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Progress In Electromagnetics Research B, Vol. 16, 311331, 2009

DISPERSION OF ELECTROMAGNETIC WAVES GUIDEDBY AN OPEN TAPE HELIX I

N. Kalyanasundaram and G. Naveen Babu

Jaypee Institute of Information Technology UniversityA-10, Sector-62, Noida 201307, India

AbstractThe dispersion equation for free electromagnetic wavesguided by an anisotropically conducting open tape helix is derivedfrom the exact solution of a homogenous boundary value problem forMaxwells equations without invoking any apriori assumption aboutthe tape-current distribution. A numerical solution of the dispersionequation for a set of typical parameter values reveals that the tape-helixdispersion curve is virtually indistinguishable from the correspondingdominant-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 betterapproximation to the slow-wave structure of a TWT amplifier thanthe sheath-helix model over the entire frequency range of operation.Moreover, there is no possibility of simulating the input and theoutput 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 TWTcommunity.

An indepth study of electromagnetic wave propagation on helicalconductors has been performed by Samuel Sensiper way back in1952 [1]. He has outlined essentially two approaches for analyzingthe tape-helix problem. Using the first approach, he has demonstratedthe 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 determiningthe detailed character of the solutions preferring instead a second

Corresponding author: N. Kalyanasundaram ([email protected]).

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312 Kalyanasundaram and Naveen Babu

approach that involved an apriori assumption about the currentdistribution on the tape as a result of which it was possible to

satisfy the boundary conditions on the tangential electric field onlyapproximately. Nevertheless, it is this latter approach that has beenendorsed by the majority of later generations of research workersin the TWT area mainly because of its tractability. All variantsof this second simplified approach are characterized invariably by acommon assumption, namely, that the tape current density componentperpendicular to the winding direction may be neglected without mucherror. A notable exception to the practice of satisfying the tangentialelectric field boundary condition only along the centerline of the tapeis the variational formulation developed by Chodorow and Chu [2] forcross-wound twin helices wherein the error in satisfying the tangentialelectric field boundary condition is minimized for an assumed tape-current distribution by making the average error equal to zero. Therationale behind the approach of Chodorow and Chu for single-wirehelix has been outlined by Watkins in his book [3] assuming that (i)the tape current flows only along the winding direction, (ii) it doesnot vary in phase or amplitude over the width of the tape, and (iii)its phase variation is according to 0z for z corresponding to a pointmoving along the centerline of the tape.

The method adopted for the solution of the cold-wave problem forthe tape helix in this paper derives from the following fact: If one iswilling to neglect in any case the contribution of the perpendicularlydirected current density component on the tape then there is neither aneed for any apriori assumption regarding the tape-current distributionnor is there any difficulty in satisfying the tangential electric fieldboundary condition over the entire width of the tape. The hypothesisthat the transverse component of the tape-current density is zeromay be incorporated explicitly into the model by assuming thatthe tape helix is made out of an anisotropic material exhibitinginfinite conductivity in the winding direction but zero conductivityin the orthogonal direction. This anisotropically conducting modelfor the tape helix leads to considerable simplification of the solutionof the boundary value problem for the guided modes supported byan open helical structure. First of all, the boundary conditions giverise to only a single infinite set of linear homogeneous equationsfor 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 theinfinite set of equations, to zero will be in the form of a series whoseterms decrease rapidly in magnitude with the order of truncation.

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Progress In Electromagnetics Research B, Vol. 16, 2009 313

This last feature of the truncated secular equation is quite attractivefrom a computational point of view since it then becomes possible

to secure a fairly accurate estimate of the dispersion characteristicwith a reasonably low order of truncation. The entire analysis is ofcourse based on the premise that the transverse component of thetape-current density does not have any significant effect on the valueof 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. Thehelix is assumed to be made of an anisotropic material exhibitinginfinite conductivity in the direction of the tape winding but zeroconductivity in the orthogonal direction.

Since the formulation of the cold-wave problem for the tapehelix has become quite standard, we make use of the notation andterminology employed in one of the conventional treatments followingSensiper [1] of the problem as presented in [4] except that we use w,instead of , to denote the width of the tape in the axial direction.Accordingly, we take the axis of the helix along the z-coordinate of a

cylindrical coordinate system (,,z). The radius of the helix is a, thepitch is p, and cot = 2a/p (Fig. 1).

Periodicity of the infinite helical structure in the z and variables permits an expansion of the phasor representation F(,,z)(corresponding to a radian frequency ) of any field component in a

Figure 1. Geometrical relations in a developed tape helix.

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double infinite series

F (,,z) = ej0z

=

n=

Fn () ejej2nz/p (1)

in view of Floquets theorem [4, 5] where 0 = 0() is the guidedwave propagation constant at the radian frequency . Moreover, theinvariance of the infinite helical structure under a translation z in theaxial direction and a simultaneous rotation by 2z/p around the axisimply that Fn() are non zero only if = n. Thus, the double-seriesexpansion (1) for any field component degenerates to the single-seriesexpansion

F(,,z) =

n=

Fn () ej(nnz) (2)

wheren = 0 + 2n/p (3)

Each term in the series-expansion has to satisfy the Helmholtz equationin cylindrical coordinates. Hence, the Borgnis potentials [4] for guided-wave solutions, at the radian frequency , may be assumed in the form

U =

n=

[An + (Cn An)H( a)] Gn(n)ej(nnz), (4a)

V =

n=

[Bn + (Dn Bn)H( a)] Gn(n)ej(nnz), (4b)

where

Gn(n) In(n) for 0 < a,Kn(n) for > a,

In and Kn are nth order modified Bessel functions of the first andsecond kind respectively, and H is the Heaviside function and where

2n 2n = k20 2 (5)

with the permeability and the permittivity of the ambient space.The explicit expressions for the field components become [4]

Ez =2U

z2+ k20U =

n=

2nn()Gn(n)ej(nnz) (6a)

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E =2U

z j

V

=

n=

jnnn()Gn(n)+nn()Gn(n)/ej(nnz) (6b)E =

2U

z j

V

=

n=

nnn()Gn(n)+jnn()G

n(n)

ej(nnz) (6c)

Hz =2V

z2+ k20V =

n=

2nn()Gn(n)ej(nnz) (7a)

H =2V

z+

j

U

=

n=

nn()Gn(n)/ +jnnn()G

n(n)

ej(nnz) (7b)

H = 2

Vz

j

U

=

n=

jnn()Gn(n)+nnn()Gn(n)/ ej(nnz) (7c)In the expressions (6) and (7) for the field components, Gn denotes thederivative of the function Gn with respect to its argument and

n()An + (Cn

An)H(

a) (8a)

n()Bn + (Dn Bn)H( a) (8b)and where An, Bn, Cn and Dn, n Z, are (complex) constants to bedetermined by the tape helix boundary conditions.

In order for (6) and (7) to correspond to guided waves (as opposedto radiation modes) supported by the open helix, we need n > 0 forall n Z, that is

| n |> k0 for all n Z (9)which, for n = 0, becomes

|0

|> k0 i.e., only slow guided waves are

supported by the helical structure. The condition (9) for other valuesof n translates, in view of (5), to

| 0 + n cot /a |> k0

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which is equivalent to either

0 > k0 n cot /a (10a)or 0 < k0 n cot /a (10b)

If 0 > k0, the inequality (10a) is automatically satisfied for all n 0.We then need the condition (10b) to be satisfied for the remainingvalues of n, i.e., for n 1, that is,

| n | cot /a < (0 + k0) for all n 1 (11)In order for the inequality (11) to hold for all n

1, it is sufficient

to have cot /a < (0 + k0)or

cot /a > (0 + k0) > 2k0which implies the cut-off condition

k0a < (1/2) cot (12)

If, on the other hand, 0 k0 0 for all n 1for which it is sufficient to have

cot /a > (k0 0) > 2k0that is, k0 < (1/2)cot which is again the cut-off condition (12). Thus,

pure guided modes do not exist on an open tape helix for a (radian)frequency (c/2a)cot , where c = 1/ is the speed of light inthe ambient space.

The boundary conditions at = a for the anisotropicallyconducting model of the tape helix are

(i) The tangential electric field is continuous for all values of andz.

(ii) The tangential component of the magnetic field parallel to thewinding direction is continuous for all and z.

(iii) The discontinuity in the tangential component of the magneticfield perpendicular to the winding direction is equal to the surfacecurrent density on the tape.

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(iv) The tangential component of the electric field parallel to thewinding direction is zero on the tape surface.

Thus

Ez(a, , z) Ez(a+, , z) = 0 (13a)E(a, , z) E(a+, , z) = 0 (13b)[Hz(a, , z) Hz(a+, , z)]sin + [H(a, , z) H(a+, , z)]cos = 0 (13c)

[Hz(a, , z) Hz(a+, , z)]cos [H(a, , z) H(a+, , z)]sin = Js(, z) (13d)

[Ez(a,,z)sin E(a,,z)cos ]g(, z) = 0 (13e)where Js(, z) is the surface current density component supportedby helix and the function g(, z), defined in terms of the indicatorfunctions of the disjoint (for the same value of ) intervals[(l + /2)p w/2, (l + /2)p + w/2] , l Z, by

g(, z)

l=1[(l+/2)pw/2,(l+/2)p+w/2](z)

will be equal to 1 on the tape surface and 0 elsewhere on the surfaceof the (infinite) cylinder = a. In (13a)(13d),

F(a, , z) lim0

F(a ,,z)

for any field component F(,,z). In the case of an helical conductorof nonzero thickness, = a and = a+ correspond respectively tothe inner and the outer surface of the tape. The functional form of

the surface current density component Js(, z), which is confined onlyto the two-dimensional region occupied by the tape-helix material, isrestricted by the periodicity and the symmetry conditions imposed bythe helix geometry. Accordingly, Js(, z) admits the representation

Js(,,z) =

n=

Jnej(nnz)

g(, z) (14)

where the (complex) constants Jn, n Z/{0}, in the expansion (14)are to be determined in terms of the arbitrary constant J0 by theboundary conditions. An explicit representation for the surface currentdensity on the helix as in (14) does not appear to have been made useof in any of the previous analysis of the cold wave problem for the tape

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helix. In the terms of the new (independent) variables and , definedby

p2 + (2a)2/2, z p/2on the cylindrical surface = a containing tape helix, we have

ej(nnz) = ej0 sin ejn

since z = + sin and = 2 sin /p. Thus, the surface currentdensity component Js(, z), when expressed in terms of the variables and , becomes

Js(, ) = ej0(+ sin )

f() = ej0z

f() (15)

where

f() =

l=

n=

Jnej2n/p

1[lpw/2,lp+w/2]() (16)

The function f, being periodic in with period p, may be expandedin a Fourier series

f() =

k=

Jkej2k/p

where the Fourier coefficients Jk, k Z, are given by

Jk = (1/p)

p/2p/2

f()ej2k/pd = w

n=

Jn sinc(n k)w (17)

In (17), ww/p and sincX sin X/X. Thus

Js(, ) = wej0(+ sin )

k=

n=

Jn sinc(n k)w

ej2k/p

Reverting back to the original variables and z, we have

Js(, z) =

n=

nej(nnz) (18)

where

n w

k=

Jk sinc(k n)w (19)

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We are now ready to tackle the boundary conditions. First, weintroduce the following convenient abbreviations for the modified

Bessel functions and their derivatives evaluated at = a:

Ina In(na), Kna Kn(na),

InaIn(na), KnaKn(na),The boundary conditions (9a)(9c) immediately give the followingrelations among the four sets of coefficients An, Bn, Cn and Dn, n Z:

Cn = (Ina/Kna)An (20a)

Dn = (Ina/Kna)Bn (20b)

Bn = (janKna cos )An/Kna(a

2n sin nn cos ) (20c)

The fourth boundary condition (13d) together with the relations (20)and the expression (18) for Js(, z), in turn, relates An to n as

An = Kna[a2n sin nn cos ]n/j2n (20d)

Finally, the enforcement of the homogeneous boundary condition on

the tangential electric field component parallel to the winding directionleads to the set of equations

ej0z

l=

n=

nnej2n/p

1[lpw/2,lp+w/2]() = 0 (21)

on cancellation of the non-zero constant factor (j/a) where

n

KnaIna(a

2n sin

nn cos )

2/2n + (k0a)2InaK

na cos

2 (22)

Since ej0z = 0, (21) implies that each Fourier coefficient of theperiodic function

h()

l=

n=

nnej2n/p

1[lpw/2,lp+w/2]()

of (with period p) must vanish, that is,

n=

nnsinc(n k)w = 0 for k Z (23)

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Substituting for n from (19), the condition (23) may be put in theform

q=

kqJq = 0 for k Z (24)

where

kq =

n=

nsinc(k n)w sinc(q n)w (25)

For a nontrivial solution of the infinite set of linear homogenousequations (24) for Jq, q Z, it is necessary that the determinantof the coefficient matrix A

[

kq], k, q

Z is zero, that is,

|A| = 0 (26)The determinantal equation (26) gives the dispersion relation for thecold wave modes supported by an open anisotropically conductingtape helix. It is to be emphasized that, unlike similar treatmentsof the cold-wave problem that neglect the transverse component ofthe tape current density, the present derivation of the dispersionequation is based neither on any apriori assumption regarding the tape-

current distribution nor on any approximation of the helix boundaryconditions. In this sense, the derivation is exact within the assumedmodel for the tape helix.

It may be observed from (25) that the diagonal entries of thecoefficient matrix A are given by

kk =

n=

nsinc2(k n)w, k Z (27)

Since the function sinc2X decreases fairly rapidly with |X|, an estimateof kk to any required order of accuracy may be obtained bytruncating the infinite series (27) to the neighborhood Ak, of k where

Ak,

n Z : nsinc2(k n)w >

However, for a given order of accuracy , the cardinality of the setAk, decreases only slowly with k since n is only of order 1/|n| for|n| 1. Unfortunately, this also means that a fairly large number ofdiagonal entries needs to be retained in any truncation of the coefficientmatrix if an accurate estimate of the dispersion characteristic is desired.However, a redeeming feature is the presence of the factor sinc(kn)wsinc(q n)w in the expression for the matrix entry kq ; this implies

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Under this truncation, the dispersion equation reduces to

00 =

n=

n sinc2nw = 0 (30)

In spite of the drastic nature of this truncation, the resulting grosslyapproximate dispersion equation (30) is similar in appearance to theapproximate eigenvalue equation

n=

n Rn = 0 (31)

derived in the literature [35] on the basis of an assumed tape-current distribution which forces the tangential electric field boundarycondition to be satisfied only along the centerline of the tape. Thevalue of the multiplying factor Rn in (31) is decided by the typeof assumption made regarding the tape-current distribution, andirrespective of the particular assumption made, the decay of Rn withrespect to n turns out to be no better than |n|1 except for the caseof the one-term approximation to the tape-current distribution madeby Chodorow and Chu [2] and Watkins [3]. In fact, the approximatedispersion equations derived by Chodorow and Chu and Watkins have

a form identical to that of (30). Thus, the terms of the series in (30)and those in [2] and [3] decrease rapidly enough with |n| (due tothe presence of (sinc)2 factors) to enable the infinite series to besymmetrically truncated to a low order without appreciable error.Even though such a truncated series does not appear to be anythinglike a reasonable approximation to the actual dispersion equation, itwill be shown in the sequel that the zeroth term 0 in the infinite-seriesrepresentation (30) of 00 does indeed serve as the leading order termin a numerical scheme for getting better and better approximations by

successive addition of higher order correction terms.In order to test how good (30) is as an approximation for thedispersion equation for narrow tapes, the infinite-series representationof00 is symmetrically truncated to the order Wp/w so as to retainall the terms falling within the main lobe of the (sinc)2 functions inthe truncated series. The truncated version of (30) is then put in thefixed-point format

k0a =

20a/Qsa + F(W)0 /QsaI0a sin

2 1/2

(32)

where

k0a = k0a, 0a = 0a, F(W)0

Wn=1

(n + n)sinc2nw

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and Qsa1 I0aK0acot2/I0aK0a. An attempt to solve (32) for thechoice of w = 0.1/ (prototypical value used in the literature for a

narrow tape) and = 10

leads to the unexpected conclusion that thetruncated version of (30), which has been projected for so long in theliterature as a reasonable approximation to the dispersion equation fornarrow tapes, does not possess any real solution for k0a(0a) beyond0a = 1.543. An increase of the truncation order does not improvethe situation; in fact the interval of existence shrinks slightly withan increase in the order truncation beyond W for a fixed value of w.For larger values of w, however, the truncated version of (30) is seento possess real solutions for k0a(0a) for progressively larger values of0a in the complement of the forbidden regions. In the limit w

1,

equation (30) (and its truncated version) degenerates to the dominant-mode sheath-helix dispersion equation which is of course known topossess a unique real solution k0a(0a) for every real value of 0a.

3. A NUMERICAL SCHEME

We now present an approach for improving upon the approximatedispersion equation (30) resulting from the single-entry truncation ofthe coefficient matrix A on the basis of the decay properties of the

matrix entries kq . We start out with a symmetric truncation ofthe infinite-order coefficient matrix A to the (2N + 1) (2N + 1)matrix [kq ]Nk, qN. Our objective is to study, for a specific valueof the ratio w = w/p, the behavior of the dispersion characteristic withrespect to the truncation order N, and arrive at a compromise value ofN that gives a reasonably good approximation to the actual dispersioncurve within the confines of the assumed model for the tape helix. It isreadily seen from (25) that only the main lobes of the sinc functionscontribute significantly to the value of kq . We may therefore restrict

the range of values of n in the summation for kq tomax(k, q) p/w < n < min(k, q) +p/w (33)

For the specific choice of w/p = 1/2, (33) becomes

max(k, q) 1 n min(k, q) + 1 (34)Since (34) implies that 0 |k q| 2, the (2N + 1) (2N + 1)coefficient matrix reduces to a banded symmetric matrix with nonzeroentries only along the main diagonal and the four subdiagonals (twoeach on either side of the main diagonal) adjacent to the main diagonal.

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Thus, the infinite series for kq gets truncated to

akqkq =min(k,q)+1

n=max(k,q)1

nsinc(k n)

2sinc (q n)

2,

N k, q N, 0 |k q| 2 (35)It will be demonstrated in the sequel how this banded structure ofthe coefficient matrix can be exploited to give an effective algorithm(which may be programmed easily on a computer) for the computationof its determinant. Denoting the (2N + 1) (2N + 1) truncatedcoefficient matrix by

A and the corresponding (2N + 1)-dimensional

null-space vector by J = [JN, JN+1, . . . , J1, J0, J1, . . . , JN1, JN]T

the truncated version of (24) becomes

AJ = 0 (36)

The convention of denoting a negative index by an overbar will beadopted in the sequel in the interests of brevity; thus, Jn, for

example, will be denoted by Jn. When the contributions from themain lobes of the sinc functions only are retained in the expressionfor akq , N k, q N, there will only be three types of non-zeroentries in the (2N + 1) (2N + 1) symmetric matrix A for the choicew = w/p = 1/2, viz.,

akk = k + (2/)2(k1 + k+1) N k N,

ak,k+1 = ak+1,k = (2/)(k + k+1) N k N 1,ak,k+2 = ak+2,k = (2/)

2k+1

N

k

N

2.

Assume N > 2. Since JN and JN appear respectively in the firstthree and the last three equations in the set (36), we may solve for

JN (respectively JN) from the first (respectively the last) equation in

terms of JN1 and JN2 (respectively JN1 and JN2) and substitute

for JN and JN in the next two equations to eliminate JN and JN from

the set (36) thereby reducing the order of the matrix A by two from(2N + 1) (2N + 1) to (2N 1) (2N 1) at the first stage of thereduction process. Continuing this process of successive substitutionand elimination on the resulting matrix (of the same symmetry andband structure), we see that the order of the coefficient matrix getsreduced by two at each of the succeeding stages. At the ith stage (count

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the starting stage as the 0th stage), we will have a set of 2(N i) + 1equations of the form

A(i)

J(i)

= 0where

J(i) = [JNi, . . . . . . JNi]T

and the entries in the 2 2 submatrix in the right bottom corner(respectively in the left top corner) of A(i) are given in terms of the

corresponding entries of A(i1) by

a(i)Ni,Ni = a

(i1)Ni,Ni a

(i1)Ni,Ni+1

2/a

(i1)Ni+1,Ni+1 (37)

a(i)Ni,Ni1 = a(i)Ni1,Ni = aNi,Ni1

a(i1)Ni,Ni+1aNi+1,Ni1/a(i1)Ni+1,Ni+1a

(i)Ni1,Ni1 = aNi1,Ni1 (aNi1,Ni+1)2 /a(i1)Ni+1,Ni+1

and by an identical set of three relations with an overbar over thesuffixes so long as 1 i N 2. The remaining entries of A(i1) arenot affected by the reduction process. At the (N 2)th stage we havethe following set of five equations:

a(N2)

22J2 + a

(N2)

21J1 + a20J0 = 0

a(N2)

12J2 + a

(N2)

11J1 + a10J0 + a11J1 = 0

a02J2 + a01J1 + a00J0 + a01J1 + a02J2 = 0 (38)

a11J1 + a10J0 + a(N2)11 J1 + a

(N2)12 J2 = 0

a20J0 + a(N2)21 J1 + a

(N2)22 J2 = 0

After eliminating

J2 and

J2 from (38), we have a set of three equationsfor J1, J0 and J1 at the (N 1)th stage:a

(N1)

11J1 + a

(N1)

10J0 + a11J1 = 0

a(N1)

01J1 + a

(N1)00 J0 + a

(N1)01 J1 = 0 (39)

a11J1 + a(N1)10 J0 + a

(N1)11 J1 = 0

Solving for J1 and J1 from the first and the third equations in (39) in

terms of

J0 and substituting for

J0 in the second equation of (39), wefinally have at the Nth stagea

(N1)00 a(N1)01 a

(N)

10 a(N1)01 a(N)10

J0 = 0 (40)

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where

a

(N1)

00 = a00 (a20)2

/a

(N2)

22 (a20)2

/a

(N2)

22 , (41)a

(N)

10=

a(N1)

10a

(N1)11 00a(N1)10

/

a(N1)

11a

(N1)11 20 20

,

a(N)10 =

a(N1)10 a

(N1)

11 00a(N1)10

/

a(N1)11 a

(N1)

11 20 20

and where 0 = (2/)2. The first two relations in (37) and their two

counterparts without the overbar over the suffixes continue to be validfor i = N 1 also. Thus, the approximate dispersion equation for atruncation order of N becomes

0 + 0 (1 + 1)

a

(N1)11

a

(N1)

10

2+ aN1

11

a

(N1)01

2 200a(N1)10 a(N1)01

a

(N1)

11a

(N1)11 20 20

20

21/a(N2)

22+ 21/a

(N2)22

= 0 (42)

Once a solution 0 of the approximate dispersion equation (42) for0 is obtained it is a simple matter to work backwards expressingsuccessively Ji and Ji for i = 1, 2, 3, . . . , N in terms ofJ0 to determine

the (2N + 1) 1 mode vector J, corresponding to 0, for the tapecurrent density. A documentation of the mode-vector expression is notattempted here since an open tape helix is only of academic interest as aslow-wave structure for travelling wave tubes. However, a field analysisof the cold-wave (including the contribution of the transverse tape-current density if found significant) problem for a dielectric-loaded tape

helix enclosed in a perfectly conducting coaxial cylindrical shell, whichserves as a good model for the TWT slow-wave structure, is proposedto be taken up in the near future.

The approximate dispersion equation of the tape helix for anyorder of truncation N is of the form

0 + 0FN(1, 2, . . . , (N+1)) = 0 (43)

where FN = GN/HN is a symmetric rational function of the 2(N+ 1)arguments

1, 2

, . . . , (N+1)

. The expressions for the GN

and theHN, which are homogeneous functions of their arguments, are given

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below:

G0 = 1 + 1G1 =

(1)1

(1)

1

1 + 1 0

21/(1)1 +

21

/(1)

1

G2 =

(2)2

(2)

2

1 + 1 0

21/(2)1 +

22

/(2)

1

(2)2

(2)

3

2 (2)2

(2)3

2(44)

GN = (N)2N2

(N)

2N2

1 + 1 0

21/(N)2N4 +

21

/(N)

2N4

(N)2N2

(N)

2N12 (N)2N2 (N)2N12 for N 3

H0 = 1

H1 = (1)1

(1)

1 20

1(1)

1+ 1

(1)1

+20

(1 + 1)

(1)1

(1)

1

(1 1)2

H2 = (2)2

(2)

2 20

(2)2

(2)

3+

(2)

2

(2)3

2

0

(2)

3 (2)

32

+20

(2)2 +

(2)

2

1 +10

21/(2)1 +

21

/(2)

1

(45)

HN = (N)2N2

(N)

2N2 20

(N)2N2

(N)

2N1+

(N)

2N2

(N)2N1

20

(N)2N1 (N)2N12

+ 20

(N)2N2 +

(N)

2N2

1 + 1 0

21/(N)2N4 +

21

/(2)

2N4for N 3

The recursive formulae for the 2K1, 2K1, 2K2, and 2K2, 1 K N, appearing in (44) and (45) are

(N)1 = N + 0(N+1 + (1 0,N1)N1) for N 1

(N)2 = N1 + 0(N + (1 0,N2)N2)

0(N + N1)2/(N)1 for N 2

(N)3 = N1 + (1

0,N2)N2)

0(N + N1)N1/(N)1 for N 2

(N)4 = N2 + 0(N1 + (1 0,N3)N3)

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20 2N1/(N)1 0

(N)3

2/

(N)2 for N 3

(N)2K1 = NK+1 + (1 0,NK)NK0NK+1(N)2K3/(N)2K4 for N K 3

(N)2K2 = NK+1 + 0(NK+2 + (1 0,NK)NK)

20 2NK+2/(2)2K6 0

(N)2K3

2/

(N)2K4

for N K 4together with a corresponding set of formulae with an overbar over the

suffixes.For the purpose of studying the behavior of the dispersion

characteristic as a function of the truncation order, it is convenientto introduce the nondimensional parameter

nana =

(0a + n cot )2 k20a1/2

in addition to 0a0a, and k0ak0a. In terms of the nondimensionalquantities, the expression (22) for n becomes

n =

20a k20a

sin + n0a cos 2

InaKna/2na

+k20aInaK

na cos

2 , n Z (46)The relation 0 = 0 may be recognized as the sheath-helix dispersionequation for the dominant mode (n = 0). Making use theexpression (46) for 0, the dispersion equation (43) may be put inthe fixed-point format

k0a = G(k0a; 0a)

20a/Qsa +0FN

1, . . . , (N+1)

QsaI0aK0a sin2

1/2(47)

The right member of (47) may be viewed as an operator G that mapsk0a into G(k0a; 0a) for a fixed 0a. The symmetric dependence ofthe function FN on i, 1 i N, implies that if 0a satisfies thedispersion equation (47) for a given k0a so does 0a for the same k0a.Therefore, solution of the dispersion equation (47) for k0a(0a) needonly be sought for nonnegative values of 0a. Thus, equation (47)may be solved numerically for k0a(0a), 0a 0, by the method ofsuccessive substitutions to find any fixed point of the operator G for

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k0a

0a

2.5

2

1.5

1

0

0.5

0 2 4 6 8 10 12 14 16

Sheath HelixTape Helix

Figure 2. Dispersion character-istic of tape helix for truncationorder N = 2.

Sheath HelixTape HelixSheath HelixTape Helix

0a

k0a

2.5

2

1.5

1

0.5

00 2 4 6 8 10 12 14 16

Figure 3. Dispersion character-istic of tape helix for truncationorder N = 3.

Sheath HelixTape HelixSheath HelixTape Helix

0a

0 2 4 6 8 10 12 14 16

k0a

2.5

2

1.5

1

0

0.5

Figure 4. Dispersion characteristic of tape helix for truncation orderN = 4.

k0a in the range 0 < k0a < (1/2) cot . The resulting family of tape-helix dispersion curves for the choice of the pitch angle = 10 areplotted in Figs. 24 for truncation orders of 2, 3 and 4 respectively. Thedominant mode dispersion curve of the sheath helix (for the same valueof = 10) is also plotted in the figures for comparison. Tape-helixdispersion curves for the truncation orders of 0 and 1 are not shownbecause the iterations for these two cases could not be continued (toconvergence) beyond 0a = 4.64 to yield real values for k0a(0a) in thecomplement of the forbidden regions. A portion of Fig. 2 magnifiedseveral fold to make the minute difference between the dispersioncurves discernible is shown in Fig. 5. It may be seen from Fig. 5that the phase speed for the tape-helix model is lower than that forthe sheath-helix model for the same value of in the complement ofthe forbidden regions. However, the dispersion curves of Fig. 5 are so

closely spaced that it may not be appropriate to draw any inferencebased on an inspection of Fig. 5.

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

4. CONCLUSION

It may be observed from the plots of Figs. 24 that the tape-helixdispersion curves for N = 2, 3 and 4 follow the dominant-mode sheathhelix dispersion curve very closely everywhere within the complementof the forbidden regions (shown shaded in the figures). The solutionfor k0a(0a) within the forbidden regions acquires a small imaginary

part (on the order of 105) on account of1a, 2a and 3a becomingpurely imaginary within the 1st, the 2nd and the 3rd forbidden regionrespectively where the nth forbidden region for n Z, is taken to bethe portion of the 0a-k0a plane inside the (inverted) triangle formedby the straight lines k0a = 0a + n cot , k0a = 0a n cot and k0a = (1/2)cot . It is thus seen that the mode constant naof the nth space-harmonic contribution to the total field becomesimaginary in the nth forbidden region, and that the resultant Poyntingvector acquires a small radial component in order to account for theradiation of power from the

nth space harmonic. Since the dispersion

curves for N = 2, 3 and 4 are virtually indistinguishable from oneanother, a truncation order as low as 2 is adequate to deliver anaccurate estimate of the tape-helix dispersion characteristic (at leastwithin the validity limits of the assumed model for the parametervalues used in the numerical computations). However, a fairly largenumber of modal amplitudes Jn, |n| 0, is needed in the infinite-series representation (14) for the surface current density componentJs(, z) so as to ensure a reasonably good approximation for the tape-current density, and hence for the electromagnetic field vectors. The

main conclusion that may be drawn from the present study is thefollowing: The dominant-mode dispersion characteristic of the sheathhelix is an excellent approximation to that of the tape helix in the

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complement of the forbidden regions provided that the neglect of thetransverse component of the tape-current density does not give rise to

any appreciable error even for tapes which are not narrow. Whethersuch an hypothesis is true or false can be ascertained only throughan analysis of guided electromagnetic wave propagation that fullyaccounts for the transverse component of the tape-current density.

Based on the outcome of such a study (which is currentlyunder progress), it is proposed to extend the method adopted forthe derivation of the tape-helix dispersion equation to a full fieldanalysis of the practically important case of a dielectric-loaded helicalslow-wave structure enclosed in a coaxial metal cylindrical shell andsupported by azimuthally symmetrically placed dielectric rods. Theeffect of the dielectric support rods will have to be modeled bya homogeneous dielectric the effective dielectric constant of whichcan be determined in terms of the geometric arrangement of thesupport rods and the actual dielectric constant of the rod material.Thisprocess of homogenization is equivalent to replacing the azimuthallynonhomogeneous dielectric constant of the annular region between thehelix and the outer conductor by its azimuthal average, which becomesa constant independent of the radial coordinate, for azimuthallysymmetrically placed wedge-type support rods. In any case, it is

necessary to smooth out any kind of azimuthal nonhomogenity beforeattempting a solution of the cold-wave problem by an extension ofthe method introduced in this paper because any axial asymmetryof the slow-wave structure would be inconsistent with the propertyof geometrical invariance under simultaneous translation and rotationexhibited by an infinite helical structure.

REFERENCES

1. Sensiper, S., Electromagnetic wave propagation on helicalconductors, Sc.D. Thesis, Massachusetts Institute of Technology,Cambridge, Mar. 1951.

2. Chodorov, M. and E. L. Chu, Cross-wound twin helices fortraveling-wave tubes, J. Appl. Phys., Vol. 26, No. 1, 3343, 1955.

3. Watkins, D. A., Topics in Electromagnetic Theory, John Wiley &Sons, New York, 1958.

4. Zhang, K. A. and D. Li, Electromagnetic Theory for Microwavesand Optoelectronics, 2nd Edition, Springer-Verlag, Berlin-

Heidelberg, 2008.5. Basu, B. N., Electromagnetic Theory and Applications in Beam-wave Electronics, World Scientific, Singapore, 1996.

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Errata to DISPERSION OF ELECTROMAGNETIC WAVES

GUIDED BY AN OPEN TAPE HELIX I by N. Kalyanasun-daram and G. Naveen Babu, in Progress In Electromagnetics ResearchB, Vol. 16, pp. 311331, 2009

(i) Page 315, 3rd line from top: Correct 2U

z j

V to

12Uz +

jV .

(ii) Page 315, 4th line from top: Insert / after Gn(n).

(iii) Page 315, 7th line from top: Insert between

n=

and [.

(iv) Page 315, 8th line from top: Correct 2V

z j

U to

12Vz

jU .

(v) Page 316, 12th line from top: Remove

before cot/a.(vi) Page 316, 14th line from bottom: Remove before cot /a.

(vii) Page 316, 13th line from bottom: After that is, replace k0 byk0a.

(viii) Page 317, 10th line from top: Change sign to + sign.

(ix) Page 317, 6th line from bottom: Correct Js(,,z) to Js(, z).

(x) Page 319, 6th line from top: Replace (9a)(9c) by (13a)(13c).

(xi) Page 322, Equation (32): Insert k0a between I0a and sin2 in

the 2nd term within the curly brackets.(xii) Page 325, 5th line from bottom: Replace the coefficient a11 by

a11.

(xiii) Page 326, 6th line from top: After counterparts replace withoutby with.

(xiv) Page 327, 3rd line from top: Delete the 0 in front of theparenthesis.

(xv) Page 327, 4th line from top: Replace 22

by 21

.

(xvi) Page 327, 10th line from top: Insert + in between (1)1 and

(1)

1.

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