Outline: Motivation The Mode-Matching Method Analysis of a simple 3D structure Outlook Beam Coupling Impedance for finite length devices N.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.

Representation of a circular cross section subdivided in subsets. 3 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 cavity

In a more compact form the static modes may be considered as dynamic modes with k n =0. 4 ode Matching Method for Beam Coupling Impedance Computation Cavity Eigenvector properties Divergenceless Eigenvectors (dynamic modes): Irrotational Eigenvectors (static modes): 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 α p is the p th zero of J 0 ( α p )=0 With: Mode Matching Method for Beam Coupling Impedance Computation cavity

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

7 Explicit expressions for waveguide modes Where is the waveguide admittance of the p-mode and k 0 is the free space propagation constant. Remark: The unknown quantities are the vectors V 1p and V 2p Mode Matching Method for Beam Coupling Impedance Computation waveguides

8 Explicit expressions for the radial waveguide Where: ε T is the dielectric constant in the torus, Remark: The unknown quantity is the vect or A s 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 Explicit expressions of the primary fields Mode Matching Method for Beam Coupling Impedance Computation

10 Matching conditions on the Magnetic fields 1 Matching on the ports S 1 and S 2. 2 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 I ps coefficients. 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 0L z S2S2 S1S1 S3S3 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=5cm c=30cm L=20cm ε r =8 1- Use of thick wall formula as a benchmark

2- Crosscheck with CST – Varying σ (1/6) b=5cm c=30cm L=20cm ε r =1 σ=10^-4

b=5cm c=30cm L=20cm ε r =1 σ=10^-3 2- Crosscheck with CST – Varying σ (2/6)

b=5cm c=30cm L=20cm ε r =1 σ=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=5cm c=30cm L=20cm ε r =1 σ=1 2- Crosscheck with CST – Varying σ (4/6)

b=5cm c=30cm L=20cm ε r =1 σ=10^3 2- Crosscheck with CST – Varying σ (5/6)

b=5cm c=30cm L=20cm ε r =1 σ=10^4 2- Crosscheck with CST – Varying σ (6/6)

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

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

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

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

b=5cm c=30cm σ=10^-2 ε r =1 L=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