frequency-domain study of acceleration & beam

loading based on a circuit model

by raquel fandos

Outline• Motivation• Introduction• Scheme of the analysis• From structure parameters to circuit elements• Information extracted from the circuit model• RF response calculation• Beam response calculation• Example: G241

– Phase advance– Power and electric field– S parameters– Group delay– RF response– Beam loading

Motivation

Low vgstructures

Dispersion isnot negligible

Accurate modelBeam Loading& Acceleration

IntroductionAn accelerating structure

Matching elements: Zin=Zmatch

cell 1 cell 2 cell 3

…

Input matching cell Output matching cell Tapered structure

- vg, Q & R/Q vary R, C, L & k vary.

- Input & output have different matching

parameters (Rt & Lt)

…

A series of coupled resonant circuits

Scheme of the analysis

Due to insufficient accuracy in PSPICE the analysis was performed using scripts that work with all the signals in the frequency domain.

S-params.Filling time

Power & Grad

RF responseBeam loading

Struct.params.

Circuitparams.

PSPICE

Cell to CellTransferFunction

ijH

)

.(,

)(

elements

matchLR

couplingk

C

L

R

ttQR

Q

v

f

g

0

Signal proc.

Beam&

RF pulseparams.

From structure parameters to circuit elements

R

QC

Q

RL

0

0

2

kvQf g ,,, 0

When the structure is tapered, vg, Q and R/Q vary along the structure, and so do R, C, L and k from cell to cell.

(circuit differential equations)

Cell i

(from PhD thesis of C.D. Nantista, SLAC)

Information extracted from the circuit model

g

tin

in

tin

in

V

Vouts

RIV

RIV

s

1021

1011

log20

log20

…

Directly in PSPICE we can measure:

- Filling time- Voltage (prop. to electric field) and power flow along the structure- S parameters

…

tR : matching impedance

Information extracted from the circuit model

Voltage Amplitude & Phase as functions of frequency at the output of every cell n

Transfer Functions from input cell i to output cell j

… …

i

jij V

VH

nnn jVphaseVampV expHij(f)

Frequency

11.88GHz 11.90GHz 11.92GHz 11.94GHz 11.96GHz 11.98GHz 12.00GHz 12.02GHz 12.04GHz 12.06GHz 12.08GHzV(R27:2)/v(r01:2)

0

0.50

1.00

1.46

f(GHz)

Working in the frequency domain

0f0f

))(()( 1 fVFTtv

f

V(f)

In order to have a reasonable number of samples in the pass band, we need to store a lot of zeros

Solution: Work in baseband

f

AV(f)

)))((Re()( 1 fAVFTtv

min0 ff max0 ff

minf maxf

RF pulse response calculation

1nINi HRFRF

11 V

VH n

n

1RFcell1 cell2 cell3

t

Envelope of the Input RF pulse

…

Transfer function from the input to cell n

2RF 3RFINRF

nnn jVphaseVampV exp

FT

Voltage signal at the output of cell n

1RFcell1INRF

cell1 cell22RFINRF

21H

31H

The beam in the time domain can be assumed to be a Dirac train, therefore its FT is a sinc signal centered in f0 and with

- a width that depends on the number of bunches ( ) and the bunch spacing

- an amplitude Vbeam that depends on the charge per bunch. The voltage amplitude that corresponds to a certain bunch charge is estimated in the PSPICE circuit model from the response in voltage to a current Dirac signal of value

Beam response. The beam signal.

0T

0T

0T

…t

Beam signal (v)

FT

NbunchesT

0

1

Nbunches

Te

0

qNI

VbeamBEAM

f(Hz)

Beam response

in

i

nni

ii

HBEAMRESP

c

pjBEAMBEAM

1

1 exp

3RESP2RESP

1BEAM

cell1 cell2 cell3 …2BEAM

3BEAM

4RESP

4BEAM

1BEAM cell1 cell2 cell3

cell2 cell3

cell3

2BEAM 4RESP

3BEAM

cell4

5RESP

5BEAM

Example: G241

– f0 = 11.994GHz

– = 120deg

– 26 cells

Cell First Middle Last

vg/c[%] 1.66 1.19 0.83

Q 6100 6177 6265

R’/Q[Linac kOhm/m]

14.6 16.2 17.9

Parameters:

G241 phase advance

Nominal =120deg. Matched to 119deg.

Very sensitive to changes in matching elements. Example: 0.01% change in the output Lt

f(GHz)

Ph

ase

ad

van

ce (

de

gre

es)

Cell number

Cell number

Phase value @ nominal frequency

Phase value @ nominal frequency

Ph

ase

ad

van

ce (

de

gre

es)

Ph

ase

ad

van

ce (

de

gre

es)

G241 S-parameters

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

11.8 11.85 11.9 11.95 12 12.05 12.1

s11 s21 s22

f(GHz)

S p

aram

s (d

B)

G241. Group delay

Frequency

11.800GHz 11.850GHz 11.900GHz 11.950GHz 12.000GHz 12.050GHz 12.100GHz 12.150GHz11.752GHzVG(R27:2)

0s

40.0ns

80.0ns

120.0ns

158.4ns

65.15ns @ f0 (62 ns from difference model)

G241. RF pulse response

RF pulse at cell

70ns

40ns

G241. Unloaded Power and Electric Field along the structure.

0

50

100

150

0 5 10 15 20 25

63.8

29.5

127.7118

Electric field (MV/m)

Power (MW) HFSS data for the first, last and middle cell were available. 2nd order polynomial interpolation used for the rest.

cell number

Circuit Model Difference model based on HFSS results

cell number

300 bunches separated by 6 cycles

G241.Beam loading

Difference model based on HFSS results Circuit model

G241.Beam loading

cell number cell number

Ele

ctric

fie

ld (

MV

/m)

Ele

ctric

fie

ld (

MV

/m)

Loaded and unloaded electric field along the structure

70ns

7ns

Filling time=65.15ns

G241.Beam loading compensation

G241.Beam loadingRF pulse and beam response along the structure

Thanks

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