RF FUNDAMENTALSBEAM LOADING

Jean Delayen

Thomas Jefferson National Accelerator FacilityOld Dominion University

Joint Accelerator School Indore January 2008

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 L

Ê ˆ∫ = = = Á ˜Ë ¯ω ω

ω

1

00

0

1Z R iQ-

È ˘Ê ˆ= + -Í ˙Á ˜Ë ¯Î ˚

ωωω ω

1

00 0

0

1 2Z R iQ-

È ˘Ê ˆ-ª ª +Í ˙Á ˜Ë ¯Î ˚

ω ωω ωω

,

1-Port System

2 2 00 0

01 2

1 2

gg

c

c

kV RI V kVR Qk Z R k Z iQi

= = + + + ∆ω ω + ∆ω

ω

20

01 2Total impedance:

c

Rk Z Qi ωω

∆+

+

1-Port System

( )

( )

2 20

22 20

222 4 2 20 0 0

2

0

20 0

02 22

0

1 12 212

4

8

4 11 21

1

Energy content

Incident power:

Define coupling coefficient:

g

C

ginc

inc C

C

QU CV VR

Q Rk VR

R k Z k Z Q

VP

ZR

k ZQU

P Q

ω

ω ωω

β

βω β ω

β ω

∆

∆

= =

=Ê ˆ

+ + Á ˜Ë ¯

=

=

=+ Ê ˆÊ ˆ+ Á ˜ Á ˜Ë ¯+ Ë ¯

1-Port System

( )

( )

2 220 0

2

4 11 21

1

0, 1 :

4 111 21

Power dissipated

Optimal coupling: maximum or

critical coupling

Reflected power

diss inc

C

diss incinc

ref inc diss mc

UP PQ Q

U P PP

P P P P

ω ββ ω

β ω

ω β

ββ

∆

∆

= =+ Ê ˆÊ ˆ+ Á ˜ Á ˜Ë ¯+ Ë ¯

=

fi = =

= - = -+

+2

01 C

Q ωβ ω

∆

È ˘Í ˙Í ˙Í ˙Ê ˆÍ ˙

Á ˜Í ˙+Ë ¯Î ˚

1-Port System

( )

( )

02

2

2

4141

11

At resonance

incC

diss inc

ref inc

QU P

P P

P P

βω β

ββ

ββ

=+

=+

Ê - ˆ= Á ˜Ë ¯+

Dissipated and Reflected Power

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

β

Equivalent Circuit for a Cavity with Beam

• Beam in the rf cavity is represented by a current generator. • Equivalent circuit:

(1 )sh

LRR

β=

+

0

0-21

produces with phase (detuning angle) produces with phase

tan

b b

g g

c g b

i Vi V

V V V

Q

ψψ

ωψβ ω

∆

= -

=+

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

β ψβ

ψβθ

θ

θ

=+

=+

=

Equivalent Circuit for a Cavity with Beam

( ) [ ]{ }2

2 211 (1 ) tan tan

4c

gsh

VP b b

Rβ β ψ φ

β= + + + + -

0 02cos cosPower absorbed by the beam =

Power dissipated in the cavityc sh

cc

sh

V i R ibVV

R

φ φ= =

2

(1 ) tan tan

1

1 (1 )2

opt opt

opt

opt cg

sh

b

b

b bVPR

β ψ φ

β

+ =

= +

+ + +=

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 Qδω δω δω

ψ ψω ω

δωδω

±= - = -

-10-8-6-4-202468

10

90 95 100 105 110 115 120

Time (sec)

Freq

uenc

y (H

z)

Probability DensityMedium β CM Prototype, Cavity #2, CW @ 6MV/m

400000 samples

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6 8

Peak Frequency Deviation (V)

Prob

abili

ty D

ensi

ty

Qext Optimization with Microphonics2

20

0

222

00

( 1) 2

( 1) ( 1) 22

mopt

opt c mg

sh

b Q

VP b b QR

δωβω

δωω

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

δωβω

δωω

δω δω

Ê ˆ= + Á ˜Ë ¯

È ˘Ê ˆÍ ˙= + + Á ˜Í ˙Ë ¯Î ˚

Example7-cell, 1500 MHz

0

2

4

6

8

10

12

14

16

18

20

1 10 100

Qext (106)

P (k

W)

21.0 MV/m, 460 uA, 50 Hz 0 deg

21.0 MV/m, 460 uA, 38 Hz 0 deg

21.0 MV/m, 460 uA, 25 Hz 0 deg

21.0 MV/m, 460 uA, 13 Hz 0 deg

21.0 MV/m, 460 uA, 0 Hz 0 deg

Example3-spoke, 345 MHz, β=0.62

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

1.0 10.0 100.0

Qext (10^6)

P (k

W)

10.5 MV/m, 400 uA, 10 Hz 20 deg

10.5 MV/m, 300 uA, 10 Hz 20 deg

10.5 MV/m, 200 uA, 10 Hz 20 deg

10.5 MV/m, 100 uA, 10 Hz 20 deg

10.5 MV/m, 0 uA, 10 Hz 20 deg

Example

• ERL Injector and Linac: δfm=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 βopt=385 ⇒ QL=2.6x107 ⇒ Pg = 4 kW per cavity.

• ERL Injector: I0=100 mA and βopt= 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