JMD 4DESY-TEMF DRM · 2019. 12. 9. · Title: JMD_4DESY-TEMF_DRM Author: dohlus Created Date:...
Transcript of JMD 4DESY-TEMF DRM · 2019. 12. 9. · Title: JMD_4DESY-TEMF_DRM Author: dohlus Created Date:...
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
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
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
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
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*:
ph
ase
/deg
ph
ase
/deg
6 kHz
3 MHz1.5 GHz
phase of reflection coefficient:
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
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
phase of reflection coefficient (again):
phase/d
eg
phase/d
eg
phase/d
eg
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
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
deviation from flat top
spectrum of “pickup” signal and stimulation
a,b
“pickup” signal
spectrum of “pickup” signal and LP filter
a,b
“pickup” signal after LP filter
a,b
spectrum of “pickup” signal and BP filter“pickup” signal after BP filter
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
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
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
= −
= −
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
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
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
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%!
simulation for network with tolerances
correlation between field flatness and relative spectral deviation
RS
D
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.
( )( )
( )( )
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= =
-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
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