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