Shuichi Noguchi, KEK 6-th ILC School, November 2011 1Shuichi Noguchi, KEK 6-th ILC School, November 2011 1
RF Basics; Contents Maxwell’s Equation Plane Wave Boundary Condition Wave Guide Cavity & RF Parameters Normal Mode Analysis Perturbation Theory Equivalent Circuit Coupled Cavity
Part-1
Shuichi Noguchi, KEK 6-th ILC School, November 2011 2Shuichi Noguchi, KEK 6-th ILC School, November 2011 2
Literatures
J. C. Slator “Microwave Electronics”
Rev. Mod. Phys. 18,(1946)
Shuichi Noguchi, KEK 6-th ILC School, November 2011 3Shuichi Noguchi, KEK 6-th ILC School, November 2011 3
Maxwell’s Equation ( MKS )
EJHBED
DdivJt
DHrot
Bdivt
BErot
,,
,
,0,0
Not a Beam Current
Faraday
Ampere
Shuichi Noguchi, KEK 6-th ILC School, November 2011 4Shuichi Noguchi, KEK 6-th ILC School, November 2011 4
Pointing Vector & Power Flow
SdHEVdHEdiv
VdErotHHrotEVdHEt
VdJE
nSV
VV V
xx
2
1 22
From Maxwell’s Equation
Energy Loss + Change of Electric and Magnetic Energy= Power Flow at Boundary S
HEP
HES
xRe2
1;FlowPower
x ;Vector Pointing
Shuichi Noguchi, KEK 6-th ILC School, November 2011 5Shuichi Noguchi, KEK 6-th ILC School, November 2011 5
Maxwell’s Equation - Wave Equation
EEdivEErotrotJt
Et
Et
Jt
Hrott
Hrott
Erotrot
222
2
2
2
,
;11
;
2
2
2
2
2
2
2
2
2
2
22
Azrr
rrr
Azyx
A
Cartesian Coordinate
Cylindrical Coordinate
= 0
Shuichi Noguchi, KEK 6-th ILC School, November 2011 6Shuichi Noguchi, KEK 6-th ILC School, November 2011 6
Wave Equation
01
,01
2
2
22
2
2
22
t
H
cH
t
E
cE
0,0 2222 HkHEkE
222,,
ckeHE tj
0,
Shuichi Noguchi, KEK 6-th ILC School, November 2011 7Shuichi Noguchi, KEK 6-th ILC School, November 2011 7
Wave Equation Helmholtz Equation
022 EkE
Mode Magnetic / Electric Transverse;TEM;0,0
Mode Magnetic Transverse;TM;0,0
Mode Electric Transverse;TE;0,0
zz
zz
zz
HE
HE
HE
Particular Solution for our Application
No TEM Modes in one closed Conductor
Shuichi Noguchi, KEK 6-th ILC School, November 2011 8Shuichi Noguchi, KEK 6-th ILC School, November 2011 8
Maxwell’s Equation in Cartesian Coordinates
0,
,
,
,
z
H
y
H
x
H
z
E
y
E
x
E
EjJy
H
x
HHj
y
E
x
E
EjJx
H
z
HHj
x
E
z
E
EjJz
H
y
HHj
z
E
y
E
zyxzyx
zzxy
zxy
yyzx
yzx
xxyz
xyz
Shuichi Noguchi, KEK 6-th ILC School, November 2011 9Shuichi Noguchi, KEK 6-th ILC School, November 2011 9
Maxwell’s Equation in Cylindrical Coordinates
011
,11
11,
11
,
1,
1
z
HH
rHr
rrz
EE
rEr
rr
EjJH
rHr
rrHj
E
rEr
rr
EjJr
H
z
HHj
r
E
z
E
EjJz
HH
rHj
z
EE
r
zr
zr
zzr
zr
zrzr
rrz
rz
Shuichi Noguchi, KEK 6-th ILC School, November 2011 10Shuichi Noguchi, KEK 6-th ILC School, November 2011 10
Plane Wave in Uniform Medium
0,0
,
,
,
z
H
y
H
x
H
z
E
y
E
x
E
EjJy
H
x
HHj
y
E
x
E
EjJx
H
z
HHj
x
E
z
E
EjJz
H
y
HHj
z
E
y
E
zyxzyx
zzxy
zxy
yyzx
yzx
xxyz
xyz
0;exp;exp tjzHHtjzEE
Shuichi Noguchi, KEK 6-th ILC School, November 2011 11Shuichi Noguchi, KEK 6-th ILC School, November 2011 11
Plane Wave in Uniform Medium
0;exp;exp tjzHHtjzEE
0,0
,
,
zz
xy
xy
yx
yx
HE
Ejz
HHj
z
E
Ejz
HHj
z
E
Frequency Time Dependence exp( jt )
No Boundary
TEM Mode
Shuichi Noguchi, KEK 6-th ILC School, November 2011 12Shuichi Noguchi, KEK 6-th ILC School, November 2011 12
Plane Wave
,112
,112
,
expexp,
2
1
22
2
2
1
22
2
212
2
jjj
zEzEEEjjzd
Edxx
x
Propagation Constant
Attenuation Constant( Real Part )
Phase Constant( Imaginary Part )
Shuichi Noguchi, KEK 6-th ILC School, November 2011 13Shuichi Noguchi, KEK 6-th ILC School, November 2011 13
Impedance ; E / H
jj
jjZ
zEzEZzd
Ed
jH
i
i
xy
,expexp11
21
Intrinsic Impedance
i
i
Zj
jZj
,,Dielectric;
21,
21Conductor,;
Shuichi Noguchi, KEK 6-th ILC School, November 2011 14Shuichi Noguchi, KEK 6-th ILC School, November 2011 14
Boundary Condition221121112221 ,,, nnSttSnntt HHJHHEEEE
t
ldE
Medium 11, 1, Z1
Medium 22, 2, Z2
Medium 1 Medium 2
sJs
Et1 Et2 Ht1 Ht2
En1 En2 Hn1 Hn2
SJldH
E = H = 0 in Perfect Conductor ; Et =Hn = 0
Faraday Ampere
0
Shuichi Noguchi, KEK 6-th ILC School, November 2011 15Shuichi Noguchi, KEK 6-th ILC School, November 2011 15
Reflection & Transmission
zjkii
zjki e
Z
EHeE 11
1
,
zjktt
zjkt e
Z
EHeE 22
2
,
Medium 11, 1, Z1
Medium 22, 2, Z2
z
x
zjkrr
zjkr e
Z
EHeE 11
1
,
Dielectric Boundary
Shuichi Noguchi, KEK 6-th ILC School, November 2011 16Shuichi Noguchi, KEK 6-th ILC School, November 2011 16
iitiir
tritri
zjkty
zjkrzjkiy
zjktx
zjkr
zjkix
ETEZZ
ZEEE
ZZ
ZZE
Z
E
Z
EEEEE
eZ
EHe
Z
Ee
Z
EH
eEEeEeEE
12
2
12
12
21
22
111
21
2,
,
,
,
211
211
Shuichi Noguchi, KEK 6-th ILC School, November 2011 17Shuichi Noguchi, KEK 6-th ILC School, November 2011 17
Metallic Boundary
111111
1
,
Z 21,21 2222222
22
jjjZ
zjziy
zjzix
iyix
itir
eZ
EHeE
Z
ZE
zkZ
EHzkEjE
EZ
ZEEEZZ
2222
12
1
22
11
111
1
221
2,2
cos2
,sin2
2,,
Shuichi Noguchi, KEK 6-th ILC School, November 2011 18Shuichi Noguchi, KEK 6-th ILC School, November 2011 18
Metallic Boundary
111111
1
,
Z 21,21 2222222
22
jjjZ
z
x
DepthSkin ;21
222
Dielectric Metallic
E
H
Shuichi Noguchi, KEK 6-th ILC School, November 2011 19Shuichi Noguchi, KEK 6-th ILC School, November 2011 19
Power Loss & Surface Impedance
LossPower ;2
1
Resistance Surface;1
2
Impedance Surface;
22
1Re
2
1Re
2
1
2
22
2
2
222
2
tS
S
SSS
tttt
HRP
R
XjRZ
HZHHEP
Shuichi Noguchi, KEK 6-th ILC School, November 2011 20Shuichi Noguchi, KEK 6-th ILC School, November 2011 20
Wave Guide
Coaxial LineParallel ConductorStrip LineCircular Wave GuideRectangular Wave Guide Ridged Wave Guide
Shuichi Noguchi, KEK 6-th ILC School, November 2011 21Shuichi Noguchi, KEK 6-th ILC School, November 2011 21
Traveling Wave Mode
0,
,
,
,
z
H
y
H
x
H
z
E
y
E
x
E
EjJy
H
x
HHj
y
E
x
E
EjJx
H
z
HHj
x
E
z
E
EjJz
H
y
HHj
z
E
y
E
zyxzyx
zzxy
zxy
yyzx
yzx
xxyz
xyz
0,;exp,;exp, zjtjyxHHzjtjyxEE
Shuichi Noguchi, KEK 6-th ILC School, November 2011 22Shuichi Noguchi, KEK 6-th ILC School, November 2011 22
Traveling Wave Mode 0,;exp,;exp, zjtjyxHHzjtjyxEE
0,0
,
,
,
zyx
zyx
zxy
zxy
yz
xyz
x
xz
yxz
y
Hjy
H
x
HEj
y
E
x
E
Ejy
H
x
HHj
y
E
x
E
Ejx
HHjHj
x
EEj
Ejy
HHjHj
y
EEj
Shuichi Noguchi, KEK 6-th ILC School, November 2011 23Shuichi Noguchi, KEK 6-th ILC School, November 2011 23
TE-Modes ; Ez = 0
0,0
0,
,
,
zyxyx
xyz
xy
yz
xyx
xz
yxy
Hjy
H
x
H
y
E
x
E
y
H
x
HHj
y
E
x
E
Ejx
HHjHjEj
Ejy
HHjHjEj
Shuichi Noguchi, KEK 6-th ILC School, November 2011 24Shuichi Noguchi, KEK 6-th ILC School, November 2011 24
2222222
2
2
2
2222
2222
,0
,
,
kkHky
H
x
H
x
H
k
jE
y
H
k
jE
y
H
k
jH
x
H
k
jH
czczz
zy
zx
zy
zx
number wavecutoff / critical;ck
Shuichi Noguchi, KEK 6-th ILC School, November 2011 25Shuichi Noguchi, KEK 6-th ILC School, November 2011 25
TE-mn Modes in Rectangular WG
22
21
222
2222
22
12
121
2
22
22
22
12
121
,
011
,
kk
Hyd
HdH
xd
Hd
kyd
Hd
Hxd
Hd
HyHxHH
c
zz
zz
cz
z
z
zzzz
x
z
y
a
b
1,,
expcoscos
21
nmb
n
a
m
zjyb
nx
a
mH z
From Boundary Condition
Shuichi Noguchi, KEK 6-th ILC School, November 2011 26Shuichi Noguchi, KEK 6-th ILC School, November 2011 26
;12
;22
;22
;
2
22
2222
cg
cc
cc
bnamk
k
kkbnamk
Wave Length in Medium
Critical Wave Length
Guide Wave Length
If k < kc ( c ) wave can not propagate.
Shuichi Noguchi, KEK 6-th ILC School, November 2011 27Shuichi Noguchi, KEK 6-th ILC School, November 2011 27
TE-mn Modes
zjz
zj
cy
zj
cx
z
zj
cy
zj
cx
eyb
nx
a
mH
eyb
nx
a
m
b
n
k
jH
eyb
nx
a
m
a
m
k
jH
E
eyb
nx
a
m
a
m
k
jE
eyb
nx
a
m
b
n
k
jE
sincos
,sincos
,cossin
,0
,cossin
,sincos
2
2
2
2
Shuichi Noguchi, KEK 6-th ILC School, November 2011 28Shuichi Noguchi, KEK 6-th ILC School, November 2011 28
TM-Modes ; Hz = 0
0,0
,0
,
,
y
H
x
HEj
y
E
x
E
Ejy
H
x
H
y
E
x
E
EjHjHjx
EEj
EjHjHjy
EEj
yxz
yx
zxyxy
yxyz
x
xyxz
y
Shuichi Noguchi, KEK 6-th ILC School, November 2011 29Shuichi Noguchi, KEK 6-th ILC School, November 2011 29
TM-mn Modes
0
,sincos
,cossin
,cossin
,cossin
,sincos
2
2
2
2
z
zj
cy
zj
cx
zjz
zj
cy
zj
cx
H
eyb
nx
a
m
a
m
k
jH
eyb
nx
a
m
b
n
k
jH
eyb
nx
a
mE
eyb
nx
a
m
b
n
k
jE
eyb
nx
a
m
a
m
k
jE
Shuichi Noguchi, KEK 6-th ILC School, November 2011 30Shuichi Noguchi, KEK 6-th ILC School, November 2011 30
Power Loss
dSHRPS tSloss
2
2
1
Shuichi Noguchi, KEK 6-th ILC School, November 2011 31Shuichi Noguchi, KEK 6-th ILC School, November 2011 31
TEM-Modes ; Ez, Hz = 0
0,0
0,0
,
,
y
H
x
H
y
E
x
E
y
H
x
H
y
E
x
E
EjHjHjEj
EjHjHjEj
yxyx
xyxy
yxyx
xyxy
0,,
Cutoff No , WavePlane;
2
22
yE
xE yx
Shuichi Noguchi, KEK 6-th ILC School, November 2011 32Shuichi Noguchi, KEK 6-th ILC School, November 2011 32
Maxwell’s Equation in Cylindrical Coordinates
011
,11
11,
11
,
1,
1
z
HH
rHr
rrz
EE
rEr
rr
EjJH
rHr
rrHj
E
rEr
rr
EjJr
H
z
HHj
r
E
z
E
EjJz
HH
rHj
z
EE
r
zr
zr
zzr
zr
zrzr
rrz
rz
Shuichi Noguchi, KEK 6-th ILC School, November 2011 33Shuichi Noguchi, KEK 6-th ILC School, November 2011 33
Traveling Wave Modes 0,;exp,;exp, zjtjrHHzjtjrEE
011
,11
11,
11
,
1,
1
z
HH
rHr
rrz
EE
rEr
rr
EjJH
rHr
rrHj
E
rEr
rr
EjJr
H
z
HHj
r
E
z
E
EjJz
HH
rHj
z
EE
r
zr
zr
zzr
zr
zrzr
rrz
rz
Shuichi Noguchi, KEK 6-th ILC School, November 2011 34Shuichi Noguchi, KEK 6-th ILC School, November 2011 34
011
,011
11,
11
,
1,
1
zrzr
zr
zr
zr
zr
rz
rz
HjH
rHr
rrEj
E
rEr
rr
EjH
rHr
rrHj
E
rEr
rr
Ejr
HHjHj
r
EEj
EjHjH
rHjEj
E
r
0,;exp,;exp, zjtjrHHzjtjrEE
Traveling Wave Modes
Shuichi Noguchi, KEK 6-th ILC School, November 2011 35Shuichi Noguchi, KEK 6-th ILC School, November 2011 35
TM-Modes ; Hz = 0
011
,011
11,0
11
,
,1
H
rHr
rrEj
E
rEr
rr
EjH
rHr
rr
E
rEr
rr
EjHjHjr
EEj
EjHjHjEjE
r
rzr
zrr
rz
r
rrz
Shuichi Noguchi, KEK 6-th ILC School, November 2011 36Shuichi Noguchi, KEK 6-th ILC School, November 2011 36
22222
2
2
2222
,011
1,
,
kkEkr
Er
rr
E
r
E
rk
jE
r
E
k
jE
EHEH
czczz
zzr
rr
Shuichi Noguchi, KEK 6-th ILC School, November 2011 37Shuichi Noguchi, KEK 6-th ILC School, November 2011 37
Function Bessel;
,cos
,011
,
;
1
2
122
21
21
2
22
22
2
21
rkJE
mE
Erk
m
rkd
Ed
rkrkd
Ed
Emd
Ed
ErEE
cmz
z
zcc
z
cc
z
zz
zzz
Shuichi Noguchi, KEK 6-th ILC School, November 2011 38Shuichi Noguchi, KEK 6-th ILC School, November 2011 38
Boundary Conditionr = a
z
22
,
,0
,0
a
y
yak
akJ
arE
mn
mnc
cm
z
m n 1 2 3 4
0 2.4048 5.5201 8.6537 11.7915
1 3.8317 7.0156 10.1735 13.3237
2 5.1356 8.4172 11.6198 14.7960
3 6.3802 9.7610 13.0152 16.2235
4 7.5883 11.0647 14.3725 17.6160
ymn
Shuichi Noguchi, KEK 6-th ILC School, November 2011 39Shuichi Noguchi, KEK 6-th ILC School, November 2011 39
TM-man Modes
0
,cos
,1
sin
,cos
,1
sin
,cos
'
2
2
'
z
zjmnm
mn
zjmnm
mn
r
zjmnmz
zjmnm
mn
zjmnm
mnr
H
era
yJm
ayjH
era
yJ
rm
ay
mjH
era
yJmE
era
yJ
rm
ay
mjE
era
yJm
ayjE
Shuichi Noguchi, KEK 6-th ILC School, November 2011 40Shuichi Noguchi, KEK 6-th ILC School, November 2011 40
TE-Modes
011
,011
011
,11
,
1,
zrr
rz
r
zrr
rz
r
HjH
rHr
rr
E
rEr
rr
H
rHr
rrHj
E
rEr
rr
Ejr
HHjHjEj
EjHjH
rHjEj
Shuichi Noguchi, KEK 6-th ILC School, November 2011 41Shuichi Noguchi, KEK 6-th ILC School, November 2011 41
TEM-Modes ; Ez = Hz = 0
011
,011
011
,011
,
,
H
rHr
rr
E
rEr
rr
H
rHr
rr
E
rEr
rr
EjHjHjEj
EjHjHjEj
rr
rr
rr
rr
off-Cut No,0,22 ck
Shuichi Noguchi, KEK 6-th ILC School, November 2011 42Shuichi Noguchi, KEK 6-th ILC School, November 2011 42
Coaxial Waveguide
Impedance sticCharacteri;In2
A2,InA
A1,
A
0,0
,0,0
0
2
0
a
bZ
I
VZ
eZ
daJIea
bdrEV
erZ
Her
E
Hrr
Err
HE
i
zj
iS
zjb
a r
zj
i
zjr
r
r
a b
Shuichi Noguchi, KEK 6-th ILC School, November 2011 43Shuichi Noguchi, KEK 6-th ILC School, November 2011 43
Power
baZ
PR
a
b
baZ
PR
baZRbdHadHRP
a
b
ZrdrdHEP
fS
i
fS
iSbrarSloss
i
b
a rf
11
2In
11
11A
2
1
InA
Re2
1
0
2
22
0
2
0
22
22
0
Shuichi Noguchi, KEK 6-th ILC School, November 2011 44Shuichi Noguchi, KEK 6-th ILC School, November 2011 44
Resonator / Cavity
Shuichi Noguchi, KEK 6-th ILC School, November 2011 45Shuichi Noguchi, KEK 6-th ILC School, November 2011 45
Can be solved Analyticallyor by Computer Codes
Boundary Condition Short-Circuited Plane S Open-Circuited Plane S’
S'on 0,0x
Son0,0x
EnHn
HnEn
・
・
S
S’ S’n
Media ;
wall
Cavity
; Perfect Conductor
Shuichi Noguchi, KEK 6-th ILC School, November 2011 46Shuichi Noguchi, KEK 6-th ILC School, November 2011 46
Analytic Solution, Example
L
a
L
lLzz
at ;0sin
Shuichi Noguchi, KEK 6-th ILC School, November 2011 47Shuichi Noguchi, KEK 6-th ILC School, November 2011 47
TM-01l Modes
0
,cos
,1
sin
,cos
,1
sin
,cos
'
2
2
'
z
zjmnm
mn
zjmnm
mn
r
zjmnmz
zjmnm
mn
zjmnm
mnr
H
era
yJm
ayjH
era
yJ
rm
ay
mjH
era
yJmE
era
yJ
rm
ay
mjE
era
yJm
ayjE
0
,cos
,0
,cos
,0
,sin
'
01
010
01'0
01
z
mnm
r
z
r
H
zL
lr
a
yJ
ayjH
H
zL
lr
a
yJE
E
zL
lr
a
yJ
ayjE
Shuichi Noguchi, KEK 6-th ILC School, November 2011 48Shuichi Noguchi, KEK 6-th ILC School, November 2011 48
Cavity RF Parameters
22
2
0
2
2
2
00
222
0
,
,
22,
2
Cavityacc
Cavityacc
S
S
LW
E
Q
RL
P
ER
dSH
dVHG
R
G
P
WQ
dVEdVHWdSHR
P
dzztrzEL
EL
zCavity
acc cos0,1
0
Geometric Factor
Shuichi Noguchi, KEK 6-th ILC School, November 2011 49Shuichi Noguchi, KEK 6-th ILC School, November 2011 49
Transit Time Factor ( TTF )
La
aQ
E
E
E
EL
c
L
L
c
dzrzE
dzztrzE
acc
Sp
Sp
acc
L
z
L
z
1
Cavity ngAccelerati;2
,2
TTF,2
,2
sin2
0,
cos0,TTF
0
0
TM010 Mode in Cylindrical Cavity
Shuichi Noguchi, KEK 6-th ILC School, November 2011 50
Calculate Skin Depth & Surface Resistance
using following Values.
m
mm
/mho10x6
/H10x4;/F10x36
1
7
79
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