impedances and wake fields: they are forty, but they don’t look it vittorio giorgio vaccaro...
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Impedances and Wake Fields:They are forty, but they don’t look it
Vittorio Giorgio Vaccaro
Università degli Studi di Napoli “Federico II” and
INFN Sezione di Napoli
Impedances and Wake Fields: Why?
On 1966 the design of a large Intersecting proton Storage Ring (ISR) started. It was supposed to accumulate at 30GeV the largest current ever reached. Bad news were coming from USA. At MURA in the electron storage ring, the accumulator current could not go beyond a limit. Above this threshold there was evidence of transverse instabilities, which were interpreted as the presence of electrodes. This evidence produced in the scientific community the conviction that the in the beam dynamics the interaction with the surrounding equipment should not be neglected (e.m. field produced by image currents).
In the ISR a lot of additional equipment was present in the ring (for instance, almost 600 clearing electrode plates). It was not clear what type of interaction the whole complex might have with the beam. In addiction to this, a large number of cavities were foreseen and many other equipment, such as bellows, had to be studied in this respect.
Impedances and Wake Fields: Why?
An example of beam-equipment interaction
Beam-discontinuity interaction
RF e.m. field
Particle bunches
8
Considering a Fourier harmonic in the charge distribution of a beam. It will generate a current, interacting with the surrounding medium, will generate electromagnetic fields acting back on beam itself. This action may consist into an enhancement of the perturbation itself. This may trigger an avalanche effect that may limit the beam.
This occurrence will be more strong as more high is the charge of beam bunches.
The Devices
9
The DevicesThe variation of the cross section in accelerators cannot be avoided. Two examples: the accelerating cavity gap and the bellows.The first one is by definition an essential device for carrying out the function itself of the accelerator. The second one is strictly necessary to absorb the thermal stresses and mechanical fabrication tolerances. Hence, the wake fields are unavoidable.
10
A DAΦNE accelerating cavity example:
Waveguides literally “absorb” HOMs from the cavity
The Devices
11
MITSUBISHI example:
C-Band choke mode type accelerating structure (Shintake cavity). The “choke” influence cavity modes, damping the High Order Modes (HOMs). It appears to the fundamental mode as a closed cavity (short circuit)
The Devices
Instabilities for pedestrians
The lowest modes which may occurs
The starting point was the study of instabilities: the beam in a circular accelerator behaves (because it is circular!) as a feedback device. Circularity is the core of the feedback device indeed (main difference with respect to linacs).
1) This device, likewise all feedback devices, may be unstable when some conditions are fulfilled.
2) The stability conditions are dictated not only by the dynamical properties of the beam (charge/mass, energy, magnetic focusing, etc.), but also by the type of electromagnetic interaction with the environment (response of the machine, since 1966 called Coupling Impedance).
Instabilities for pedestrians
Instabilities for pedestriansThe gedanken experiment.
An ideal device, fed by a voltage ΔV(Ω), produces a small longitudinal density perturbations in the current I at a frequency Ω
)(0 III
Suppose that there is no interaction with surrounding medium and measure the perturbation by means of a reciprocal device which may define the perturbed current ΔI(Ω)
)()()( VYI B
where the BEAM ADMITTANCE YB(Ω) depends on the beam properties:
charge/mass, focusing properties etc. Of course the perturbed current is proportional to the d.c. current I0
Instabilities for pedestriansTaking into account the e.m. interaction on one turn of the beam with the environment, this interaction will produce e.m. forces having the same space and time distribution as the perturbation. These forces can be represented by an equivalent voltage ΔV1’(Ω) which can be meant as produced by the current ΔI(Ω),
loading an IMPEDANCE. This impedance, ZM(Ω), represents the overall
interaction with surrounding equipment.
)()()('1 IZV M
This voltage now acts back again on the beam, producing an additional perturbation, and so on. After m turns we have:
)()()()('1 VYZV BM
)()]()([)(' VYZV m
BMm
Instabilities for pedestrians
The process is circular:
1. The forcing source ΔV(Ω), acting on the particles, produces the effect ΔI(Ω). (the effect is governed by the particle dynamics)
2. The effect ΔI(Ω) becomes the forcing source of the electromagnetic fields, namely ΔV1’(Ω), which is the integrated forces over one turn.(the effect is governed by Maxwell’s equations with the appropriate boundary conditions)
3. The effect ΔV1’(Ω) becomes the forcing source…….and so on.
Instabilities for pedestrians
Intuition suggests that there will be no amplification, if
1)()( BM YZ or )()( 1 BM YZ
In previous equations we see that the stability is possible if, at certain frequency, the e.m. interaction with the machine (l.h.s.) is lower than a quantity, which depends on the properties of the beam (r.h.s.)
Remember
)()()( VYI B
Instabilities for pedestriansstability charts for mono-energetic beams
Stability chart below transition energy Stability chart above transition energy
Instabilities for pedestriansstability charts for real beams (spread in energy)
Landau damping
A realistic distribution function Stability chart for realistic distribution function
Instabilities for pedestriansstability charts for real beams (spread in energy)
A stability criterion for longitudinal stability:
0
222
0|| )/(||
Ie
ppcmF
n
Z
Stability chart for realistic distribution function
n = harmonic number
e = elementary charge
I0 = stored current
Δp = momentum spread
η = slippage
Instabilities for pedestriansThe concept of impedance, originally conceived for longitudinal dynamics was
extended according to the phenomenon to be studied. Therefore we have:
1. Transverse impedance
2. Longitudinal wake field
3. Transverse wake field
4. Loss factor
etc.
dzezwZ v
zi
||||
dzez v
zi
wZ
dfZK2
)(||
The Impedances and their estimation
In parallel to the studies been equipment interaction, various problems were tackled.
1. Transverse stability charts
2. Bunched beam stability criteria: (single bunch, multi-bunch, etc.)
etc.
Since their birth an intensive effort was done in order to calculate (BEIC-Beam Equipment Interaction Committee) and to measure the coupling impedances. This effort started at CERN around 1968 and is still lasting.
Huebner, Zotter, Palumbo, Ruggiero and many others were active in the calculation tecniques.
In the measurement techniques Sands and Rees, Sacharer and Nassibian and many others were active.
Elliptic Vacuum Chamber
sourcesk
t
2
22
In the case of relativistic particles the EM fields can be described resorting to a quasi-static representation
Therefore all the EM-fields can be derived by the following equation
Flatness parametershw
hwq
24t
setoffeq
t
setoffeq
Xjba
ZRjZ
n
Xj
a
bZj
n
Z
22220
220
11
ln2
1
2220
20
11
2
1
'
'
ln
eqt
eq
ba
ZRjZ
a
bZj
n
Z
A new definition of impedances
)|()|()|( 00
0
0
2
0
20
2
0
2
0 PPZc
PPZ
yyxy
yxxxcPPZ t
Elliptic Vacuum Chamber
n
n
nn
n
imimself
q
qnnnn
q
qnnnn
where
hwPPPPPP
Zj
n
PPZ
11
82
24
12
10000
000200
sinsinhsinsinhcoscoshcoscosh
;ln)|(;)|()|()|(
We report the sentence by Morse and Feschbach about the sum "As all these Green's function expansions, the series is only conditionally convergent and should not differentiated unless the poorly convergent part can be condensed into a closed function". Anyway the series has been condensed in a closed form as a linear combination of Jacobi Θ-functions .
Elliptic Vacuum Chamber
)(
1
)(
1
sin2
)(
sin2
)(ln
,2
ln24
1)|(
442444
31
14
11
0
qqL
whereLLhw
PPim
)|()|(
0200 2
PPZ
jn
PPZ
0
011
0220
0 )()()|(2
)|(
t
tt PhPhPPRZ
jPPZ
The result of these computations is given, for the image potential, by the following formula,
Elliptic Vacuum Chamber
Kk
qhwbdove
K
a
bZj
n
PZeq
eq
2
4
2sin
2sn
lnln2
1)(
0
0
200
a
bZj
n
Z eqln2
1)0(2
0
',
2cncoshln
)0()(002
00 kKZ
jn
Z
n
PZ
Elliptic Vacuum Chamber
L. Palumbo, V.G. Vaccaro, “Coupling Impedance Between Circular Beam and a LossyVacuum Chamber in Particle Accelerators”, Il
NuovoCimentoVol. 89 (1985).
Vittorio G. Vaccaro Green function’s for a lossy elliptical vacuum chamber. Seminar at CERN (1992)
Francesco Ruggiero. “Resistive wall impedance as derivative of the electric capacitance for a beam pipe of arbitrary cross section.”
Phys. Rev. E 53, 2802 - 2806 (1996)
14
24
12
12
21
11)(
2
2222
2
22220
0
K
q
K
q
k
hwb
Xjba
ZRjPZ
eq
tsetoff
eqt
Elliptic Vacuum Chamber
Measurements: Hera Vertex Chamber
Impedance measurement setup of “Mazinga”
Measurements: Hera Vertex Chamber
Measurements: Hera Vertex Chamber
MEASUREMENT OF THE LONGITUDINAL COUPLING IMPEDANCEOF THE HERA-B VERTEX DETECTOR CHAMBERV.G. Vaccaro & alii, EPAC96
32
Analytic EM characterization of some
structures
The fields induced by the beam passing through discontinuities and changes of the vacuum chamber cross section (irises, accelerating cavities, bellows, etc.) may be particularly
critical for the beam stability. Therefore in the design of an accelerating machine it is necessary to have simple and accurate
numerical tools in order to evaluate the effects of these fields.
33
The adopted method
1. Wave Matching Technique (WMT): a well tested technique which describes the em-field making use propagating guide modes with orthogonality properties in the cross section (twofold domain).
2. Mode Matching (MMT): beside the guide modes, it resorts to resonant modes with orthogonality properties in a threefold domain. It is possible to insert losses.
34
An examle of MMT: Shintake the cavity.
A
A
A
A
B
B
B
B
B
B
B
B
A zones: cavities
B zones: waveguides
A
A
A
B B
B
B
B
B
Measurements: Loss Factor of a DUT
Measurements: Loss Factor of a DUT
Cavity longitudinal loss factor measurement by means of a beam test facility. V.G. Vaccaro & alii Phy. Rev. S. T. - volume 3, 2000
Impedance calculations (FMT)
An instructive particularexample: abrupt junction.
Longitudinal coupling impedance of an abrupt junction in avacuum chamber V. G. Vaccaro & Al. Nuovo Cimento A 1999
Impedance calculations (FMT)
Impedance calculations (FMT)
Impedance calculations (FMT)
Impedance calculations (FMT)
Impedance calculations (FMT)
Impedance calculations (FMT)
44
Mode Matching Technique
The expansion on a complete set
VPPhGPHIPH
PeFPEVPE
n mmmnn
mmm
nnn
)()()(
)()()(
00
0
0
0
0
22
22
nn
nnn
nnn
nnnn
nnnn
HnEnPECforBC
HkH
EkE
HHkE
EEkH
ˆ;ˆ: 00
0
0
0
0
22
22
mm
mmm
mmm
mmm
mmm
nPECforBC
hh
ee
ˆ;:
The definition of a complete set
22
22
n
s nS nn
n
n
nnnn
kk
SdnExHjkSdnHxEkI
kk
Sdns ExHkSdnS HxEjkV
..
..
Mode Matching Technique
46
Mode Matching TechniqueAll methods subdivide the space affected by EM field in subspaces in which the field is representable through a linear combination of orthogonal configurations
•The numerical codes based on Mode Matching method are more simple to be implemented, but sometimes they show slowness to reach convergence. •The numerical codes based on Modal Matching are more difficult to be implemented, but the convergence is reached more quickly.
On the separation surface between subspaces, the field continuity equations are turned in linear system of infinite equations of infinite unknowns. The coefficients of the linear system are the unknowns of the problem.
47
How to verify the numerical methods
•In order to verify on a real device the goodness of simulations, one needs to measure the fundamental parameters as the Loss Factor or the Coupling Impedance: not easy to realize in small labs.•In Loss factor case, a relativistic bunch generator is required in order to shot particles through the device and measure the kinetic energy loss. •Moreover, any measure of energy loss must be done as a difference between two entities very similar to each other, leading to a variance in the same order of the measure or bigger. •An indirect measure is required in order to achieve reliable measurements.
48
The studied models :The Choke mode cavity
(Shintake).
bunch
49
1999 2008
The models to be studied :The Shintake cavity array
It is used for an High performer LINAC allocated as F.E.L. feeder (according to MITSUBISHI)
The maquillage is suspicious: remark the absence of the fundamental mode…
50
Wire method as a testGoodness of numerical methods is achieved “modifying” the Device
Under Test (D.U.T.) : a wire, which substitutes the beam, is inserted on the DUT axes . The response to a known signal is measured by means of
the scattering matrix (reflection and transmission response).
Then the measured scattering parameters are compared with the results of a numerical algorithm applied on the same D.U.T. It is necessary to
underline that this method is not useful to determine the parameters of the original device, but it is useful to validate numerical results only. In
fact it perturbs the D.U.T. modifying its parameters.
51
The simulation method adopted is the Modal Matching technique. It is visible the agreement between
simulations and measures.
Wire method as a test
52
The studied models :The Choke mode cavity
(Shintake).
A
A
A
A
B
B
B
B
B
B
B
B
A zones: cavities
B zones: waveguides
The wire is taken in account only for central cavity
calculus
53
Wire method as a test
-40
-35
-30
-25
-20
-15
-10
-5
0
2 3 4 5 6 7 8 9
S21 misurato S21 teorico
freq [GHz]
Shintake cavity: good agreement even in this case
S21
[dB
]
S21 measured
S21 simulated
54
As a preliminary study. A typical case of discontinuity which generates wake fields
Thick Iris.
Devices excited by a beam:
55
Good convergence at the resonant frequency already with few modes.
REMARK: the real part (as expected) is exactly null below the waveguide cutoff frequency!
Insensitive to variation of resistivity from zero to 107
Waveguide radius= 15mm
Iris radius= 3 mm
Iris length= 7.5mm
= 10
Devices excited by a beam: Thick Iris.
Normalized k
56
The most simple accelerating structure.
We have verified the agreement between M.M. and Mo.M. methods for an ideal structure first (PEC boundaries), then we
tested the same structure inserting losses.
Losses can be inserted for modal matching technique only
Devices excited by a beam: Pill Box.
57
In PEC case, we can see the cutoff frequency of the waveguide and the resonances of the cavity separated.
Waveguide radius= 10mm
Cavity radius= 60mm
Cavity length= 80mm
= 10
Devices excited by a beam: Pill Box.
58
Waveguide radius= 10mm
Cavity radius= 60mm
Cavity width = 80mm
= 10
Copper: =5.98•107S/m
Steel: =6.00•10 5 S/m
new peak in real part, depending on the conductivity
Devices excited by a beam: Pill Box.
Devices excited by a beam: Pill Box.
Waveguide radius= 15 mm
Cavity radius= 41 mm
Cavity width= 28 mm
= 10
Copper: =5.98•107S/m
Steel: =6.00•10 5 S/m
new peak in real part, depending on the conductivity. Strong influence
60
The work goes on
• Electromagnetic characterization of cavity prototypes based on PBG structures with increased performances
• Apply Modal Matching technique to more complicate structures
• Insert losses to those ideal structures
• Use the same theory to calculate transverse coupling impedance
61
Mode Matching Technique: a hybrid procedure
bunch
Numerical technique to solve in the cavity Maxwell’s equations with homogeneous boundary condition, in order to find inside the cavity the modes and the relevant eigenvalues. Resort to MMT to match on the dashed boundaries to match the waveguide modes with the cavity modes.
62
END
Panorama of Naples Gulf
mm V
m
V
mmm
S
mm
mm
V
m
EdVEJjkZdVHJkdSEEnkk
edVeJjk
ZE
022
0
)ˆ(1
mm S
m
V
mm
V
mmm
mm
V
mm
S
m
HdSHEnjkYdVHJjkYdVEJkkk
hdVhJdShEnjk
YH
)ˆ(1
)ˆ(
0022
0
Impedance calculations
An instructive particularexample: abrupt junction.
Longitudinal coupling impedance of an abrupt junction in avacuum chamber V. G. Vaccaro & Al. Nuovo Cimento A 1999
Instabilities and Impedances
Instabilities and Impedances
Instabilities and Impedances
Instabilities and Impedances
Instabilities and Impedances
Instabilities and Impedances
71
Motivations
Collective instabilities limit the current stored in a modern high-energy and high-intensity accelerator. They could arise either from direct electromagnetic interaction of the particles in the same bunch or indirectly. In fact there is an EM interaction between beam and surrounding equipment that implies an induced current. This current allows production of EM field acting back on the beam.
72
The models to be studied :The Fermi Linac intrasections.The LINAC for FERMI project will be used as feeder of a F.E.L. Knowing the characteristics of a beam used for this application, it is fundamental to study wake fields and minimize their effect. In fact they will be enhanced by the high currents generated for this extreme application