GENERAL FORMULA FOR THE RESISTIVE WALL IMPEDANCE … · Elias Metral, LCE meeting, 20/02/2004 6...

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Elias Metral, LCE meeting, 20/02/2004 1 E. E. Metral Metral GENERAL FORMULA FOR THE RESISTIVE WALL IMPEDANCE DERIVED FROM L. VOS’ FORMALISM

Transcript of GENERAL FORMULA FOR THE RESISTIVE WALL IMPEDANCE … · Elias Metral, LCE meeting, 20/02/2004 6...

Page 1: GENERAL FORMULA FOR THE RESISTIVE WALL IMPEDANCE … · Elias Metral, LCE meeting, 20/02/2004 6 2000 4000 6000 8000 10000 w 0.2 0.4 0.6 0.8 1 ImHZTL 2000 4000 6000 8000 10000 w 0.005

Elias Metral, LCE meeting, 20/02/2004 1

E.E. MetralMetral

GENERAL FORMULA FOR THE RESISTIVEWALL IMPEDANCE DERIVED FROM L. VOS’

FORMALISM

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THEORY (1/3)THEORY (1/3)

General formula for 2 layers ( )

( )112

1

111

2

2

0

20

Tanh1

Tanh1

2

12

sZZ

sZZ

Zbj

bZzulT

γ

γµω

π

+

+×+−

×=

ulï length of 1 m

( )111

2

1 εωσµωβωγ jjc

++

=

ii

ii j

jZσεω

µω+

=

radiuspipeBeam=b1layerofThickness1 =s7

0 104 −×= πµπ1200 =Z

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THEORY (2/3)THEORY (2/3)

Consider the case where the layer 2 is vacuum

111 >>sγ

( )

21

12

110

20

ωσµπbj

bZjzulT

++−×+=

⇒ The thick-wall formula with inductive bypass is recovered

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THEORY (3/3)THEORY (3/3)

111 <<sγ

2

12 110

20

ωσµπ sbjbZzulT

+−×=

⇒ The thin-wall formula with inductive bypass is recovered

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NUMERICAL APPLICATIONNUMERICAL APPLICATION

The layer 2 is vacuum

The impedance parameters are mm2=b

m141 Ω= µρ

cm2.51 =s

01 µµ =

cZ001

1== εε

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Elias Metral, LCE meeting, 20/02/2004 62000 4000 6000 8000 10000

w

0.2

0.4

0.6

0.8

1

Im HZTL 2000 4000 6000 8000 10000w

0.005

0.01

0.015

0.02

0.025Re HZTL

Full = General formula

Dashed = Thin-wall formula

Dotted = Thick-wall formula

Remark : The plots are normalized by

20

2 bZπ

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20000 40000 60000 80000 100000w

0.010.020.030.040.050.060.07

Re HZTL

1depthskin s= rad/s35651≈wï

20000 40000 60000 80000 100000w

0.2

0.4

0.6

0.8

1

Im HZTL

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50000 100000150000200000250000300000w

0.02

0.04

0.06

0.08

0.1Re HZTL

50000 100000150000200000250000300000w

0.2

0.4

0.6

0.8

1

Im HZTL

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200000 400000 600000 800000 1µ106 w

0.1

0.2

0.3

0.40.5

Re HZTL

200000 400000 600000 800000 1µ106 w

0.2

0.4

0.6

0.8

1

Im HZTL

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2µ106 4µ106 6µ106 8µ106 1µ107 w

0.1

0.2

0.3

0.40.5

Re HZTL

2µ106 4µ106 6µ106 8µ106 1µ107 w

0.2

0.4

0.6

0.8

1

Im HZTL

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2µ107 4µ107 6µ107 8µ107 1µ108 w

0.1

0.2

0.3

0.40.5

Re HZTL

2µ107 4µ107 6µ107 8µ107 1µ108 w

0.2

0.4

0.6

0.8

1

Im HZTL

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THEORETICAL PREDICTIONS COMPARED TO MEASUREMENTS THEORETICAL PREDICTIONS COMPARED TO MEASUREMENTS ((CaspersCaspers et al. in CERNet al. in CERN--ABAB--20032003--051 (RF)) (1/2)051 (RF)) (1/2)

Stainless steel pipemm50=b

mm1.51 =s

1161 m101.3 −−Ω×=σ

We will compare the transverse impedances (not normalized) per unit length in a LogLog plot

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THEORETICAL PREDICTIONS COMPARED TO MEASUREMENTS THEORETICAL PREDICTIONS COMPARED TO MEASUREMENTS ((CaspersCaspers et al. in CERNet al. in CERN--ABAB--20032003--051 (RF)) (2/2)051 (RF)) (2/2)

Measurements

1000 10000 100000. 1.µ106f

5001000

500010000

50000100000.

Re HZTL

1000 10000 100000. 1.µ106f

5001000

500010000

50000100000.

Im HZTLGeneral formula 1depthskin s=

Hz86600≈f

Green dots from Burov-Lebedev

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CONCLUSION (1/2)CONCLUSION (1/2)

A general formula has been found by L. Vos for any multi-layer vacuum chamberFor the case of 2 layers, with the vacuum for the second layer :

The general formula reduces to the “thin-wall formula with inductive bypass” for low frequencies. This formula can be derived from L. Vos’ formalism and is the same as the one derived by B. Zotter. It is the same as the one used by Caspers et al., which is a corrected version of the one derived by Nassibianand Sacherer. Burov and Lebedev say that they also obtain the same formula in this approximationThe general formula reduces to the “thick-wall formula with inductive bypass” (which can be derived from L. Vos’ formalism) for high frequenciesThe general formula has a smooth transition between the “thin-wall formula with inductive bypass” and the “thick-wall formula with inductive bypass” for intermediate frequencies

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CONCLUSION (2/2)CONCLUSION (2/2)

Low and high frequencies correspond to frequencies much smaller than or bigger than the frequency where the skin depth is equal to the thickness of the layer 1

Intermediate frequencies stand for frequencies close to the frequency where the skin depth is equal to the thickness of the layer 1

As concerns the comparison between the theoretical predictions of the new general formula and the measurements by Caspers et al. (in CERN-AB-2003-051 (RF))

The agreement is very good over the whole range of frequenciesThe new general formula seems to be very close the one from Burov-Lebedev