outline: motivation the mode-matching method analysis of a simple 3d structure outlook beam coupling...

Outline: Motivation

The Mode-Matching Method

Analysis of a simple 3D structure

Outlook

Beam Coupling Impedance for finite length devicesN.Biancacci, B.Salvant, V.G.Vaccaro

Motivation

1- Why finite length models?

• Real life elements are finite in length

• Usually 2D models are supposed to be enough accurate to obtain a quantitative evaluation of the beam coupling impedance

• In case of segmented elements, or where the length becomes comparable with the beam transverse distance, this hypothesis could not work any more.

• We suppose the image currents are passing through the surface of our element.

3

Representation of a circular cross section subdivided in subsets.

radial waveguide

radial waveguide

Expansion of e.m. field in the cavity by means of an orthonormal set of eigenmodes

Expansion of e.m. field in the waveguides by means of orthogonal wave modes

Expansion of e.m. field in the radial waveguide by means of radial wave modes

Mode Matching Method for Beam Coupling Impedance Computation

waveguide

waveguide

cavity

4

In a more compact form the static modes may be considered as dynamic modes with kn=0.

0;0 nnnnnnnn eekhhhke

nnn

nnn

hIH

eVE

ode Matching Method for Beam Coupling Impedance Computation

nnn

nnn

nnn

nnn

gGhIH

fFeVE

Cavity Eigenvector properties

Divergenceless Eigenvectors (dynamic modes):

Irrotational Eigenvectors (static modes):

0;0 nnnnnn ggff

The Eigenvectors are always associated to a homogeneous boundary condition

The boundary condition are relevant to the tangential component of the Electric field.


5

Explicit expressions for the cavity eigenvectors

ˆ)(cosˆ),(),(

ˆcosˆsin1

ˆˆ

1

0122

rzkL

zrhzrh

zrzkkrrzkkL

ε

kkzzr,erzr,ezr,e

bps

spsps

bpsp

bpss

s

sp

zpspsps

)(

)/()(

1

11

q

qbq

Jb

brJr

)(

)/()(

1

00

q

qbq

Jb

brJr

bk

L

sk p

ps

;

αp is the pth zero of J0(αp)=0

With:


cavity

6

Explicit expressions of the e.m. fields in the cavity

Lz

rzkL

zrH

rzkkL

ZzrE

rzkLk

kjZzrE

sp

bps

s

sp

bpss

sr

sp

bps

spz

0

)(cos,

)(sin,

)(cos,

,1

,10

,0

00

ps

ps

ps

I

I

IRemark:The unknown quantity is the matrix Ips


7

0

exp,

exp,

exp,

2201

01

22011

22001

z

kkjzrZ

YzrH

kkjzrzrE

kkjzrk

YkjzrE

pp

bp

bp

pp

bpr

pp

bp

bpp

z

1p

1p

1p

V

V

V

zL

kkLzjrZ

YzrH

kkLzjrzrE

kkLzjrk

YkjzrE

pp

bp

bp

pp

bpr

pp

bp

bpp

z

2201

02

22012

22002

exp,

exp,

exp,

2p

2p

2p

V

V

V

Explicit expressions for waveguide modes

Where is the waveguide admittance of the p-mode and k0 is the free space propagation constant.

bpY

Remark:The unknown quantities are the vectors V1p and V2p


waveguides

8

Explicit expressions for the radial waveguide

zkrkHckH

ckHrkHA

kjH

zkrkHckH

ckHrkHA

k

kE

zkrkHckH

ckHrkHAE

ss

s

ss

ss

s

Ts

ss

s

ss

ss

s

srs

ss

s

ss

sszs

cos)()(

)()(

sin)()(

)()(

cos)()(

)()(

)1(1)1(

0

)2(0)2(

1

)1(1)1(

0

)2(0)2(

1

)1(0)1(

0

)2(0)2(

0

22sTs kkk

TTk

Where:εT is the dielectric constant in the torus,

Remark:The unknown quantity is the vect or As

In the above equation the boundary conditions are already satisfied at r=c and z=0,L.


Radial waveguide

9

zkjbγβ

kK

bγβ

kI

rγβ

kI

rγβ

kK

πγβ

qkr,zH

zzkjbγβ

kK

bγβ

kI

rγβ

kI

rγβ

kK

πγβ

kqZr,zE

βzkjbγβ

kK

bγβ

kI

rγβ

kI

rγβ

kK

βπγ

kjqZr,zE

r

z

00

0

00

01

01

00

00

0

00

01

012

000

00

0

00

00

0022

000

exp2

,exp2

exp2

Explicit expressions of the primary fields


10

Matching conditions on the Magnetic fields

brLzrHLzrH

brzrHzrH

0,,

00,0,

1Matching on the ports S1 and S2.

2 LzzbrHzbrHzbrH 0,,,0

Matching on the torus inner surface.


More involved is the matching of the Electric field because of the boundary conditions associated to the cavity eigenvectors. Namely the tangential component of the Electric field is strictly null on the entire close surface S.

However we may obtain that the matching can be reached by means of non-uniform convergence of the eigenvector expansion.

This can be achieved by an ad-hoc algebra of the expansion of the e.m fields.

This will lead to the following expressions for the Ips coefficients.

321

,,,,0,0,222

0

0

S

*pstot

S

*pstot

S

*pstot

sp

dSzbhzbEdSLrhLrEdSrhrEkkk

jkYpsI

We have therefore 4 vector equations in 4 unknowns. The problem is in principle solvable.


12

b

Analysis of simple 3D Model

c

L

0 Lz

S2S1

S3

A simple torus insertion will be used to study the effect of a finite length device on the impedance calculation.

The Mode matching Method was applied to get the longitudinal impedance.

To proof the reliability we set up 3 benchmarks:

1- Comparison with the classic thick wall formula for high values of sigma.

2- Comparisons with CST varying the conductivity.

3- Comparisons with CST varying the length.

b=5cmc=30cmL=20cmεr=8

1- Use of thick wall formula as a benchmark

c

Z

b

LjZ long

0

22

1

2- Crosscheck with CST – Varying σ (1/6)

b=5cmc=30cmL=20cmεr=1σ=10^-4


Increasing the scan step and magnifying, with the mode matching we can easily detect very high resonances which may not look as they really are.


~1700Ω?

~21000Ω!

b=5cmc=30cmL=20cmεr=1σ=1


b=5cmc=30cmL=20cmεr=1σ=10^3


b=5cmc=30cmL=20cmεr=1σ=10^4


b=5cmc=30cmσ=10^-2εr=1L=20cm

3- Crosscheck with CST – Varying L (1/5)

cut off

Outlook

● Extend the model to compute the transverse impedances, dipolar and quadrupolar.● Compare the case of complex permittivity in CST. ● Investigate the mismatch between CST and the Mode Matching code for high values

of conductivity.

The Torus model

● We presented an application of the Mode Matching Method on a simple 3D model, a torus.

● We successfully computed the longitudinal coupling impedance by means of this method.

● A series of benchmarks, with analytical formulae and simulations, let us consider our analysis enough reliable: the comparison with the thick wall formula is in close agreement for high values of σ; the comparison with CST is in well agreement for the low conductivity case, increasing the conductivity they exhibit a different behavior.

Conclusions

outline: motivation the mode-matching method analysis of a simple 3d structure outlook beam coupling...

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