Impedance Minimization by Nonlinear Tapering
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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
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.
• 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.
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
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)
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.
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
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
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
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.
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
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
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
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
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
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
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
• 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
•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
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
)
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