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. MetralMetral
GENERAL FORMULA FOR THE RESISTIVEWALL IMPEDANCE DERIVED FROM L. VOS’
FORMALISM
Elias Metral, LCE meeting, 20/02/2004 2
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
Elias Metral, LCE meeting, 20/02/2004 3
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
Elias Metral, LCE meeting, 20/02/2004 4
THEORY (3/3)THEORY (3/3)
111 <<sγ
2
12 110
20
ωσµπ sbjbZzulT
+−×=
⇒
⇒ The thin-wall formula with inductive bypass is recovered
Elias Metral, LCE meeting, 20/02/2004 5
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
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
Elias Metral, LCE meeting, 20/02/2004 7
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
Elias Metral, LCE meeting, 20/02/2004 8
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
Elias Metral, LCE meeting, 20/02/2004 9
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
Elias Metral, LCE meeting, 20/02/2004 10
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
Elias Metral, LCE meeting, 20/02/2004 11
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
Elias Metral, LCE meeting, 20/02/2004 12
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
Elias Metral, LCE meeting, 20/02/2004 13
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
Elias Metral, LCE meeting, 20/02/2004 14
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
Elias Metral, LCE meeting, 20/02/2004 15
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