Simple Formulas for Stopband Attenuation Characteristics of Asymmetric Helical Slow-Wave Structures...

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 57, NO. 6, JUNE 2010 1447

Simple Formulas for Stopband AttenuationCharacteristics of Asymmetric Helical Slow-Wave

Structures of Traveling-Wave TubesSubrata Kumar Datta, Vemula Bhanu Naidu, P. Raja Ramana Rao, Lalit Kumar, and Baidyanath Basu

Abstract—Simple closed-form formulas for the estimation of theπ-mode stopband and the stopband attenuation in an azimuthallyasymmetric helical slow-wave structure (SWS) are developed,following the coupled-mode analysis of multiple reflections. Theformulas are simple and amenable to easy computation, and alsoallow the use of the dispersion characteristics of the structureobtainable from any standard electromagnetic modeling, therebyaccruing the accuracy of 3-D electromagnetic analysis. The analy-sis is benchmarked against published results with close agreement.Based on the analysis, SWS configurations that would reduce thedeleterious effects of asymmetry and also would improve TWTefficiency are suggested.

Index Terms—Angularly asymmetric helix, coupled-modeanalysis, helix slow-wave structure (SWS), helix traveling-wavetube (TWT), metal segment asymmetry, periodic structures,support-rod asymmetry.

I. INTRODUCTION

THE EXISTENCE of a stopband in an asymmetric helicalslow-wave structure (SWS) around the π-point frequency

[1] is a concern in the design of wideband traveling-wavetubes (TWTs). This is because the stopband created by suchasymmetry limits the device capability by causing band-edgeinstabilities and associated in-band power holes [1], [2]. Ahelical SWS, typically, consists of a helix made of a metallictape supported by a number of dielectric rods symmetricallyarranged around the helix in a metal envelope. Often, theenvelope contains metal segments (vanes) projecting radiallyinward from the envelope (Fig. 1). The bandwidth of theTWT can be widened by controlling the dispersion of theSWS by optimizing the geometry of the metal segments [1],[3]–[5]. The asymmetry of such a structure limiting the devicecapability can be caused by variation in one or more of thefollowing:

1) angular positions of the dielectric helix-support rodsaround the helix (angular offset of the rods);

Manuscript received November 25, 2009; revised February 22, 2010;accepted March 17, 2010. Date of current version May 19, 2010. The review ofthis paper was arranged by Editor W. Menninger.

S. K. Datta, V. B. Naidu, P. R. R. Rao, and L. Kumar are with the MicrowaveTube Research and Development Centre, Defence Research and DevelopmentOrganization, Bangalore 560 013, India (e-mail: [email protected]).

B. N. Basu is with the College of Engineering and Technology, Moradabad244 001, India (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2010.2047069

Fig. 1. Schematic of a helix supported by dielectric rods in a metal envelopewith metal segments projecting radially inward.

Fig. 2. Qualitative dispersion characteristics of the fundamental (0) and thebackward (−1) space harmonics for (solid line) the uncoupled structure and for(broken line) the structure in the presence of coupling of the space harmonicsthat causes the stopband.

2) widths of the dielectric helix-support rods;3) permittivity values of the dielectric helix-support rods;4) angular positions of the metal segments (angular offset of

the segments);5) angular widths of the metal segments;6) helix-to-segment radial gaps.The SWS supports a number of spatial harmonics. Among

these, the fundamental forward space-harmonic wave andthe first backward space-harmonic wave crossing over at theπ-point frequency (Fig. 2) play a major role in the beam-wave interaction mechanism at the high-frequency operatingend of the device. The structural discontinuities about the axis,i.e., one at the support rods and the other at the segments,give rise to multiple reflections of these space-harmonic waves.Furthermore, asymmetry at the discontinuities also causes adifference in the relative phases of the reflected waves alongdifferent reflection paths that have different electrical lengths.This difference in relative phases of the reflected waves, inturn, couples the forward and backward space-harmonic waves(which degenerate around the π-point frequency) and thus

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1448 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 57, NO. 6, JUNE 2010

becomes responsible for the generation of the coupled hybridmodes and a stopband in the dispersion characteristics of theSWS (Fig. 2) [6]–[12].

The behavior of the π-mode stopband in an asymmetric he-lical SWS has been studied in the past by both electromagneticanalysis [6]–[11] and simulation [12]. Hinson [6] analyzed thestructure by a simple multireflection coupled-mode analysis.He, however, did not consider a specific geometry of the SWSand assumed that the phase velocity vp of the structure wassimply given by the reduction factor vp/c = tan ψ, where ψis the helix pitch angle [6]. Lien [7] analyzed a helix supportedby dielectric rods in a metal envelope without metal segmentloading, using a rigorous cascaded-network analysis [7], whichis an approach that was also used by Onodera for the structurewith metal segment loading [8]. Datta et al. [9] analyzed thestructure by a rigorous multireflection coupled-mode analysisextending the work of Hinson [6] for the structure without metalsegment loading.

These analyses [6]–[8] predict the attenuation constant of thestructure in addition to predicting the width of the stopband.Moreover, the approach of [9], unlike [6]–[8], allows one to usethe phase velocity or the propagation constant data from anyof the available sources of analysis or simulation. Joo et al.[10] analyzed the asymmetric structure, without consideringthe segment loading of the envelope, using an elegant modelin which the region between the helix and the metal envelopeis azimuthally divided, in general, into a number of dielectricregions of different angular widths and relative permittivity val-ues, including the unity value corresponding to the free-spaceregions between the dielectric-support regions. In this model[10], the azimuthal asymmetry is defined by the appropriatechoice of values for the relative permittivity and angular widthof the regions. Antonsen et al. [11] also analyzed only theasymmetries in the dielectric support rods, without consideringthe metal segment loading, and treated the effects of asymmetryas coupling between forward and backward space-harmonicwaves through a polarization current superimposed with thebeam convection current. Although the analyses of Onodera[8], Joo et al. [10], and Antonsen et al. [11] are rigorous,they are rather complex and highly computationally intensive,whereas the analysis of Datta et al. [9] provides a simpleformula for the bandwidth of the stopband and the attenuationconstant.

As in [9], in the present analysis, the simple formulas forthese quantities have been found; however, the effect of thesegment loading, variation in support rod width, and variationin the dielectric constant of support rod, which were ignored in[9], are now included in the analysis (Section II). Furthermore,as in [9], the approach allows the use of the phase velocityor the propagation constant data from any of the availablesources of analysis or simulation. In Section III, the presentanalysis is used to study the effects of the asymmetry inthe segment parameters (angular positioning, helix-to-segmentgap, and segment angular width) and in the dielectric helix-support parameters (angular positioning, relative permittivity,and angular width) on the stopband attenuation characteristicsaround the π-point frequency of the structure (bandwidth of thestopband and attenuation constant).

Fig. 3. Ladder circuit representation of a single period of the helical SWSshowing the phase difference at the π-point frequency ωπ .

II. ANALYSIS

The analysis follows the coupled-mode approach of multiplereflections of the forward and backward space-harmonic wavesaround the π-point frequency according to Pierce [13], [14]and Watkins [15]. In order to model the coupling of spaceharmonics, we represent a single-turn helical SWS as a laddercircuit by slitting the helically conducting tape at the mean-helix radius along the direction normal to the helix windingdirection and flattening it out, as shown in Fig. 3. For this modelof the helix, near the π-point frequency (ωπ), a 2π-radianazimuthal displacement along the helical path (one helical turn)results in an axial phase shift of π radians. The model allowscontinuous coupling between the spatial harmonics generatedin the structure. The propagation constants of the coupledhybrid modes around ωπ , arising from multiple reflections atthe ladder discontinuities of the model (Fig. 3), are given by thefollowing coupled-mode dispersion relation [15]:

β =(

β0 + β−1

2

)±

{(β0 − β−1

2

)2

− K2c

} 12

(1)

where β is the coupled hybrid-mode propagation constant, andβ0 and β−1 are the uncoupled propagation constants of theforward and backward space-harmonics, respectively. Here, Kc

is a coupling coefficient that characterizes the formation ofhybrid-mode waves [13]–[15]. As a special case, in the absenceof coupling of waves (Kc = 0), as one would expect in anideally symmetric structure, (1) indicates the existence of twoindependent uncoupled space-harmonic waves.

Watkins [15] followed the coupled-mode analysis of Pierce[13], [14] and carried out the analysis of the coupling of forwardand backward space-harmonic waves in a periodic structureusing a voltage-current transmission matrix (ABCD-matrix)description of the ladder equivalent network of one period ofthe structure. He defined the coupling coefficient (Kc) as thereflection coefficient per unit length of the SWS. FollowingWatkins [15], the coupling coefficient may be described interms of the magnitude of the reflection coefficient per period(|S11|) and the structure periodicity (p) as

pKc =|S11|

(1 − |S11|2)1/2

which, for small reflections at the periodic discontinuities(|S11| � 1) in a periodic structure, may be approximated as

Kc ≈ |S11|p

. (2)

DATTA et al.: FORMULA FOR STOPBAND ATTENUATION OF ASYMMETRIC HELICAL SWS 1449

The uncoupled propagation constants of the forward and back-ward space harmonics β0 and β−1, respectively, around the π-point frequency are given by [9]

β0 =ω/m

β−1 = (2π/p) − ω/m (3)

where m is the slope of the phase velocity versus frequency(dispersion) for the fundamental forward space-harmonic wavesupported by the structure near ωπ (Fig. 2), representing thegroup velocity of the uncoupled RF waves.

The uncoupled propagation constants β0 and β−1 in (1) areobtainable from the dispersion relation of the structure given byfield analysis (based on a sheath-helix or a tape-helix model)[1], [3]–[5], [11], [16] or 3-D simulation [17]. The mode-coupling coefficient Kc in (1) is obtainable from (2), for whichone has to know the magnitude |S11| of S11. The latter, inturn, may be obtained, using a small-reflection approach to theproblem [18], [19] as follows:

S11 = ρr + (ρr + Δρr2) exp{−2jπ/3 − jΔθr2}+ (ρr + Δρr3) exp{−4jπ/3 − jΔθr3}+ ρs exp{−jθ0}+ (ρs + Δρs2) exp{−jθ0 − 2jπ/3 − jΔθs2}+ (ρs + Δρs3) exp{−jθ0 − 4jπ/3 − jΔθs3}. (4)

Here, ρrn (n = 1, 2, 3) represent the reflection coefficientsat the dielectric support rods 1, 2, and 3, respectively, andρsn (n = 1, 2, 3) represent the corresponding quantities at themetal segments 1, 2, and 3, respectively (Fig. 3). Furthermore,it is implied that

ρr1 = ρr

ρr2 = ρr + Δρr2

ρr3 = ρr + Δρr3

ρs1 = ρs

ρs2 = ρs + Δρs2

ρs3 = ρs + Δρs3

where Δρr2,3 is the incremental value of ρr2,3, which iscaused by the asymmetries of the parameters of rods 2 and 3,respectively, with reference to the parameters of the referencerod 1, and Δρs2,3 is the incremental value of ρs2,3, which iscaused by the asymmetries of the parameters of segments 2and 3, respectively, with reference to the parameters of thereference segment 1 (Fig. 3). Δθr2 and Δθr3 are the azimuthalasymmetries of rods 2 and 3 with respect to their referencesymmetrical positions. Similarly, Δθs2 and Δθs3 are the az-imuthal asymmetries of segments 2 and 3 with respect to theirreference symmetrical positions. θ0 is the angle between rod 1and segment 1.

One can then obtain from (4), ignoring the second- andhigher order incremental terms of negligible contributions,

the following expression for the magnitude of the reflectioncoefficient per period:

|S11| ≈((Δρr2)2 + (ρrΔθr2)2 + (ρrΔθr3)2 + (Δρr3)2

+ (Δρs2)2 + (ρsΔθs2)2 + (ρsΔθs2)2 + (Δρs3)2)1/2

. (5)

The reflection coefficients ρr(= ρr1) and ρs(= ρs1) at refer-ence rod 1 and reference segment 1 discontinuities, in termsof the corresponding structure parameters, are given by (seeAppendix)

ρr =∣∣∣∣(√

εr,eff − 1√

εr,eff + 1

)(1 − exp{−jwr/a})

∣∣∣∣ (6)

ρs =∣∣∣∣(

Z0 − Z0,s

Z0 + Z0,s

)(1 − exp{−jws/b})

∣∣∣∣ (7)

respectively, where

Z0 =120π

(wtape

c−a+1.98

(wtape

c−a

)0.172)−1

Z0,s =120π

(wtape

b−a+1.98

(wtape

b−a

)0.172)−1

εr,eff =k

(1+

(εr−1

2

)(1+

1√1+10(c−(a)/wtape

)).

Here, a is the mean radius of the helix, b is the segment tipradius, and c is the radius of the metal envelope. wtape is thewidth of the helix tape, wr is the angular width of a support rod,and ws is the angular width of a segment. εr is the relative per-mittivity of the support rod material, and k is a field-screeningfactor (k = 0.5 for εr �= 1) defined by Lien [7]. From (6) and(7), the special case of the absence of a dielectric rod (wr = 0)or a metal segment (ws = 0) results in reflection coefficientsρr,s = 0.

The bandwidth Δω(= ω2 − ω1) of the stopband is the dif-ference between the lower edge frequency ω2 of the upperdispersion curve (above ωπ) and the upper edge frequency ω1

of the lower dispersion curve (below ωπ) (Fig. 2), with thetwo dispersion curves resulting from the two solutions of thecoupled-mode dispersion relation (1). Moreover, the imaginarypart of the solution of (1) gives the attenuation constant of thestructure. Hence, the closed-form solutions for the bandwidthof the stopband relative to the π-point frequency and the max-imum stopband attenuation (with the latter corresponding tothe propagation constant of the coupled hybrid-mode becomingpurely imaginary at ωπ) may be obtained with the help of(1)–(3), respectively, as

Δω

ωπ=

ω2 − ω1

ωπ=

(2π

)pKc =

(2π

)|S11| (8)

αmax =Kc =|S11|

p. (9)


Fig. 4. Relative stopband and maximum stopband attenuation of SWS-I dueto an asymmetry in the placement of one of the support rods.

Fig. 5. Relative stopband and maximum stopband attenuation of SWS-I dueto variation in the dielectric constant εr of one of the support rods.

III. RESULTS AND DISCUSSION

We consider two structures for which the dimensions andthe stopband characteristics are published in the literature, i.e.,1) a segment-less structure (SWS-I) from Hobrecht [20] and2) a segment-loaded structure (SWS-II) from Onodera andRaub [5]. SWS-I consists of a helix made of a tape of thickness0.28 mm and width 0.90 mm. The helix is wound with a meanradius of 2.73 mm and a pitch of 3.18 mm, and it is supportedin a metal envelope of radius 5.2 mm with three rectangu-lar Beryllia (εr = 6.3) support rods each of width 0.64 mm.SWS-II is similar to SWS-I but with a metal envelope thatis segment loaded (Fig. 1). SWS-II has a tape thickness of0.25 mm, a tape width of 0.89 mm, a helix pitch of 1.8 mm,a mean helix radius of 1.52 mm, a metal envelope radiusof 2.64 mm, and anisotropic pyrolytic boron nitride (APBN)(εr = 5.12) helix-support rods with a width of 0.51 mm. Thethree segments of SWS-II are wedge shaped, each with a tipradius of 2.1 mm and an angular width of 35◦.

For the segmentless structure SWS-I, the relative stopbandand the maximum stopband attenuation of the structure due toasymmetry of one of the support rods (Fig. 4) and that due tothe variation in the dielectric constant of one of the supportrods (Fig. 5) are compared against available published resultsby Lien [7] and Onodera [8]. While validating these character-istics, we consider the support rod to be a wave reflector of zerothickness [7], [8]. However, the effect of the variation in theazimuthal width of the support rods (Fig. 6) cannot be validated

Fig. 6. Relative stopband and maximum stopband attenuation of SWS-I dueto variation in the azimuthal width wr of one of the support rods.

Fig. 7. Relative stopband and maximum stopband attenuation of SWS-II dueto asymmetry in the placement of one of the metallic (vane) segments.

Fig. 8. Relative stopband and maximum stopband attenuation of SWS-II dueto variation in the segment-to-helix gap (g = b − a) of one of the metallicsegments.

against any such results, which, to the best of our knowledge,are not available in published literature.

For the segment-loaded structure (SWS-II), the relative stop-band and the maximum stopband attenuation due to structureasymmetry caused by variation in 1) the angular positioningof the segments (Fig. 7), 2) the helix-to-segment gap (Fig. 8),and 3) the azimuthal width (Fig. 9) of one of the segmentsare shown. The validation with respect to the variation of thehelix-to-segment gap is compared with the published results ofOnodera [8].

Severe effects are seen on the stopband characteristics due tothe asymmetry of the azimuthal placement of dielectric supportrods and that of the metal segments. The asymmetry in the


Fig. 9. Relative stopband and maximum stopband attenuation of SWS-II dueto variation in the azimuthal width ws of one of the metallic segments.

azimuthal placement of the support rods has a pronouncedeffect on the stopband characteristics, compared with that of thesegments. On the contrary, the variation in the support rod widthand that in the segment width have negligible effects on thestopband characteristics. For instance, a typical ±20 μm varia-tion in the support rod width of 0.64 mm increases the stopbandby only approximately 0.1% of ωπ and increases the associatedmaximum attenuation by only approximately 0.01 dB/turn.Similarly, a typical ±20 μm variation in the segment width of1.28 mm increases the stopband by only approximately 0.08%of ωπ and increases the associated maximum attenuation byonly approximately 0.008 dB/turn. Variation in the helix-to-segment gap also has a negligible effect on the stopband charac-teristics. A typical variation of ±20 μm in the helix-to-segmentgap increased the stopband by approximately 0.30% of ωπ andincreased the maximum attenuation by close to 0.04 dB/turn.However, variation of the dielectric constant of the support rodmaterial was found to have a noticeable effect. Consideringa typical variation of ±10% in the dielectric constant [17],one may expect the band opening to increase by 1.45% ofωπ with an associated increase in the maximum attenuationof 0.25 dB/turn. This necessitates strict quality control of thematerial property of the dielectric support rods in an assembly,suggesting the use of support rods from the same batch ofproduction, so that the rod-to-rod variation in the dielectricconstant is minimized. Notwithstanding, accepting the unavoid-able stopband troubles due to the allowable manufacturingtolerances, utmost care has to be exercised in maintaining theazimuthal symmetries in the placement of the dielectric supportrods and the segments while assembling the helical SWS.

Moreover, in a metal-segment-loaded SWS, the asymmetryeffects can be severe in that all six aspects of asymmetries (thethree with respect to the dielectric support rod parameters andthe other three with respect to the metal segment parameters)may have simultaneous contribution. The effect is additive,which may be appreciated from (5) for the reflection coefficientper period. The twofold uncertainties with respect to indepen-dent placements of the support rods and the segments can bereduced to only one uncertainty if one considers a compositesupport-rod–segment assembly (Fig. 10), which is a structurein which the support rods and the segments form a singleassembly. Such composite configurations are described in theliterature, i.e., the configuration in which 1) the support rod

Fig. 10. Schemes to reduce the effects of angular asymmetry. (a) Embeddedrod U-segment configuration. (b) Shielding of fields in a U-segment configura-tion. (c) Studded rod-segment configuration. (d) Segments coated/formed ontothe rods.

is embedded inside a U-segment [21], 2) the support rod isstudded at the tip of the segment by brazing/forming [22], [23],or 3) the thin segments/vanes are formed at the sides of thesupport rods by metallic coating [16], [17]. These configura-tions have the proven capability of reducing the dispersion ofthe structure and even providing some amount of negative dis-persion by controlling the helix-to-segment gap (g) [16], [17],[21]–[23] and widening the bandwidth of the TWT [1]. Forthe case of the U-segment geometry, the electric field radiallypenetrates into the groove region of the segmented structurebefore it is effectively shielded by a “virtual electric wall” atan effective height heff from the tip of the segment [shown inbroken line in Fig. 10(b)], instead of at the metal envelope (atradial height = h from the segment tip). We have observed by3-D electromagnetic analysis using CST-studio that, for typicalstructures having a small helix-to-segment gap (g/a ∼ 1/3)and segment width ws > 2wr, the electric field decreases bymore than two orders of magnitude beyond heff = 1.5g. Thiseffective change in the height of the support rod further reducesthe reflection from the support rod interface. The analysis of theasymmetry of the studded rod-segment geometry [Fig. 10(c)] isidentical to that of a conventional segment asymmetry with thecharacteristic impedance at the discontinuity region reduced bya factor of 1/

√εr,eff due to dielectric loading. The shielding

of the electric field inside the groove region of the structureis valid for the coated-segment configuration [Fig. 10(d)] aswell; however, the analysis of such a structure is simplifiedby using an alternate approach in which one considers thecoated segment as a capacitive discontinuity of zero thickness[24]. The complete analysis of the various asymmetries in thesetypes of configurations (Fig. 10) is outside the scope of thisstudy; however, one can carry out a recurrence computation andcascade the effects of the discontinuities using the formulationgiven in the Appendix (general composite discontinuity) todetermine how the variation in the material properties andgeometries would affect the stopband characteristics of suchstructures.

IV. CONCLUSION

Simple closed-form solutions have been proposed for theestimation of the width of the stopband and the stopbandattenuation in an azimuthally asymmetric helical SWS. Theseparameters are important from the standpoint of the design of abroadband TWT employing such a structure. The analysis has


Fig. 11. Equivalent circuit model of the support-rod discontinuity.

been validated against published results with close agreementand furthered to show the effects of the structure parameterson the stopband attenuation characteristics. The formulas aresimple and also allow the use of dispersion characteristicsobtainable from 3-D electromagnetic analysis, thereby accruingbetter accuracy. Based on the analysis, SWS configurationsthat would reduce the deleterious effects of asymmetry aresuggested. It is hoped that the simplicity and accuracy of thepresent approach arouses interest among the TWT community.

APPENDIX

REFLECTION COEFFICIENTS AT THE DISCONTINUITIES

A. Helix-to-Rod Discontinuity

Following the microstripline equivalent circuit model(Fig. 11), the characteristic impedance at the free-space regionZ0 and that at the support rod region Z0,rod may be writtenas [19]

Z0 = 120π

(wtape

c − a+ 1.98

(wtape

c − a

)0.172)−1

(A.1)

Z0,rod =Z0√εr,eff

. (A.2)

Here, εr,eff is the effective relative permittivity of the helix-support-rod region, including the field-screening effects (k =0.5 for εr �= 1 [7]). Given in terms of εr [19]

εr,eff =k

(1+

(εr−1

2

)(1+

1√1+10(c−a)/wtape

)).

(A.3)

The characteristic impedance expressions (A.1) and (A.2) maybe used to find the reflection coefficient at the support rodinterfaces ρr,face, using the following expression:

ρr,face =Z0 − Z0,rod

Z0 + Z0,rod. (A.4)

One may then use (A.4) to obtain the total reflection coefficientof the support rod discontinuity, following the theory of smallreflections [18], i.e.,

ρr = |ρr,face (1 − exp(−jθr))| . (A.5)

Here, θr(= wr/a) is the angle in radians subtended by asupport rod of azimuthal width wr at the helix (Fig. 11).

Fig. 12. Equivalent circuit model of the segment discontinuity.

B. Helix-to-Segment Discontinuity

The helix-to-segment discontinuity has also been analyzed,following the microstripline equivalent circuit model [18](Fig. 12). Each face of the segment discontinuity has a dis-continuity capacitance Cd cascaded to the segment regionof characteristic impedance Z0,s both expressed, followingRamo et al. [19], as

Cd =[(

1 + χ2

χ

)(ln

{1 + χ

1 − χ

})

−2 ln{

4χ

1 − χ2

}](ε0

π

)wtape (B.1)

Z0,s = 120π

(wtape

b − a+ 1.98

(wtape

b − a

)0.172)−1

(B.2)

where χ = (b − a)/(c − a).One may express the reflection coefficient at the segment

interfaces ρs,face as

ρs,face =1

Z0− 1

Z0,s− jωCd

1Z0

+ 1Z0,s

+ jωCd

(B.3)

which may be used for finding the total reflection coefficientof the segment discontinuity, following the theory of smallreflections [18]:

ρs = |ρs,face (1 − exp(−jθs))| (B.4)

where θs(= ws/b) is the angle in radians subtended by thesegment tip of azimuthal width ws. Incidentally, for structuresof practical relevance, the discontinuity admittance ωCd wasfound to be negligibly small (two orders of magnitude less),compared with the line admittances Z−1

0 and Z−10,s, and can be

ignored without loss of significant accuracy.

C. General Composite Discontinuity

Following the preceding analyses for special cases of dis-continuities, one may now express the equations for a generalcomposite discontinuity (as in Fig. 10) using a recurrencerelation as follows: Let Z

(n)0 be the characteristic impedance

of the nth discontinuity, ρ(n)face be the reflection coefficient at the

incident face adjoining the (n − 1)th and nth discontinuities,h

(n)d be the radial height of the electric wall from the helix in the


nth discontinuity region, he(= c − a) be the radial height of themetal envelope from the helix, and w

(n)d be the azimuthal width

of the nth discontinuity region. Using these nomenclatures, onemay express the total reflection coefficient at the compositediscontinuity as

ρ = ρ(n)face + ρ

(n+1)face exp

(−jw

(n)d /a

)+ · · · (C.1)

with

ρ(n)face =

Z(n−1)0 − Z

(n)0

Z(n−1)0 + Z

(n)0

, ρ(n+1)face =

Z(n)0 − Z

(n+1)0

Z(n)0 + Z

(n+1)0

, . . .

Z(n)0 =

120π√

εr,eff

⎛⎝wtape

h(n)d

+ 1.98

(wtape

h(n)d

)0.172⎞⎠

−1

εr,eff = k

⎛⎝1 +

(εr − 1

2

)⎛⎝1 +

1√1 + 10h

(n)d /wtape

⎞⎠

⎞⎠ .

In the case of field screening due to segments, h(n)d has to

be interpreted as the location of the “effective electric wall”from the helix. For the case of a thin/coated segment, one mayalternatively consider a zero-thickness discontinuity having acapacitive susceptance Bs, instead of a segment of finite width,expressed as [24]

Bs

Y(n)0

=(

4he

λ0

)ln csc

πg

2he. (C.2)

Here, λ0 is the free-space wavelength, and Y(n)0 (= 1/Z

(n)0 )

is the admittance of the constituent line accommodating thediscontinuity.

ACKNOWLEDGMENT

The authors would like to thank Mr. S. UmamaheshwaraReddy for the many valuable suggestions to improve the work.

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[8] T. Onodera, “An analysis of the π-mode stopband caused by asymmetriesof a metal-segment loaded helix as used in broadband traveling-wavetubes,” IEEE Trans. Electron Devices, vol. 35, no. 10, pp. 1758–1759,Oct. 1988.

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[24] C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles ofMicrowave Circuits. Cambridge, MA: MIT Press, 1948, (page 172,section 6.8, equation (17)—-discontinuity due to capacitive diaphragmin a parallel plate TEM-line using quasi-static approximation), ser. MITRadiation Series.

Subrata Kumar Datta received the B.E. degree inelectronics and telecommunication engineering fromBengal Engineering College, Shibpur, India, in 1989and the M.Tech. and Ph.D. degrees in microwave en-gineering from the Institute of Technology, BanarasHindu University, Varanasi, India, in 1991 and 2000,respectively.

He is currently a Scientist with the MicrowaveTube Research and Development Centre, DefenceResearch and Development Organization (DRDO),Bangalore, India. His current research interests in-

clude computer-aided design and development of helix and coupled-cavitytraveling-wave tubes, Eulerian and Lagrangian analysis of the nonlinear effectsin traveling-wave tubes, and the studies on electromagnetic wave propagationin slow-wave structures.

Dr. Datta is a Fellow of the Institution of Electronics and TelecommunicationEngineers of India and a Member of the Magnetics Society of India. He is thefounder General Secretary and a Fellow of the Vacuum Electronic Devices andApplications Society, India. He was the recipient of the Sir C. AmbashankaranAward of Indian Vacuum Society for Best Paper in 1998, the INAE YoungEngineer Award in 2001, the Sir C. V. Raman Young Scientist Award in 2002,the DRDO Agni Award for Excellence in Self Reliance in 2003, the DRDOScience Day Silicon Medal in 2004, the DRDO Technology Day TitaniumMedal in 2007, and the DRDO Laboratory Scientist of the Year Award in 2009.


Vemula Bhanu Naidu received the B.E. degreein electronics and communication engineering fromthe Army Institute of Technology, Pune University,Pune, India, in 2002.

He is currently a Scientist with the MicrowaveTube Research and Development Centre, De-fence Research and Development Organization,Bangalore, India. His research interests includecomputer-aided design and development of helixtraveling-wave tubes.

Mr. Naidu is a member of the Vacuum ElectronicDevices and Applications Society.

P. Raja Ramana Rao received the B.E. degree inelectronics and communication engineering from theV. R. Siddhartha Engineering College, NagarjunaUniversity, Guntur, India, in 1997.

He is currently a Scientist with the MicrowaveTube Research and Development Centre, DefenceResearch and Development Organization (DRDO),Bangalore, India. His research interests includecomputer-aided design and development of helixtraveling-wave tubes.

Mr. Rao is a member of the Vacuum ElectronicDevices and Applications Society (VEDAS), India, and the Magnetics Societyof India. He was the recipient of the DRDO Young Scientist of the Year Awardin 2005.

Lalit Kumar received the M.Sc. degree in physicsfrom Meerut University, Meerut, India, and the Ph.D.degree in physics from Birla Institute of Technologyand Science, Pilani, India.

He is the Director of the Microwave Tube Re-search and Development Centre, Defence Researchand Development Organization, Bangalore, India.His current interests include TWTs, microwavepower modules, and ultrabroadband high-power mi-crowave devices.

Dr. Kumar is a Fellow of the Institution of Elec-tronics and Telecommunication Engineers (IETE) and a Member of the IndianPhysics Association, Indian Vacuum Society, Indo-French Technical Associa-tion, and Magnetics Society of India. He is the Founder President and a Fellowof the Vacuum Electronic Devices and Applications Society (VEDAS), India.He was the recipient of the JC Bose Memorial Award of IETE for Best Paperin 1993, the Best Project Award of CEERI in 1993, and IETE-IRSI (83)-2001Award, and the DRDO Agni Award for Excellence in Self-Reliance 2003.

Baidyanath Basu received the M.Tech. and Ph.D.degrees from the Institute of Radiophysics and Elec-tronics, Calcutta University, Calcutta, India, in 1966and 1976, respectively.

He is the Director of the College of Engineeringand Technology, Moradabad, India. He was a Profes-sor and Coordinator with the Centre of Research inMicrowave Tubes, Department of Electronics Engi-neering, Banaras Hindu University (BHU), Varanasi,India. He was associated with the Central Electron-ics Engineering Research Institute (CEERI), Pilani,

Council of Scientific and Industrial Research (CSIR), India, as a DistinguishedVisiting Scientist of CSIR. He was with the University of Lancaster under theAcademic Link and Interchange Scheme (ALIS) between CSIR and the BritishCouncil. In addition, he was with Seoul National University, Seoul, Korea, asa Visiting Scientist. He has authored or coauthored about 100 research papersin peer-reviewed journals and two books entitled Electromagnetic Theory andApplications in Beam-Wave Electronics (World Scientific: Singapore, 1996)and Technical Writing (Prentice-Hall of India: New Delhi, 2007). His researchinterests and publications include helix-TWTs, gyrotrons, and gyro-TWTs.

Dr. Basu was a Member of the Technical Committee on Vacuum ElectronDevices of the IEEE Electron Devices Society. He is the present Presidentand a founder Fellow of the Vacuum Electronic Devices and ApplicationsSociety, India. He is a Fellow and the recipient of the S. V. C. Aiya MemorialAward for outstanding contributions in motivating research of the Institution ofElectronics and Telecommunication Engineers, India.

Simple Formulas for Stopband Attenuation Characteristics of Asymmetric Helical Slow-Wave Structures...

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