1. Beam Coupling Impedance for finite length devices
N.Biancacci, B.Salvant, V.G.Vaccaro
Outline:
Motivation
The Mode-Matching Method
Analysis of a simple 3D structure
Outlook

2. 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.
Mode Matching Method for Beam Coupling Impedance Computation
Representation of a circular cross section subdivided in subsets.
Expansion of e.m. field in the cavity by means of an orthonormal set of eigenmodes
Expansion of e.m. field in the radial waveguide by means of radial wave modes
radial waveguide
Expansion of e.m. field in the waveguides by means of orthogonal wave modes
waveguide
cavity

4. Mode Matching Method for Beam Coupling Impedance Computation
Cavity Eigenvector properties
Divergenceless Eigenvectors (dynamic modes):
The Eigenvectors and Eigenvalues are always associated boundary condition:
Cauchy, Dirichlet or mixed.
Irrotational Eigenvectors (static modes):
The boundary condition are relevant to the tangential component of the Electric field.
In a more compact form the static modes may be considered as dynamic modes with kn=0.

5. Mode Matching Method for Beam Coupling Impedance Computation
Explicit expressions for the cavity eigenvectors
With:
cavity
αp is the pth zero of J0(αp)=0

6. Mode Matching Method for Beam Coupling Impedance Computation
Explicit expressions of the e.m. fields in the cavity
Remark: The unknown quantity is the matrix Ips

7. Mode Matching Method for Beam Coupling Impedance Computation
Explicit expressions for waveguide modes
Remark: The unknown quantities are the vectors V1p and V2p
waveguides
Where is the waveguide admittance of the p-mode and k0 is the free space propagation constant.

8. Mode Matching Method for Beam Coupling Impedance Computation
Explicit expressions for the radial waveguide
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.
Where: εT is the dielectric constant in the torus,
Radial waveguide

9. Mode Matching Method for Beam Coupling Impedance Computation
Explicit expressions of the primary fields

10. Mode Matching Method for Beam Coupling Impedance Computation
Matching conditions on the Magnetic fields
Matching on the ports S1 and S2. 1
Matching on the torus inner surface. 2

11. 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.
We have therefore 4 vector equations in 4 unknowns. The problem is in principle solvable.

12. Analysis of simple 3D Model
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. c b r0 L z0 φ0 L
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. S3 S1 S2

13. 1- Use of thick wall formula as a benchmark
b=5cm
c=30cm
L=20cm εr=8

14. 2- Crosscheck with CST – Varying σ (1/5)
b=5cm
c=30cm
L=20cm
εr=1
σ=10^-4

15. b=5cm
c=30cm
L=20cm
εr=1
σ=10^-4

16. 2- Crosscheck with CST – Varying σ (2/5)
b=5cm
c=30cm
L=20cm
εr=1
σ=10^-3

17. 2- Crosscheck with CST – Varying σ (3/5)
b=5cm
c=30cm
L=20cm
εr=1
σ=1

18. 2- Crosscheck with CST – Varying σ (4/5)
b=5cm
c=30cm
L=20cm
εr=1
σ=10^3

19. 2- Crosscheck with CST – Varying σ (5/5)
b=5cm
c=30cm
L=20cm
εr=1
σ=10^4

20. 3- Crosscheck with CST – Varying L (1/5)
b=5cm
c=30cm
σ=10^-2
εr=1
L=20cm
cut off

21. 3- Crosscheck with CST – Varying L (2/5)
b=5cm
c=30cm
σ=10^-2
εr=1
L=40cm

22. 3- Crosscheck with CST – Varying L (3/5)
b=5cm
c=30cm
σ=10^-2
εr=1
L=60cm

23. 3- Crosscheck with CST – Varying L (4/5)
b=5cm
c=30cm
σ=10^-2
εr=1
L=80cm

24. 3- Crosscheck with CST – Varying L (5/5)
b=5cm
c=30cm
σ=10^-2
εr=1
L=100cm

25. Conclusions 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.
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.