 = Total flux over coil: 𝜙𝑡𝑜𝑡 = Self of one coil from flux-linking: 𝐿= 𝑟 𝑟𝐵𝑧 𝑟 𝑑𝑟 0 𝑟 𝑟𝐵𝑧 𝑟 𝑑𝑟 0 2𝜋𝑁 𝐴 2𝜋𝑁 𝐴𝐼 𝐴(𝑐𝑜𝑖𝑙) 𝐴(𝑐𝑜𝑖𝑙) 𝑟𝐴𝜃 𝑑𝑟𝑑𝑧 𝑟𝐴𝜃 𝑑𝑟𝑑𝜃 2nd method allows to find difference between upper and lower coil Can be calculated directly in Opera2D Protect, Enhance and Save Lives - 10 - Self-inductance from stored energy Calculated with Opera2D static solver S2C2 Stored energy and self-inductance 400 14 stored energy 350 12 static self 300 10 8 200 6 150 4 100 2 50 0 00 0 Protect, Enhance and Save Lives 250 windings/coil=3145 100 200 300 400 500 PSU current (Amps) - 11 - 600 700 800 Self (Henri) stored energy (MJ) 16 Static self from flux-linking Asymmetry may induce a quench? => probably not; DV=0.3 mV is too small Introduces a voltage difference between upper and lower coil during ramp Small vertical symmetry in the model Asymmetry in self-inductance of upper and lower coil 10 400 L(upper coil)-L(lower coil) self 320 windings/coil=3145 6 240 0.3 mV 4 160 2 80 0 0 0 Protect, Enhance and Save Lives - 12 - 100 200 300 400 500 PSU current (Amps) 600 700 800 Ltot (Henri) Lup-Llow (mH) 8 What do we get from the 2D transient solver? Eddy currents and related losses Losses Current density profiles Losses in former, cryostat walls and yoke iron during a ramp 1.8 1.6 ramp-rate=2.7 Amps/min 1.4 former Losses (Watt) 1.2 iron yoke 1.0 cryo-walls 0.8 0.6 0.4 0.2 0.0 0 Apply a constant ramp rate of 2.7 Amps/min to the coils Protect, Enhance and Save Lives - 13 - 50 100 Magnet current (Amps) 150 200 Eddy current losses during ramp up and quench During ramp-up 1. 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 2. 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 Protect, Enhance and Save Lives - 14 - Opera2d transient solver coupled to external circuit PSU drive programmed as in real live Protect, Enhance and Save Lives - 15 - 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 Protect, Enhance and Save Lives - 16 - A full ramp-up and ramp-down cycle Coil current compared to dump current coil Dump (x10) Protect, Enhance and Save Lives - 17 - 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 60 -10 50 -20 40 Fx -30 30 Fy total -40 Larger forces during down ramp However: total (kN) Fx,Fy (kN) Horizontal forces on cold mass (040613) 0 20 -50 10 -60 0 200 300 Protect, Enhance and Save Lives 400 500 PSU-current (Amps) 600 700 - 18 - 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/m3) 2 𝑑𝐵 𝑃𝑓 = 𝜆𝑠𝑢𝑝 𝐽𝑐 (𝐵)𝑑𝑓 3𝜋 𝑑𝑡 Coupling losses (W/m3) 𝑑𝐵 ( )2 𝑝 𝑃𝑒 = 𝜆𝑤𝑖𝑟𝑒 𝑑𝑡 ( )2 𝜌𝑡 2𝜋 Tool developed in Opera2D-Transient solver that integrates above expressions in coil area Protect, Enhance and Save Lives - 19 - Jc(B) df lsup lwire rt 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 𝐶0 𝐽𝑐 = 𝑏𝛼 (1 − 𝑏)𝛽 (1 − 𝑡 𝑛 )𝛾 𝐵 𝑇 𝑡 = 𝑇 (reduced temperature) 𝑐0 𝐵 𝑐2 (𝑇) 𝑏=𝐵 (reduced field) critical field at zero current 𝐵𝑐2 𝑇 = 𝐵𝑐20 1 − 𝑡 𝑛 a,b,g,C0 => fitting coefficients Protect, Enhance and Save Lives - 20 - Critical surface => Bottura formula (2) Critical surface at T=4 K (Bottura-formula) Critical field as function of temperatue at zero current (Bottura) 7 16 Normalized to unity at 5 Tesla/4.2 K 5 Spencer Somerkoski critical field (Tesla) Critical current (normalized) 6 Green 4 Morgan Hudson 3 2 1 Spencer 14 Somerkoski 12 Green Morgan 10 Hudson 8 6 4 2 0 0 0 2 Protect, Enhance and Save Lives 4 6 8 Magnetic field (Tesla) 10 0 12 - 21 - 1 2 3 4 5 6 Temperature (K) 7 8 9 10 Critical surface => S2C2 wire Critical surface of S2C2 wire (3500 Amps @ 5Tesla/4.2 K) 18000 Critical current (normalized) 16000 T=3 K 14000 T=4K 12000 T=5 K 10000 T=6 K 8000 T=7 K 6000 4000 2000 0 0 Protect, Enhance and Save Lives 2 4 6 8 Magnetic field (Tesla) - 22 - 10 12 14 AC losses obtained with OPER2D transient solver Initial results => maybe can be improved Protect, Enhance and Save Lives - 23 - 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 Primary circuit 𝑑𝐼 𝐿 𝑅 𝑐 𝑅𝑑 𝐼𝑝 − 𝐼𝑐 + 𝑅𝑑 𝑑𝑡 2 − 𝑅𝑐 𝐼𝑐 𝑑𝐼𝑐 𝑓 𝑁 = 𝑅 𝑅 1 𝑑𝑡 𝐿𝑐 1 + 2 (1 + 𝑅 𝑐 ) 𝑅𝑑 𝑁 𝑑 𝑓 SOLVED IN EXCEL Protect, Enhance and Save Lives - 24 - Secondary circuit PSU Coil self-inductance Coil resistance (only with quench) Dump resistor 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 Protect, Enhance and Save Lives - 25 - The concept of dynamic self-inductance Important for non-linear magnets Definition of self-inductance: 𝜙 = 𝐿𝐼 Faraday’s law: Combine: 𝑉 = 𝑑𝑡 𝐿𝐼 = 𝐿 𝑑𝑡 + 𝑑𝜙 𝑑𝑡 𝑑𝐿 𝐼 𝑑𝑡 = 𝑉= 𝑑 𝑑𝐼 𝑑𝐼 𝑑𝐿 𝑑𝐼 𝑑𝐼 𝐼 𝑑𝐿 𝐿 𝑑𝑡 + 𝐼 𝑑𝐼 𝑑𝑡 = 𝐿 𝑑𝑡 (1 + 𝐿 𝑑𝐼 ) For a non-linear system the dynamic self must be used in lumped element circuit simulations 𝐼 𝑑𝐿 𝐿𝑑𝑦𝑛 = 1 + 𝐿 𝑑𝐼 𝐿𝑠𝑡𝑎𝑡 Protect, Enhance and Save Lives - 26 - S2C2 self-inductance A large difference between static and dynamic self Protect, Enhance and Save Lives - 27 - 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 Protect, Enhance and Save Lives - 28 - Compare both circuit-models Resistive losses in the former during ramp-up Blue:OPER2D transient-circuit model Red: analytical lumped element model Protect, Enhance and Save Lives - 29 - Very good agreement between both models 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 𝑅(𝑇)𝐼 2 𝑚𝐶𝑣 (𝑇) 𝑑𝑇 𝑑𝑡 R(T) => resistance => 𝑅 𝑇 = r(T) => resistivity Cv(T) => specific heat Protect, Enhance and Save Lives = ℎ𝑒𝑎𝑡 𝑠𝑜𝑢𝑟𝑐𝑒 = 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 - 30 - 𝜌 𝑇 𝐿 𝐴 Specific heat of copper and aluminium Very accurate fitting is possible 𝐶𝑝 = 𝑎 + 𝑏𝜃 + 𝑑𝜃 2 + ⋯ + 𝑖𝜃 8 𝜃 = log(𝑇) Protect, Enhance and Save Lives - 31 - Electrical resistivity of copper and aluminium Same kind of fitting is possible Protect, Enhance and Save Lives - 32 - A qualitative model for quench behavior Based on (« multi-physics ») lumped element model Five different zones with four different temperatures in the cold mass Upper coil superconducting zone (T0) Upper coil resistive zone heated by resistive loss (T1) 1. 2. Resistive former heated by eddy current losses (T2) Lower coil superconducting zone (T0) Lower coil resistive zone heated by resistive loss (T3) 3. 4. 5. expanding due to longitudinal and transverse quench propagation 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 Protect, Enhance and Save Lives - 33 - Model for quench propagation From Wilson course Introduce the fraction f=fl*ft of the coil that has become resistive 1. fl => Longitudinal propagation (fast 10 m/sec): 𝑑𝑙 𝐽 𝐿0 𝜃0 = 𝑣𝑙𝑜𝑛𝑔 ⇒ 𝑣𝑙𝑜𝑛𝑔 = 𝑑𝑡 𝛾𝐶𝑣 𝜃𝑡 − 𝜃0 2. ft =>Transverse propagation (slow 20 cm/sec): 𝑑𝑟 = 𝑣𝑡𝑟𝑎𝑛𝑠 ⇒ 𝑣𝑡𝑟𝑎𝑛𝑠 = 𝛼𝑣𝑙𝑜𝑛𝑔 𝑑𝑡 Protect, Enhance and Save Lives - 34 - J => current density G => mass density Cv => specific heat q0 => base temperature qt => contact temperature L0 => Lorentz number Maximum temperature in the coil Occurs at position where the quench started Resistive loss per m3 equals increase of enthalpy per m3 𝐽2 𝑡 𝜌 𝑇 𝑑𝑡 = 𝛾𝐶𝑣 𝑇 𝑑𝑇 𝑑𝑇𝑚𝑎𝑥 𝐽2 𝜌 = 𝑑𝑡 𝛾𝐶𝑣 Where J is current density and g is mass density Allows to calculate Tmax also from a measured decay curve Protect, Enhance and Save Lives - 35 - Solution of quench module in Excel Several differential equations are integrated in parallel 1. 2. 3. 4. 5. 6. 7. 8. 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 Protect, Enhance and Save Lives - 36 - Current decay and quench propagation 700 0.35 600 0.30 500 0.25 400 0.20 300 0.15 Icoil upper coil fraction 200 0.10 lower coil fraction 100 0.05 0 0.00 0 20 Protect, Enhance and Save Lives 40 60 Time after quench (sec) 80 - 37 - fraction of coil that is quenched (-) Main coil current (Amps) Main coil 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 Cold mass temperatures during the quench Lower coil quenches about 0.1 seconds later Protect, Enhance and Save Lives - 38 - Tmax 170 K Tcoil 120 K Tform 40 K Ohmic losses during the quench Iron losses may be obtained from Opera2D transient solver Protect, Enhance and Save Lives - 39 - Voltages during the quench Large internal voltages in resistive zones may occur Protect, Enhance and Save Lives - 40 - 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 ? Protect, Enhance and Save Lives - 41 -