Machine:Impedance Budget

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Analytic Calculations

History of Analytic Studies of Taper Impedance

Yokoya[1] first estimated the impedance of tapers in the low frequency regime and concluded it is imaginary:

[2] [3] [4] [5] [6] [7]

Vacuum Chamber Impedance

The standard dipole and straight section vacuum chambers will be round, made of copper and coated with NEG (see Table 1). We model them as cylindrical tubes using the formulas presented in reference [8]. Curvature and radiation exit effects are neglected and the effect of NEG coating is considered.

Besides the standard chamber, Sirius will have special chambers for the fast corrector magnets that will operate at a rate of 10 kHz. These chambers will be made of thin foil of stainless steel (SS316L), brazed to the copper chamber and also coated with NEG.

Table 1: Main parameters used to estimate the vacuum chamber wall impedance.
NEG thickness 1.0 um
NEG conductivity 4.0 MSm-1
Chamber inner radius 12 mm
Conventional chamber
Chamber thickness 1 mm
Copper conductivity 59 MSm-1
Copper relaxation time 27 fs
Length 480 m
Fast Corrector chamber
number of correctors 80
Chamber thickness 0.3 mm
SS316L conductivity 1.3 MSm-1
Length 0.1 m

Undulators

We considered three types of undulator for Sirius, as seen in Table 2. In this initial budget, the impedance of the Superconducting Wiggler (SWC) was not taken into account, because this device is shorter, just one will be installed and the larger gap makes its impedance negligible compared to the others.


Table 2: Sirius operation phases for beam lifetime and instability threshold calculations.
Phase 0
(commissioning)
Phase 1
(initial user mode)
Phase 2
(final user mode)
Maximum total current 100 100 350 mA
Current/bunch (uniform fill) 0.116 0.116 0.405 mA
Single bunch current - - 2 mA
RF Cavities 1 NC (Petra 7-cell) 2 SC 2 SC + 3HC
Natural emittance* 0.25 0.21 0.15 nm rad
Coupling 3 3 3  %
IDs - 6 IVUs, 2 EPUs 12 IVUs, 8 EPUs
Natural energy spread* 0.085 0.084 0.083  %
Natural bunch length* 2.4 2.3 11.6 mm
* considering the zero current case only.

Table 3 shows the main parameters used in the undulator impedance modeling. There are two types of IVU with different gaps because they will be installed in sections with different betatron functions.

For the undulators it is possible to separate the impedance in two main contributions: the geometric impedance from the tapered transitions and the chamber wall impedance from the low gap sections. The first one is difficult to calculate, because it depends on the detailed design of the component and is not available yet. For this reason it will be included in the budget as a contribution to the broad band model.

The second contribution, the wall impedance, was estimated using the mulitlayer formulas for the round chamber and applying the Yokoya factors [9]. to correct the results for the flat chamber. This approach is well justified for the EPUs, where the vaccum chamber will be made of copper with an elliptical cross-section with large eccentricity.

For the IVUs, the magnetic poles will be made of permanent magnet (NdFeB) array, with different polarization directions, and a thin sheet of copper will be placed to shield the beam and reduce heating and other impedance issues. These ferromagnetic materials restrict the applicability of the multilayer formulas, because they were derived under the assumption of linear materials with longitudinal symmetry. However, considering that the fields generated by the magnets themselves were already taken into account on the beam dynamics and that the strength of the electromagnetic fields generated by the beam in the materials is small, it is possible to model this magnets as linear materials with some unknown relative magnetic permeability . This parameter was varied, showing a weak dependence on the impedance, which led to the assumption presented in Table Table 3, as a worst case estimate.

Table 3: Main parameters used on the undulator model.
Elliptically Polarizing Undulators
Length 2.7 m
Chamber thickness 1 mm
Chamber eccentricity > 0.4
Chamber minor axis 10 mm
In-Vacuum Undulators
Full gap 4.5 and 7.8 mm
Copper sheet thickness 75 um
of NdFeB 100
of NdFeB 0.625 MSm-1
Length 2 m

Injection Kickers

The Sirius injection kicker will have the window frame architecture and will be placed outside vacuum. Its ceramic vacuum chamber will be coated with Ti. Figure 1 shows a simplified scheme of the magnet.

Figure 1: Transverse cross section of the Sirius injection ferrite kickers.


The modeling of its impedance considers two main contributions: the coupled flux, driven by the coupling of the beam with the external circuits, via the endplates of the window frame, and the uncoupled flux, generated by the interaction of the electromagnetic fields of the beam with the lossy materials that compose the magnet.

Regarding the coupled flux, based in references [10] [11], a simplified model of the external circuit was considered. In this model, the generator impedance is which represents a RC circuit in parallel. Together with the inductance of the window frame, this impedance generates a beam impedance with peak centered at ~50 MHz. If needed, this model can be improved with results from measurements with magnets, once the prototypes are produced.

The contribution from the uncoupled flux involves a study of the effect of the coating and the ceramic chamber to the impedance. To evaluate this effect we modeled the kicker as a round chamber with several layers: Ti coating, ceramic, vacuum, ferrite and copper. Even though all the information about the geometry was lost with this approximation, it was possible to determine approximately the frequency in which the coating begin to damp the impedances, the shape of the impedance around this frequency and confirm that it is low enough to damp all the resonances introduced by the ceramic.

Differently from the high frequency behavior, the low frequency impedance may depend very strongly on the geometry and properties of the magnet beyond the coating. In this range, the Tsutsui model for the uncoupled flux [12] [13] applies and was used to fit the low frequency value for the beam impedance of the magnet.

References

  1. K. Yokoya, Tech. Rep. SL/90-88 (AP) (CERN, 1990).
  2. B. Podobedov and S. Krinsky, PRST-AB 10, 074402 (2007)
  3. G. Stupakov, PRST-AB 10, 094401 (2007)
  4. G. Stupakov, et al. PRST-AB 10, 054401 (2007)
  5. K. Bane, et al. PRST-AB 10, 074401 (2007)
  6. G. Stupakov, Preprint SLAC-PUB-7039 (SLAC, 1995)
  7. G. Stupakov and B. Podobedov, Preprint SLAC-PUB-14093 (SLAC, 2010)
  8. Mounet, N. and Metral, E., CERN-BE-2009-039, Nov. 2009.
  9. Yokoya, K., Part. Accel. 41 221-248, April 1993
  10. Nassibian, G. and Sacherer, F., Nucl. Inst. and Methods 159 21-27 (1979).
  11. Davino, D. and Hahn, H., Phys. Rev. ST Accel. Beams 6, 012001, January 2003.
  12. Tsutsui, H., CERN-SL-2000-004, January 2000.
  13. Tsutsui, H. and Vos, L., CERN-SL, LHC Project Note 234, September 2000.

Electromagnetic Designs

The electromagnetic design of the Sirius components results from simulations of time and frequency domains and wakefield analysis, aiming the impedance optimization in order to reduce heating and increase collective instabilities thresholds. ECHO code and GdfidL softwares are used for the analysis, through the LNLS local cluster (uv100) and SINAPAD/LNCC infrastructure. Figure 2 describes the electromagnetic designs workflow.

Figure 2: Electromagnetic designs workflow.

After optimizing the strucure, the design is sent for the mechanical project design and prototyping phases, involving Projects, Vacuum and/or Materials groups. Mecanical constraints noticed from the former or the later phases may require modifications and further EM simulations, until this whole iterative process converges. The final part of the workflow consists of simulations with finer meshes and lower bunch length for a precise impedance evaluation to be added to the machine impedance budget.

Beam Position Monitor

One of the main concerns in the BPM Button design, especially for the storage rings with short, high-current circulating bunches, is heating due to Higher Order Modes (HOM’s) [1]. The BPM Button geometry and the materials choice need to be optimized from impedance and heat transfer points of view to avoid BPM Button overheating due to the resonance modes generated between the housing and the BPM Button itself by a passing bunch [2].

Impedance Optimization

The work has been started from analysis of already existed BPM Button geometries and from their optimization process. Blednykh, Ferreira and Krinsky [2] have shown the possibility of resonance modes suppression due to changes in geometric dimensions of housings for the BPM Button. We were interested in further impedance optimization of the BPM Button geometry and applied different dielectric materials to understand their effect on the broad-band and narrow-band impedances. Several dielectric materials as the vacuum insulator have been considered with high thermal conductivity to provide good heat transfer between the BPM Button and the BPM housing. Aluminum Nitride (AlN) and Boron Nitride (BN) composite materials have been chosen for analysis as shown in Table 4 below.

Table 4: Dielectric materials considered for simulations.
Considered Properties Value (AlN - BN)
Permittivity, 9 - 4
Thermal Conductivity, 160 - 46 W/K.m
Loss Tangent, , @1 MHz 0.003 - 0.0034

Several button geometries have been analysed. The process of choosing the three candidates (Figure 3) was based on keeping the best two geometries (Bell-Shaped Button and Flat Button) from the electromagnetic analysis point of view and having the Step-Shaped one as a safe plan. The last one (Figure 3b) is based on a well-known button geometry implemented at ALBA [5]. The Flat Button geometry (Figure 3c) has been designed in a way to hide the vacuum insulator from the beam. The Bell-Shaped geometry (Figure 3a) has a special form, which not only allows us to also hide the insulator, but to increase the button cut-off frequency, without losing its sensitivity, since its bottom face area was kept the same as the other geometries.

Figure 3: Engineering drawing of the BPM Buttons for: a) Bell-Shaped BPM Button. b) Step-Shaped BPM Button. c) Flat BPM Button.

The BPM Button diameter is taken to be 6 mm for all geometries to provide about 100 nm position resolution at 100 mA. This resolution considers a 15 dB noise figure (cables + electronics) integrated in a 2 kHz bandwidth. The gap is 0.3 mm for each button. The instrumentation feedthroughs are designed for 50 Ω SMA connectors.

The longitudinal narrow-band impedance is shown in Figure 4. In the BPM Button geometries with BN insulator the lowest resonant modes exist at higher frequencies than in geometries with AlN. It agrees with the frequency dependence of . The frequency shift of the lowest mode is varied within 1-2 GHz for considered geometries. The lowest mode is the superposition of the electromagnetic fields in the gap and in the ceramics material.

Figure 4: Real part of the longitudinal impedance for: a) Bell-Shaped BPM Button. b) Step-Shaped BPM Button. c) Flat BPM Button. Bunch profiles are in gray color for σS = 2.65 mm (solid) and σS = 5.3 mm (dotted).

As a conservative approach we can assume that the bunch length in the Sirius storage ring with passive third-harmonic Landau cavities will be extended at least by a factor of two, making all geometries as HOM’s free ones. The loss factor due to single bunch was calculated as a function of bunch length for all three geometries using the GdfidL simulated or . Summary results of are shown in Table 2 for different . In the BPM Buttons design and in their optimization process, the results and experience at other synchrotron radiation facilities [6-8] were taken into account. As can be seen from Table 5 two new optimized BPM Buttons, Bell-Shaped and Flat, have the advantage over the standard Step-Shaped BPM Button at any .

Table 5: Summary results of the loss factor for 2.65, 4.5, and 6 mm bunch lengths, for the BPM Button geometries including the ceramics materials, AlN and BN.
, mV/pC
Geometry Bell-Shaped Step-Shaped Flat
, mm 2.65–4.5–6.0 2.65–4.5–6.0 2.65–4.5–6.0
AlN 4.5–0.9–0.3 8.9–2.5–0.8 5.6–0.9–0.2
BN 3.6–0.8–0.4 8.3–2.3–0.7 4.6–0.7–0.2

The power loss goes from 1.9 to 4.4 W, for = 2.65 mm, at = 500 mA withn M = 864 bunches and 1.73 μs revolution period :

Mechanical Aspects

The the current BPM Assembly Project of the Bell-Shaped button is shown in Figure 5 in a perspective view, so as the first prototype made found in Figure 6:

Figure 5: Perspective view of the Bell-Shaped BPM mechanical design.
Figure 6: First BPM prototype.

Based on experience at other facilities [4], the coaxial cavity effect of the BPM Housing with the Body needs to be eliminated. The idea of using threads came up to obtain more benefits beyond the RF shielding: a better thermal contact between the housing and the body is required. After tightened with appropriated torque, the housing can be welded to the BPM body. However, it is well known that such threads can cause virtual leaks that can violate vacuum requirements. In order to reduce this problem, some valleys were designed to work as gassing channels, highlighted in Figure 7.

Figure 7: Housing of the Bell-Shaped Button, with highlighted gassing channels: a) Side channel. b) Bottom channel.

Other mechanical aspects are briefly discussed:

• Reverse polarity connectors: the button pin can be a male conductor, avoiding the need of BeCu spring fingers that can be easily damaged. The price is a drawback;

• Constant pin diameter: brings improvements in mechanical resistance and in the thermal contact area between the pin and the ceramics. Also it allows to have the button and the pin as a single piece;

• Tiny dimensions require very small tolerances and precise alignment. It makes the brazing process harder (small filler sites, short circuit risks, etc.);

• Materials choice: The higher the button electrical conductivity, the lower the button power dissipation [8]. Brazing and welding process play also a big role on materials choice;

• Vacuum insulation between the BPM body and housing: brazing and laser welding are the considered alternatives under evaluation.

Flanges

The mechanical design of the Sirius flanges, shown by Figure 8, was based on KEK MO-flange.

Figure 8: Modified MO flange mechanical design.

The electromagnetic analysis have shown that the loss factor and the HOMs magnitudes are neglibible.

Synchrotron Radiation Masks

In order to understand the impact of different profiles, several nomenclatures and cases were analyzed. Figure 9 cases of combinations between the longitudinal profiles, called as soft (1/10 tapered transition), hard (1/1 tapered transition) and step.

Figure 9: Examples of longitudinal mask profiles: soft-hard, hard-hard and hard-step.

Combined with several transverse configurations these profiles were used as basis for creating the studied 3D cases, shown by Figure 10.

Figure 10: Different analyzed mask profiles.

Some conclusions were made for the studied cases:

• Axisymmetric: free of HOM’s, but very large broadband impedance;


• Single ridge, 5mm width:

- Approx. 8x lower broadband impedance than axisymmetric;

- Suffers from several quite strong HOM’s;


• Single ridge, increasing to 10mm width:

- Broadband impedance: ~20% increase;

- HOM’s: from 25% to complete damping;


• 10 mm width, adding other ridge (2-ridge type):

- Broadband impedance: 2x higher;

- HOM’s: less HOM’s, but the remaining ones were more than 3x higher!


• Keeping with single ridge, w10: Parabola shapes were not succesfull. Full width was slightly better!


And the conclusions regarding the geometric parameters were made as well:

• Taper in: a steeper taper causes a more defined stair-shape of the broadband impedance;

• Taper out: a steeper taper increases the slope of the broadband impedance – does not affect the narrowband impedance;

• Iris height (h): broadband impedance increases linearly with it and narrowband one increases quadratically – actually h is the only parameter which affects narrowband impedance


A final sketch was made in order to guide VAC group to determine the best possible configurations considering their ray-tracing analysis. Figure 11 shows the isolines (in orange) that give the best trade-off between narrowband (pink) and broadband (purple) impedance isolines.

Figure 11: Parameters sweep isolines for mechanical design.

Gate Valves

The analyzed gate valve from VAT was evaluated in impedance point of view. The RF-shielding fingers and the flanges were optimized as much as the mechanical constraints have allowed.

RF-shielded Bellows

Figure 12: Comb-type bellows.
Figure 13: Finger-type (standard) bellows.
Figure 14: Omega-strip bellows.
Figure 15: Loss factor comparison between standard (finger-type) and omega strip bellows.
Figure 16: Real part of the longitudinal impedance comparison among different stripe widths (or slot widht): 8 mm (2.2 mm slot), 9.5 mm (150 um slot) and without slots.
Figure 17: Loss factor comparison among different stripe widths (or slot width): 8 mm (2.2 mm slot), 9.5 mm (150 um slot) and without slots.

Dipole Chambers

The dipole chamber profile (see Figure 18) with 6 mm slot width has proved that does not significantly contribute with the impedance budget. A preliminary design of the infrared extraction chamber (see Figure 19) was made in order to make an initial comparison with the standard one.

Figure 18: Standard dipole chamber vacuum profile.
Figure 19: Infrared extraction dipole chamber vacuum profile.

Initial comparison shows that if the extraction mirror is far enough (24 mm or more) from the electrons path, the IR chamber behaves almost the same as the standard one.

Striplines

Tapered Cavity vs. Capacitive Gap Types

The tapered cavity stripline (NSLS-2-like) was compared with the capacitive gap (SOLEIL-like) one.

Figure 20: Tapered cavity stripline model.


Figure 21: Parallel plates capacitive gap stripline model (left) and its transverse-cut profile (right).


Figure 22: Real part of longitudinal impedance comparison between tapered cavity and capacitive gap stripline types.

Despite lower shunt impedance, we opt for the capacitive gap model in order to save the feedthroughs from the incoming electromagnetic high power from the beam.

Alternative Capacitive Gap Types

Alternative gaps in order to achieve options not only in increasing the gap capacitance but also in mechanical assembly possibilities.

Figure 23: Standard-type (parallel plates) gap.
Figure 24: Upper gap type.
Figure 25: Sliding-type gap.
Figure 26: Comb-type gap.
Figure 27: Loss factor comparison among the studied capacitive gap types.
Figure 28: Real part of longitudinal impedance comparison among the studied capacitive gap types.

The comb-type stripline, which has shown better EM performance, is being analyzed and possibly designed by the Projects Group.

Pumping Station

The pumping station slots configuration is going to be optimized in terms of electromagnetic performance. The initial design is shown by Figure 29.

Figure 29: Pumping station: a) Entire view and b) Vertical cut.