Impedance Minimization by Nonlinear Tapering

Transcription

1 Impedance Minimization by Nonlinear Tapering Boris Podobedov Brookhaven National Lab Igor Zagorodnov DESY PAC-7, Albuquerque, NM June 7, 7 BROOKHAVEN SCIENCE ASSOCIATES

2 Outline Motivation Review of theoretical results Optimal boundary EM solvers used Results for impedance reduction Conclusion Z-optimization by non-linear tapering is not that new. Apart from acoustics, used for gyrotron tapers, mode converters, antennae design, etc.

3 Examples of Accelerator Tapers & Motivation and Scope of This Work hmax/hmin~ 9 (gap open) or 7 (gap closed) NSLS-II transition to SC RF cavity A. Blednykh preliminary 88 mm 3.3 mm (ops) mm (inj.) NSLS MGU Taper for X3, X5, X9 & X9 hmax/hmin~ Focus on large X-sectional variations and gradual tapering; study transverse, broadband geometric low frequency inductive regime Goal to reduce Z to avoid instabilities (TMCI) 3 in rings, or H degradation in linacs...

4 Axially symmetric taper ( ) Theory Review iz r' ( z) rect π y ( ) ( ) Z k dz r z iz h' ( z) ( ) ell Zx ( k) = dz 4π h z Flat rectangular taper w h, w>>h iz iz w h' ( z) ( ) Z k = dz 4 h z K. Yokoya, 99 G. Stupakov, 995 Elliptical x-section taper w h, w>>h π w h' ( z) ( ) ell Zy ( k) = dz 6 h z B. Podobedov & S. Krinsky, Z x : h<<l, k<~/h min Z y : w<<l, k<~/w min These are inductive regime impedances. Tapers are gradual to be effective. Functionals lend themselves to simple boundary optimization. 4

5 Optimizing boundaries hz ( ) = h ( + ( h / h ) z/ L) min max min h max linear exponential optimal ver. linear hz h h h / ( ) = min ( max / min ) z L exponential to minimize Z x (or Z h->r) h min hz ( ) = h min / ( + (( hmin / hmax ) ) z/ L) L optimal vertical to minimize Z y Reduced small h(z); big difference when h max /h min >> At h max /h min = predict factor of reduction for Z or Z x, factor of ~3 for Z y 5

6 What Was Done We attempted to check the accuracy of theoretical predictions for impedance reduction by non-linear tapers in axially-symmetric, elliptical, and rectangular geometry using EM field solvers ABCI (axially symmetric) ECHO (axially symmetric & 3D) GDFIDL (3D) 6

7 Wakefield code ECHO (TU Darmstadt / DESY) moving mesh bunch Electromagnetic Code for Handling Of Harmful Collective Effects Zagorodnov I, Weiland T., TE/TM Field Solver for Particle Beam Simulations without Numerical Cherenkov Radiation// Physical Review STAB,8, 5. 7

8 Wakefield code ECHO (TU Darmstadt / DESY) zero dispersion in z-direction staircase free (second order convergent) moving mesh without interpolation in.5d stand alone application accurate results with coarse mesh Model and mesh in CST Microwave Studio Preprocessor in Matlab ECHO 3D Solver Postprocessor In Matlab in 3D only solver, modelling and meshing in CST Microwave Studio allows for accurate calculations on conventional single-processor PC To be parallelized 8

9 Impedance Reduction for Axially Symmetric Tapers Z _exp / Z _lin Z reduction for exponential tapering.8.6 Theory ECHO ABCI r max / r min -Im[Z ],kω/m Re[Z ],kω/m Z (f ) for r max /r min = 8, r min = cm frequency, GHz linear taper exponential taper 5 5 frequency, GHz Z [kω/m] and reduction due to exponential taper agree well with theory Impedance reduction extends through inductive regime (k~/r min ) & beyond 9

10 Geometry for Rectangular Taper Calculations Linear taper Optimal taper L hmax hmin

11 Geometry for Elliptical Taper Calculations cm variable confocal geometry w(z) -h(z) =const. 4: or 8: aspect min X-section x y z Gradual tapers in convex geometry Long straight pipes to avoid interaction between two tapers Each taper subdivided into 4 linearly tapered pieces to approx. nonlinear boundary.

12 Impedance Reduction for Elliptical X-Section Tapers Z x_exp / Z x_lin Z x reduction for exponential tapering Theory GDFIDL, w min /h min = 4 GDFIDL, w min /h min = 8 Z y_opt / Z y_lin Z y reduction for optimal vert. tapering Theory GDFIDL, w min /h min = h max /h min h max /h min Z x [kω/m] and reduction due to exponential taper agree well with theory Z y [kω/m] is less than theory; Z y gets reduced due to optimal taper less than predicted

13 Impedance Reduction vs. Frequency for Elliptical X-Section -Im[Z y ], kω/m Re[Z y ], kω/m frequency, GHz frequency, GHz linear taper optimal vert. taper h max /h min = 8 Z y reduction extends through inductive regime (k~/w min ) & beyond Z x reduction extends through inductive regime (k~/h min ) & beyond 3

14 Impedance Reduction for Rectangular X-Section Tapers Z x_exp / Z x_lin.8.6 Z x reduction for exponential tapering Theory w=h max Z y_opt / Z y_lin.8.6 Z y reduction for optimal vert. tapering Theory w=h max w=*h max.4 w=*h max h max /h min h max /h min Z x [kω/m] and reduction due to exponential taper agree well with theory Z y [kω/m] is less than theory; Z y gets reduced due to optimal taper less than predicted Results are very similar to elliptical structure 4

15 Conclusion For gradual tapers with large cross-sectional changes substantial reduction in geometric impedance is achieved by nonlinear taper. Theoretical predictions for impedance reduction are confirmed by EM solvers for axially symmetric structures and for Z x of flat 3D structures. The vertical impedance gets reduced less than predicted, but the linear taper Z y is lower as well. Optimal tapering for Z x reduces Z y as well and vice versa. Impedance reduction holds with frequency through the entire inductive impedance range and beyond. For fixed transition length, the h(z) tapering we consider appears to be the only knob to reduce transverse broadband geometric impedance of tapered structures. 5

16 Acknowledgements Many thanks to S. Krinsky and G.V. Stupakov for insightful discussions and to W. Bruns, A. Blednykh, and P.J. Chou for help with GDFIDL. We thank CST GmbH for letting us use CST Microwave Studio for mesh generation for ECHO simulations. Thanks to the PAC7 Program Committee for selecting this work for oral presentation. Work supported by DOE contract number DE-AC- 98CH886 and by EU contract 935 EUROFEL. 6

17 EXTRAS 7

18 mm FLASH (DESY) collimators. Tapering Geometry of the step+taper collimator d mm mm 7 mm Bunch length σ z =.5 mm V / pc Loss factor L P ( Δ W P ) V / pc / mm 5 5 d / mm d / mm Collimator geometry optimization. Optimum d ~ 4.5mm Kick factor L ( ΔW )

19 Wakefield code ECHO (TU Darmstadt / DESY) zero dispersion in longitudinal direction. Δz : σ,inecho 3 σ,inmafia L staircase free (second order convergent) W V/pC ECHO MAFIA ( Vz = σ /5) ( Vz = σ /) cells s /cm O(h) O(h ) moving mesh without interpolation C Ωout C indirect integration I Ω z s s z z

20 Convex vs. Concave Not a real ID chamber but rather somethnig to test the theory For fixed mesh size ABCI and GDFIDL are more precise for convex structures Same BB impedance, same theory, but easier EM calculations

21 Results: Horizontal Impedance Z Dx /Z round Ellipse aspect ratio GDFIDL Theory Good agreement between GDFIDL and theory Flat value of Z round / at aspect ratios >~