BEAM IMPEDANCE CALCULATION AND ANALYSIS OF HIGHER
ORDER MODES (HOMS) DAMPED RF CAVITIES USING MAFIA IN
TIME DOMAIN
∗
Derun Li
†
, Robert A. Rimmer
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, USA
Abstract
A time domain method using MAFIA code has been de-
veloped to calculate narrow band beam impedance in RF
cavities over a wide range of frequency spectrum. The
impedance is obtained through Fast FourierTransformation
(FFT) of computed wakefields by the MAFIA. Analysis of
the calculated impedance spectrum will be presented. Ap-
plication of the method to a known RF cavity design (PEP-
II cavity) has showngood agreements with bench and beam
measurements. The methodhas been applied to theRF cav-
ity design of of Damping Rings for Next Linear Collider
(NLC).
1 INTRODUCTION
It is well known that higher order modes (HOMs) in
RF cavities has been one of the main sources of longitu-
dinal and transverse impedance which could contribute to
coupled bunched beam instability in storage rings and syn-
chrotron light sources, and become a limit factor for beam
intensity. Properly damping the HOMs while leaving the
fundamental one intact has been main efforts for cavity de-
signs for decades for high intensity electron/positron stor-
age rings and light sources[1]. HOMs exist in any RF cav-
ity, once they are excited(forinstance by beam underaccel-
eration), they oscillate and eventfully decay with their nat-
ural frequencies and time constants which are determined
by the cavity geometry, surface resistance and couplings
to surroundings. In addition to interaction with fundamen-
tal mode in the cavity, the beam bunch interacts with the
HOMs as well. Subsequently two things may happen, 1)
the beam excites HOMs within its spectrum; 2) the result-
ing EM fields, which is called wakefield, from the HOMs
act back on the beam. If the natural time constant of a
HOM is long (or in another word, Q value is high), the EM
field of this mode will last for a long time and act back
on the trailing beam bunch, usually harms the beam. It is
important, but not easy to damp HOMs efficiently (keep
the Q below a certain value) while leaving the fundamen-
tal mode intact. Ways to damp HOMs include, for exam-
ple, adding damping ports of HOMs on the cavity body or
putting antennas inside the cavity, or on the beam pipe (su-
perconducting cavity). Discussions on the damping mech-
∗
This Research Work is supported by the Director, Office of En-
ergy Research, Office of High Energy and Nuclear Physics, High Energy
Physics Division, of the U.S. Department of Energy, under Contract No.
DE-AC03-76SF00098
†
Email: [email protected]
anism can be found in many literatures. In this paper, we
discuss how to calculate beam impedance exhibited by the
HOMs in a RF cavity using the MAFIA in the time do-
main. Traditionally HOMs are computed in the frequency
domain, HOMs impedance are studied and conducted ex-
perimently on cold-test cavities, which is time-consuming,
expensiveand limited to a few measurable modes. Simulat-
ing a cavity with HOMs couplers in the frequency domain
has been difficult due to limited computational capability
of available codes. With the development of advanced 3-D
computer codes in recent years, for instance the MAFIA
code, broadband waveguide boundary conditions become
available in the time domain, many of these experiments
now can be carried out on computers. We developed a
method using the MAFIA in the time domain to calculate
narrow band beam impedance in a wide frequency range.
The method has been applied to a RF cavity design of the
Damping Ring (DR) for the Next Linear Collider (NLC),
and cavity designs for light sources [2].
2 THE TIME DOMAIN METHOD
The beam impedance is a description of the wakefield
in the frequency domain. In principle the beam impedance
of the HOMs can be calculated in the frequency domain.
However, once a cavity has HOM coupling ports, couplings
of these ports with outside make the boundary conditions
on the port interfaces difficult to treat mathematically in the
frequency domain. Hence without broadband waveguide
boundary conditions, the frequency domain simulation re-
sults are good and limited only to trapped or nearly trapped
modes. For a heavily HOMs damped cavity in particular
(PEP-II cavity for instance), the frequency domain model-
ing simply does not represent the real physics in the cav-
ity. Special method has been developed successfully for
calculating external coupling [3] using the MAFIA in the
frequency domain before the waveguide boundary condi-
tion became available in the time domain. We developed a
method using the simulation results of the MAFIA in the
time domain where the external coupling is dealt by the
waveguide boundary condition. The wakefield in the cav-
ity is excited by a Gauss distribution beam, and computed
and recorded. This computed wakefield is then being Fast
Fourier Transformed (FFT) to obtain its spectrum in the
frequency domain. The beam impedance of the cavity is
yielded by normalizing the wakefield spectrum to the spec-
trum of the excitation beam. The Gauss distribution beam
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Proceedings of the 2001 Particle Accelerator Conference, Chicago
(line charge) used in the calculation is given by,
ρ(s)=
Q
√
2πσ
s
e
(s−s
0
)
2
2σ
2
s
(1)
where Q is total charge of the beam bunch, σ
s
is bunch
length.
The resulting wake function
1
may be expressed as a sum
of all modes being excited in the cavity, where each mode
is represented by an index of n
W (s)=
∞
n=1
−
ω
n
2
R
n
Q
n
e
j
ω
n
c
s
e
−
ω
n
2Q
n
s
(2)
where W (s) represents the wake function, ω
n
=2πf
n
with f
n
as resonant frequency of mode n; R
n
2
and Q
n
are shunt impedance and quality factor of mode n, respec-
tively; s = vt with v as the speed of beam bunch and s
starts from the head of the beam. The beam impedance is
obtained by the FFT of the wake function,
Z(ω)=
1
c
∞
0
W (s) e
−j
ω
c
s
(3)
where we have denoted the beam impedance with Z(ω)
and assumed v = c. Apparently the beam impedance
depends on the frequency ω. For a RF cavity, the beam
impedance can be expressed as a sum of all narrow band
shunt impedance which it can be deduced directly from
Equation (3),
Z =
∞
n=1
z
n
=
∞
n=1
R
n
. (4)
It is worth to point out that the wakefield simulation is al-
ways computed and recorded in a finite time period τ or
length s
max
= cτ , the integral in Equation (3) does not ex-
tend to ∞. We therefore yield a calculated beam impedance
z
n
for mode n,
z
n
= R
n
1 − e
−
τ
τ
n
(5)
where τ
n
=
2Q
n
ω
n
is the natural time constant of mode n.
The calculated beam impedance does not always equal
to the shunt impedance of the cavity as it should be. This
is due to the truncation of the wakefield calculation. There-
fore the beam impedance calculated based on the FFT of
the calculated wakefield may or may not give the correct
beam impedance depending on the mode is resolved (no
truncation) or not resolved (truncated) during the finite time
1
The wake function can be obtained mathematically or numerically by
computer codes by,
W (s)=
1
Q
∞
0
E(r, z,
z + s
v
)+v ×
B(r, z,
z + s
v
)
dz,
where
E and
B are excited by charge Q
2
R
n
is defined as
V
2
t
P
n
period of the wakefield calculation. This can be further cat-
egorized as the following,
Z
n
≈
R
n
if
τ
τ
n
>> 1
τ
2
ω
n
R
n
Q
n
if
τ
τ
n
<< 1
needs two runs if
τ
τ
n
∼ 1
(6)
A single time domain simulation of the wakefield can not
get the beam impedance of high Q modes which is quan-
tified in Equation (6) by τ
n
. The simplest solution is to
make two simulations with the second simulation doubling
the time record of the first wakefield calculation. Two cal-
culated beam impedance z
n
(τ
1
) and z
n
(τ
2
) (τ
2
=2τ
1
) are
then obtained. We define a parameter ξ =
z
n
(τ
1
)
z
n
(τ
2
)
, which
is the beam impedance ratio of the two simulations. It is
evident that a mode is fully resolved if ξ ≈ 1; not resolved
if ξ ≈ 0.5 (e.g., fundamental mode). At ξ ≈ 0.5 , external
coupling is very weak, thus the frequency domain results
are good and can be used directly (e.g., R, Q, and
R
Q
).
At 0.5 ≤ ξ ≤ 1, the mode is partially resolved, both z
n
(τ
1
)
and z
n
(τ
2
) are needed for calculating R
n
. It is not difficult
to find that R
n
is given by,
R
n
=
ξ
2ξ − 1
z
n
(τ
1
)=
z
n
(τ
1
)
2
2z
n
(τ
1
) − z
n
(τ
2
)
. (7)
Typically z
n
(τ
1
) and z
n
(τ
2
) can be measured quite accu-
rately from the calculated beam impedance spectrum, but
sometimes need to be fitted and measured carefully due to
limited resolution of the spectrum in the frequencydomain.
Better resolution can always be gained by increasing the
calculated wakefield length or decreasing the time step in
sacrifice of the CPU time and computer memory.
3 SIMULATION PROCEDURES
We list the procedures to compute the beam impedance
using the MAFIA in the time domain,
• generate 3-D model in x-y-z coordinate (many built-in
functions in the MAFIA post processor do not support
the cylindrical coordinate)
• based on the 3-D model, slice out 2-D waveguideports
with the same mesh configuration in the mesh genera-
tor
• use the 2-D eigen-value solver to solve for the waveg-
uide modes (TW) at each port
• load the calculated waveguide mode solutions to each
port for using in the waveguide boundary conditions
in the time domain solver (T3)
• define an excitation source as either beam or current,
and calculate record the induced wake fields in T3
• post process (FFTs, etc.) the calculated wakefields to
obtain the beam impedance in the post-processor (P3)
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Proceedings of the 2001 Particle Accelerator Conference, Chicago
4 SIMULATION RESULTS AND
DISCUSSIONS
we use simulation results conducted for the PEP-II RF
cavity as an example to detail the discussions on the beam
impedance calculation. The PEP-II cavity is shown in Fig-
ure 1.
Figure 1: The PEP-II RF cavity: three HOM ports are
equally spaced on the cavity body
A 3-D MAFIA model is established for the PEP-II cav-
ity, good agreements between the MAFIA time domain
simulations, measured and beam induced spectrum are
achieved and reported in [1]. To illustrate the time domain
method, only the calculated beam impedance of the dipole
modes is shown in Figure 2 where the wakefields are com-
puted and recorded for 100, 200, 400 and 800 meters long,
respectively. The beam impedance is plotted versus fre-
quency in Figure 2.
Figure 2: Calculated beam impedance of the dipole modes
in the PEP-II cavity is plotted versus frequency.
Notation X in Figure 2 denotes the beam impedance
calculated using Equation (7) from the z
n
of 100 and
200 meters long wakefields. As a cross-check, the beam
impedance is also calculated using the z
n
from 200 and
400 meters long wakefields. They agree very well with the
former calculation. As indicated in Figure 2, some of the
modes are fully resolved, and some of them are partial re-
solved and their amplitudes increase with the time record of
the wakefield. A close view at a mode frequency near 1.2
GHz is shown in Figure3 exhibitingthis amplitudechanges
explicitly,
Figure 3: A close view of the beam impedance of the PEP-
II RF cavity near 1.2 GHz
where the calculated z
n
is indicated again by X which
has slightly higher value than that obtained from 800 me-
ters long wakefield implying that this mode is nearly re-
solved by the 800 meter wakefield.
5 CONCLUSION
The time domainmethod has beenverified with the beam
induced HOM spectrum measurement and bench measure-
ment of the PEP-II cavity, and used successfully for the
NLC DR RF cavity design and RF cavity designs in light
sources. It can be easily extended for applications of any
coupler design of RF cavities.
6 REFERENCES
[1] R. Rimmer, J. Byrd, D. Li, “Comparison of Calculated, Mea-
sured and Beam Sampled Impedance of a HOM-damped RF
Cavity”, Physical Review Special Topics - Accelerators and
Beams, Volume 3, 102001 (2001)
[2] F. Marhauser, et al, “HOM Damped 500 MHz Cavity Design
for 3rd Generation SR Sources”, this conference.
[3] N. Kroll, D. Yu,”Computer Determination of the External
Q and Resonant Frequency of Waveguide Loaded Cavities”,
SLAC-PUB-5171
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Proceedings of the 2001 Particle Accelerator Conference, Chicago
