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## Erice Lecture 2 Padamsee

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**Topics for Today**• Why Elliptical Shape? • Multi-cells • Couplers • Input power • Higher Order Mode • Tuners**Multipacting in Nearly Pill-Box Shaped CavitiesThe Folly of**Youth! Early SRF cavity geometries frequently limited by multipacting, usuallyat Eacc< 10 MV/m H. Padamsee**Multipacting as Seen in Q vs E curve**H. Padamsee**Multipacting in Nearly Pill-Box Shaped Cavities**Thermometers show heating in barriers H. Padamsee**MP is due to an exponential increase of electrons under**certain resonance conditions Multipacting Low Field High Field H. Padamsee**Multipacting**Cyclotron frequency Resonance condition: Cavity frequency (g) = n x cyclotron frequency Possible MP barriers given by H. Padamsee**Field Levels for Barriers**H. Padamsee**Simulated Trajectories**H. Padamsee**MP only active for these impact energies**Multipacting, Secondary Emission Coefficient • Not all potential barriers are active because electron multiplication has to exceed unity. H. Padamsee**Electrons drift to equator**• Electric field at equator is 0 • MP electrons don’t gain energy • MP stops 350-MHz LEP-II cavity (CERN) Multipacting Solution • Solved multipacting by adopting a spherical, (later -elliptical) shape. H. Padamsee**Two Point Multipacting**H. Padamsee**Many MP Simulation Codes Exist**H. Padamsee**Two Side Multipacting Simulation**H. Padamsee**Q: Why is two point MP not as harmful as One point was?**H. Padamsee**Multicells**• One of the parameters to vary • Number of cells • A large number makes for structure economy but entails • trapped HOMs, • field flatness sensitivity to tuning errors, • and calls for high power input per coupler. H. Padamsee**Multicell Cavity**Modes 9-cell cavity H. Padamsee**Dispersion Relation**H. Padamsee**Simplified Circuit Model of MultiCells**H. Padamsee**Solve the circuit equations for mode frequencies**Dispersion Relation Mode spacing increases with stronger cell to cell coupling k Mode spacing decreases with increasing number of cells N H. Padamsee**Aperture and Cell-Coupling**H. Padamsee**Field Flatness**• Stronger cell-to-cell coupling (k) and smaller number of cells N means • Field flatness is less sensitive to mechanical differences between cells H. Padamsee**Mechanical Properties and Cavity Design**Cavity should not collapse or deform too much under atmospheric load Shape avoid flat regions Elliptical profile is stronger Choose sufficient wall thickness Use tuner to bring resonance to right frequency Differential thermal contraction due to cool-down induces stress on the cavity walls. 24 H. Padamsee**Mechanical Design**To avoid plastic deformation the cumulative mechanical stress on the cavity walls must not exceed the cavity material yield strength, including some engineering margin. The frequency shifts due to these stresses must be taken into account for targeting the final frequency or tuner settings and tuner range. Stresses due to the operation of the tuner mechanism should not exceed yield strength while cold. The mechanical requirements may be dealt with by proper choice of cavity wall thickness or by adding stiffening rings or ribs at locations of high strain. H. Padamsee 25 H. Padamsee**Stress Calculations**• Codes such as ANSYS or COSMOS determine structural mechanical properties and help reduce cavity wall deformations in the presence of mechanical loads and vibrations by choosing the appropriate wall thickness or location of stiffening rings or ribs.**Beta 0.65**Mechanical design • Von Mises stresses for 1.5 bar @ 300K < 50 MPa with 4mm 46 MPa Cavity walls = 4mm Niobium cost ~70 k€ beam axis H. Padamsee**COSMOS stress calculation results for the b = 0.5, 700 MHz**elliptical cavity. • Without conical stiffener, the maximum stress is 54 MPa. • (b) With conical stiffener at the optimum location, the maximum stress drops to 11.8 MPa**ANSYS stress calculations for the triple-spoke resonator,**350 MHz, = 0.4. The peak stress is 15 MPa [2.98]. • FNAL single spoke resonator β=0.22 and a 30 mm aperture β=0.22 and 325 MHz diameter [2.99]. Each end wall of the spoke resonator is reinforced by two systems of ribs: a tubular rib with elliptical section in the end wall outer region and six radial daisy-like ribs in the inner region (nose).**Ponderomotive effects**• Ponderomotive effects: changes in frequency caused by the electromagnetic field • – Static Lorentz detuning (CW operation) • – Dynamic Lorentz detuning (pulsed operation) • Microphonics: changes in frequency caused by connections to the external world • – Vibrations • – Pressure fluctuations • Note: The two are not completely independent. When phase and amplitude feedbacks are active, the ponderomotive effects can change the response to external disturbances. • The electromagnetic fields in a cavity exert Lorentz forces on the cavity wall. The force per unit area (radiation pressure) is given by**Lorentz-force detuning**Coupling parameter b • The Lorentz forces near the irises try to contract the cells, while forces near the equators try to expand the cells. • The residual deformation of the cavity shape shifts the resonant frequency of the accelerating mode from its original value by where DV is the small change in the cavity volume. • In the linear approximation, the steady-state Lorentz-force frequency shift at a constant accelerating gradient is • The quantity KL is called the Lorentz-force detuning constant. • The 9-cell TESLA cavities have KL = 1 Hz/(MV/m)2.**Lorentz-force detuning can be evaluated using a combination**of mechanical and RF codes (e.g., SUPERFISH and Microwave Studio). H. Padamsee 32 H. Padamsee**H. Padamsee**33 H. Padamsee**H. Padamsee**34 H. Padamsee**The resonant frequency shifts with the square of the field**amplitude distorting the frequency response. Typical detuning coefficients are a few Hz/(MV/m)2. A fast tuner is necessary to keep the cavity on resonance, especially for pulsed operation. A large LF coefficientcan generate “ponderomotive” oscillations, where small field amplitudeerrors initially induced by any source (e.g. beam loading), cause cavity detuning through Lorentz force and start a self-sustained mechanical vibration which makes cavity operation difficult. H. Padamsee 35 H. Padamsee**Stiffeners**Stiffeners must be added to reduce the coefficient But these increase the tuning force. For the TESLA-shape 9-cell elliptical structure the LF detuning coefficient is about 2 - 3 Hz/MV/m2 resulting in a frequency shift of several kHz at 35 MV/m, much larger than the cavity bandwidth (300 Hz) chosen for matched beam loading conditions for a linear collider (or XFEL). Stiffening rings in the 9-cell structure reduce the detuning to about 1 Hz/MV/m2 at 35 MV/m pulsed operation. H. Padamsee 36 H. Padamsee**Feedforward techniques can further improve field stability.**• In cw operation at a constant field the Lorentz Force causes a static detuning which is easily compensated by the tuner feedback, but may nevertheless cause problems during start-up which must also be dealt with by feedforward in the rf control system.**Microphonics**• External vibrations couple to the cavity and excite mechanical resonances which modulate the rf resonant frequency - microphonics. • => Amplitude and phase modulations of the field becoming especially significant for a narrow rf bandwidth. H. Padamsee**H. Padamsee**Examples of vibration modes of a 7-cell, 1.3 GHz cavity. The active length of the cells is 80 cm. Modes from top to bottom are: transverse, longitudinal, and breathing (ANSYS simulations) 39 H. Padamsee**Input and HOM Couplers**H. Padamsee**Input Power Coupler - Functions**- Provides power to make up for wall losses at Eacc - Provides beam power = beam current x Vgain Definition of Coupling Strength in terms of Q Defines Qhole or Qexternal R/Q comes up again and again !**Coupler Types**• Waveguide • Can carry more power, lower power density • Only one conductor needs cooling • Large • Coaxial • Compact • Easier to make variable • Two conductors • Cooling is more complex**Design Aspects**• Microwave transmission properties • Standing wave and travelling wave patterns • Cooling of high power carrying regions • Minimization of static heat • Interception of static heat • Variable coupling • HOM vulnerability • Antimultipactor geometry • Windows • Number • Placement, warm or cold or both • Antimultipactorstrategies: simulations, coatings, bias…**Fabrication issues, assembly, cryomodule interface**• Vacuum ports • High power testing, conditioning • Diagnostics**TTF3 Coupler Description**PMT Pump-out port 70 K Cold window 4.2 K 1.8 K Warm window e- probes • Designed for 5kW average power, 500 kW pulsed power , 1% duty factor • Variable Qext range: 1106 to 2107 (calculated) for 15mm antenna movement • Cylindrical RF windows made of 97.5% Al2O3 with TiN coating • Cold coaxial line: 70Ohm, 40mm OD Warm coaxial line: 50Ohm, 62mm OD • All s.s. parts are made of 1.44 mm thick tubes • Copper plating is 30mm thick on inner conductor and 10mm thick on outer conductor • There are two heat intercepts: at 4.2K and at 70K**RF simulation of TTF-III input coupler in standing wave**operation. • Windows are placed at the electric field minimum**3D CAD rendering of the variation of TTF-III coupler for 75**kW CW operation**S11 parameter of the Cornell ERL injector coupler for a**range of coupling values (due to different bellows’ extension/compression). The value of dl corresponds to the antenna travel relative to the middle position. (b) Calculated temperature profile [8.54Vadim].