Coupled Helices
By J. S. COOK, R. KOMPFNER and C. F. QUATE
(Received September 21, 1955)
An analysis of coupled helices is presented, using the transmission line
approach and also the field approach, with the objective of providing the
tube designer and the microwave circuit engineer with a basis for approxi-
mate calculations. Devices based on the presence of only one mode of propa-
gation are briefly described; and methods for establishing such a mode are
given. Devices depending on the simultaneous presence of both modes, that
is, depending on the beat wave phenomenon, are described; some experi-
mental results are cited in support of the view that a novel and useful class of
coupling elements has been discovered.
CONTENTS
1 . Introduction 129
2. Theory of Coupled Helices 132
2.1 Introduction 132
2.2 Transmission Line Equations 133
2.3 Solution for Synchronous Helices 135
2 4 Non-Synchronous Helix Solutions 137
2.5 A Look at the Fields 139
2.6 A Simple Estimate of b and x 141
2.7 Strength of Coupling versus Frequency 142
2.8 Field Solutions 144
2.9 Bifilar Helix 146
2.10 Effect of Dielectric Material between Helices 148
2.11 The Conditions for Maximum Power Transfer 151
2.12 Mode Impedance 152
3. Applications of Coupled Helices 154
3.1 Excitation of Pure Modes 156
3.1.1 Direct Excitation 156
3.1.2 Tapered Coupler 157
3.1.3 Stepped Coupler 158
3.2 Low Noise Transverse Field Amplifier ' 159
3.3 Dispersive Traveling Wave Tube 159
3.4 Devices Using Both Modes 161
3.4.1 Coupled Helix Transducer 161
3.4.2 Coupled-Helix Attenuator 165
4. Conclusion 167
Appendix
I Solution of Field Equations 168
II Finding r 173
III Complete Power Transfer 175
127
128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
GLOSSARY OF SYMBOLS
a Mean radius of inner helix
b Mean radius of outer helix
b Capacitive coupling coefficient
Bio, 20 shunt susceptance of inner and outer helices, respectively
Bi, 2 Shunt susceptance plus mutual susceptance of inner and outer
helices, respectively, Bio + B m , B 2 o + B m
B m Mutual susceptance of two coupled helices
c Velocity of light in free space
d Radial separation between helices, b-a
D Directivity of helix coupler
E Electric field intensity
F Maximum fraction of power transferable from one coupled helix
to the other
F(ya) Impedance parameter
7i, 2 RF current in inner and outer helix, respectively
K Impedance in terms of longitudinal electric field on helix axis
and axial power flow
L Minimum axial distance required for maximum energy transfer
from one coupled helix to the other, X 6 /2
P Axial power flow along helix circuit
Radial coordinate
Radius where longitudinal component of electric field is zero for
transverse mode (about midway between a and b)
Return loss
Radial separation between helix and adjacent conducting shield
Time
RF potential of inner and outer helices, respectively
Inductive coupling coefficient
Series reactance of inner and outer helices, respectively
Series reactance plus mutual reactance of inner and outer helices,
respectively, X M + X m , X 20 + X m
Mutual reactance of two coupled helices
Axial coordinate
Impedance of inner and outer helix, respectively
Attenuation constant of inner and outer helices, respectively
General circuit phase constant; or mean circuit phase constant,
Free space phase constant
Axial phase constant of inner and outer helices in absence of
r
f
R
s
t
Fl.2
X
-A-10, 20
Xi, 2
X m
Z
Zl, 2
Oil, 2
/3
ft
/3l0. 20
coupling, V-BioXio , V-B20X
20
COUPLED HELICES 129
/3i , 2 May be considered as axial phase constant of inner and outer
helices, respectively
ft, Beat phase constant
fi c Coupling phase constant, (identical with /3& when /3i = /3 2 )
/3 c e Coupling phase constant when there is dielectric material be-
tween the helices
/3d Difference phase constant, | j8i — ft |
fit Axial phase constant of single helix in presence of dielectric
(3 t , i Axial phase constant of transverse and longitudinal modes, re-
spectively
7 Radial phase constant
7,, ( Radial phase constant of transverse and longitudinal modes,
respectively
T Axial propagation constant
T t , c Axial propagation constant for transverse and longitudinal
coupled-helix modes, respectively
e Dielectric constant
e' Relative dielectric constant, e/e
Co Dielectric constant of free space
X General circuit wavelength; or mean circuit wavelength, \AiX 2
X Free space wavelength
Xi, 2 Axial wavelength on inner and outer helix, respectively
Xfc Beat wavelength
X c Coupling wavelength (identical with X& when /3i = /3 2 )
\f/ Helix pitch angle
\j/i_ 2 Pitch angle of inner and outer helix, respectively
u> Angular frequency
1. INTRODUCTION
Since their first appearance, traveling-wave tubes have changed only
very little. In particular, if we divide the tube, somewhat arbitrarily,
into circuit and beam, the most widely used circuit is still the helix, and
the most widely used transition from the circuits outside the tube to the
circuit inside is from waveguide to a short stub or antenna which, in
turn, is attached to the helix, either directly or through a few turns of
increased pitch. Feedback of signal energy along the helix is prevented
by means of loss, either distributed along the whole helix or localized
somewhere near the middle. The helix is most often supported along its
whole length by glass or ceramic rods, which also serve to carry a con-
ducting coating ("aquadag"), acting as the localized loss.
We therefore find the following circuit elements within the tube en-
velope, fixed and inaccessible once and for all after it has been sealed off:
■'\
130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
1. The helix itself, determining the beam voltage for optimum beam-
circuit interaction;
2. The helix ends and matching stubs, etc., all of which have to be
positioned very precisely with relation to the waveguide circuits in
order to obtain a reproducible match;
3. The loss, in the form of "aquadag" on the support rods, which
greatly influences the tube performance by its position and distribution.
In spite of the enormous bandwidth over which the traveling-wave
tube is potentially capable of operating — a feature new in the field of
microwave amplifier tubes — it turns out that the positioning of the tube
in the external circuits and the necessary matching adjustments are
rather critical ; moreover the ovefall bandwidths achieved are far short
of the obtainable maximum.
Another fact, experimentally observed and well-founded in theory,
rounds off the situation: The electro-magnetic field surrounding a helix,
i.e., the slow wave, under normal conditions, does not radiate, and is
confined to the close vicinity of the helix, falling off in intensity nearly
exponentially with distance from the helix. A typical traveling-wave
tube, in which the helix is supported by ceramic rods, and the whole
enclosed by the glass envelope, is thus practically inaccessible as far as
RF fields are concerned, with the exception of the ends of the helix,
where provision is made for matching to the outside circuits. Placing
objects such as conductors, dielectrics or distributed loss close to the
tube is, in general, observed to have no effect whatsoever.
In the course of an experimental investigation into the propagation of
space charge waves in electron beams it was desired to couple into a long
helix at any point chosen along its length. Because of the feebleness of
the RF fields outside the helix surrounded by the conventional sup-
ports and the envelope, this seemed a rather difficult task. Nevertheless,
if accomplished, such a coupling would have other and even more im-
portant applications; and a good deal of thought was given to the
problem.
Coupled concentric helices were found to provide the solution to the
problem of coupling into and out of a helix at any particular point, and to
a number of other problems too.
Concentric coupled helices have been considered by J. R. Pierce,
who has treated the problem mainly with transverse fields in mind.
Such fields were thought to be useful in low-noise traveling-wave tube
devices. Pierce's analysis treats the helices as transmission lines coupled
uniformly over their length by means of mutual distributed capacitance
and inductance. Pierce also recognized that it is necessary to wind the
COUPLED HELICES 131
two helices in opposite directions in order to obtain well denned trans-
verse and axial wave modes which are well separated in respect to their
velocities of propagation.
Pierce did not then give an estimate of the velocity separation which
might be attainable with practical helices, nor did anybody (as far as we
are aware) then know how strong a coupling one might obtain with such
helices.
It was, therefore, a considerable (and gratifying) surprise ' to find
that concentric helices of practically realizable dimensions and separa-
tions are, indeed, very strongly coupled when, and these are the im-
portant points,
(a) They have very nearly equal velocities of propagation when un-
coupled, and when
(b) They are wound in opposite senses.
It was found that virtually complete power transfer from outer to
inner helix (or vice versa) could be effected over a distance of the order
of one helix wavelength (normally between \{ o and 3-^0 of a free-space
wavelength.
It was also found that it was possible to make a transition from a co-
axial transmission line to a short (outer) helix and thence through the
glass surrounding an inner helix, which was fairly good over quite a con-
siderable bandwidth. Such a transition also acted as a directional coupler,
RF power coming from the coaxial line being transferred to the inner
helix predominantly in one direction.
Thus, one of the shortcomings of the "conventional" helix traveling-
wave tube, namely the necessary built-in accuracy of the matching
parameters, was overcome by means of the new type of coupler that
might evolve around coupled helix-to-helix systems.
Other constructional and functional possibilities appeared as the
work progressed, such as coupled-helix attenuators, various types of
broadband couplers, and schemes for exciting pure transverse (slow) or
longitudinal (fast) waves on coupled helices.
One central fact emerged from all these considerations: by placing
part of the circuit outside the tube envelope with complete independence
from the helix terminations inside the tube, coupled helices give back to
the circuit designer a freedom comparable only with that obtained at
much lower frequencies. For example, it now appears entirely possible
to make one type of traveling wave tube to cover a variety of frequency
bands, each band requiring merely different couplers or outside helices,
the tube itself remaining unchanged.
Moreover, one tube may now be made to fulfill a number of different
132 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
functions; this is made possible by the freedom with which couplers
and attenuators can be placed at any chosen point along the tube.
Considerable work in this field has been done elsewhere. Reference
will be made to it wherever possible. However, only that work with
which the authors have been intimately connected will be fully reported
here. In particular, the effect of the electron beam on the wave propaga-
tion phenomena will not be considered.
2. THEORY OF COUPLED HELICES
2.1 Introduction
In the past, considerable success has been attained in the under-
standing of traveling wave tube behavior by means of the so-called
"transmission-line" approach to the theory. In particular, J. R. Pierce
used it in his initial analysis and was thus able to present the solution
of the so-called traveling-wave tube equations in the form of 4 waves,
one of which is an exponentially growing forward traveling wave basic
to the operation of the tube as an amplifier.
This transmission-line approach considers the helix — or any slow-
wave circuit for that matter — as a transmission line with distributed
capacitance and inductance with which an electron beam interacts.
As the first approximation, the beam is assumed to be moving in an RF
field of uniform intensity across the beam.
In this way very simple expressions for the coupling parameter and
gain, etc., are obtained, which give one a good appreciation of the
physically relevant quantities.
A number of factors, such as the effect of space charge, the non-uniform
distribution of the electric field, the variation of circuit impedance with
frequency, etc., can, in principle, be calculated and their effects can be
superimposed, so to speak, on the relatively simple expressions deriving
from the simple transmission line theory. This has, in fact, been done and
is, from the design engineer's point of view, quite satisfactory.
However, physicists are bound to be unhappy over this state of
affairs. In the beginning was Maxwell, and therefore the proper point to
start from is Maxwell.
So-called "Field" theories of traveling-wave tubes, based on Maxwell's
equation, solved with the appropriate boundary conditions, have been
worked out and their main importance is that they largely confirm the
results obtained by the inexact transmission line theory. It is, however,
in the nature of things that field theories cannot give answers in terms of
COUPLED HELICES 133
simple closed expressions of any generality. The best that can be done
is in the form of curves, with step-wise increases of particular param-
eters. These can be of considerable value in particular cases, and when
exactness is essential.
In this paper we shall proceed by giving the "transmission-line" type
theory first, together with the elaborations that are necessary to arrive
at an estimate of the strength of coupling possible with coaxial helices.
The "field" type theory will be used whenever the other theory fails, or
is inadequate. Considerable physical insight can be gotten with the use
of the transmission-line theory; nevertheless recourse to field theory is
necessary in a number of cases, as will be seen.
It will be noted that in all the calculations to be presented the presence
of an electron beam is left out of account. This is done for two reasons:
Its inclusion would enormously complicate the theory, and, as will
eventually be shown, it would modify our conclusions only very slightly.
Moreover, in practically all cases which we shall consider, the helices are
so tightly coupled that the velocities of the two normal modes of propaga-
tion are very different, as will be shown. Thus, only when the beam
velocity is very near to either one or the other wave velocity, will
growing-wave interaction take place between the beam and the helices.
In this case conventional traveling wave tube theory may be used.
A theory of coupled helices in the presence of an electron beam has
been presented by Wade and Rynn, 4 who treated the case of weakly
coupled helices and arrived at conclusions not at variance with our views.
2.2 Transmission Line Equations
Following Pierce we describe two lossless helices by their distributed
series reactances X i0 and A"n and their distributed shunt susceptances
Bid and B> . Thus their phase constants are
fto = Vb 10 x 10
020 — V -^20^20
Let these helices be coupled by means of a mutual distributed reac-
tance X m and a mutual susceptance B m , both of which are, in a way
which will be described later, functions of the geometry.
Let waves in the coupled system be described by the factor
«*V r "
where the r's are the propagation constants to be found.
134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
The transmission line equations may be written:
r/i - jBiVi + jB m V 2 = o
TVi - jXih + jX m I 2 =
17, - jB»V» + jB m Vi =
TVi - jX 2 I a + jXJl = o
where
B x = J5io + 5 m
Xi = Xm "T Am
J5 2 = B w + £„'
-A2 = -^-20 ~\- X. m
' h and h are eliminated from the (2.2.1) and we find
F_ 2 = +(r 2 + l,Bi + iX)
7i
(2.2.1)
F 2
X\B m + BiX m
+ (r 2 + x 2 # 2 + XA)
(2.2.2)
(2.2.3)
X 2 B m + B\X m
These two equations are then multiplied together and an expression for
r of the 4th degree is obtained :
,2
r 4 + (XiBi + x 2 J5 2 + 2X m 5 m )r 2
+ (XiX 2 - X m *)(BiB* - Bj) =
We now define a number of dimensionless quantities:
bJ
(2.2.4)
B\Bi
Y *
= b~ = (capacitive coupling coefficient)
= af = (inductive coupling coefficient)"
X\X-i
B x Xx = ft 2 , £2X2 = i3 2 2
X1B1X2B2 = P* = (mean phase constant) 4
With these substitutions we obtain the general equation for r
_ (2.2.5)
COUPLED HELICES 135
If we make the same substitutions in (2.2.2) we find
Yi = A /h f r 2 + ft 2 + fbx \ , 2 2 6)
where the Z's are the impedances of the helices, i.e.,
z n = VxjW n
2.3 Solution for Synchronous Helices
Let us consider the particular case where /3i = ft = /3. From (2.2.5)
we obtain
r 2 = -&[1 + a* ± (x + 6)] (2.3.1)
Each of the above values of r characterizes a normal mode of propaga-
tion involving both helices. The two square roots of each r 2 represent
waves going in the positive and negative directions. We shall consider
only the positive roots of T", denoted r ( and Yi , which represent the
forward traveling waves.
Tt.t = i/3\/l + xb ± (x + b) (2.3.2)
If x > and b >
| iY| > |/3 1, | r,| < |/3 1
Thus T t represents a normal mode of propagation which is slower than
the propagation velocity of either helix alone and can be called the
"slow" wave. Similarly I\ represents a "fast" wave. We shall find that,
in fact, x and b are numerically equal in most cases of interest to us; we
therefore write the expressions for the propagation constants
r, - Ml + V 2 (x + b)]
(2.3.3)
Tt = Ml - H(x + b)]
If we substitute (2.3.3) into (2.2.6) for the case where ft = ft = |8 and
assume, for simplicity, that the helix self-impedances are equal, we find
that for T = I\
for r = r<
F, ^
136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
Thus, the slow wave is characterized by equal voltages of unlike sign on
the two helices, and the fast wave by equal voltages of like sign. It fol-
lows that the electric field in the annular region between two such coupled
concentric helices will be transverse for the slow wave and longitudinal
for the fast. For this reason the slow and fast modes are often referred
to as the transverse and longitudinal modes, respectively, as indi-
cated by our subscripts.
It should be noted here that we arbitrarily chose b and x positive. A
different choice of signs cannot alter the fact that the transverse mode is
the slower and the longitudinal mode is the faster of the two.
Apart from the interest in the separate existence of the fast and slow
waves as such, another object of interest is the phenomenon of the simul-
taneous existence of both waves and the interference, or spatial beating,
between them.
Let Vi denote the voltage on the outer helix; and let Vi , the voltage
on the inner halix, be zero at z = 0. Then we have, omitting the common
factor e"* ,
v>-y« r "+r>*- T ; (2 . 3 . 4)
Since at 2 = 0, Vi = 0, Va = - V tl ■ For the case we have considered we
have found V a = - Va and v " = V n • We can write ( 2 - 3 - 4 ) as
Vl = I (.-'* - f T n
(2.3.5)
y 2 = I ( e - T " + e- r n
Vo can be written
m Ve -MW*» cos [-jl/ 2 (T t - V t )z]
In the case when x = b, and /Si = ft — ^
7 2 = Ve~ j&z cos \V 2 {x + b)fa] (2.3.6)
Correspondingly, it can be shown that the voltage on the inner helix is
y x = jVe-*" sin [V 2 (x + b)fr] (2.3.7)
The last two equations exhibit clearly what we have called the spatial
beat phenomenon, a wave-like transfer of power from one helix to the
COUPLED HELICES 137
other and back. We started, arbitrarily, with all the voltage on the outer
helix at 2 = 0, and none on the inner; after a distance, z', which makes
the argument of the cosine tt/2, there is no voltage on the outer helix
and all is on the inner.
To conform with published material let us define what we shall call
the "coupling phase-constant" as
ft = 0(b + x) (2.3.8)
From (2.3.3) we find that for 0i = /3* = 0, and x = b,
r, - r< = jp e
2.4 Non-Synchronous Helix Solutions
Let us now go back to the more general case where the propagation
velocities of the (uncoupled) helices are not equal. Equation (2.2.5) can
be written :
p- - -0- [1 + (1/2)A + xb ±
V(i + xb)A + (i/4)a 2 + (b + x y]
where
-[^1
(2.4.1)
In the case where x = b, (2.4.1) has an exact root.
T,, t = jfi W\ + A/4 ± 1/2 VA 4- (x + by] (2.4.2)
We shall be interested in the difference between F< and I\,
r ( - r, = j|8 Va + (x + by- (2.4.3)
Now we substitute for A and find
I\ - T t = j V(/3i - ft? TWW+ xf (2.4.4)
Let us define the "beat phase-constant" as:
ft - V(ft - ft) 2 + F-(b + x y
so that
r< - r, = ift (2.4.5)
Further, let us define
ft - I ft - ft j
138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
and call this the "difference phase-constant," i.e., the hase constant cor-
responding to two uncoupled waves of the same frequency but differing
phase velocities. We can thus state the relation between these phase
constants :
ft 2 = ft 1 + ft, 2 (2.4.6)
This relation is identical (except for notation) with expression (33) in
S. E. Miller's paper. 6 In this paper Miller also gives expressions for the
voltage amplitudes in two coupled transmission systems in the case of
unequal phase velocities. It turns out that in such a case the power trans-
fer from one system to the other is necessarily incomplete. This is of
particular interest to us, in connection with a number of practical
schemes. In our notation it is relatively simple, and we can state it by
saying that the maximum fraction of power transferred is
A* 1
F-fJjJ (2-4.7)
f?{h + xY
or, in more detail,
v _ & _
tf + & 2 (Pi - ft) 2 + W + xY
This relationship can be shown to be a good approximation from (2.2.6),
(2.3.4), (2.4.2), on the assumption that b tt x and Zi « Z 2 , and the
further assumption that the system is lossless; that is,
| y t | ' + | Vi 1 2 = constant (2.4.8)
We note that the phase velocity difference gives rise to two phenomena:
It reduces the coupling wavelength and it reduces the amount of power
that can be transferred from one helix to the other.
Something should be said about the case where the two helix imped-
ances are not equal, since this, indeed, is usually the case with coupled
concentric helices. Equation (2.4.8) becomes:
LM + LM 2 = constant (2.4.9)
Zi Z\
Using this relation it is found from (2.3.4) that
r l \/Yr ± ^l {l±vr ^ 1) (2 - 410)
When this is combined with (2.2.6) it is found that the impedances drop
out with the voltages, and that "F" is a function of the 0's only. In other
COUPLED HELICES 139
words, complete power transfer occurs when ft = ft regardless of the
relative impedances of the helices.
The reader will remember that ft n and ft , not ft and ft , were defined
as the phase constants of the helices in the absence of each other. If the
assumption that b ^ x is maintained, it will be found that all of the de-
rived relationships hold true when ft is substituted for ft, . In other
words, throughout the paper, ft and ft may be treated as the phase con-
stants of the inner and outer helices, respectively. In particular it should
be noted that if these quantities are to be measured experimentally each
helix must be kept in the same environment as if the helices were coupled ;
only the other helix may be removed. That is, if there is dielectric in the
annular region between the coupled helices, ft and ft must each be
measured in the presence of that dielectric.
Miller also has treated the case of lossy coupled transmission systems.
The expressions are lengthy and complicated and we believe that no
substantial error is made in simply applying his conclusions to our case.
If the attenuation constants on and a 2 of the two transmission systems
(helices) are equal, no change is required in our expressions; when they
are unequal the total available power (in both helices) is most effectively
reduced when
^j*"! (2.4.11)
This fact may be made use of in designing coupled helix attenuators.
2.5 A Look at the Fields
It may be advantageous to consider sketches of typical field distribu-
tions in coupled helices, as in Fig. 2.1, before we go on to derive a quanti-
tative estimate of the coupling factors actually obtainable in practice.
Fig. 2.1(a) shows, diagrammatically, electric field lines when the
coupled helices are excited in the fast or "longitudinal" mode. To set up
this mode only, one has to supply voltages of like sign and equal ampli-
tudes to both helices. For this reason, this mode is also sometimes called
the "(+ + ) mode."
Fig. 2.1(b) shows the electric field lines when the helices are excited in
the slow or "transverse" mode. This is the kind of field required in the
transverse interaction type of traveling wave tube. In order to excite
this mode it is necessary to supply voltages of equal amplitude and
opposite signs to the helices and for this reason it is sometimes called the
"(H — ) mode." One way of exciting this mode consists in connecting one
140 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
helix to one of the two conductors of a balanced transmission line
("Lecher "-line) and the other helix to the other.
Fig. 2.1(c) shows the electric field configuration when fast and slow
modes are both present and equally strongly excited. We can imagine
the two helices being excited by a voltage source connected to the outer
• ONE "COUPLING" WAVELENGTH H
(C) FAST AND SLOW WAVES COMBINED SHOWING SPATIAL "BEAT" PHENOMENON
Fig. 2.1 —Typical electric field distributions in coupled coaxial helices when
they are excited in: (a) the in-phase or longitudinal mode, (I)) the out-of-phase or
transverse mode, and (c) both modes equally.
COUPLED HELICES 141
helix only at the far left side of the sketch. One, perfectly legitimate,
view of the situation is that the RF power, initially all on the outer helix,
leaks into the inner helix because of the coupling between them, and then
leaks back to the outer helix, and so forth.
Apart from noting the appearance of the stationary spatial beat (or
interference) phenomenon these additional facts are of interest:
1) It is a simple matter to excite such a beat-wave, for instance, by
connecting a lead to either one or the other of the helices, and
2) It should be possible to discontinue either one of the helices, at
points where there is no current (voltage) on it, without causing reflec-
tions.
2.6 4 Simple Estimate of b and x
How strong a coupling can one expect from concentric helices in prac-
tice? Quantitatively, this is expressed by the values of the coupling fac-
tors x and b, which we shall now proceed to estimate.
A first crude estimate is based on the fact that slow-wave fields are
known to fall off in intensity somewhat as e~ fir where /3 is the phase con-
stant of the wave and r the distance from the surface guiding the slow
wave. Thus a unit charge placed, say, on the inner helix, will induce a
charge of opposite sign and of magnitude
on the outer helix. Here b = mean radius of the outer helix and a =
mean radius of the inner. We note that the shunt mutual admittance
coupling factor is negative, irrespective of the directions in which the
helices are wound. Because of the similarity of the magnetic and electric
field distributions a current flowing on the inner helix will induce a simi-
larly attenuated current, of amplitude
-0(6-a)
on the outer helix. The direction of the induced current will depend on
whether the helices are wound in the same sense or not, and it turns out
(as one can verify by reference to the low-frequency case of coaxial
coupled coils) that the series mutual impedance coupling factor is nega-
tive when the helices are oppositely wound.
In order to obtain the greatest possible coupling between concentric
helices, both coupling factors should have the same sign. This then re-
quires that the helices should be wound in opposite directions, as has
been pointed out by Pierce.
When the distance between the two helices goes to zero, that is to say,
142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
if they lie in the same surface, it is clear that both coupling factors b and x
will go to unity.
As pointed out earlier in Section 2.3, the choice of sign for b is arbi-
trary. However, once a sign for b has been chosen, the sign of x is neces-
sarily the opposite when the helices are wound in the same direction, and
vice versa. We shall choose, therefore,
b = +6-**-*
(2.G.1)
x = Te-^
the sign of the latter depending on whether the helices are wound in the
same direction or not.
In the case of unequal velocities, ft the propagation constant, would
be given by
(3 = Vftft ( 2 - 6 - 2 )
2.7 Strength of Coupling versus Frequency
The exponential variation of coupling factors with respect to frequency
(since /3 = u/v) has an important consequence. Consider the expression
for the coupling phase constant
ft = Kb + x)
or
| ft | = 2/3<r /,(6-0)
The coupling wavelength, which is defined as
\c =
2v
P.
(2.3.8)
(2.7.1)
(2.7.2)
is, therefore,
or
\ _ A .(2r/X)(b-«)
(2.7.3)
where X is the (slowed-down) RF wavelength on either helix. It is con-
venient to multiply both sides of (2.7.1) with a, the inner helix radius,
in order to obtain a dimensionless relation between ft and ft
fta = 20a<f /ta<<Wo) - 1)
(2.7.4)
This relation is plotted on Fig. 2.2 for several values of b/a.
COUPLED HELICES
143
/3c a
/
/
/
/
/
f
4=<y
/
/
/
/
/
/(£c*)max
/
/
d
/
/
J
/
/
/
/
«• — 7
/
/
b
= 1.5
/
/
/
f/j
/
/
//
/
/
^*^i
75
1/7
/
/
2.0
1/
3.0
2.5
/3a
Fig. 2.2 — Coupling phase-constant plotted as a function of the single helix
phase-constant for synchronous helices for several values of b/a. These curves
are based on simple estimates made in Section 2.7.
There are two opposing tendencies determining the actual physical
length of a coupling beat-wavelength:
1) It tends to grow with the RF wavelength, being proportional to it
in the first instance;
2) Because of the tighter coupling possible as the RF wavelength
increases in relation to the helix-to-helix distance, the coupling beat-
wavelength tends to shrink.
Therefore, there is a region where these tendencies cancel each other,
and where one would expect to find little change of the coupling beat-
wavelength for a considerable change of RF frequency. In other words,
the "bandwidth" over which the beat-wavelength stays nearly constant
can be large.
This is a situation naturally very desirable and favorable for any
device in which we rely on power transfer from one helix to the other by
144 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
means of a length of overlap between them an integral number of half
beat-wavelengths long. Obviously, one will design the helices in such a
way as to take advantage of tins situation.
Optimum conditions are easily obtained by differentiating ft with
respect to /3 and setting dp c /d0 equal to zero. This gives for the optimum
conditions
'opt —
b — a
(2.7.5)
or
2<r : -i
/W = r = 2e ovt
o — a
(2.7.6)
Equation (2.7.5), then, determines the ratio of the helix radii if it is re-
quired that deviations from a chosen operating frequency shall have
least effect.
2.8 Field Solutions
In treating the problem of coaxial coupled helices from the transmis-
sion line point of view one important fact has not been considered,
namely, the dispersive character of the phase constants of the separate
helices, ft and ft . By dispersion we mean change of phase velocity with
frequency. If the dispersion of the inner and outer helices were the same
it would be of little consequence. It is well known, however, that the
dispersion of a helical transmission line is a function of the ratio of helix
radius to wavelength, and thus becomes a parameter to be considered.
When the theory of wave propagation on a helix was solved by means of
Maxwell's equations subject to the boundary condition of a helically
conducting cylindrical sheath, the phenomenon of dispersion first made
its appearance. It is clear, therefore, that a more complete theory of
~ T
*$z/
A
Fig. 2.3 — Sheath helix arrangement on which the field equations are based.
COUPLED HELICES 145
coupled helices will require similar treatment, namely, Maxwell's equa-
tions solved now with the boundary conditions of two cylindrical heli-
cally conducting sheaths. As shown on Fig. 2.3, the inner helix is specified
by its radius a and the angle ipi made by the direction of conductivity
with a plane perpendicular to the axis; and the outer helix by its radius
b (not to be confused with the mutual coupling coefficient b) and its
corresponding pitch angle ^2 . We note here that oppositely wound helices
require opposite signs for the angles \f/i and ^ 2 ; and, further, that helices
with equal phase velocities will have pitch angles of about the same
absolute magnitude.
The method of solving Maxwell's equations subject to the above men-
tioned boundary conditions is given in Appendix I. We restrict our-
selves here to giving some of the results in graphical form.
The most universally used parameter in traveling-wave tube design is
a combination of parameters:
/3 a cot \f/i
where /3 ( , = 2x/X , Xo being the free-space wavelength, a the radius of
the inner helix, and \J/\ the pitch angle of the inner helix. The inner helix
is chosen here in preference to the outer helix because, in practice, it will
be part of a traveling-wave tube, that is to say, inside the tube envelope.
Thus, it is not only less accessible and changeable, but determines the
important aspects of a traveling-wave tube, such as gain, power output,
and efficiency.
The theory gives solutions in terms of radial propagation constants
which we shall denote 7^ and yi (by analogy with the transverse and
longitudinal modes of the transmission line theory). These propagation
constants are related to the axial propagation constants f3 t and /3< by
7,. = V$n 2 ~ /3(f
Of course, in transmission line theory there is no such thing as a radial
propagation constant. The propagation constant derived there and de-
noted T corresponds here to the axial propagation constant jfi. By
analogy with (2.4.5) the beat phase constant should be written
ft = Pt - Pt
However, in practice /3 is usually much smaller than and we can there-
fore write with little error
/3& = it — it
for the beat phase constant. For practical purposes it is convenient to
146 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G
3.60
COT ^ 2
COT^,
-0.90.
\.
-0.82.
s
X
CO
jD
<£ 2.40
8
ro
s?
,{,2.00
A
///
i
//
f
5 = 1.25
a
1.60
/
1.20
4
V
L
0.80
3
5
1
i
5
2
k a
2
:ot
5
3
3.5
4.0
4.5
Fig. 2.4.1 — Beat phase-constant plotted as a function of /3oa cot i£i . These
curves result from the solution of the field equations given in the appendix. For
b/a = 1.25.
normalize in terms of the inner helix radius, a:
Pta = 7*a — 7/a
This has been plotted as a function of |3o a cot ^i in Fig. 2.4, which
should be compared with Fig. 2.2. It will be seen that there is considerable
agreement between the results of the two methods.
2.9 Bifilar Helix
The failure of the transmission line theory to take into account dis-
persion is well illustrated in the case of the bifilar helix. Here we have
two identical helices wound in the same sense, and at the same radius.
If the two wires are fed in phase we have the normal mode characterized
by the sheath helix model whose propagation constant is the familiar
Curve A of Fig. 2.5. If the two wires of the helix are fed out of phase we
have the bifilar mode; and, since that is a two wire transmission system,
we shall have a TEM mode which, in the absence of dielectric, propa-
gates along the wire with the velocity of light. Hence, the propagation
constant for this mode is simply /3 a cot \p and gives rise to the horizontal
COUPLED HELICES
147
0.60
4.5
Fig. 2.4.2 — Beat phase-constant plotted as a function of j3 a cot \f/i .These
curves result from the solution of the field equations given in the appendix. For
b/a = 1.5.
line of Curve B in Fig. 2.5. Again the coupling phase constant (3 C is given
by the difference of the individual phase constants:
p c a = ft,a cot^ - ya (2.9.1)
which is plotted in Fig. 2.6. Now note that when /3 « y this equation is
accurate, for it represents a solution of the field equations for the helix.
From the simple unsophisticated transmission line point of view no
coupling between the two helices would, of course, have been expected,
since the two helices are identical in every way and their mutual capacity
and inductance should then be equal and opposite.
Experiments confirm the essential correctness of (2.9.1). In one experi-
ment, which was performed to measure the coupling wavelength for the
bifilar helices, we used helices with a cot \J/ = 3.49 and a radius of 0.030
cm which gave a value, at 3,000 mc, of /3 n a cotrj/ = 0.51 . In these experi-
ments the coupling length, L, defined by
(0oa cot \p — ya) - = it
a
was measured to be 15.7a as compared to a value of 13.5a from Fig. 2.0.
At 4,000 mc the measured coupling length was 14.0a as compared to
148 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
1.20
b_
a "
1.75
/%-
>«*
Si
^
^
j
'/
s
^
\
\_
/
7
^
X
/
//
-"^0.82
i
8
COT^,
0.90
m 0.60
is*
1
* 0.40
)
5
1
1
5
2
/2
ac
5
ft
3.0
3.5
4.0
4
Fig. 2.4.3 — Beat phase-constant plotted as a function of o a cot ^i .These
curves result from the solution of the field equations given in the appendix. For
b/a = 1.75.
12.6a computed from Fig. 2.6, thus confirming the theoretical prediction
rather well. The slight increase in coupling length is attributable to the
dielectric loading of the helices which were supported in quartz tubing.
The dielectric tends to decrease the dispersion and hence reduce (3 e . Tliis
is discussed further in the next section.
2.10 Effect of Dielectric Material between Helices
In many cases which are of interest in practice there is dielectric ma-
terial between the helices. In particular when coupled helices are used
with traveling-wave tubes, the tube envelope, which may be of glass,
quartz, or ceramic, all but fills the space between the two helices.
It is therefore of interest to know whether such dielectric makes any
difference to the estimates at which we arrived earlier. We should not be
surprised to find the coupling strengthened by the presence of the di-
electric, because it is known that dielectrics tend to rob RF fields from
the surrounding space, leading to an increase in the energy flow through
the dielectric. On the other hand, the dielectric tends to bind the fields
closer to the conducting medium. To find a qualitative answer to this
question we have calculated the relative coupling phase constants for
two sheath helices of infinite radius separated by a distance "d" for 1)
COUPLED HELICES
149
2.0 2.5
/3 a cot ifjy
4.5
Fig. 2.4.4 — Beat phase-constant plotted as a function of /3 a cot ^ . These
curves result from the solution of the field equations given in the appendix. For
b/a = 2.0.
the case with dielectric between them having a relative dielectric con-
stant e' = 4, and 2) the case of no dielectric. The pitch angles of the two
helices were \p and —if/, respectively; i.e., the helices were assumed to be
synchronous, and wound in the opposite sense.
Fig. 2.7 shows a plot of the ratio of &«/& to c /0 versus /3 {(1/2) coti//,
1.00
Fig. 2.4.5 — Beat phase-constant plotted as a function of /3 « cot ^i . These
curves result from the solution of the field equations given in the appendix. For
b/a = 3.0.
150
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
2.4
2.2
2.0
<nN
1.4
1.2
1.0
0.8
\
— }
\A
R
/J a cot i//
Fig. 2.5 — Propagation constants for a bifilar helix plotted as a function of
/3 a cot ^i . The curves illustrate, (A) the dispersive character of the in-phase
mode and, (B) the non-dispersive character of the out-of -phase mode.
where /3 ce is the coupling phase-constant in the presence of dielectric,
j8 € is the phase-constant of each helix alone in the presence of the same
dielectric, 0« is the coupling phase-constant with no dielectric, and is
the phase constant of each helix in free space. In many cases of interest
0o(d/2) cot ip is greater than 1.2. Then
MP
V + 1
\_2e' + 2 J
e -(V2e '+2-2)0o (dl2) cot *
(2.10.1)
Appearing in the same figure is a similar plot for the case when there is a
conducting shield inside the inner helix and outside the outer, and
separated a distance, "s," from the helices. Note that
d = b — a.
It appears from these calculations that the effect of the presence of
dielectric between the helices depends largely on the parameter ft (d/2)
cot yp. For values of this parameter larger than 0.3 the coupling wave-
length tends to increase in terms of circuit wavelength. For values smaller
than 0.3 the opposite tends to happen. Note that the curve representing
(2.10.1) is a fair approximation down to po(d/2) cot\f/ = 0.6 to the curve
representing the exact solution of the field equations. J. W. Sullivan, in
unpublished work, has drawn similar conclusions.
COUPLED HELICES
151
2.11 The Conditions for Maximum Power Transfer
The transmission line theory has led us to expect that the most efficient
power transfer will take place if the phase velocities on the two helices,
prior to coupling, are the same. Again, this would be true were it not for
the dispersion of the helices. To evaluate this effect we have used the
field equation to determine the parameter of the coupled helices which
gives maximum power transfer. To do this we searched for combinations
of parameters which give an equal current flow in the helix sheath for
either the longitudinal mode or the transverse mode. This was suggested
by L. Stark, who reasoned that if the currents were equal for the indi-
vidual modes the beat phenomenon would give points of zero RF current
on the helix.
The values of cot ^ 2 /cot fa which are required to produce this condi-
tion are plotted in Fig. 2.8 for various values of b/a. Also there are shown
values of cot^ 2 /cot \f/i required to give equal axial velocities for the helices
before they are coupled. It can be seen that the uncoupled velocity of the
inner helix must be slightly slower than that of the outer.
A word of caution is necessary for these curves have been plotted
without considering the effects of dielectric loading, and this can have a
rather marked effect on the parameters which we have been discussing.
The significant point brought out by this calculation is that the optimum
0.24
0.20
■0
^0.16
h-
o
0.12
flj
o
<Q
0.08
0.04
/
/
1
1.2 1.6 2.0 2.4- 2.8 a2 3.6 4.0
Aj a cot ^,
Fig. 2.6 — The coupling phase-constant which results from the two possible
modes of propagation on a bifilar helix shown as a function of P^a cot ^i .
152 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
2.6
/i f COT^
Fig. 2.7 — The effect of dielectric material between coupled infinite radius
sheath helices on their relative coupling phase-constant shown as a function of
Pud/2 cot y]/\ . The effect of shielding on this relation is also indicated.
condition for coupling is not necessarily associated with equal velocities
on the uncoupled helices.
2.12 Mode Impedance
Before leaving the general theory of coupled helices something should
be said regarding the impedance their modes present to an electron beam
traveling either along their axis or through the annular space between
them. The field solutions for cross wound, coaxially coupled helices,
which are given in Appendix I, have been used to compute the imped-
ances of the transverse and longitudinal modes. The impedance, K, is
defined, as usual, in terms of the longitudinal field on the axis and the
power flow along the system.
COUPLED HELICES
153
K =
5l
1/3
©"•
F( 7 a)
In Fig. 2.9, F(ya), for various ratios of inner to outer radius, is plotted
for both the transverse and longitudinal modes together with the value
of F(ya) for the single helix (b/a = <*>). We see that the longitudinal
mode has a higher impedance with cross wound coupled helices than
does a single helix. We call attention here to the fact that this is the
same phenomenon which is encountered in the contra wound helix 6 , where
the structure consists of two oppositely wound helices of the same radius.
As defined here, the transverse mode has a lower impedance than the
single helix. This, however, is not the most significant comparison; for
it is the transverse field midway between helices which is of interest in
the transverse mode. The factor relating the impedance in terms of the
transverse field between helices to the longitudinal field on the axis is
E r ~(f)/Ez~(Q), where f is the radius at which the longitudinal component
of the electric field E t , is zero for the transverse mode. This factor,
plotted in Fig. 2.10 as a function of ( >a coti/v , shows that the impedance
in terms of the transverse field at f is interestingly high.
1.6 2.0 2.4
/j a cot y
Fig. 2.8 — The values of cot ^ 5 /cot \pi required for complete power transfer
plotted as a function of /3 n fl cot \pi for several values of b/a. For comparison, the
value of cot ^ 2 /cot \j/\ required for equal velocities on inner and outer helices is also
shown.
154
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
7.0
6.5
5.5
F(ra)
4.0
3.5
3.0
2.5
LONGITUDINAL WAVE
COT Xfl,
^ = -0.90
COT ^,
\
\
s
\
H
\ <
\
\ 2
\
\
\
\
\
\
\
\
\
l.5>
N
\
\
\
\
\
\
\
\
\
\
,.25\
vs.
\
\
\
\
\
V
\
\
\
i
l\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\ >
\
\
\
\
\
\
Sj-5
,2.0
V
\
\
\
#-<
25\
"\^
«
5 10 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
/5 a cot f,
Fig. 2.9 — Impedance parameter, F(ya), associated with both transverse and
longitudinal modes shown for several values of b/a. Also shown is F(ya) for a
single helix.
It is also of interest to consider the impedance of the longitudinal
mode in terms of the longitudinal field between the two helices. The
factor, En(f)/E*(0), relating this to the axial impedance is plotted in
Fig. 2.11. We see that rather high impedances can also be obtained with
the longitudinal field midway between helices. This, in conjunction with
a hollow electron beam, should provide efficient amplification.
3. APPLICATION OF COUPLED HELICES
When we come to describe devices which make use of coupled helices
we find that they fall, quite naturally, into two separate classes. One
COUPLED HELICES
155
class contains those devices which depend on the presence of only one of
the two normal modes of propagation. The other class of devices depends
on the simultaneous presence, in roughly equal amounts, of both normal
modes of propagation, and is, in general, characterized by the words
"spatial beating." Since spatial beating implies energy surging to and
fro between inner and outer helix, there is no special problem in exciting
both modes simultaneously. Power fed exclusively to one or the other
3 4
/3 a COT j£,
Pig. 2.10 — The relation between the impedance in terms of the transverse
field between coupled helices excited in the out-of-phase mode, and the impedance
in terms of the longitudinal field on the axis shown as a function of o a cot \p\ .
156 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
helix will inevitably excite both modes equally. When it is desired to
excite one mode exclusively a more difficult problem has to be solved.
Therefore, in section 3.1 we shall first discuss methods of exciting one
mode only before going on to discuss in sections 3.2 and 3.3 devices
using one mode only.
In section 3.4 we shall discuss devices depending on the simultaneous
presence of both modes.
3.1 Excitation of Pure Modes
3.1.1 Direct Excitation
In order to set up one or the other normal mode on coupled helices,
voltages with specific phase and amplitudes (or corresponding currents)
Ez 2 (r)
COT
W-?
" '—= -0.90
COT Ipi
1
1
*-w
1
1
//
//
/
L
/
/
1
/
//
u
//
/
/ y
/
/ ^
2 3 4
/3 a COT^,
Fig. 2.11 — The relation between the impedance in terms of the longitudinal
field between coupled helices excited in the in-phase mode, and the impedance in
terms of the longitudinal field on the axis shown as a function of /3 a cot ^i .
COUPLED HELICES 157
have to be supplied to each helix at the input end. A natural way of doing
this might be by means of a two-conductor balanced transmission line
(Lecher-line), one conductor being connected to the inner helix, the other
to the outer helix. Such an arrangement would cause something like the
transverse (H — ) mode to be set up on the helices. If the two con-
ductors and the balanced line can be shielded from each other starting
some distance from the helices then it is, in principle, possible to intro-
duce arbitrary amounts of extra delay into one of the conductors. A delay
of one half period would then cause the longitudinal (++) mode to be
set up in the helices. Clearly such a coupling scheme would not be
broad-band since a frequency-independent delay of one half period is not
realizable.
Other objections to both of these schemes are: Balanced lines are not
generally used at microwave frequencies; it is difficult to bring leads
through the envelope of a TWT without causing reflection of RF energy
and without unduly encumbering the mechanical design of the tube plus
circuits; both schemes are necessarily inexact because helices having
different radii will, in general, require different voltages at either input
in order to be excited in a pure mode.
Thus the practicability, and success, of any general scheme based on
the existence of a pure transverse or a pure longitudinal mode on coupled
helices will depend to a large extent on whether elegant coupling means
are available. Such means are indeed in existence as will be shown in the
next sections.
3.1.2 Tapered Coupler
A less direct but more elegant means of coupling an external circuit
to either normal mode of a double helix arrangement is by the use of the
so-called "tapered" coupler. 8, 9 " 10 By appropriately tapering the relative
propagation velocities of the inner and outer helices, outside the inter-
action region, one can excite either normal mode by coupling to one
helix only.
The principle of this coupler is based on the fact that any two coupled
transmission lines support two, and only two, normal modes, regardless
of their relative phase velocities. These normal modes are characterized
by unequal wave amplitudes on the two lines if the phase velocities are
not equal. Indeed the greater the phase velocity difference and /or
the smaller the coupling coefficient between the lines, the more their
wave amplitudes diverge. Furthermore, the wave amplitude on the line
with the slower phase velocity is greater for the out-of-phase or trans-
verse normal mode, and the wave amplitude on the faster line is greater
158 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
for the longitudinal normal mode. As the ratio of phase constant to
coupling constant approaches infinity, the ratio of the wave amplitudes
on the two lines does also. Finally, if the phase velocities of, or coupling
between, two coupled helices are changed gradually along their length
the normal modes existing on the pair roughly maintain their identity
even though they change their character. Thus, by properly tapering the
phase velocities and coupling strength of any two coupled helices one
can cause the two normal modes to become two separate waves, one
existing on each helix.
For instance, if one desires to extract a signal propagating in the in-
phase, or longitudinal, normal mode from two concentric helices of equal
phase velocity, one might gradually increase the pitch of the outer helix
and decrease that of the inner, and at the same time increase the diameter
of the outer helix to decrease the coupling, until the longitudinal mode
exists as a wave on the outer helix only. At such a point the outer helix
may be connected to a coaxial line and the signal brought out.
This kind of coupler has the advantage of being frequency insensitive;
and, perhaps, operable over band widths upwards of two octaves. It
has the disadvantage of being electrically, and sometimes physically,
quite long.
3.1.3 Stepped Coupler
There is yet a third way to excite only one normal mode on a double
helix. This scheme consists of a short length at each end of the outer helix,
for instance, which has a pitch slightly different from the rest. This
has been called a "stepped" coupler.
The principle of the stepped coupler is this: If two coupled transmis-
sion lines have unlike phase velocities then a wave initiated in one line
can never be completely transferred to the other, as has been shown in
Section 2.4. The greater the velocity difference the less will be the maxi-
mum transfer. One can choose a velocity difference such that the maxi-
mum power transfer is just one half the initial power. It is a characteristic
of incomplete power transfer that at the point where the maximum trans-
fer occurs the waves on the two lines are exactly either in-phase or out-of-
phase, depending on which helix was initially excited. Thus, the condi-
tions for a normal mode on two equal-velocity helices can be produced
at the maximum transfer point of two unlike velocity helices by initiating
a wave on only qne of them. If at that point the helix pitches are changed
to give equal phase velocities in both helices, Avith equal current or volt-
age amplitude on both helices, either one or the other of the two normal
modes will be propagated on the two helices from there on. Although the
COUPLED HELICES 159
pitch and length of such a stepped coupler are rather critical, the re-
quirements are indicated in the equations in Section 2.4.
The useful bandwidth of the stepped coupler is not as great as that
of the tapered variety, but may be as much as an octave. It has however
the advantage of being very much shorter and simpler than the tapered
coupler.
3.2 Low-Noise Transverse-Field Amplifier
One application of coupled helices which has been suggested from the
very beginning is for a transverse field amplifier with low noise factor.
In such an amplifier the RF structure is required to produce a field which
is purely transverse at the position of the beam. For the transverse mode
there is always such a cylindrical surface where the longitudinal field is
zero and this can be obtained from the field equation of Appendix II.
In Fig. 3.1 we have plotted the value of the radius f at which the longi-
tudinal field is zero for various parameters. The significant feature of
this plot is that the radius which specifies zero longitudinal field is not
constant with frequency. At frequencies away from the design frequency
the electron beam will be in a position where interaction with longitudinal
components might become important and thus shotnoise power will be
introduced into the circuit. Thus the bandwidth of the amplifier over
which it has a good noise factor would tend to be limited. However, this
effect can be reduced by using the smallest practicable value of b/a.
Section 2.12 indicates that the impedance of the transverse mode is
very high, and thus this structure should be well suited for transverse
field amplifiers.
3.3 Dispersive Traveling-Wave Tube
Large bandwidth is not always essential in microwave amplifiers. In
particular, the enormous bandwidth over which the traveling-wave tube
is potentially capable of amplifying has so far found little application,
while relatively narrow bandwidths (although quite wide by previous
standards) are of immediate interest. Such a relatively narrow band, if
it is an inherent electronic property of the tube, makes matching the
tube to the external circuits easier. It may permit, for instance, the use
of non-reciprocal attenuation by means of ferrites in the ferromagnetic
resonance region. It obviates filters designed to deliberately reduce the
band in certain applications. Last, but not least, it offers the possibility
of trading bandwidth for gain and efficiency.
A very simple method of making a traveling-wave tube narrow-band
160 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
1.8
COT ^
COT lf/ {
82^
4-o
-0.90
-0.9
8
x.
a
COT 1ft 2
^- = -0.82
COT ^, _—
*=«
po.q
8
1.2
HT
t-u.
COT Ipz
COT l//i
,' 4
= -0.82 -
-0.90
' /
/
•
1.0
-0.98
1
r-"""
5
1
1
5
2
2
/
5
3 a
3
:otj
ft
3
5
4
4
5
b
Fig. 3.1 —The radius f at which the longitudinal field is zero for transversely
excited coupled coaxial helices.
is by using a dispersive circuit, (i.e. one in which the phase velocity varies
significantly with frequency). Thus, we obtain an amplifier that can be
tuned by varying the beam voltage; being dispersive we should also
expect a low group velocity and therefore higher circuit impedance.
Calculations of the phase velocities of the normal modes of coupled
concentric helices presented in the appendix show that the fast, longitu-
dinal or (+ + ) mode is highly dispersive. Given the geometry of two
such coupled helices and the relevant data on an electron beam, namely
current, voltage and beam radius, it is possible to arrive at an estimate
of the dependence of gain on frequency.
Experiments with such a tube showed a bandwidth 3.8 times larger
than the simple estimate would show. This we ascribe to the presence
COUPLED HELICES 161
of the dielectric between the helices in the actual tube, and to the neglect
of power propagated in the form of spatial harmonics.
Nevertheless, the tube operated satisfactorily with distributed non-
reciprocal ferrite attenuation along the whole helix and gave, at the
center frequency of 4,500 mc/s more than 40 db stable gain.
The gain fell to zero at 3,950 mc/s at one end of the band and at
4,980 mc/s at the other. The forward loss was 12 db. The backward
loss was of the order of 50 db at the maximum gain frequency.
3.4 Devices Using Both Modes
In this section we shall discuss applications of the coupled-helix princi-
ple which depend for their function on the simultaneous presence of both
the transverse and the longitudinal modes. When present in substantially
equal magnitude a spatial beat-phenomenon takes place, that is, RF
power transfers back and forth between inner and outer helix.
Thus, there are points, periodic; with distance along each helix, where
there is substantially no current or voltage; at these points a helix can be
terminated, cut-off, or connected to external circuits without detriment.
The main object, then, of all devices discussed in this section is power
transfer from one helix to the other; and, as will be seen, this can be ac-
complished in a remarkably efficient, elegant, and broad-band manner.
3.4.1 Coupled-Helix Transducer
It is, by now, a well known fact that a good match can be obtained
between a coaxial line and a helix of proportions such as used in TWT's. A
wire helix in free space has an effective impedance of the order of 100
ohms. A conducting shield near the helix, however, tends to reduce the
helix impedance, and a value of 70 or even 50 ohms is easily attained.
Provided that the transition region between the coaxial line and the
helix does not present too abrupt a change in geometry or impedance,
relatively good transitions, operable over bandwidths of several octaves,
can be made, and are used in practice to feed into and out of tubes em-
ploying helices such as TWT's and backward-wave oscillators.
One particularly awkward point remains, namely, the necessity to lead
the coaxial line through the tube envelope. This is a complication in
manufacture and requires careful positioning and dimensioning of the
helix and other tube parts.
Coupled helices offer an opportunity to overcome this difficulty in the
form of the so-called coupled-helix transducer, a sketch of which is
shown in Fig. 3.2. As has been shown in Section 2.3, with helices having
162 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
the same velocity an overlap of one half of a beat wavelength will result
in a 100 per cent power transfer from one helix to the other. A signal in-
troduced into the outer helix at point A by means of the coaxial line will
be all on the inner helix at point B, nothing remaining on the outer helix.
At that point the outer helix can be discontinued, or cut off; since there
is no power there, the seemingly violent discontinuity represented by the
'open" end of the helix will cause no reflection of power. In practice, un-
fortunately, there arc always imperfections to consider, and there will
often be some power left at the end of the coupler helix. Thus, it is de-
sirable to terminate the outer helix at this point non-reflectively, as, for
instance, by a resistive clement of the right value, or by connecting to it
another matched coaxial line which in turn is then non-reflectively ter-
minated.
It will be seen, therefore, that the coupled-helix transducer can, in
principle, be made into an efficient device for coupling RF energy from
a coaxial line to a helix contained in a dielectric envelope such as a glass
tube. The inner helix will be energized predominantly in one direction,
namely, the one away from the input connection. Conversely, energy
traveling initially in the inner helix will be transferred to the outer, and
made available as output in the respective coaxial line. Such a coupled-
helix transducer can be moved along the tube, if required. As long as the
outer helix completely overlaps the inner, operation as described above
should be assured. By this means a new flexibility in design, operation
and adjustment of traveling-wave tubes is obtained which could not be
achieved by any other known form of traveling- wave tube transducer.
Naturally, the applications of the coupled-helix transducer are not
restricted to TWT's only, nor to 100 per cent power transfer. To obtain
Fig. 3.2 — A simple coupled helix transducer.
COUPLED HELICES 103
power transfer of proportions other than 100 per cent two possibilities
are open: either one can reduce the length of the synchronous coupling
helix appropriately, or one can deliberately make the helices non-syn-
chronous. In the latter case, a considerable measure of broad-banding
can be obtained by making the length of overlap again equal to one half
of a beat-wavelength, while the fraction of power transferred is deter-
mined by the difference of the helix velocities according to 2.4.7. An
application of the principle of the coupled-helix transducer to a variable
delay line has been described by L. Stark 12 in an unpublished memo-
randum.
Turning again to the complete power transfer case, we may ask:
How broad is such a coupler?
In Section 2.7 we have discussed how the radial falling-off of the RF
energy near a helix can be used to broad-band coupled-helix devices
which depend on relative constancy of beat-wavelength as frequency
is varied. On the assumption that there exists a perfect broad-band match
between a coaxial line and a helix, one can calculate the performance of
a coupled-helix transducer of the type shown in Fig. 3.2.
Let us define a center frequency a>, at which the outer helix is exactly
one half beat-wavelength, \ b , long. If cj is the frequency of minimum
beat wavelength then at frequencies wi and o> 2 , larger and smaller,
respectively, than cu, the outer helix will be a fraction 8 shorter than
YiKb , (Section 2.7). Let a voltage amplitude, V« , exist at the point where
the outer helix is joined to the coaxial line. Then the magnitude of the
voltage at the other end of the outer helix will be | TV sin (ir5/2) | which
means that the power has not been completely transferred to the inner
helix. Let us assume complete reflection at this end of the outer helix.
Then all but a fraction of the reflected power will be transferred to the
inner helix in a reverse direction. Thus, we have a first estimate for the
"directivity" defined as the ratio of forward to backward power (in db)
introduced into the inner helix:
D =
10 !..^ >\vr [^
(3.4.1.1)
We have assumed a perfect match between coaxial line and outer helix;
thus the power reflected back into the coaxial line is proportional to
sin 4 (x5/2). Thus the reflectivity defined as the ratio of reflected to
incident power is given in db by
R = 10 log sin 4 0^\ (3.4.1.2)
164 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
For the sake of definiteness, let us choose actual figures: let /3a = 2.0.
and b/a = 1.5. And let us, arbitrarily, demand that R always be less than
-20 db.
This gives sin (tt5/2) < 0.316 and tt5/2 < 18.42° or 0.294 radians,
5 < 0.205. With the optimum value of fi c a = 1.47, this gives the mini-
mum permissible value of &a of 1.47/(1 + 0.205) = 1.22. From the
graph on Fig. 2.2 this corresponds to values of /3a of 1.00 and 3.50.
Therefore, the reflected power is down 20 db over a frequency range of
W2 / Wl = 3.5 to one. Over the same range, the directivity is better than
10 to one. Suppose a directivity of better than 20 db were required.
This requires sin (irS/2) = 0.10, 5 = 0.0638 and is obtained over a fre-
quency range of approximately two to one. Over the same range, the
reflected power would be down by 40 db.
In the above example the full bandwidth possibilities have not been
used since the coupler has been assumed to have optimum length when
/3 c a is maximum. If the coupler is made longer so that when /3 c a is maxi-
mum it is electrically short of optimum to the extent permissible by
the quality requirements, then the minimum allowable /3 c a becomes even
smaller. Thus, for b/a =1.5 and directivity 20 db or greater the rea-
lizable bandwidth is nearly three to one.
When the coupling helix is non-reflectively terminated at both ends,
either by means of two coaxial lines or a coaxial line at one end and a
resistive element at the other, the directivity is, ideally, infinite, irrespec-
tive of frequency; and, similarly, there will be no reflections. The power
transfer to the inner helix is simply proportional to cos (ir8/2). Thus,
under the conditions chosen for the example given above, the coupled-
helix transducer can approach the ideal transducer over a considerable
range of frequencies.
So far, we have inspected the performance and bandwith of the
coupled-helix transducer from the most optimistic theoretical point of
view. Although a more realistic approach does not change the essence
of our conclusions, it does modify them. For instance, we have neglected
dispersion on the helices. Dispersion tends to reduce the maximum at-
tainable bandwidth as can be seen if Fig. 2.4.2 rather than Fig. 2.2 is
used in the example cited above. The dielectric that exists in the annular
region between coupled concentric helices in most practical couplers
may also affect the bandwidth.
In practice, the performance of coupled-helix transducers has been
short of the ideal. In the first place, the match from a coaxial line to a
helix is not perfect. Secondly, a not inappreciable fraction of the RF
power on a real wire helix is propagated in the form of spatial harmonic
COUPLED HELICES
165
m 14
o
V
A /
i
\
\
I
1 i /
1 * /
fl
r
\
V
1
1
' V
1 I
1 ■
u
^
1
I
I
1
1
1
• >
1
• t /"
1 1 /
/
/
i
I
I
1
1
1
1 /
1 /
V"f^
/
/
/
\
<
1
1
1
A
V
1
/ \
\
f\
1
!
I
1
1
i
\
\
\
\
i
— COUPLER DIRECTIVITY
RETURN LOSS
\
\
'
j
\
\
V
V,
(.5 2 2.5 3 4 5 6 7 B
FREQUENCY IN KILOMEGACYCLES
Fig. 3.3 — The return loss and directivity of an experimental 100 per cent
coupled-helix transducer.
wave components which have variations with angle around the helix-
axis, and coupling between such components on two helices wound in
opposite directions must be small. Finally, there are the inevitable me-
chanical inaccuracies and misalignments.
Fig. 3.3 shows the results of measurements on a coupled-helix trans-
ducer with no termination at the far end.
3.4.2 Cowpled-Hclix Attenuator
In most TWT's the need arises for a region of heavy attenuation
somewhere between input and output; this serves to isolate input and
output, and prevents oscillations due to feedback along the circuit. Be-
cause of the large bandwidth over which most TWT's are inherently
capable of amplifying, substantial attenuation, say at least 60 db, is
166 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
required over a bandwidth of maybe 2 octaves, or even more. Further-
more, such attenuation should present a very good match to a wave on
the helix, particularly to a wave traveling backwards from the output
of the tube since such a wave will be amplified by the output section of
the tube.
Another requirement is that the attenuator should be physically as
short as possible so as not to increase the length of the tube unneces-
sarily.
Finally, such attenuation might, with advantage, be made movable
during the operation of the tube in order to obtain optimum performance,
perhaps in respect of power output, or linearity, or some other aspect.
Coupled-helix attenuators promise to perform these functions satis-
factorily.
A length of outer helix (synchronous with the inner helix) one half of a
beat wavelength long, terminated at either end non-reflectively, forms a
very simple, short, and elegant solution of the coupled-helix attenuator
problem. A notable weakness of this form of attenuator is its relatively
narrow bandwidth. Proceeding, as before, on the assumption that the
attenuator is a fraction 8 larger or smaller than half a beat wavelength
at frequencies coi and w 2 on either side of the center frequency u, we find
that the fraction of power transferred from the inner helix to the attenu-
ator is then given by (1 - sin 2 (t8/2)). The attenuation is thus simply
A = sin (^
For helices of the same proportions as used before in Section 3.4.1, we
find that this will give an attenuation of at least 20 db over a frequency
band of two to one. At the center frequency, w , the attenuation is in-
finite ; — in theory.
Thus to get higher attenuation, it would be necessary to arrange for a
sufficient number of such attenuators in tandem along the TWT. More-
over, by properly staggering their lengths within certain ranges a wider
attenuation band may be achieved. The success of such a scheme largely
depends on the ability to terminate the helix ends non-reflectively. Con-
siderable work has been done in this direction, but complete success is
not yet in sight.
Another basically different scheme for a coupled-helix attenuator rests
on the use of distributed attenuation along the coupling helix. The diffi-
culty with any such scheme lies in the fact that unequal attenuation in
the two coupled helices reduces the coupling between them and the more
they differ in respect to attenuation, the less the coupling. Naturally, one
COUPLED HELICES 167
would wish to have as little attenuation as practicable associated with
the inner helix (inside the TWT). This requires the attenuating element
to be associated with the outer helix. Miller 5 has shown that the maxi-
mum total power reduction in coupled transmission systems is obtained
when
«1 — CC2
fa
where t*i and a-> are the attenuation constants in the respective systems,
and /3ft the beat phase constant. If the inner helix is assumed to be loss-
less, the attenuation constant of the outer helix has to be effectively equal
to the beat wave phase constant. It turns out that 60 db of attenuation
requires about 3 beat wavelengths (in practice 10 to 20 helix wave-
lengths). The total length of a typical TWT is only 3 or 4 times that,
and it will be seen, therefore, that this scheme may not be practical as
the only means of providing loss.
Experiments carried out with outer helices of various resistivities and
thicknesses by K. M. Poole (then at the Clarendon Laboratory, Oxford,
England) tend to confirm this conclusion. P. D. Lacy 11 has described a
coupled helix attenuator which uses a multifilar helix of resistance
material together with a resistive sheath between the helices.
Experiments were performed at Bell Telephone Laboratories with a
TWT using a resistive sheath (graphite on paper) placed between the
outer helix and the quartz tube enclosing the inner helix. The attenua-
tions were found to be somewhat less than estimated theoretically. The
attenuator helix was movable in the axial direction and it was instructive
to observe the influence of attenuator position on the power output from
the tube, particularly at the highest attainable power level. As one might
expect, as the power level is raised, the attenuator has to be moved nearer
to the input end of the tube in order to obtain maximum gain and power
output. In the limit, the attenuator helix has to be placed right close to
the input end, a position which does not coincide with that for maximum
low-level signal gain. Thus, the potential usefulness of the feature of
mobility of coupled-helix elements has been demonstrated.
4. CONCLUSION
In this paper we have made an attempt to develop and collect together
a considerable body of information, partly in the form of equations,
partly in the form of graphs, which should be of some help to workers
in the field of microwave tubes and devices. Because of the crudity of the
assumptions, precise agreement between theory and experiment has not
168 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
been attained nor can it be expected. Nevertheless, the kind of physical
phenomena occurring with coupled helices are, at least, qualitatively
described here and should permit one to develop and construct various
types of devices with fair chance of success.
ACKNOWLEDGEMENTS
As a final note the authors wish to express their appreciation for the
patient work of Mrs. C. A. Lambert in computing the curves, and to
G. E. Korb for taking the experimental data.
Appendix i
i. solution of field equations
In this section there is presented the field equations for a transmission
system consisting of two helices aligned with a common axis. The propa-
gation properties and impedance of such a transmission system are dis-
cussed for various ratios of the outer helix radius to the inner helix radius.
This system is capable of propagating two modes and as previously
pointed out one mode is characterized by a longitudinal field midway
between the two helices and the other is characterized by a transverse
field midway between the two helices.
The model which is to be treated and shown in Fig. 2.3 consists of an
inner helix of radius a and pitch angle \f/i which is coaxial with the outer
helix of radius h and pitch angle ^ 2 . The sheath helix model will be
treated, wherein it is assumed that helices consist of infinitely thin sheaths
which allow for current flow only in the direction of the pitch angle \{/.
The components of the field in the region inside the inner helix, be-
tween the two helices and outside the outer helix can be written as
follows — inside the inner helix
H Zl = BMyr) (1)
E 2% = BMyr) (2)
H Pi = j^BMyr) (3)
7
H ri =^B 1 h(v) (4)
7
E fl = -j^Bihiyr) (5)
7
E rt = ^ BMyr) (6)
COUPLED HELICES 169
and between the two helices
H, t = BJ (yr) + B<K (yr) (7)
E. t = BMyr) + B*K (yr) (8)
H V6 = j — WMyr) - BAM] (9)
7
H rt = & [BMv) - BAM] (10)
7
E» = - 3 - [BMv) - BAM] (ID
7
E Ti = J l [BMir) - BAM] (12)
7
and outside the outer helix
H, v = BAM (13)
E H = BAM (14)
B n -= - j — W:i(tt) (15)
7
ff„ = ^ MTiOyr) (16)
tf„ = j 2e BAM (17)
7
#r, = ^ BbKjCtt) (18)
7
With the sheath helix model of current flow only in the direction of wires
we can specify the usual boundary conditions that at the inner and outer
helix radius the tangential electric field must be continuous and per-
pendicular to the wires, whereas the tangential component of magnetic
field parallel to the current flow must be continuous. These can be written
as
E z sin yf, + E 9 cos $ = (19)
E z , E v and (H z sin \f/ + H v cos $) be equal on either side of the helix.
By applying these conditions to the two helices the following equations
are obtained for the various coefficients.
170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
First, we will define a more simple set of parameters. We will denote
h(ya) by hi and h(yb) by J M , etc.
Further let us use the notation introduced by Humphrey, Kite and
James 11 in his treatment of coaxial helices.
P 0l = I01K01 P02 — I02K02 Ro = h\K 2
P u = I n Kn Pu = M12 Ri = InKia
and define a common factor (C.F.) by the equation
< , .- . ,"(^otj^ PoiPo2 _ (ftacotfr) 2 cot*_ 2 E[! ,
(20)
3.F. = - [!
hay
(7a) 2 cot yp\
+ Ro' — P01P01
(21)
With all of this we can now write for the coefficients of equations 1
through 18:
B 2 J
ya
L
S--VI
/x /3 a cot t/'i 7o2
e 0oa cot i/'! 7oi^i2 fOoq cot li) 2
/' 7a
/3oa cot \p
C.F. L (Ta) 2 cot^i
1 /oi/u r
" C.F. L
(j3oa cot ^ 2 ) 2
ft
#2
7a < .1 . L (t«) 2
(/3oa cot i/'i) cot i/'
P12 - Pi
COt l/'
1 J
= -^r (7^7
5 6 /01 2 [(gj g cot ^ 2 ) 2 nl
E - "of. L (yay Pl2 " Po2 J
fl = • /£ *»cat* 1 U T p ^ _ <*£ j
£ 2 J y m 7a C.F. Z12 L cot «£i J
- Pio/?o"|
1 J
7?8 (/3pa cot i/'Q" cot \p2 hi'
K = (7a) 2 cot Vi C.F.Eo
Po 2 Ki -
COt \(/2
COt l^i
(22)
(23)
(24)
(25)
(26)
(27)
(28)
The last equation necessary for the solution of our field problem is the
transcendental equation for the propagation constant, 7, which can be
COUPLED HELICES
171
written
[" _ (go a cot fr) 2 cot fa R T
L ° (to) 2 cot ^1 J
f p (/3o a cot r£ 2 ) 2 plf" p (fta cot iAi)" ,, 1
= L /o2 " " (yay /l2 J L Po1 " (7a) 2 ' "J
(29)
The solutions of this equation are plotted in Fig. 4.1.
There it is seen that there are two values of 7, one, y t , denoting the
slow mode with transverse fields between helices and the other, 7/ ,
denoting the fast mode with longitudinal fields midway between the two
helices.
»—
y/
/
COT V 2
COT 1fl s
0.90
0.98
/*
f
A
/
7
>t a
7a
//
/
A
•
y /y
Y
V
A
[1
^
y
J ^gtf^-'*'
0.5 1.0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
/j a cot yi
Fig. 4.1.1 — The riidi.il propagation constants associated with the transverse
and longitudinal modes on coupled coaxial sheath helices given as a function of
0,)<i cot i/m for several values of b/a = 1.25.
172 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
These equations can now be used to compute the power flow as defined
by
P = Y 2 Re f E XH*dA
which can be written in the form
RW-iGK-
(30)
(31)
where
\F(ya yb)\- 3 = ^! *°L <
'(
(/3 a cot ft) 2 r
7 « + (yaY '
')
(/n 2 - /01/21XC.F.) 2
9 / fog cot ft V
z « 7n l^o— ;
/„ 2 . (fta cot ft) 2 A/ (/3 a cot ft) 2 cot ft
\ (7a) 2 A (7a) 2 cot ft
[(!)'
(/02/22 — /12 ) + Un" ~~ I oil 21)
]
, . ; , _ (goo cot ft) 2 cot ft B Y fp 02 _ ^° a cot J*!?. p 12
(7a) 2 cot ft
(7a) 2
["(-) (2ZuKu + IwKa + /22K02) - (2/nK u + I u K a + /21K01)
(32)
(floa cot ft) 3 p
^02 — 7 — T7 — * 12
r 2 , (/3 acotft) 2 T 2 ir
" I/ 01 + (7a) 2 /u J [/ Ui (7a) 2
[7-Y (^02/^22 - KW) - (K 0l K 21 - A',, 2 ) I
+
a cot ft) 2 /6 Y f p
ayKifRf \aj L C
(0
(7«) :
+
(/Soo cot ft)"
(7a) 2
/01'A
>]
\p«Ri - ^ PuBoT [£02*22 - Kn 2 )
|_ cot ft J
In (32) we find the power in the transverse mode by using values of
COUPLED HELICES
173
Fig. 4.1.2 — The radial propagation constants associated with the transverse
and longitudinal modes on coupled coaxial sheath helices given as a function of
/3oO cot yj/i when b/a = 1.50.
y t obtained from (29) and similarly the power in the longitudinal mode is
found by using values of yi .
II. finding r
When coaxial helices are used in a transverse field amplifier, only the
transverse field mode is of interest and it is important that the helix
parameters be adjusted such that there is no longitudinal field at some
radius, f, where the cylindrical electron beam will be located. This condi-
tion can be expressed by equating E z to zero at r = f and from (8)
BJo(yf) + BsKoh?) =
(33)
174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
which can be written with (25) and (26) as
(/3 a cot ^i) 2 cot ^ 2
7Cn 2 R Q -
(7a) 2 cot \j/\ J
= /01
/a(7*0
(fioa cot \f/ 2 ) 2 D
1 02 — ; — 77 ■* 12
(ya)-
(34)
Ko(7f)
This equation together with (29) enables one to evaluate f/a versus /3oa
cot i^i for various ratios of b/a and cot ^ 2 /cot ^1 . The results of these
calculations are shown in Fig. 3.1.
Fig. 4.1.3 — The radial propagation constants associated with the transverse
and longitudinal modes on coupled coaxial sheath helices given as a function of
6qu cot \pi when b/a = 1.75.
COUPLED HELICES
175
2.0 2.5 3.0
/3 d COT #,
Fig. 4.1.4 — The radial propagation constants associated with the transverse
and longitudinal modes on coupled coaxial sheath helices given as a function of
Q a cot \p\ when b/a = 2.0.
III. COMPLETE POWER TRANSFER
For coupled helix applications we require the coupled helix parame-
ters to be adjusted so that RF power fed into one helix alone will set up
the transverse and longitudinal modes equal in amplitude. For this
condition the power from the outer helix will transfer completely to the
inner helix. The total current density can be written as the sum of the
current in the longitudinal mode and the transverse mode. Thus for the
inner helix we have
—ffllM
J a = Jaie~ 3P ' Z + Jafi
„-#*■
(35)
176 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
7* 2.5
Fig. 4.1.5 — The radial propagation constants associated with the transverse
and longitudinal modes on coupled coaxial sheath helices given as a function of
Pod cot \j/i when b/a = 3.0.
and for the outer helix
•h = Ju>** + JtiT*»
For complete power transfer we ask that
Jbt = Jbt
when J a is zero at the input (z = 0)
J at = —J at
or
Jbt _ _Jbt
J at J at
(36)
(37)
COUPLED HELICES 177
Now J at is equal to the discontinuity in the tangential component of
magnetic field which can be written at r = a
J a i = (// z3 cos to — //„ 6 sin to) — {H 2 \ cos to - H v * sin to)
which can be written as
Jat- - (II:i - Ha)at(cot to + tan to) sin to (38)
and similarily at r = b
J hi = — (Hgi — H z3 )bi(cot to + tan to) sin to (39)
Equations (38) and (39) can be combined with (37) to give as the condi-
tion for complete power transfer
At = -A t (40)
where
(J K X T K \ (p (ffoa cot to) 2 p \
U ISA-OS T I02K-I2) I "oi — 7 T7 / 11 1
A = \ (TO)- / / 41)
(/oiA'n + /„*„) (R - <**"**)' 2*A fi \
\ (7a)- cot to /
In (40) 4 1 is obtained by substituting y ( into (41) and /l t is obtained by
substituting y t into (41).
The value of cot to/cot to necessary to satisfy (40) is plotted in Fig.
2.8.
In addition to cot to/cot to it is necessary to determine the interference
wavelength on the helices and this can be readily evaluated by consider-
ing (36) which can now be written
J„ = Me*" + e - * 1 *)
or
J b = J^-WW™ cos (/3f " Pt)
(48)
and
•h = j ht r*«*«*i* cos y 2 fa (49)
where we have defined
0,,a = (y t a - y t a) (50)
This value of fi b is plotted versus /3 a cot to hi Fig. 2.4.
178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956
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