SLAC-TN-63403
E. Westbrook
December 1963
MICROWAVE IMPEDANCE MATCHING OF FEED WAVEGUIDES TO THE
2n/3 MODE
DISK-LOADED ACCELERATOR STRUCTURE OPERATING IN THE
I.
An interesting
problem
ordinary
rectangular
periodic
structure
operating
of electric
field
velocity
of capturing
close
in this
velocity
disk
looked
upon as a series
interest
forming
and stop
wave
accelerator.
circular
is a longitudinal
be useful
for
component
accelerating
of the wave could
use in the
when an
traveling
from uniform
short
be reduced
initial
and accelerating
disks
with
a center
process
them to very
aperture,
The aperture
the
of the
strength
wave structure
of cascaded
a bandpass
bands.l
the phase
as desired.
amount of power available.
traveling
coupled
filter,
Only the
resonant
with
lowest
electric
With
is created,
size
and'
field
equally
spaced
which
can also
cavities
or two-
an infinite
frequency
be
number of
passband
is of
here.
The combination
the
purpose
the
structure,
It
of aperture
of obtaining
was found
with
that
In early
for
The "mode" of
the phase shift
this
the
the best
and disk
lowest
such a periodic
per cavity,
disk
its
the
for
of
and filling
be 3-l/2
four
microwave
disks
disk
properties
spacing.3
structure
in radians.
would
at Stanford,
of
on the axis
energy
of disks
the measurement
was studied
field
stored
spacing
particular
spacing
accelerating
accelerators
was chosen because
was simplified
size
the highest
compatible
per wavelength.2
structure
loading
a given
a periodic
passbands
arises
of light).
disks,
networks,
would
electrons
can be used to control
for
There
for
of the wave can be reduced
spacing
time.
lower,
the
inductive
on the axis,
port
TMol mode.
(or
and bunching
By adding
electron
the phase velocity
of light
to the velocity
linear
to the
by starting
mode which
if
matching
is to be connected
is derived
in the
of electrons,
impedance
in a microwave
structure
waveguide
to the
in microwave
waveguide
used
The periodic
bunches
INTRODUCTION AND HISTORICAL REVIEW
is customarily
The definition
defined
as
can be applied
in two senses.
-l-
---_--.--
_”
___.____-^
_ .._.
--l~--.ll”-
.. ..-.--_ ..-.. ..-.I.-- .-
.
,
, _,_.--_- . . -..
__.__
.-,-... ,...- .-..
First
is the
is equal
this.
to the
(2n)/n,
where
disk
(per
disks,
it
periodic
at
(zero)
because
shift
per
wavelength
ever)
the
shift
per
velocity
disks
twice
mode is
the
cavity,
spaced
per
phase
velocity
is fixed
to assure
the
space
free
mode,
free
is
simply
space wavelength
Thus the
is
is
(2n)/4,
at
operating
or
in studying
development
matching.
n/2,
is constant
high
mode
and for
the
disk
(27r)/2,
or
'moder'
although
only
with
velocity
the
no further
end of the
varies
at the
design
than
is zero
frequency
over
two disks
per
free
the
phase
does the
then
guide
what-
0 to 7[ radians
and only
phase
the
spacing
frequency
in-
mode is 0
passband,
the
from
of light,
apart
the
any disk
JY. By varying
actually
becomes
and there
(for
of
such as microwave
thus
everywhere
spacing
properties
wavelength
passband,
frequency
the
work
The guide
end of the
At the
coincide
are
to
or the
frequency.
and in doing
phase
cavity.
range,
cavity,
definition
low frequency
and the
frequency
is equal
structure
the
(2fi)/3.
is
use of the
the
per
wavelength)
becomes
passband
shift
and impedance
finite
wavelength
operating
structure
measurements
and the
number of disks
fixed
The second
the
phase
is the
n
predetermined
three
guide
in which
condition,
of light,
case the
and the
of a four
operating
velocity
In this
wavelength,
the
normal
if
phase
the
space wave-
length.
Further
tage
experimental
in accelerating
disks
was chosen
Stanford
two-mile
enough
is customarily
a given
over
group
that
use in several
showed an appreciable
velocity,
of the
four
commercial
for
disks,
advanthe
three
and this
accelerators
new
and the
THE PROBLEM AND A TECHNIQUE FOR MEETING IT
The peak power
transmission
Stanford
machine.
II.
is great
spacing
for
at
for
field,
per wavelength
spacing
work
required
to require
line
between
a standard
mode, and is either
for
high
adequate
power
accelerating
sources
of microwave
a microwave
source
rectangular
waveguide,
pressurized
or evacuated.
-2-
field
and the
periodic
operating
strength
energy.
The
structure
in the
TElC
the
Matching
problem
for
microwave
moving
junction
which
standard
impedance
the
amounts
This
would
between
structure
(Fig.
transmission
period
known,
effect
successive
would
be obtainable.
However,
in actual
which
is
oversize
because
ished
Section
of the
be made relatively
mode structure
partially
in the
first
which
tion
it
for
modifications
for
disk
(or
coated
with
is
impedance
possible
proper
data
is
a detuning
taken
what
knowledge
of the
unfin-
failure
resonated.
can still
described
be
above can
structure,5
a coupler
in
less
to a n/2
and a dielectric
These are positioned
the
in the
of the
to be given
problem
plunger
material.
steps
out-
r(/2)
Match.ing
(including
to determine
coupling
technique
resistive
two or more cavities
while
of the
&-C/J) one.
mechanical
one needtobe
the
slightly
on the
procedure
(or
its
most
in
the
the first
disk
the
can be adjusted
pushing
such that
a four
has been done using
sliver
junction)
is
diameter,
cavity
causes
In the
after
unfinished.
of holding
of exact
what
to above.
cavities
easily
a three
just
situation
and workable
for
is
to its
is manufactured
or otherwise
lack
also
cavity
(An oversize
But this
none of the
The nature
readily
impossibility
referred
is
the
appear
on a uniform
attached
structure
practical
squeezing,
would
per
to match an unfin-
integrally
sense that
as desired.
phase shift
it
desired
to phase shift,
wall.)
technique
often
the
not a uniform
known amounts
respect
phase shifts
III,
attacked
of the
cavity
cavities'
matching
with
by denting,
of the
is
design
line.
periodic
is
cavities,
a short
periodic
in the
as perfectly
electrically
it
the
unfinished
dimension
side
practice
while
critical
tolerance
to moving
matching
structure
while
by
transmission
for
a
to make
circuit)
accelerator
the
has been manufactured
structure,
The latter
the periodic
and by detuning
line
coupler
and the
structure,
equivalent
uniform
is
line,
a short
usable
In a periodic
the
periodic
waveguide
even though
that
ished
be directly
rectangular
l),
but
is
transmission
(usually
nonstandard
lines
One approach
standard
reflection
in the
tranSmi.SSiOn
exist.4
in the
apparently
line.
is
techniques
of a total
known discrete
junction
two different
measurements
position
technique
between
one with
input
to take
the matching
waveguide,
to adjust
from
the
junc-
and tuning.
-3-
..
___..._*_... ^ ,.
..---.
.
,--
~. ..-.-
-
_._.-- --..
.,.
STANDARDAECT4NGULAR
WA VEGUm?
kF.SOh!A/vT
cm/TV WAf 1s
FIG.
1 --
Cutaway view of junction
periodic
structure.
between
-4-
rectangular
waveguide
and the
It
should
be realized
any particular
only
once,
junction
after
The need to tune
the
III.
exactly
tuned
determination
proper
coupling
for
to the
finished
dimensions
first
cavity
of the
should
couplers
other
succeeding
remains,
but
tuned
for
matching
accelerating
with
sufficient
this
is
The input
during
the
this
input
determine
the
cavity
with
"detuned
the
short'
the
is
need to have
the
only
cavity
The frequency
for
the
2n/3
b.
The frequency
for
the
c/2
resonances
usual
metallic
plunger,
in the
input
and
waveguide
6
a.
from
an undercoupled
cavity
positions
frequencies:
C.
same
procedure.
each of three
test
accuracy.
done in the
eliminates
coupling
for
mined
sections
of R. L. Kyhl)
and tuning
to a 2r(/3 mode structure
cavities.
Detune
of
cavities.
procedure
junction
dimensions
have to be determined
MATCHING AND TUNING PROCEDURE (Method
at any time
1.
the
the
the
The following
waveguide
for
which
can be manufactured
way as for
that
mode (operating
mode in the
in a short
frequency),
structure
section
(as deter-
of exactly
tuned
cavities),
The arithmetic
mean of the
above
frequencies
in
"a"
and "b",
above.
2.
Retract
the
detuning
plunger
so as to detune
the
following
cavity.
3.
At the
form
the
quarter
Leaving
at the
were
frequency
first
cavity
of a guide
position
4.
average
for
the
input
away from
plunger
~(/2 mode and the
the
data
2.
undercoupled
order
the
aperture),
1 (c),
mechanically
impedance
the
de-
point
"detuned
one-
short"
frequency.
detuning
(in
step
the
wavelength
this
correct,
in
to bring
as shown in Fig.
ling
given
in position,
take
impedance
If
25r/3 mode frequencies.
points
Since
data
it
would
is
on a Smith
assumed that
to be able
points
fall
the
to remove metal
will
actually
fall
the
data
coupling
chart
plot
cavity
is
at the
coup-
as in Fig.
3.
-5“.-.-^_.---~-
--._ . ._.._....-. ..._... --. .--- _.-...
_
I______.___
*_
.
_ ..- .
_
.-..- I_ . -
FIG.
2--Smith
chart plot,
properly
coupled
first
cavity
and tuned.
SECOND
/
FIG.
3--Smith
tuned
chart plot,
first
but undercoupled.
-
6
-
cavity
/
DEJLJAfED
5.
The aperture
the
cavity
Repeat
the
coupling
is
tuned
and matched.
sions
and record
filter
is
the
coupling
capacitance
between
constant
over
to the
to be correct,
accuracy
the
other
coupling
-.l
.-....
-..-
the
coupling.
of step
is
4,
now both
aperture
dimen-
reactance
frequency
of an equivalent
lumped
cavity
of the
over the
Its
cavities.
The first
disk
in the
circuit
of microwave
periodic
structure.
and is
as the
study
operating
representation.7'8'g,'0
passband,
cavities.
the
structure
at the
C
fl/2
is
between
dimensions
the
mode frequency
cavities
are held
is
assumed to be
assumed to be essentially
The coupling
aperture
arm, and is
series
susceptance
evaluated
is
linear
passband.11T12
to
is
idenassumed
sufficient
in manufacturing.
,
7-
*
_._.
the
The coupler
input
from
-
--.._
detune
some repetition
of the periodic
total
with
tical
use.
network
assumed to be linear
definiteness.
future
h--Equivalent
accelerator
4, Xser
for
the
can be determined
FIG.
at
Measure
will
to increasing
to be correct.
properties
bandpass
In Fig.
for
This
and
BASIS FOR TRE FOREGOING PROCEDURE
The electrical
circuit
in addition
found
amount by machining,
and recorded.
3, 4, and 5, until,
steps
passband
measured
slightly,
IV.
lowest
be opened a small
new dimensions
first
6.
should
-...
--
I.
.._
^_
..-_
-
.,
“..
..-.
_
,
I
-.
Including
former
first
a means of coupling,
because
cavity13
FIG.
If
the
effects
hole
of magnetic
field
the
becomes
circuit
5--Input
coupler
of fringing
for
in this
the
electron
added to the
network
coupling
coupling
cavity
is
fields
beam,
between
transformer
at the
the
waveguide
added to Fig.
and to represent
coupling
frequency
aperture
insensitive
the
reactances
6.
zl
I
I
I
FIG.
6--Periodic
structure
with
input coupling
network.
-8
-
and the
4.
modifying
and at the
I
I
IN
by a trans-
5.
as shown in Fig.
detuned,
as in Fig.
case represented
arbitrary
entrance
can be
The input
three
network
arbitrary
actual
shown between
elements,
frequency
and is
insensitive
vertical
dashed
therefore
coupling
capable
system
lines
6
in Fig.
of representing
used with
the
has
any
periodic
structure.
The next
by choosing
capacitor
input
step
to view
C placed
coupling
general
is
to modify
the
two-port
periodic
at the
network
the
circuit
structure
network
input.
is modified
network
in Fig.
for
mathematical
halfway
At the
accordingly,
through
same time
and is
convenience
a coupling
the
arbitrary
represented
by a
7.
t
FIG.
The circuit
able
for
T--Periodic
suitable
of the
analysis
structure
and input network,
for mathematical
analysis.
periodic
structure
as a standard
bandpass
more basic
start
can now be considered
filter,
viewed
at the
as suitmid-shunt
terminals.14
It
structure
is
perhaps
and input
The mid-shunt
is
given
network
characteristic
to
the
analysis
when operating
in the
admittance
by l5
- ,9-
Y.
by considering
the
7r/2 mode.
of the
periodic
network
is
yO
at the
a pure
low end to
Xser(0)
is
representing
mid-shunt
over
the passband,
zero
high
frequency
zero
series
The high
cavity.
conductance
network
at the
at the
low frequency
resonance
frequency
section
of the
jX
end of the
ser
passband
sensitive
passband
8) becomes series
(Fig.
from
infinity
end.
frequency
end of the
varying
(O-mode),
elements
(R-mode)
occurs
resonant,
of the
when the
that
is,
when
-j4X,=O
or
xser(n) = 4x, = $
C
where
it
should
be recalled
that
= WC
Xc
is
constant
over
the
pass-
band.
/Y
Set-
a. General
mid-shunt
b. Mid-shunt
periodic
section
FIG.
At the
g
frequency,
8
section
structure
of accelerator
as it
the
will
be recalled
passband.
that
The x/2
X
varies
ser
mode frequency
is
linearly
halfway
with
across
frequency
the
over
passband.
‘
2
yc
ILyc
=J- r 1--=2
2
Properly
structure
series
placing
has the
a detuning
effect
arm from the
plunger
in the
preceding
XHES
in a cavity
equivalent
section
2
circuit
of the
periodic
of disconnecting
of the network.
SeeFigs.
ARM DISCONAhW-EL’
/
FIG.
g--Periodic
structure
with
- 11 -
coupler
cavity
detuned.
the
9 and 10.
ARM DISCONNECTED
FIG.
lo--Periodic
The input
admittance
structure
In Fig.
given
detuned.
by
yc
=j,.
in
cavity
10,
y.
by definition
= 4-7
in
of the mode of the
We now have data
will
second
9 is evidently
in Fig.
y.
with
normalize
points
admittances
yc
2
network,
which
to the
and verified
by calculation.
can be plotted
on a Smith
Chart.
We
characteristic
admittance
of the
net-
= yJ2 J and view the system at the input to the coupling
maintaining
normalized
admittances
in the input transmission
line.
work,
Y
we obtain
the
Fig.
complex
tours
We are
structure
12,
considering
reflection
of pure
in Fig.
network,
0
coefficient
conductance
now prepared
12 if
we wish
designed,
the Smith
plane
and pure
tuned,
to be the
unit
(Fig.
ll),
admittance
susceptance
to ascertain
to perform
Chart
the
the
overlaid
- 12 -
for
operation
circle
on the
change in position
same impedance
and matched
with
Thus
in
con-
circle.
of the points
measurements
on a
in the 2rr/3 mode.
/MAGI/VARYAXIS
Fig.
ll--Unit
the
circle
complex
of reflection
plane.
Basis
coefficients
in
of Smith chart.
FOR DE 7UW-P
JECOA’D CAVIT Y
FOR DE 7-UNED
IAIPU 7 CAV/T Y
FIG.
12--Smith
chart plot
and characteristic
of admittance
admittance
- 13 -
in ~r/2 mode,
across passband.
We have first
the
n/2
calculate
and in the
with
tures,
to
the
ratio
The usual
2fl(/3 modes.
low group
velocity
shape and can be represented
which
is
admittances
of accelerator
thin
very
disks
nearly
in
struc-
widely
sinusoidal
spaced,
in
by
= fl;)
f
design
and reasonably
or cu - @ diagram
has a Brillouin
of characteristic
- [f(z)
- f(Ol
cos Sd
where
is
the
frequency
is
the
frequency
for
the
x/2
is
the
frequency
for
the
0 mode,
is
the
phase
unit
is
d
We also
note
that
length
the
in Fig.
the
conditions
in the
that
is
Chart
existing
plot
of Fig.
when the
211/3 mode of operation.
necessary
is
in
radians
periodic
of the
per
structure,
periodic
spacing.
10
(I
Smith
of the
length
2n
Y.1n 3
Now the
shift
mode,
to multiply
= 0.
12 can be renormalized
input
network
To accomplish
all
admittances
- 14 -
is
this
adjusted
to represent
for
normalization,
on the
chart
a match
all
of Fig.
12
by fi,
which
When Fig.
will
bring
Yo(21x/3)
2 is
rotated
60'
be seen to coincide
Y(S)
Second
with
those
to the
center
counter-clockwise,
of Fig.
of the
the
data
chart,
Fig.
points
13.
will
13.
= -
cavity
detuned
Second cavity
/y(s)=jfi
Input
FIG.
cavity
detuned
lj--Admittance
plot for structure
matched in the 2rr/3 mode.
Arc cos (-l/4)
is the mode for the mean frequency
between the n/2 mode and the 2~r/3 mode frequencies.
- 15 -
detuned
If
the
input
distributed
coupler
and symmetry
cavity
is
tuned,
frequencies,
at the
(shifted)
second
cavity
remain
symmetrically
average
admittance
input
It
was mentioned
plete
periodic
dimension.
cavity
However,
walls
impedance
preting
the
in the
frequencies
able
from
of Fig.
coupling
modes,
as the
sitive
the
coupling.
phase
This
would
will
This
veri-
its
they
a comfinal
tuned
of loose
yield
the
de-
in a multi-cavity
together
in a suitable
to the
successive
the
the
set
jig.
of cavities
cavities
and inter-
or in
small
fact
a
to any
enough to avoid
of experimentally
corresponding
of resonant
test
be between
modes are to the
a very
to matching
one of them.
structure)
admittance
for
method
remains
a pair
modes should
input
Kyhl
21-r/3 mode,
cavities
select
chosen
call
to
a set
that
be clamped
length
these
of the
pieces
as in
a n/2
the
held
to produce
when detuning
short
closer
frequencies
to manufacture
diameter
to adapt
to
matching
&3 shows that
detuned
mentioned.
cavities
a relatively
the
procedure.
when resonated
the
between
be preferable
(for
impractical
by detuning
be possible
between
would
taken
r(/2 mode as well
the
oppo-
diameter.
matching
such accuracy
could
as previously
should
to this
be possible
to be matched
data
mode in which
It
shift
data
5[/2 and 2~/3
cavity
made by clamping
and the
interaction
the
cavity
coupler
coupler
with
is
per
Then the
it
it
of
arrangement
Also,
that
disks
test
point,
of the
and loading
of phase
lc/2 and 23713
admittance
proof
should
value
the
cavity
respect
it
sign
Then, if the input
between
the
the, con-
ADDITIONAL, DISCUSSION
above
structure
transformation.
diametrically
with
the
Consider
point
for
disposed
13 are re-
of Fig.
admittance
input
points
v.
points
factor.
frequency
detuned
3 and completes
Fig.
of the
the
the
multiplication
properties
to produce
to the
fies
undercoupled,
by a new mismatch
formal
site
is
is
long
to modes obtainInspection
section.
the
fi/s
r(/2 mode,
to deviation
test
determined
section
and 271/j
the
from
for
more senproper
modes close
to Jr/2.
As a practical
will
have,
example,
among others,
a 12 cavity
resonances
test
at the
- 16 -
section
5lr/l2
could
be set
and 7~~/l2 modes.
up.
The
It
respective
normalized
be -23.146
will
first
cavity
spectively
from
is
input
admittances
and -jO.318
is properly
when the
coupled.
clounter-clockwise
the -jl.O
detuned
susceptance
and the
input
was worked
Voltage
approach
out by Dr.
Engineering
second
point
cavity
is
input
is measured
frequencies
detuned
points
by 0.0760
seen at the
admittance
to the
corresponding
These admittance
and clockwise
VI.
The basic
at the
will
guide
lie
re-
wavelength
when the
at the
and the
second
cavity
~t/2 mode frequency.
ACKNOWLEDGMENT
matching
R. L. Kyhl
while
procedure
the
author
for
the
2~(/3 structure
was employed
by High
Corporation.
- 17 -
---_--
-.----..
_ ._- . . -_-..._,.._-,_I......-.-_ ... ....--_.
_
I.,-_,..... .--. ..-----------_-.__"._..._"_-..
_--.. _.._
*_ . _,.ll"
-
REFERENCES
1.
J.
C. Slater:
1950,
New York,
2.
Microwave
W. W. Hansen,
Accelerator,"
D. Van Nostrand
Company,
Inc.,
Chap. 8.
E. L. Ginzton,
Review
Good pasic
Electronics,
W. R. Kennedy,
of Scientific
discussion
"A Linear
Instruments,
of determination
Electron
2, 1948, p. 98.
&No.
of accelerator
,
structure
geometry.
3..
M. Chodorow,
ator
(Mark
1.c.3,
Section
4.
E. L.
"Stanford
High-Energy
Linear
et al.,
-Review of Scientific
Instruments
III),"
Ginzton,
Microwave
1957,
5.
6.
Reference
3, Section
Reference
4, Fig.
7.
S. A. Schelkunoff,
8.
11.
R. G. E. Hutter,
International
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series
I type,
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Company,
New York,
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1953,
D. Van Nostrand
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E. M. Purcell,
(Znl)
Principles
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1948,
Series,
4-9,
Section
Circuit
McGraw-Hill
esp.
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4.30(b).
Tubes,
New Jersey,1960,
Corporation,
7-9.
Section
Reference
1956, p. 172,
New York,
curves
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in Microwave
Princeton,
edition,
Company,
Data
-3-element
for
and formulas.
Theory,
John Wiley
& Sons,
Inc.,
P. 305, Eq. (153).
Microwave
Circuit
New York,
Reference
1, Section
Reference
10.
15.
E. A. Guillemin,
New York,
Inc.,
Introductory
13.
14.
Inc.,
Eq.
and Telegraph
mid-shunt
Inc.,
Waves,
10,
397.
p.
Beam and Wave Electronics
E. A. Guillemin,
R. N. Ghose,
9.lb,
Radiation
Telephone
Radio
New York,
Chap.
R. H. Dicke,
Inc.,
Inc.,
p* 161.
Electromagnetic
8, M.I.T.
Vol.
D. Van Nostrand
10.
D.5,
II
903 and Section
C. G. Montgomery,
Book Company,
6.5.
1943,
Book Company,
9.
No. 2, 1955,
3,
McGraw-Hill
Measurements,
Section
New York,
Circuits,
Acceler-
p. 138.
New York,
Inc.,
Electron
7.6,
1963,
Theory
p. 265, Eq.
p. 157,
Communication
and Analysis,
first
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1935, p. 179, Eq. (427).
- 18 -
McGraw-Hill
Book
(10.80).
sentence
Vol.
II,
of
second
John Wiley
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& Sons,