1288 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 7, JULY 1994

Design of Traveling Wave Tubes Based on Field Theory

Norman R. Vanderplaats, Mary Anne Kodis, and H. P. Freund

Abstract-A method is described for the design of helix travel- ing wave tubes (TWT) which is based on the linear field analysis of the coupled beam-wave system. The dispersion relations are obtained by matching of radial admittances at boundaries instead of the individual field components. This approach provides flex- ibility in modeling various beam and circuit configurations with relative ease by choosing the appropriate admittance functions for each case. The method is illustrated for the case of a solid beam inside a sheath helix which is loaded externally by lossy dielectric material, a conducting cylinder, and axial vanes. Extension of the analysis to include a thin tape helix model is anticipated in the near future. The TWT model may be divided into axial regions to include velocity tapers, lossy materials and severs, with the helix geometry in each region varied arbitrarily. The relations between the ac velocities, current densities, and axial electric fields are used to derive a general expression for the new amplitudes of the three forward waves at each axial boundary. The sum of the fields for the three forward waves (two waves in a drift region) is followed to the circuit output. Numerical results of the field analysis are compared with the coupled-mode Pierce theory. A method is suggested for applying the field analysis to accurate design of practical TWT’s that have a more complex circuit geometry, which starts with a simple measurement of the dispersion of the helix circuit. The field analysis may then be used to generate a circuit having properties very nearly equivalent to those of the actual circuit.

I. INTRODUCTION HE current design of traveling wave tubes is based on T the coupled-mode theory originally reported in 1947

by J. R. Pierce et al. [I]-[3] and subsequently refined by various authors [4]-[5]. The interaction is described in terms of coupling between the forward wave of the circuit and the space charge waves on the electron beam, where maximum growth of the circuit wave occurs when the circuit velocity is approximately equal to the velocity of the slow space charge wave. The space charge waves are typically calculated with the circuit, usually a helix, replaced by a conducting cylinder.

The implementation of the Pierce theory for design requires knowledge of the dispersion of the cold helix circuit and the “interaction impedance” defined by 2 = Ep/(2p;P), where E, is the axial electric field averaged over the beam, ,Bo is the propagation constant and P is the axial power flow for the cold circuit. While dispersion is readily determined by measurement, the interaction impedance is difficult to measure

Manuscript received September 1, 1993. The review of this paper was arranged by Associate Editor R. Temkin. This work was supported by the Office of Naval Research.

N. Vanderplaats and M. A. Kodis are with the Vacuum Electronics Branch, Naval Research Laboratory, Washington, DC 20375 USA.

H. P. Freund is with the Science Applications Intemational Corp., McLean, VA 22102 USA.

IEEE Log Number 9401534.

w: EI = 1 0, r < b - ~ 1 = 1 0 , b < r < a

w > l O , a < r < v ~ 1 > 1 0 , v < r < s

Fig. 1. matenal and axially conducting vanes enclosed in a conducting shell.

accurately for a helix. This has led to various approximate methods to determine this parameter based on correction factors applied to the sheath helix model to account for the effects of the helix tape and dielectric or shell loading of the helix [6]-[SI.

Chu and Jackson [9] provided a more complete theory based on the electromagnetic analysis of Maxwell’s equations applied to a sheath helix with no external loading; however the utility of their analysis was limited by the simplicity of their model and by approximations necessary to obtain limited numerical results. The present authors have reported results for the case of a thin annular beam inside a shell-loaded sheath helix [lo] and have recently extended the analysis to include the tape helix [ I l l with a solid beam of uniform current density.

In this paper we develop the dispersion relation for the sheath helix TWT from Maxwell’s equations and the rela- tivistic force equation by matching the transverse admittances at the boundaries between four (or more) radial regions. The boundary condition at the helix radius is expressed in terms of four admittance functions which vary individually as a function of the particular geometry considered. This basic approach makes it rather easy to accommodate a wide variety of beam and circuit configurations within the context of the basic formulation. This is illustrated for the case of a solid beam inside a sheath helix which is enclosed by a conducting shell and loaded by axial conducting vanes and by materials having arbitrary complex dielectric constant or magnetic permeability so that the effect of circuit losses is included. The configuration is illustrated in Fig. 1.

In Section I1 the expressions relating the axial a-c com- ponents of the velocity and current to the electric field are developed from the fundamental equations using essentially the approach described by Chu and Jackson [9] except that the only approximation made is the neglect of transverse beam

Model of the beam inside a sheath helix surrounded by dielectric

0018-9383/94$04.00 0 1994 IEEE

VANDERPLAATS et al.: DESIGN OF TRAVELING WAVE TUBES 1289

motion. The latter assumption implies that the beam is confined by a strong focusing system and does not interact with the TE components of the field.

The form of the fields for each radial region and the admittance functions are given in Section IT1 while the working equations for the cold circuit, the coupled beam-wave system, and the space-charge waves for a sever drift region are developed in Section IV. The coupled dispersion equation is stated simply by equating the TM admittance function calculated at the beam outer radius looking inward toward the beam to that looking outward toward the helix. A negative conductance (or positive resistance) for the admittance at the beam edge indicates transfer of power from the beam to the circuit, and vice versa.

The procedures and working equations for matching at boundaries between successive axial regions are described in Section V. A simple perturbation analysis which treats the presence of the beam as a perturbation of the cold circuit, is presented in Section VI. This yields a simple cubic polynomial for the approximate roots of the three forward- wave propagation constants which is used to accelerate the convergence for the iterative method used to obtain the precise roots. This procedure also requires evaluation of the axial power flow for the cold circuit which is developed in Section VI1 and is used in the remainder of this paper to compare the results of the field theory with the conventional Pierce theory.

11. BASIC APPROACH

k = w / c where c is the velocity of light in vacuum Pe = W/UO where uo is the dc beam velocity

P, = 2 where wp = [$$$-I ' is the relativistic plasma fre uency

1

-9 y = [ 1 - $ ] .

Here the dc current density J, is a positive quantity and the space charge density po = -en0 (where no is the density of electrons in the beam) is a negative quantity for electrons. Using (1) through (5) the axial components of (1) become in the beam region

111. FORM OF SOLUTIONS TO THE WAVE EQUATION The axial view shown in Fig. 1 illustrates the configuration

to be analyzed, consisting of a sheath helix enclosing a solid beam of uniform beam density and surrounded by dielectric material and axial vanes enclosed in a conduction shell.

For the symmetric mode of the sheath helix the axial fields have the general form

From Maxwell's equations, we obtain the vector wave equations for the electric and magnetic fields with space charge and current density source terms included.

where B = D = 0 for regions which include the origin. The form of the Bessel function argument x takes on one of the forms given by (8)-(10) below depending on the type and region of the solution, as follows. For E, inside the beam region (T < b ) the argument x in (7) is replaced by

d2 V p aJ V ~ E - p E - ~ = - + p-

at2 E at d2 V2H - pe-H = -V x J. a t 2 (1) r2 = p 2 [ 1 - ] . (8)

(P - P e l 2 The relations for conservation of charge, charge continuity,

and the Lorentz force equation are also required and are as follows

For the region behveen the beam and the helix ( b < T < a) the argument x is replaced by p , where

a P at V . J = -- Charge conservation (2) p 2 = k 2 - P2 where I C 2 = w2p0q. (9) ..

J = pv

Y

Charge continuity (3) Similarly from (6) x is replaced by p in the expressions for H , everywhere inside the helix radius (r < a) . Outside the

(10)

dv av - dt - dt + (' . = ' E) + B] helix (T > a ) , x is replaced by p, , where

p: = ETpTk2 - p2 = p2 + ( E T & - l)P Lorentz force equation (4)

where 90 = e /mo , m = ymo, and q = -e for electrons. For the linear analysis of the azimuthally symmetric mode, it is assumed that ac quantities vary as ej (wt-pz) where P is a complex axial propagation constant. In the analysis that follows we constrain the beam motion to the z-direction, J = J,z. By linearizing (2)-(4) we obtain the following basic expressions that relate the ac components of the current density and velocity to the axial electric field.

where

for both E, and H,, where E = E O E I and p = popT are the complex dielectric constant and magnetic permeability, respectively.

The general solution could be obtained by field matching at the boundaries between each region to eliminate the constants and obtain the dispersion equation. However, it is more convenient for our purpose to use the equivalent procedure of matching admittances at radial boundaries, which are defined by the quantities

I290 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 7, JULY 1994

From (7) and (1 1) the transverse fields and admittances have the form

The admittances defined in (1 1) are continuous at all radial boundaries except at the helix radius, where the helix current leads to a discontinuity in the magnetic fields. To obtain the hot dispersion equation, we will determine the beam admittance at the outer edge of the beam (at T = b) and set this admittance equal to the circuit admittance calculated at the same radius. Since the beam is assumed to flow only in the axial direction, it couples only to the TM component of the circuit field, although the waves must in general contain TE components in order to satisfy the boundary conditions at the circuit.

At the helix radius (T = a) the components of the electric fields are continuous while the total field in the direction of current flow on the helix is zero. The helical angle $ for the sheath helix model is defined by $ = tan-1(JZ/J6), where J, and J+ are the components of the current density at the helix radius. The magnetic field in the direction of the helical angle must be continuous, since current flow is zero perpendicular to the helical angle. These boundary conditions at T = a are expressed as follows.

E,, = E,; E,#,* = E4; E4i cos $ + E,; sin $ = 0

H,i sin $ + Hbi cos $ = H,, sin $ + H4,, cos $.

Then using (1 1) it is readily shown that the boundary conditions at the helix radius are equivalent to the following admittance matching condition

(13) where the subscripts i and o refer to the admittances inside and outside the helix, respectively. The ratio of the TM to TE field components may be determined from the helical boundary condition

Y H ~ + YE( Cot2 $ = Y H ~ + YE^ Cot2 $ at T = a

E, + Ebcot$ = 0 at r = a. (14)

We normalize the constants in (12) such that E, = A at r = a and utilize the following mixed Bessel function of two arguments which are encountered frequently in matching the

We also find that the following coefficients and constants appear in the admittance functions that partially cancel out in the complete dispersion equations

where ZO = && 1207r and E, and pr are the rela- tive dielectric constant and permeability which are generally complex quantities. It is convenient to introduce the following modified admittance functions

WEi = Y Y E i WE0 = % Y E o

. (17) WHi = Y y H i WHO = * y H o

IV. THE DISPERSION EQUATIONS

At the vane radius (r = w) E, = 0 while E4 is continuous. Both E, and E4 vanish at the conducting shell (T = s). On the axis the fields must remain finite. Using (12) and (17) with the appropriate boundary conditions on the axis and at the vane and shell radii, it is readily determined that the four admittance functions in (1 3) for the cold circuit corresponding to the geometry of Fig. 1 are as follows

Substitution of (18) into (13) yields the dispersion equation for the cold circuit

tan2$ = 0 (19)

where 00 denotes the cold circuit propagation constant. In general the relative dielectric constant E, and the relative magnetic permeability p, will be complex quantities for lossy materials.

The hot dispersion equation with the beam present may be written

where the case of a solid beam, by

is the beam admittance function given, for

where

VANDERPLAATS et al.: DESIGN OF TRAVELING WAVE TUBES 1291

and WE)^^^ is the circuit admittance function at the beam outer radius, given by

where, from (19)

1 P O p 2 IC2 [ Ppr P WE^ = - E , W E ~ + - tan2 $J W H ~ - - W H ~ . (23)

If the helix is replaced by a conducting cylinder at T = a, then WE^ goes to infinity in (22) and the dispersion equation for the space charge waves becomes

where PS denotes a space charge propagation constant. This expression is used to obtain the unattenuated space charge waves for circuit sever regions.

The dispersion equations are evaluated numerically using an iterative Muller procedure, starting with the cold circuit dispersion equation (19). Then the procedure is repeated using (20) to obtain the hot circuit roots, using initial estimates obtained from a cubic polynomial for the three forward waves which is derived in Section VI. After the first root is found, the dispersion function is divided by @ - P I ) to find the second root, and the resulting function is in tum divided by (p - pz) to find the third root. If desired, the unattenuated backward wave may readily be obtained by extending the process one more time by using -PO obtained from (19) as the initial estimate; however it is not needed for our three-wave analysis.

v. MATCHING AT AXIAL BOUNDARIES

Backward waves produced by reflection at axial disconti- nuities in the circuit are ignored in the three-wave analysis. We assume that the circuit is always terminated in the forward direction so that no backward waves are excited. For small reflections the error in neglecting backward waves will be small. High gain TWT's are designed to minimize reflections by avoiding any abrupt changes in circuit parameters and by using drift regions, called severs, with tapered loss attenuators at each end to provide isolation. The backward wave does not interact with the beam and its primary effect is to provide a feedback signal along the circuit which can result in periodic gain variations with frequency, or feedback oscillations if the net loop gain in any circuit section exceeds unity. These effects can be determined readily from knowledge of the reflection coefficients of the cold circuit.

We normalize the propagation constant according to the relation

which is similar to the Pierce normalization [ 3 ] except that Pierce's parameter C6 is replaced by 6. Letting a be the beam area, the total ac and dc components of beam current are given by i, = aJ, and IO = a&. Then equations (5) become

where V, is the effective beam voltage, taken as the average dc beam energy accounting for potential depression. It is con- venient to introduce the following dimensionless normalized parameters for the wave amplitudes

where the subscript k is used to denote one of the inde- pendently propagating waves so that equations (26) become simply

For the three-wave analysis, the three forward waves travel independently through the uniform circuit regions of the TWT and the total field, velocity, and current given by

3

ET = Ek k = l

3 3 .

must be continuous in each circuit or sever region. New wave amplitudes at circuit or sever discontinuities are obtained by solving (29) for the values of Ek in terms of the total E-field, velocity, and current just beyond the discontinuity. Then (28) determines the ac velocity and current for each wave. The solution of (29) for El is

which is a general expression that may be used to compute the launching loss [3] defined as E ~ / E T with ZIT = i~ = 0, the sever loss (for a short sever) with ET = 0, or new wave amplitudes at any circuit discontinuity. The wave amplitudes for E2 and E3 are obtained by permuting the subscripts in (30).

After the initial amplitudes are determined from (30), the new circuit fields for each wave at a distance z beyond the discontinuity are given by

E k ( Z ) = Ek(0)e-j@e ( l + j 6 k ) z (31)

and the new total circuit field, velocity and current are given by

k = l 3 -

For the case of a very short sever, the energy on the circuit is assumed to be suddenly absorbed in a perfect termination, and this is followed by a new circuit. In this case the new wave amplitudes are determined from (30) with ET = 0 and with i~ and VT equal to their values before the termination.

1292 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 7, JULY 1994

More commonly, the sever consists of a drift region with a tapered loss circuit termination at each end so that ET = 0 at the start of the drift region but WT and i~ are nonzero and equal to their values at the end of the preceding circuit section. For a drift region consisting of a perfectly conducting cylinder at radius a, the dispersion equation (24) has only two well-behaved low-order roots, which are real and correspond to slow and fast space charge waves.

As before we normalize the space charge propagation con- stants according to

with tapering of circuit characteristics to optimize gain flatness or other desired operating characteristics.

VI. THE PERTURBATION THEORY FOR OBTAINING APPROXIMATE ROOTS

For the perturbation analysis, the presence of the beam is treated as a perturbation of the uncoupled cold circuit. Maxwell’s equations for the coupled beam-wave system are

V x E = -jwpH = -jw(pCLI - jp,”)H V . B = 0 V - D = p V x H WEE+ J -jd’)E+ J

(36) and for the cold circuit without a beam, indicated by zero

where we note that y2 + -y1 in general. Here 6 k = xk + j y k in general but 21, = 0 in a drift region. The velocities and

fields for each wave by (28). We set the sum of the currents and

subscripts, Maxwell’s equations are V x EO = -jwpHo = - jw (p l - jp”)Ho V . Bo = 0

currents for the space charge waves are related to the electric V x Ho = ~ W E E O = j w ( & - j ~ ” ) E o V . Do = 0. (37)

velocities for the two space charge waves equal to the total initial current and velocity, respectively to obtain the initial field amplitudes, given by

From (36) and (37) we Obtain the relation

V . (ET, x H) + V . (E x H;) + E; . J El(()) = 6: [ u T ( o ) - j 62 iT(O) ] + 2w(~”E. ET, + p”H . HT,) = 0 (38)

We assume lossless materials for simplicity. Then by in- (38) Over a volume and applying the divergence

(34)

where 6 k = j y k . The wave amplitudes at the end of a sever of length L are then determined from Ek(L) =

E2(0) = 6; [UT(()) - j&iT(o)]

theorem we obtain Ek(0) exp( -P&L) and the total field, velocity, and current at the end of the sever region are determined from - S, ET,. ~d~ = 1 (ET, x H + E x HT,) . d ~ . (39)

S

The integration is performed over a volume element of axial length dz extending to the conducting shell at T = s. The terms at r = s vanish due to the boundary conditions and the condition f(z + dz ) - f (z ) = g d z leads to a term j ( P - PO) for the axial boundaries so that (39) becomes

(35) rE;Jrdr = j ( P - P o ) z.(E;xH+ExHT,)rdr (40) C

The section of circuit at the end of the sever is terminated in a short lossy attenuator, so that ET = 0 at the start of the following circuit section. The initial wave amplitudes are then determined from (30) with ET = 0.

It is apparent from this development that the total complex amplitude for the sum of the forward propagating waves is obtained by applying a series of simple transformations as the wave progresses through the TWT. At the TWT input coupler, the initial wave amplitudes are obtained from (30) with WT = 0 and i~ = 0. Then (29) is applied to obtain the total circuit field, current, and velocity at the end of the initial uniform circuit region. If this is followed by another circuit region with different characteristics, then (29) is reapplied to obtain total amplitudes at the end of the next circuit region. If it is followed instead by a sever, then (35) is used to determine the total current and velocity at the end of the sever which provides the initial condition for the following circuit region with ET = 0. This process is easily programmed to include any arbitrary number of circuit sections and severs and to compute the total gain or phase delay at any position within the TWT. Tapering of velocity or RF loss may be included in a piecewise linear fashion. This provides considerable flexibility to design TWT’s

where J # 0 for r in the range c < r < b and c = 0 for a solid beam. Next we note that the average axial power flow for the cold circuit is given by

~ = q L ’ z . ( E ; x H o + E O x H T , ) r d r

M 1’ z . (ET, x H + E x HT,)rdr (41)

where the approximation follows because we are assuming that the cold circuit is only slightly perturbed by the electron beam. Now we assume that J = J z and E: are approximately con- stant over the beam region and invoke the relation previously obtained between J and E,

where JO is the dc beam current density and VO is the beam energy. Relating the dc current density to the total dc current by 10 = r [b2 - c2]Io, (41) and (42) yields the dispersion relation

4Y(Y + 1)VoPfo(P - P e ) 2 ( P - Po) + PeIoE:oE* = 0 (43)

VANDERPLAATS et al.: DESIGN OF TRAVELING WAVE TUBES 1293

which shows similarities to the Pierce theory. We take E,*,E, M IE,o12and apply the Pierce definitions for the interaction impedance and gain parameter C (modified for relativistic velocity)

This yields a simple cubic polynomial for the three forward propagating waves which is used to provide an initial estimate of the roots of the dispersion equation when applying Muller’s method to solve the hot dispersion equation given by (20).

(P - P e l 2 ( P - P O ) + p e p , 2 c 3 = 0. (45)

VII. EVALUATION OF POWER FLOW

In solving for the field expressions to determine the power flow in (44) the constants were adjusted so that E, = A at T = a. Then the Pierce interaction impedance at r = a is given by

where PT = Re(P) is used because the Pierce definition of beam interaction impedance assumes that the circuit has zero loss while the actual circuit may have low loss, leading to complex P. The average beam interaction impedance is obtained by evaluating the average value of IEZl2 over the beam region

b Re [ IEZI2rdr. E2 2Z(r = a)

Z(Beam) = I Z ( r = U ) = A2 b2 - c2

Y L

(47) We define an impedance reduction factor Fb by the expres-

sion

Z(Beam) = FbZ(r = U ) (48)

where for the case of a solid beam

The axial power flow may be written as follows for the symmetric mode of the sheath helix

- P, = :Re lZT dq5 I’ T(E x H*) . zdr

where PT is the real part of P and E: and pk are the real parts of the relative dielectric constant and the relative magnetic permeability respectively. This suggests writing Z ( r = U ) in the form

where

VIII. NUMERICAL RESULTS AND COMPARISON TO THE PIERCE THEORY

(55 )

The model for the numerical calculations is based on a tape helix circuit used in a current experiment for which the cold circuit dispersion has been measured. The circuit is supported by cylindrical posts of aluminum oxide material and loading ridges were used to reduce the variation of phase velocity with frequency. The Pierce interaction impedance was previously derived from the equivalent circuit analysis by Paik [12]. The “effective dielectric constant” external to the helix was determined in the manner prescribed by Paik [12]. Then the helix and vane radii in Fig. 1 for the sheath helix model were adjusted to match the measured phase velocity approximately over the frequency range from 0.2 to 1.6 GHz, with the results shown in Fig. 2.

The calculations for the conventional coupled-mode analy- ses were based on the circuit and electronic equations (10-4) and (10-74) of Hutter [4] except for substitution of relativistic velocity into Hutter’s equations. The space charge reduction factors were determined by replacing the helix by a conducting cylinder at the same radius. In order to obtain a direct com- parison with the conventional Pierce analysis, the interaction impedance used for the Pierce calculation was calculated from the field analysis. The result of this comparison is shown in Fig. 3 where the gain/cm for the growing wave is shown for both cases. The agreement between the field theory and the modified Pierce theory is found to be surprisingly close for this particular case. For both calculations the effective beam energy at VO = 16 keV is 15.773 keV which was determined at the beam radius enclosing one-half the total beam current.

The dispersion equations for the Paik equivalent circuit model are based on the field theory and are equivalent to those used here. However the Pierce interaction impedance for Paik’s model relies on quasi-static assumptions and does not agree with the rigorous field theory. The discrepancy is shown in Fig. 4, which compares the Pierce interaction impedance calculated using the field analysis with that using the Paik equivalent circuit analysis, where Paik’s result is seen to be

I294

1 I --7

I40 I8kVBeam 1 0 2 6 - - -

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 7, JULY 1994

_ - r . , , \

3s 24

0 23

0 22

021

0 20

Fig.

Reld .\. Calculation

a = 1486cm v = 3 OlOcm s = 3 533 cm &=IO3

02135

14 kV Beam

40 1 . 1 1 4 i _ L L A i _ _ L 1

0 0 0 4 0 8 1 2 1 6 2 0 I - l-i L I L__>

0.5 I V= 16kV

a = 1.486 cm v = 3.010 cm s = 3.533 cm

\ \ 0 1 L 3 L l - I 2 . U

0 0 0 4 08 1 2 1 6 2 0 Frequency (GHz)

3. Gainkm versus frequency for the sheath and modified Pierce models.

about 12-18% too high over the 0-2 GHz frequency range for this case.

Distributed loss is commonly applied to the dielectric sup- port structure in helix TWT's in order to prevent oscillations or reduce the gain ripple. The effect of increasing the circuit loss was studied by increasing the loss tangent, tan 6 ~ , where tan6L M -E:. Fig. 6 shows the RF loss/cm and for the cold circuit and the gain/cm of the growing wave for the hot circuit dispersion as a function of the dielectric loss tangent. The gainkm approaches zero in the high loss limit but does not disappear completely. The gain and the cold circuit loss are equal with a value of 0.38 dBlcm for a loss tangent near 0.115, which corresponds to a cold circuit loss of 2.64 dB per wavelength at 1 GHz. The circuit wavelength decreases substantially for values of loss tangent greater than 1.0 and is reduced by a factor of two for a loss tangent of 9.3.

The effect of increasing the circuit loss on the forward propagation constants of the TWT are shown in Figs. 6 and 7 where the values of the real and imaginary parts of the propagation constants are plotted as a function of the loss tangent. In these figures (@.)I is the (almost) unattenuated wave while (&)z and (@a), are the growing and decaying waves, respectively. For zero loss the growing and decaying roots are complex conjugates, that is ( P u ) ~ = (Pa);. As the loss is increased the imaginary part of the growing wave decreases as expected while the magnitude of the imaginary part of the decaying wave increases; that is the decaying wave decays more rapidly with increasing loss. At still higher loss a point is eventually reached for which the decaying wave ( P U ) ~

2.5 1 V = 1 6 k V I = I . O A b = 0.762 cm a = 1.486 cm v = 3.010 cm s = 3.533 cm & = 1.03

tan w = 0.2135

1 GrowingWaveGain ,/ 1

0.01 0.10 I .00

Loss Tangent

Fig. 5. Comparison of the gain/cm for the growing wave and the loss/cm for the cold circuit as a function of the dielectric loss tangent.

can no longer be found using the iterative numerical procedure, while (@.)I and (pa)z evolve into a complex conjugate pair with very small complex part, becoming unattenuated space charge waves in the limit of high loss with real parts approximately centered around the beam propagation constant

Fig. 8 illustrates the evolution of the cumulative three- wave gain versus axial distance with a stepwise tapered attenuation located between z = 40 and 70 cm with a maximum attenuation rate of 0.654 dB/cm over the 22 cm long central region of the attenuator. This provides 35.5 dB gain at z = 90 cm with 25.5 dB of distributed attenuation. We note that the total gain initially decreases. This cannot be explained by the net attenuation of the three waves because the attenuation of pa)^ is extremely small while pa)^ and (Pa)3 are complex conjugates, which provides a hyperbolic cosine type of growth with distance for the latter pair of waves. Instead the initial decrease in gain occurs because Re(Pa)l < R e ( p ~ ) ~ so that the difference in the propagation velocities causes destructive interference to occur initially. For the case of zero loss this would superimpose a cyclical variation with a periodicity of L = 27r/(Pz - P I ) in the total gain versus z .

The gain is shown as a function of the beam current at several frequencies in Fig. 9. Also shown for reference is the average beam energy versus the beam current, determined at the radius enclosing one-half the total beam current. Referring to Fig. 2 we note that at 0.6 GHz the circuit velocity is greater than the beam velocity, which is referred to as an

P e a .

VANDERPLAATS et al.: DESIGN OF TRAVELING WAVE TUBES 1295

1.10 L - d ’ ’ ’ -

1wj 10” 10-1 Id IO’ Loss Tangent

0.6 0’7 f

0.2

0.1 1 ’,, 1

1 1.4 GHz

16

1s.s

9 IS * 8

14.5 - b I‘ B m 13.5

Fig. 6. function of the dielectric loss tangent.

Real part of the propagation constants for the forward waves as a Fig. 9. of the beam current.

Gaidcm at several frequencies and average beam energy as a function

Fig. as a

0.00 ; -

Loss Tangent

Imaginary part of the propagation constants for the forward waves 7. function of the dielectric loss tangent.

25

? 20 C .- 3 15

g 10 U

5

Cumulative Gain

t

0

0 10 20 30 40 50 60 70 80 90 (cm)

Fig. 8. tributed loss located 40-70 cm from the input.

Cumulative three-wave gain versus distance with 25.5 dB of dis-

“undervoltaged” operating condition. As the beam current is increased, the beam depression causes the TWT to become increasingly undervoltaged, which results in the slow increase in the growth rate with current observed at 0.6 GHz. At 1.0 GHz the beam velocity is higher than the circuit velocity by a moderate amount, which is reflected in a greater initial increase in gain with increasing beam current. At 1.4 GHz the circuit is severely “overvoltage&’ at low beam current and the gain begins to increase very rapidly for beam current above 0.6 A. The growth rates reach a maximum in the vicinity of 3.5 to 5.0 A of beam current over the 0.6 to 1.4 GHz frequency range in spite of the large differences in the beam-wave synchronism condition. These results illustrate the widely observed phenomena that (1) beam-wave synchronism

Fig. 10.

0.6

0.5 h

E

d 0.4 .B 0.3 0

0.2

0.1

f a = 1.486cm I v=3.010cm - s=3.533cm

0 t .~ I- _ . - L L - L I I - - l L i 0.0 0.4 0.8 1.2 1.6 2.0

Frequency (GHz)

Gaidcm versus frequency for several beam voltages at I = 1.0 A.

is much more critical at low beam current (usually expressed in terms of perveance = Ib/V:’2) than at high beam current and (2) the space-charge debunching forces eventually dominate over the circuit bunching forces at sufficiently high values of beam current. Further analysis is needed to quantify the effects of space charge at high values of current in the context of the field theory.

The gain/cm for the growing wave versus frequency is shown in Fig. 10 for various beam voltages at Ib = 1 A. At low frequency the growing wave gain is relatively insensitive to the beam voltage and proportional to the frequency. Setting 61 = 2 1 + jy, in (31) the growing wave gain is 2010g10(e)x~,&2 decibels, which is proportional to the number of wavelengths. The growing wave gain is seen to be very sensitive to the beam-wave synchronism conditions at the higher frequencies.

Ix. SUMMARY AND DISCUSSION In this paper we have provided a new formulation of the

field theory for traveling wave tubes in terms of matching admittance functions at the helix and beam radii and have included multiple axial regions in a form that may be readily applied to the practical design of TWT’s. The capability to simulate a variety of practical beam and helix circuit configurations simply by inserting the appropriate admittance function for the particular geometry into (19) or (20) is a significant advantage to this approach. For example, for the case of a hollow beam of constant current density and inner radius c, the beam admittance function (21) for the solid beam

1296 IEEE TRANSACI’IONS ON ELECTRON DEVICES, VOL. 41, NO. 7, JULY 1994

is replaced in (20) by

where

(57)

For situations in which the beam current density is nonuni- form, or for very high beam current where substantial beam depression occurs, the beam may be divided into several annular regions with the current in each annular region is specified independently. Conductor losses at the shell may be incorporated by specifying the admittances at the shell radius in terms of the wall resistivity and translating those admit- tances to the helical boundary to determine the admittance functions YE^ and YH, to be substituted into (19).

However, there remain several technical issues to be in- vestigated in order to extend the field theory approach to include the full range of configurations and beam focusing methods used in modern TWT’s. First is the question of how accurately the actual circuit, which is often very complex, may be modeled by any idealized circuit model. Based on the observation that the transverse dimensions of practical helix structures are usually quite small in comparison with the wavelength over the usual range of TWT operation, we might expect that dimensional adjustments to the model to match the measured dispersion will also enable the model to predict the hot circuit performance accurately. However the range of validity for this hypothesis remains to be fully tested.

A second issue is the effect of the helix tape configuration used in most TWT’s as compared to the sheath helix model. One effect of the tape is the presence of spatial harmonics, only one of which interacts cumulatively with the beam. To first order, the Pierce interaction impedance at the helix for the sheath helix may be multiplied by a tape correction factor F(tape) given by

where w/p is the ratio of the tape width to the helix period. This expression is derived from the simple assumption that E, is constant in the gaps at the helix radius. This effect by itself might be expected to reduce the gain by approximately the one-third power of F(tape); however this effect may be partially offset by the increased inductance of a tape structure, which will also modify the dispersion.

Another important issue concerns the effect of the type of beam focusing system. The assumption of zero transverse beam velocity is valid only for strong (immersed flow) focus- ing in which individual electrons follow the magnetic lines of flux from their origin at the cathode, with only a small component of beam rotation resulting from radial space charge forces. At the opposite extreme of “Brillouin” focusing, which is utilized in most lightweight TWT’s, the individual electrons

rotate at the cyclotron frequency and pass through the axis while the beam as a whole rotates about the axis at one-half the cyclotron frequency. For this case the assumption of zero transverse beam velocity may result in significant error for the case of high perveance beams. In addition the polarity of the magnetic field in relation to the helical angle may be a significant factor for high values of tan +. The self-consistent treatment of the interaction with periodic magnetic focusing and beam ripple poses a further challenge.

The authors are continuing to extend the field theory and nu- merical procedures in order to expand its utility as an accurate design tool over a wider range of parameters. Replacement of the sheath helix by a thin tape helix model is considered to be the most significant near-term improvement. In the interim the present field theory method may be usefully employed, either by itself or as an adjunct to the Pierce theory, to explore the effect of varying parameters such as vane loading, distributed loss, velocity tapering, etc. for optimizing the TWT performance to achieve the desired operating characteristics.

REFERENCES

[ l ] J. R. Pierce and L. M. Field, Proc. IRE, vol. 35, p. 108, 1947. [2] J. R. Pierce, Proc. IRE, vol. 35, p. 111, 1947. [3] J. R. Pierce, Traveling-Wave Tubes. [4] R. G . E. Hutter, Beam and Wave Electronics in Microwave Tubes.

York Van Nostrand, 1960. [5] A. H. W. Beck, Space-Chrge Waves. [6] P. K. Tien, Proc. IRE, vol. 41, p, 210, 1953. [7] B. J. McMurtry, Proc. IRE, vol. 9, no. 2, p. 210, 1962. [8] T. Hashimoto, ‘‘Electronics and communications in Japan,” vol. 52-B,

no. 10, 1969. [9] L. J. Chu and J. D. Jackson, Proc. IRE, vol. 36, p. 853, 1948.

[lo] H. Freund, M. A. Kodis, and N. R. Vanderplaats, IEEE Trans. P l a s m Science, vol. 20, p. 543, 1992.

[ 1 I] H. Freund, N. Vanderplaats, and M. A. Kodis, “IEEE Trans. Plasma Science, vol. 21, p. 654, 1993.

[12] S . F. Paik, IEEE Trans. Electron Devices, vol. ED-16, Dec. 1969.

New York Van Nostrand, 1950. New

New York: Pergamon, 1958.

Norman R. Vanderplaats was bom in Modesto, CA, on January 24, 1937. He received the B.S.E.E. degree from the University of California, Berkeley, in 1959 and the M.S.E.E. degree from Stanford University in 1964.

He was with Varian Associates from 1959 to 1967 and the Watkins-Johnson company from 1967 to 1972, where he was engaged in the development of high power microwave tubes. Since 1972 he has been employed at the Naval Research Laboratory, currently as an Electronics Engineer in the Vacuum

Electronics Branch. His research interests are focused primarily on high power microwave and millimeter-wave RF amplifiers.

Mary Anne Kodis received the B.S. degree in electrical engineering from the University of Califomia, Davis, CA, in 1978. She received the M.S in 1980 and the Ph.D. in 1987 for work on the parametric interaction of microwave fields with plasma density gradients.

She joined the Naval Research Laboratory in 1987 where she has been working on emission-gated microwave amplifiers.

H. P. Freund, photograph and biography not available at time of publication.