1. Differential inductance of superconducting magnets: the role of coupling currents
Vittorio Marinozzi
CERN, 14/06/2016

2. Experimental motivation Differential inductance
Vittorio Marinozzi, CERN 14/06/2016
Outline:
Experimental motivation
Differential inductance
IFCC electromagnetic model
Implementation in QLASA
Experimental comparison
Effects on quench protection
Conclusions

3. 1.1 – Experimental motivation
Vittorio Marinozzi, CERN 14/06/2016
1.1 – Experimental motivation
Larp tests in 2013
Fast extraction on dump resistor 𝑅 𝑑
𝐼= 𝐼 0 𝑒 − 𝑅 𝑑 𝑡 𝐿
Using measured inductance, it is impossible to fit the decay,
even at the very beginning!

4. 1.2 – Experimental motivation
Vittorio Marinozzi, CERN 14/06/2016
1.2 – Experimental motivation
The HQ inductance from exponential fit of the decay beginning is about 30% lower than the one measured at low ramp rate

5. Can the inductance actually change at high ramp rate?
Vittorio Marinozzi, CERN 14/06/2016
1.3 – Experimental motivation
Same phenomenon seen in LQ tests
Impossible to justify with quench back, AC losses, current redistribution….
Too fast
Can the inductance actually change at high ramp rate?
If yes, why?
I want to justify this behavior with coupling currents effect on differential inductance

6. 𝐿 𝑠 = 𝜙(𝐵) 𝐼 𝑒.𝑚.𝑓.= −𝐿 𝑠 𝑑𝐼 𝑑𝑡 What is the inductance?
Vittorio Marinozzi, CERN 14/06/2016
2.1 – Differential inductance
𝐿 𝑠 = 𝜙(𝐵) 𝐼
𝑒.𝑚.𝑓.= −𝐿 𝑠 𝑑𝐼 𝑑𝑡
What is the inductance?
𝐿= 𝜇 0 𝑛 2 𝑙𝜋 𝑟 2
The inductance of a device depends only on the geometry!
What happens when the magnetic flux is not linear?
𝑒.𝑚.𝑓.≠− 𝐿 𝑠 𝑑𝐼 𝑑𝑡
Inductance depends on the current!

7. 𝐿 𝑑 = 𝐿 𝑠 + 𝑑 𝐿 𝑠 𝑑𝐼 𝐼= 𝑑𝜙(𝐵) 𝑑𝐼
Vittorio Marinozzi, CERN 14/06/2016
2.2 – Differential inductance
𝑉=− 𝑑𝜙 𝐵 𝑑𝐼 𝑑𝐼 𝑑𝑡 =− 𝑑 𝐿 𝑠 𝐼 𝑑𝐼 𝑑𝐼 𝑑𝑡
Faraday-Neumann-Lenz:
𝑉=− 𝑑𝜙 𝐵 𝑑𝑡
𝑉=− 𝐿 𝑠 + 𝑑 𝐿 𝑠 𝑑𝐼 𝐼 𝑑𝐼 𝑑𝑡
It is useful to define a “differential inductance”
𝐿 𝑑 = 𝐿 𝑠 + 𝑑 𝐿 𝑠 𝑑𝐼 𝐼= 𝑑𝜙(𝐵) 𝑑𝐼
𝑉=− 𝐿 𝑑 𝑑𝐼 𝑑𝑡
Such that
“Standard” inductance and differential inductance are quantities conceptually different!
Standard inductance is related to the field flux
Differential inductance is related to the flux change (voltage)

8. Example: iron saturation
Vittorio Marinozzi, CERN 14/06/2016
2.3 – Differential inductance
Example: iron saturation
“…remember that the inductance is lower at high current because of the iron saturation…”
“…remember that the inductance is larger at low current because of the iron saturation…”
Because of the iron, the standard inductance is larger at high current.
Instead, the differential inductance is larger at low current, while iron does not affect it at all at high current!

9. Example: magnetized materials
Vittorio Marinozzi, CERN 14/06/2016
2.4 – Differential inductance
Example: magnetized materials
𝐵= 𝜇 0 𝑀+𝐻 = 𝜇 0 𝐻 1+𝜒
𝑑𝑢=𝐻𝑑𝐵
𝑀=𝜒𝐻
𝑑𝑢= 𝜇 0 𝐻𝑑𝐻+ 𝜇 0 𝜒𝐻𝑑𝐻 +𝜇 0 𝜒 𝐻 2
𝐿 𝑑 = 𝑑𝜙(𝐵) 𝑑𝐼
𝐿 𝑑 = 1 𝐼 𝑑𝑈 𝑑𝐼
𝑉=− 𝐿 𝑑 𝑑𝐼 𝑑𝑡
𝐿 𝑑 = 𝜇 0 𝐻 𝐼 𝑑𝐻 𝑑𝐼 + 𝜇 0 𝜒𝐻 𝐼 𝑑𝐻 𝑑𝐼 + 𝜇 0 𝐻 2 𝐼 𝑑𝜒 𝑑𝐼 𝑑𝑉
𝑑𝑈 𝑑𝑡 =−𝑉𝐼
Known the susceptivity of the material, one can compute
the inductance from the integration of the field

10. 𝜏= 𝜇 0 2 𝜚 𝑒 𝑝 2𝜋 2 𝐵 𝑖 = 𝐵 𝑒 − 𝑑 𝐵 𝑖 𝑑𝑡 𝜏 𝜒=− 1 1+ 1 2𝜆 1 1− 𝐵 𝑒 𝐵 𝑖
Vittorio Marinozzi, CERN 14/06/2016
3.1 –IFCC electromagnetic model
Extracted from M. N. Wilson “Supercoducting magnets”
𝐵 𝑖 = 𝐵 𝑒 − 𝑑 𝐵 𝑖 𝑑𝑡 𝜏
𝜏= 𝜇 0 2 𝜚 𝑒 𝑝 2𝜋 2
𝜏 is the decay time constant of the IFCC
𝜚 𝑒 is the effective transverse resistance between filaments
𝑝 is the filament twist pitch lenght
From direct integration of the current, one can compute the magnetization associated to IFCC
From the known relations 𝐵= 𝜇 0 𝑀+𝐻 and 𝑀=𝜒𝐻, one can compute the susceptivity associated to IFCC
𝜒=− 𝜆 1 1− 𝐵 𝑒 𝐵 𝑖
𝑀= 2𝜆𝜏 𝜇 0 𝑑 𝐵 𝑖 𝑑𝑡
𝜆 is the packing factor

11. It is the same phenomenon that we see with the iron!
Vittorio Marinozzi, CERN 14/06/2016
3.2 –IFCC electromagnetic model
A qualitative explanation
We have seen that IFCC can be seen as a magnetization of the coils
It is the same phenomenon that we see with the iron!
Iron makes the differential inductance higher at low current. Then it saturates and it does not affect the inductance at all
IFCC are basically screening currents. They try to keep the flux constant. This is why they make the differential inductance lower!

12. 𝜏 𝑒 is the transport current decay time
Vittorio Marinozzi, CERN 14/06/2016
3.3 –IFCC electromagnetic model
Exponential approximation
𝜏 is the IFCC decay time
𝜏 𝑒 is the transport current decay time
𝐵 𝑖 = 𝐵 𝑒 − 𝑑 𝐵 𝑖 𝑑𝑡 𝜏 𝐵 𝑒 = 𝐵 0 𝑒 − 𝑡 𝜏 𝑒 𝐵 𝑖 0 = 𝐵 𝑒 (0) =𝐵 0
𝜏 𝑒 =𝐿/𝑅
𝐵 𝑖 = 𝐵 0 𝜏− 𝜏 𝑒 𝜏 𝑒 − 𝑡 𝜏 − 𝜏 𝑒 𝑒 − 𝑡 𝜏 𝑒
With a simple assumption, one can find a solution for the field produced by inter-filament currents
With a simple assumption, one can find an analytical formula for the susceptivity related to the inter-filament currents
𝜒(𝑡)= 2𝜆𝜏 𝑒 − 𝑡 𝜏 𝑒 − 𝑒 − 𝑡 𝜏 𝜏 𝑒 − 𝑡 𝜏 − 𝜏 𝑒 𝑒 − 𝑡 𝜏 𝑒 −2𝜆𝜏 𝑒 − 𝑡 𝜏 𝑒 − 𝑒 − 𝑡 𝜏
V. Marinozzi et al., "Effect of coupling currents on the dynamic inductance during fast transient in superconducting magnets", Physical Review Special Topics – Accelerators and beams, 2015.

13. We do not want to calculate the new inductance, we want to use it!
Vittorio Marinozzi, CERN 14/06/2016
3.4 –IFCC electromagnetic model
Main issue of the “exponential” model:
We do not want to calculate the new inductance, we want to use it!
𝜏 𝑒 =𝐿/𝑅 depends on the inductance, therefore it cannot be used as input.
𝐵 𝑒 = 𝐵 0 𝑒 − 𝑡 𝜏 𝑒
The current decay depends on the magnet inductance
BUT
The magnet inductance depends on the decay!
First conclusion: differential inductance of superconducting magnets is not a defined value, but it depends on the current decay, and vice-versa.

14. Very useful for a simple implementation of the IFCC model
Vittorio Marinozzi, CERN 14/06/2016
4.1 –Implementation in QLASA
QLASA is a software for the simulation of quench propagation in superconducting magnets.
Very useful for a simple implementation of the IFCC model
It is semi-analytical
Magnet inductance is an input (not from a BEM-FEM or similar)
Inductance can be changed using a simple analytical formula
I will use the exponential approximation, in an iterative and step-by-step way

15. 𝐿 𝑑 = 𝜇 0 𝐻 𝐼 𝑑𝐻 𝑑𝐼 + 𝜇 0 𝜒𝐻 𝐼 𝑑𝐻 𝑑𝐼 + 𝜇 0 𝐻 2 𝐼 𝑑𝜒 𝑑𝐼 𝑑𝑉= 𝐿 ′ +Δ𝐿
Vittorio Marinozzi, CERN 14/06/2016
4.2 –Implementation in QLASA
Important assumption: we assume that the coupling currents do not affect the magnetic field
𝐿 𝑑 = 𝜇 0 𝐻 𝐼 𝑑𝐻 𝑑𝐼 + 𝜇 0 𝜒𝐻 𝐼 𝑑𝐻 𝑑𝐼 + 𝜇 0 𝐻 2 𝐼 𝑑𝜒 𝑑𝐼 𝑑𝑉= 𝐿 ′ +Δ𝐿
We can talk of
inductance variation
Δ𝐿= 𝜇 0 𝜒 2𝐼 𝑑 𝑑𝐼 𝐻 2 𝑑𝑉 + 𝜇 0 𝐼 𝑑𝜒 𝑑𝐼 𝐻 2 𝑑𝑉
Computation of the field integral can be done numerically

16. Circuit solution 𝑅 𝑑 + 𝑅 𝑞 𝐼+𝐿 𝑑𝐼 𝑑𝑡 =0 4.3 –Implementation in QLASA
Vittorio Marinozzi, CERN 14/06/2016
4.3 –Implementation in QLASA
Circuit solution
𝑅 𝑑 + 𝑅 𝑞 𝐼+𝐿 𝑑𝐼 𝑑𝑡 =0
At each time step, the quantities are constant
Inductance variation is computed through the susceptivity at each time step, and used in the following step as input
The exponential approximation can be used!

17. Analytical solution for a costheta quadrupole
Vittorio Marinozzi, CERN 14/06/2016
4.4 –Implementation in QLASA
Analytical solution for a costheta quadrupole
𝑎 1
𝑎 2
Δ𝐿= 𝜇 0 𝜒 2𝐼 𝑑 𝑑𝐼 𝐻 2 𝑑𝑉 + 𝜇 0 𝐼 𝑑𝜒 𝑑𝐼 𝐻 2 𝑑𝑉
The field integral is limited to the magnetized material volume, i.e. to the coils
For a quadrupole, the field in the coils is analytical and can be integrated
𝐻 2 𝑑𝑉 = 𝑙 𝐽 0 2 𝜋 𝑎 2 4 − 𝑎 ln 2 𝑎 2 𝑎 1 + ln 𝑎 2 𝑎 𝑎 𝑎
With the exponential approximation, and for cos 𝑛𝜗 magnets, the inductance reduction can be computed analytically, and can be implemented very easily in any software

18. We expect an exponential decay 𝐼= 𝐼 0 𝑒 − 𝑡 𝜏 𝑒
Vittorio Marinozzi, CERN 14/06/2016
4.5 –Implementation in QLASA
Example: discharge on a dump resistor
We can use:
Exponential approximation
Analytical magnetic field
We do not have the complication of quench resistance
We assume that a quadrupole is short-circuited on a dump resistor, and that quench resistance is null
We expect an exponential decay 𝐼= 𝐼 0 𝑒 − 𝑡 𝜏 𝑒
From excel sheet!

19. Very good agreement! 5.1 –Experimental comparison
Vittorio Marinozzi, CERN 14/06/2016
5.1 –Experimental comparison
Comparison with HQ test on exponential decay (60 mΩ dump resistor)
IFCC 𝜏=15 ms (average value)
The model can describe the decay at the beginning
In order to describe the whole decay, one should add coupling currents power dissipation and subsequent quench back
Very good agreement!

20. 5.2 –Experimental comparison
Vittorio Marinozzi, CERN 14/06/2016
5.2 –Experimental comparison
We can describe a current decay. Is that a proof?
A current decay is a basically combination of dump resistor, magnet inductance and quench resistance
We know very well the dump resistor
We have understood that we do not know well the inductance, even if we have measured it at low ramp rate
Therefore, we do not know the quench resistance!
We have a model that predicts the inductance, but when quench resistance begins to become important, the model cannot be validated
We know only that the decay time is good if we describe well the decay, i.e. that the ratio L/R is good, but we do not have any hint on them separately
We need to measure inductance and resistance during a quench,
both of them separately

21. Inductance measurement during a quench
Vittorio Marinozzi, CERN 14/06/2016
5.3 –Experimental comparison
Inductance measurement during a quench
You need at least two coils of the magnet with no resistance
You can extract the inductive voltage
You can subtract the inductive voltage to the other coils, in order to obtain the resistive voltage
𝑅 𝑐𝑜𝑖𝑙 = 𝑉 𝑅 𝑐𝑜𝑖𝑙 − 𝐿 𝑐𝑜𝑖𝑙 𝐼 𝐼
𝐿 𝑐𝑜𝑖𝑙 = 𝑉 𝑆𝐶 𝑐𝑜𝑖𝑙 𝐼
Why two coils?
You should have one, but you have to be sure that it has only inductive voltage. The easiest way to be sure is to compare to another one. If voltages are identical, you are sure that coils are superconductive.
It can be proved that the magnet inductance is 𝐿= 𝑁 𝑐𝑜𝑖𝑙 𝐿 𝑐𝑜𝑖𝑙
This method works only for identical coils!

22. 5.4 –Experimental comparison
Vittorio Marinozzi, CERN 14/06/2016
5.4 –Experimental comparison
Another HQ test
No dump resistor
Quench induced by heaters in two coils
Again, dynamic effects needed to describe well the decay
Coil 15 and 16 are clearly resistive
Coil 17 and 20 have identical voltages until about 100 ms
You can compute inductance and resistance!

23. Vittorio Marinozzi, CERN 14/06/2016
5.5 –Experimental comparison
Good agreement between simulated and experimental resistance until 100 ms.
After 100 ms, resistance cannot be measured!
Important conclusion: you can measure magnet resistance only if you have one (better two) superconducting coils!

24. Vittorio Marinozzi, CERN 14/06/2016
5.6 –Experimental comparison
The inductance measured during the quench is very different from the inductance measured at low ramp rate (factor 2!)
The model of the IFCC allows to describe well the experimental inductance!
Important conclusion: you have to compute the differential inductance of a magnet case by case. Every quench has conceptually a different inductance!

25. The dynamic effects on differential inductance are needed!
Vittorio Marinozzi, CERN 14/06/2016
5.7 –Experimental comparison
Recent test of QXFS1 at Fermilab
The dynamic effects on differential inductance are needed!

26. CLIQ exploit coupling currents in order to induce quench
Vittorio Marinozzi, CERN 14/06/2016
5.8 –LEDET
Ledet is the program developed by Emmanuele Ravaioli for the CLIQ simulation
CLIQ exploit coupling currents in order to induce quench
Courtesy of Emmanuele Ravaioli
Coupling currents cannot be added as external source power in the code!
Similar results with independent, completely different approach!
Coupling currents are described with lumped elements (small L-R circuits), which takes energy from the main circuit in order to be generated
E. Ravaioli et al., "Lumped-Element Dynamic Electro-Thermal model of a superconducting Magnet", Cryogenics.

27. Nominal (IL+OL PH) MQXF hot spot temperature [K]
Vittorio Marinozzi, CERN 14/06/2016
6.1 – Quench protection
Does this study really affects quench protection analysis?
Nominal (IL+OL PH) MQXF hot spot temperature [K]
Including inductance reduction
Neglecting inductance reduction
~260 K
~285 K
Neglecting the inductance reduction gives ~25 K more in the MQXF quench simulation.
MQXF has high inductance, no dump resistor
In smaller magnets, with dump resistor, it can be more
For example, in the first slide quench we have ~40 K
This model could strongly affect protection study of magnets

28. Inductance is not the one measured at low ramp rate
Vittorio Marinozzi, CERN 14/06/2016
7.1 –Conclusions
Experimental evidences show that the magnet inductance measured at low ramp rate cannot be used to describe the current decay.
An electromagnetic model shows that the IFCC affects the magnet differential inductance.
The model can be implemented in a simple analytical way in a quench software.
The model can describe very well current decays, and it has been experimentally validated on LARP tests.
The magnet inductance has been experimentally measured during a quench, showing that:
Inductance is not the one measured at low ramp rate
Inductance is lower than expected
IFCC model can describe the inductance behavior
Resistance measurement is possible only using reduced inductance (measured, or even simulated). Using nominal inductance you get wrong.
This method really affects protection studies, and gives more realistic and accurate simulations.
Future steps:
Implement the method with FEM (not average 𝜏, magnetic field…)
Add inter-strand coupling currents
New inductance measurements with more accurate signal analysis