1. Equivalent lumped-element
IFCC/ISCC model
Emmanuele Ravaioli
thanks to
B. Auchmann, A. Verweij
4 December 2014

2. Magnet / Cables / Strands / Filaments
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

3. Inter-filament coupling loss
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

4. Inter-strand coupling loss
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

5. What is the inductance?
Inductance is the property of a conductor by which a change in current flowing through it "induces" (creates) a voltage (electromotive force) in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance).
Faraday’s law of induction asserts a relation between voltage and magnetic flux.
For a fixed system, you can also view the inductance as the ratio between a change in the magnetic flux and a change in the current, L=dφ/dI
The inductance of an ideal inductor only depends on its geometry (position and number of turns). For an ideal inductor, U=L*dI/dt with L=constant.
And L=dφ/dI=const for any current and ramp-rate
In reality, non-linear effects occur which change the behavior of an inductor: saturation of the iron yoke ( L=f(I) ), hysteresis effects in the superconducting filaments ( L=f(hystory) ), inter-filament and inter-strand coupling currents ( L=f(dI/dt, B, conductor properties) )
In non-linear inductors, L=dφ/dI is not constant. Depending on the conditions (hysteresis, saturation, ramp-rate, etc) a change in the current produces a different change in the magnetic flux. You can imagine that a part of the flux is lost, or stored.
We define a differential inductance Ld=U/(dI/dt) which is easily measurable but difficult to compute. Note that Ld not being constant means that a magnet needs more/less voltage to be ramped up, is discharged more/less quickly during an energy extraction, rings at unexpected frequencies in CLIQ discharges, etc
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

6. How to model the differential inductance? – Empirical approach
If measurements are available, the non-linear behavior of a magnet (impedance vs frequency) can be reproduced by means of an equivalent lumped-element model.
The used components often have little no physical meaning, but they correctly reproduce the behavior of the magnet.
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

7. Why do we need to do better?
For studying magnet protection, we need to develop models that predict the behavior of a magnet before it’s actually built!
Also, we want a model with physical meaning so that we can assess the impact of system parameters (current level, B, RRR, filament and strand twist pitch, etc) on the magnet performance (hot-spot temperature, etc)
Also, we want to predict the impact of non-linear effects locally, not just as an overall behavior of the entire magnet. A typical 2D model will have a spatial resolution of 1 or a few magnet turns
Also, we want to link the effects of coupling currents/loss in the electrical (differential inductance) and in the thermal domains (heat generation, temperature increase)
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

8. Why a lumped-elements model?
Efficient to solve lumped-element problems with existing network solvers (Simulink/Matlab, PSpice, Simplorer, etc)
Target: Complete electro-thermal simulation of a quench event (CLIQ, QH, EE, etc) in less than 30 minutes.
Challenge: To find analytical relations that contain enough physics to model properly complex behaviors, but are simple enough to be implemented with a limited number of equivalent lumped elements
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

9. Approach to an equivalent model of IFCC/ISCC – 1
Local balance of magnetic field change
dBt/dt = dBa/dt + dBcc/dt
(total magnetic field change is the sum between applied magnetic field Ba, due to magnet currents), and magnetic field induced by the presence of coupling currents Bcc)
The applied magnetic field change is proportional to the magnet current change
dBa/dt = MTF * dIm/dt
For any “magnetization volume” (strand for IFCC, or cable for ISCC)
Coupling Current Bcc = gamma * Icc  dBcc/dt = gamma * dIcc/dt
Induced Field Bcc = -tau * dBt/dt  dBt/dt = -gamma/tau * Icc
Coupling Loss Pcc = beta * Volume * ( dBt/dt )^2
We can rewrite the local balance of magnetic field change as
dBt/dt = dBa/dt dBcc/dt
-gamma/tau * Icc = MTF * dIm/dt + gamma * dIcc/dt
-Rcc * Icc = Mcc * dIm/dt + Lcc * dIcc/dt
This is the equation of an RL loop mutually coupled with the magnet inductance!
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

10. Approach to an equivalent model of IFCC/ISCC – 2
dBt/dt = dBa/dt dBcc/dt
-gamma/tau * Icc = MTF * dIm/dt + gamma * dIcc/dt
-Rcc * Icc = Mcc * dIm/dt Lcc * dIcc/dt
This is the equation of an RL loop mutually coupled with the magnet inductance!
The ohmic loss in the resistor Rcc must equal the coupling loss
Pcc = Rcc * Icc^2 = beta * Volume * ( dBt/dt )^2
Rcc = beta * Volume / gamma^2
The characteristic time constant of the coupling currents is
tau=Lcc/Rcc
The other lumped-element parameters follow
Rcc = beta * Volume / gamma^2
Lcc = tau * Lcc = beta * Volume / gamma^2
Mcc = beta * Volume / gamma * MTF
beta, gamma, tau depend on the strand/cable parameters and partly on the current level, Rcc=f(I) (due to magneto-resistivity)
MTF depends on the magnet geometry and partly on the current level, MTF=f(I) (due to iron saturation)
A more rigorous derivation is available, but too long to present here.
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

11. Approach to an equivalent model of IFCC/ISCC – 3
If NE different sections of the magnet have different current (as in the case of a CLIQ discharge), the magnet is modeled by NE series inductors (with nominal inductance), each coupled with NB loops representing IFCC and ISCC in a certain volume of the magnet cable.
This electrical model is coupled with a thermal model which is composed of NB volumes, each with its own temperature, magnetic field, coupling loss, ohmic loss, etc
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

12. LEDET Lumped-Element Dynamic ElectroThermal model
Mij
dI/dt
dB/dt
ISCL Ld
IFCL U
OHM I T
T>TCS !
Rel
Cth
Rth
hHe
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

13. Example: Constant dI/dt
~3 times the average IFCC time constant
2x filament twist-pitch
4x IFCC time constant
4x filament twist-pitch
16x IFCC time constant
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

14. Example: EE discharge Only IFCC IFCC+ISCC Only ISCC No IFCC, ISCC
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

15. Example: EE discharge Only IFCC IFCC+ISCC Only ISCC No IFCC, ISCC
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

16. Example: EE discharge IFCC+ISCC
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

17. Example: CLIQ 50 strands All strands
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

18. Example: CLIQ 50 strands All strands
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

19. Example: CLIQ – Parametric studies
For I0 = 20 kA
For C = 28.2 mF
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014

20. Annex

21. Transverse resistivity across the matrix
Poor contact to filaments
Good contact to filaments J J J
where lsw is the fraction of superconductor in the wire cross section (after J Carr)
Some complications
Thick copper jacket
Copper core
include the copper jacket as a resistance in parallel
resistance in series for part of current path
Martin Wilson Lecture 3 slide21
JUAS Febuary 2012

22. Coupling Losses (Heat)
Current Change
CLIQ – Coupling-Loss Induced Quench
Magnetic Field Change
Coupling Losses (Heat)
Temperature Rise
QUENCH
LHC-CM Equivalent lumped-element IFCC/ISCC model December 2014