Post on 16-Oct-2021
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RF FUNDAMENTALSand BEAM LOADING
Jean Delayen
Thomas Jefferson National Accelerator FacilityOld Dominion University
USPAS June 2008 U. Maryland
Equivalent Circuit for an rf CavitySimple LC circuit representing an accelerating resonator
Metamorphosis of the LC circuit into an accelerating cavity
Chain of weakly coupled pillbox cavities representing an accelerating cavity
Chain of coupled pendula as its mechanical analogue
Parallel Circuit Model of an Electromagnetic Mode
• Power dissipated in resistor R:
• Shunt impedance:
• Quality factor of resonator:
212
cdiss
VPR
=
2c
shdiss
VRP
∫ 2shR Rfi =
1/20
0 0diss c
U R CQ CR RP L Lw w
wÊ ˆ∫ = = = Á ˜Ë ¯
1
00
0
1Z R iQww
w w
-È ˘Ê ˆ
= + -Í ˙Á ˜Ë ¯Î ˚1
00 0
0
1 2Z R iQw ww ww
-È ˘Ê ˆ-
ª ª +Í ˙Á ˜Ë ¯Î ˚ ,
1-Port System
2 2 00 0
0 0
0
1 21 2
gg
kV RI V kVR Qk Z R k Z iQiw
www
= =Ê ˆ+ + + DÁ ˜Ë ¯+ D
20
0
0
1 2Total impedance: Rk Z Qi w
w
++ D
1-Port System
( )
( )
2 20
22 20
222 4 2 2
0 0 00
2
0
20 0
02 22
0 0
0
1 12 212
4
8
4 11 21
1
Energy content
Incident power:
Define coupling coefficient:
g
ginc
inc
QU CV VR
Q Rk VR
R k Z k Z Q
VP
ZR
k ZQU
P Q
w
w ww
b
bw b w
b w
= =
=Ê ˆD+ + Á ˜Ë ¯
=
=
=+ Ê ˆÊ ˆ D+ Á ˜ Á ˜+Ë ¯ Ë ¯
1-Port System
( )
( )
2 220 0
0
2
4 11 21
1
0, 1 :
4 111 21
Power dissipated
Optimal coupling: maximum or
critical coupling
Reflected power
diss inc
diss incinc
ref inc diss mc
UP PQ Q
U P PP
P P P P
w bb w
b w
w b
bb
= =+ Ê ˆÊ ˆ D+ Á ˜ Á ˜+Ë ¯ Ë ¯
=
fi D = =
= - = -+
+2
0
01Q wb w
È ˘Í ˙Í ˙Í ˙Ê ˆDÍ ˙Á ˜Í ˙+Ë ¯Î ˚
1-Port System
( )
( )
02
0
2
2
414
1
11
At resonance
inc
diss inc
ref inc
QU P
P P
P P
bw b
bb
bb
=+
=+
Ê ˆ-=Á ˜+Ë ¯
Equivalent Circuit for a Cavity with Beam
• Beam in the rf cavity is represented by a current generator. • Equivalent circuit:
(1 )sh
LR
Rb
=+
0
0
tan -21
produces with phase (detuning angle)
produces with phase
b b
g g
c g b
i V
i V
V V V
Q
yy
wyb w
= -
D=+
Equivalent Circuit for a Cavity with Beam
1/21/2
0
0
2( ) cos1
cos2(1 )
sin22
2: beam rf current: beam dc current: beam bunch length
g g sh
b shb
b
bb
b
b
V P R
i RV
i i
ii
b yb
ybq
q
q
=+
=+
=
Equivalent Circuit for a Cavity with Beam
( ) [ ]{ }2
2211 (1 ) tan tan
4c
gsh
VP b b
Rb b y f
b= + + + + -
0 cosPower absorbed by the beam = Power dissipated in the cavity
sh
c
R ib
Vf
=
2
(1 ) tan tan
1
1 (1 )2
opt opt
opt
opt cg
sh
b
b
b bVP
R
b y f
b
+ =
= +
+ + +=
Minimize Pg :
Cavity with Beam and Microphonics
• The detuning is now 0 0
0 0
0
0tan 2 tan 2
where is the static detuning (controllable)
and is the random dynamic detuning (uncontrollable)
mL L
m
Q Qdw dw dw
y yw w
dwdw
±= - = -
Qext Optimization with Microphonics
22
00
222
00
( 1) 2
( 1) ( 1) 22
mopt
opt c mg
sh
b Q
VP b b QR
dwbw
dww
Ê ˆ= + + Á ˜Ë ¯
È ˘Ê ˆÍ ˙= + + + + Á ˜Í ˙Ë ¯Î ˚
Condition for optimum coupling:
and
In the absence of beam (b=0):
and
2
00
22
00
1 2
1 1 22
If is very large
mopt
opt c mg
sh
m m
Q
VP QR
U
dwbw
dww
dw dw
Ê ˆ= + Á ˜Ë ¯
È ˘Ê ˆÍ ˙= + + Á ˜Í ˙Ë ¯Î ˚
Example
Example
Example
• ERL Injector and Linac: fm=25 Hz, Q0=1x1010 , f0=1300 MHz, I0=100 mA, Vc=20 MV/m, L=1.04 m, Ra/Q0=1036 ohms per cavity
• ERL linac: Resultant beam current, Itot = 0 mA (energy recovery)and opt=385 QL=2.6x107 Pg = 4 kW per cavity.
• ERL Injector: I0=100 mA and opt= 5x104 ! QL= 2x105 Pg = 2.08 MW per cavity!
Note: I0Va = 2.08 MW optimization is entirely dominated by beam loading.
RF System Modeling
• To include amplitude and phase feedback, nonlinear effects from the klystron and be able to analyze transient response of the system, response to large parameter variations or beam current fluctuations
– We developed a model of the cavity and low level controls using SIMULINK, a MATLAB-based program for simulating dynamic systems.
• Model describes the beam-cavity interaction, includes a realistic representation of low level controls, klystron characteristics, microphonic noise, Lorentz force detuning and coupling and excitation of mechanical resonances
RF System Model