jmd 4desy-temf drm · 2019. 12. 9. · title: jmd_4desy-temf_drm author: dohlus created date:...
TRANSCRIPT
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Discrete Resonator Model
Maxwell approach:
DRM from eigenmode expansion
example: Tesla cavity → cavity signals
empiric approach:
network models for (quasi) periodic cavities
field flatness and cavity spectrum
field flatness and loss-parameter
transient detuning
summary/conclusions
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DRM from eigenmode expansion2
2t tµε µ∂ ∂∇×∇× + = −
∂ ∂E E J
2
ν ν νµεω∇×∇× =E E
( ) ( ) ( ), t tν να=E r E r
( ) ( )2
2
2t
t tν ν νµε ω α µ
∂ ∂+ = − ∂ ∂ E r J
12
dV Wν µ ν νµε δ= E E ( )( )
22 1
22t dV
t t
g t
ν ν ν
ν
ω α ∂ ∂→ + = − ∂ ∂ E J�����
( ) ( )( ) ( ) ( ) ( )( )12p p p pq t t g t q t tν νδ= − → = ⋅J r r r r E rɺ ɺ
( ) ( )2
2
0 2
1d du t i t
dt C dtω
→ + = −
C( )u t
( )i t
0ω
( ) ( ) ( )( )0
0
1cos
t
u t i t dC
τ ω τ τ= − −
beam
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0
0 0
g hd
dt
ν ν ν ν ν
ν ν ν
α ω αβ ω β
+ = − −
EoM + eigenmode approach
1 for 0ν ν ν
ν
ωω
= ∇× ≠B E
( ) ( ) ( )( ) ( ) ( )t t
t t
ν ν
ν ν
α
β
=
=
E E r
B B r
( )
( ) ( ) ( ){ }, ,
d
dt
dq t t
dt
µ µ
µ µ µ µ
=
= + ×
r v p
p E r v p B r
with
( ) ( ) ( )1
2
mW
g t dV qν
ν ν µ µδ= = −E J J v r r
( ) ( ) ,port~h t a b dAν ν ⊥− E E
,port~a b dAν νν
α ⊥+ E E port coupling
dynamic equations for
state quantities
“instantaneous” equations
all particles µ and
all modes ν
beam
port
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port modes in matrix formalism
( )
2 t
Wm
d
dt
ω
ω
+ = − −
+ = −
0 Dα α g V a - b
D 0β β 0
a b V D α
2 t
Wmd
dt
ω
ω
↓
− = −
α αVV D D
β βD 0
beam g
port a = all forward waves
b = all backward waves
modes α = all E mode-amplitudes
β = all B mode-amplitudes
all coefficients can be calculated from EM eigenmode results
(but MWS does not support it)
the lossy eigenmode problem: , = =a 0 g 0
but it does not work too well if one considers not enough modes
example: TESLA cavity with modes below 2 GHz
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example: TESLA cavity
Xport =-193
∆xpen= 8
f/GHz Q/1E6
0 0
0.721865 0.000001
1.2763 88.2703
1.27838 22.6809
1.28157 10.5396
1.28551 6.32979
1.28973 4.41263
1.29373 3.41628
1.29701 2.87407
1.29917 2.57969
1.29991 4.97314
f/GHz Q/1E6
0 0
0.833513 0.000001
1.2763 44.2134
1.27838 11.3918
1.28157 5.31692
1.28551 3.21128
1.28973 2.25295
1.29373 1.7553
1.29701 1.48468
1.29917 1.33733
1.29991 2.57894
direct calculation (modes below 2GHz):
* compare:
Dohlus, Schuhmann, Weiland: Calculation of Frequency Domain Parameters Using 3D
Eigensolutions. Special Issue of International Journal of Numerical Modelling:
Electronic Networks, Devices and Fields 12 (1999) 41–68
improved calculation*:
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ph
ase
/deg
ph
ase
/deg
6 kHz
3 MHz1.5 GHz
phase of reflection coefficient:
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the improved method
( )
2 t
Wm
jω
ω
ω −
= − −
+ = −
0 D V a bα α
D 0β β 0
a b V D α
ɶɶɶ ɶ
ɶ ɶ
ɶɶ ɶ
frequency domain (quantities with tilde)
( )( )ω
↓
+ = −a b Z a bɶ ɶ ɶɶ ɶ ( ) 2 2
2 tjWν ν ν
ν
ωωω ω
=−Z v vɶwith
( )
( ) ( )
2 2 2 21
2 2Nt t
N
k u
j jj W W
j j
ν ν ν ν ν νν νν ν
ω ωωω ω ω ω
ω ω= >
= +− − Z v v v v
Z Z
ɶ
��������� ���������ɶ ɶ
known part unknown part
this part is smooth in the frequency range of the known part;
with some additional information one can find a good approximation
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f(PMC)/GHz
0
0.8612027
1.276303833
1.278377530
1.281569543
1.285508949
1.289734390
1.293732580
1.297014913
1.299167855
1.299908996
f(PEC)/GHz
0.429000747
1.276303633
1.278376664
1.281567291
1.285503885
1.289722730
1.293696870
1.296413920
1.297209500
1.299220100
1.299932200
example: TESLA cavity
( ) ( ) 0k PEC u PECZ j Zω ω+ =ɶ ɶ( )k PMCZ jω = ∞ɶ
one-port system: Zu can be calculated
for PEC resonance frequencies
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phase of reflection coefficient (again):
phase/d
eg
phase/d
eg
phase/d
eg
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excitation of the modes* 1 to 11 by an ultra-relativistic point particle (q=1C)
( ), 0
V/m
zE z r =
these are the first 11 modes without divergence (div E) in vacuum
modes 3-11 are related to the first band of monopole modes
zq
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port stimulation
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
t [sec] 10-3
-0.2
0
0.2
0.4
0.6
0.8
1-a vs time
t/Tg==n
t/Tg==n+1/4
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
t [sec] 10-3
-1
-0.5
0
0.5
1b vs time
t/Tg==n
t/Tg==n+1/4 this is time domain, ~ 5E6 periods
stimulation ωg on resonance of “pi-mode”
blue curves are sampled at t = nTg
they can be interpreted as real part
red curves are sampled at t = nTg + Tg/4
they can be interpreted as real part
fill mode ln 2πτ τ=
a,b
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deviation from flat top
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spectrum of “pickup” signal and stimulation
a,b
“pickup” signal
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spectrum of “pickup” signal and LP filter
a,b
“pickup” signal after LP filter
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a,b
spectrum of “pickup” signal and BP filter“pickup” signal after BP filter
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empiric approach:
network models for (quasi) periodic cavities
geometry
1 resonance, 2x1 ports, electric coupling
2 resonances, 2x2 ports, magnetic coupling
K. Bane, R. Gluckstern: The Transverse Wakefield of a Detunded
X-Band Accelerator Structure, SLAC-PUB-5783, March 1992
1 resonance, 2x1 ports, magnetic coupling
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approach from network theory
a,b
c1 c2 c3 c4 c5 c6 c7 c8 c9b1 b2… … …… … … …… ……
b1, b2: boundary blocks with n respectively n+1 ports
c1, c2, … c9: cell blocks with 2n ports
simplifications: c2, … c8 (or c1, ... c9) are identical and symmetric
use periodic solutions of 3d system to characterize cell blocks
special treatment of boundary blocks
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a,b
system with beam
zq
beam ports with delay are connected in series
c1 c2 c3 c4 c5 c6 c7 c8 c9b1 b2… … …… … … …… ……
τ 2τ 3τ 4τ 5τ 7τ6τ 8τ 9τ
τia(t)
ib(t)
ua(t)
ub(t)
( ) ( )( ) ( )
b a
b a
i t i t
u t u t
ττ
= −
= −
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c… …
the general symmetric 2n-port network
1
2
3
n
n+1
n+2
n+3
2n
( ) 2 21
M jj jν
ν ν
ωω ωω ω ∞
=
= +−Z A Aɶ
( )( )
( )
( )
( )( )
( )
1 1
2 2
2 2n n
u j i j
u j i jj
u j i j
ω ωω ω
ω
ω ω
=
Z
ɶɶ
ɶɶɶ
⋮ ⋮
ɶɶ
with
( ) ( )t t t tν νν ν ν ν ν
ν ν
∞
= +
=
r sA r s s r
s r
A ⋯
and
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9
9
ii
i
f f
F F∆ = −
i∆
RSD
RSD is a measure for the change of the mode spectrum of one cavity compared
to itself
relative spectral deviation
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field flatness and mode spectrum
Is there a direct relation between the field flatness of the accelerating
mode and the resonance frequencies of modes in the same band?
test it for a simpler problem:
even simpler:1iɶ
2iɶ
1C2C
cC
1 1L = 2 1L =
1 2
1 2
1.25 1 11 1,
5 1 12 2c
C C
C
= = − → = = =
i iɶ ɶ
1 2 1 2
110 10 1 1 3, , 10 3 ,
2 211 3 9 3 3 1cC C C
−= = = → = = − − i iɶ ɶ
no: both setups have the same eigen-frequencies, but different flatness/eigenvectors
1
2
4 5
6 5
ω
ω
=
=
1
2
i
i
=
iɶ
ɶɶ
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field flatness and loss parameter
again the discrete network:
loss-parameter:( ) ( )
2 212 2
2 2 2 2
0 0
1 11 1
4 2 2tot
C i iVk
W L i C iν
ν νν ν νπ
ν ν
ωω ω
−− −= ≈ ≈
ɶ ɶ
ɶ ɶ
( ) ( )1 1i xν
ν ν− +ɶ ∼
pi-mode
deviation from flat field
1 9ν = ⋯
( )( )
2
2 2
1 1
19 1
xk
xx
ν
νν
+≈
++
∼ 0xν ≈if
for σx = 0.1 the loss parameter is reduced by only 1%!
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simulation for network with tolerances
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correlation between field flatness and relative spectral deviation
RS
D
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time dependent detuning
( )
2 t
d
dt
ω
ω
= − −
+ = −
0 Dα α V a - b
D 0β β 0
a b V α
( )( )
( )( )
( )2 t
t t
t
t
d
dt
ω
ω
= − −
+ = −
0 Dα α V a - b
D 0β β 0
a b V α
?
for instance no excitation (a = b) →( )
( )
0 0
0
0
t
t
d
d
ω
ω
τ τ
τ τ
= −
0 Dαα
ββD 0
?
This would mean all the modes are independently ringing, there is no mode
conversion.
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( )( )
( )( )
t
t t
t t
d
dt
ω
ω
= + −
0 R 0 D 0 Rz q
R 0 D 0 R 0
example:
with 1
2
0
0ω
ωω
=
D
( ) ( ) ( )( ) ( )
cos sin
sin cos
t tt
t t
ϕ ϕϕ ϕ
= −
R
system matrix is anti symmetric → energy conservation with tW z z∼
eigenvalues are time invariant { }1 2,j jλ ω ω= ± ±
but eigenvectors are time dependent
next slide:
numerical calculation for
the effect of mode conversion is weak if the time scale of ϕ(t) is long compared
to the resonances
1 21.98 , 2.02ω π ω π= =
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-100 -80 -60 -40 -20 0 20 40 60 80 100-1
-0.5
0
0.5
1
z1(t)
-100 -80 -60 -40 -20 0 20 40 60 80 100-0.1
-0.05
0
0.05
0.1
z2(t)
-100 -80 -60 -40 -20 0 20 40 60 80 100
t
0
0.02
0.04
0.06
0.08
0.1
(t)
no excitation
0=q
initial state:
ω1 resonance
is ringing
final state:
both resonances
are ringing
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summary/conclusion
• eigenmode expansion is effective to analyse cavity signals
in the frequency range of the first monopole band →signals seen by couplers and pickups
• pickups are more sensitive to non-accelerating monopole
modes than the beam
• eigenmode expansion is standard for long range effects
empiric approach: discrete network
Maxwell approach: field eigenmode expansion
• discrete network models allow qualitative insight
• it is easy to analyze discrete models and to consider random effects
• it is difficult to relate network parameters to geometric properties
and imperfections; it is in principle possible
• loss-parameter is very insensitive to field flatness; but the peak field
is sensitive!
• no sharp correlation between relative spectral deviation and flatness
• it is possible to calculate time dependent resonance, but modeling
requires (some) caution
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