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It is also advisable to retain the symmetry of the
structure in the choice of mesh densities. Otherwise
similar objects may distort into different shapes. Our
experience shows that detailed drawings of the struc-
ture projections on all reference planes are necessary.
These drawings should show the surfaces of the struc-
ture as well as the mesh lines. This procedure is quite
laborious, thus we have develororl a computer pro-
gram which produces the necessary drawings. The
use of the auto meshing routine in the M3 is not rec-
ommended for optimum placement of mesh planes in
complicated structures such as this SRFQ.
The number of mesh points required for the geom-
etry shown in figure 1 Is 347,733 (81x81x53). The
average mesh density is -1 mesh point/cm', the high-
est is 11.8/cm' and the lowest is 0.5/cm'.
III. COMPUTATION AND RESULTS
Following the mesh generation we use the eigen-
value solver E31 to compute the electromagnetic fields.
The E31 requires considerable memory space. Total
running time depends on the availability of on-line
memory. The E31 requires frequent access to large
arrays, thus a lot of virtual memory storage results
In excessive I/O activity. To run the E31 in fast
mode with ~350,000 mesh.points we need about 70
Megabytes of core memory. Since last publication [51
several improvements have been made to increase the
precision of the solution. We have described the tech-
nique by which we generate the transverse profile of
the electrodes. The application of this technique re-
quires a higher mesh density in beam region and in
electrode tip area. However, rounding the electrode
tips leads to a better simulation of the structure. As
a result, we observe an increase in t-he computed res-
onant frequency from 54.7 Miiz to 56.5 Mlz. The
CPU time of the ~350,000 mesh point problem was
about 1 hour on a CRAY 2.
The accuracy of the solution is also dependent on
E31 input parameters. In this simulation the measure
of the accuracy was V x (V x E) = 9.4 x 10-', V - D =
5.5 x 10-'' and V - B = 9.1 x 10-i' (MKS units).
This precision has been obtained by using 10 resonant
modes and optimizing the highest mode frequency in
the computation. Other techniques [6] have also been
used to improve the solution.
Table 1 lists the main electrical characteristics
computed by MAFIA. The values from bead pulling
measurement and those from approximate ex pressions
derived from a lumped circuit model [5] are also iven.
The experimental value of the capacitance is de-
rived from an axial bead-pull measurement in a given
cell of the SRFQ, using the following expression:
Cns = 2akok 1A f/f) ,
where a is the radius of the metallic bead, (A /f)..r n
is the measured fractional peak frequency deviation,
k = 21r/,A and Ain is taken as the theoretical two
term potential value. Units are MKS. A better agree-
ment should be obtained once we get A10 from the
complete analysis of the bead-pull data.
Chnaracterlktl
ct,,, (pF)
Ut') (J)
I (l)
E!,/U(') ((MV/mn7/J)
E,/E.
ED u G'/J)
Centre
Notes'
MIAFIA
30.493
10400
41
3.s
20.2
T2
1.1
0.12
T.4 x 10'
Approx.i
62.4
8480
48
3.,
17.2
e2
1.0
0.13
3.3 x 10'
Measure.
11.372
7200
aso
4,7
14.1
40
1.1
0.1'
3 x 10'
4.0 3.8
0.94 3.8
(1) For ronm temperature copper.
(2) At . designed Inter-vane voltage or v=0.419MV Il.
(3) At the ,nsidie or a SRFQ cell.
(4) Tnclde, transit tmne factor .nd fringe field efect.
IV. DISCUSSION
As we see in Table 1, the agreement between the
MAFIA and the experiment in frequency is reason-
able, considering the complexity of the structure. As
mentioned above, rounding the sharp edges over the
tips of the electrodes has increased the frequency by
1.8 Mlhz. Sharp corners lead to an anomalously high
energy density which lowers the frequency. The present
simulation still contains some sharp corners which do
not exist in the real resonator. We estimate that by
rounding the remaining electrode edges the MAFIA
frequency will go up by 0.87 MHz to 57.37 MHz, in
remarkable agreement to the measured value. This
estimate is obtained by scaling the frequency change
of 1.8 Mlz by the ratio of the length of the tips and
the electric energy density there to the length and
energy density of the remaining sharp edges.
'We can not explain the higher Q value and geo-
metric factor r in MAFIA relative to the measure-
ment. We note that similar discrepancies occur fre-
quently between simulations and measurement. This
may be the result of oxidation of the copper surface.
The electric unbalance AV/V [5] calculated from
the MAFIA field distribution shows a difference be-
tween the ends of the electrodes and the center. This
difference is due to transmission line effects along the
electrodes. The approximate calculation does not in-
clude this effect.
The approximate analytical estimate for the total
capacitance Ct,,Ing in Table 1 also includes the contri-
butions of the frinfe regions and the support struc-
ture. A better estimate of the various capacitances
has also improved the precision in the calculation of
the AV/V as compared to a previous publication [5].
We also note that MAFIA calculates higher peak
surface electric field E, and magnetic field B. than
measured. This can be explained in part by sharp
corners which appear in the simulation. There are
several reasons for the sharp corners. First, the fi-
nite mesh density results in sharp corners at the mesh
Table 1. MAFIA Remutts ve Measurements
and Approximate Expressions
