2001 IEEE

12 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 1, JANUARY 2001
Tape Helix Perturbation Including 3-D Dielectrics
for TWTs
Paul Greninger
Abstract—A perturbation technique described in a previous
paper [1] has been applied to the tape helix model of a traveling
wave tube (TWT). The perturbation technique employs two solu-
tions; the unperturbed solution is that for a helix in a conducting
sleeve, suspended in vacuum, and the perturbed solution is with
the above configuration, but with the introduction of dielectric
support rods. Usually the perturbed solution is not known and
is approximated by the homogeneous dielectric solution. Then
in three-dimensional (3-D) fashion, the perturbation technique
weights the dielectric with an energy-like term that is only
evaluated within the discrete dielectric. Therefore, the model
accounts for the distribution of dielectric material. Extending the
perturbation to the tape helix, the current density is accurately
portrayed in space. In all cases, the perturbed tape helix solution
has a lower least square error in normalized phase velocity than
the unperturbed tape helix solution. It had a lower error than
the perturbed sheath helix solution of the previous paper, which
in turn had a lower least square error than the Naval Research
Laboratory’s (NRL’s) Small Signal Gain (SSG) Program. An
initial discussion of the tape helix is presented and the calculated
dispersion relation given. This solution incorporates a conducting
backwall or shield with a homogeneous dielectric between it and
the helix, using = 6 to = +6 space harmonics. Then the
perturbation theory is applied, first for uniform support rods, and
then to include notched rods. The resultant simulations produce
phase velocity and coupling impedance data. Deviation from
theory to experiment is reported by comparing the square root of
the average sum of the squares difference, divided by an average
phase velocity, between calculated phase velocities and a second
order least squares fit of the measured data. Experimental data
on the dispersion properties of four helices came from Northrop.
The average error in phase velocity for four cases was 0.95%. For
uniform rods the perturbation theory lowers the phase velocity,
but doesn’t significantly alter the dispersion. However, for notched
rods, the perturbation theory raises the phase velocity and flattens
the dispersion, in agreement with experiment. Computations can
be performed in 1/2 min for a ten-frequency-step problem on a
Pentium II processor.
Index Terms—3-D dielectric, dispersion, impedance, notched
rods, perturbation, slow wave structure, tape helix, traveling wave
tube.
NOMENCLATURE
Electric field of the th mode.
Subscript be the component.
Subscript be the radial component.
Subscript be the azimuthal component.
Manuscript received December 23, 1999; revised August 21, 2000. The re-
view of this paper was arranged by D. Goebel.
This is General Atomics Rep. GA-A23191.
The author is with General Atomics, San Diego, CA USA (e-mail:
Grenin@GAT.com).
Publisher Item Identifier S 0018-9383(01)00675-X.
0 Superscript be the solution in the region
.
Superscript be the solution in the region
.
Subscript be the solution with no dielectric
present.
1 Subscript be the perturbed solution approximated
by the homogeneous dielectric solution.
2 Subscript = a better approximation to the per-
turbed solution.
Designates the magnetic field of the th mode.
Bessel functions of the first kind of order with
imaginary arguments.
An integral if explicitly defined.
axial propagation constant of the th mode.
Wave vector number for a plane wave, whose
spatial period varies as , ,
.
Radial constant of the th mode.
.
Pitch angle.
An integer, if a subscript of or a field compo-
nent a mode.
Number of support rods.
Permittivity of the dielectric.
Permittivity of free space.
Relative permittivity of the dielectric.
Permeability of free space.
Angular frequency.
Width of rod.
Helix mean radius.
Radius of backwall or shield.
Current density on the helix.
Ordinary Bessel function, if explicitly defined.
parallel to the helix.
th Fourier component of current density on the
helix in the direction.
th Fourier component of current density on the
helix in the direction.
Axial propagation constant in a periodic struc-
ture.
Coefficient used to describe the th component
of the electric or magnetic fields. See Appendix
(A1a), (A2a) and (A2b).
A uniform rod is made of dielectric support material which has
a constant cross sectional area, it may be rectangular or circular.
0018–9383/01$10.00 © 2001 IEEE

GRENINGER: 3-D DIELECTRICS FOR TWTs 13
I. INTRODUCTION
APREVIOUS paper published in this TRANSACTIONS [1]
presented a perturbation technique for finding phase
velocities and coupling impedances in a traveling wave tube
(TWT) for an arbitrary distribution of dielectric material. A
model of the sheath helix was presented. Previously, deviation
from theory to experiment was reported by stating the average
sum of the squares difference between theoretical calculations
and a second order least squares fit of the measured data. In
each of the four cases presented the calculated perturbed phase
velocities had the lowest average sum of the squares difference
than those calculated by the homogeneous dielectric solution, or
to the Naval Research Laboratory’s (NRL’s) computer program,
Small Signal Gain (SSG) [2]. Reference [3] treats the dielectric
in stratified layers with a uniform current on the helix. Reference
[4] treats the dielectric in stratified layers, and solves for the
exact current on the helix. Only graphs for the homogeneous di-
electric solution are provided, with no experimental data. While
the method of stratified layers can analyze inhomogeneities in
the radial direction it cannot analyze inhomogeneities in the
axial direction. Stratified layers smooth out discrete supports
azimuthally, perhaps into 20 continuous dielectric-tube regions.
This perturbation technique employs only two solutions in a
region, one without dielectric, and one with a homogeneous di-
electric solution. This paper extends the perturbation technique
to the tape helix model for the inhomogeneous dielectric in both
the and direction with a peak current distribution on the helix.
Reference [5] states that a peaked current distribution is a more
valid assumption than a uniform current distribution. Necessary
equations will be brought forward from the previous paper. For a
more complete discussion the reader is encouraged to consult the
first paper. It may be possible to extend this analysis to include
vane loading by a similar technique [6].
Three assumptions are made in solving the problem: 1) the
helix has infinitesimal thickness, 2) for computational ease, ,
the radial constant, has been approximated to be the same in
both regions, and 3) the current distribution on the helix be-
comes infinitely large in an inverse square root manner as the
tape edges are approached. The first assumption is made to
simplify the problem, and consistently with the first assump-
tion, all fields are evaluated at the helix tape mean radius. The
dielectric boundary is now virtually extended from the outer
tape helix radius to the mean tape helix radius. The dielectric
permittivity is proportionally reduced everywhere to account
for the increase in volume. In the first paper the area between
the helix outer radius and mean radius was filled with addi-
tional dielectric material, with no reduction in permittivity. A
third embodiment would terminate the dielectric at the helix
OD. In this paper, the author chose the latitude that favored the
tape helix solution. A diagram of a helix dielectric-support-rod
is illustrated in Fig. 1(a) and (b). In the second assumption
this model is approximately true when the phase velocity of
the slow wave is small compared to the speed of light, which
means that the term in the definition of may be neglected
in comparison to the term. From two possible choices of
, and , only outside has been retained. In the
third assumption, given the current density on the helix, two
Fig. 1. (a) Geometry of support rod virtually extended through the helix OD
to the mean radius. (b) Diagram of a notch.
possible determinantal equations exist. In this paper the deter-
minantal equation sought satisfies the condition along
the centerline of the helix. For the case of anisotropic pyrolytic
boron nitride the dielectric constant in the “ ” plane is 5.12,
while 3.4 in the “ ” plane. In a typical TWT the and vectors
correspond to the “ ” plane orientation of the dielectric. For the
tape helix model the author found that approximately 98% of
the energy was stored in the and components, therefore a
dielectric constant of 5.1 was chosen.
In Section II the basic perturbation technique is reviewed. In
Section III the tape helix model is discussed. In Section IV, the
basic equation which computes the phase shift in is validated.
In Section V, the technique is applied to the tape helix model
for uniform dielectric support rods. In Section VI, the analysis
is extended to notched dielectric support rods. Section VII is the
conclusion.
II. REVIEW
The equation for the change in , the axial propagation con-
stant, when dielectric rods are introduced is given by the inte-
gration of [1, Eq. (6)] over a single cell of the helix
(6) in [1]
Field variables without subscript, or with the subscript 0, repre-
sent the solution without dielectric. Field variables with the sub-
script 1 represent the perturbed solution for a distributed dielec-
tric. Exponential dependency of the form is then assumed

under the integrand for both the , and solution. The diver-
gence theorem is applied to the left-hand side. The resulting sur-
face integral has two surface area elements, each in opposite di-
rection, along the axis. On the right-hand side, the quantity
is replaced by the conjugate of the displacement current where
.
For uniform rods exponential dependency may also be as-
sumed under the integrand. Then the r.h.s. may be integrated
immediately. The result is [1, Eq. (9)].
(9) in [1]
The area element is the cross-sectional area bounded by the
backwall while is only that of the rods.
Consider the factor ( ) in front of the integral in the
numerator. It vanishes outside the discrete dielectric. Therefore
[1, Eq. (9)] accounts for the distribution of dielectric material in
the model. Usually the solution is not known and is approxi-
mated by the homogeneous dielectric solution. Outside the helix
mean radius this solution uses a dielectric that has been smeared
out as in [1, Eq. (10)] below,
(10) in [1]
where
number of rods;
cross-sectional area of a rod;
cross sectional area between the helix and
the backwall.
If we define as the difference in [1, (9)] with as the ho-
mogeneous dielectric solution, then this may be added to to
get a new axial propagation constant .
It should be mentioned that all perturbation solutions are not
exact. Here, we are approximating the actual fields at the di-
electric with the homogeneous dielectric solution. The boundary
conditions are not properly satisfied. Nevertheless, the approxi-
mation seems to work well. In any perturbation, the stronger the
action of the perturber, the less accurate the solution becomes.
The model may be extended to notched rods, typically used
to reduce the effect of dielectric loading. Such a geometry
is defined in Fig. 1(b). Again [1, Eq. (6)] is integrated over
a single pitch distance, which is our periodic cell. The l.h.s.
integrates as before for the case of uniform rods. The r.h.s. may
be broken up into two integrals symmetric about the notch for
, and from . For
the integral change the integration variable from to
. The integrand, exclusive of the exponential dependence,
is a function of only, so it acts as an even function in
since . Limits in front of the resultant term
may now be reversed along with a change
in sign from back to . The resulting exponentials may
now be combined as a term. When this
term is integrated about the notch the result is
[1, Eq. (15)] shown below.
(15) in [1]
The coordinate is integrated over the angular distribution of
dielectric material.
III. TAPE HELIX SOLUTION
Before applying the perturbation it is necessary to discuss the
tape helix solution. The basic form of the field solutions [7] are
listed in (A1a). The six coefficients solved for in (A1a) are listed
in (A1b). Unlike Tien’s model [8] a backwall is present, and the
exact determinantal equation is solved for numerically. This so-
lution requires the summation of all space harmonics in order to
meet the boundary conditions. Had the exact current distribution
been solved for on the helix the electric field would be zero ev-
erywhere along the tape. In principle Fourier components
of current could be solved for with appropriate boundary condi-
tionsat locationsonthehelix.Nowletthenumberofspace
harmonics go to infinity. The result is an infinite by infinite ma-
trix [9]. This problem is impossible to solve. A commonly used
approximation is to assume a current distribution on the helix.
The distribution used here becomes infinitely large in an inverse
square root manner as the tape edges are approached. Such a dis-
tribution is more likely to satisfy the boundary conditions at the
tape edge [5]. The determinantal equation will be extracted by
applying the condition along the centerline of the
helix. The approximation holds for narrow tapes. This yields the
determinantal equation in , listed in (A1c). The axial propaga-
tion constant is written as a sum of spatial harmonics, where
by Floquet’s Theorem [10]. For a given
the equation is solved iteratively for a self consistent and
where . The wave vector number is for a plane
wave. The perturbed solution uses , based upon the average di-
electric constant outside the helix. This minimized the error in
validating the basic equation (see Section IV). It should be men-
tioned,given thecurrent distributionon the helix,(A1c)is notthe
only determinantal equation that could be derived. It could have
been demanded that just the electric field along the center line of
the helix be zero. The criteria is viewed as a stronger
condition. This criteria produced answers more in line with ex-
perimental data and minimized the error in validation.
Both the perturbed and the unperturbed solutions presented
were checked numerically to meet the required boundary con-
ditions. For just the condition along the centerline the
determinantal equation would match Sensiper’s determinantal
equation [13] in the limit and for the case .
In solving for the determinantal equation 13 space harmonics
were employed. As a result both , and converged to four
decimal places.
Typically, in performing the Fourier transform of current on
the helix, the phase reference is chosen at the center of a helix

Fig. 2. Possible phase references: Centerline along the helix, between helix turns.
turn. For no rotation of the helix the current is real and at a max-
imum on the tape center. This would make the phase vary as
, where , corresponds to a point moving
along the centerline ( , ) of the tape. See
Fig. 2. In our case the phase reference has been chosen unity
between helix turns, so that the phase varies as ,
for a point constrained to , . The
in front of this type of current distribution (A1b) [12] is a result
of having effectively rotated our coordinate system 180 degrees
away from the center of the tape.
The current density in the homogeneous dielectric solution,
as the sum of its Fourier components, is shown in Fig. 3 for
a notched rod TWT. Four different windows, each apart
graph the current density. The helix is right-handed, advances
from zero to , while remains on the interval to .
For clarity the position of the helix is sketched in dotted lines.
All current on the helix is parallel to the tape. Fig. 3(a) depicts
the current density as nearly zero between helix turns. As the
tape edges are approached from within, the current rises to a
peak, and then quickly falls off at the edge. At a discontinuity
a Fourier transform converges to a mean value of the left and
right hand solutions. While the density is singular, approaching
the tape edge from within, and zero just outside, the transform
converges to a finite value less than the peak density at the edge.
Because the parallel current is assumed to have a dependency
, these windings are of conjugate phase. Their mag-
nitudes are equal and the current density is less than unity. In
Fig. 3(b), the coordinate has advanced 90 . The notch has
moved to the right, while the tape to the left shows again a dis-
tribution that is peaked at the edges. In Fig. 3(c) the origin now
lies on the center of tape. This current distribution has a normal-
ized density of one. In Fig. 3(d) has been rotated 270 . Here
the current distribution lies mostly in the right hand portion of
the graph.
IV. TAPE HELIX PERTURBATION AND VALIDATION
The integration of [1, Eq. (6)] will now be performed from
to with all space harmonics present. In a straight
forward manner is evaluated by taking
Fig. 3. Normalized current on the helix, TWT 3,

solution, as the sum of 13
Fourier components. (a) = 0, (b) = 90 , (c) = 180 , (d) = 270 .
cross products of corresponding scalar electric and magnetic
field components, both inside, and outside the helix. Simpson’s
rule, with fifty intervals both inside and outside the helix, was
used to converge the final answer to four decimal places. The

cross products can be written in such a way that all products are
real. Cross products in Appendix (A2a) employ some of the co-
efficients over imaginary quantities. However, from the r.h.s. of
the coefficient definitions in Appendix (A1b) they are seen to
be real quantities. If you wish to check your cross product ex-
pressions, for the case , they will match four times the
power flow listed in (A2b). In the denominator of (5), contri-
butions only exist where by orthogonality. On the l.h.s.
phase factors of can be further reduced to
by Floquet’s theorem. This term will be
written as where we have dropped leading
subscript 0 from the second and imply that it represents the
unperturbed solution. Similarly, on the r.h.s. of (5) an expres-
sion for is constructed from the electromagnetic scalar
quantities outside the helix, and is listed in Appendix (A2a). The
resultant equation is (5).
(5)
where
the magnitude of surface area element at , bound
by ;
cross-sectional element of the discrete
dielectric;
double summation in and , each
representing spatial harmonics ranging
from to .
Equation (5) can be validated in the following manner. Sup-
pose a dielectric shell of material surrounds the helix. Then the
perturbed solution should exactly equal the homogeneous di-
electric solution or . Before this question is addressed,
consider the approximations that have been made. The first ap-
proximation is the radial constant , was assumed the same both
inside and outside the helix. The plane wave vector , does have
two values, one inside the helix, where no dielectric resides,
and one outside the helix, based upon the average dielectric.
From the equation to calculate one for your
system, only one can be retained. The value of chosen was
for that outside the helix. For the second approximation the tape
helix solution assumes a current distribution along the helix. In
the determinantal equation, the condition is applied
along the centerline. Had Maxwell’s equations been satisfied
correctly, the quantity would have been zero everywhere
along the helix. Given these approximations (5) was verified to
within 99.9% or better. This error was computed in the typical
range of a TWT defined by the equation .
The angular integration over the dielectric in (5) proceeds as
follows. The dielectric is virtually extended into the helix mean
radius, and the dielectric everywhere is reduced to account for
the subsequent increase in volume. The dielectric is reduced by
the following formula
Vol 1
Vol 1 Vol 2
(6)
where
Vol 1 volume of a rod over a pitch distance (less any notches
if present—see next section);
Vol 2 virtual volume by extension of the rod through the
helix outer radius to the helix mean radius.
This is based upon a weighted integration of volumes with di-
electric strength, divided by the total volume. Hereafter, the re-
duced quantity will be substituted for of the rods.
Let be the half angle subtended by the width of a sup-
port rod. Let there be rods, spaced uniformly around the helix.
From the integration, there results a term:
, where , for equally spaced
support rods. For the quantity not equal to ,
etc. there results phasors, equally spaced in in the com-
plex plane. Their sum is zero. However, for equal to
etc., all phasors lie on the real positive axis
and must be accounted for. Therefore, this sum on the exponen-
tial acts as , where is the Kro-
necker Delta function. The resultant expression for the phase
shift in for uniform rods is in (7), shown at the bottom of
the page, where the magnitude of is the surface area element
at , bound by , is the number of rods, and
is the half angle subtended by a rod.For the case , the
expression is replaced by its limiting
value .
V. PERTURBATION TAPE HELIX SOLUTION — UNIFORM RODS
The four cases presented from the previous paper will be re-
examined. The perturbed tape helix solution , along with the
homogeneous dielectric solution , are shown in the first case
for a uniform-rod-helix, UBB1, of Fig. 4. For uniform support
rods the perturbation does not significantly alter the dispersion
of the homogeneous dielectric tape helix solution. The perturbed
answer lies below the unperturbed solution. With the errors in
the new format: the error for SSG is 5.33%, the error for the per-
turbed sheath is 1.024%, the error for the tape helix solution is
(7)

Fig. 4. Graph of phase velocity vs. frequency for UBB1 helix with uniform
rods. RHID = helix inner radius, RHOD = helix outer radius, RB = backwall
radius, TWID = tape width, WB = rod width. Tape is the homogeneous dielectric
solution.
Fig. 5. Graph of phase velocity vs. frequency for I–J band input helix.
Notation see Fig. 4.
2.74%, and the error for the perturbed tape helix is 1.017%. The
perturbed tape helix has the lowest error compared to the tape
helix, the perturbed sheath, or SSG.
Fig. 5 depicts the result of the perturbation for an – Band
input helix, again with uniform rods. Stated in the new format
the error for SSG was 4.52%, the error for the perturbed sheath
was 1.82%. The error for the tape solution is 2.00%, and the
error for the perturbed tape is 1.68%. The perturbed tape helix
has the lowest error. Again, the perturbed answer lies below the
unperturbed solution, the basic shape of dispersion has not been
altered.
Coupling impedances for the tape helix solution may now
be calculated from (8) through (10) below. Here, all solutions
represent the perturbed solution.
(8)
(9)
(10)
While only the zeroeth component of and the electric field
are used in the definition of coupling impedance, the power is
Fig. 6. Graph of coupling impedance versus frequency for UBB1 helix. At low
frequencies the perturbed solution has overlaid onto the measured data making
them indistinguishable.
written as a sum of spatial harmonics, see (A2b). Theory and
experiment are compared in Fig. 6. Experimental values are de-
rived from (transmission) measurements of phase change
for a stick rod on axis, placed all the way through the helix (be-
lieved to be the method of Lagerstrom). In this analysis there
was no smoothing of data because it did not necessarily follow
a second order power series expansion. It also would not have
made sense dividing the sum of the squares by some average
coupling impedance to arrive at an error. Instead just the sum of
the squares will be listed. Calculated coupling impedances for
SSG had a sum of the squares of 4887, calculated coupling im-
pedances for the perturbed sheath have had a sum of the squares
of 1298. Calculated coupling impedances for the tape helix have
a had a sum of the squares 210, and calculated coupling imped-
ances for the perturbed tape helix have a sum of the squares
of 28.8. The perturbed tape solution has the lowest sum of the
squares error. With some statistical variation errors ran from
1 to 5%, 4 GHz freq. 14 GHz and 5% to 17%, 14 GHz
freq. 18 GHz. The largest errors occurred with the per-
turbed tape helix calculations lower than those measured. Ref-
erence [13] reports discrepancies between Zcoup measured with
perturbing rods, using the method of Lagerstrom, and MAFIA
calculations. They explain approximations used in the theory
which contribute to the experimental result. Three graphs are
presented which show that measurement errors increase with
frequency. Their calculated results are similarly lower than the
measured results.
VI. PERTURBED TAPE HELIX SOLUTION FOR NOTCHED RODS
For notched rods three times the integration of a single rod
is slightly different than integrating over all three rods. The in-
tegration will be performed exactly over the three rods. Even
though this is a perturbation, the resultant expression on the
r.h.s. of (11), shown at the bottom of the next page, is shown
to be real, and can exactly balance the l.h.s. Let there be
rods spaced uniformly in around the helix (refer to Fig. 7). Pa-
rameters that typify a notched rod are listed in Fig. 1(a) and (b).
Within the limits , for a rotation from

Fig. 7. Geometry for integrating over the three rods.
rod #1 to rod #2 the notch advances a distance . If the corner
of the notch extends past , the remaining portion reappears
at . The rod is divided into two halves, respectively. Each
half, with indices lie to the left and right of the notch
apex respectively. When (7) is modified to include the notch ge-
ometry, and the integration performed over and the resultant
equation is shown in (11) below where the magnitude of is
a surface area element bound by , half angle
subtended by a rod, is the rod number, and are the
left and right-hand segments about the notch apex. Again, for
the case , the expression is
replaced by its limiting value . The integral of is
evaluated using 50 Simpson integration intervals.
The below iterative equation can be balanced in both mag-
nitude and phase since both the l.h.s. and the r.h.s. are real. To
see that the r.h.s. is real, rearrange the rod pieces as in Fig. 8.
The integration of rod 1 about the symmetrical notch produces
a real sine term. When differential elements from rod 2, with
a phase of and a net phase factor from the
integration of , combine with negative
differential elements from rod 3, with a phase
, and a net phase of , the resultant
term is real. A similar argument may be made for combining
the positive half of rod 3 and the negative half of rod 2. Thus the
r.h.s. of (11) shall balance the l.h.s. of (11) in both magnitude
and phase.
Fig. 9 compares normalized phase velocity plots for the
notched rod TWT 3 derived both from theory and measurement.
Fig. 8. Geometry for showing E 1 E dx is real.
Fig. 9. Graph of phase velocity versus frequency for TWT 3 with notched
rods. RHID = helix inner radius, RHOD = helix outer radius, RB = backwall
radius, TWID = tape width. Notch dimensions see Fig. 1(a), Fig. 1(b). Tape is
the homogeneous dielectric solution.
In these notched rod cases, SSG’s input could not adequately
describe the geometry. SSG phase velocities are consistent with
those presented in the previous paper, they use a dielectric con-
stant averaged in both and . The rod height is the difference
between the helix outer radius and the backwall radius. The
volume in a pitch distance is a multiplication of the height, the
width, and the pitch distance, less the volume of the notch. Then
the effective cross sectional area calculated by dividing through
by the pitch distance. The dielectric is then smeared by [1, Eq.
(11)

Fig. 10. Graph of phase velocity versus frequency for I–J band output helix.
Notation see Fig. 9.
(10)] where is the effective cross sectional area calculated
above. Inthe next two cases the dielectric in[1]hasbeen changed
from 6.75 to 6.7 because that is in the middle of the acceptable
range. The dielectric constant used is for dry-pressed-fired BeO
powder [14].For thiscase, inthe last papertherewasan omission
of two data points in the least squares calculation. The corrected
errors will be given in the new format, with the new permittivity.
The average error for SSG is 3.12%. The average error for the
sheath helix is 5.46%. The average error for the perturbed sheath
2.46%. In this paper, the results are as follows. The average error
for the tape helix is 6.78%, the average error for the perturbed
tape helix is 0.57%. The perturbed tape solution has the lowest
average sum of the squares. As a result of notching the rods
the perturbed phase velocity lies above the unperturbed phase
velocity. It has a flattened dispersion.
Fig. 10 depicts results for an – Band notched rod TWT. In
the previous paper, the last data point for the perturbed sheath
helix was misplaced 0.5% in error, making the errors even lower
than stated. Stated in the new format the average error for SSG
was 1.90%, for the sheath helix 2.35%, and for the perturbed
sheath was 0.95%. For the tape helix the average error is 2.21%,
for the perturbed tape helix the average error is 0.55%. The per-
turbed tape helix has the lowest average sum of the squares.
Again, the perturbed phase velocity lies above the unperturbed
phase velocity, with a flattened dispersion.
VII. CONCLUSION
The perturbation provides a mathematical tool for the anal-
ysis of TWT dispersion and coupling impedances. The physical
reason the technique works better than previous models is that
rather than average the dielectric the technique accounts for the
distribution of dielectric material. Certain approximations are
made, i.e., an infinitesimally thin tape, is constant both inside
and outside the helix, and an assumed peaked current distribu-
tion. With the extension of the model to include the tape helix the
current is correctly distributed in space. In all cases presented
the perturbed tape helix had a lower error in phase velocity than
the unperturbed solution or SSG. In all cases presented the per-
turbed tape helix had a lower error in phase velocity than the
perturbed sheath helix of the previous paper. For uniform rods
the perturbation does not significantly alter the shape of the dis-
persion. The perturbed solution is lower in phase velocity, with a
lower least square error. However, for notched dielectric rods the
perturbation raises the phase velocity, with a lower least square
error. Here the dispersion is flattened, in agreement with exper-
iment. A rationale for the shape of the notched-rod-dispersion
was given in the previous paper. This prediction is again con-
firmed numerically and by experiment.
For uniform dielectric support rods phase velocities can be
calculated where the error 1.68%. NRL’s program SSG was
5.33% by comparison. For notched dielectric support rods phase
velocities can be calculated where the error is 0.57%. SSG
was 3.12% by comparison (the average dielectric had to speci-
fied via special input). The average error in phase velocity for
four cases was 0.95%. The perturbed tape helix has the lowest
sum of the squares error in coupling impedance over all other
methods of computation. Coupling Impedance errors progres-
sively increased from 1 to 17% over a range of 4 to 18 GHz.
The errors are believed to be largely due to approximations in
the theory inherent with the experimental result. The perturbed
tape-coupling-impedance results are equal or lower than those
measured and a concurring reference has been provided.
APPENDIX
These are the tape helix solutions for the th mode:
In (A1a)–(A1c) coefficients for the unperturbed solution have
the subscript 0, let and .
In (A1a)–(A1c) coefficients for the homogeneous dielectric
solution have the subscript 1, let , where ,
and .
Outside:
Inside:

(A1a)
where:
These are the th mode coefficients used in the field solution
of (A1a).
Let and be the th components of current in the helix
along the and direction.
, , in (A1b), shown at the
bottom of the page, without subscript is .
Fourier Transform of the Current on the Helix
Assume the current distribution on the helix becomes infin-
itely large in an inverse square roots manner as the tape edge
is approached. Let the total current on the helix be the same
as if it were a uniform unit distribution. Then the normalized
current distribution on the interval
would be
where is the tape width.
Let the phase and magnitude of the current on the helix vary
as for a point moving along the center line. The parallel
component of current is , where is constrained
to the path .
The parallel current can be represented as the sum of its
Fourier components
Multiply both sides above by
Integrate both sides from to . The limits on the
l.h.s. represent the range where the current lay on the helix.
With a change in variable the r.h.s. integral may be integrated
from 0 to with a phase factor out in front containing ( ).
With another change in variable it may be integrated from 0 to
to yield . After summing on the delta function only
terms where remain, so the r.h.s. . With a change
in variable the l.h.s. may be rewritten as
(A1b)

Replacing in front of the term in the exponential with
and simplifying
l.h.s.
The odd part integrates to zero. The remaining can be integrated
with the relation
where is an ordinary Bessel function.
Combining everything and solving for results in
which is the same transform as in [12] except for the difference
in argument because we have assumed the phase of the current
varies as for a point moving alone the center line.
If the origin is now between helix turns this corresponds to a
rotation in . Phase factors of become . The th
parallel transform becomes
The th Fourier component along the and direction can be
written as
For the condition along the center line of the helix,
the following determinantal equation exists as in (A1c), shown
at the bottom of the page.
The quantity
(A2a)
is evaluated by Simpson’s rule where
where above by orthogonality from the integration in .
The quantity
(A1c)

is evaluated by Simpson’s rule. In the definition of the terms
substitute for the inside coefficients
where above by orthogonality from the integration in .
The quantity
is evaluated by Simpson’s rule where
Below all coefficients represent the perturbed solution.
(A2b)
where
From the identity
it can be integrated with 5.54 2. Gradshteyn [15].
Similarly integrated using 5.54 2. Gradsheteyn [15].
is solved with the linear combination
and noting that it satisfies recursion relations
for both an . Then with the relation
substitute, group like terms in powers of and , and finally
with and the substitution of and above, solve
for .

2001 IEEE

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