Beam Loading Effects in EMMA and Commissioning Strategy

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Version: 1.1

Date: 11 January 2008







Beam Loading Effects in EMMA and Commissioning
Strategy




Shinji Machida





With a simple assumption of a point like charge, beam loading effects are calculated. It turns
out that bea
m loading is negligible with nominal operation. Without the rf power source,
energy loss is only 1.5% after 1,000 turns if the cavity is detuned +1 MHz that is within
hardware specifications. It becomes 0.38% if the detuning of +2 MHz is allowed. In realit
y,
debunching occurs and the effects should be even smaller than that with a point like charge
model. If that is the case, we do not need rf cavities switched on to measure basic lattice
parameters such as transverse tune and time of flight in the early st
age of commissioning.
This can eliminate tedious tuning of rf frequency and injection timing whenever incoming
beam momentum varies. Tune shift due to energy loss is simulated.




















Stored energy and energy taken out by a beam

Let us first co
mpare the stored energy in an rf cavity and energy taken out by a beam when it
is accelerated from 10.5 to 20.5 MeV/c. The stored energy is calculated as

!

W

QP
"
R
,










(1)

where
!

Q
is the quality factor,
!

P
is the rf power put in a cavity, and
!

"
R
is the resonant
frequency. The rf power is determined by the necessary rf voltage and the shunt impedance of
the rf cavity as,

!

P

V
0
2
2
R
s
,










(2)

where
!

V
0
is the voltage and
!

R
s
is the shunt impedance. Notice that the factor 2 in the
denominator which defines the shunt impedance in the conventional manner.

Using the parameters in Table 1, the stored energy is

!

W

V
0
2
2
"
R
Q
R
S

1.1
#
10
$
3
[J].








(3)

On the other hand, energy taken out by a beam is

!

W
b

qV

q
E
ex
"
E
in
N
cav

4.2
#
10
"
5
[J],






(4)

where
!

q
is the bunch charge and we assume
!

5
"
10
8
particles (80 pC),
!

E
in
and
!

E
ex
are
extraction and injection energy, respectively,
!

N
cav
is the number of rf cavities. 3.8% of the
stored energy is taken out by a beam. Voltage drop is therefore 1.9% and it is negligible.


Table 1. rf parameters.

Shunt imp
edance:
!

R
s

3.4 M
!

"

Quality factor:
!

Q

2.3 x 10
4

Resonant frequency:
!

"
R

2
!

"
x 1.3 GHz

Voltage:
!

V
0

51 kV (when a=1/12)

Harmonic
number:
!

h

72

Beam induced voltage

Another way to see the effect of beam loading is to calculate the beam induced voltage.


Formula and approximation

When a point like charge crosses an rf cavity, the beam induced voltage becomes

!

0
t

0
"
R
s
q
t

0
2
"
R
s
e
#
"
t
cos
$

t
#
"
$

sin
$

t
%

&

'

(

)

*

q
t

0
+

,

-

-

.

-

-








(5)

where
!

"

#
R
2
Q
and
!

"


"
R
2
#
$
2

"
R
1
#
1
4
Q
2
[1]. Since
!

Q
is large, the voltage is
approximated as

!

0
t

0
"
R
2
R
s
Q
q
t

0
"
R
R
s
Q
e
#
"
R
t
2
Q
cos
"
R
t


q
t

0
$

%

&

&

&

'

&

&

&








(6)

the voltage at
!

t

0
means the voltage a
beam sees in its own induced one. With periodic
passing of a beam, the beam induced voltage which a beam sees becomes,

!

V
b

V
b
0
1
2

exp
"
#
R
nT
rev
2
Q
$

%

&

'

(

)

cos
#
R
nT
rev


n

1
*
$

%

&

'

(

)

,





(7)

where
!

V
b
0

"
R
R
s
Q
q
and
!

T
rev
is bunch spacing. In EMMA,
!

T
rev
is revolution time since there
is only one bunch. If there is enough number of passages, the summation can be replaced
using a formula

!

e
"
n
#
n

0
$
e
jn
%

1
1
"
e
"
#
e
j
%
,








(8)

therefore

!

V
b

Re
V
b
0
1
1
"
e
"
#
e
j
$
"
1
2
%

&

'

(

)

*

+

,

-

.

/

0


V
b
0
1
"
e
"
2
#
2
1
"
2
e
"
#
cos
$

e
"
2
#


,




(9)

where
!

"

#
R
T
rev
2
Q
and
!

"

#
R
T
rev
.


Beam loading after many turns

Using the parameters in Table 1, let us calculate the beam induced voltage when the rf power
source is switched off. We introduce detuning
!

"
#
of the rf cavity as

!

"
R

2
#
h
T
rev

$
"
,









(10)

where
!

h
is harmonic number. With this,

!

"

2
#
h
T
rev

$
%
&

'

(

)

*

+

T
rev

2
#
h

$
%
,
T
rev
,







(11)

and

!

"

2
#
h

$
%
&
T
rev
2
Q
.









(12)

The induced voltage after many passages and becoming asymptotic is depicted in Fig. 1. We
assumed
that the bunch charge is
!

5
"
10
8
particles (80 pC).


Figure 1. Induced voltage vs. detuning.


When the detuning is 1 MHz, the induced voltage becomes 7.9 V, and when 2 MHz, it is 2.0
V. Therefore, if a beam circulates for 1,000 turns, the
total energy loss is 150 keV and 38
keV, respectively assuming there is 19 rf cavities. It is 1.5% and 0.38% of the injection
energy.


Transient beam loading

In nominal operation of EMMA, the induced voltage does not reach the asymptotic value
because the
beam circulates for only 10 turns. Also, the revolution time is not constant in
reality when a beam loses energy. Beam loading effects are embedded a tracking code
“SCode” [2] to see the transient phenomena with self
-
consistent energy calculation. It did n
ot
assume the large Q value, but the charge is still point like and de
-
bunching is not included. In
the model lattice, the total of 21 rf cavities, not 19, are distributed evenly around the ring.

First, to check the code and compare with the calculation ab
ove, asymptotic voltage is
calculated as shown in Fig. 2. Except in the detuning of +1 MHz, the voltage calculated with
the tracking agrees with the previous results. When the detuning is +
1 MHz
, the voltage is
lower than expected. This is because energy l
oss due to the self
-
induced voltage makes the
revolution frequency lower and the detuning becomes larger.






(a)






(b)

Figure 2. Asymptotic voltage per a cavity after 1,000 turns. Detuning frequency is taken as a
parameter. (b) is magnified figure o
f (a) in y axis.


Secondly, we looked at momentum oscillations due to the self
-
induced voltage for the first 20
turns. They become noticeable when the detuning is +1 MHz as shown in Fig. 3. It is,
however, 0.3% at most.







(a)






(b)

Figure 3. Momen
tum oscillations due to transient beam induced voltage for the first 20 turns.
(b) is a magnified figure of (a) in x axis.

Tune measurement

Tune measurement in the commissioning without rf is simulated. Because of the finite
chromaticity and energy loss du
e to beam loading, transverse tune changes. We assume that
the transverse beam position can be measured at every cell. Then, we make a Fourier analysis
for the last 1,024 position data every turn. Figure 4 shows horizontal and vertical tune
evolution for 1
,000 turns. The finite chromaticity per cell of 0.2 agrees well with the
measured data.






(a)






(b)

Figure 4. Tune shift during a measurement for 1,000 turns. (a) horizontal and (b) vertical.
Detuning frequency is taken as a parameter.

Acceleration

Finally, the rf power source is switched on. The induced voltage and the longitudinal phase
space with and without beam loading effects are calculated. Figure 5 shows the induced
voltage and Fig. 6 shows the phase space trajectory when a beam is accelerat
ed. The beam
loading effects with 5 x 10
8
particles are negligible.



Figure 5. Beam induced voltage which a beam sees during acceleration.



Figure 6. Longitudinal phase space both for zero and nominal current.

Summary

With a simple assumption of a poin
t like charge, beam loading effects are calculated. It turns
out that beam loading is negligible during nominal operation. Without the rf power source,
energy loss is only 1.5% after 1,000 turns if the cavity is detuned +1 MHz that is within
hardware speci
fications. It becomes 0.38% if detuning of +2 MHz is allowed. In reality,
debunching occurs and the effects should be even smaller

than that with a point like charge
model. If that is the case, we do not need rf cavities switched on to measure basic lattic
e
parameters such as transverse tune and time of flight in the early stage of commissioning.
This can eliminate tedious tuning of rf frequency and injection timing whenever incoming
beam momentum varies.
Tune shift due to energy loss is simulated.

Acknowle
dgements

I would like to thank Dr. J. S. Berg for valuable comments and corrections. I also appreciate
many helps of Dr. D. J. Kelliher and Dr. G. H. Rees.

References

1.

A. W. Chao,
Physics of Collective Beam Instabilities in High Energy Accelerators
,
John Wi
ley & Sons, Inc., p. 73, 1993.

2.

S. Machida, to be published, 2008.