This document provides a tutorial on beam impedance that includes: 1. Goals of getting acquainted with beam impedance concepts and understanding the effect of different materials and geometries. 2. The tutorial plan includes introductions, experiments with a network analyzer and simulations, covering topics like surface impedance, measurements of impedance for different materials, and simulations of impedance for different chamber geometries. 3. Impedance concepts are explained like surface impedance, square resistance, and beam coupling impedance and how they relate to materials, surfaces and beam currents.

2. Tutorial:
Impedance
Sergio Calatroni & Benoît Salvant- CERN

3. Tutorial
• Goals
- Get acquainted with the main concepts related to beam impedance
- Understand the effect of different materials, and different geometries
• Means
- Experimental measurements with Vector Network Analyser (VNA)
- Sincere thanks to Jean-Jacques Gratier from Keysight / Computer
Controls for kindly providing a E5080A VNA for free for the school
- Computational exercises with CST Studio Suite and ImpedanceWake2D
(IW2D)
- Sincere thanks to Monika Balk from CST / 3DS for providing several free
full licenses for Studio Suite for the duration of the school
- Short explanations of theoretical background
• The students group will be divided in two for the experimental work
• Computer work will be done in teams of 2.
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 3

4. Plan of the tutorial
• Introduction
- Surface impedance (10’ Sergio)
- Recap on beam impedance, introduction to CST studio Suite and IW2D (20’ Benoît)
• Work with network analyser 1 (60’ Sergio)
- Measurement of induction of different materials with solenoid
- Identification of materials and coatings
• And in parallel: first practice with CST Studio Suite (60’ Benoît)
- Different runs on predefined models: tubes, cavities, bellows…
• Guided simulations with IW2D (30’)
- Dependence of impedance on chamber diameter, bunch length, coating, …
• Work with network analyser 2 (30’)
- Measurement of resonances (with/without RF fingers)
• And in parallel: simulations à la carte (30’)
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 4
Ice-cream break

5. What is impedance?
• A term with many meanings, depending on context
- Impedance in electrical circuits
- Impedance of materials
- Surface impedance
- Beam coupling impedance
• There is a common rationale:
-
- Voltage and current flowing in the circuit, the material, the
surface
- The beam current, and the potential it generates on the
vacuum pipe
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 5
I
V
Z 

6. Square resistance and surface resistance
Consider a square sheet of metal and calculate its resistance to a
transverse current flow:
This is the so-called square resistance often indicated as 𝑅∎
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 6
current a
d
a


R
d
a
a
d



7. Square resistance and surface resistance
And now imagine that instead of DC we have RF, and the RF current is
confined in a skin depth:
This is a (simplified) definition of surface resistance Rs
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 7

2
0





s
R
current a
d
a


R
d
a
a
d






0
2


8. Surface impedance
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 8
 
)
0
(
)
0
(
)
0
(
)
0
(
)
0
(
0
x
z
z
z
z
z
s
s
s
H
E
J
E
dy
y
J
E
iX
R
Z 








I
V
Zs 
 




0
;
)
0
( dy
y
J
d
I
x
dE
V z
z ;
 



0
)
0
(
1
dy
y
J
J
z
z

S=d2
V~
)
0
(
z
E
)
(y
Jz
y
z
x
2
2
2
2
1
2
1
)
( x
s
s
tot H
d
R
I
R
t
P 

2
2
2
1
2
1
/ 










o
rf
s
rf
s
rf
B
R
H
R
P
S
P

)
0
(
z
H

9. Normal metals in the local limit
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 9
  
y
z
z e
J
y
J

 )
0
(
o



2

)
1
(
2
)
1
(
)
0
(
)
0
(
i
i
J
E
iX
R
Z o
z
z
n
n
n 



















2
2
1 o
o
s
s X
R 



 )
( 

n
R
  








t
i
e
J
t
J 
)
0
(
d
Rsquare

 (in DC)

10. Surface impedance practice
• For a bulk metal:
• Thin film over a substrate (metal)?
- Depends on substrate
- Depends on thickness wrt skin depth
• Practice: open two-layers.xlsx
- Thanks to Patrick Krkotić and Uwe Niedermayer for the paper
“Iterative Model for Calculating the Surface Impedance of
Multilayers”, 2016
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 10
2

o
s
s X
R 

d
o





2
o



2


11. Coil measurement from 100 kHz to 2 MHz
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 11

12. How the measurement is done
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 12
• We’ll measure S11
• (i.e. S11) power from port 1 going back into port 1
• 𝑆11 =
𝑃𝑅𝐸𝐹
𝑃𝐹𝑊𝐷
• If no power dissipated by the coil: 𝑆11 = 1
• Output from VNA is always in dB = −10 × 𝑙𝑜𝑔10 𝑆11
• What happens when we put a material (with or without
coating!) facing the coil?

13. Cavity measurement with Keysight VNA up to 1.5 GHz
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 13

14. How the measurement is done
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 14
• We’ll measure S21
• (i.e. S21) power from port 1 going into port 2
• 𝑆21 =
𝑃𝑇𝑅𝐴𝑁𝑆
𝑃𝐹𝑊𝐷
• Output from VNA is always in dB = −10 × 𝑙𝑜𝑔10 𝑆21
• Compare modules with perfect and damaged fingers
• How to identify a resonance?

15. Impedance in accelerators
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 15

16. Impedance: image charges
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 16
Image charges flow on the surface of the beampipe
The wakefields potential is proportional to surface impedance
of the vacuum chamber surface b
s
I
V
Z 
 = 2𝑏

17. Bunch frequency spectrum: LHC case
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 17
Gaussian bunches of 1011 protons, 8 cm long
Beam instantaneous image current
Line spacing:
revolution frequency
(11 KHz for LHC)
From: E. Métral
Frequency spectrum (Fourier transform)
Real bunches
Power dissipation from wakes is
where is a summation of
over the bunch frequency spectrum
( is the accelerator circumference)
( is the vacuum chamber circumference)
)
Re(
2 eff
s
b
loss Z
MI
P 
  s
Z
b
R 
 2
2
eff
s
Z
R

2
b

2

18. Transverse impedance
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 18
 
]
/
[
2
0
m
I
ds
B
c
E
j
Z
R





 


F. Sacherer in: Proceedings of the First Course of the International School of Particle Accelerators, CERN 77-13, p. 198

19. From wakefields to impedance
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 19
Wakefields have an effect on beam stability, in particular the transverse plane
Risetime of resistive-wall instability depends on the transverse impedance
)
Re(
1 eff
T
b
Z
EL
M
I

 s
T Z
b
c
R
Z



3
2

with at the frequency of the unstable mode

20. Betatron motion – simplified model
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 20
0
2
0
, 
 x
x 



)
(
0
0
,
)
(

 

t
j
e
x
t
x
Unperturbed motion of single particle with and
R
c

0

0
0
, 
 x
Q
 x
x R
Q 

Adding a perturbation to the beam that changes focussing:
Fx
x
x 

 2
0
,




Quadrupoles give the restoring (focussing) force -> harmonic oscillations
    0
,
2
0
,
2
0
,
2
2 


 




 F
F 











 





Im
)
Re
(
0
)
( e
e
x
t
x
t
j




 Im
1
Growth rate of instability

21. Anomalous skin effect
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 21

22. Limits for conductivity and skin effect
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 22



1. Normal skin effect if: e.g.: high temperature, low frequency
0
2


 

1
~

2. Anomalous skin effect if: 

 e.g.: low temperature, high frequency
Note: 1 & 2 valid under the implicit assumption 1


1 & 2 can also be rewritten (in advanced theory) as:
  4
3
2
2
1 





It derives that 1 can be true for and also for
1

 1



23. Mean free path and skin depth
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 23
Skin depth
Mean free path




0
2


1
~

.
2
0
const

 


 





F
e
eff
F
e v
m
ne
v
m
ne 


2
2
0 





0
n
neff 

24. Anomalous skin effect
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 24
Understood by Pippard, Proc. Roy. Soc. A191 (1947) 370
Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A195 (1948) 336
Normal skin effect
Anomalous skin effect
Asymptotic value

25. Two-layers
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 25

26. Iterative model
Tutorial: Impedance CAS Vacuum 2017 - S.C. & B.S. 26