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