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.™I«S
COMPUTER CALCULATIONS OF TRAVELING-WA1
PERIODIC STRUCTURE PROPERTIES'"
SLAC-PU3-2295
March 1979
(A)
Locw, R.H. Miller, R.A.
and R.L. Bane
Early
3
?_>
J infrtnirprionly en
Stanford Linear Accelerator Center
Stanford Unlveralty, Stanford, California 94305
Introduction
The versatility and accuracy of programs such as
LALA
1
and specially SUPERFlSH
2
to calculate the tf prop-
erties of standing-wave cavities for llnacs and storage
rings Is by now well established. Such rf properties
include the resonant frequency, the phase shift per pe-
riodic length, the E- and H-flcld configurations, the
shunt Impedance per unit length and
r
- While other pro-
grams such os TWAP
3
have existed for sume time for travel-
ing-wave structures, the wide availability of SUPERFISH
makes it desirable to extend the use of this program to
traveling-wave structures as well. That is the putpose
of this paper. In the process of showing how the con-
version from standing waves to traveling waves can be ac-
complished and how the group velocity can be calculated,
the paper also attempts to clear up some of the common
ambiguities between the properties of these two types of
waves.
Good agreement is found between calculated re-
sults and experimental values obtained earlier.
Space Harmonics, Standing and Traveling Waves
To illustrate our problem, let us review the case
of the classical cylindrlcally symmetric disk-loaded
waveguide for which LALA and SUPERFISH can yield exact
field solutions. It is well known
1
* that in the lowest
pass-band (accelerating TMnx-type
node),
the traveling-
wave E field can be expressed as
wave snapshots of E are shown for two instants of time,
ut-0 and ut-tr/2. Notice that E
z
Is plotted on axis
(r"0) but E
r
and IU are zero on axis and thus are pJot-
ted for 0<r<a. The units are arbitrary. The field
patterns that are shown have for many years been knoto
approximately from bead measurements, paraxial approxi-
mations of Maxwell's equations and general symmetry
arguments. However, some of the subtleties In Fig. la
can only be gotten from a complete computer solution,
as shown later IK this paper. Notice also that since
the fields are sketched at an instant of time, they are
not at thtir maxima, except for selected symmetry planes.
HA
travels in phase with E
r
to preserve a net power flow
(ExH)
z
-E
r
H*. Fig. lb showB E
z TW
max at r-0 vs z and
the corresponding phaBe variation, as governed by Eq. (l).
The standing waves are shown in Fig. lc. The snap-
shots of E are given for two different boundary condi-
tions:
Neuraan (Ej - 0) on the left, and Dirichlet (Hj = 0)
on the right. E
z
and E
r
which are shown at their maxi-
mum values in time are in time-phase, H* leads them in
time quadrature and there is no power propagation: the
energy simply switches back and forth between the elec-
tric and magnetic fields. On the axis
(r-0),
the axial
electric fields can be expressed BB:
>t
£ 2a cos 6 z (Neuman)
(3)
where a
n
Is the amplitude of the Bgace harmonic of In-
dex n, e
n
z-
6
Q
z+
2rrnz/d, k^, - k
2
- 6g , k-u/c and d la
the periodic length. Let a be the radius of the Iris
and b the radius of the cylinder. On axis r-o,
J
o
{0)
n
1 and the amplitudes all reduce to the a
n
,
s.
Furthermore, the fundamental (n - 0) field amplitude at
any r, for a structure where
S
c
-k-u/c
Is equal to
a
o
J
o
(0),
which indicates that a synchronous electron
undergoes the same average acceleration independently
of radial position. If one chooses the origin at a
point of symmetry of the structure (in the middle of a
cavity or a disk) the a 's are all real. Notice that
for r = 0, expression (1) assumes a special form when
z-0 and when z-d/2:
>t
a>t-Be4
°7>Sa
(2)
°-l
i.e., the axial traveling-wave E-field goes through an
er.remum where all the space harmonica are collnear.
This Is also how at r • a the space harmonics "conspire"
to make the tangential E-field at the disk edge equal
to zero, I.e., how they fulfill the function for which
they were invented in the first place, namely to match
periodic boundary conditions. Notice also that if the
phase shift per cell is an exact sub-multiple of 2TF,
i.e.,
P
0
d-2n/m,
then 6
n
- B
0
(l +mn)
.
In what follows,
we will focus on the so-called 2ir/3 mode (m •
3)
which
1B
easy to represent schematically and for which there
1B
a large amount of experimental data from the SLAC
llnac and many others. The results, however, are quite
general and apply to any 0
o
d except v. Fig. la Illus-
trates the behavior of E , E and H.: two traveling-
Work, supported by the Department of Energy under
contract number EY-76-C-03-0515.
£ 2a sin B z (Dirichlet)
(4)
where the factor of 2 comes from the summation of two
traveling waves of amplitude a
n
. These and the corre-
sponding Ej and HA are the components calculated by LALA
and SUPERPISH. Notice that the snapshots of E
2
^ and
Ej.
sw
at the Instants chosen are indistinguishable but
H^'is different.
Croup Velocity
The group velocity for a traveling wave can be ob-
tained from the dispersion diagram (v -dw/dfl) or from
the energy velocity (Vg-P/W^) where P is the power
flow and W^y is the energy stored per unit length. In
order to calculate v
R
with some accuracy from the first
expression, which 1B generally done for the standing-
wave case, one needs to compute several frequencies on
the u-e
0
d diagram, typically for 8
0
d "0, n/3,
TT/2,
2n/3
and
IT,
and then fit the data to some smooth curve. If
however we want to obtain v_ by calculating the fields
at only one frequency, namely the operating frequency,
then the second expression is to be used. For a given
z, we have:
8 "n
*Unit'
length
DISTSUHrm
(5)
unit
length
ITJ>
Contributed to the 1979 Particle Accelerator Conference, San Francisco, CA., March 12 - 14, 1979.
I
to) Trovtling vov*, ^Qd'2ry3
l__-2»
J
Snrpshot
f i E at ul "0
Snapshot
of E ot wt> w/2
Iniiantoneous amplitudes
of E
t
,
Instantaneous amplitudes
of E,,
E
r
,
ond
H* ot
ul
*
0
E
f
,
and
HA ol
-»1
•
W2
tt>) Maximum ottumable omplituoe
j
of
o»ioi
E
t<
T« at any i and
phase
ai
o
function
1
distance
(c) Standing waves,
^QO'ZIT/S,
for
boundaries
ot z'O and 7'3d
1*0 1=0
trv~7t
-
Nlumon Boundaries {E
T
'D}-
1
' ^Dinchlet Boundaries (Hy'O)-
Snapshots
of E max
E7*
V"
Mgnmum amplitudes
of E, ond E, I in
prose)
and H
#
(leading
i
Hme quadroluie)
Id) Standing
WOYM,
^d-2*73,
lor
boundaries shihod
by t*~d
i»-d
z'O z'-d i=0
Plgure
1
It turns
out
that LALA
and
SUPERFISH already give
W
sw
,
the energy stored
for the SW
case.
The
denominator
Wpy
is
simply
Hgy/2:
this
can be
shown rigorously
or
seen
by
superposition since over
a
wavelength,
the
energy stored from
a TW
coming from
the
left added
to
that Jrora
a TU
coming from
the
right results
in
twice
Che energy stored.
The
expression
in the
numerator
can
in
principle
be
calculated
at any
cross-sectional
plane
(S)
since,
by
continuity, energy cannot accumu-
late
and the net
power flow over
a
period must
be In-
dependent
of the
plane
of
integration. What
we
need
to know
are the
simultaneous values
of £
r TU
and HA ^y
at their time maxima
in one
plane. These quantities
can
be
extracted from
the SW
plots.
To do
so, a"trick"
Is needed.
If two
traveling waves
of the
proper phase
add
up to a
standing wave
(Eqs.
(3), (4)),
there must
conversely
be two
standing waves which
add up to a
traveling wave. Referring
to
Fig.
Id, we see
that
If
for example
we
shift
the
diagram
of
Fig.^lc
to
the left
by
z
»-d,
we
have
a
second
SW
solution
\uj
which looks
just like
the
first
one
(A);
..
„J»f-
5
2a
cosS
(z+d)
both
of
which
are
made
up of one TU
going left
and one
going right.
The
"trick"
la to add
them with
the
proper
phases
to
have
the TW's
going left cancel
and
those
go-
ing right
add.
This
can be
achieved
by
multiplying
lbyeJ
(
M-»«>
and % by e^'
2
.
Then:
Ae
J<M-
2>+
B
e
j
2
- 2 sin B,
dY*
a e
>
L* n
and
it
follows that
the
amplitude
and
phase
of the TW
are:
|TU|',
tan 9
(z)
B
-
A cos
B
d
(8)
where
A and B are
functions
of z.
Eqs.
(7),
(6)
are
gen-
eral
and
apply
to any
field component,
E
r
, E
z
or H*, at
any
z.
Hence, given exact
SW
field values,
e.g., as
shown
in
Fig.
2a and 2b, one can now
obtain exact
TW
plots
as in
Fig.
lb. Eq,
(7) gives
the
maximum
TW am-
plitude
at any z and
thus yields
the
Ej.
and H^'s
needed
for
Eq.
(5). Notice furthermore that
Eqa.
(7)
and (8)
can
be
obtained from
A and 6
plots
in
either
the
Ncuraan
or Dirichlet configurations.
In
what follows,
we
shall
narrow down
the
discussion
to
planes
of
symmetry half-
way through
a
cavity
or a
disk where
Eqs.
(7)
and
(8)
are
simplified.
Neuman case: With
the
Neutron boundaries
of
Eifi.
1c,
we
see
that E
rj
sw
=
0 at z-0 and 3d/2 but has
finite
values
at z-d/2 and d. At z =
d/2,
B = 0 and E
r
TW *
Er.SwW
2
)'/
5
"-
At z =
3d
/2.
B3
-
A
and
E
r(T
y
=»
E
r
jswOd/2W^
Similar observations
can be
made
for
HA.
For
example,
at
Z =
0,
B =
AcosB
0
d
and
Kj,
^y
3
HA,
gg(0)/2
and at z = d,
B«A
and
IU
(TW
=
HA
SW(^).
Tne
results
are
summarized
in Table
I.
Since
the
tabulated values
are the
maxima
o.
c
the
fields,
the
results must
be
self-consistent
and
independent
of
which mid-cavity
or
disk
one
considers.
For
the
power calculation,
we can
take
the
power flow
at z-d/2,
i.e.,
tr
iTW
^
pTW
=E
rjSW
(d/'2)H
ff sw
(d/2)/s/3~
or
at z - d, i.e.,
E
riTW
H
+>TH
=
E
r>Sw
{d)H
A
;
sw
(d)/V3.
Dirichlet case: Table
II
shows very similar
re-
sults
for the
Dirichlet case shown
in
Fig.
lc.
Results
Table
III
shows
the
results that have been obtained
by computing
the
properties
of
four SLAC-type cavities
and
by
comparing them with results obtained experimen-
tally^
in the
early 1960's.
The
four cavities whose
2b
and
2a
dimensions
are
shown
are
equally spaced along
a
constant-gradient 3,05msection.
The
computed values
of
r/Q,
Q and r are
obtained from
the
standing-wave SUPER-
FISH calculations.
The
values
of r/Q for the TW
case
are simply twjee those
for the SW
case.
All
values
of
r/Q
and r
have been corrected
for the a
Q
(velocity
of
light) space harmonic amplitude.
The
values
of Q are
the same
for the SW and the TW
cases.
The
assumed con-
ductivity
of
copper
Is 5.8 x 10
mhos/m.
We see
that
in
general,
agreement between computed
and
experimental
re-
sults
is
excellent.
For
reasons
not
understood,
the
resonant frequency
is
almost systematically high
by
1 MHz.
Most other differences including those
for the
group
Table 1
Haxlaurt Values of
t
r
and H
for
Neman
Boundarlaa
Hld-
Hld-
Location
Cavltv
0
Dl.k
Cavltv
d
Dlak
z
Cavltv
0
d
2
Cavltv
d
3d
2
E
r.SV
0
Finite
Finite
0
E
r.TV
E
r.SW<!>
E
r.SW<">
E
r.TV
yr yr
H
*.sw
Finite Finite Finite
Finite
H
*,T«
H
tl
sw
(
°)
2
Ysw
(d
'
W>
2
Table II
Maxima Value* of E
and H for
Dirlchlat
Boundarlaa
Location
I
Mid-
Cavity
0
Dlak
Mld-
Cavltv
d
Dlak
Location
I
Mid-
Cavity
0
d
2
Mld-
Cavltv
d
3d
2
E
r.SV
Finite Finite Finite
Finite
E
r.TV
E
r,SW«'
2
E
r.S«
(
t>
E
r.SW
(d
>
E
r.sw
(
2>
2
Vsw
0
Finite Finite
0
H
»,TW
H
tl
SW
(d
>
yr
Table III
Campari
son of Computed
and Experimental Reaulte for Four SLAC Cavities
Neumai?
Cavity
Boundaries
NO.
,
2
\
(cm)
2a
(cm)
f
exp
MHz
2856
*copp
MHz
(r/Q)
fl/cm
38.13
«"»comp
n/cm
Q
^exp
14160
q
comp
13780
r
exp
ttl/m
54
r
comp
Hil/in
53.7
(v /c)
0.0202
(v /c)
K comp
0.0204
1
8.3442
2.6201
f
exp
MHz
2856
2857.04
(r/Q)
fl/cm
38.13
38.99
Q
^exp
14160
q
comp
13780
r
exp
ttl/m
54
r
comp
Hil/in
53.7
(v /c)
0.0202
(v /c)
K comp
0.0204
28
8.2960
2.4506
2856 2857.74 40.40
40.70
13860
13760 56
56
0.0157
0.0161
57
8.2393
2.21B5
2856
2857.40 42.77
43.0B
13560
13736
58
59.2
0.0111
0.0113
84
8.1773
1.9171
2856
2857.15 45.45
46.07 13200 13710 60 63.2
0.0067
0.0073
Dirlchlet
Boundarle
B
1
8.3442
2.6201 2856 2857.01
38.13
38.70 14160
13780 54
53.4
0.0202
0.0204
2B
8.29f>0
2.4506
2856
2857.28 40.40
40.40 13B60 13759 56
55.6
0.0157
0.0162
57
8.2393
2.2185
2856
2856.83
42.77
42.76
13560
13734
58 58.8
0.0111
0.0114
84
8.1773
1.9171
2856
2856.56 45.45 45.79
13200 13708 60 62.80
0.0067
0.0066
velocity, are within 1 or 2Z. It ahould alao be remem-
bered that the experimental results were certainly - t
accurate to more than 2%. Slight discrepancies between
the Neuman and Dirlchlet results can be used as final
checks to verify the ultimate reliability of the field
calculations. Pigs. 2a and b give actual computer
plotB
of the maximum amplitude standing-wave snapshots shown
in Fig. le. Both examples were computed for the di-
mensions of the first cavity in Table III. The periodic
length d is 3.5 cm and the disk thickness
0.584
cm. All
field amplitudes are in arbitrary units, E being on
axis,
E and H off axis.
r $
References
1, H.C. Hoyt, "Designing Resonant Cavities With the
LALA Computer Program," Proc. of the 1966 Linear
Accelerator Conf., Los Alamos Scientific Labora-
tory, New Mexico, Oct. 3-7, 1966, pp. 119-124.
2.
K. .Halbach, et al, "Properties of the Cylindrical
RF Cavity Evaluation Code SUPERFISH," Proc. of the
1976 Proton Linear Accelerator Conf., Chalk River
Nuclear Laboratories, Chalk River, Ontario,
Sept.
14-17,
1976, pp. 122-128,
R.H. Helm, "Computation of the Properties of Travel-
ing-Wave Linac Structures," Proc. of the 1970 Pro-
ton Linear Accel. Conf., National Accelerator
Laboratory, Batavia, Illinois, Sept 28 - Oct. 2,
1970,
Vol. I, pp. 279-291.
^or earlier discussions on the subject treated in
this paragraph, see P.M. Lapostolle and A.L. Septler
"Linear Accelerators," North-Holland Pub. Co.,
Amsterdam
(1970),
pp. 40-47 and 88-107.
R.3,
Neal, D. W. Dupen, H.A. Hogg, G.A. Loew, "The
Stanford Two-Mile Accelerator," W.A. Benjamin, Inc.,
New York-Amsterdam
(1968),
p. 130, Fig. 6-22.
(a) NEUMAN- NEUMAN BOUNDARIES
ti>) OIRICHLET-DIRICHLET BOUNDARIES
Fig. 2. Standing-wave amplitudes of E
z
, E
r
and H in cavity (1) (Bee Table III) as calculated by SUPERFISH.
E
z
is on-axlB, E
r
and H are off-axis.
