These slides present the results from the thesis with the same name as the presentation title. It deals with operational aspects of superconducting (SC) accelerating cavities related to charging the same. It proposes an optimal scheme to charge the SC cavities such that the total reflected energy is minimised. Following which it presents a way of extending the well known Q-circle method to SC cavities for accurate estimation of cavity quality factor (Q0). The measurements reveal the effect of cool down on cavity Q0. Present literature proposed the effect of trapped magnetic field in surface impurities. In the thesis I use the Ginzburg-Landau model of superconductivity to theoretically explain the effect of magnetic field, impurity and temperature gradient. This allows a theoretical model that explains the experimental results.

1. From Macroscopic to Microscopic
Dynamics of the Superconducting
Cavities
Anirban Krishna Bhattacharyya
FREIA Laboratory/High Energy Physics,
Department of Physics and Astronomy,
Uppsala University

2. FREIA
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 2

3. FREIA
UU-ESS-IPNO-CERN &
Industry Collaboration
(Thales, Electrosys, DB Elettronica,
Siemens, NXP, ESRF, CERN)RF Source Development
(vacuum tube amplifier, solid-
state amplifier, SSA module &
combiner optimization)
High-power Spoke Cavity Testing
(tuning system, dynamic load, electron
emission, mechanical parameters and
multipacting)
Digital LLRF
Combined THz/X-ray
source
Present & Future
 Acceptance testing of spoke
cryomodule
 Test of prototype spoke
cryomodule valve box
 Test of prototype Single spoke
and Double Spoke
superconducting cavities✓
 Test of elliptical cavity
 Testing of crab cavities for
LHC upgrade
 ESS neutrino super beam.
Liquifier (He, N)
FREIA
33/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities

4. Outline
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 4
𝐸+
𝐸−
Wikimedia commons

5. Outline
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 4
𝐸+
𝐸−
Reflection

6. Outline
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 4
𝐸+
𝐸−
Cavity
Quality
factor (𝑄0)
Reflection

7. Step Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
RF
Source
Circulator Cavity
Load
Spoke Cavity
(super - conducting)
Courtesy of P. Duthil
5

8. Step Charging (Frequency domain)
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
(MHz)
6

9. Optimal Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
• Instantaneous cavity voltage
• Reflected current
• Reflected energy
Filling time Loaded Q Generator
current
Loaded
impedance
External Q Bare cavity Q
7

10. Minimum Action: Example from classical
mechanics
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 8
𝑡1
𝑡2
η(𝑡)
𝑆 = 𝑡1
𝑡2
(𝐾. 𝐸. −𝑃. 𝐸. )𝑑𝑡 = 𝑡1
𝑡2
𝐿 𝑑𝑡 = 𝑡1
𝑡2
[
1
2
𝑚
𝑑𝑥
𝑑𝑡
2
− 𝑉(𝑥)]dt

11. Minimum Action: Example from classical
mechanics
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 8
𝑡1
𝑡2
η(𝑡)
𝑆 = 𝑡1
𝑡2
(𝐾. 𝐸. −𝑃. 𝐸. )𝑑𝑡 = 𝑡1
𝑡2
𝐿 𝑑𝑡 = 𝑡1
𝑡2
[
1
2
𝑚
𝑑𝑥
𝑑𝑡
2
− 𝑉(𝑥)]dt

12. Optimal Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
• The concept of minimum action,
Optimal charging profile
Find optimal
and such
that is
minimum.
9

13. Optimal Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
• The concept of minimum action,
Optimal charging profile
9
Free parameter

14. Effect of Optimal filling
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Step filling Optimal filling
10

15. Effect of Optimal filling
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Step filling Optimal filling
No free lunch!!!
10

16. Effect of Optimal filling
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Step filling Optimal filling
More peak power
10

17. Effect of charging time ( )
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Peak generator powerRelative reflected energy
11

18. Practical sources
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Gain characteristics Efficiency characteristics
/ IOT / IOT
12

19. Practical sources
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Source efficiency during filling
RF
Gain
Source
loss
13

20. Practical sources
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Tetrodes can be run in
Doherty architecture
14

21. Effect of Transit time factor
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Variation Along Spoke LINAC
15

22. Effect of Transit time factor
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Peak power Charging time
Beam injection time,
16

23. Medium and High 𝛽 cavities
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 17
KlystrodeSolidstateDohertyamplifier
Energy saved for = 𝑡 𝐹

24. Savings for medium and high β cavities
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 18
Medium Beta Cavities High Beta Cavities
Source type IOT SSDA IOT SSDA
Energy saved/pulse (J) 27 17 47 40
Energy saved/sec (J) 380n 240n 650n 560n
Energy saved in lifetime
(MWhrs)
121n 76n 211n 179n
Number of cavities 36 84
Total savings (MWhrs) 4355n 2742n 17687n 15053n
SEK saved (Millions) 7.8n 4.9n 31.8n 27.1n
Operation: 14n Hz pulse rate
Total operation time: 40 years, 8000hours/year
At an electricity price in Sweden of 1.8 SEK/kWhr

25. Cryogenic considerations
• Losses on cavity surface 𝑃𝑑 =
ω 𝑐 𝑊
𝑄0
=
ω 𝑐
μ0
2 0
𝑡 𝑖 |𝐵|2 𝑑𝑉
𝑄0
•|𝐵|2
∝ |𝑉|2
• Ratio of energy loss
between step and
optimal filling
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 19
𝑄0
Intrinsic quality of the resonant
structure

26. Cavity Quality factor
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
• To measure Quality factor of
bare ESS superconducting
spoke cavities
• Vertical tests in horizontal
cryostat
http://newsline.linearcollider.org/2013/11/21/a-little-dirt-never-hurt/cavity-performance/
20
𝑄0 =
ω 𝑐 𝑊
𝑃𝑑

27. Cavity resonator and self-excited loop
𝑉 +
ω 𝑐
𝑄 𝐿
𝑉 + ω 𝑐
2 𝑉 = 2ω 𝑐(𝑅/𝑄) 𝐼𝑖 𝑄 𝐿 = 𝑄0
−1
+ 𝑄 𝑒𝑥𝑡
−1 −1
21
θ 𝑐 ω + θ𝑙 ω = 2𝜋𝑛
Self-excited loop
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
adapted from paper III © 2016 IEEE
adapted from paper III © 2016 IEEE

28. Reflection coefficient
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 22
𝐸+ 𝐸−
Γ =
𝐸−
𝐸+
=
κ − 1 + 𝑖𝑄0δ
κ + 1 − 𝑖𝑄0δ
δ =
𝑄 𝑒𝑥𝑡 + 𝑄0
𝑄 𝑒𝑥𝑡 𝑄0
tan θ 𝑐
Wikimedia commons
κ =
𝑄0
𝑄 𝑒𝑥𝑡
critical coupling κ = 1
over coupling κ > 1
under coupling κ < 1

29. Superconducting cavity
Γ′
+
1
1 + κ
2
+ Γ′′2
=
κ
1 + κ
2
233/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
κ =
𝑄0(𝑉)
𝑄 𝑒𝑥𝑡

30. Reflection co-efficient, Q-circle and surface
Γ′ +
1
1 + κ
2
+ Γ′′2
=
κ
1 + κ
2
233/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities

31. Experimental setup
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 24
Change θ𝑙 ω
Measure: Forward power (𝑃𝐹)
Reflected power (𝑃𝑅)
Transmitted power (𝑃 𝑇)
adapted from paper III © 2016 IEEE

32. Q-surface
25
Cavity voltage from 𝑃 𝑇and 𝑄 𝐿 and
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
𝑄 𝐿 measurement
κ =
1
1
𝑟
− 1

33. Q-slope
Double-spoke cavity
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34. Q-slope
273/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
adapted from paper III © 2016 IEEE

35. Q-slope
28
Effect of cooling
rate
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
adapted from paper III © 2016 IEEE

36. Effect of cooling rate on 𝑄0
Residual resistance depends on trapped magnetic field
• Helmholtz Zentrum Berlin (HZB): magnetic field are generated by thermal
currents.
• HZB & Cornell: Slow cooling to reduce thermal currents.
• Fermilab: Magnetic field (ambient) can be expelled by large temperature
gradients creating a quick propagating super-conducting phase front.
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 29
adapted from Dependence of the residual surface resistance of superconducting radio frequency cavities on the cooling dynamics around Tc

37. Effect of cooling rate on 𝑄0
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 30
Cavity warming up

38. Models of Superconductivity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 31
London Model:
conditions:
London penetration length
adapted from cmms.triumf.ca/theses/Sonier/MSc/

39. Models of Superconductivity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 32
Ginzburg-Landau Model:
Order parameter
Density of superconducting electrons:
Helmholtz free energy:
Total free energy:
= 0 => normal conducting state
> 0 => superconducting state

40. Models of Superconductivity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 33
Ginzburg-Landau (G-L) Model
Total free energy:
is the effective Lagrangian. Minimized w.r.t and
If =0

41. Time dependent G-L Equations
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 34
London penetration length
Coherence length
Gor’kov and Eliashberg
Order parameter normalized by
Independent of T, completely defines type of superconductor.

42. Superconducting materials
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 35
Type - I Type - II
𝑀 = 𝐵 − μ𝐻
Wikimedia commons

43. Superconductor in temperature gradient
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 36

44. Superconductor in temperature gradient
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 37

45. Superconductor in temperature gradient
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 38
Nucleation
vortex velocity

46. Superconductor in temperature gradient
3/17/2018 39

47. Superconductor with impurity
Parameter Value
ξ(0) 100 𝐴
𝑇𝑐 10K
κ 2
𝐻 0.0005𝐻𝑐2(0)
𝐿 150ξ(0)
𝜕𝑇
𝜕𝑥
10𝐾/𝑚
200𝐾/𝑚
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 40
3 impurity zones. Impurity modeled
by keeping temperature above 𝑇𝑐.

48. Superconductor with impurity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 41
Magnetic field at temperature gradient of 200𝐾/𝑚

49. Superconductor with impurity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 42

50. Superconductor with impurity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 43
Magnetic field at temperature gradient of 10𝐾/𝑚

51. Pinning force
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 44
Free energy of flux line
Gradient of free energy gives the interaction force between the flux and the
pinning center.

52. Conclusions
• Reflected energy can be reduced by the proposed filling profile.
• R&D on high power sources for high efficiency at low output powers.
• The proposed 𝑄0 measurement technique provides a means of
accurately characterising the gradient dependent character of the super-
conducting cavities.
• G-L equations can be used to theoretically explain experimental
observations.
• Both the order parameter and the pinning force play important roles in
flux trapping.
• Use G-L- equations for analytical expression of vortex velocity in the
presence of temperature gradient.
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 45

53. 3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Thank You

54. Cavity Parameters
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 4

55. Step Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
RF
Source
Circulator
Cavity
Load
Spoke Cavity
(super - conducting)
Courtesy of P. Duthil
5

56. Step Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
RF
Source
Circulator
Cavity
Load
Charging time
Spoke Cavity
(super - conducting)
Courtesy of P. Duthil
Beam injection
5

57. Optimal Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
• Instantaneous cavity voltage
Ib
Ig
Ir
Filling time
(Natural time
scale)
Loaded
Quality
factor
Generator
current
Loaded
cavity
impedance
(Nominal cavity voltage)
7

58. Optimal Charging
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
• Instantaneous cavity voltage
• Reflected current
Ib
Ig
Ir
Filling time
(Natural time
scale)
Loaded
Quality
factor
Generator
current
Loaded
cavity
impedance
External
Quality factor
Bare cavity
Quality factor
7

59. Practical sources
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
Efficiency characteristics
/ IOT
W. Doherty, A new high efficiency power amplifier for modulated waves,
Radio Engineers, Proceedings of the Institute of 24 (9) (1936) 1163–1182.
doi:10.1109/JRPROC.1936.228468.
B. Kim, J. Kim, I. Kim, J. Cha, The doherty power amplifier, Microwave
Magazine, IEEE 7 (5) (2006) 42–50. doi:10.1109/MW-M.2006.247914.
R. Pengelly, N-way rf power amplifier with increased backoff power and
power added efficiency, wO Patent App. PCT/US2003/002,365 (Aug. 7
2003).
URL http://www.google.com/patents/WO2003065573A1?cl=en
P. Colantonio, F. Giannini, R. Giofr, L. Piazzon, The doherty power ampli
fier, INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL
TECHNOLOGY 5 (6) (2010) 419–430.
G. Ahn, M. su Kim, H. chul Park, S. chan Jung, J. ho Van, H. Cho,
S. wook Kwon, J.-H. Jeong, K. hoon Lim, J. Y. Kim, S. C. Song, C.-S.
Park, Y. Yang, Design of a high-efficiency and high-power inverted doherty
amplifier, Microwave Theory and Techniques, IEEE Transactions on 55 (6)
(2007) 1105–1111. doi:10.1109/TMTT.2007.896807.
NXP Semiconductors, AN10967 BLF578 demo for 352 MHz 1kW CW
power, 2nd Edition, application note (November 2012).
D. Rees, D. Keffeler, W. Roybal, P. Tallerico, Characterization of a
Klystrode as a RF Source for High-Average-Power Accelerators, Conf.Proc.
C950501 (1995) 1521.
E. Montesinos, Tetrode power amplifiers, in: TIARA Workshop on RF
Power Generation for Accelerators, Uppsala University, A˙ ngstr¨om Laboratory,
2013.
N. Pupeter, Significant increase of efficiency of solid state amplifiers due
to improved ac/dc conversion and adaption of p1 point to actual operating
power, in: EnEfficient RF Sources, Cockcroft Institute, 2014.
12

60. Effect of Transit time factor
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities
ESS operation: 14 Hz pulse rate
140 J/sec saved
per cavity
Assuming 8000 hours/year
of operation 1.12 MWhrs
saved per cavity
2 SEK/kWhr => 2240 SEK
10 J/pulse
26 Spoke cavities, 40
years of operation, =>
2.32 MSEK
17

61. Experimental setup
293/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities

62. Digital loop delay
303/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities

63. Normal conducting cavity
Γ′′
Γ′
263/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities

64. Cavity conditioning
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65. Reflection coefficient
Before cable compensation After cable compensation
323/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities

66. 𝑄 𝐿 measurement
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 66
back

67. Superconductor with impurity
3/17/2018 Anirban, ESS/AD Seminar - From Macroscopic to Micrsoscopic dynamics of SC cavities 67