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. S11) 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. S21) 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?
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
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
