Outline: Motivation The Mode-Matching Method Analysis of a simple 3D structure Outlook Beam Coupling...
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Transcript of 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.
Mode Matching Method for Beam Coupling Impedance Computation
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:
Mode Matching Method for Beam Coupling Impedance Computation
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
Mode Matching Method for Beam Coupling Impedance Computation
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
Mode Matching Method for Beam Coupling Impedance Computation
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.
Mode Matching Method for Beam Coupling Impedance Computation
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
Mode Matching Method for Beam Coupling Impedance Computation
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.
Mode Matching Method for Beam Coupling Impedance Computation
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.
Mode Matching Method for Beam Coupling Impedance Computation
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
b=5cmc=30cmL=20cmεr=1σ=10^-3
2- Crosscheck with CST – Varying σ (2/6)
b=5cmc=30cmL=20cmεr=1σ=10^-3
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.
2- Crosscheck with CST – Varying σ (3/6)
~1700Ω?
~21000Ω!
b=5cmc=30cmL=20cmεr=1σ=1
2- Crosscheck with CST – Varying σ (4/6)
b=5cmc=30cmL=20cmεr=1σ=10^3
2- Crosscheck with CST – Varying σ (5/6)
b=5cmc=30cmL=20cmεr=1σ=10^4
2- Crosscheck with CST – Varying σ (6/6)
b=5cmc=30cmσ=10^-2εr=1L=20cm
3- Crosscheck with CST – Varying L (1/5)
cut off
b=5cmc=30cmσ=10^-2εr=1L=40cm
3- Crosscheck with CST – Varying L (2/5)
b=5cmc=30cmσ=10^-2εr=1L=60cm
3- Crosscheck with CST – Varying L (3/5)
b=5cmc=30cmσ=10^-2εr=1L=80cm
3- Crosscheck with CST – Varying L (4/5)
b=5cmc=30cmσ=10^-2εr=1L=100cm
3- Crosscheck with CST – Varying L (5/5)
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