See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/242901167
Field Theory of Planar Helix Traveling-Wave Tube
Article in IEEE Transactions on Microwave Theory and Techniques · January 1983
DOI: 10.1109/TMTT.1983.1131432
CITATIONS
READS
22
254
3 authors:
Devi Chadha
Sheel Aditya
Indian Institute of Technology Delhi
Nanyang Technological University
52 PUBLICATIONS 174 CITATIONS
235 PUBLICATIONS 1,721 CITATIONS
SEE PROFILE
SEE PROFILE
R.K. Arora
Florida State University
47 PUBLICATIONS 330 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
RF&Microwave Measurement View project
Development of Microfabricated Travelling-wave Tube Amplifers (TWTAs) at Ka-band (Office for space technology and industry) View project
All content following this page was uploaded by Sheel Aditya on 25 December 2014.
The user has requested enhancement of the downloaded file.
IEEE TRANSACTIONS
ON MICROWAVE
5) The feasibility
when
adipose
large
lateral
radiators
tioned,
At
dimensions
the applicator
915 MHz
with
would
frequency.
of 433 MHz,
These results
from
statements
because
with
and pork
than these values
depth,
thus losing
the required
system, like, for instance,
affect
blood
only
frequency
by surface cooling
and heat
of the various
to occur
flow.
for
However,
depth
to the interior.
systems should
remain
the in oioo
of a living
it is assumed that
penetration
effects of the blood
conduction
by experi-
Fig.
1.
The planar traveling-wave
y“-directions
of conduction
0 —helix angle.
of the used
stream and heat
The
principal
Field
theory
presence
is applied
to anafyze
of a flat electron
indicate
the presence
attenuation
are shown for
current
of three
constant,
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
A W. Guy, J. F. Lehmarm, and J. B. Stonebndge “Therapeutic
applications of electromagnetic power, Proc. IEEE, vol. 62, pp. 55-75, 1974.
J. W. Hand and G. ter Haar, “Heating techniques in hyperthermia~
Brit.
J. Radloi., vol. 54, PP. 443-466, 1981.
D. A, Christensen and C. H. Dumey, “ Hyperthermia
production
for
cancer therapy: A review of fundamentals and methods,” J. Microwaoe
Power, vol. 16, pp. 89-105, 1981.
G. Kantor, “Evaluation
and survey of microwave and radio-frequency
applicators,” J. Microwrzoe Power, vol. 16, pp. 135-150, 1981.
G, M. Samaras, A. Y. Cheung, and S. F. Weinmann, “ Focussed microwave radiation therapy for deep tumors,” in Hvpertherwua as an A ntzneoplastlc Treatment Modality, NASA Publ. 2051, 1978, pp. 67-68,
A. Y, Cheung, W. M. Gelding, and G. M. Samaras, “Direct contact
applicators fOr microwave hyperthermia,”
J, Microwave Power, VO1. 16,
pp. 151-159, 1981.
F. A. Chbbs, “’Clinical evaluation of a microwave/radio-frequency
system (BSD Corporation) for induction of local and regional hyperthermia;’
J. Mtcrowuoe Power, vol. 16, pp. 185-192, 1981.
M, Melek and A. P. Anderson, “A throned cylindrical array for focussed
in Proc. 11th European Microwave Conf.
microwave hyperthemna,”
(Amsterdam), pp. 427-432, 1981.
H. P. Schwan, “Survey of rmcrowave absorption characteristics of body
tissue,” in Proc. Trl-Sem. Conf. on B1ologzcal Effects of Microwave Energp, NTIS Dec. AD 131477 and 220124, pp. 126-145, 1958.
M. A. Stuchly and S. S. Stuchly, “Dielectric
properties of biological
substances-Tabulated,”
J. Mzcrowaoe Power, vol. 15, pp. 19-26, 1980.
J. L. Guerquin-Kem,
L, Palas, M. ,Gautherie, C. Foumet-Fayas,
E.
Gimonet, A. Priou, and S. Samsel, “Etude comparative d’ apphcateurs
hyperfrequences
(2450 MHz, 434 MHz) sur frmt~mes et sur pi~ces
opiratoire,
en vne d’une utilisation
therapeutique
de l’hyperthermie
micro-onde en cancirologie,”
in Proc. URSI Symp, (Electromagnetic
Waoes and Bmkrgp) (Jouy-en-Josas), pp. 241-247, 1980.
modes,
a typicaf
proposed
Abstract
—A
as a slow-wave
constitute
Traveling-Wave
structure
conducting
a planar helix.
for application
one mode having
TWT.
Also.
a negative
TWT.
the effect
Curves
of beam
The planar
TWT
considered
of UC screens (Fig.
tion.
As
shown,
directions
the y-axis.
cally
located
beam
the
top
An
by a distance
the
beam
screens
of thickness
extent
the structure
the wave in regions
here will
consists of a pair
2a in the x-direc-
be applicable
conduct
make angles 0 and
the two screens. Both
to be of infinite
not disturb
bottom
respectively,
electron
out in [5], limiting
will
and
which,
between
are assumed
here for analysis
1) separated
y’ and y“
with
CONFIGURATION
in
– O
2 b is symmetri-
the screens and the
in the y-direction.
in the transverse
remote
from
As
direc-
the ends,
to a structure
several
wavelengths wide. A practicaf structure can be terminated
in the
y-direction
by closing the “helix”
by conducting
wires. The beam
Tube
current
density
z-component.
SHEEL ADITYA, MEMBER,IEEE,AND
K. ARORA, SENIORMEMBER,IEEE
pair of unirfirectiottafly
directions,
planar
in the
INTRODUCTION
II.
by means
applied
different
with
helix
the two screens. Remdts
The analysis of a helix-type traveling-wave tube (TWT) has
been carried out by Pierce [1], Chu and Jackson [2], and others
[3], [4]. In a TWT, a slow-wave structure such as a helix or
coupled cavity is used to slow down the electromagnetic wave to
the velocity of the electron beam so that a strong interaction
between the two can take place. A slow-wave structure in planar
geomet~, consisting of a pair of parallel unidirectionally
conducting (UC) screens conducting in different directions and separated by some distance, has been studied by Arora [5], Aditya
[6], and Aditya and Arora [7]. It appears that this planar slow-wave
structure can be used in a TWT. In view of this, a planar helical
TWT is analyzed in the present text, Field equations are derived
and the modaf solution of the problem is obtained. The variation
of the complex propagation constants of different modes with
beam voltage is studied.
tion
DEVI CHADHA,
RAJENDIb4
of the planar
between
as in the case of the usuaf helix-type
and results obtained
of Planar Helix
.v’ and
is indicated.
pointed
Field Theory
of propagation;
UC screens, resPectiveI~
unchanged.
WFEfLf3NCES
[2]
the behavior
beans present
1,
[1]
tube: z —duection
of top and bottom
tissue. Deviations
of the dynamics
the effective
I
are valid
applicator
fair accuracy
adipose
are likely
of the presence
this could
features
would
of 14 cm
The same remarks
have been checked
the above
convection,
depth
smaller
where
men-
be 30 cm.
ments using muscle phantom
situation,
or three
diameter
to a penetration
dimensions
two
5 cm at 2450 MHz.
decrease of penetration
of the lower
the frequency
diameter
only
fat, the applicator
the applicator
result in a rapid
realistic
Because of the
size. As already
size is approximately
and irradiating
the benefits
to be only
is irradiated.
reasonable
13 cm (corresponding
[1]). Making
would
appears
cancer)
of the applicators,
can be grouped
be about
for
of a focaf point
tissue (breast
73
AND TECHNIQUES, VOL. 31, NO. 1, JANUARY 1983
THEORY
is constant
In practice,
of the focusing
over the cross section
this assumption
action
and has onlly a
is very nearly
of a strong
realized
dc magnetic
field
in the z-direction.
screens, conducting in
III.
The planar helix is proposed
in a traveling-wave
tube (’ITVT).
The boundary
present.
Manuscnpt recewed March 29, 1982; revised August 4, 1982.
The authors are with the Department
of Electrical Engineering,
Institute of Technology, Harrz Khas, New Delhi, 1I 0016, India.
Indian
WAVE EQUATIONS AND ELECTRON DYNAMICS
and are constant
equations
0018-9480/83/0100-0073$01.00
conditions
require
both TE and TM modes to be
If the fields are assumed to be varying
in the y-direction,
can be written
01983
IEEE
as exp (jut
– ~z),
i.e., ~/ dy = O, then the field
in the following
form.
74
IEEE TRANSACTIONS
ON MICROWAVE
THEORY
AND
TECHNIQUES,
x
4
For the TE mode
rEy +
japoHX
rHx+(dH:/dx)+
,UC
– ~
L__-- ____-----T___
\
(2)
r = a + j/3 is the propagation
and .TZis the z-component
from
the field
presence
equations
that
constant
ilong
of the current
the TE mode
the z-direc-
density.
For
the transverse-symmetric
nents in the different
by the
B1eu(x-
H =
b)~B2e-u(-’
x<b
x>a
{ Nsinhux,
u~=–rz–k;
by the following
1) Tangential
components
2) Tangential
k;
However,
of current
components
changes into
The
With
the help
of
the
z
L/
—Jz
of electric
and magnetic
field
are
of electric
field
are continuous
at
(4)
= O.
field
above
acteristic
components
are continuous
boundary
equation
are zero and the magnetic
field
along y’ at x = a.
conditions
result
in the following
for the transverse-symmetric
char-
modes:
J(JC~
continuity
and
force
equations
for
between .T2and E, can be obtained
charges, the relationship
components
3) Electric
because of the presence of the
and the wave equation
#Ez
—–UZE.
8.X2
These coefficients
conditions:
x=a;
= U2/.Loco.
the TM mode is affected
z-component
coefficients.
boundary
at x = b;
continuous
and
(9)
x<a
where A, B,, B2, etc., are constant
are related
(8)
b<x<a
-b),
Me-.(x-a)
,
(3)
compo-
in the form
Csinhpx,
and is given as
where
the longitudinal
can be expressed
x>a
(
E, =
for it remains
82H,
–uzH=o
p:
modes,
regions
~e-u(x-a)
It is seen
is unaffected
of the beam and hence the wave equation
unchanged
U C Screen
Fig. 2. Configuration
considered for theoretical analysis, The structure is of
infirute extent in .v- and z-directions. The top and bottom UC screens
conduct in dmectlons making angles 8 and — @,respectively, with the v-axis.
+ jmpoHY=O
tion,
Screen
(1)
~Hy– jwcOEX=O
where
1983
jotoEy=o
mode
–~Ex
1, JANUARY
= O
13EY
— + jupOHz = O.
ax
For the TM
31, NO.
VOL.
~cothpb+tanhu(a-b)
k;
the
(l+tanhua)tan20=~
1+
u.
[1] as
l+~cothpbtanh
u(a-b)
1
I
(10)
(5)
For
In (5), C$ = pOe/m CO and i?,= U/OO, where PO is the average
charge
density,
e/m
is the charge-to-mass
and on is the average value
the wave equation
f&
of electron
the TM
ratio
velocity.
transverse-antisymmetric
modes,
of the electron,
Thus
using
(5),
the characteristic
equation
is
1+
u
l+~tanhpbtanh
u
mode becomes
1
‘tanhpb+tanhu(a-b)
k;
(l+cothua)tan20=7
[
u(a-b)
(6)
(11)
Only
where
the transverse-antisymmetnc
present
P’=-(r’+
+(:)’(+)’”
context
axis. Solution
(7)
propagating
BOUND.ARY CONDITIONS AND CHARACTERISTIC
As
EQUATION
Fig. 2 shows the structure
in cross section.
The symmetry
about
the plane x = O suggests the generaf solution
to be separable
(even) and 2) trans-
components
dinal
components
the latter
that
only
analysis.
1) transverse-symmetric
(odd).
In the former
are symmetrical
type, the transverse
respect
to x, while
E, and H, are antisymmetrical.
type of modes,
one half
with
the opposite
of the structure
the
interaction
two types of modes:
verse-antisymmetric
in
of (10) and (11) gives the various
into
field
the longitu-
need be considered
electromagnetic
normal
imaginary
in
are plotted
parameters.
in
the
a circular
those
in
the beam
of electrons
In
forward
of
and
in the
E, along the
possible
In
order
helix,
the
When
DRIFT
TWT
modes
is close
presence
of
In
Figs.
facilitate
parameters
the
difference
is a strong
3
breaks
and
for
Fig.
between
4,
for
beam,
the
the
chosen
phase
the
three
real
three
different
with
4 are
when
of the
up into
the
two
comparison
in
wave
velocity
an electron
constant
of beam voltage
to
there
to the phase
in the structure
modes.
VELOCITY
[2],
the electromagnetic
of the propagation
as functions
[2].
the
of propagation
parts
WITH
a helix-type
wave.
mode
independent
VARIATION
of
velocity
ensures
Similarly,
is true. Symmetry
MODE
case
between
the drift
are of interest
in the structure.
V.
IV.
modes
since only these modes have nonzero
and
modes
sets of
the
case
of
similar
to
velocity
of
IEEE
TRANSACTIONS
ON MICROWAVE
THEORY
AND
TECHNIQUES,
,req”e.cy
.:075
75
31, NO. 1, JANUARY 1983
VOL.
: ,0(3,,
mm,
b/.
:os
0:5”
0 0s
006
/—/r-
‘/
.:/
/“,2
15000.
‘1, , “’”,&
0 OL
2600
002
3000
~3400
/
‘\
/
.00L
‘.
I
I
00
.=-’
I
I
2500
2100
I
1
2703
2900
j
7
-008
-002
-012
I
!
230
21p
:
/
.0 OL
1
-006
21
-008
I
[
(a)
25t
19
I
13
11 L1l’
1400
1800
2600
2200
3000
17
-L
3400
2s00
,,s~
I
8EAM VOLTAGE(volts) —
(b)
1700
Fig, 3. (a) Attenuation
constant and (b) phase constant versus beam voltage
for b/a = 0.5, a = 0.75 mm, frequency=
10.0 GHz, O = 5°, for three different vatues of beam density, .70= 350 A/mz, Jo = 3500 A/mz, and Jo = 35000
A/mz.
have
constants
no
velocity,
electromagnetic
tion
constants
constant
when
with
amplification;
(~,, &,
it is close
to the phase
The mode with
as it advances
the increase
amplification
mum attainable
the propagation
~3 ). Over a firrite range of
along
VI.
resulting
NUMERICAL
cal structure
for
a structure
the order
the velocity
for comparison,
of a helix-type
at the input.
order
modes
to be equally
In
TWT
general,
TWT
OF GAIN
An attempt
the propagation
be other
modes
all
of 31831
A/m2,
the maximum
attainable
that
the
for TWT
constant
the structure
electromagnetic
for application
microwave
TWT.
in TWT,
integrated
The char-
as a function
is capable
wave.
the circular-helix
helix could be
applications.
The
amplification
The planar
with
circuits
of
of providing
structure
advantages
and
of
ease of
REFERENCES
[8].
[3]
For
is 11.5
fabrication.
[1]
[2]
wavelengths.
set of parameters
is
is given as
in
of
well with
there
in the three forward
of tube
compares
show
with
gtin
=‘9”54+54”575
()E ‘dB
N is the length
depicting
voltage
compatibility
power
Fig. 4. It
in the case
, CONCLUSIONS
acteristics
of gain in a practi-
and the input
from
has been made to show that a planar
promise
will
as derived
for a corresponding
sthctnre
amplification
TWT
calculation,
long,
that the gain obtained
used as a slow-wave
thus holds
In the present
are neglected
distributed
The gain of the planar
density
1
2900
—
dB [2].
range over
mode in the planar
of magnitude
ten wavelengths
may be noted,
VII.
and so does the maxi-
FOR THE ORDER
can be calculated.
also present
these higher
EXAMPLE
of the increasing
has been established,
where
]
in
amplification.
Once the presence
taken
2500
(volts
(b)
beam
modes
VOLTAGE
Fig. 4, (a) Attenuation
constant and (b) phase constant versus beam voltage
for b/a = 0.2, a = 5 mm, frequency = 1.875 GHz, 8 = 5°, for three different
vrdues of beam density, Jo = 318.3 A/mz, Jo = 3183 A/mz, and Jo = 31 830
A/mz.
attenuation
the structure,
increases
of the
propaga-
parts but equal
the negative
of beam current,
cart occur
velocity
have complex
the same value of imaginary
real parts.
increases
gain. With
or
imaginary
wave, two of the modes
and opposite
which
attenuation
are purely
electron
2300
BEAM
and the drift velocity of electrons is large, the three
the wave
modes
I
2100
1’300
a current
gain is 12 dB
[4]
[5]
J. R. Pierce, Traoelling- Waoe Tube. New York: Van Nostrand, 1950.
L. J. Chu and J. D. Jackson, “Field theory of traveling-wave tubes,” F’roc.
IRE, VOI. 36, pp. 853-863, kdy 194$
“The traveting-wave-tube
as amplifier at microwaves,”
R. Kompfner,
PrW, IRE, VO1. 35, pp. 124-128, Feb. 1947.,
M, Chodorow and N. J. Nalos, “The design of higt-power traveting-wave
tubes,” Proc. IRE., vol. 44, pp. 649:659, May 1956.
R. K. kora, “Surface wave &r a pair of parallel unidirectionally
conduct-
,
76
[6]
[7]
[8]
IEEE TRANSACTIONS
ON MICROWAVE
mg screens,” IEEE Trans Antennas Propagat., vol. AP- 14, pp. 795–797,
NOV. 1966
S. Adltya, “Studies of a planar helrcal slow-wave structure,” Ph.D. dissertation, Indian Institute of Technology Dellu, India, 1979
S Aditya aud R K. Arora, “Guided waves on a plarmr helix,” IEEE
pp 860-863, Ott 1979
Trans. MKrowuoe
Theory
Tech., vol MIT-27,
R E Collin,
Foundations
for Mzcrowave Enemeermz.
New
York,
McGraw-Hall,
1966.
THEORY
AND
TECHNIQUES,
31, NO. 1, JANUARY
VOL.
1983
(n
w
u
~
0
+“
+
~
%
Choosing Line Lengths for Calibrating
Analyzers
:
<
g
Network
I
—
OPERATING
FREQUENCY
RANGE —1
I
f2
o
CLETUS A. HOER, MEMBER.IEEE
o
f,
f, - fz
FREQUENCY
,Osfract
—Equations,
examples,
best length for a precision
for calibrating
about
a network
and a table are given to help choose the
transmission
analyzer.
line which is used in some methods
One line will cover a frequency
range of
10:1. Two lines will cover a range of about 65:1.
I.
INTRODUCTION
Some techniques
for calibrating
a network
analyzer
use a
length of precision transmission
line as the standard. Examples of
such techniques
are the “thru-reflect-line”
“thru-short-delay”
length
technique
[2]–[5].
of line as the standard
be too near multiples
characterizing
electrical
length
problem
electrical
One problem
is that its electncaf
analyzer
of 90 °+1800nz
length
methods
technique
of 180”, or the solution
the network
Another
Three
Flg 1 Phase shift through a single transmission lme designed to cover the
frequency range ~1 to j’l. The optimum length is that which gwes a phase
skft of 180° at ~1 + ~:.
is that
becomes
is ideal,
the physical
[1] and the
with
length
using
a
must not
for the constants
ill-conditioned.
where
length
An
rn =0,1,2...
of
a line
This
is also the effective
shift
at
broken
f,
into
two
intermediate
ranges,
frequency.
fz /f, = f, /f I = ~.
chosen as described
calculated
from
isless’than
180° may be too short to be practical.
of choosing
the line lengths
(3) for
~z. If ~,i~l
electrical
performance
one or two longer
length
absolute
phase shift
difference
between
through
~1 +
be the same at both ~1 and
f2,
provide
range
f2, as shown
is defined
phase shift
effective
be from ~1 to
frequently
operating
the line
the actual
of 0° or 180°, then the minimum
line will
analyzer
the whole
is 180° at the frequency
1. If the effective
to ~ and
f,
practicaf
choice
the smaller
through
if its
is the
of the coaxiaf
an air-dielectric
frequency
in
some
that
then all be equaf
be extended
to
length
Then
1 calculated
one must
from
use either
below.
phase shifts
the
for
coaxial
frequencies
be equally
at
is derived
frequencies
distance
having
from
multi-
spaced a distance
changes in phase, Ar#J, from
Af
(1) will
Aq=121Af,
If the line length
line
operation
to avoid associated
spaced
minimum
The corresponding
to use a longer
although
of 180°. In this section
equally
are a specified
be 90°
transmission
range,
must be excluded
choosing
Let the calibration
apart.
frequency
too near multiples
an expression
ples of 180°.
and the closer
wide
be
(4)
or the frequencies
are chosen as shown
in Fig.
(1)
For the phase shift
(5)
gigahertz
and
I is the length
to be 180° at ~1 +
f2, the
in
length
~=
f,+f,
from
0° and 180°, then the effective
cies of measurement
line must be
the phase shifts
15
for
cm’
(2) in (1), the phase shift
fin
‘fl = 1 +
(2)
gigahertz.
f2/f ,
from
180 °=2@m1n+n
?r=o,
A@,
A+ from
(6)
(5) and (6) leads to
90
+~,~ is restricted
Eliminating
Ar#I from
(7)
to values
of 90°,
45°,
22.5°,
18°,
etc.
(4) and (5) gives
(8)
CO 80303
001 8-9480/83/0100-0076$01
1,2, . . .
r#—
mm=~+n.
Thus
Mamrscrlpt rccelved April 30, 1982; revmed August ’23.1982
The author N with the National Bureau of Standards, Boulder,
at all frequen-
than ~ml~. Adding
0° to 180° in Fig. 2(a) gives
(3)
deg.
phase shifts
will be equal to or greater
where n = 4 in Fig. 2(a). Eliminating
at ~1 is
180
View publication stats
is
2(a) so that the phase skufts nearest to 0° and 180° are a distance
centimeters.
Using
f,
is such
ONE LONG LINE
bands of frequencies
as the
phase shift through
and the phase will
certain
phase angles which
@=12 fldeg
f
the physical
to the above, it is possible
an arbitrarily
in Fig.
line is [6]
where
phase
to ~z may be
where
for f,
ranges will
to be practical.
at the center of the band.
The phase shift
f2,
to
This idea can obviously
lines as discussed
As an alternative
10, one line will
over
f,
toavoidtheseprob-
SHORT LINES
is less than about
satisfactory
f,
III.
range of the network
the effective
range
A
frequencies,
(2) may be too small
over
Let the frequency
f2. If
more than two ranges.
lems are given in this short paper.
II.
at
the frequency
If the line lengths for each range are
above, the phase angles at f 1, f,, ~d fz
and as large as possible.
At the higher
whose
phase shift
and f2 is too small,
.00 Q1983 IEEE