Dispersion of Electromagnetic Waves Guided by an Open Tape Helix I

TL;DR: The dispersion equation for electromagnetic waves guided by an open tape helix for the standard model of an inflitesimally thin and perfectly conducting tape is derived from an exact solution of a homogeneous boundary value problem for Maxwell's equations as mentioned in this paper.

Abstract: The dispersion equation for electromagnetic waves guided by an open tape helix for the standard model of an inflnitesimally thin and perfectly conducting tape is derived from an exact solution of a homogeneous boundary value problem for Maxwell's equations. A numerical analysis of the dispersion equation reveals that the tape current density component perpendicular to the winding direction does not afiect the dispersion characteristic to any signiflcant extent. In fact, there is a signiflcant deviation from the dominant-mode sheath- helix dispersion curve only in the third allowed region and towards the end of the second allowed region. It may be concluded that the anisotropically conducting model of the tape helix that neglects the above transverse-current contribution is a good approximation to the isotropically conducting model that takes into account this contribution except at high frequencies even for wide tapes.

Figures

Figure 5. Blown-up portion from Fig. 2.

Figure 1. Geometrical relations in a developed tape helix.

Figure 4. Dispersion characteristic of tape helix for truncation order N = 4.

Figure 3. Dispersion characteristic of tape helix for truncation order N = 3.

Figure 2. Dispersion characteristic of tape helix for truncation order N = 2.

Full text

Progress In Electromagnetics Research B, Vol. 16, 311–331, 2009
DISPERSION OF ELECTROMAGNETIC WAVES GUIDED
BY AN OPEN TAPE HELIX I
N. Kalyanasundaram and G. Naveen Babu
Jaypee Institute of Information Technology University
A-10, Sector-62, Noida 201307, India
Abstract—The dispersion equation for free electromagnetic waves
guided by an anisotropically conducting open tape helix is derived
from the exact solution of a homogenous boundary value problem for
Maxwell’s equations without invoking any apriori assumption about
the tape-current distribution. A numerical solution of the dispersion
equation for a set of typical parameter values reveals that the tape-helix
dispersion curve is virtually indistinguishable from the corresponding
dominant-mode sheath helix dispersion curve except within the tape-
helix forbidden regions.
1. INTRODUCTION
It is now well established that the tape-helix model gives a better
approximation to the slow-wave structure of a TWT amplifier than
the sheath-helix model over the entire frequency range of operation.
Moreover, there is no possibility of simulating the input and the
output ports of the amplifier with the sheath-helix model. Thus, a
field-theoretical analysis of the dispersion characteristics of the tape-
helix slow-wave structure will be of immense interest to the TWT
community.
An indepth study of electromagnetic wave propagation on helical
conductors has been performed by Samuel Sensiper way back in
1952 [1]. He has outlined essentially two approaches for analyzing
the tape-helix problem. Using the first approach, he has demonstrated
the feasibility of an exact solution for the tape helix; unfortunately,
he chose to eschew this approach on the ground that “it is of no
practical use for obtaining useful numerical results or for determining
the detailed character of the solutions” preferring instead a second
Corresponding author: N. Kalyanasundaram (n.kalyanasundaram@jiit.ac.in).

312 Kalyanasundaram and Naveen Babu
approach that involved an apriori assumption about the current
distribution on the tape as a result of which it was possible to
satisfy the boundary conditions on the tangential electric field only
approximately. Nevertheless, it is this latter approach that has been
endorsed by the majority of later generations of research workers
in the TWT area mainly because of its tractability. All variants
of this second simplified approach are characterized invariably by a
common assumption, namely, that the tape current density component
perpendicular to the winding direction may b e neglected without much
error. A notable exception to the practice of satisfying the tangential
electric field boundary condition only along the centerline of the tape
is the variational formulation developed by Chodorow and Chu [2] for
cross-wound twin helices wherein the error in satisfying the tangential
electric field boundary condition is minimized for an assumed tape-
current distribution by making the average error equal to zero. The
rationale behind the approach of Chodorow and Chu for single-wire
helix has been outlined by Watkins in his book [3] assuming that (i)
the tape current flows only along the winding direction, (ii) it does
not vary in phase or amplitude over the width of the tape, and (iii)
its phase variation is according to β
0
z for z corresp onding to a point
moving along the centerline of the tape.
The method adopted for the solution of the cold-wave problem for
the tape helix in this paper derives from the following fact: If one is
willing to neglect in any case the contribution of the perpendicularly
directed current density component on the tape then there is neither a
need for any apriori assumption regarding the tape-current distribution
nor is there any difficulty in satisfying the tangential electric field
boundary condition over the entire width of the tape. The hypothesis
that the transverse component of the tape-current density is zero
may be incorporated explicitly into the model by assuming that
the tape helix is made out of an anisotropic material exhibiting
infinite conductivity in the winding direction but zero conductivity
in the orthogonal direction. This anisotropically conducting model
for the tape helix leads to considerable simplification of the solution
of the boundary value problem for the guided modes supported by
an open helical structure. First of all, the boundary conditions give
rise to only a single infinite set of linear homogeneous equations
for determining the modal amplitudes of the tape-current density.
Moreover, the approximate secular equation, for determining guided-
mode propagation constant, resulting from setting the determinant of
the coefficient matrix, corresponding to a symmetric truncation of the
infinite set of equations, to zero will be in the form of a series whose
terms decrease rapidly in magnitude with the order of truncation.

Progress In Electromagnetics Research B, Vol. 16, 2009 313
This last feature of the truncated secular equation is quite attractive
from a computational point of view since it then becomes possible
to secure a fairly accurate estimate of the dispersion characteristic
with a reasonably low order of truncation. The entire analysis is of
course based on the premise that the transverse component of the
tape-current density does not have any significant effect on the value
of the propagation constant even for tapes which are not narrow.
2. DERIVATION OF THE DISPERSION EQUATION
A tape helix of infinite length, constant pitch, constant tape width and
infinitesimal thickness surrounded by free space is considered. The
helix is assumed to b e made of an anisotropic material exhibiting
infinite conductivity in the direction of the tape winding but zero
conductivity in the orthogonal direction.
Since the formulation of the cold-wave problem for the tape
helix has become quite standard, we make use of the notation and
terminology employed in one of the conventional treatments following
Sensiper [1] of the problem as presented in [4] except that we use w,
instead of δ, to denote the width of the tape in the axial direction.
Accordingly, we take the axis of the helix along the z-co ordinate of a
cylindrical coordinate system (ρ, ϕ, z). The radius of the helix is a, the
pitch is p, and cot ψ = 2πa/p (Fig. 1).
Periodicity of the infinite helical structure in the z and ϕ
variables permits an expansion of the phasor representation F (ρ, ϕ, z)
(corresponding to a radian frequency ω) of any field component in a
Figure 1. Geometrical relations in a developed tape helix.

314 Kalyanasundaram and Naveen Babu
double infinite series
F (ρ, φ, z) = e
−jβ
0
z
∞
X
υ
=
−∞
∞
X
n
=
−∞
F
υn
(ρ) e
jυϕ
e
−j2πnz/p
(1)
in view of Floquet’s theorem [4, 5] where β
0
= β
0
(ω) is the guided
wave propagation constant at the radian frequency ω. Moreover, the
invariance of the infinite helical structure under a translation ∆z in the
axial direction and a simultaneous rotation by 2π∆z/p around the axis
imply that F
υn
(ρ) are non zero only if υ = n. Thus, the double-series
expansion (1) for any field component degenerates to the single-series
expansion
F (ρ, φ, z) =
∞
X
n=−∞
F
n
(ρ) e
j(nϕ−β
n
z)
(2)
where
β
n
= β
0
+ 2πn/p (3)
Each term in the series-expansion has to satisfy the Helmholtz equation
in cylindrical coordinates. Hence, the Borgnis potentials [4] for guided-
wave solutions, at the radian frequency ω, may be assumed in the form
U =
∞
X
n=−∞
[A
n
+ (C
n
− A
n
)H(ρ − a)] G
n
(τ
n
ρ)e
j(nϕ−β
n
z)
, (4a)
V =
∞
X
n=−∞
[B
n
+ (D
n
− B
n
)H(ρ − a)] G
n
(τ
n
ρ)e
j(nϕ−β
n
z)
, (4b)
where
G
n
(τ
n
ρ) 4 I
n
(τ
n
ρ) for 0 ≤ ρ < a,
K
n
(τ
n
ρ) for ρ > a,
I
n
and K
n
are nth order modified Bessel functions of the first and
second kind respectively, and H is the Heaviside function and where
β
2
n
− τ
2
n
= k
2
0
4 ω
2
µε (5)
with µ the permeability and ² the permittivity of the ambient space.
The explicit expressions for the field components become [4]
E
z
=
∂
2
U
∂z
2
+ k
2
0
U =
∞
X
n=−∞
−τ
2
n
Λ
n
(ρ)G
n
(τ
n
ρ)e
j(nϕ−β
n
z)
(6a)

Progress In Electromagnetics Research B, Vol. 16, 2009 315
E
ρ
=
∂
2
U
∂ρ∂z
−
jωµ
ρ
∂V
∂ϕ
=
∞
X
n=−∞
£
−jβ
n
τ
n
Λ
n
(ρ)G
0
n
(τ
n
ρ)+ωµnΩ
n
(ρ)G
n
(τ
n
ρ)/ρ
¤
e
j(nϕ−β
n
z)
(6b)
E
φ
=
∂
2
U
∂ϕ∂z
−
jωµ
ρ
∂V
∂ρ
=
∞
X
n=−∞
£
nβ
n
Λ
n
(ρ)G
n
(τ
n
ρ)+jωµτ
n
Ω
n
(ρ)G
0
n
(τ
n
ρ)
¤
e
j(nϕ−β
n
z)
(6c)
H
z
=
∂
2
V
∂z
2
+ k
2
0
V =
∞
X
n=−∞
−τ
2
n
Ω
n
(ρ)G
n
(τ
n
ρ)e
j(nϕ−β
n
z)
(7a)
H
ρ
=
∂
2
V
∂ρ∂z
+
jω²
ρ
∂U
∂ϕ
=
∞
X
n=−∞
£
ωεnΛ
n
(ρ)G
n
(τ
n
ρ)/ρ + jβ
n
τ
n
Ω
n
(ρ)G
0
n
(τ
n
ρ)
¤
e
j(nϕ−β
n
z)
(7b)
H
φ
=
∂
2
V
∂ϕ∂z
−
jω²
ρ
∂U
∂ρ
=
∞
X
n=−∞
£
−jωετ
n
Λ
n
(ρ)G
0
n
(τ
n
ρ)+nβ
n
Ω
n
(ρ)G
n
(τ
n
ρ)/ρ
¤
e
j(nϕ−β
n
z)
(7c)
In the expressions (6) and (7) for the field components, G
0
n
denotes the
derivative of the function G
n
with respect to its argument and
Λ
n
(ρ)∆A
n
+ (C
n
− A
n
)H(ρ − a) (8a)
Ω
n
(ρ)∆B
n
+ (D
n
− B
n
)H(ρ − a) (8b)
and where A
n
, B
n
, C
n
and D
n
, n ∈ Z, are (complex) constants to be
determined by the tape helix boundary conditions.
In order for (6) and (7) to correspond to guided waves (as opposed
to radiation modes) supported by the open helix, we need τ
n
> 0 for
all n ∈ Z, that is
| β
n
|> k
0
for all n ∈ Z (9)
which, for n = 0, becomes | β
0
|> k
0
i.e., only slow guided waves are
supported by the helical structure. The condition (9) for other values
of n translates, in view of (5), to
| β
0
+ n cot ψ/a |> k
0

Frequently asked questions

Q1. What have the authors contributed in "Dispersion of electromagnetic waves guided by an open tape helix i" ?

Kalyanasundaram et al. this paper derived the dispersion equation for free electromagnetic waves guided by an anisotropic conducting open tape helix without invoking any apriori assumption about the tape-current distribution. 

Q2. What are the future works mentioned in the paper "Dispersion of electromagnetic waves guided by an open tape helix i" ?

Based on the outcome of such a study ( which is currently under progress ), it is proposed to extend the method adopted for the derivation of the tape-helix dispersion equation to a full field analysis of the practically important case of a dielectric-loaded helical slow-wave structure enclosed in a coaxial metal cylindrical shell and supported by azimuthally symmetrically placed dielectric rods. The effect of the dielectric support rods will have to be modeled by a homogeneous dielectric the effective dielectric constant of which can be determined in terms of the geometric arrangement of the support rods and the actual dielectric constant of the rod material. 

Q3. What is the way to solve the tape helix problem?

The hypothesis that the transverse component of the tape-current density is zero may be incorporated explicitly into the model by assuming that the tape helix is made out of an anisotropic material exhibiting infinite conductivity in the winding direction but zero conductivity in the orthogonal direction. 

Q4. What is the helix of the tape?

A tape helix of infinite length, constant pitch, constant tape width and infinitesimal thickness surrounded by free space is considered. 

Q5. What is the effect of the dielectric support rods?

The effect of the dielectric support rods will have to be modeled by a homogeneous dielectric the effective dielectric constant of which can be determined in terms of the geometric arrangement of the support rods and the actual dielectric constant of the rod material. 

Q6. What is the way to estimate the tape-helix dispersion characteristic?

Since the dispersion curves for N = 2, 3 and 4 are virtually indistinguishable from one another, a truncation order as low as 2 is adequate to deliver an accurate estimate of the tape-helix dispersion characteristic (at least within the validity limits of the assumed model for the parameter values used in the numerical computations). 

Q7. What is the symmetric dependence of the function FN on i?

The symmetric dependence of the function FN on σ±i, 1 ≤ i ≤ N , implies that if β0a satisfies the dispersion equation (47) for a given k0a so does −β0a for the same k0a. 

Q8. What is the reason behind the tape-helix approach?

The rationale behind the approach of Chodorow and Chu for single-wire helix has been outlined by Watkins in his book [3] assuming that (i) the tape current flows only along the winding direction, (ii) it does not vary in phase or amplitude over the width of the tape, and (iii) its phase variation is according to β0z for z corresponding to a point moving along the centerline of the tape. 

Q9. What is the effect of the nth space harmonic on the tape-helix dis?

It is thus seen that the mode constant τ−na of the −nth space-harmonic contribution to the total field becomes imaginary in the nth forbidden region, and that the resultant Poynting vector acquires a small radial component in order to account for the radiation of power from the −nth space harmonic. 

Q10. What is the dispersion equation for k0a?

Tape-helix dispersion curves for the truncation orders of 0 and 1 are not shown because the iterations for these two cases could not be continued (to convergence) beyond β0a = 4.64 to yield real values for k0a(β0a) in the complement of the forbidden regions. 

Q11. what is the boundary condition of the nth sheath helix?

In this caseαkq → ∞∑n=−∞ σnδknδqn = σkδkqand the dispersion equation (26) degenerates to∞∏k=−∞ σk = 0that is, σn = 0, n = 0,±1,±2, . . . (28)The relation σn = 0 may be recognized as the dispersion equation for the nth mode of a sheath helix made of the same anisotropically conducting material as the tape helix provided βn, as a whole, is interpreted as the propagation constant of the nth sheath-helix mode. 

Q12. What is the truncation order of the sheath-helix model?

5. It may be seen from Fig. 5 that the phase speed for the tape-helix model is lower than that for the sheath-helix model for the same value of ω in the complement of the forbidden regions.