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Impedance Minimization by Impedance Minimization by Nonlinear Tapering Nonlinear Tapering Boris Podobedov Boris Podobedov Brookhaven National Lab Brookhaven National Lab Igor Zagorodnov Igor Zagorodnov DESY DESY LER-2010, CERN, Geneva, LER-2010, CERN, Geneva, January 14, 2010 January 14, 2010 BROOKHAVEN SCIENCE ASSOCIATES

Impedance Minimization by Nonlinear Tapering

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BROOKHAVEN SCIENCE ASSOCIATES. Impedance Minimization by Nonlinear Tapering. Boris Podobedov Brookhaven National Lab Igor Zagorodnov DESY LER-2010, CERN, Geneva, January 14, 2010. Acknowledgements. - PowerPoint PPT Presentation

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Page 1: Impedance Minimization by Nonlinear Tapering

Impedance Minimization by Impedance Minimization by Nonlinear TaperingNonlinear Tapering

Impedance Minimization by Impedance Minimization by Nonlinear TaperingNonlinear Tapering

Boris PodobedovBoris PodobedovBrookhaven National LabBrookhaven National Lab

Igor ZagorodnovIgor ZagorodnovDESYDESY

LER-2010, CERN, Geneva,LER-2010, CERN, Geneva, January 14, 2010January 14, 2010

BROOKHAVEN SCIENCE ASSOCIATES

Page 2: Impedance Minimization by Nonlinear Tapering

Acknowledgements Acknowledgements Acknowledgements 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.

•Work supported by DOE contract number DE-AC02-98CH10886 and by EU contract 011935 EUROFEL.

Page 3: Impedance Minimization by Nonlinear Tapering

• Motivation

• Review of theoretical results

• Optimal boundary

• EM solvers used

• Results for impedance reduction

• Conclusion

OutlineOutlineOutlineOutline

Z-optimization by non-linear tapering is not that new ….

Apart from acoustics, used for gyrotron tapers, mode converters, antennae design, etc.

Page 4: Impedance Minimization by Nonlinear Tapering

Examples of Accelerator Tapers Examples of Accelerator Tapers & Motivation and Scope of This & Motivation and Scope of This

WorkWork

Examples of Accelerator Tapers Examples of Accelerator Tapers & Motivation and Scope of This & Motivation and Scope of This

WorkWork

hmax/hmin~ 9 (gap open) or 27 (gap closed)

hmax/hmin~10

NSLS-II transition to SC RF cavityNSLS MGU Taper for X13, X25, X29 & X9

Focus on large X-sectional variations and gradual tapering; study transverse, broad-band geometric impedance @ low frequency inductive regime

Important for LS, i.e. at NSLS-II ~3/4 of Z┴

come from ID chambers (RW and tapers)

Goal to reduce Z to avoid instabilities (TMCI) in rings, or degradation in linacs ...

A. Blednykh

3.3

mm

(ops

)

10 m

m (i

nj.)

88 m

m

Page 5: Impedance Minimization by Nonlinear Tapering

0 5 10 15-1

0

1

2

3

4Low frequency impedance of a taper

frequency

Z

Re[Z]

-Im[Z]

Inductive Impedance Inductive Impedance Regime Regime

Inductive Impedance Inductive Impedance Regime Regime

In inductive regime (low f) Re[Z]~0, Im[Z]~const vs. frequency. We concentrate on this regime and attempt to minimize Z by optimizing the taper profile.

Inductive regime

At high frequencies, Im[Z]~0 Re[Z]~const and independent of taper length (optical regime).

Tapers are ineffective, see i.e. Stupakov, Bane, Zagorodnov, 2007

k~1/rmin, 1/hmin (hor),

1/wmin(ver)

Page 6: Impedance Minimization by Nonlinear Tapering

Theory Review Theory Review Theory Review Theory Review

2

02

'

2

r ziZZ k dz

r z

2

03

'

4recty

h ziZ wZ k dz

h z

2

03

'

16elly

h ziZ wZ k dz

h z

2

02

'

4ellx

h ziZZ k dz

h z

Axially symmetric taper Flat rectangular taper 2w × 2h, w>>h

K. Yokoya, 1990 G. Stupakov, 1996, 2007

Elliptical x-section taper 2w × 2h , w>>h

B. Podobedov & S. Krinsky, 2007

Zx: h<<L, k<~1/hmin

Zy: w<<L, k<~2/wmin

These are inductive regime impedances. Tapers are gradual to be effective.

Functionals lend themselves to simple boundary optimization.

Page 7: Impedance Minimization by Nonlinear Tapering

Optimizing Boundaries Optimizing Boundaries Optimizing Boundaries Optimizing Boundaries

0 L0

hmin

hmax

linearexponentialoptimal ver.

min max min( ) (1 ( / 1) / )h z h h h z L

/min max min( ) ( / )z Lh z h h h

min

21/ 2min max

( )1 (( / ) 1) /

hh z

h h z L

linear

exponential to minimize Zx (or Z┴ h->r)

“optimal vertical” to minimize Zy

Reduced slope @ small h(z); big difference when hmax/hmin>>1

At hmax/hmin=20 predict factor of 2 reduction for Z┴ or Zx, factor of ~3 for Zy

Page 8: Impedance Minimization by Nonlinear Tapering

Can We Trust the Theory that Can We Trust the Theory that Ignores Corners? Ignores Corners?

Can We Trust the Theory that Can We Trust the Theory that Ignores Corners? Ignores Corners?

For gradual tapers corners add small corrections to the inductive impedance.

• Optimizations assume smooth boundaries, i.e. ignore “corners”

• Taper with corners can be thought of as a limit of a sequence of smooth structures =>

• Z┴ was found for cornered taper as

limit a->0 0.5 1 1.5

0.4

0.6

0.8

1

z

r(z)

a=0.3a=0.1a=0.03a=0

“smoothness” parameter

B. Podobedov & S. Krinsky, 2006

• Corrections due to corners were found on the order of

Z┴/ Z

┴ ~ rav/L , Z

x/ Z

x ~ hav/L , Z

y/ Z

y ~ wav/L (small for gradual tapers)

rav =(rmin+ rmax)/2

Page 9: Impedance Minimization by Nonlinear Tapering

Summary of Numerical Summary of Numerical Calculations Calculations

Summary of Numerical Summary of Numerical Calculations Calculations

• ABCI (axially symmetric)

• ECHO (axially symmetric & 3D)

• GDFIDL (3D)

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

Page 10: Impedance Minimization by Nonlinear Tapering

Wakefield code ECHO (TU Darmstadt / DESY)

Wakefield code ECHO (TU Darmstadt / DESY)

bunch

moving meshElectromagnetic

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, 2005.

Page 11: Impedance Minimization by Nonlinear Tapering

Wakefield code ECHO (TU Darmstadt / DESY)

Wakefield code ECHO (TU Darmstadt / DESY)

zero dispersion in z-direction accurate results withstaircase free (second order convergent) coarse meshmoving mesh without interpolation in 2.5D stand alone application

in 3D only solver, modelling and meshing in CST Microwave Studioallows for accurate calculations on conventional single-processor PC To be parallelized …

Preprocessorin Matlab

Model and mesh in CST

MicrowaveStudio

ECHO 3DSolver

PostprocessorIn Matlab

Page 12: Impedance Minimization by Nonlinear Tapering

Impedance Reduction for Impedance Reduction for Axially Symmetric TapersAxially Symmetric TapersImpedance Reduction for Impedance Reduction for Axially Symmetric TapersAxially Symmetric Tapers

0 5 10 15 200.4

0.6

0.8

1

rmax

/ rmin

Z

_e

xp /

Z

_lin

TheoryECHOABCI

0 5 10 150

2

4

frequency, GHz

-Im

[Z

],k

/m

0 5 10 150

2

4

frequency, GHzR

e[Z

],

k/m

linear taperexponential taper

Z┴[k/m] and reduction due to exponential taper agree well with theory

Impedance reduction extends through inductive regime (k~1/rmin) & beyond

Z┴ reduction for exponential tapering Z

┴(f ) for rmax/rmin= 18, rmin =1 cm

Page 13: Impedance Minimization by Nonlinear Tapering

Geometry for Rectangular Geometry for Rectangular Taper Calculations Taper Calculations

Geometry for Rectangular Geometry for Rectangular Taper Calculations Taper Calculations

Linear taper “Optimal” taper

max2h min2h

L

Page 14: Impedance Minimization by Nonlinear Tapering

Geometry for Elliptical Taper Geometry for Elliptical Taper Calculations Calculations

Geometry for Elliptical Taper Geometry for Elliptical Taper Calculations Calculations

varia

ble

1 cm

4:1 or 8:1 aspect ratio @ min X-section

zy

x

confocal geometry w(z)2-h(z)2=const.

Each taper subdivided into 4 linearly tapered pieces to approx. nonlinear boundary.

Gradual tapers in convex geometry

Long straight pipes to avoid “interaction” between two tapers

Page 15: Impedance Minimization by Nonlinear Tapering

Impedance Reduction for Impedance Reduction for Elliptical X-Section TapersElliptical X-Section TapersImpedance Reduction for Impedance Reduction for Elliptical X-Section TapersElliptical X-Section Tapers

1 5 9 13 17 200.4

0.5

0.6

0.7

0.8

0.9

1

hmax

/hmin

Zx_

exp

/ Z

x_lin

TheoryGDFIDL, w

min/h

min= 4

GDFIDL, wmin

/hmin

= 8

1 5 9 13 17 200

0.2

0.4

0.6

0.8

1

hmax

/hmin

Zy_

op

t / Z

y_lin

TheoryGDFIDL, wmin/hmin= 4

Zx reduction for exponential tapering Zy reduction for optimal vert. tapering

Z x[k/m] and reduction due to exponential taper agree well with theory

Z y[k/m] is less than theory; Zy gets reduced due to optimal taper less than predicted

Page 16: Impedance Minimization by Nonlinear Tapering

Impedance Reduction vs. Impedance Reduction vs. Frequency for Elliptical X-Frequency for Elliptical X-

Section Section

Impedance Reduction vs. Impedance Reduction vs. Frequency for Elliptical X-Frequency for Elliptical X-

Section Section

0 2 4 6 8 10-2

0

2

4

6

frequency, GHz

Re[

Zy],

k

/m

0 2 4 6 8 100

2

4

6

8

-Im

[Zy],

k

/m

frequency, GHz

linear taperoptimal vert. taper

Zy reduction extends through inductive regime (k~1/wmin) & beyond

Zx reduction extends through inductive regime (k~1/hmin) & beyond

hmax/hmin= 18

Page 17: Impedance Minimization by Nonlinear Tapering

Impedance Reduction for Impedance Reduction for Rectangular X-Section Tapers Rectangular X-Section Tapers

Impedance Reduction for Impedance Reduction for Rectangular X-Section Tapers Rectangular X-Section Tapers

2 4 6 8 100.4

0.6

0.8

1

hmax

/hmin

Zx_

exp

/ Z

x_lin

Theoryw=h

max

w=2*hmax

2 4 6 8 100.4

0.6

0.8

1

hmax

/hmin

Zy_

opt /

Zy_

lin

Theoryw=h

max

w=2*hmax

Zx reduction for exponential tapering Zy reduction for optimal vert. tapering

Z x[k/m] and reduction due to exponential taper agree well with theory

Z y[k/m] is less than theory; Zy gets reduced due to optimal taper less than predicted

Results are very similar to elliptical structure

Page 18: Impedance Minimization by Nonlinear Tapering

• 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 Zx of flat 3D structures. The vertical impedance gets reduced less than predicted, but the linear taper Zy is lower as well.

• Optimal tapering for Zx reduces Zy 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 is the only “knob” to reduce transverse broadband geometric impedance of tapered structures. Replacing true optimal profile with just a few linear pieces works quite well.

Conclusion Conclusion Conclusion Conclusion

Page 19: Impedance Minimization by Nonlinear Tapering

•B. Podobedov, I. Zagorodnov, PAC-2007, p. 2006•G. Stupakov, K.L.F. Bane, I. Zagorodnov, PRST-AB 10, 094401 (2007)•K. Yokoya, CERN SL/90-88 (AP), 1990•G.V. Stupakov, SLAC-PUB-7167, 1996•G.V. Stupakov, PRST-AB 10, 094401 (2007)•B. Podobedov, S. Krinsky, PRST-AB 9, 054401 (2006)•B. Podobedov, S. Krinsky, PRST-AB 10, 074402 (2007)

ReferencesReferencesReferencesReferences

Page 20: Impedance Minimization by Nonlinear Tapering

z , cm

Various Impedance Various Impedance Regimes: vertical kick factor Regimes: vertical kick factor

Various Impedance Various Impedance Regimes: vertical kick factor Regimes: vertical kick factor

Vertical kick factor for elliptical transition in pg. 14 ( bmin=1 cm, bmax=4.5 cm, 2L=20 cm)

For large a/b, y~1/z (inductive regime) down to z~ amin, then, for shorter

bunch, y~z-1/2 ( intermediate regime )

For small a/b, y becomes independent of z at z~ bminb’ (optical regime)

B. Podobedov & S. Krinsky, 2007

10-2

10-1

100

101

102

z, cm

,

V/(

pC

m)

amin

/bmin

=1

z = b

min b'

y ,

V/(p

C m

)