Short-Range Wakes in Elliptical Pipe Geometry

Igor Zagorodnov

Beam Dynamics Group Meeting

05.09.05

1w mm width of the slot 5L m length of the slot

Pumping slot

25w m

Theoretical results (Bethe theory) only for .w

Pumping slot

Effect of the slot is small.

•Accuracy estimation of the numerical results.•Wake scaling for geometry parameters.

Used tools: ECHO (time-domain), CST Microwave Studio (modeling, meshing), Matlab (pre- and postprocessing).

0 lnhi

Z bZ

a 2

hiloss

z

Z ck

250 m

25 m

1( ln( / ))lossk O b a

37.2 /lossk V pC

372 /lossk V pC

in outW W W

Out In

2( ) /(2 )L b a

ba

[m]

dx

[dy,dz

[Loss,2D

[V/pC]

Error, 2D

[V/pC]

Loss, 3D

[V/pC]

Error, 3D

[V/pC]

250 1/40 1/40 36.3097 ref.

250 1/20 1/20 36.2932 0.016

250 1/5 1/5 36.2369 0.07 36.2265 0.1

25 1/20 1/20 370.77 ref.

25 1/10 1/10 370.60 0.17

25 1/5 1/5 369.96 0.8

25 1/5 2 369.78 1

outlossk

1( ln( / ))outlossk O b a

Accuracy of 3D calculations

-5 0 5-15

-10

-5

0

5x 10-3

out V mW

pC

/s

250 m

25 m

1( )outW O

[m]

dx

[dy,dz

[Loss,2D

[V/pC]

Error, 2D

[V/pC]

Loss, 3D

[V/pC]

Error, 3D

[V/pC]

250 1/40 1/40 -0.8606 ref.

250 1/10 1/10 -0.8678 0.001

250 1/5 1/5 -0.8802 0.012 -0.8478 0.013

25 1/20 1/20 0.8639 ref.

25 1/10 1/10 -0.8619 0.002

25 1/5 1/5 -0.8859 0.02

25 1/5 2 -0.9380 0.17

inlossk

1( )inlossk O a

in outloss lossk k

Accuracy of 3D calculations

-5 0 50

0.5

1

1.5

2in V

WpC

/s

250

(3 )

m

D

(1), 0inW O

25

(3 )

m

D

25

(2 )

m

D

1 lnoutloss

bk O

a

1inlossk O

a

Scaling laws

Out In

2( ) /(2 )L b a

ba

( )w

1 lnoutloss

bk O

a

inlossk O

a

, ?

Scaling laws

w

a

b

w10h mm

4a mm

Out In

w10h mm

4a mm

w

[mm]

[m]

Loss

[V/pC]

7 0.339 250 3.04+-0.1

4 0.167 250 0.91+-0.1

2 0.083 250 0.23+-0.1

7 0.339 25 31.8+-1

4 0.167 25

2 0.083 25 2.8+-1

outlossk

1( )outlossk O

1.8( ) ?!outlossk O

w

-5 0 5-1.5

-1

-0.5

0

0.51.8 1.8

outW V

pC rad

/s

0.3392

1.8( ), 0,outW O w

0.1672

0.0832

250 m

w10h mm

4a mm

w

[mm]

[m]

Loss

[V/pC]

7 0.339 250 -0.162+-0.013

4 0.167 250 -0.065+-0.013

2 0.083 250 -0.01+-0.013

7 0.339 25 -0.247+-0.17

4 0.167 25

2 0.083 25

inlossk

( ) ?!inlossk O

(1), 0inlossk O

in outloss lossk k

w

-5 0 5-0.05

0

0.05

0.1

0.15

0.2inW V

pC rad

/s

0.3392

( ), 0,inW O w

0.1672

0.0832

250 m

?error

w

1 1.8 lnoutloss

bk O

a

inlossk O

a

w

[mm]

a,

[mm]

[m]

Loss

[V/pC]

7 4 0.339 250 -0.16

1 7.5 0.021 25 -0.005

0.021 40.16 0.005

0.339 7.5inloss

Vk

pC

w

[mm]

b,

[mm]

a,

[mm]

[m]

Loss

[V/pC]

7 10 4 0.339 250 3

1 20 7.5 0.021 25 0.2

1.820

ln0.021 7.5

3 10 0.2100.339

ln4

outloss

Vk

pC

0.2loss

Vk

pC

Scaling laws and extrapolation

in outW W W 2out in eW W W

2 2 2 , 0in e e inW W W W if W

in outloss lossk k

( )ez ellipse ellipse slot zW w w w

(K. Zotter et al. , Impedances and Wakes …, 1998 andK.L.F.Bane, I.A.Zagorodnov, SLAC-PUB-11388, 2005)

w

Energy method

w

[mm]

[m]K_loss from time-domain

[V/pC]

K_loss from energy

[V/pC]

Error,

[V/pC]

7 250 2.88 3.10 0.22

7 25 31.5 31.0 0.5

2 250 0.22 0.26 0.04

2 25 2.75 2.63 0.12

Accuracy of the energy method

[mm] Mesh step in

Ellipse Ellipse with slot

0.25 2 2.89161337 2.89165659 4.32e-5

0.25 1 2.90553777 2.90560728 6.95e-5

0

2w

Z c

0.03 0.12 /lossk V pC

, 0.001 /kick xk V pC( , 0, 0)

( , 0, 0)s

yx

W s x yW s x y ds

a

xy

xa

20h mm

2 2 5000 25 3 16 [ ]h L e mm

0.2 /lossk V pC

Energy method

3mm

3.3mm3.8mm

84 248 257 /loss s sk x y V pC

, 390 46 436 /kick x s wk x x x V pC

, 348 47 301 /kick y s wk y y y V pC

Energy method

85 /lossk V pC

Round (a=3.8, b=3.3)

0.5mm

15mm

3mm

7.6yw mm

2 351 /( )Dkickk V pC m

3 359 /( )Dkickk V pC m

Direct time-domain with ECHO

x

y

9yw mm

7.8yw mm 85 /lossk V pC

91 /lossk V pC

Old pipe?

The energy loss for the new smaller pipe is a little smaller!

2%Error

x

y0.5mm

15mm

3mm

7.6 mm

250 250 /84loss s sk x y V pC

, 390 46 /436kick x s wk x x x V pC

, 348 47 /301kick y s wk y y y V pC

1200 1200 /42loss s sk x y V pC

, 280 206 /486kick x s wk x x x V pC

, 78 208 /130kick y s wk y y y V pC

Loss,

V/pC

Spread,

V/pC

Peak,

V/pC

var1 84 33 -118

var2 42 16 -58

x

yKick

, /486kick xk x V pC

, /130kick yk y V pC

xy

xa

, 0.001 /slotkick xk V pC

, ,0.0001 0.01 /slot abskick x kick xk k V pC

pro section (6.1 m)

Loss,

V/pC

Spread,

V/pC

Peak,

V/pC

absorber 1 42 16 -58

pumping slot 1 <0.2 <0.1 >-0.3

pump 1 9 4 -13

BPM 1

bellow 1 13 5 -18

flange gap 1 6 2.4 -8.5

Total geom. 70 25 -95

resistive (Cu) 6.1m 220 279 -542

resistive (Al) 6.1m 303 325 -660

Longitudinal wake for the case of the elliptical pipe (3.8mm)

-5 0 5

-300

-200

-100

0

100

200

300

-5 0 5-300

-200

-100

0

100

200

300

Shape dependence

-5 0 50

1000

2000

3000

4000

5000

Loss,

V/pC

Spread,

V/pC

Peak,

V/pC

Gauss, cu/al 36/50 46/53 -89/108

Rect, cu/al 79/93 90/72 -256/243

-1 -1: 5.8e+7[ m ], =2.46e-14[sec]cond relaxcu -1 -1: 3.66e+7[ m ], =7.1e-15[sec]cond relaxal

copper aluminum

I A

VW

pC

VW

pC

/ zs

/ zs / zs

25z m

4000 [ ]bN bunches

10 [ ]repf Hz

1[ ]q nC

2b rep lossP N f q k

0.52geom

WP

m

, 1.4res Cu

WP

m

, 2res Al

WP

m

Power Loss (Gaussian bunch)

Conclusion

• Loss and kick of the pumping slot are negligible • Wake of the absorber is reduced by factor 2• Resistive wake is dominant• Non-smooth bunch shape (rectangular) increases

the energy spread less than by factor 2