4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Irina Petrushina 4/26/2013

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Half wave coaxial cavity Frequency, MHz E_surf_max, MV/m5.2 E_y_center, MV/m3.04 Power losses (average), kW2.9 Q12575 Kick voltage, MV1.07 Eff. deflecting voltage, kV262 Proton β (23.5 MeV)0.22 βλ/2, mm135 Deflecting angle, mrad5.0 Geometry and parameters were taken from G. Romanov’s presentation “30 mm gap MHz RF deflector” July 30,2012 Initial geometry Main tasks: Multipacting simulations; Geometry optimization; Coupler tuning; Thermal & cooling simulations; Deformations & frequency shift; Frequency tuning

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Field distribution Electric field Magnetic field

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Optimization The transverse momentum: The Panofsky-Wenzel theorem: Normalized electric field:

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Optimization R 1, mmR 2, mmR sh /Q, ΩP loss, kW Best parameters R 1, mmR 2, mmR sh /Q, ΩP loss, kW Initial parameters

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Higher modes ModeFrequency, MHz Mode 2 (953.5 MHz) TM 011 Mode 3 (1376 MHz) TM 021

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Beam loading Electric field distribution E z (z) at x = 10 mm Deflecting angle If a bunch is shifted from the axis : Ez, V/m Consider the beam coming through the cavity. It will undergo not only the deflecting field: z, mm Power losses due to the beam loading are negligible in comparison with thermal losses

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Multipacting SEY for the copper as received PIC solver TRK solver Multipacting simulations were carried out by CST PS PIC solver & Trk Solver for 2 kinds of the copper surface treatment: copper as-received (dark-blue SEY curve) and copper after the electron bombardment (next slide turquoise one). For the copper as received the particles growth was obtained at levels MV/m. Secondary particles locate in the “corners” of structure and the middle part of the cavity. PIC solver TRK solver

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Multipacting PIC solverTRK solver SEY for copper after the electron bombardment There is no particle growth for the 0-4 MV/m :

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Multipacting An exponential increase in particle number can be described as follows: N ≈ N 0 exp(αt), α – particle number growth rate. Copper as received Copper after the electron bombardment If we don’t take into account oscillatory and just consider the average value – we roughly get the same result for both solvers. Conclusion: there is no multipacting for the copper after the electron bombardment at 0-4 MV/m (α is negative). The probability of multipactor discharge for copper as-received presents, but it’s very low (α<0.05).

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Input coupler Input coupler geometry was taken from the buncher cavity without changes : Frequency, MHz S 11, dB S 21, dB R 3 =29.95 mm R 1 =8.45 mm R2=13.05mm L=71 mm dc=3.9 mm 1 2 Part of coupler with the magnetic loop was cut away and coaxial part with ceramic window was checked for the S-parameters for both frequencies (buncher frequency MHz, deflector frequency MHz)

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE If losses in the walls are considered : Including the roughness of the surface (20%) and angle of the loop rotation 0°: H-plane E-plane First step of the coupler tuning was done using the Frequency domain solver for the structure with 4 couplers (it allows to use the symmetry planes). Loop rotation angle 90°. Coupler loop submerged length (xshift) was varied. Input coupler

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Loop rotation45°90° Q-factorQ0Q0 Q load Q0Q0 Eigenmode solver Frequency domain solver Simulations for the 1 coupler structure were carried out by Eigenmode solver and Frequency domain solver for 2 angles of the loop rotation. Example of the results: Finally tuned structure: Q 0 = , Q load = 5500 (45° loop rotation, 20 % roughness is included ) L coupler Input coupler

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Thermal loss calculations The copper skin depth : Power losses in the walls: Inner surface Outer surface Top & bottom surfaces Cavity walls cooling Outer walls cooling : copper tube loops with diameter Flow rate v – velocity of water in the channel Reinold’s number μ – dynamic viscosity Nusselt coefficient Temperature rise between input and output water Heat transfer coefficient λ – thermal conductivity of water, Prandtl number Friction number Temperature rise of the tube walls

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Inner conductor cooling: coaxial channel. Hydraulic diameter of the coaxial channel: Cavity walls cooling T, °Cv, m/sG, l/sh, W/m 2 KΔT in-out, °CΔT w, °C T, °Cv, m/sG, l/sh, W/m 2 KΔT in-out, °CΔT w, °C Parameters of water in the loops Parameters of water in the coaxial channel These parameters were used for all simulations

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Cavity walls cooling 2 loops cooling Max. temperature rise ΔT = 163°C 2 loops + coaxial channel Max. temperature rise ΔT = 28°C ΔT in_part ≈38°C 4 loops + coaxial channel Max. temperature rise ΔT = 16.8 °C ΔT out ≈ 2.5°C Simulations: Various combinations of the cooling channels were considered. Temperature of water T=20°C, water velocity v = 6 m/s Thermal conductivity for copper:Wall thickness Temperature rise of the inner and outer parts of the cavity was calculated by formulas:

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Coupler cooling Without coupler cooling Max. temperature rise ΔT=21°C With a cooling loop at the coupler Max. temperature rise ΔT=16.7°C

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Mechanical simulations for the cavity without coupler Temperature map ABS displacement Max. displacement: ΔR ≈ 26.7 μm ΔL ≈ 26.7 μm

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Temperature map ΔT long =4.3 K ΔT R =11 K Mechanical simulations for the cavity with coupler Thermal expansion coefficient for copper Max. displacement in radial and longitudinal directions were calculated using formulas. ABS Displacement ΔL = 23.5 μm R ΔRx = 23.3 μm ΔRz = 21.6 μm X Z ABS bottom displacement ABS top displacement Temperature map (top) L Simulation results: Estimations:

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Frequency shift calculations The resonant frequency shift can be induced by: thermal deformations fabrication inaccuracy. Frequency shift range can be calculated from: 1.CST MWS simulations: simulations for the original and deformed structures. 2.CST MWS field integration: simulations for the original structure, field integration (Template based post processing) + Slater perturbation theorem 3. Mathcad: CST MWS field export -> Field integration&Slater perturbation theorem using Mathcad 4. Slater perturbation theorem estimations (“by hands”) The Slater perturbation theorem describes the shift of the resonant frequency, when a small volume ΔV is removed from the cavity of volume V: W – storage energy

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Parameter Sensitivity, MHz/mm CST MWS simulations Slater Perturbation Theorem Mathcad CST MWS Estimation ΔRΔR ΔLΔL ΔR1ΔR ΔR2ΔR ΔR 1 & ΔR ΔR fillet_ ΔR fillet_ ΔL coupler ΔL gap Expecting frequency shift due to the temperature rise, kHz ΔR=23 μm ΔL=23 μm Frequency shift calculations Max. frequency shift due to the thermal deformations is about 50 kHz. If we suppose that the fabrication accuracy will be 0.1 mm max. frequency shift will be 250 kHz R fillet_1 R fillet_2

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Frequency tuning Slater perturbation theorem Tuning range has to be about 250 kHz (frequency shift due to the thermal deformations and the fabrication accuracy). Two tuners are located in the bottom area of the cavity (coupler is in the upper one). Tuning cylinder has a 5 mm rounding radius for the electrical discharge and breakdown prevention. Frequency shift vs. tuner submerged length for various cylinder radiuses (slater perturbation theorem)

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Frequency tuning Frequency shift vs. tuner submerged length for various cylinder radiuses

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Shape of the deflecting plates Round platesRectangular plates 2 mm 8 mm ParameterRectangular plateRound plates Q Power loss, kW R sh /Q E_surf_max, MV/m

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE z y x Deflecting voltage Shape of the deflecting plates

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Alternative cooling channel Inner conductor of the cavity was made hollow for the total cavity mass reducing. It allows to reduce the mass of copper from 6 kg to 3 kg. Alternative cooling channel for the hollow conductor allows effective cooling of the deflecting plates.

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Full structure Frequency, MHz E_surf_max, MV/m6.65 E_y_center, MV/m2.88 Power losses (average), kW2.69 Q Proton β (23.5 MeV)0.22 βλ/2, mm135 Deflecting angle, mrad5.0

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Conclusions

4/26/2013 Irina PetrushinaDeflecting cavity MHz for PXIE Thank you for your attention!