[Advances in Microwaves] Advances in Microwaves Volume 6 Volume 6 || Electron Dynamics and Energy...

Electron Dynamics and Energy Conversion in O-Type Linear-Beam Devices*

Harry K. Detweilerf and Joseph E. Rowe

ELECTRON PHYSICS LABORATORY

DEPARTMENT OF ELECTRICAL ENGINEERING

THE UNIVERSITY OF MICHIGAN

ANN ARBOR, MICHIGAN

I. Introduction . . . . . . II. Equations of Motion

A. General Force Equations . . . . B. Magnetic Field Components C. Combined Form of the Equations of Motion

III. Generalized Two-Dimensional Analysis . A. Large-Signal Interaction Equations for the Traveling-Wave Amplifier B. Closed-Form Solutions of the Radial Wave Equations . C. Expressions for Gain, Efficiency, and Harmonic Currents D. Conservation of Power and Momentum Equations

IV. Approximate Two-Dimensional Analysis . . . . A. A Set of Simplified Two-Dimensional Nonlinear Interaction Equa

tions . . . . . . . . B. A Deformable-Disk Model for the Electron Stream C. Approximate TWA Equations Based on the Deformable-Disk Model

V. Computer Solutions of the DDM Equations A. General . . . . . . . . B. TWA Solutions for a Uniform Magnetic Field C. TWA Solutions for Spatially Varying Magnetic Fields D. Special Topics . . . . . .

VI. Computer Solutions of the General Equations and Comparisons with the DDM Results A. General . . . . . . . . . . B. TWA Solutions of the General Equations . . . . C. Comparison with the DDM Results . . . . .

VII. Experimental Investigation of a TWA Employing a Uniform Magnetic Focusing Field . . . . . . . . . A. Introduction . . . . . . . . . B. Experimental Results . . . . . . .

30

32 33 35 39

41 41 50 55 55

58

58 62 63

69 69 70 90 93

96 96 99

108

111 111 111

* This work was supported by the Rome Air Development Center under Contract AF 30(602)-3569.

t Present address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California.

29

30 Harry K. Detweiler and Joseph E. Rowe

VIII. Conclusions .

List of Symbols

References

118

119

123

I. INTRODUCTION The traveling-wave tube has proven its versatility in a wide variety of

applications ranging from low-noise receivers to high-power radar systems. It is particularly impressive that noise figures as low as 3 dB have been achieved and efficient traveling-wave amplifiers are in operation at various power levels ranging from milliwatts to megawatts. In some applications weight is of primary concern, while in others overall energy conversion efficiency must be optimized. In such sophisticated applications for the traveling-wave amplifier as a transmitter tube in multichannel communica-tions satellites, both minimum weight and high conversion efficiency are of the utmost importance and thus it is desirable to examine in detail the electron dynamics and bunching process in tubes with minimum-weight focusing systems.

A typical traveling-wave amplifier configuration is schematically illustrated in Fig. 1. In essence, this device consists of an electron gun, an RF circuit

ELECTRON GUN

CATHODE COLLECTOR

FIG. 1. Schematic illustration of a traveling-wave amplifier.

capable of supporting the propagation of a slow electromagnetic wave having a longitudinal electric field component, transducers for coupling the RF signal onto and off of the circuit, and an electron collector. Electrons are emitted from the cathode and focused into a beam in the gun region such that their motion is predominantly parallel to the axis of the slow-wave circuit. They next traverse the circuit region where an interaction with the RF wave propagating on the circuit takes place. Subsequently, the electrons enter the

O-TYPE LINEAR-BEAM DEVICES 31

collector and are removed from the device. When conditions are proper, the beam-wave interaction which occurs in the circuit region results in a net transfer of energy from the beam to the wave with resultant amplification of the RF signal. The increase in power carried by the circuit wave comes about as a result of a conversion of the kinetic energy of electrons in the beam. This interaction also produces a radial perturbation of the electron trajectories. In a magnetically focused tube, a static magnetic field is employed to con-strain the electron beam as it passes through the circuit region ; it may be a uniform axially directed field or spatially varying (e.g., periodic or tapered).

A general nonlinear interaction theory is used to investigate the effects of transverse fields, i.e., radial circuit fields and radial space-charge fields, in traveling-wave amplifiers for a variety of beam-focusing conditions. Magnetic focusing fields which are periodic or tapered (increased) with distance along the device are considered in addition to uniform magnetic fields. In particular, results are presented for Brillouin flow and near-Brillouin flow (partially immersed flow at fields near the Brillouin value), and the minimum magnetic-field strength required to effectively constrain the electron beam is determined as a function of the operating parameters for the various focusing systems. Confined flow is also considered for the uniform-field case in order to have a basis of comparison from which the effects of radial motion of the beam electrons can be determined. The results of such investigations indicate the importance of transverse effects and further yield information on the stability of strongly modulated cylindrical electron beams.

The generalized two-dimensional nonlinear interaction theory developed herein is based upon that of Rowe (7), suitably modified to account for spatially varying magnetic focusing fields. A Lagrangian formulation is used in which the electron beam is subdivided into representative charge groups that are followed through the interaction region. For the type of device being considered, the electron beam is modeled in terms of axisymmetric charge rings which are allowed to expand and contract in response to the RF circuit field, space-charge field, and static magnetic field; interception on the RF structure is accounted for in a rigorous manner. This Lagrangian approach accounts for nonlinear effects and permits the specification of arbitrary input conditions on the distribution of charge density or current and velocity. Such a technique is thus readily adaptable to a consideration of the effects of the thermal velocity of the beam electrons. This general analysis system can be used to investigate the dc behavior of the beam if desired by simply deleting the RF circuit equations and circuit-field terms in the force equations and specifying the beam input conditions. A klystron analysis can also be per-formed by appropriately reducing the interaction equations. As indicated above, spatially varying magnetic focusing fields are included. Other focusing systems, such as electrostatic focusing, can be treated by modifying the force

32 Harry K. Deîweiler and Joseph E. Rowe

equations. While the theory presented here is based on the assumptions that the cold-circuit phase velocity does not vary with axial distance and that there is no applied dc potential gradient in the interaction region, the basic method is applicable to any configuration; the interaction equations are easily altered to allow for the spatial change of the device parameters that is characteristic of tapered-circuit or velocity-jump tubes.

A useful simplification of the general theory is developed by eliminating the effect of the backward-traveling wave, neglecting the RF space-charge forces in the beam and reformulating the problem on the basis of a disk-model representation of the electron beam. This analysis is referred to as the "deformable-disk-model (DDM) analysis" and has been shown to be quite accurate (2). By using this simplified approach, solutions can be obtained in considerably less computer time than is required for the general theory. The design engineer thus has the opportunity to use the two analysis techniques in a complementary fashion in designing efficient light-weight traveling-wave amplifiers.

One of the significant results obtained from the general theory is that the RF current in a traveling-wave amplifier is carried primarily on the beam edge for both Brillouin flow and confined flow. This contrasts with the klystron in which the RF current is on the periphery for Brillouin-flow condi-tions, but is rather uniformly distributed throughout the beam under confined-flow conditions. Confirmation of this situation, and of other operating characteristics predicted by the theory, has been obtained by performing experiments on a power traveling-wave amplifier. One part of the experi-mental investigation was carried out by mounting this device in a "beam analyzer" and measuring the fundamental and harmonic RF current dis-tributions over the beam cross section through the use of a coaxial RF beam-sampling collector. The agreement between the theoretical and experimental results is excellent and indicates the general utility of the nonlinear theory.

II. EQUATIONS OF MOTION

The model used to represent an electron beam passing through the inter-action region of an O-type device is illustrated in Fig. 2. A right-handed cylindrical coordinate system (r, φ, ζ), with the origin located at the input plane of the interaction region, is used throughout. A cylindrical electron beam enters the RF circuit region with an initial radius b'. The RF circuit, whose radius is taken to be a, may assume a number of different configura-tions. Circuits used for a traveling-wave tube generally fall within two categories: nonresonant structures such as the helix and ladder line, and resonant structures like the disk-loaded waveguide or coupled cavities. In a klystron, the circuit may be a series of modulating gaps or resonators and


drift tubes. Some focusing system is required to obtain good transmission of the beam through the interaction region ; a magnet system producing a finite

FIG. 2. Model representing an electron beam passing through the interaction region of an O-type device.

magnetic field is considered here. In the analysis which follows, the system is assumed to be axially symmetric, i.e., the electron beam, circuit field, and magnetic field are all assumed to possess axial symmetry.

A. GENERAL FORCE EQUATIONS

The motion of charged particles in the presence of electric and magnetic fields is described by the Lorentz force equation. The beam velocity and wave phase velocity are assumed to be small compared to the velocity of light, i.e., (vlc)2<^ 1. As a result, nonrelativistic mechanics are applicable, forces due to RF magnetic fields and the magnetic field generated by the motion of the electrons are negligibly small, and the equations of electrostatics can be used to calculate the space-charge fields. Under these conditions, the Lorentz force equation may be written for electrons in the interaction region as

^ = - h | [ ( E s c + Ec)+vxB] (1)

where -r)=e\m, the charge-to-mass ratio for the electrons (e is a negative number and m is the rest mass), v is the velocity vector, and B is the magnetic field vector. The electric field vector has been expanded into its component parts in this expression; Esc is the space-charge field and Ec is the electric field resulting from the RF wave on the circuit. By virtue of the assumed axial symmetry, there is no functional dependence on the angular position (φ is an ignorable coordinate). The electric fields acting upon the electrons (Esc and Ec) then have only radial and axial components which are, in general, functions of r, z, and t (time). Since nonrelativistic velocities have been assumed, the effects of the self-magnetic field of the electrons and the


RF magnetic fields are negligible. Therefore, the total magnetic field vector can have only radial and axial components (determined by the applied static magnetic field) which are functionally dependent only upon r and z in this axially symmetric system. The electric and magnetic field vectors are written below in terms of their components with functional dependencies indicated :

Esc = ÏE8C -r(r, z91) + z Esc - z(r, z, t) (2a) Ec = PEc_r(r,z,t) +zEC-z(r,z,t) (2b) B=tB,(r9z)+iB,(r9z) (2c)

where P and Î are, respectively, the unit vectors in the radial and axial directions.

Equation (1) can be expanded to give the following component equations:

£-'(*)'~'"(*~+Ä-*+"$) <3°> ;« ( "$ ) - " i ( *Ä - ' - a ) w

g = - h | ( £ s c _ 2 + £ c _ 2 - 5 r r ^ (3c)

where the functional dependencies as given previously have been omitted for brevity and d<p\dt is the electron angular velocity.

A general solution to Eq. (3b) can be obtained through the application of the divergence theorem and Leibnitz's rule. The result is

r2 _9 = | η | Γ rßz dr + (constant) (4) dt Jo

which is recognized as Busch's theorem in its general form. Although Eq. (3b) was presented to describe the angular motion of the electrons in the inter-action region, it applies in the gun region as well, provided that the gun is axially symmetric. The constant therefore can be evaluated in terms of the initial conditions at the cathode. Assuming that the thermal velocities of the electrons are negligible, which is usually true in power tubes, dy\dt = 0 at the cathode where r = rc and Bz = Bc. With very little loss in generality, the magnetic flux density at the cathode, 2?c, can be assumed to be independent of radius since, in an actual device, it is usually possible to adjust conditions so that this is the case. The possibility of a radial variation of the axial magnetic field component in the interaction region is, however, retained.

Under the initial conditions mentioned above, Eq. (4) takes on the form

0-TYPE LINEAR-BEAM DEVICES 35

where </rm= 2-nrBzdr, which is the total magnetic flux linking the disk Jo

defined by the electron radius r. This result is quite general and can be applied to cases in which the axial magnetic field varies with radius as well as with axial distance.

The closed-form expression for αφ/ώ, given by Eq. (5), can be used to eliminate its appearance in the radial and axial force equations. When this is done, Eqs. (3a) and (3c) become

£~I,I«..,+*.>-*[£(».-£)

and

2 [irr2 \rj

d2Z , , , „ „ , 7|2,

dt2'

(6a)

(6b)

It remains to evaluate these force equations for the magnetic-field distribu-tions which are of particular interest for O-type linear-beam devices.

B. MAGNETIC FIELD COMPONENTS

When the dc beam behavior for spatially varying magnetic focusing fields is considered, it is common practice to assume a particular axial-magnetic-field distribution, generally neglecting any radial variation of the axial com-ponent. The electron motion is then described by the force equations appro-priate to the assumed distribution. However, if radial variations of the axial component are neglected, the assumed field may not satisfy Maxwell's equations. The approach followed here is to develop the magnetic-field distributions which are of interest directly from Maxwell's equations. This procedure yields the form of the radial variation associated with a particular axial field variation, and the conditions under which the radial variation can be safely neglected are easily determined.

Consider the following two Maxwell's equations:

V - B = 0 , V x B = 0

which, under the assumed conditions, the magnetic field must satisfy. For an axially symmetric system these equations yield the following differential equation for Bz,

! ί ( ^ · ) + ^ - 0 (7) r dr\ dr I dz2

A direct consequence of Eq. (7) is that the axial magnetic field component can be independent of radius only if it varies linearly with z or is constant.


For any other type of magnetic field, Bz necessarily has a radial dependence. When Bz does not vary with radius, Br is independent of z; it is zero when Bz is constant and is linearly proportional to r when Bz varies linearly with z.

By applying the method of separation of variables, in which it is assumed that Bz can be written in the form

Bz(r,z) = B0 + h(r)g(z)

where B0 is the constant or uniform part of the axial magnetic field, the following equations are obtained:

h"{r) + (l/r)A'(r) -ß*h(r)=0 (8) and

g"(z)+ß*g(z) = 0 (9)

where ß is an undetermined constant. In these equations, primes are used to indicate differentiation with respect to the independent variable of the function.

There are three types of axial magnetic field variations which are of particular interest here. These are (1) a linear taper of the axial magnetic field from one constant value to a new constant value (usually in the output region), (2) a periodic component of axial magnetic field superimposed on a constant axial magnetic field, and (3) an axial magnetic field with an expo-nentially increasing component. The first case includes the uniform-field case since the taper can be taken as zero, and can also represent an approximate step change in field or field reversal by appropriately tapering the field over a short distance. The magnetic field components for these three cases will be developed next.

1. Uniform and Linearly Tapered Magnetic Fields

If ß is taken as zero in Eqs. (8) and (9), the resulting general solution is

Bz(r, z) = B0 + (ολζ + c2)(c3 In r + c4)

Since from physical considerations the magnetic field must remain bounded as r -> 0, c3 must be zero, with the result that Bz is independent of radius. The magnet structure can be arranged so that Bz varies as shown in Fig. 3(a). The actual variation will be described mathematically by a straight-line approximation given by

B, = B0 + [U.2(z -z0) - U_2(z - Zl)][BJiz, - z0)] (10)

where U_2 is the unit ramp function having the properties

TT , Λ /O for z<0 U-^=\z for z>0


Bx can be either positive or negative so that Eq. (10) applies for both linearly increasing and decreasing magnetic fields. Combining Eq. (10) and the diver-gence relation, and applying the requirement that Br be bounded as r -> 0 yields

Br = -M*i / (* i -*o)][*/-i(* -*o) - t/-i(* -*i)l 01)

where U _x is the unit step function having the properties

fO for z < 0 tf-i(*) = 1 for z > 0

It is seen that Br is zero when Bz is constant and is linearly proportional to radius when Bz varies linearly with z as illustrated in Fig. 3(b).

(a)

B2 APPROXIMATE

(b)

- Br FOR z 0 < z < z

Br = 0 FOR z < z Q , z > z f

FIG. 3. Axial and radial magnetic field variations for the linear-taper case.

Strictly speaking, Br and Bz are not correctly given by Eqs. (10) and (11) since both contain discontinuities in slope at the points z0 and zl9 which is


physically impossible. In the actual case, the change takes place over a finite distance and the discontinuities do not occur.

When the magnetic field is uniform throughout the interaction region, Eqs. (10) and (11) reduce to Bz = B0 and Br = 0.

2. Periodic Magnetic Field

For ß2 nonzero and positive, it follows from Eqs. (8) and (9) that the axial component of the magnetic field can be expressed as

B,(r,z) = B0+2BN cos (Nßmz)I0(Nßmr) N=l

(N odd)

when the peak value of the axial field occurs at z = 0. If the magnet structure is arranged so that only the N = 1 component

exists, then Bx(r, z) = B0 + Bx cos (ßmz)I,{ßmr) ( 12)

where j3mê27r/L, L=the magnet period, and Z^êthe peak value of the varying component of the axial field on the axis (r = 0).

Unlike the case of a linearly tapered magnetic field, it is seen from Eq. (12) that Bz for a periodic magnetic field is dependent on radius. For some magnet structures this may have an important effect and should be taken into account.

From Eq. (12) and the divergence relation, and requiring boundedness on the axis, the radial magnetic field is found to be

£,(/·,z) = Bx sin (ßmzy.ißmr) ( 13)

For the types of devices of interest here, ßmr is usually sufficiently small (approximately 0.6 or less) that it is reasonable to use the following approxi-mations in the magnetic field expressions:

7oÛ8mr)sl, ΛΟΜ^ίΑηΓ

When these approximations apply, Eqs. (12) and (13) reduce to

BZ = B0 + Bx cos (j8mz), Br = \ßmrB1 sin (ßmz) (14)

These expressions should be quite adequate for nearly all cases in which periodic magnetic focusing fields are utilized.

3. Exponentially Increasing Magnetic Field

If ß2 is a negative number (nonzero), it is convenient to replace ß2 by - a 2

in Eqs. (8) and (9). Proceeding as before, the magnetic field components for an exponentially increasing axial magnetic field are :


Bz(r, z) = B0 + Βγ exp (amz)/0(amr)

£,(/·, z) = - Βλ exp (amzj/^amr)

where am=the axial magnetic field growth rate and Z^âthe value of the varying portion of the axial field on the axis (r = 0) at z = 0. As in the case of a periodic magnetic field, Bz is seen to have a radial dependence.

Considering only moderate magnetic field growth rates, amr will be small enough so that

/0(amr) £ 1, /i(amr) £ £amr

With these approximations, the magnetic field components are given by

Bz = B0 + B1 exp (amz), Br = - ^(xmrB1 exp (amz) ( 16)

The exponentially increasing magnetic field has been included here so that the effects of a field which varies at a rate slightly greater than a linearly increasing magnetic field can be investigated. Equations (16) will be sufficiently accurate for such an application.

C. COMBINED FORM OF THE EQUATIONS OF MOTION

Expressions for the magnetic field components accounting for radial variations in the axial magnetic field were derived in the preceding section. From the mathematical forms describing the magnetic fields, it can be seen that for most O-type linear-beam-device configurations, the variation over the diameter of the electron stream is small enough so that it can be safely neglected. Under these conditions, the equations of motion for each of the three cases can be conveniently combined into a single set of equations where-in the terms are defined appropriately for each particular case.

If the radial variations of the axial magnetic field are sufficiently small, the axial and radial components of the magnetic field can be written as

Bzk = B0 + Β,Μζ), Brk = -^B, [dfk(z)ldz] (17)

where fk{z) is a function denoting the axial-field variation. It should be pointed out that the magnetic field components, as given by

Eqs. (17) may not exactly satisfy Maxwell's equations. However, these expressions are obtained from the magnetic fields which are solutions by pas-sing to the limit of small radial variation of the axial magnetic field compon-ent. They should be sufficiently accurate for most practical applications.

Upon substitution of the above expressions for the magnetic field compo-nents, the radial and axial force equations become :

(Prldt*- - h | ( ^ s c - r + c_r)-i^cV{[^+yf c/ f c(z)]2-(r0/r)%4 (18a)

d*z/dt* = - \η | (Esc_2 + Ec_,) - io&rV* [dMz)ldz] [ξ, +ykfk(z) -(rJr)*Ktf\ (18b)

40 Harry K. Dei we Her and Joseph E. Rowe

and the equation for the angular velocity becomes

dfp/dt = \œck[L· +y *Λ(ζ) - {ΦΥΚΙ£] ( 18c)

Each parameter and function in Eqs. (17) and (18) which bears a subscript k takes on a particular meaning that depends on the form of magnetic field under consideration. These are summarized in Table I. Definitions for the parameters which are introduced in this table are: ^co=\v\^ γ^Β^Β^ K00t(rclr0Y(BcIB0)\ ων1*\η\Βΐ9 and Κ^{φ,)\Β,ΙΒλ)\ where r0*the electron radius at z = 0.

Table I

PARAMETER AND FUNCTION DEFINITIONS FOR THE VARIOUS TYPES OF MAGNETIC FIELDS

Type of magnetic

Uniform

Linearly tapered

Periodic

field

Exponentially increasing

™vk

ω,·ο

"\ :0

ω,ι ωΓΐ

(t

1

1

l/yo i/yo

y*

0

yo

1 1

JÄZ)

0

(Zi - Z«) COS ßmZ

exp (amz)

Kok

^ 0 0

^ 0 0

^ 0 1

^ 0 1

K00 (a cathode-flux parameter) is the square of the ratio of the magnetic flux linking a disk defined by the radius of the electron at the cathode (rc) to that linking a disk defined by its radius at the input plane (injection radius, r0) of the interaction region. This definition differs from that sometimes used in the analysis of perturbed dc flow where the cathode-flux parameter is defined instead in terms of the equilibrium radius. The force equations given above will ultimately be used in the solution of the large-signal interaction problem which is solved as an initial-value problem. Thus, it is more con-venient to use a cathode-flux parameter defined on the basis of the initial radius since the equilibrium radius is not generally known a priori for a perturbed flow. Of course, if K00 is adjusted to give space-charge-balanced flow, r0 will be the equilibrium radius.

Another cathode-flux parameter, K01, has been defined for the periodic and exponentially increasing field cases in terms of Bx rather than the constant part, B0. The obvious reason for this choice is that B0 can be zero, i.e., the axial magnetic field may be purely periodic or exponentially increasing with no superimposed constant component. The equations of motion in the form given above will be incorporated into the large-signal interaction equations in the next section.

In the event that a consideration of the device geometry indicates that the radial variation of the magnetic field will have an important effect, the more

0-TYPE LINEAR-BEAM DEVICES 4)

exact forms of the magnetic field components should be used in developing the equations of motion. This has been done elsewhere (2).

III. GENERALIZED TWO-DIMENSIONAL ANALYSIS

A fluid-flow analysis in which spatial position and time are the independent variables is inapplicable in the large-signal regime since electron overtaking and trapping can occur, which result in the velocity and current being multivalued functions of position. This difficulty is successfully resolved through the use of a Lagrangian or particle-dynamics treatment. In this type of treatment the entering beam charge is subdivided into representative charge groups and these individual charge groups are followed through the interaction region. For the two-dimensional, axially symmetric problem con-sidered here, the appropriate charge groups are axially symmetric rings (also referred to as electron rings or simply electrons) possessing the same charge-to-mass ratio as electrons. Their radial, angular, and axial motions are dictated by the equations of motion presented previously. The remaining equations required to describe the electron-wave interaction are obtained from an RF circuit equation, the conservation of charge, Poisson's equation, and the wave equation. The equations given below have been developed by Rowe (/) and Detweiler (2) and are only summarized. Modification of these equations for other device configurations which incorporate tapered-velocity circuits and voltage jumps is easily accomplished.

A. LARGE-SIGNAL INTERACTION EQUATIONS FOR THE TRAVELING-WAVE AMPLIFIER

The new independent variables used are related to the longitudinal position of an electron ring z, the initial radius of a ring r0, and its entrance time t0 (or alternatively its initial longitudinal position z0). The spatial quantities are normalized with respect to the initial average electron velocity w0, and the charge-ring entering time is transformed to an entry phase relative to the applied RF wave. The normalized independent Lagrangian variables have been defined as

yt (Γω/Μ0)ζ, xQt (Cco/w0)r0, Φ0Ε (ω/Μ0)ζ0 = - ωί0

where ω is the radian frequency of the RF signal and C is the usual gain parameter. It is clear that y is a normalized axial distance, x0 is the normalized initial radius of a charge ring, and Φ0 is its entrance phase with respect to the RF wave at the input, i.e., y = 0.

The normalized radius is defined by x=(Ca>/w0)r. (When applied to field quantities, JC is treated as an independent variable, whereas when used to


designate an electron's radial position it is treated as a dependent variable.) A dependent phase variable has been defined according to

Φ=(ωΙΐ40)ζ-ωί-θν(γ) = (ylQ-œt-ey(y)

Figure 4 is a flight-line diagram indicating the way in which the phase-variable relation has been developed. Four flight lines are shown in this figure. The

o

=>

7 yi

z UJ Έ UJ u < _l Q. ω û

wt OR PHASE

FIG. 4. Large-signal flight-line diagram.

dashed line represents a reference trajectory of an electron which moves through the interaction region with the initial average beam velocity u0

starting at zero position and time. As a result of the large-signal interaction, all electrons are actually accelerated or decelerated and a representative perturbed trajectory is shown to indicate this fact. The remaining curves depict the flight lines for the unperturbed and the perturbed circuit wave. The phase velocity of the circuit wave differs from the cold-circuit (unperturbed) phase velocity as a result of the stream loading and under large-signal conditions is a function of axial displacement. dy(y) denotes the phase differ-ence between the perturbed RF wave and the reference electron traveling at u0. It is negative, indicating a phase lag, and in the nonlinear case it varies with displacement as a result of the interaction.

The dependent variable Φ is seen to denote the phase position of charge


groups at a particular displacement plane relative to the RF wave. Each charge group has a particular value of Φ dependent upon its initial co-ordinates; thus

Φ=Φ(γ,χ0,Φ0)

It can be seen that Φ has a dual nature; in addition to indicating the phase position of individual charge groups at any displacement plane, it denotes the phase of the traveling RF circuit wave at any y plane. The actual circuit-wave phase velocity is obtained (άΦ/ώ = 0 when moving with the wave) as

v(y) = ^ (19) Vy) \-C[dey(y)ldy]

The undisturbed circuit phase velocity is

O0*Uol(l+Cb) which defines the relative injection velocity parameter b.

It can be easily seen that the time variable Φ is based on a coordinate system which moves at the initial average electron velocity. Thus, the co-ordinate transformation which has been used results in a shift from a sta-tionary coordinate system to one moving at the velocity w0.

The RF potential in the interaction region arising from the wave on the circuit is defined as the product of slowly varying amplitude and phase functions by

V(z^t) = V{y,x&) t Re{(Z01 /01 IC)A(yMx) exp ( -]Φ) exp [j%(x)]} (20)

where Z0 is the characteristic impedance of the circuit at the frequency of the wave and I0 is the dc beam current (a negative number). The gain parameter is given by

C3êZ0|/0 | /4K0

where V0 is the dc potential on the axis. In Eq. (20), A(y) is the normalized circuit-voltage amplitude and represents

the axial amplitude variation of the RF circuit voltage. Φ takes on the meaning of the phase of the traveling RF wave on the circuit and accounts for its axial variation in phase. The functions φ(χ) and θχ(χ), which are assumed to be functions of radius only, represent the radial variation in the interaction region of the RF fields associated with the wave on the circuit; φ(χ) accounts for the amplitude variation and ^(JC) the phase variation. These are denoted, respectively, as the radial amplitude coupling function and the radial phase coupling function. On the circuit, which is located at

0 | r = a = l, ftjr=a = 0


The remaining dependent variables to be defined are the electron velocities, which are given by

dzldttuQ[\+2Cuy(y,x^)} drldt*u0[2Cux(y9xM] (21) d9ldttuM2Clr)U(p(y,xM}

where 1 + 2 Cuy, 2 Cux, and 2 Cu9 are, respectively, the normalized axial, radial, and circumferential electron velocities.

A generalized velocity-phase equation (a relation between dependent variables) has been obtained as

d<P(y,x0,<P0) +ddy(y) = 2^Çy,xo,0o) dy dy l+2Cw? /(^xo,0o)

The normalized radius of an electron ring is found by evaluating

\X dx = \\dxldy)dy J xa J o

along its trajectory, which gives

x(y,*M =*o + 2 c \y ^ ^ ο > φ ο ) dy> (23)

where the prime is used to designate the variable of integration. In the region r < a, the RF potential must satisfy the homogeneous wave

equation, V2 V(z, r,t)-c "2 [a2 V(z, r, t)/dt2] = 0

if the stream RF charge is neglected. Substituting the appropriate derivatives of the potential and separately equating the coefficients of sin (Φ -0 r ) and cos (Φ -6X) to zero yields the following two radial wave equations:

„ ,d2A(y) Ai w / J \ dey(y)Y Λ( .ά2ψ(χ) ίάθχ(χ)\2

^ W ~JV - ΑΜΦ(χ)[ c - ~~- I + A(y) -^y - l·—- J Λ(γ)φ(χ)

+ ^ ) ^ (24a) x dx \C J

-Μο-άΨΥ-ψ dy2 dx dx

+ A(y)Kx) *°** + ^ i W ^ - 0 (24b) dx2 x dx

where kc=u0lc. Since nonrelativistic flow has been assumed, ke2<^\, and


the term containing ke can be safely neglected. Closed-form solutions to these radial wave equations will be obtained later.

Rowe (7) has shown that an equivalent-circuit transmission-line model can be used to determine the effects of the stream on the RF wave. In this method, a second-order partial differential equation relating the RF potential along the circuit to the charge induced on this circuit by the electron stream serves as a substitute for Maxwell's field equations in solving the RF interaction problem. This technique is valid for any number of circuit space harmonics providing that the appropriate phase velocities and impedances are used.

The circuit equation for the variation of the RF voltage along the equivalent transmission line, located at r = a, is (for a forward-wave interaction)

The parameters v0 and Z0 are, respectively, the characteristic phase velocity and characteristic impedance of the one-dimensional transmission line and are chosen to be equal to the corresponding quantities of the actual circuit. The axial circuit loss is given by the loss parameter d9 which is defined by

dt 0.01836 IjC

where / is the series loss expressed in dB per undisturbed wavelength along the circuit. The terms appearing on the right-hand side of Eq. (25) are the driving terms, with σ(ζ, /) representing a linear charge density induced on the circuit by electrons in the stream. The axial circuit equations have been derived as (7) :

_ 2(1 + Cb)Γ r">·/ΐ*"φ(χ') cos Φ'x0' dx0' d0o' irCxl Uo Uo l+2Cuy(y,xo',0o') '

r>*<Kx')sm0'xo'dxo>d0o'\-)

Jo l+2Ca^,Xo',*o') /J

_ 2(1 + Cb)f r*V/pepsinΦ'x0'dx0' d0o' TTCX\. UO UO l+2Ct/„(>',xo')0o')

- ICd Γ2'^*') c o s φ' χο dxo <MV\ 1 Jo 1 +2 Cuy(y,Χο',Φο') ) \

(26b)


It remains to develop the equations of motion given previously in terms of the normalized Lagrangian variables. In doing so, the transformation

a-as-«'+204 which is the particle derivative written following the motion of the electrons, will be used. Using this transformation and the definition of the electron velocities given by Eqs. (21), the following component force equations have been developed (2).

Radial force equation:

p(l+2Cuv) By

= CA(y) ί ^ cos [Φ - θχ{χ)] + φ(χ) ^ sin [Φ - 0,(*)]]

X

8C~3\

1,1 fic-r (27a) 2C2œu{

Axial force equation:

p(l+2 Cuv) dy

ϋφ{χ) ^ cos [Φ - θχ(χ)] -(\-C α^^Α(γ)φ(χ) sin [Φ - θχ(χ)]

-£(Έ)»Ψ[Μ*>-{?)**] ^ ' Esc-z (27b)

2C2œu0

Direct substitution of the new variables into Eq. (18c) yields the following expression for the normalized angular velocity :

2C\ _ 1 ίω0Λ ~x~ 2C\œ)

^+ηΛ-ω-^0)2«] (27c)

The function fk(y) can be determined for the various types of magnetic focusing fields from Table I with the aid of the definition y = (Ccu/w0)z. Kok

can be determined in terms of the normalized radii at the cathode and input


planes by following a similar procedure with χ = (Οω/«0>; it remains numerically the same as when expressed in terms of these radii before normalization.

Explicit expressions for the space-charge fields can be obtained through the solution of Poisson's equation within the interaction region by using Green's function techniques. Rowe (7, pp. 95-97) has carried out this procedure by finding the space-charge potential determined from the Green's function for a delta-function ring of charge located in a perfectly conducting drift tube. Expressions for the radial and axial space-charge fields are then found by differentiating this space-charge potential function. Utilizing the results, the space-charge terms appearing in the radial and axial force equations [Eqs. (27a) and (27b)] can be written

" | 7 ?Us c_ r ( ;c50) 2C2œu0

= f e ) T X°' αχο\2/2-Λχ,χΊΦ,ΦΊ d<P0' (28a)

2C2wt/0

= 2 ^ ( ä ) X ^ ° ' ^° j/2-,Ο^',Φ,Φ') sgn (Φ-Φ')αΦ0' (28b)

where ωρ is the radian plasma frequency for electrons defined by

.„ 2Δ hlpo CUp = —

«0

and ( 1 for Φ>Φ'

8 Εη(φ_φ')Δ o f o r φ=φ> \-\ for Φ<Φ'

The functions F2_r and F2_z are the space-charge weighting functions which are defined by

F2_r(x,x'&&')

A fJMx'lxa)ViMxg)]exp Γ_ / Ç\ \Φ-Φ'\ zéi UM)]2 I \ xj

F2_z(x,x'&&')

i- i [Λ(μ012 L \ l xJ J

where μι is determined from the successive zeros of J0(pi), i.e., Λ0*ί) = 0.


A few comments pertinent to the space-charge-field expressions are appropriate at this point. Since the Green's function method gives the total potential, the space-charge fields obtained therefrom are the total fields and include the effects of the dc, as well as the RF, space charge. The space-charge forces have a short-range nature, that is, the forces between charge groups fall off rapidly as their spatial (or phase) separation increases. For phase separations approaching ±π, the mutual forces are essentially zero; thus, it is only necessary to integrate over a 2π phase range. The space-charge forces are obtained from these equations by performing the indicated integrations in a straightforward manner; this is readily accomplished once the weighting functions are found and the positions (in radius and phase) of all charge groups are known. However, some difficulties are encountered with this formal procedure when performing space-charge calculations for a finite number of discrete charge groups, as is done in a computer simulation of the interaction. The modifications used for the discrete case are discussed in Appendix A of Detweiler (2)\

In summary, the system variables are as follows :

Independent y9 x0, and Φ0

Dependent A(y) ey(y) ux(y,xo,0o) φ(χ) θχ{χ) uy(y,xO90o) x(y,x0&o) Φ&,*ο>φο) Μφ&,χ0,Φ0)

Equations (22)-(24), (26), and (27) form a system of nine continuous equations in terms of the nine continuous dependent variables. These equations, together with the space-charge-field expressions constitute a complete set of large-signal interaction equations for the magnetically focused traveling-wave amplifier which must be solved to determine its performance: a prodigious task to say the least.

In solving the above system of equations, the problem could be treated as a boundary-value problem in which the conditions at the input and output planes are specified. However, since the output conditions are not known beforehand, an iterative solution procedure would be necessary. Alterna-tively, the problem can be solved as an initial-value problem by specifying the values of the dependent variables (and their derivatives where required) at the input and then integrating along particle trajectories through the interaction region until saturation is reached. This procedure yields the optimum device length as well as the saturated output. A requisite condition for the validity of this approach is that only forward-traveling waves exist in the device. This condition is satisfied provided the circuit structure is terminated everywhere in its characteristic impedance and no reflections occur at the output. Since


good results are obtained by using the initial-value method and its use reduces the complexity of the solution procedure, it has been used exclusively in obtaining the solutions presented here.

For solution of the interaction problem as an initial-value problem, the following initial values and parameters must be specified.

RF signal:

(1) A(y)\y=0tA0, the normalized amplitude of the input RF signal. (2) dA{y)jdy \ y =0, the rate of change of the normalized RF signal amplitude

at the input. When the beam is initially unmodulated, dA(y)ldy\y=0 = -A0-(1 + Cb) d, which is zero for a lossless circuit (d = 0).

(3) Oy(y)\y=0= 0 since the RF signal is applied and the beam enters at y=0.

(4) dey(y)ldy\y=0. Again when the beam is initially unmodulated, dey(y)ldy\y=0=-b.

These initial values of the derivatives of A(y) and 6y(y) can be obtained directly from Eqs. (26) by using the fact that the entering beam is unmodu-lated so that the beam has no effect on the wave. Thus, the wave phase velocity and amplitude are constant (for d = 0) leading to the second deriva-tives of the circuit variables being zero. Also, the induced-current integrals (the terms appearing on the right-hand side of these equations) are zero because of the absence of initial modulation.

Beam input conditions:

(1) Input electron velocities for an unmodulated beam are

ι/„(0, χ0,Φ0) = Uy(09 x0\ ux(0, χ0,Φ0) = ux(0, x0)

The electron velocities are specified in this manner to allow for the possibility of a radial variation of the initial axial electron velocities and for initial radial electron velocities resulting from improper beam entrance conditions.

(2) Initial phase positions for an unbunched beam are specified according to

0 O J =27r(y- l ) / (m- l ) y = l, 2, ...,m

where m is the number of charge groups injected at each initial radius during one cycle of the RF wave. Through this specification the injected electrons are uniformly distributed over phase.

Parameter specifications:

(1) C, the gain parameter. (2) d9 the circuit-loss parameter which may vary with axial distance.

50 Harry K. Deîweiler and Joseph E. Rowe

(3) b = (w0 - v0)ICv0, the injection velocity parameter. (4) γα. (5) b'la, the ratio of the initial beam radius to the circuit radius. (6) ωρ/ω, the normalized plasma frequency. (7) <*Wa>, ξfc, yfc, and/*(>>) to describe the strength and type of magnetic

focusing field. (8) Kok = Kok(x0), the cathode-flux parameter which, in general, can be a

function of initial radius as indicated.

The initial-value problem is thus completely specified by the nonlinear interaction equations mentioned previously and the initial conditions and parameters given above. Unfortunately, exact analytical solution of the problem cannot be carried out due to the nonlinear character of the equations. Solutions can be obtained on a case-by-case basis using a high-speed digital computer. A number of representative computer solutions are presented later.

B. CLOSED-FORM SOLUTIONS OF THE RADIAL WAVE EQUATIONS

It is possible to obtain solutions in closed form (neglecting RF space-charge effects) for the radial amplitude and phase coupling functions, φ(χ) and θχ(χ), which are valid for most cases of interest. These solutions are obtained from the radial wave equations [Eqs. (24)] and take their place in the set of large-signal equations, thereby reducing by two the number of equations which must be solved simultaneously.

Since A(y) and φ(χ) are nonzero in the interaction region, Eqs. (24) can be written as

and

where

d^^Am+^)Yx)=o (29)

*0JLx) + /J_ #W + Γ) d^x) =My) (30) dx2 \ψ(χ) dx xj dx

f(V\äJL-i}i-^M\ dA(y) J2°y(y) JlKy)~A{y)\C dy ) dy dy*

Recall that the RF voltage on the circuit is defined as the product of slowly varying amplitude and phase functions. The functions fx(y) and f2(y) then should also vary relatively slowly. Thus, for the purpose of determining the coupling functions, φ(χ) and θχ(χ), they may be evaluated at a particular displacement plane, yi9 and then treated as constants in the solution of Eqs.


(29) and (30). This procedure is not exact, but proves to be a satisfactory means for taking into account the radial variation of the fields.

By proceeding in this manner, Eq. (30) can be integrated once to yield

ËjM=Uyi)_rx,nx,)dx, (31) dx χφ2(χ) J o

In obtaining the above result, the boundedness of d6x{x)jdx at x = 0, neces-sary from physical considerations, has been utilized.

Note that Eq. (29) is nearly a Bessel's equation if [ddx(x)ldx]2 is small compared to/2(^,·) for all x and y{. This has been shown to be true for most cases of practical interest (2), so that

φ(χ) = BJfax) + B2K0(oLiX) (32)

where α,·=[/20\·)]1/2· By applying the boundary conditions that ψ(χ) is bounded at the origin and is unity at the circuit, Eq. (32) becomes

/ο(«Λ)

This coupling function is seen to have the same form as that for a thin hollow stream, which is a consequence of neglecting the RF charge in the stream. Admittedly, neglect of the RF stream charge in determining the coupling functions is an approximation. However, the overall effect of this approximation should be small and seems reasonable, especially in view of the considerable complexity of solving the problem exactly. The coupling is seen to change as the electrons move radially. In the large-signal case the coupling varies with axial distance as well since at varies as a result of the stream loading the RF circuit.

In computing the radial forces on the electrons, the derivative of the coupling function is also required. From Eq. (33) this is found to be

#W = ^ Λ(«<*) ( 3 4 ) dx */0(αζ.χα)

When the expression for φ{χ) given by Eq. (33) is substituted into Eq. (31), the integral on the right-hand side can be evaluated to yield

dO. Âx)_xfi(yù\{_(iMiX)Y be 2 I WoLiX)) dx 2 L \/0(α,·χ)/

This can, in turn, be integrated with the result

(35a)

A(*)=^(*/#^4^!) (35b) 2at \ /o(atxa) l0((XiX)/


where the value of θχ at the circuit has been set equal to zero, i.e.,

#*(*«)= 0

The solutions for the coupling functions presented above are fairly general. The assumptions required for their derivation are relatively nonrestrictive and are easily satisfied for most cases of interest. A comparison of results obtained using the closed-form solutions of the coupling functions to those obtained when the radial wave equations are solved numerically has shown a very close agreement between them, which serves as sufficient justification for the approximation employed. For cases in which the assumptions are violated, the radial wave equations will be coupled to the other large-signal equations and the entire set must be solved simultaneously; a separate solution, such as given above, will not be accurate.

The coupling functions and their derivatives, as given by Eqs. (33)-(35), are in forms convenient for use in obtaining computer solutions, i.e., having solved for the circuit variables at a particular displacement plane, fY{yt) and Myt)9 and therefore ψ(χ)9 θχ{χ\ and their derivatives, can be evaluated and used for the subsequent solutions at the next displacement plane. The way in which the coupling functions vary with radius and axial displacement is illustrated in Figs. 5 and 6. The data for plotting these curves were obtained from a solution of the large-signal interaction equations.

i.o

0.9

0.8

0.7

-0.6

0.5

0.4

0.3

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

NORMALIZED RADIUS, x

FIG. 5. Radial amplitude coupling function vs normalized radius.


0.10

0.09

0.08

0.07I

Ί αοεΙ

[ 0.05

0.04

0.03

0.02

0.01

0

C = 0.1 b= 1.0 ke = 0

* a : 0.15,(yOS1.65) b'/o = 0.7

y = 3.2

y= 1.2 y =9.0 INITIAL

STREAM J EDGE " Π

J L J L 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

NORMALIZED RADIUS, x

FIG. 6. Radial phase coupling function vs normalized radius.

The initial location of the stream edge is indicated in these figures so that the degree of radial variation of these functions can be seen. It is obvious that, for a beam filling factor as large as in this case (b'\a = 0.7), the variation of the coupling functions across the stream is not negligible; thus, it must be taken into account if accurate results are to be obtained.

A simplified treatment of the large-signal interaction problem is of con-siderable interest because of the complexity of the general nonlinear equa-tions. In a simplified treatment, evaluating the coupling functions and their derivatives to the degree of accuracy obtainable from Eqs. (33)—(35) is unnecessary. Approximate forms of these equations, which are sufficiently accurate, are easily obtained.

It has been shown that (2)

and

where

2vi dA{yt) / ι ( Λ ) £

A(yd dy

1 C dy

(36)

(37)


Utilizing these expressions, Eqs. (33)—(35) become

Xx)JJp* m„vJbp* (3g)

_ θχ(χ) = (dAildldn Lh^}_ _ Jj^Î]

άθχ(χ) = ldA{yt)ldy\ Γ /I^x)\ \ A(yt) / * L WviX))

(39)

dx

These approximate expressions retain the eflFects of the variation in the phase velocity of the RF wave, which is the primary contributor to the variation in coupling with axial distance at a fixed radius. It is easily seen that a change in coupling necessarily results when the phase velocity changes, since the electrical length corresponding to a given spatial distance is dependent upon the phase velocity. The relationship of vt to the wave phase velocity is

v-t = UolCviyj)

In many cases the variation of the wave phase velocity is sufficiently small that in determining the coupling functions it can be assumed to be constant. A first-order approximation is obtained by assuming that the phase velocity does not deviate markedly from its unperturbed value. A new parameter (a constant) is defined by

vk(\+Cb)IC

which is related to the unperturbed wave phase velocity by v = u0ICv0

In order that the wave phase velocity remain close to its unperturbed value it is necessary that

\-C[dey(y)ldy]^\+Cb (40)

To determine the order of approximation involved in Eq. (40) the following expansion is carried out:

1 - C [dey(y)ldy] = \+Cb-C{b + [dey(y)ldy]}

- ( l + C A ) ( l - * ± W ^ )

The term multiplying 1 + Cb in this equation generally does not deviate much from unity since it is found that

\b + [dey(y)ldy) \<l

i " I

when C = 0.1. Thus, the appropriate expressions when the variation of the


phase velocity is to be neglected are Eqs. (38) and (39) with v, as defined above, replacing v{. Using v in the coupling functions is least accurate for small values of b. If the greater accuracy is necessary, v{ should be retained.

C. EXPRESSIONS FOR GAIN, EFFICIENCY, AND HARMONIC CURRENTS

The determination of gain, efficiency, and harmonic-current content of the electron stream is of fundamental interest in evaluating traveling-wave-amplifier performance. Expressions appropriate for the two-dimensional case are summarized here.

The gain is calculable from

gain(^=201og10[/i(^)Mo] (41)

The total RF power along the structure is obtained by a calculation of the Poynting vector and since the beam power is | /01 V0l the efficiency is given by

V(y) - 2CAHy){l:^l^) - 2 W (42)

It can be seen that the ratio of 1 - C d9y(y)ldy to 1 + Cb is typically close to unity so that

v(y) s 2C[A\y) - /f02], Pc(y) = 2C \ I01 V0A\y) (43)

For an initially unmodulated beam with no initial radial variation of the axial velocity, the magnitudes and phases of the harmonic currents are given by

in(y)\ 4- =A[(P ' Γ^ηΦ'χ0'άχ0'άφΛ IQ TTXhWU Jo /

+ if V ['"sin ηΦ'χ0' dx0' άφλ (44a)

f b' f sin ηΦ'χο dx0'd<P0' phase IW) = arctan ^-^° (44b)

\ T I cxh' r2,r V ° 7 Ί cos ηΦ' χ0' dx0' άΦ^

Jo Jo where AI = the harmonic number.

D. CONSERVATION OF POWER AND MOMENTUM EQUATIONS

The concept of conservation of power, in addition to providing useful information on the energy flow in a traveling-wave amplifier, can serve a very practical purpose, namely that of checking the accuracy of solutions to the large-signal equations obtained by computer means.

56 Harry K. De t we Her and Joseph E. Rowe

Considering the stream-circuit system as a whole, there is a total flow of energy across any section of the system, which includes both potential energy flow and kinetic energy flow of the electrons, where the former is made up of the total flow of RF potential energy and the electron potential energy flow. The conservation of power requires that when moving from any particular axial displacement plane to a plane farther along the tube, the net loss in kinetic plus potential energy flow must exactly equal the loss due to electrons striking the circuit and resistive heating.

Assuming no loss (i.e., no interception or circuit loss), negligible space charge and small kinetic energy associated with the radial and angular electron motion, the conservation of power equation has terms remaining which involve (1) the average power on the circuit, (2) the average power in the stream-circuit coupling field, and (3) the change in the average axial electron kinetic energy. These assumptions are in addition to those made earlier.

An expression for the total average RF power along the circuit is obtained by a calculation of the Poynting vector as

/> e W = 2 C | / 0 | M ^ ) ( 1 - « | > ^ ) (45)

The average power in the coupling field is expressed by

where Vc is the circuit potential, i.e., V(r = a), and iljC(y) is the fundamental component of the current induced on the circuit by the stream. The following result can be obtained by evaluating the terms in Eq. (46):

1 + Cb \irxi'J Jo Jo 1 + cos Φ' XQ dx0' αΦ0'

2 Cu^y9xo\0o')

This term, which is of order C2, is small compared to the circuit power and can be approximated by

/>„ _cO0 3 4σ| / . | M w f ~ i e ) (47)

The average kinetic power, designated by Pb(y), is obtained by integrating the product of the stream current density and the kinetic voltage [i.e., {dz\dtf\21 η | ] over the stream cross section and taking a time average. Upon performing these operations and introducing the normalized variables, the result is found to be

P*(y) = I /.I K 0 ( - L ) f *"' Γ [1 + 2 Cuv(y,x0\<P0'Wx0' dx0' <®0' (48) \iTXtl J « Jo


The conservation of power requires, under the assumed conditions, that

Pc(y) +Pb -c(y) +P*(y) = constant

The value of the constant can be determined from the initial conditions. It is easily seen from Eqs. (45) and (47) that

/> c (0 )=2C | / 0 |Mo 2 , Pb_c(0)=0

For no initial velocity modulation, Eq. (48) yields

Ph(0) = \I0\V0

as it should. The "conservation of power" theorem for this case can then be written as

2CA

= 2CA* + 1 - ( J L ) J*ft' J** [1 + 2 Cuy(y,x0\<P0')Yx0' dx0' d<P0' (49)

There is another relation between the circuit variables and the stream velocity which resembles the above conservation theorem. It is obtained directly from the general interaction equations in the following manner. After invoking the assumptions used in deriving the conservation-of-power theorem, the axial force equation, Eq. (27b), is divided by 1 + 2 Cuy, θχ(χ) is neglected, and the resulting equation is integrated over initial radius and phase to yield

[2"duy(y,x0',<t>0') r v r2,r

Jo Jo x0' dxo d<P0'

[2π φ(χ') cos Φ' XQ dx0' d<P0' dy Jo Jo \+2Cuy(y,x0',<P0')

Λ r dey(y)\ Λ(Λ Γ V Γ2" ψ(χ') sin Φ' x0'dx0'd<P0' " l 1 " C " * " r W J o Jo 1 + 2 C ^ * O ' , 0 O ' ) ( 5 0 )

Examining Eqs. (26) (for d = 0), it is recognized that the terms remaining on the right-hand side of these equations also appear in Eq. (50). Making the appropriate substitutions, integrating in y, and applying the same initial conditions used above yields

2C A\y){{\ - C[dey{y)ldy]f + (1 + Cbf) + {C[dA(y)ldy]f

2(1 +0>)2 ] - 2 C A ° * - ( Î 4 ^ ) ( 4 ) Γ Γ^>χ°>φ°)χ°dx«άφ« (5ΐ)


This is a "conservation of momentum" equation. It is particularly useful for making accuracy checks and revealing programming errors since it is derived directly from the equations which are being solved.

IV. APPROXIMATE TWO-DIMENSIONAL ANALYSIS

Solution of the general nonlinear interaction equations for the traveling-wave amplifier (TWA) is a considerable undertaking even if the closed-form solutions of the radial wave equations are employed. A general solution in closed form of the remaining equations is not possible. Solutions can be obtained for individual cases with the aid of a high-speed digital computer. However, the required amount of computer time per case is fairly large, especially when RF space-charge calculations are included. It should also be pointed out that programming the general equations for solution on the computer is no small task. A simplified approach to the problem is certainly in order.

Reduction of the general equations to a more tractable form proceeds in two stages. First, a set of simplified two-dimensional nonlinear interaction equations is obtained through the application of a number of judiciously chosen assumptions. The problem is then reformulated on the basis of a disk-model representation of the electron beam in which the radial motion is described according to laminar-flow theory, thus incorporating some aspects of an Eulerian treatment into a Lagrangian analysis. The details of this development are given in Detweiler. (2).

A. A SET OF SIMPLIFIED TWO-DIMENSIONAL NONLINEAR INTERACTION EQUATIONS

In the simplified equations presented below, the following assumptions are employed in addition to those already mentioned :

(1) The interaction parameter C is limited to moderate values, i.e., C ^ O . l .

(2) The effects of RF space charge are neglected. (3) The combined effect of the average axial space-charge force and the

axial force due to the radial magnetic field component produces a negligibly small change in the axial velocity of the electrons.

(4) The radial phase variation of the RF wave, given by θχ(χ), is small and negligible. [ddx{x)jdx is, however, retained in the radial force equation so that its effect can be determined.]

(5) The circuit-loss parameter d is taken as zero.

These assumptions are justified for many cases of practical interest (2).


The assumptions specified above do not alter the velocity-phase or radial-position equations, so they remain as given by Eqs. (22) and (23).

The following pair of simplified circuit equations is obtained by applying assumptions (1) and (5) to Eqs. (26).

Longitudinal circuit equations:

A(y)(b + -^-)'^X Jo l+2Cu/y,x.W - (52a)

dA(y) = J _ Γ V p* ψ(χ') sin Φ' x0' dx0' d0o' ( 5 2 b )

dy Ttx\, Jo Jo 1 +2 Cuy(y,x9',0o')

Force equations:

Applying assumptions (2) and (4) to Eq. (27a), the radial force equation reduces to the following:

radial

dy è \ \ + 2 Cuy) = CA(y)(d-*W cos Φ + φ(χ) ^ sin φ)

\ dx dx i

££)*[[&♦»«*-(*)*. %σ\

-äSS*- <53a)

where E0 _r is the radial space-charge field due to the average space charge of the stream.

Upon application of assumptions (l)-(4) to Eq. (27b), an approximate axial force equation is obtained as

axial du^(\+2Cuyy= - (I+Cb)A(y)*Kx) sin Φ (53b) dy

The angular velocity expression is not modified by the assumptions so that :

Angular velocity equation:

2Cu{ -έ(?)[&+^>-(?)'** (53c)

It is appropriate to use, in conjunction with these equations, the approxi-mate expressions for ψ(χ), άψ(χ)1άχ, and ddx{x)jdx as given by Eq. (38) and the second part of Eq. (39) with either v, or v.


In solving these equations by treating the problem as an initial-value problem, the same quantities are required as are listed in Section III.A, with the exceptions that the initial values of the derivatives of A(y) and 0y(y) need not be specified (the circuit equations are now first-order equations) and d is taken as zero.

A considerable reduction in the complexity of the interaction equations, which is readily apparent in the simplified forms given above, has resulted from the assumptions which have been employed. The circuit equations are now first-order integrodifTerential equations and in these, as well as in the axial force equation, the number of driving terms has been reduced to one. The simplification in the radial force equation rests primarily in the absence of the RF space-charge term ; the average space-charge field is easily calcu-lated.

It is of some interest to determine whether any special meaning can be ascribed to the forms the circuit equations and the axial force equation take on when assumption (1) is used as it has been here. A specific meaning is revealed when the simplified equations of this section are compared to some one-dimensional equations which have appeared in the literature. To facilitate comparison, the simplified two-dimensional equations are reduced to an equivalent set for the one-dimensional case by assuming (1) the electron stream is in confined flow so that no radial or angular motion of the electrons occurs, (2) there is no radial variation of the circuit field or the electron variables across the stream, and (3) unity coupling between the stream and the circuit wave exists, i.e., φ(χ)= 1 for all x. The radial position, radial force, and angular velocity equations are thereby eliminated from consideration and the remaining equations become as follows for the one-dimensional case.

Velocity-phase equation:

^(y&o) + ddjy) = 2uy{y&0) ( 5 4 )

dy dy \+2Cuy{y&,)

Circuit equations:

yy,\ dy J 2 T J 0 l+2Cuy(y,4>0')

dA{y) _ 1 Γ2* sin 0(y,0„') d<P0' dy 2ITJO Ί +2 Cuy(y,<P0')

Axial force equation:

Suv(y,0o) _ (1 +Cb)A{y) sin <%,Φ„) Sy l+2Cuv(y,0o)

(55)

(56)


If the same assumptions used here (moderate C, lossless circuit, and no RF space charge) are applied to Rowe's one-dimensional equations (5), the resulting equations are identical to those presented above. This is a natural consequence of the fact that he uses basically the same method in deriving the axial force and circuit equations in the general theory (which has been used to obtain the above equations) as he does in the one-dimensional case except that, in the general theory, radial variations are taken into account.

Equations (54)-(56) are seen to bear a resemblance to Nordsieck's " small-C" equations (4) (which also neglect loss and space charge). These equations are, however, not "small-C" equations; the effects of the stream nonlinearities, which are important for C as large as one tenth, are accounted for by the 1 + 2Cuy factor included in the right-hand side of the equations.

Tien (5) has presented a set of large-C, one-dimensional equations in which space-charge effects are included. Although these equations and Rowe's equations are different in form, their equivalence has been clearly demon-strated [cf., Rowe (6) and Hess (7)]. The basic difference in Tien's approach is to separate the total circuit voltage into two components, forward- and backward-traveling waves, which satisfy first-order differential equations. If the effect of the backward-traveling wave is entirely neglected in Tien's equations (written for the space-charge-free case), a particularly interesting event occurs, viz., the resulting equations are identical to Eqs. (54)-(56). Thus, the "moderate-C" assumption [assumption (1)] used in obtaining the simplified equations in this section has the effect of neglecting the backward-traveling wave. This could have been anticipated since the second-order circuit-wave variations that have been neglected are those related to the backward-traveling wave. Since the use of assumption (1) only results in neglecting the effect of the backward-traveling wave, which should not be significant unless C is fairly large (C> 0.1), while the stream nonlinearities are accounted for, the simplified equations should be reasonably valid for CÛ0A.

It is appropriate to make the "moderate-C" assumption in the conserva-tion theorems in order to put them in forms consistent with the simplified equations. When this is done, Eq. (49) becomes

2CA\y) s 2CAJ + 1 - ( - L ) Γ6' Γ [1 + 2 α^,χ0',Φ0')]2χ0' dx0' άΦ0' (57) \πχρ/ Jo Jo

and Eq. (51) becomes

2CA\y) s 2CV - ( γ ^ Χ ^ ) \]b \*"Α?>Χ*>φο')Χο d*o W (58)

The simplified two-dimensional equations presented in this section could be programmed for computer solution without too much effort and the


amount of computer time required to obtain solutions would most likely not be excessive. In modeling these equations for the computer, the electron stream would be divided radially into a number of concentric annular layers and the charge contained within each layer would be subdivided into repre-sentative charge groups [cf., Rowe (8)]. For accuracy, the number of charge groups needed in each layer would be the same as in a one-dimensional analysis. Since the computer solution time necessarily depends upon the total number of charge groups used, it is seen that the execution time for the two-dimensional problem would exceed that required for the one-dimensional problem, typically by a factor approaching the ratio of the number of charge groups used for the two cases. If comparable execution times are to be achieved, the two-dimensional equations must be modified in a way that will allow the use of the same number of charge groups. A particular way in which this can be done is presented next.

B. A DEFORMABLE-DISK MODEL FOR THE ELECTRON STREAM

For an approximate treatment of the two-dimensional interaction problem, it is appropriate to seek a stream model which will allow the use of fewer charge groups and will still be capable of showing the effects of radial electron motion and radial RF field variations. Such a model evolves naturally from the disk model employed in one-dimensional analyses if the "one-dimen-sional" restriction (no radial field variations and confined flow) is removed. With this restriction removed, radial electron motion and its effects can then be determined by using some aspects of fluid-flow theory; the effects of radial RF field variations can be obtained by an averaging process.

The disk model to be described here is illustrated in Fig. 7 for a discrete number of charged disks. These disks, which are assumed to be axially

CIRCUIT Λ.

v/r d CHARGED

DISKS

Bz(z) — *

FIG. 7. Illustration of the deformable-disk model.

symmetric, are injected into the interaction region uniformly distributed over phase, having initial radii equal to the initial stream radius given by xb>. The axial velocity and phase position of a disk are determined by averaging the


axial RF forces acting upon it across its diameter. Each electron in the disk is assumed to execute this average axial motion; thus, no axial shearing of the disks is permitted. The disks are allowed to vary in diameter in response to the radial forces acting upon them and the electrons contained within a disk (internal electrons) are assumed to behave according to laminar-flow theory. Thus, the radial and angular equations of motion need only be solved for an electron at the disk edge. (The radial and circumferential velocities and radial position of an internal electron are related to those of an electron at the disk edge by a constant which is the ratio of the initial radius of the internal electron to %.) Consistent with the laminar-flow assumption, the cathode-flux parameter is assumed to be the same for all electrons (independent of initial radius). Also, the charge density at any axial plane is assumed to be independent of radius. Since there is a net transfer of energy from the stream to the RF wave in the interaction, the average stream velocity decreases with axial distance, which acts to increase the average charge density; this effect is taken into account in the calculation of the average radial space-charge force.

The type of model postulated above precludes obtaining from the analysis information relative to radial velocity distribution effects which result from a variation of the RF fields across the stream, thermal velocities, or magnetic flux linking the cathode. However, these effects primarily concern the micro-structure of the stream to which the overall performance of the device (of chief interest here) is not particularly sensitive. Provided the assumptions made in developing this model are reasonably well founded for a practical TWA beam, it should adequately predict the gross behavior of the stream and thus serve the intended purpose.

The use of this disk model seems reasonable for an approximate analysis. It ignores effects associated with the detailed electronic motion, but accounts for (in an approximate manner) radial motion of the electrons and the resultant change in coupling between the stream and circuit wave, as well as the variation of the RF circuit field across the stream diameter. With this model, the same number of charge groups (disks) can be used as are used for a one-dimensional analysis. As mentioned, the disks are capable of changing shape, i.e., they can be deformed. This model therefore is called the "deform-able-disk model," abbreviated DDM, and the approximate equations based on this disk model will be referred to at times as the "DDM equations."

C. APPROXIMATE TWA EQUATIONS BASED ON THE DEFORMABLE-DISK MODEL

The simplified two-dimensional nonlinear interaction equations presented in Section IV. A will now be reformulated on the basis of the DDM represen-tation of the electron stream. All assumptions made in developing the simplified set of equations, as well as those mentioned in the previous section


pertaining to the disk model, are considered to be in effect. The symbols used previously to designate the dependent stream variables for the ring model (x, Φ, Uy, ux, and ηψ) will be retained. However, when applied to the disk model, they acquire different meanings and these are summarized as:

x(y,0o)=the normalized radius of a disk, 0(y,0o)tthe phase position of a disk relative to the RF wave,

1 + 2 Cuy(y,<P0)=ihe normalized axial velocity of a disk, 2 Cux(y,0o)=the normalized radial velocity of an electron at the edge of

a disk, and

2CuJy,0o)A

.-~~φΤ~ = the normalized angular velocity of a disk [2 Cu9(y,0o) is ° the normalized circumferential velocity of an electron at

the disk edge], where Φ0 is the initial phase position of the disk. Of course, there is no de-pendence on x0 in the disk-model variables.

The equations for the disk model are derived from the simplified equations for the ring model in the following manner. The axial force on a disk is obtained by averaging the force for the ring model over the stream diameter, i.e.,

' {Fz) ring *o' dx0'

(^)disK- °-j^f (59) Xo dx0' Jo

From Eq. (53b),

(^)ring = - (1 + Cb)A(y)t[x(y9xM] sin 0(y,xo,0o)

s - (1 + α)Α(γ)φ^ χ(γ,Φ0ή sin 0(y,0o) (60)

where in the approximation, the disk-model assumptions are invoked, i.e.,

x(y,x&&o) = (xolXb-My&o) since laminar flow (geometrically similar) is assumed and 0(y,xo,0o) is replaced by 0(y,0o). Substituting Eq. (60) into Eq. (59) and performing the indicated integrations, the result is

(^)disk = - (1 + Cb)A(y) t[x(y,0o)] sin 0(y,0o) (61) where

ψ[χ&,0ο)]=(5χο{ν'Φο)Φ(η)η dw)/(ix^ w dw) (62)

By applying the same assumptions as above, the driving terms on the right-


hand side of the longitudinal circuit equations, Eqs. (52), become for the disk model

TTxh J o J 0 1+2 CuJ V, Χη,Φη)

27rJo 1 + 2 C W 2 / ( 7 , 0 O/ ) ^ j

The average radial space-charge field can be evaluated using the con-servation of charge, which for the disk model requires

7rb'2p0 dz0 = πήρ dz or

(64)

where b' is the initial disk radius and p0 its charge density, and r and p are the disk radius and charge density at z. dz^dz can, in general, be a multi-valued function and therefore all charge at this plane must be taken into account; the absolute value signs are used to indicate this.

If a Fourier expansion is made of the charge density, the first term repre-sents the average value which is

2TT J _„. 1 + 2 Cuy(y,<P0 )

Assuming the stream to be substantially smooth, it then follows from Gauss's law and the laminar-flow assumption that

= Pov* i r άΦ^ °~' 2€0r 2TTJ- , \+2Cuv(y,0o') '

Then, in terms of the normalized variables, the average radial space-charge force at the disk edge is given by

where

^Α^-ΙΓ ^ ^ (68) Po 2π)-π \+2Cuy(y,0o')

which is a normalized, average charge density. The above expression for the radial space-charge force gives some measure

of RF space-charge effects in that it accounts for the change in the average

66 Harry K. Detweiter and Joseph E. Rowe

charge density resulting from a change in the average stream velocity. It is, however, only an approximation to the true radial space-charge force.

Drawing upon the above results where needed, the approximate TWA equations for the disk model can be written as follows.

Velocity-phase equation:

3Φ(γ,Φ0)+ddy(y) _ 2uy(y90o) dy dy 1+2C«M)

Disk-radius equation:

«y&o) = *» - + 2 C Γ y f f A' (70) Jo \+2Cuy(y ,Φ0)

Longitudinal circuit equations:

λττ Jo

<#,ω\ _ -1 r Ά[^,ΦΟ')]ΟΟ8Φ(^,ΦΟ ,)^Φ,'

2 Οφ,Φο')

dA(y) _ 1 f2" frpS sin Φ(γ,Φ0') άΦ0' dy 2πϋ0 1+2 αφ\Φ0 ' )

Force equations:

radial (for the disk edge)

axial

Angular velocity equation:

2C\ = 1 x 2C

(71)

dU-\\ + 2C*) - < W ^ cos Φ + # c ) *&> sin φ ) dy \ dx dx J

-Μ?)'[Ά*Μ*-(Τ)'Κ' -ΐί-ήψ-ή'Μ] (72a)

Wfc/ \ x I J

e"*( 1 + 2 Cuy) = - ( 1 + Cb)A(y)W) sin Φ (72b) dy

(?) [f*+ηΛω " (?νίκ™\ (72c)

Functional dependencies have been omitted in some of the above equations to condense their form. The system variables are summarized below.


Independent variables:

Dependent variables:

y, Φο

A(y) 6v(y) x(y,4>o) Φ( ,Φο) Ux(y,&o) Uv(y&<,) Uyiy&o)

It is appropriate to use Eq. (38) and the second part of Eq. (39), rewritten in terms of v, in conjunction with the above equations. These are

ψ(χ) = Ι<^χϊ, dM = vL^- (73) h(vxa) dx φχα)

and //0 (Y\ /ΛΑ(Λ>\ΙΛΛ,\ Γ / Γ Λ , ν \ \ 2 ΐ

(74) άθχ{χ) _ldA{yi)ldy\ Γ / / ^ V

Combining Eq. (62) and the first part of Eq. (73), the average coupling function is found to be

?W"T^S (75)

This average coupling function is illustrated for a typical set of parameters in Fig. 8. Two other curves are also plotted in this figure. One is ψ(χ) (which is a thin-beam coupling function) as given by the first part of Eq. (73). The other is an average coupling function related to the root-mean-square value of the axial electric field such as that used by Rowe (6). In normalized variables it is expressed by

i » » - ^ , ; ' ^ <76)

This root-mean-square coupling function is seen to correspond closely to the average coupling function used here, which is a weighted average over the stream diameter.

The approximate TWA equations given above can be solved as an initial-value problem with specification of the initial RF and stream conditions and device parameters as discussed previously. The method used to obtain solu-tions on a digital computer has been described elsewhere (2). Representative results are presented in the next section.

Calculations of gain, efficiency, and harmonic currents are also carried out during computer solution of the approximate interaction equations. The gain is determined from Eq. (41), while the efficiency is calculated from

V(y)=2C[A*(y)-A0*] (77)


0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 NORMALIZED RADIUS, x

FIG. 8. Thin-beam, root-mean-square and average coupling functions vs normalized radius.

The harmonic current amplitudes and phases for the disk model are obtained from

!'-#! =- ίί Γ c o s ηφ(^φο) ««vV | / θ I 7Γ L \ J 0 /

+ (J sin ηΦ(γ,Φ0') αΦ0' \

p h a s e d = a r c t a n ^ « ^ ^ ) ^ ' \ I0 J ft" cos ηΦ(γ,Φ0')αΦ0'

1/2

(78a)

(78b)

The approximate power and momentum conservation theorems [Eqs. (57) and (58), respectively] can be written for the disk model as

ICAHy) = 2CV + 1 - ^ Γ [1 + ΙΟφ,Φ^* αΦ0' 2π Jo

(79)


2CA\y) = 2GV - {j^j ~ J* itf**·') *V (80)

In the computer solution procedure, the two sides of each of these equations are calculated and compared. This serves several purposes: it shows whether the equations being solved are consistently formulated and correctly pro-grammed; it also gives a measure of the accuracy being achieved in the integration process.

V. COMPUTER SOLUTIONS OF THE DDM EQUATIONS

A. GENERAL

This section contains the results of digital computer solution of the DDM equations for various device operating conditions. Primary emphasis is placed on determining the effects of the magnetic focusing conditions and the device parameters on TWA performance. Operation of the device as an energy converter is also investigated, as is the phenomenon of RF slow-wave focusing of the electron stream. As a result of the assumptions made in deriving the DDM equations, all solutions presented in this section neglect the effects of RF space charge and circuit loss.

These solutions have been obtained by numerically integrating the inter-action equations along the device in y, using a finite number of representative charge groups (disks). This was accomplished by making a first-order difference-equation approximation and obtaining piecewise-linear solutions with a digital computer, using very small increments of the linear segment.

A distribution of 32 "electrons" over one cycle of the RF wave yields sufficient accuracy in one-dimensional calculations when RF space-charge effects are negligible (3). Since the effects of RF space charge are neglected in the approximate two-dimensional analysis, a similar subdivision of the stream charge is adequate for the calculations. Thus, 32 charge disks were injected into the interaction region during one cycle of the RF wave and subsequently followed through the device. These disks were injected carrying equal amounts of charge and equally spaced in phase, which corresponds to an unmodulated entering beam. In addition, the beam electrons were assumed to enter with zero radial velocity.

The integration increment, Ay, should be small compared to the cyclotron wavelength (normalized) in order that the beam variables be accurately calculated (/). Accordingly, a value Ay = 0.01 was chosen for the calculations. This choice yields sufficient accuracy for the circuit variables as well since the RF circuit voltage characteristic is less sensitive to having the proper value of Ay than are the beam variables.

While the input RF signal A0 must be specified for the calculations, it is


convenient and meaningful to designate the results according to the input power level corresponding to the specified value of A0. Thus, a parameter φ0 is defined by

φ0*10\οίΜ(-Ά-) (81a) \ci/0|K0; which gives the RF input power level in dB relative to C times the beam power. The relationship between φ0 and A0 is given by

0ο = 1Ο1οβιο(2Λ2) (81b)

A value A0 = 0.0225 was used for the TWA calculations. This corresponds approximately to ψ0 = - 30 dB, i.e., an input power level approximately 30 dB below C|/0|K0.

The values of b, C, xfl, xh>, and ωρ/ω are specified according to the particular device to be considered. In evaluating C it should be noted that it has been defined in terms of the interaction impedance at the circuit. The decrease in the RF fields with distance from the circuit, which in effect reduces the interaction impedance "seen" by the stream electrons, is taken into account through the use of radial amplitude coupling functions [the average coupling function ψ(χ) in the DDM equations]. The values of xb. and xa can be determined from the commonly used device parameters by means of

x>.-°ïb'*W ^ - ^ e Ä ^ (82) w0 1 + Cb u0 1 +Cb

where y is the undisturbed radial propagation constant given by

yt[(œlv0Y-(œlcWl^œlv0

The approximation applies for nonrelativistic velocities. The value of ων/ω is found from

ων/ω= (3.04 ΡμΥ'η(\0 ßebf) (83)

where Ρμ = (\Ι0\ΐνf/2) χ 106, the beam microperveance.

B. TWA SOLUTIONS FOR A UNIFORM MAGNETIC FIELD

The one-dimensional results of Rowe (5, 9) and Tien (5) indicate that the optimum value of the injection velocity parameter b (i.e., the value of b which gives maximum saturation gain and efficiency) is approximately 1.5-2.0. The optimum b value, which generally is larger than that for maxi-mum small-signal gain, was found to depend on the particular values of C and QC. In later studies Rowe (/) found that accounting for radial variations considerably reduces the value of b for optimum large-signal performance


from that predicted by the one-dimensional analyses, a result which is in agree-ment with experimental observations. The DDM program was used to invest-igate this effect and to determine the differences in performance for a TWA employing a Brillouin-focused beam as opposed to a beam in confined flow.

Parameter values which are typical of an efficient high-power TWA were selected for these calculations. These are C = 0.1, œjœ =0.1414, ßeb' = xh.jC = 1.05, and b'ja =0.7, which correspond to Ρμ =0.725 and when 6 = 1, ya = 1.65 (which is near optimum from the standpoint of maximum band-width). Using Brewer's results (70), the value of QC = 0.064 is calculated when the beam is Brillouin focused, while QC = 0.123 for confined-flow conditions. For QC values as low as these it can be expected that RF space-charge effects will not be too important and reasonable results should be obtained from the DDM theory. The accuracy of these calculations will be evaluated later by comparison with results obtained from the general large-signal theory.

Figure 9 illustrates the variation of the normalized RF voltage amplitude A(y) along the circuit for several values of the injection velocity parameter. The results for the limiting case of confined flow are shown in addition to those obtained for Brillouin flow. Also included for comparison are some

FIG. 9. Variation of the normalized RF voltage amplitude along the circuit as a function of the injection velocity parameter. (ωρ/ω = 0.1414, K00 = Khy C = 0.1, xa = 0.15, xh> = 0.105, <Ao=-30dB.)


one-dimensional results. The DDM program was used to obtain all of these solutions. The two-dimensional confined-flow calculations were performed with the DDM program modified by program overrides which, in effect, prohibited radial motion of the electrons. For the one-dimensional calcula-tions, an additional modification was employed to specify unity stream-circuit coupling, i.e., φ(χ) = 1 for all x\ thus ψ(χ) is also identically equal to unity.

The parameter M0 is used to express the ratio of the strength of the uniform axial magnetic field to the Brillouin value, i.e.,

M ä^L = ω^ω ^ #Br \ / 2 ωρ/ω

The designation K00 = Kh indicates that the cathode-flux parameter is specified as that value which will result in space-charge-balanced flow under dc condi-tions (smooth flow when the electrons enter with zero initial radial velocity), which is

From these relationships, it can be seen that the specification M0 = 1 and ^οο = ^i) corresponds to Brillouin flow (for which Kh = 0), while the specifica-tion M0 -» oo and K00 =Kh denotes confined flow (for which Kh -> 1). The results shown in Fig. 9 are designated accordingly.

Since there is no radial motion of the electrons when the beam is in con-fined flow, the effects on device performance due solely to the radial variation of the RF fields and the resultant reduced stream-circuit coupling can be evaluated by comparing the one-dimensional and two-dimensional confined-flow results shown in Fig. 9. From the cases for which b = 1 it is seen that the saturation gain and efficiency are less (since AmSLX is less) when the trans-verse variation of the RF wave is taken into account, as has been found by Rowe (6). It is also seen that the device must be appreciably longer in order to reach saturation for the same input signal power than is indicated by the one-dimensional result because of the reduction in the stream-circuit coup-ling. In addition, the saturation characteristic is somewhat broader as a result of the additional velocity spread in the stream which is induced by the radial RF field variation.

When b = 1.5, the one- and two-dimensional results are strikingly different. The signal growth rate is seen to be significantly less for the two-dimensional case. In fact the two-dimensional result indicates that this value of b is very close to the drop-off point, i.e., the b value at which growing-wave gain ceases to exist. This is appreciably lower than the drop-off value of b £ 2.2 which is predicted by one-dimensional calculations. Thus, the effect of the reduced stream-circuit coupling is to make the stream appear to be more out of synchronism with the wave. Since the best combination of gain, efficiency, and device length is usually obtained for b somewhat less than the drop-off value,


it is clear that taking into account radial RF field variations does significantly reduce the optimum injection velocity from that predicted by one-dimensional analyses. The two-dimensional calculations indicate that a value of b = 1 or slightly larger should yield near-optimum large-signal performance, which is in agreement with experimental findings.

The electrons execute no radial motion under space-charge-balanced-flow conditions in a finite uniform magnetic field when there is no RF signal since the inward Lorentz force and the outward dc space-charge and centrifugal forces are in balance for all radii. However, when an RF signal is present the radial electric field associated with the RF wave propagating along the slow-wave circuit disturbs this balance and radial motion of the electrons occurs. As a result of the traveling-wave interaction the RF signal, and therefore the radial RF electric field, builds up along the circuit and an expansion of the beam with interaction distance can be expected. Since the coupling between the circuit wave and the electrons depends upon the proximity of the electrons to the circuit, the signal growth rate and device gain will be altered as a result of the radial motion of the electrons. Under balanced-flow conditions, the radial motion is greatest when the beam is Brillouin focused since the trans-verse force which acts to restore a perturbed electron to its equilibrium trajectory is least for this case, i.e., the beam "stiffness" as defined by Palmer and Susskind (77) has the smallest value when the beam is in Brillouin flow. The major effects on device performance of radial motion of the electrons can therefore be evaluated by comparing the extreme cases of Brillouin flow (largest radial motion) and confined flow (no radial motion).

The two-dimensional results illustrated in Fig. 9 show that the small-signal gain for a Brillouin-beam amplifier is greater than for a corresponding confined-flow device, which is in agreement with the predictions of Rigrod and Lewis (72). The mechanism of this gain increase will be considered later. The results of Rigrod and Lewis indicate that the small-signal-gain increase for Brillouin flow over confined flow would be only a few per cent for the y a values used in these calculations, which agrees quite well with the results which were obtained. It is also seen in Fig. 9 that the gain at saturation is larger for the Brillouin-flow case as well. However, the increase in large-signal gain for Brillouin flow is not very great, just as is the case under small-signal conditions. It will be shown later that a significant amount of circuit current interception usually occurs for a Brillouin-focused beam under large-signal conditions and this modifies the increase in saturated gain. Under some conditions this interception can actually result in a lower saturation level for Brillouin flow than is obtained with confined flow.

Some of the two-dimensional results for confined flow and Brillouin flow are summarized in Fig. 10. The power level and efficiency at saturation (Psat and 77sat, respectively) are shown in order to indicate the increase in the


FIG. 10. Efficiency and power level at saturation and average gain vs b for Brilloiiin flow and confined flow. (ων/ω = 0.1414, K00= Kh, C= 0.1, xa = 0.15, xh> = 0.105, φ0= - 30dB.)

absolute saturation level relative to C | / 0 | F0 and in efficiency with increasing values of b. The average gain per unit ^-distance computed on the basis of the saturation gain and length (<7sat and ^sat) is also shown to illustrate the decrease of gain per unit length and the general lengthening of the amplifier to achieve saturation at the larger values of b. It can be seen that the optimum injection velocity is not significantly altered by the radial motion of the electrons. Thus, b = 1 was chosen for subsequent TWA calculations which are presented later in this section. Based only on the quantities shown in this figure, it is clear that an improvement in device performance, even though it may be rather small, can be achieved by operating with Brillouin flow rather than with confined flow. However, a final judgment must await a considera-tion of the current intercepted on the circuit under Brillouin-flow conditions. While these results indicate an improvement for Brillouin flow, the relative


improvement becomes less at the larger values of b. This is a direct conse-quence of the circuit current interception as will be demonstrated later.

It is interesting to examine the magnitude of the fundamental component of RF current in the stream since this is a measure of how effectively the stream has been bunched by the RF wave. If the stream were perfectly bunched, i.e., if the stream electrons were formed into ideal delta-function bunches, the value |/i//0 | =2 would be obtained as pointed out by Pierce (13). In fact |/n//01 ^ 2, i.e., the maximum amplitude of any harmonic current is two. This is easily seen from Eq. (78a). However, since the fundamental current results directly from the stream-wave interaction, whereas the harmonic components are generated by the stream nonlinearities, it can be expected that the fundamental component will be largest in amplitude and the higher harmonics will have successively smaller amplitudes, at least until the stream behavior becomes highly nonlinear. This will be examined later in connection with the results obtained from the general theory.

The variation with interaction distance of the fundamental RF stream current (normalized with respect to the dc beam current I0) is illustrated in

FIG. 11. Variation of the amplitude of the fundamental RF stream current with normal-ized axial distance for Brillouin flow and confined flow as a function of the injection velocity parameter. (ωρ/ω = 0.1414, K00= Kh, C = 0 . 1 , xa = 0.15, AV = 0.105, φ0= - 30 dB.)


Fig. 11 for both Brillouin-flow and confined-flow conditions with different relative injection velocities. It is seen that the values of \ii(y)II0\, and the maximum values which they ultimately reach, are larger for smaller values of b, which is indicative of the better bunching that is achieved at low relative injection velocities. This is observed for both types of flow conditions. The results for Brillouin flow and confined flow are virtually identical until interception commences, which occurs at y = 7.15, 6.97, and 7.61 for b = 0, 0.5, and 1, respectively. Thereafter, the Brillouin-flow results are lower due to the removal of a portion of the stream charge. It is therefore clear that the axial bunching is not significantly different for the two types of flows, at least until interception occurs for the Brillouin-flow cases. Thus, the greater gain for the Brillouin-beam amplifier must be due primarily to the changes in coupling which result from the radial motion of the electrons.

The "electron" trajectories for one of the Brillouin-flow cases discussed previously are shown in Fig. 12. These trajectories depict the normalized radii of the 32 representative charge disks used in the computer simulation as they move through the interaction region ; each line represents the trajec-tory of one particular disk. The initial radius of each disk is equal to xh>.

1.50r

0.60 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Y

FIG. 12. Normalized disk radius vs normalized axial distance. (ωρ/ω = 0.1414, M0= 1, K00=Kb=0, C=0.1 ,6= 1, xa = 0.15, xb> = 0.105, 0O= - 30 dB.)


Note that the axial and radial scale increments have been chosen in the ratio of 100:1, which is approximately the ratio of the device length to the initial stream radius. The electrons enter with no initial radial velocity and the magnetic focusing conditions are adjusted for space-charge-balanced flow at the Brillouin value of magnetic-field strength (i.e., M0 = 1, which corresponds to ωοο/ω = 0.2 for ωρ/ω = 0.1414, and K00 = K\> = 0). Thus the electron flow is initially smooth and would remain so if no RF signal were applied.

As mentioned earlier, the radial RF forces cause a perturbation of the trajectories. Near the input where the signal level is relatively small, the perturbations are of a small magnitude. As the signal grows along the circuit, the radial RF forces increase, and near the output (i.e., the saturation plane) where the forces are quite large, the stream is significantly spread. The increase in the average stream diameter in this region, which results in a stronger average stream-circuit coupling and thereby an increase in gain over that obtained for confined flow, is easily seen. However, the gain is larger in the small-signal region as well and it is not immediately obvious from this figure whether there has been any change in the average stream diameter in this region. It will be necessary to employ other means to explore the mechan-ism of this gain increase further.

The darkened area which occurs in this figure just before saturation as a result of the overlapping of a number of disk trajectories indicates the formation of the bunch in the decelerating phase of the RF wave. This similarity of radial motion of the bunch electrons is to be expected since their phase positions are nearly the same and they are therefore acted upon by similar radial RF forces. The bunch formation is less distinct in these trajec-tory plots (for b = 1) than for smaller relative injection velocities because the axial bunching of the electrons is not as good in this case, i.e., there is a greater spread in phase and axial velocity among the bunch electrons.

It is apparent from the trajectories in Fig. 12 that the Lorentz force supplied by a magnetic field equal in strength to the Brillouin value is in-sufficient to effectively confine the stream under large-signal conditions for the particular device parameters used in these calculations. Electrons begin to strike the circuit before saturation, and by the time the saturation plane has been reached the per cent of the dcbeam current intercepted by the circuit is /intCVsat) =21.3. Large amounts of current were also intercepted in the Brillouin-flow cases at lower relative injection velocities: /intOW = 19.6 and 23.4 for b = 0 and 0.5, respectively.

Those electrons which strike the circuit dissipate their remaining kinetic energy on the circuit as heat. The fraction of the dc beam power dissipated on the circuit by these electrons is approximately the same as the fraction of the dc beam current intercepted by the circuit. (It is generally somewhat less since the electrons which reach the circuit usually have a lower velocity and


0.120 h

0.090 7Γ/2

rad

FIG.

- 0.1414, M0

DISK PHASE, Φΐγ,Φ0)

FIG. 13 (a)

13. Normalized disk radius vs disk phase as a function of axial distance. (ωρ/ω

K00=Kh=0, C=OA,b= 1, JC„ = 0.15, x6. = 0.105, <£<>=-30 dB.)

therefore less kinetic energy than they had initially.) Thus, interception can lead to serious problems with heating of the circuit if the circuit does not have a sufficient heat-dissipation capability. However, it is not readily apparent from the information considered thus far whether or not the interception has an adverse effect on the device gain in these cases. Even though an appreciable fraction of the stream which is interacting with the circuit wave is being removed from the interaction, the important consideration is whether the electrons which strike the circuit, and which are thus removed, are located in phase positions such that they are giving energy to or extracting energy from the wave. If an electron is located in the decelerating phase of the RF wave when it hits the circuit, it will be removed before it has given up as much energy to the wave as it would have had the interception not occurred. Such an interception has a detrimental effect on the device gain. When an electron


0.15

0.12

0.09

0.15 '

<j? 0.12

w' 0.09

ö 0.15 < cr £ 0.12 a

£0.09

^ 0.15 < i 0.12 o ■z.

0.09

0.151

0.12

0.09

0 0 7 - 7 T -77-/2 0 7Γ/2 7Γ

DISK PHASE,Φ (y^0),rad

FIG. 13 (b). (Legend on p. 78)

in the accelerating phase strikes the circuit, the converse is true. Thus, if the majority of the electrons being intercepted are in the accelerating phase, or are about to pass from the decelerating into the accelerating phase, the device gain will not be degraded. In fact, an enhancement of gain can actually result under these circumstances, as will be seen from results presented later.

It is appropriate at this point to examine the "space-phase" diagrams given in Fig. 13 for one of the Brillouin-flow cases. These are plots of the radii of the disks as a function of their phase positions (normalized to the range - 7Γ to π) at several axial displacement planes. These plots reveal the nature of the electron motion under the combined influences of the RF wave and the uniform axial magnetic field and why larger gains result when the electron flow is not fully confined. The case for b = 1 was selected for this purpose since the saturation gain and efficiency were larger than for the other cases investigated.

Electrons which occupy a normalized phase position <P(y,<P0) in the interval 0 to π are acted upon by a retarding axial electric field due to the RF wave on the circuit; these electrons are decelerated in the direction of increas-ing y and deliver energy to the RF wave. On the other hand, electrons with


normalized phases in the interval - π to 0 are accelerated in the direction of increasing y by the axial electric field and thus extract energy from the wave. The maximum of the retarding field occurs at Φ =π/2, while the accelerating field is maximum at Φ = - π/2.

For the case illustrated, the electrons enter with a larger axial velocity than the phase velocity of the RF wave on the circuit (b is nonzero). The average axial velocity of the electrons remains greater than the wave phase velocity for some distance and hence the electrons advance continually in phase, alternately passing through some combination of accelerating and decelerat-ing regions in the process, until the majority of them are slowed down and become trapped in the negative-potential wells of the RF wave.

The advancement of the disks in phase over much of the interaction distance produces a variation in the radial RF forces which they experience. The radial RF force has two components as can be seen from Eq. (72a); one is due to the radial amplitude variation of the RF wave, while the other results from its radial variation in phase. A comparison of the relative magnitudes of these two components reveals that the dominant term is the one which involves dip(x)ldx, i.e., the component which arises from the radial amplitude variation of the wave. Some calculations were carried out using the DDM program with the term involving d6x{x)jdx eliminated in order to substantiate this. It was found that neglecting this term produced no significant change in the results even though the trajectories were slightly altered. Thus, this compo-nent could be safely neglected in the equations and can be ignored when discussing the variation of the radial RF force with phase.

In light of the above discussion, it is seen from Eq. (72a) that electrons having normalized phases in the range -πβ to π/2 experience an outward radial RF force. The maximum of the outward force occurs at Φ = 0. Con-versely, an inward radial RF force acts upon electrons whose normalized phases are in the intervals -π to -π /2 and π/2 to π and these forces are largest for \Φ\ =π. The radial and axial RF forces are obviously in time (phase) quadrature as well as space quadrature.

Consider first the motion of a disk which enters the interaction region having an initial phase position Φ0 =π/2 (No. 9). The retarding axial force is a maximum for such a disk while the radial RF force is zero. As it advances in phase due to its axial velocity being greater than the wave phase velocity, it is acted upon by a steadily increasing inward radial RF force. Clearly, when this disk reaches Φ =π, where the inward RF force is a maximum and the axial field changes from decelerating to accelerating, its radius will be less than it was initially. A continued advance in phase results in a reduction of the inward RF force until it reaches zero at the normalized phase position Φ = - 7Γ/2 and then the radial RF force acting upon this disk begins increasing in the outward direction.

OTYPE LINEAR-BEAM DEVICES 81

A disk entering with Φ0 = - πβ (No. 25) exhibits an opposite type of behavior. At this phase position the radial RF force is zero also, but the axial field is accelerating. As this disk advances in phase, it experiences a steadily increasing outward radial RF force. When it reaches the boundary between accelerating and decelerating axial fields at Φ = 0 its radius will have been increased. The outward RF force on this disk decreases with a continued phase advance and becomes zero at Φ =πβ.

The other disks will change in radius in a manner similar to the examples cited above when they occupy similar phase positions. However, the details of their motion will differ because of their different initial phases. It is apparent from the previous discussion that the radial RF electric field associated with the wave on the circuit acts to increase the radii of electrons which advance in phase from an accelerating region into a decelerating region, while those which advance from a decelerating region into an ac-celerating region have their radii reduced by this field. Thus, the electrons which are giving energy to the wave are, on the average, more strongly coupled to the wave than are those which are extracting energy from it. This obviously is the mechanism which results in greater small-signal as well as large-signal gains when the magnetic-field strength is not so large as to prevent radial motion of the electrons.

This behavior is clearly shown by the curves in Fig. 13(a). At y = 2, it is seen that the majority of the favorably phased disks (those in the decelerating field region) have radii larger than the initial stream radius, xh>, while most of those in the accelerating region have smaller radii than they had initially. When y = 3 is reached, the disks have advanced in phase through nearly one-half cycle of the wave. Numbers 21, 25, and 29, which entered during the accelerating phase of the RF wave, are now in a retarding phase and have increased in radius. The converse is true for Nos. 5, 9, and 13 which entered during the decelerating phase of the wave. This preferential radial motion of the electrons as they advance into the retarding-field region is seen to continue at the larger axial distances. However, the pattern of the curves is altered as a result of the bunching and the "dc" restoring forces acting upon the electrons. (The net radial force due to the dc space-charge force, the centri-fugal force, and the Lorentz force act to restore an electron which is perturbed by the RF field to its equilibrium radius, i.e., its initial radius under balanced-flow conditions.)

The ensuing developments at larger signal levels are illustrated in Fig. 13(b). At y = 7.6, the majority of the disks are in the retarding-field region and the average radius of these disks is greater than xb>. The relative positions of disk Nos. 9, 13, and 17 indicate the formation of a bunch near Φ = π/2. It can be seen that as the bunch is slowed by the wave and subsequently falls back in phase, the electrons (disks) which are in the bunch have similar radii and


execute similar radial motions. This behavior was observed in the trajectory plots presented previously.

Disk No. 1 is about to strike the circuit, which is located at xa =0.15, at y = 7.6. As the interaction proceeds this disk, and many other favorably phased disks as well, strikes the circuit. (The roughness in the line connecting the radii of disks which have hit the circuit is the result of the radius of a disk being discontinuously reduced in the computer simulation when it reaches the circuit.) By the time y = 8.4 has been reached an appreciable number of electrons which would have given energy to the wave have been removed from the interaction as a result of interception. This definitely has a detri-mental effect on the device gain. The disks in the bunch (e.g., Nos. 13 and 17) have increased in radius at this displacement plane as a result of the increasing outward radial RF forces they experience as they fall back in phase toward the boundary between the decelerating- and accelerating-field regions. The disks in the bunch begin to strike the circuit shortly after they pass into the accelerating region and thus some of the stream electrons are removed before they have had the opportunity to extract much energy from the wave. While this has a beneficial effect on the interaction, in this case it is not sufficient to counteract the earlier interception of favorably phased electrons. The condi-tions just before saturation (which occurs at y = 8.93 for this case) are illustrated by the plot for y = 8.8; the majority of the disks have hit the circuit by this point.

If the results for the Brillouin-flow cases with b = 0 and 0.5 are examined in the same manner as done above for the case with b = 1, it is found that the general behavior is much the same even though the detailed motions of the charge groups are somewhat different because the electrons do not advance in phase as rapidly at the lower relative injection velocities. There is a similar preferential radial motion (i.e., increased radii for electrons which advance into a retarding-field region and decreased radii for those which advance into an accelerating-field region). The process which results in greater gain when the electrons are allowed to move radially in response to the radial RF field is entirely the same. It is also found for these cases that an early interception of favorably phased electrons occurs. However, this interception precedes the interception of electrons which have dropped back into an accelerating region (i.e., the bunch electrons) by lesser amounts for lower values of b and thus the adverse effects on the gain are correspondingly less. This will be demon-strated later. Since the saturated output in each of these cases is larger than that obtained for the corresponding confined-flow case, the interception does not seriously affect the energy conversion process. However, some adverse effects on gain are noted in all cases and, in addition, heating of the circuit by the intercepted electrons can cause problems as mentioned earlier.

It is therefore appropriate to consider means for reducing the interception.


One possibility is to increase the magnetic-field strength near the output to counteract the defocusing of the stream by the wave. This method will be treated later. In the uniform-field case the interception can be reduced by increasing the beam "stiffness," i.e., using a stronger magnetic field over the entire length of the device. Of course, the correct amount of magnetic flux must be allowed to link the cathode if initially smooth flow is to be obtained. The effects of using a magnetic field stronger than the Brillouin field will be considered next.

The disk trajectories for a magnetic field 1.5 times the Brillouin value, with b = 1 and the cathode-flux parameter adjusted to give balanced flow, are shown in Fig. 14. By comparing these trajectories to those for the correspond-

0.60 0.00 1.00 200 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Y

FIG. 14. Normalized disk radius vs normalized axial distance. (ωρ/ω = 0.1414, M0= 1.5, Kw=Kh= 0.55556, C = 0 . 1 , £ = 1, xa= 0.15, xb* = 0.105, φ0= - 30 dB.)

ing Brillouin-flow case (Fig. 12), it is easily seen that the radial motion of the electrons and the amount of spreading of the stream by the RF forces are reduced through the use of a stronger uniform magnetic field. Thus, a "stiffer" beam is more resistant to perturbations by the RF wave. This reduction in radial motion causes the signal to grow more slowly with distance and increases the saturation length. The onset of interception is also seen to be delayed until further along the axis (which corresponds to a larger signal level) and the amount of interception is reduced. In this case interception


commences at y = 8.98, which is just beyond saturation. The interception could be eliminated entirely by using a magnetic field slightly stronger than 1.6 times the Brillouin value for the device parameters used in these calcula-tions.

The radial motion of the bunch electrons near the saturation plane is easily detected in this figure, as was true for the Brillouin-flow case. The details of the radial motion are somewhat different in this case where the magnetic field is stronger and magnetic flux is allowed to link the cathode. A reduction in radial velocity results from the use of a stronger magnetic field. The maximum normalized radial velocity achieved by the disks under Brillouin-flow condi-tions was 2 Cux ^0 .1 , whereas in this case the maximum is 2 Cux s 0.05. It is interesting to note that even in the extreme case of Brillouin flow, the kinetic energy associated with the radial motion of the electrons is, on the average, less than one per cent of their initial kinetic energy; this is reduced to less than 0.25% for M0 = 1.5.

Figure 15 summarizes the efficiency and gain at saturation as a function of magnetic-field strength. The per cent current interception, which is 7int evaluated at an axial distance 10% past the saturation plane, is also shown. This is an approximation to the interception which would occur in a physical

4 0 r —

25

—\ 70

H60;

JL GAIN

X EFFICIENCY

—uo*

CURRENT INTERCEPTION

501

30O

20 !

10

1.0 1.25 1.50 1.75 2.0

FIG. 15. Efficiency, gain, and current interception as a function of magnetic-field strength. (ωρ/ω= 0.1414, Kw= Kh, C = 0.1, 6 = 1 , xa = 0.15, xb> = 0.105, φ0= - 30 dB.)

OTYPE LINEAR-BEAM DEVICES 85

device which is necessarily longer than the calculated saturation length due to the finite length of the RF transitions.

When the magnetic-field strength is twice the Brillouin value the results are virtually identical to those obtained under confined-flow conditions. As the magnetic field is reduced from this value, the efficiency and gain increase because the beam expands radially and is, therefore, more strongly coupled to the circuit wave. At approximately 1.5 times the Brillouin field, interception commences. The gain and efficiency are maximum for a magnetic-field strength approximately 1.25 times the Brillouin value. At first glance, it might seem strange that the gain and efficiency would be a maximum when the current interception is as high as it is in this case. However, most of the electrons which are being intercepted have already passed from the decelerat-ing phase of the RF wave into the accelerating phase; thus, their removal aids somewhat in the interaction. A decrease of magnetic field below 1.25 times the Brillouin value results in a drop of gain and efficiency and an appreciable increase in interception. This drop in gain and efficiency occurs because the majority of the electrons which contribute to the increase in interception are located in the decelerating phase of the RF wave and their removal adversely affects the interaction.

The effects of interception on gain can be determined by a calculation of the change in the rate of increase of the RF signal that results from the interception, which is proportional to the fundamental component of RF current which the intercepted charge would have induced on the circuit if it had not been intercepted. The intercepted RF current is positive when a favorably phased electron strikes the circuit, which causes a reduction in the slope of A(y); thus, this interception is detrimental to the interaction. When an unfavorably phased electron is intercepted, the intercepted RF current is negative so that the change in the slope of A(y) is positive and the interception is beneficial. If the individual contributions of the various electrons intercepted within a small interval in y are added, the incremental RF interception current is obtained. This can be integrated in y, linearly weighting the interception current according to the y plane at which it occurs, to reveal the overall effect of the interception on the gain.

The results of some computations carried out as described above are shown in Fig. 16 for several of the cases considered previously. The incremental RF current intercepted in a 0.1 interval in y is illustrated by bars above and below the zero line. Those above the line indicate the interception of favorably phased electrons (detrimental to the interaction), while those below the line represent the interception of unfavorably phased electrons (beneficial to the interaction). The integrated intercepted RF current is shown by the curves and the location of the saturation plane has been designated on these curves.

The plots shown in the lower part of Fig. 16 clearly demonstrate what was


ELECTRONS INTERCEPTED IN A FAVORABLE PHASE

ELECTRONS INTERCEPTED IN AN UNFAVORABLE PHASE

68 7.2 7.6 8.0 8.4 8.0 9.2 9.6

FIG. 16. Intercepted fundamental RF current as a function of normalized axial distance. (ωρ/ω= 0.1414, K00=Kh, C= 0.1, ΛΓβ= 0.15, xb> = 0.105, φ0= - 30 dB.)

mentioned previously. In the case for M0 = 1 and b = 1, the early interception of favorably phased electrons is not offset by the later interception of the bunch electrons which have passed into the accelerating-field region, and therefore the integrated effect of the interception is adverse to the interaction when the saturation plane is reached. For M0 = 1.25 and b = 1, the intercep-tion is clearly beneficial, which is the reason a larger saturated output was obtained for this case. From an inspection of the plots in the upper part of


this figure, which are for Brillouin flow at lower relative injection velocities, it can be seen that interception also has an adverse effect on the gain in these cases. However, it becomes relatively less detrimental as b is reduced. This is true in spite of the fact that nearly the same percentage of the dc beam current is intercepted by the circuit at saturation in each of these Brillouin-flow cases (it is actually largest for b = 0.5). This type of behavior is, of course, due to the differences in the current interception patterns along the circuit, as can be seen in this figure.

A magnetic-field strength corresponding to M0 = 1.5 for ωρ/α> =0.1414 and b = 1 with C = 0.1 was seen to be sufficient to keep the interception at a tolerable level when b'ja = 0.7. It appears that a practical device would operate satisfactorily under such conditions. It is appropriate to determine what magnetic-field strength is required to retain the same degree of beam confinement when the device parameters are changed, in particular, when C is changed. This can be estimated by comparing the relative magnitudes of the forces acting to restore a perturbed electron to its equilibrium radius with the radial RF force which causes the perturbation.

Neglecting the radial force due to ddx(x)ldx, the maximum radial RF force (i.e., assuming cos Φ = 1) acting upon an electron at the edge of a disk is of the order of

FRFxA(y)(\+Cb) Iffya)

Assuming the disk to be slightly perturbed in radius, i.e., x = xb{\ +§(;;)] where | 8(y) | < 1, and taking p(y) = 1, it is found that the restoring force for a balanced-flow beam is of the order of

Fr Ä — —(S/Cœ)* 8(y) (85) 2(1+Ce)

where

5/ω=(ωρ/α>)(2Λ/02-1)1/2

which is a normalized stiffness parameter. The ratio of the perturbing force (for a small perturbation) to the restoring

force is therefore

FRF ^ 2A(y)(\ + Cb)\I1{ybW<fya)\ Fr yb\SICœf 8(y)

(86)

Thus, if all other device parameters remain the same, it is seen from Eq. (86) that, as a first-order approximation, the magnetic-field strength should be adjusted to keep SjCœ constant when C is changed if the degree of beam perturbation is to remain the same. Some calculations carried out with the DDM program confirmed this relationship.


A smooth-flow de beam, such as assumed for the previous calculations, is never achieved in practice. Initial radial velocities of the electrons, which result from thermal velocities and improper beam entrance conditions, and an incorrect amount of magnetic flux linking the cathode invariably give rise to a perturbed, or rippled, dc flow.

When the cathode-flux parameter deviates from the balanced-flow value (ATb), the beam is perturbed outward or inward depending upon whether

oo > ^b or #00 < K\). The average beam diameter, and therefore the average stream-circuit coupling, changes as a result and the growth rate of the RF signal along the circuit is altered accordingly. This affects the device gain in the manner shown in Fig. 17. For the cases illustrated the magnetic-field

FIG. 17. Gain vs normalized axial distance for unbalanced flow. (ωρ/ω= 0.1414,

Λ/0= 125, Kw*Khi C= 0.1, b= 1, x0 = 0.15, xb> = 0.105, φ0= - 30dB.)

strength was 1.25 times the Brillouin value, but the values of the cathode-flux parameter were such that the dc beam would ripple approximately 20%. For Kon > K\) the beam was perturbed outward, while for K00 < K^ it was per-turbed inward· In the former case this produces a larger gain at a particular y plane (short of saturation) than is obtained in the corresponding balanced-flow case, while in the latter case the gain is lower. As can be seen in the figure, the difference in gain for the two perturbed-flow cases increases with distance as a result of the integrated effects of the differences in average stream-circuit coupling, and at y = 7.5 the gain difference is approximately 2.5 dB. These beam perturbations have resulted in a change in gain from


that obtained in the balanced-flow case of approximately ±1.2 dB. The outward perturbation of the beam is seen to result in an earlier saturation at a lower gain. The lower gain is the result of early, adverse interception. This does not occur when the beam is perturbed inward. In fact, the interception pattern in this case is such that the interception actually enhances the gain slightly.

The above results indicate that larger small-signal gains are achieved for a device of fixed length when the beam is perturbed outward as the result of too much magnetic flux linking the cathode. Conversely, lower small-signal gains result when there is insufficient cathode flux. Although larger small-signal gains result when K00 > Κ^, the interception which occurs generally reduces the large-signal gain. Operation under space-charge-balanced-flow conditions therefore appears to result in the best overall device performance. This is verified by the experimental findings presented in Section VII.B.

The beam entering with a radial velocity content was also found to alter the device characteristics, especially when it resulted in an appreciable per-turbation of the electron flow. Generally, the effect of the electrons entering with a nonzero radial velocity was found to be undesirable since it resulted in either early and increased circuit current interception or a longer saturation length. A reduction in saturated power output was also noted in some cases.

It has been mentioned previously that the equations of conservation of power and momentum can be used to check the accuracy of the computer solutions. Power and momentum checks, defined as the percentage difference between the right- and left-hand sides of Eqs. (79) and (80), respectively, were used for this purpose. Figure 18 illustrates the variation of these quantities with axial distance for a typical case. The error indicated by the momentum check is quite small. Since this check is derived directly from the interaction equations, this shows that the integration increment has been chosen small enough for the piecewise-linear integration process to be sufficiently accurate. The approximate power check shows a somewhat larger error. This is due in large part to the fact that the average power stored in the coupling field has been neglected in the DDM power check (its effect is, of course, included in the interaction equations). If the approximate power check is corrected to account for this power, the indicated error is considerably reduced. The corrected power check then has nearly the same value as the momentum check. The magnitudes of both checks are small. Thus, the equations have been consistently formulated and correctly programmed. When any input parameter is changed such that the interaction proceeds more quickly, e.g., if A0 and/or C are made larger, these checks indicate a larger error, showing the necessity of using a smaller integration increment to achieve the same relative accuracy in the solution. Of course, larger errors are indicated for the same input parameters when Ay is increased.


- 6

APPROXIMATE POWER CHECK (NEGLECTS POWER STORED IN COUPLING FIELD)

8 10

FIG. 18. Power and momentum checks for a typical TWA solution. (ωρ/ω = 0.1414, Λί0=1.5, K00=Kb= 0.55556, C=0.1, b=\, *« = 0.15, xb> = 0.105, 0o=-3OdB, Ay =0.01.)

C. TWA SOLUTIONS FOR SPATIALLY VARYING MAGNETIC FIELDS

Spatially varying magnetic fields are often employed for beam confinement in a traveling-wave amplifier since they provide some practical advantages over uniform-magnetic-field focusing. One particularly popular focusing scheme involves the use of periodic magnetic fields. This type of field can be supplied by a magnetic structure which requires no electrical power (when permanent magnets are used) and is considerably lighter than that necessary to provide a uniform magnetic field having equivalent focusing properties. However, a dc electron beam focused in a periodic magnetic field is known to exhibit instabilities under certain conditions. While the magnetic-field parameters can be selected to avoid these instabilities, the electron flow is appreciably altered when a large RF signal is present and the question of stability must be re-examined. The effects on overall device performance of the radial motion of the electrons which is inherent to a periodically focused flow must also be considered.

Magnetic-field tapers (i.e., a magnetic field which increases with distance along the device) are sometimes employed in a practical TWA. The reason


for this is perfectly obvious from the results of the preceding section; the magnetic field need be strong only where the RF signal is large, which is near the output end of the device. By tapering the magnetic field, a strong con-fining force can be supplied only where it is necessary, thus effecting a reduc-tion in the total magnetic-field requirement. In practice, the placement of the taper region and the degree of taper are usually determined by empirical methods. The DDM analysis provides a means for selecting a near-optimal taper and for determining the effects of the taper on the RF performance of the device. Some representative results for the above focusing schemes are presented next.

A comparison of the device behavior for various types of magnetic focusing fields is given in Fig. 19. Results are shown for magnetic focusing by a uniform field, a periodic field (PPM), and a linearly tapered field.

1.2

1.0

0.6

0.4

ω ρ / ω =0.1414 γα = 1.65

C = 0.1 b'/o = 0.7

b = I.O *o = - 3 0 dB

TYPE

UNIFORM

PPM

LINEAR TAPER

M

1.5

1.5 (RMS)

1.25 (INITIAL) 1.5 (FINAL)

A max

1.292

1.293

Ι.2ΘΘ

W ' y s a t )

5.6

8.2

1.8

UNIFORM -

GAIN S 35.2 dB

EFFICIENCY S 33.3%

-r 1 8 9 10 II

FIG. 19. Comparison of TWA solutions for uniform, periodic, and linearly tapered magnetic fields.

The uniform-magnetic-field case is for a magnetic-field strength 1.5 times the Brillouin value, while for the PPM case the root-mean-square value of the magnetic field is 1.5 times the Brillouin value. Comparing the curves for these two cases, it is seen that the RF signal increases with distance at a lower rate when PPM focusing is used. This is a result of the beam being injected into the interaction region at the same radius in each of these cases. For PPM


focusing, the beam ripples inward from this initial radius and the average beam-circuit coupling is, therefore, a little lower. Nevertheless, these focusing methods are nearly equivalent in terms of maximum achievable gain since almost the same saturation value is obtained in each case. Of course, the PPM-focused tube must be a little longer to achieve this gain if all other conditions are the same. It should also be noted that when PPM focusing is used there is more interception because the beam is slightly less stable. Even so, at this value of the stability parameter a (a = 0.3), the focusing properties of the periodic magnetic field are essentially equivalent to those of the uniform magnetic field. Some additional calculations showed that this is true provided a ^ 0.3. For larger values of a, a periodically focused beam is considerably less stable under an RF perturbation and the interception is much larger than occurs when a uniform magnetic field is employed.

The third curve in this figure is for a magnetic field which is linearly increased over the output of the tube. For the case illustrated, the magnetic-field strength is 1.25 times the Brillouin value in the region between the input and the location of the first arrow on the abscissa. Between the arrows, the magnetic field increases linearly to 1.5 times the Brillouin value at the second arrow; it remains at this value over the rest of the tube. The region in which the magnetic field is increased constitutes a magnetic compression region. The reason for magnetically compressing the beam is to offset the effects of the radial RF forces which spread the beam. It was found to be possible to reduce the amount of circuit interception while leaving the RF characteristics of the tube essentially unchanged. For the case shown, a reduction in the intercep-tion did occur while a slight decrease in the maximum value of the circuit amplitude was observed. Thus, with this magnetic-field configuration, less total magnetic field is required than for the uniform-field case and, at the same time, the amount of circuit interception is reduced; there is, however, a small sacrifice in gain. This type of behavior has been observed experimentally by Brewer and Anderson (14).

Figure 20 shows the trajectories for the PPM-focusing case. The rippling of the beam which is inherent to this type of magnetic focusing field is evident. The spreading of the beam by the RF forces and the formation of the bunch can be perceived. As saturation is approached, the bunch moves from the decelerating phase of the RF wave into the accelerating phase, passing through the zero-phase position where the RF force is a maximum. The bunch is then accelerated outward quite rapidly and interception on the circuit results. This behavior is similar to that shown by the trajectories for the corresponding uniform-field case which were given in Fig. 14. In this PPM case, however, the flow is slightly less stable; interception commences just before saturation and is of a larger magnitude.

The trajectories for the linearly tapered magnetic field case are shown in


ELECTRON SYMBOL ]

1 · I 0.04+ 9 · I

17 I 25 ♦ I

0.02+ I

SATURATION PLANE »"■

000» » > t 1 1 < » 1 »—J · Ό.00 1.00 2.00 3.00 4.00 5.00 6.00 700 8.00 9.00 10.00

Y

FIG. 20. Normalized disk radius vs normalized axial distance. (ων/ω= 0.1414, A/ r ms= 1.5, K01 = 0.173, l / y 0 = 0 , a = 0 . 3 , C= 0.1, 6 = 1, xa = 0.15, xb> = 0.105, 0 o = - 3 O d B . )

Fig. 21. Recall that the taper region is located between the two arrows on the abscissa. When the electrons enter this tapered-field region, the increasing magnetic force tends to deflect them toward the axis as can be seen. The bunch is again formed and, as it moves outward, it is deflected away from the circuit by the magnetic force. The interception for this case is seen to be delayed until after saturation and is of a lower magnitude.

D. SPECIAL TOPICS

The large-signal interaction equations which have been presented are not restricted in applicability to the traveling-wave amplifier. They apply in general to any O-type device in which a single-frequency forward wave is interacting with an electron stream. Thus, they can be used to analyze the operation of an RF to dc energy converter and RF slow-wave focusing, which are considered in this section.

1. O-Type Traveling-Wave Energy Converter

Rowe (/) has analyzed the operation of an O-type Traveling-Wave Energy Converter (O-TWEC) using a one-dimensional theory. In such a device, the


0.16

0.14+ CIRCUIT LINEARLY TAPERED MAGNETIC FIELD

0.12-

0.10

0.08

0.06

0.04

0.02

nnn

ELECTRON SYMBOL 1 a 9 17 25

, SATURATION PLANE -

, L_ _

ΥΨΚΝΑΚΙΚ Λ

- - — ■

-M , 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Y

FIG. 21. Normalized disk radius vs normalized axial distance. (ωρ/ω = 0.1414, <oco = 0.25, 00 = 0.25,70 = 0 . 2 , ^ = 6 . 8 , ^ = 8 . 8 , 0 = 0 . 1 , 6 = 1 , ^ = 0.15,^^ = 0.105, 0o=-3OdB.)

beam voltage is adjusted so that the stream velocity is less than the wave phase velocity. Under these conditions, the wave accelerates the stream electrons, thereby increasing their kinetic energy. Thus, RF energy is converted into dc energy. It is of interest to determine how radial variations of the RF wave and radial motion of the electrons affect the behavior of an O-TWEC.

The results of some calculations made with the DDM program are shown in Fig. 22. These correspond to a few of the cases investigated by Rowe (/, Fig. 2, p. 388). Two types of differences are noted. The first is that the energy exchange takes place at a slower rate in the two-dimensional case as a result of the reduced average coupling. Thus, the device would characteristically be longer than predicted by the one-dimensional treatment. The second is that the behavior of A(y) near "saturation," which in this device is where A(y) reaches a minimum, is considerably different. This is the result of radial motion in the following way.

The magnetic-field strength employed in these cases was equal to the Brillouin value. The radial motion is then quite large since A(y) is relatively large. This can be seen from the trajectory plots for one of these cases shown in Fig. 23. As "saturation" is approached, the majority of the electrons have been accelerated and advance in phase from the accelerating region into the decelerating region. They experience a large outward radial force when they approach and pass through the Φ = 0 position. Thus, upon entering the


decelerating region, they are more strongly coupled to the circuit than those still in the accelerating region. They contribute energy to the wave and, since they are more strongly coupled, their effect predominates. A(y) then begins to increase. Some distance later, the majority of the electrons are again in

1.2

1.0

0.8

< 0.6

0 4

J L J L J I I L 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

y

FIG. 22. Variation of the normalized RF voltage amplitude along the circuit for an O-TWEC. (ω11/ω= 0.1414, M0= 1, K00=Kh=0, C= 0.1, xa= 0.15, xb> = 0.075.)

016

014

0.10

008+

0.04

0 02

0.00 050 1.00 150 200 2 50 3 00 3 50 4 00 4.50 5.00 5.50 6.00

FIG. 23. Normalized disk radius vs normalized axial distance for an O-TWEC. (ωρ/ω = 0.1414, MQ= \,KO0=Kh=0, C= 0.1, b= - 1.305, xa= 0.15, x6, = 0.075, A0= 0.8.)


accelerating phase positions and the conversion of RF to dc energy resumes. Subsequently, the above behavior is seen to repeat. Thus, the radial motion of the electrons which was found to result in better dc to RF energy conver-sion in a TWA is seen to adversely affect the operation of an O-TWEC. If efficient conversion of RF to dc energy is to be achieved, it is necessary to strongly confine the stream.

2. RF Slow- Wave Focusing

Birdsall and Rayfield (75) have studied the phenomenon of RF slow-wave focusing both theoretically and experimentally. Some calculations were carried out with the DDM program to determine how well their experimental results could be predicted.

The case chosen for the calculations is that represented by Curve 1 in their Fig. 11 (75, p. 618). The device employs no magnetic field (M0=0) and the following parameter values were determined from their specified data: ωρ/ω = 0.0764, C = 0.0425, b = -8 .5 , γα^\, and b'la =0.26. The trajec-tories for input power levels of 5 and 15 W are shown in Fig. 24. [For no input power, all of the electrons simply follow the universal beam spreading curve, which is designated by UBS in Fig. 24(a), (b).] It can be seen that RF slow-wave focusing is indicated and the results show an improvement in focusing with increased input power, just as found experimentally by Birdsall and Rayfield (75). The computer results showed good agreement with the experimental data, e.g., the DDM program predicted a 90% beam trans-mission for an input power of 10 W. Their results show 87% beam trans-mission at this power level.

Although the results show the possibility of focusing a beam by a slow RF wave, it does not appear that this would be particularly useful for an accelera-tor or a traveling-wave tube. Relatively large power levels and large negative values of the injection velocity parameter, b9 are necessary for effective focusing and these are far from the conditions for which these devices operate efficiently. In other words, it does not seem that these devices could be made self-focusing and still operate well. While large negative values of b were found to be necessary for effective focusing, some degree of RF slow-wave focusing occurs for less negative values of b and it was observed in the experimental investigation to be discussed in Section VII.

VI. COMPUTER SOLUTIONS OF THE GENERAL EQUATIONS AND COMPARISONS WITH THE DDM RESULTS

A. GENERAL

Results for representative TWA cases which were obtained by using the general theory are presented and discussed in this section. Computer solutions


0.70 T

FIG. 24. Normalized disk radius vs normalized axial distance for RF slow-wave focusing. (a) Input power, 5 W; (b) input power, 15 W. (ωρ/ω = 0.0764, M0 = 0, K00 = 0, C= 0.0425, b= - 8 . 5 , y a = l , b'/a= 0.26.)

98 Harry K. DeWeiler and Joseph E. Rowe

were obtained both by accounting for the RF space-charge forces and by assuming that these forces are negligible compared with the magnetic-field and RF circuit-field forces. These results reveal not only the effects of RF space charge when a finite magnetic focusing field is employed, but also the conditions under which the RF space-charge forces can be omitted from the calculations without significantly affecting the accuracy of the results.

In the latter part of this section, the general results are compared to those obtained with the DDM program. The effects of the various assumptions employed in deriving the DDM equations are thereby evaluated and the conditions for which this approximate treatment yields reasonably accurate results are determined.

The general equations are formulated on the basis of the electron stream being represented by a continuous distribution of charged rings over radius and phase. In order to solve these equations on a digital computer, the stream charge must be subdivided into a finite number of representative charge groups (rings). This is accomplished through a radial division of the stream into annular layers of charge and a subdivision of these layers into discrete charge groups.

The model used here for the radial segmentation of the stream is the same as that employed by Rowe (8). In this model the stream is divided into N annular streams (layers) and it is assumed that each of these streams carries the same current, IJN. If it is further assumed that all electrons enter the interaction region having the same axial velocity and that the charge density is independent of radius at the input, then the cross-sectional area of each of the layers is the same. Thus, the normalized radii at the layer boundaries are given by

xb>r(llN)ll2xb>, / = 1,2,...,7V (87)

Each of these layers is then subdivided into m representative charge rings per cycle of the RF wave which are injected at the mid-radii of the individual layers and subsequently followed through the interaction region. This equal-current-per-layer model was selected because in this model each charge group represents the same fraction of the total stream charge and the layers are more closely spaced near the stream edge where the RF fields vary more rapidly. This should result in the best overall accuracy.

In obtaining the solutions to the general equations which are presented in this section, the stream was divided into three layers, i.e., TV = 3. The results of Rowe (7) indicate that this degree of radial segmentation will yield sufficient accuracy for the values oïyb' used in these calculations. The number of charge groups per layer was specified as m = 33. With this specification, electron rings are injected in each layer at equally spaced phase intervals of π/16. Thus, the initial phase spacing of the charge rings in each layer is the same as the


initial disk spacing in the DDM calculations where m = 32. The additional charge group per layer is needed for the general calculations so that Simpson's rule, which requires an odd number of data points, can be used to evaluate the integrals over phase which appear in the interaction equations.

When RF space-charge effects are taken into account, the weighting-function method (/), which is described in detail in Appendix A of Detweiler (2), is employed. When the effects of RF space charge are assumed to be negligible, the axial space-charge force is taken to be zero and the radial space-charge force is evaluated by using

Esc-r=-(o>OW)l(2\v\r) (88)

which is obtained by assuming that the stream is in laminar flow. The interaction equations are solved using basically the same method as

utilized for the DDM equations, i.e., a difference-equation approximation of the first order is employed and then a piecewise-linear solution is con-structed using very small increments of the linear segment. However, the details of the solution process differ because of the additional terms which appear in the general equations and the different stream model which is employed. The procedures which were utilized are described in Detweiler (2).

The integration increment was specified for the calculations as Ay = 0.02, which is a coarser step size than was used in obtaining the DDM results. The use of this larger increment was necessary to limit the computing time per solution to a reasonable amount. The resulting loss in accuracy is not signi-ficant, as was demonstrated by performing some calculations with Ay = 0.01 and comparing the results to those obtained with Ay = 0.02. Only minor differences were observed; the relative errors were less than one to one and one-half per cent, even in the beam variables which are more sensitive to a nonoptimum value of the integration increment than are the circuit variables.

In all of the results which follow, the circuit was assumed to be lossless (d = 0) and the electrons were injected into the interaction region having the same value of axial velocity and no radial velocity component. The cathode-flux parameter was assumed to be independent of radius and was specified in such a way that minimum-ripple dc electron flow (space-charge-balanced flow in the uniform-magnetic-field cases) would be achieved. The normalized amplitude of the input RF signal was set at A0 = 0.0225, which corresponds approximately to ψ0 = - 30 dB.

B. TWA SOLUTIONS OF THE GENERAL EQUATIONS

Figure 25 illustrates the variation of the normalized RF voltage amplitude A(y) along the circuit for several values of magnetic-field strength, with and without RF space charge. The parameter values ωρ/ω = 0.1414, C = 0.1,


1.2

1.0

0.8

0.6

0 4

0.2

-

—

1 1 1

yun

NO

H RF SPACE CHARGE

RF SPACE

J

CHARGE

J L_

M ° = l ^ \

/ // fi // A / A / A /

As AS

1 1 L

jr—%-2

i i

FIG. 25. Variation of the normalized RF voltage amplitude along the circuit as a function of magnetic-field strength. (ων/ω= 0.1414, K00= Ku, C=0.1, 6 = 1 , i/=0, *α = 0.15, *b' = 0.105, «Ao=-30dB.)

b = 1, ya = 1.65, and b'\a = 0.7, which the DDM results indicate would yield satisfactory device performance, were used in these calculations.

The results for the two cases in which RF space-charge effects were neglected show the same type of behavior that was predicted by the DDM calculations, viz., the RF signal grows with distance along the circuit at a somewhat lower rate when the strength of the uniform magnetic field is increased above the Brillouin value because the stream is then spread less by the radial RF circuit field and the average stream-circuit coupling is therefore lower. As can be seen, the predicted change in gain for an increase in magnetic-field strength to 1.5 times the Brillouin value is not very large when RF space-charge forces are neglected. Specifying an even stronger magnetic field for the calculations would produce essentially no further gain change.

When RF space-charge forces are included in the calculations a more pronounced dependence on magnetic-field strength is indicated by the results. The A(y) curves, and therefore the gain, are seen to be appreciably lower for M0 = 1.5 and 2 than when the beam is Brillouin focused, at least until near saturation where the interception which occurs in the Brillouin-flow case adversely affects the device gain. Interception commences at y = 6.9 when M0 = 1 (Brillouin flow) and this results in a significant reduction in the growth rate of the RF signal. Little or no interception occurs in those cases where a stronger magnetic field is employed and no such effect is observed. At the


displacement plane where interception begins in the Brillouin-flow case, the gain is approximately 2.5 dB lower for M0 = 1.5 than for M0 = 1, and when M0 = 2 it is 2.8 dB less. These are the maximum gain differences which occur in these cases.

A comparison of the results shown in Fig. 25 for the two cases with M0 = 1.5 (with and without RF space charge) reveals the effects of RF space charge on gain. The space-charge forces tend to prevent the beam from bunching as tightly and it is seen that the RF signal grows more slowly with distance as a result. Also, the saturation gain is lower since the space-charge forces tend to destroy the bunches once they have been formed. The radial component of the RF space-charge force acts to increase the radii of those charge rings which are located in the bunch. Thus, an increased spreading of the stream should occur when RF space-charge forces are included in the calculations. The trajectory plots for these two cases shown in Figs. 26 and 27

0.00 0.00 1.00 2.00 3.00 4.00 5.00 600 700 800 9.00 10.00 11.00

Y

FIG. 26. Normalized ring radius vs normalized axial distance. (ωρ/ω = 0.1414, M 0 = 1.5, K00=Kh= 0.55556, C = 0 . 1 , b= l , < / = 0 , *α = 0.15, xb< = 0.105, φ0 = - 30 dB, RF space-charge forces neglected.)

demonstrate this effect. These trajectory plots depict the normalized radius of each of the representative charge rings used in the calculations (3 layers with 33 charge rings per layer) as a function of axial distance. The greater stream expansion which takes place when the RF space-charge forces are taken into account is easily seen when the trajectory plots for the two cases are compared. The bunch formation is readily apparent in the trajectory plots for no RF space charge. When RF space-charge forces are included, the


bunch formation can still be seen but it is not as distinct because of the spreading by these forces.

0.14]

0.12-

0.10-

0.08

0.06

0.04

0.02

0 0 0

CIRCUIT

LAYER

3

2

1

J

NO.

WITH RF SPACE CHARGE

SATURATION

JE&xaQ&y

Jm mm

jäte PLANE «η

■ I

w

Λ >

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 Y

FIG. 27. Normalized ring radius vs normalized axial distance. (ωρ/ω = 0.1414, M 0 = 1.5, Kw = Kh= 0.55556, C = 0.1, b= l , < / = 0 , Jtn = 0.15, xb> = 0.105, φ0= - 3 0 dB, RF space-charge forces included.)

Inclusion of the RF space-charge forces in the calculations produces an even greater increase in the stream expansion when the beam is Brillouin focused than was observed for M0 = 1.5. In fact, the stream expansion when RF space-charge forces are included is such that there is very little change in charge density when the stream is modulated by the RF circuit field. The effect of the small axial space-charge field which exists is counterbalanced by the stronger coupling that results from the stream expansion and the RF field grows with distance, until the point is reached where interception has a significant effect, at virtually the same rate as when RF space-charge forces are neglected. This behavior is indicated by the curves shown in Fig. 25. The use of a stronger magnetic field reduces the radial expansion of the stream. An axial space-charge field then develops which reduces the gain below that predicted by the "space-charge-free" calculations. The results shown in Fig. 25 for MQ = 1.5 illustrate this effect. Since there is still a significant stream expansion for M0 = 1.5, increasing the magnetic-field strength to twice the Brillouin value causes a further reduction in device gain when the RF space-charge forces are taken into account.


Some of the numerical results for the cases discussed above are summarized in Table II. The trends described previously are evident in these data, namely,

Table II

TWA RESULTS FOR UNIFORM MAGNETIC FOCUSING FIELDS'1

M0

1.0 1.0 1.5 1.5 2.0

RF space charge

Neglected Included Neglected Included Included

Cwt(dB)

35.3 34.4 35.0 34.5 34.3

^ a t ( % )

35.0 28.1 32.8 29.4 28.0

• sat

9.10 £10 9.05 9.56 9.72

/int(%) at ^ 8 a t

14.6 55.2

0 3.1 0

« ω ρ /ω = 0.1414, K00=Kh, C = 0 . 1 , b = 1, d= 0, * a = 0.15, xb>= 0.105, <A 0 =-30dB.

the predicted device gain and efficiency are reduced while the circuit current interception an<^ optimum device length are increased when the RF space-charge forces are included in the calculations. The most significant change in saturation characteristics occurs when Brillouin focusing is employed because of the large amount of interception that takes place.

The differences in predicted device performance (with and without RF space-charge forces in the calculations) are less when a magnetic field strong enough to restrict the interception to a tolerable amount is employed. For M0 = 1.5 the gain and efficiency are seen to be reduced by only 0.5 dB and 3.4 percentage points, respectively, and the device must be 5.6% longer when RF space-charge forces are taken into account. Although some interception takes place while none occurred in the space-charge-free calculations, this discrepancy is not significant.

It is clear from these results that a magnetic field approximately 1.5 times the Brillouin value should be adequate to effectively confine the beam in a large-signal TWA whose device parameters correspond to those used in the calculations. The use of a stronger magnetic field is neither necessary, because the interception is quite small, nor desirable, since this would reduce the gain and require a larger solenoid or magnet. When the beam is effectively confined, the space-charge-free calculations predict the device behavior with reasonable accuracy for moderate values of ωρ/ω. The space-charge-free calculations also provide good accuracy in predicting the small-signal behavior of the Brillouin-beam TWA. However, they are not sufficiently accurate when this device is operating under large-signal conditions. The results clearly indicate that RF space-charge effects are smaller when the beam is Brillouin focused than when it is in confined flow, but only under small-signal conditions. Thus, even though the Brillouin-beam TWA would provide more small-signal gain than the corresponding confined-flow device, it would not perform satisfactorily as a high-power amplifier. Under large-signal conditions an


appreciable fraction of the beam electrons would strike the slow-wave circuit, causing a reduction in gain and problems with circuit heating.

A series of calculations were made to evaluate RF space-charge effects, and their relative importance, at different injection velocities. The magnetic-field strength was specified as 1.5 times the Brillouin value for these calcula-tions because the previous results for b = 1 indicate that this would provide adequate beam confinement. As before, the results indicated that the RF space-charge forces lower the signal growth rate, thereby increasing the interaction length required to reach saturation, and reduce the saturation gain and power output. However, the effects of RF space charge were found to be relatively larger for low injection velocities, which can be seen from the data given in Table III. This is to be expected since the RF wave tends to

Table III

TWA RESULTS FOR VARIOUS INJECTION VELOCITIES'1

b

0 0 0.5 0.5 1.0 1.0

RF space charge

Neglected Included Neglected Included Neglected Included

1?,at(%)

17.9 15.9 25.4 23.3 32.8 29.4

Pmi (dB rela-tive to

c|/0| K0)

2.52 2.01 4.05 3.67 5.16 4.69

Jsat

8.02 9.72 8.06 9.20 9.05 9.56

*int(°/o) a t y*t

0 11.5 0

13.5 0 3.1

"ων/ω= 0.1414, Λ/0=1.5, K00=Kh= 0.55556, C=0.1, cf=0, *„= 0.15, xh> = 0.105, <A„=-30dB.

cause tighter bunching of the electrons at low relative injection velocities, and therefore the RF space-charge forces are larger and have a greater effect. The larger RF space-charge forces result in a greater amount of current intercep-tion for b = 0 and 0.5 than occurs for b = 1 even though the RF power levels are lower. Since the electrons located in the bunch are acted upon by the largest space-charge forces, it would be expected that the increase in inter-ception results primarily from the collection of bunch electrons on the circuit, and this was found to be true. These results indicate that interception under large-signal conditions is less of a problem when the injection velocity is adjusted to maximize the saturated power output rather than the small-signal gain, which is fortunate since the former is generally the operating condition chosen for a power tube.

An examination of the distribution of electron velocities with phase reveals the effects of radial RF circuit-field variations and RF space-charge forces on the electron bunching. Figure 28 shows the velocity-phase characteristics for

0.50' -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00

RING PHASE,Φ (y,χ0,Φ0), rod

FIG. 28. Velocity-phase characteristics as a function of axial distance. (ωρ/ω = 0.1414, Mo =1.5, #00=^=0.55556, C=0.1, 6 = 1 , d=0, *o = 0.15, AV = 0.105, <A0=-30dB, RF space-charge forces included.)


the case in which b = 1 and M0 = 1.5 with RF space charge included. The normalized axial distance from the input, y, is specified for each set of plots, and for each value of y the RF power on the circuit is given in decibels referenced to both the saturation power level (designated by dBs) and to C | /01 V0 (designated by dB). The data is displayed at the saturation plane and at y planes where the RF power is approximately 1, 3, and 6 dB below the power level at saturation.

The radial variation of the RF fields is seen to cause a variation in modula-tion in the different stream layers. Also, there is an increase in charge density in the vicinity of the bunches and this produces an additional potential depression in the beam. When magnetic flux links the cathode, as it does in this particular case, this results in a radial variation in the axial component of electron velocity. The bunch electrons in the outer part of the stream then have slightly larger axial velocities than do those in the core of the stream and hence advance in phase somewhat more rapidly. This produces an inversion in the relative phase positions of the bunches in the various stream layers from those indicated by space-charge-free calculations. An inversion takes place only when magnetic flux links the cathode and did not occur in the Brillouin-flow case discussed earlier.

The plots at y = 7.62 show that, in spite of the inversion, the outer layer has given more energy to the wave than have the inner layers due to its stronger coupling to the circuit. At y = 8.22 it is seen that a cusp has developed in the velocity plots for each of the stream layers. This is clearly the result of a folding of the velocity lines and is caused by space-charge forces. Subsequent developments show that the merged loci are stable. This behavior has also been observed experimentally by Cutler (16).

As the interaction proceeds, the main parts of the bunches remain in nearly fixed phase positions and there is a " bleeding" of electrons from these bunches into the accelerating-field region. While the electrons in the inner layers enter this region prior to the outer-layer electrons, they do so at higher velocities and therefore have contributed less of their energy to the wave. The various stream elements saturate at different displacement planes. As a result, the device saturates at a lower power level and at a somewhat longer distance than would be predicted by one-dimensional analyses which fail to account for radial variations. This behavior also produces a broad saturation characteristic. Some of the charge groups originally in the outer layer have been intercepted on the circuit by the time the saturation plane is reached and thus do not appear in the plot at saturation. These velocity-phase diagrams and the behavior that they indicate are strikingly similar to Cutler's experimental observations.

The effects of radial variations and RF space-charge forces on the stream modulation can also be determined by examining the variation of RF current in the stream with interaction distance. Figures 29 and 30 indicate the RF


FIG. 29. Variation of the fundamental RF layer-current amplitudes with normalized axial distance. (ωρ/ω= 0.1414, M0= 1.5, K00= Kh= 0.55556, C = 0 . 1 , b=\, i / = 0 , ΛΓα=0.15,Λ:6' = 0.105, </»„=-30 dB.)

FIG. 30. Variation of the stream RF current amplitudes with normalized axial distance. (ωρ/ω= 0.1414, M 0 = 1.5, #oo= Kb= 0.55556, C = 0.1, b= 1, d= 0, xa = 0.15, xb> = 0.105, ^ 0 = - 3 0 d B . )


current amplitudes for the cases with b = 1 and M0 = 1.5 (with and without RF space charge). The one-dimensional layer currents (fundamental RF component) given in Fig. 29 show that the current modulation develops within the core of the stream at a slower rate than in the outer portion, which is the result of its weaker coupling to the RF wave on the circuit. When RF space-charge forces are neglected, all parts of the stream attain essentially the same peak current amplitude, but at different axial distances. RF space charge is seen to cause not only a reduction of the current modulation in the small-signal region but also in the peak currents achieved under large-signal conditions. This latter effect is most pronounced in the inner layers and clearly shows that the core of the stream contributes relatively little to the interaction in a power amplifier. The higher-harmonic currents were found to be similarly affected when RF space-charge forces were included in the calculations. Thus space charge, because it prevents the electrons from bunching as tightly, inhibits the generation of harmonic currents.

The fundamental and harmonic stream RF currents (total) illustrated in Fig. 30 show that the harmonics reach lower peak amplitudes earlier in the interaction than the fundamental and subsequently exhibit more irregular behavior. As would be expected, the total RF currents are affected by space charge in the same manner as the layer currents, viz., the RF space-charge forces retard the development of RF currents in the stream (fundamental and harmonic) and reduce their peak amplitudes.

Additional calculations performed with the general large-signal program clearly show that it is desirable to design for low ωρ/ω in order to maximize the output and minimize the device length and interception. It was also found that the concept of beam stiffness is useful for estimating RF defocusing only when RF space-charge effects are small. Of those parameter values considered in the investigation, the combination ωρ/α> =0.1414, b = 1, M0= 1.5, and b'\a = 0.7 was found to yield the best overall performance when C = 0.1 and ya = 1.65. This is precisely what was indicated by the DDM results. For this "optimum" set of parameters, it was also found that RF space charge does not significantly affect the gross characteristics of the device. Thus a space-charge-free analysis, such as the DDM analysis, could be used to predict the device performance with reasonable accuracy. It will now be shown that the DDM results do agree quite well with those obtained from the general program.

C. COMPARISON WITH THE DDM RESULTS

Some comparisons were made between one-dimensional results obtained from the general and approximate theories to determine how the moderate-C assumption, which in effect neglects the backward-traveling wave, affects the


results. It was found that the use of this assumption has virtually no effect on the predictions of small-signal device performance. Some errors do arise in the predictions (optimistic) for large-signal conditions, but these are not significant for C ^ 0.1. For C = 0.1, which is the upper limit of the postulated region of validity for the approximate theory, the largest difference in the saturation value of A(y) was less than two per cent, and hence these theories agree in predicting the device efficiency to within four per cent. In fact, reasonably accurate results are obtained with the approximate theory for C ^ 0.2. Thus, the effect on the interaction of the second-order circuit-wave variations which result from the backward-traveling wave is relatively un-important. The stream nonlinearities have a greater effect and these have been taken into account in the DDM equations.

Figure 31 compares the two-dimensional results obtained with the approxi-mate treatment (disk model) to those obtained with the general treatment (ring model) for the particular uniform-magnetic-field-focused TWA which both indicated would yield the best overall performance. It is seen that excellent agreement is obtained between the results for the disk model and those obtained with the ring model for no RF space charge. The DDM theory predicts a slightly optimistic saturation amplitude which, as discussed previously, results from the use of the moderate-C assumption in this approxi-mate treatment. RF space-charge forces are seen to have an adverse effect on

1.4

1.2

S 0.8

0.2

MODEL

DISK

RING (NO S-C)

RING (WITH S-C)

Gsat(dB)

35.2

35

34.5

^sot

334

32.8

29.4

Iintd 1 Ysat)

5 6

0

3.1

DISK MODEL -

RING MODEL NO RF

SPACE CHARGE

RING MODEL WITH RF

SPACE CHARGE

FIG. 31. Comparison of TWA solutions obtained with the DDM and general large-signal programs for a uniform magnetic focusing field. (ωρ/ω = 0.1414, M0— 1.5, K00= Kh

= 0.55556, C = 0 . 1 , 6 = 1, d= 0, γα= 1.65, b'la= 0.7, 0O = - 3 0 d B . )


the interaction. A reduction in saturation gain and efficiency, along with an increase in device length, is seen to result from the space-charge forces.

By comparing the trajectory plots given in Fig. 14 to those in Figs. 26 and 27, it is seen that the radial motion predicted by the approximate theory is quite similar to that predicted by the more exact theory. There was also found to be a strong similarity between the stream RF currents. The fundamental RF current in the stream predicted by the approximate theory agreed quite well with that predicted by the general theory for no RF space charge. However, the relative errors were larger for the higher harmonics as a result of the approximations made in accounting for the radial variations of the RF circuit fields and electron variables in the DDM theory.

A comparison of other results showed that the DDM solutions are rela-tively less accurate for low values of b. This results from neglecting the varia-tion of the circuit-wave phase velocity in the coupling-function calculations. The accuracy is improved if this variation is taken into account.

The neglect of RF space-charge forces is by far the most restrictive assump-tion employed in deriving the DDM equations. However, reasonably accurate results are obtained from these equations for ωρ/ω as large as 0.1414 and this is an appropriate value for an efficient high-power tube.

Figure 32 gives a comparison of the solutions for a periodically focused

l.4i

1.2

10

0.8

0 6

0.4

0.2

DISK MODEL-

MODEL

DISK

RING (NO S-C)

RING (WITH S-C)

Gsat (dB)

35.2

349

34.3

^sat

333

323

27.5

W ' y s a t )

2.1

0

0 RING MODEL

WITH RF SPACE CHARGE

(AXIAL)

RING MODEL NO RF y

SPACE CHARGE-^ / "

ro

FIG. 32. Comparison of TWA solutions obtained with the DDM and general large-signal programs for a periodic magnetic focusing field. (ων/ω= 0.1414, Mrm8 = 1.5, KQl = 0.228, l/7o=0, a=0.15, C=0.1 ,6=l ,</=0,y*=1.65,£7e=0.7 , 0o="3OdB.)


TWA which both theories indicated would function satisfactorily as a high-power device. The agreement between the disk-model and space-charge-free ring-model results is reasonably good, although the relative errors in the DDM results are somewhat larger than occurred in the corresponding uniform-field case. Similar results were also obtained for the trajectories and the other stream and circuit-wave variables.

On the basis of the above comparisons, it can be concluded that the approximate theory yields results substantially in agreement with those obtained from the general theory when RF space-charge forces are not the predominant factor in determining the device performance. Since a practical device will generally be designed to minimize space charge because of its adverse effects, the DDM theory can be used to predict the performance of such a device with sufficient accuracy. The real value of the approximate theory lies in the ease and speed with which solutions can be obtained. Thus, the DDM theory can be conveniently employed to evaluate a design and to indicate the modifications necessary to achieve near-optimum performance. The general theory can then be used to improve the accuracy of the solution and finalize the design.

VIT. EXPERIMENTAL INVESTIGATION OF A TWA EMPLOYING A UNIFORM MAGNETIC FOCUSING FIELD

A. INTRODUCTION

A traveling-wave amplifier which employs a uniform magnetic focusing field was built in order to obtain some correlative experimental evidence. The experiments were performed with this device supported in an electron-beam analyzer which has provisions for measuring the radial distribution of the average and RF currents in the spent beam. Measurements of gain, helix current interception, and current distributions as a function of RF drive level, magnetic-field strength, and cathode flux were performed. A detailed descrip-tion of the experimental apparatus and procedures, and the results are given in Detweiler (2).

B. EXPERIMENTAL RESULTS

The experimental results were found to exhibit the general behavior expected on the basis of the theoretical investigations. In particular, it can be seen in Fig. 33 that it is necessary to operate with a magnetic field somewhat in excess of the Brillouin value in order to achieve optimum performance. The performance of the experimental device was found to be best for a mag-netic field approximately 1.6 times the Brillouin value. For a stronger magnetic field the gain is less because the greater beam confinement results in relatively


25 30 35 40 POWER INPUT, dBm

FIG. 33. Power output and helix current vs power input for space-charge-balanced flow at different values of main magnetic field. (K0 ^ 2160 V, /0 ^ 150 mA, ΑΌ 0 = Kh.)

larger space-charge effects and a lower average beam-circuit coupling. When a magnetic field near the Brillouin value is used, a significantly larger fraction of the beam is intercepted on the helix as a result of RF defocusing and this adversely affects the device gain. It can also be seen in this figure that the


helix current decreases slightly as the drive power is increased from small-signal levels, which indicates that some RF slow-wave focusing is taking place. There is, however, a net defocusing under large-signal conditions in all cases; it is seen that the degree of defocusing increases with decreasing magnetic-field strength.

The amount by which the helix current changes from its zero-drive value (i.e., its value when no RF is applied) as a function of the RF power level is a more sensitive indicator of the degree of beam perturbation caused by the RF signal. The normalized change in helix current, ^/heiix//0> *s plotted for balanced flow at several magnetic-field strengths as a function of output power in Fig. 34. The helix-current change is negative at small power levels

0.06

0.05

004

0.03

J? 0.02

ZJ

< 0.01

0

-0.01

- 0 . 0 2

- 0 . 0 3

30 35 40 45 POWER OUTPUT, dBm

FIG. 34. Variation of normalized helix-current change with power output for space-charge-balanced flow at different values of main magnetic field. (V0 £ 2160 V, /<> = 150 mA, K00 = Kb.)

in all cases as a result of the RF slow-wave focusing mentioned previously. The degree of focusing is greatest (i.e., J/heiix//0 is most negative) when the magnetic-field strength is least and decreases as the beam is made stiifer by


using stronger magnetic fields. The beam is defocused in all cases under large-signal conditions and the most extreme behavior is observed for the case in which the magnetic field is weakest.

These curves also show that the output power level at which the helix current begins to significantly increase from its minimum value becomes progressively larger as M0 is increased. The net defocusing (measured by a net change in dlheuxll0 from its minimum value to its value at saturation) steadily decreases with increasing M0, which is the type of behavior expected from the computer calculations. However, the value of ^l/heiix//o under saturated conditions is not much different for values of M0 equal to 1.38 and larger. This is due to a redistribution of current in the beam as the magnetic field is increased. Even though the outward perturbation of the beam by RF forces is reduced by increasing the magnetic-field strength, an increased concentration of current at the beam edge results in nearly the same value of ^/heiix//0· At any rate, the experiments indicate that a magnetic field approxi-mately 40% larger than the Brillouin value is sufficient to reduce interception due to RF forces to an insignificant amount. This is in good agreement with the theoretical predictions.

Figure 35 illustrates the effect of the drive level on the average and RF

K 0 o s K b

AVERAGE FUNDAMENTAL SECOND HARMONIC THIRD HARMONIC

CURRENT RF CURRENT CURRENT CURRENT POWER DENSITY DENSITY DENSITY DENSITY INPUT, dBm

DISTANCE ACROSS THE BEAM, inch

FIG. 35. Variation of the average and RF current-density profiles with input power for balanced flow. (V0= 2160 V, /„= 150 mA, M0= 1.61.)


current distributions across the beam for the case with M0 = 1.61. As the RF drive level is increased, the average-current-density profile is seen to spread, indicating the defocusing of the beam by the RF forces. The fundamental RF current is on the beam edge for small-signal conditions and, even though there is some penetration into the beam as the drive level is increased, it remains predominantly on the beam edge under large-signal conditions. Similar behavior was observed at magnetic-field strengths as large as twice the Brillouin value where space-charge-wave theory predicts that the RF current should be uniformly distributed throughout the beam. These results show that the interaction in a TWA tends to concentrate the current on the beam edge and hence the electrons in the core of the beam do not contribute much to the interaction. The second- and third-harmonic currents start out at small drive levels on the edge of the beam. However, as the operation becomes nonlinear, they appear throughout the beam since they are generated by the beam nonlinearities.

Figure 36 shows the effect on the power output and helix current vs the drive level of varying the cathode flux with the magnetic-field strength held constant at 1.28 times the Brillouin value. Adjusting the cathode-flux para-meter away from the balanced-flow value produces a perturbed or rippling dc flow. When the cathode-flux parameter was greater than the balanced-flow value, a higher small-signal gain was observed as a result of the beam rippling outward from its initial radius. This outward rippling not only increases the average beam-circuit coupling, but reduces the adverse effects of RF space charge as well because the average space-charge density in the beam is lower. However, as can be seen from the lower portion of the figure, the helix interception for this case is higher and this results in an early saturation at a lower value. With the cathode-flux parameter less than that required for balanced flow, the beam ripples inward and, as can be seen, this results in lower gain due to the reduced coupling and increased space-charge effects. In all cases the helix current increased for high drive levels and this is because the RF forces spread the beam. This type of behavior is in agreement with the predictions of the approximate theory and clearly indicates that it is desirable to operate under balanced-flow conditions.

The effect of the drive level on the average and fundamental RF current-density distributions across the beam is illustrated in Fig. 37 for a magnetic field 1.28 times the Brillouin value. The curves on the right-hand side of the figure are for the cathode-flux parameter equal to that required for balanced flow. The average-current-density profile is quite uniform for this case. As the RF drive level is increased, the profile is seen to spread more than for the case with M0 = 1.61 because of the weaker magnetic field. The fundamental RF is still predominantly on the edge of the beam but less appears within the beam than did for the stronger magnetic field. Again, the fundamental RF


POWER INPUT, dBm

FIG. 36. Power output and helix current vs power input for a fixed value of main magnetic field with different amounts of magnetic flux linking the cathode. ( V0 = 2140 V, /0 = 148 m A, Λ?0=1.28.)

current penetrates into the beam for large drive signals. For the curves on the left-hand side, the cathode-flux parameter was greater than the balanced-flow value. The average-current-density profile is seen to be less uniform. This is because the nonoptimum focusing conditions resulted in a rippled flow which was less laminar. Again, the average-current-density profile is seen to spread with increasing drive. For this case the RF current is also on the edge of the beam for small signals. However, it penetrates further into the beam, especi-


Koo^b K oo = K b /VERAGE FUNDAMENTAL AVERAGE FUNDAMENTAL CURRENT RF CURRENT CURRENT RF CURRENT POWER DENSITY DENSITY DENSITY DENSITY INPUT, dBm

DISTANCE ACROSS THE BEAM, inch

FIG. 37. Variation of the average and fundamental RF current-density profiles with input power for unbalanced and balanced flow. {V0= 2140 V, I0= 148 mA, M0= 1.28.)

ally at large drive levels. This is apparently due to nonlaminarities in the flow. Some calculations were carried out with both the DDM and the general

large-signal programs for specific experimental cases in order to determine how well they predict the device behavior. The device parameters for the experimental TWA used in the calculations are ωρ/ω = 0.202, C = 0.094, b^ -0.2 to -0.12, γα = 1.65, and a\b' = 1.6.

The calculations made with the DDM program predicted very optimistic values of gain for the experimental device. This is because the value of oup/ω for this TWA is fairly large and the experiments were performed at beam voltages slightly below synchronism. Under these conditions, RF space-charge effects are very important and a space-charge-free analysis cannot be expected to give accurate results.

Computer runs made with the general large-signal program, including the effects of RF space charge, predicted the device gain to within 0.2 dB. In addition, good qualitative predictions of the distribution of RF currents (fundamental and harmonics) and their variation with signal level were obtained. The calculations were found to be deficient in only two respects, namely, in accurately predicting the saturation length and the amount of current interception. The calculated saturation length was approximately 87% of the actual interaction length. However, the nonlaminar, rippling


nature of the electron flow was not taken into account in the calculations, nor was the small amount of circuit loss which was present. These could easily account for the optimistic prediction of the saturation length and the dis-crepancies in predicting the interception accurately.

VIII. CONCLUSIONS The validity and usefulness of the nonlinear Lagrangian analysis as applied

to such linear-beam devices as the klystron and the traveling-wave amplifier has been well established previously on the basis of one-dimensional con-siderations and subsequent comparison with experiments. The treatment of transverse circuit-field and space-charge-field effects through the use of a multiply periodic circuit potential function and a ring model for the electron beam (which can be solid or hollow) has been shown previously to yield more accurate results with regard to saturation gain, efficiency, and phase shift. The results presented and discussed here relate primarily to the question of the stability of strongly modulated beams in a variety of focusing systems such as uniform magnetic fields (Brillouin and near-Brillouin flow), linearly tapered magnetic fields, and periodic-permanent-magnet field systems. De-tailed comparisons with experimental results have further verified the basic method.

The importance of investigating the effects of transverse fields is apparent when one considers the problem of designing light-weight, high-efficiency, medium-power tubes for space applications and very-high-power tubes for radar applications. Here one must effectively control the beam dynamics with a minimum-strength focusing field. The results indicate that the important space-charge effects include the radial dc force and the axial RF force, whereas the radial RF space-charge force is relatively unimportant. A con-siderable asset in understanding the nature of the interaction process and in assessing the importance of various operating parameters is the approximate nonlinear analysis, denoted as the DDM analysis. In this treatment, the backward-traveling wave is neglected and the beam is described in terms of disks which are allowed to expand and contract in response to the RF fields. Although approximate, this analysis yields quite accurate results and requires considerably less computing time in comparison to the generalized theory which is based on rings of charge. Both analyses are easily simplified to investigate dc flow in the interaction region for arbitrary focusing systems. Thermal effects are also easily incorporated in the ring-model analysis.

A particularly interesting result for the traveling-wave amplifier relates to the distribution of fundamental and harmonic RF currents over the beam cross section for different magnetic focusing conditions. The space-charge-wave theory of klystron amplifiers indicates that the RF current is carried


on the edge of the beam under Brillouin-flow conditions and is rather uniformly distributed over the beam cross section under confined-flow conditions. On the contrary, in the traveling-wave amplifier the fundamental RF current has been found to be carried principally on the outer radius of the beam under all focusing conditions. This is due to the basically different nature of a distributed interaction system in which the beam-circuit wave coupling is strongest on the beam edge so that the modulation is continually enhanced in this region. Further, the RF space-charge forces in the interior of the beam tend to destroy the modulation in the inner region. This is also true for the harmonic currents at low levels; however, since they arise from beam nonlinearities it is reasonable that they would become more uniformly distributed over the beam at high levels, as was experimentally observed.

The efficiency of the traveling-wave amplifier may be optimized by design-ing for smooth electron flow and operating with approximately 1.5 times the Brillouin field (for C = 0.1) to prevent excessive beam spreading due to both dc and RF space-charge forces and the RF circuit wave. The principal limitation on efficiency is the velocity spread in the beam and this can be minimized, leading to increased efficiency, by employing severed circuits with voltage jumps in the output region. The maximum efficiency will result from the utilization of a combination of voltage jumps and tapered-velocity circuits.

It is suggested that the design engineer can effectively combine the use of the two analysis programs, i.e., the DDM analysis and the more general ring-model analysis, to optimize the design of high-efficiency, minimum-weight traveling-wave amplifiers. The DDM computer program is efficient and fast and may be used to evaluate device performance over a wide range of operating parameters. Once an optimum set is found, detailed information on such characteristics as beam expansion, phase variations, and harmonic current amplitudes can be determined using the more general and accurate computer program based on the ring model of the electron beam.

ACKNOWLEDGMENT

The authors wish to acknowledge support of this work by the Rome Air Development Center under Contract AF 30(602)-3569.

LIST OF SYMBOLS A(y) Normalized amplitude of RF wave along circuit

Am&x Value of A(y) at saturation A0 Normalized amplitude of RF wave impressed on circuit at the input

plane (y = 0) a Mean radius of the circuit, meters B Magnetic flux density, T

BBr Brillouin value of magnetic flux density, T

Harry K. Detweiler and Joseph E. Rowe

Bc Axial component of magnetic flux density at cathode, T Br, Bz Radial and axial components of magnetic flux density, T

Brk,Bzk Generalized radial and axial components of magnetic flux density as defined by Eqs. (17), T

B0 Spatially uniform part of the axial component of magnetic flux density, T

Bx Amplitude factor of spatially varying part of the axial component of magnetic flux density, T

b Relative injection velocity parameter, (w0 - v0)/Cv0

b' Radius of the electron beam at input plane (z = 0), meters C Gain parameter defined by C3 =Z0\ I0 \ /4V0

c Velocity of light in free space, meters/sec d Circuit-loss parameter, 0.01836 l/C

Ec Electric field intensity due to RF wave on circuit, V/meter _r,£"c _2 Radial and axial components of Ec, V/meter

Esc Electric field intensity due to space charge in the stream, V/meter r,£sc _2 Radial and axial components of E8C, V/meter

£O-r Radial component of average electric field intensity due to space charge in the stream, V/meter

e Charge of an electron (negative), C _r,F2_2 Radial- and axial-component space-charge weighting functions

Λ0') fh(z) rewritten in terms of normalized axial distance fk(z) A function, defined in Table I, representing axial variation of axial

component of magnetic flux density (y)*fi(y) Functions defined in Section III.B

G8Rt Gain at saturation, dB /houx Current intercepted by the helix, A

/intOO Per cent of the dc beam current intercepted on the circuit as a function of normalized interaction length

/0 Dc beam current, A I0(w) Zero-order modified Bessel function of the first kind of argument w Ii(w) First-order modified Bessel function of the first kind of argument w

i Index used to indicate evaluation of the variable at the /th y plane i„(y) nth component of RF current in the stream, A

J0(w) Zero-order Bessel function of the first kind of argument w Ji(w) First-order Bessel function of the first kind of argument w

j Index used to indicate evaluation of variable for theyth electron (ring or disk)

A* Value of K00 required to achieve space-charge-balanced flow, given by Kh = 1 - M0

2

Kok A generalized cathode-flux parameter defined in Table I ΛΌο A cathode-flux parameter defined by K001 (rclr0y(BcIB0)

2= (xclx0Y' (BclBoy

K01 A cathode-flux parameter defined by K01 t (rclr0Y(BclBl)2= (xc/xo)4'

(Bc/Β,Υ K0(w) Zero-order modified Bessel function of the second kind of argument w

k0 Velocity ratio, ujc L Magnet period, meters / Index used to indicate evaluation of a variable for the /th stream layer / Series loss of circuit expressed in dB per undisturbed wavelength along

circuit


M Ratio of axial component of magnetic flux density to Brillouin value Mrmt Ratio of the root-mean-square value of the axial component of

magnetic flux density to Brillouin value M0 Ratio of spatially uniform part of axial component of magnetic flux

density to Brillouin value Mo Effective value of M0 for the experimental device

m Rest mass of an electron, kg m Number of charge disks injected into interaction region during one

cycle of the RF wave. Also, number of charge rings injected into interaction region in each stream layer during one cycle of RF wave

N Number of stream layers used in the ring-model representation of the electron stream

n Harmonic number Λ,Ο) Kinetic power in the stream, W

Λ> -cO) RF power in the stream-circuit coupling field, W Pc(y) RF power on the circuit, W

Peat RF power on the circuit at saturation, dB relative to C \ I0 \ V0

Ρμ Beam microperveance defined by Ρμ t ( | h \ I Vl/2) x 106

QC Sniall-signal space-charge parameter r Radial-position variable, meters

rc Radius at which an electron leaves the cathode, meters r0 Radius of an electron at the input plane (z = 0), meters S Beam-stiffness parameter given by 5 = ωρ(2Λ/0

2 - 1)1/2

/ Time variable, sec t0 Electron entrance time, sec

t/_i(z) Unit step function i/_2(z) Unit ramp function

ux Normalized radial velocity variable defined according to (dr/dt) i2Cu0ux

uy Normalized axial velocity variable defined according to (dz/dt) = w0· (\+2Cuy)

ιΐφ Normalized circumferential velocity variable defined according to r{d<p\di)±2CuoU<p

u0 Average initial axial-velocity component of electrons in the beam, meters/sec

V(z,t) RF potential along the circuit, V V(y,x,0) RF potential in the interaction region defined as a function of the

normalized variables by Eq. (20), V V{z,r,t) RF potential in the interaction region, V

V0 dc beam voltage, i.e., the dc voltage on the beam axis, V v Velocity vector of an electron, meters/sec

v(y) Actual phase velocity of RF wave along circuit given in terms of normalized variables by Eq. (19), meters/sec

v0 Undisturbed circuit phase velocity, meters/sec x Normalized radial-position variable, (Cw/u0)r

x(y,0o) Normalized radius of an electron disk χΟ>,Χο,Φο) Normalized radius of an electron ring

xa Normalized mean circuit radius xb> Normalized radius of the electron beam at the input plane 0>= 0) xc Normalized radius at which an electron leaves the cathode x0 Normalized radius of an electron at the input plane (y = 0)

Harry K. Detweiler and Joseph E. Rowe

y Normalized axial-distance variable (Cœlu0)z ye&i Normalized axial distance at saturation

^ο, ι Normalized axial distances at which a linear magnetic-field taper begins and ends, respectively

Z0 Characteristic impedance of the RF circuit (evaluated at the circuit radius) at the frequency of the wave, Ω

z Axial-distance variable, meters Zo Initial axial position of an electron, meters

z0, z1 Axial distances at which a linear magnetic-field taper begins and ends, respectively, meters

a Stability parameter for a periodic magnetic field defined by :

"=*(££) (1+έ) oti A function defined by at. £ [f2{yt)]

1 /2

am Axial growth rate of an exponentially increasing magnetic field, Np/meter

ββ Phase constant of the electron beam, ω/ί/0, rad/meter ßm Phase constant of a periodic magnetic field defined by ßm à 2TT/L ,

rad/meter y Undisturbed radial propagation constant

yk A magnetic-field parameter defined in Table I y0 A magnetic-field parameter defined by y0= BJBo

Ay Integration increment used in piecewise-linear solution of interaction equations

b(y) Normalized radius perturbation e0 Permittivity of free space, F/meter η Charge-to-mass ratio for an electron e/m (a negative number), C/kg

>70) Conversion efficiency as a function of normalized interaction length, defined by Eq. (42)

7?8at Conversion efficiency at saturation, per cent θχ(χ) Radial phase coupling function, rad Qy{y) Phase angle of the RF wave along circuit relative to hypothetical wave

traveling at w0, rad μι Roots of ΛΟ ί) = 0 v A parameter defined by v= (1 + Cb)/C= u0/Cv0

vt A function defined by v,^ 1/C- dev(yt)ldy= u0/Cv(yi) £fc A magnetic-field parameter defined in Table I P Instantaneous charge density in the stream, C/m3

Po dc charge density in the entering beam given by p0 = I0l7rb/2u0i C/m3

Po(y) Average charge density in the stream given by Eq. (65), C/m3

p(y) Normalized average charge density in the stream defined by Eq. (68) σ(ζ,ί) Instantaneous linear charge density induced on circuit by electrons

in the stream, C/meter Φ Instantaneous phase of the RF wave along the circuit, rad

Φθν,Φο) Phase position of an electron disk relative to the RF wave, rad 0(y,xo,<Po) Phase position of an electron ring relative to the RF wave, rad

Φ0 Entrance phase of an electron, rad Ψ Angular-position variable, rad

Φ(χ) Radial amplitude coupling function


φτα Total magnetic flux linking the disk defined by the electron radius, Wb ψ0 RF input power level in dB relative to C \ I0 \ V0

[<A(*)]rm8 Root-mean-square radial amplitude coupling function given by Eq.(76)

ψ(χ) Average radial amplitude coupling function which is equal to <Α[*0,Φο)] as defined by Eq. (62)

ω Radian frequency of the RF wave, rad/sec wct A generalized radian cyclotron frequency defined in Table I, rad/sec coco A radian cyclotron frequency defined by a>co= Ul B0, rad/sec wci A radian cyclotron frequency defined by ω0ι = | -17 j Bu rad/sec ων Radian electron-plasma frequency defined by ωρ

2^ - \η\ p0/€o,

rad/sec

REFERENCES

1. Rowe, J. E. "Nonlinear Electron-Wave Interaction Phenomena." Academic Press, New York, 1965.

2. Detweiler, H. K. "Characteristics of Magnetically Focused Large-Signal Traveling-Wave Amplifiers," Tech. Rep. No. 108, Contr. No. AF 30(602)-3569. Electron Phys. Lab., University of Michigan, Ann Arbor, Michigan, 1968.

3. Rowe, J. E. A large-signal analysis of the traveling-wave amplifier: Theory and general results, IRE Trans. Electron Devices 3, No. 1, 39-56 (1956).

4. Nordsieck, A. Theory of the large signal behavior of traveling-wave amplifiers. Proc. ÏRE 41, No. 5, 630-637 (1953).

5. Tien, P. K. A large signal theory of traveling-wave amplifiers. Bell System Tech. J. 35, No. 2, 349-374 (1956).

6. Rowe, J. E. One-dimensional traveling-wave tube analyses and the effect of radial electric field variations. IRE Trans. Electron Devices 7, No. 1, 16-21 (1960).

7. Hess, R. L. Traveling-wave tube large-signal theory with application to amplifiers having D-c voltage tapered with distance. Ph.D. Dissertation, Elec. Eng. Dept., University of California, Berkeley, California, 1960.

8. Rowe, J. E. N-beam nonlinear traveling-wave amplifier analysis. IRE Trans. Electron Devices 8, No. 4, 279-283 (1961).

9. Rowe, J. E. "A Large-Signal Analysis of the Traveling-Wave Amplifier," Tech. Rep. No. 19. Electron Tube Lab., University of Michigan, Ann Arbor, Michigan, 1955.

10. Brewer, G. R. Some effects of magnetic field strength on space-charge-wave propaga-tion. Proc. IRE 44, No. 7, 896-903 (1956).

11. Palmer, J. L., and Susskind, C. Stiffness of electron beams. J. Electron. Control 10, No. 5, 365-373 (1961).

12. Rigrod, W. W., and Lewis, J. A. Wave propagation along a magnetically-focused cylindrical electron beam. Bell System Tech. J. 33, No. 2, 399-416 (1954).

13. Pierce, J. R. "Traveling-Wave Tubes." Van Nostrand, Princeton, New Jersey, 1950. 14. Brewer, G. R., and Anderson, J. R. Electron-beam defocusing due to high-intensity

RF fields. IRE Trans. Electron Devices 8, No. 6, 528-539 (1961). 15. Birdsall, C. K., and Rayfield, G. W. Focusing of an electron stream with radio-

frequency fields. / . Electron. Control 17, No. 6, 601-622 (1964). 16. Cutler, C. C. The nature of power saturation in traveling wave tubes. Bell System Tech.

J. 35, No. 4, 841-876 (1956).

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