Traveling-Wave Tube

V. M. Lopukhin

Recently, information has appeared in the literature about a new microwave amplifier.* Experiments have shown that this type of amplifier has a number of advantages in comparison with other microwave amplifiers used up to now. The operation of the new amplifier is based on the ingenious use of the interaction of an electron beam with a traveling electromagnetic wave. This justifies a fairly detailed exposition of this question, both from the fundamental and from the technical standpoint.

Fig. 1. \(a\)—input circuit of an amplifier stage; \(b\)—equivalent of the input circuit of an amplifier stage.

1. Microwave Amplification

Before setting forth the construction and theory of the traveling-wave tube, we shall briefly enumerate the principal methods of microwave amplification used at the present time, and point out the difficulties that arise. These difficulties are connected with the special features of the operation of tubes at microwaves, and also with the behavior of oscillatory circuits under the same conditions.

Let us first consider the difficulties connected with the use of ordinary electron tubes for microwave amplification. As is known, a stage of a resonant amplifier has the form shown in Fig. 1, \(a\). It contains an oscillatory circuit with inductance \(L\), resistance \(R\), and capacitance \(C\), connected into the grid circuit of the amplifying tube. The alternating voltage applied to the grid from the circuit \(L, R, C\) is amplified by the tube.

For normal operation of the device it is very important that the tube should not introduce into the circuit of the oscillatory system \(L, R, C\) any significant additional—

* Let us recall that microwaves are usually understood to mean electromagnetic waves lying in the range from \(1\ \text{cm}\) to \(1\ \text{m}\).

circuit losses, for such losses are equivalent to an increase in the damping of the oscillatory circuit and, consequently, to a decrease in the resonant voltage that can be taken from it; that is, in the final analysis, the damping introduced by the tube will reduce the gain of the stage.

Theory and experiment show that the input circuit of a tube can be represented as an active resistance \(R_{\text{in}}\) and a capacitance \(C_{\text{in}}\) connected in parallel.

The equivalent circuit of the stage is shown in Fig. 1, b. The physical meaning of the capacitance \(C_{\text{in}}\) is clear. It is the capacitance of the control grid with respect to the cathode and the other electrodes connected to the cathode.

Especially harmful for increasing the damping of the circuit is the active resistance \(R_{\text{in}}\). Indeed, if \(R_{\text{in}}\) is small, the circuit will simply be shunted almost to a short circuit and amplification will be impossible, since the signal will not reach the grid at all.

At medium and long waves, for modern tubes \(R_{\text{in}}\) has a large value, of the order of hundreds of thousands of ohms. However, on going to ultrashort waves and especially to microwave radio waves, the value of \(R_{\text{in}}\) rapidly falls, taking values measured in hundreds or even tens of ohms.

This effect is connected primarily with the current induced on the grid (and, consequently, flowing in the grid circuit) by the electron stream flying through the device from cathode to anode. Each electron passing through the grid induces in it a certain current, which depends on the electron velocity and on its distance from the grid. This current has different signs depending on whether the electron is approaching the grid or moving away from it. The total induced current on the grid is the integral of the electron currents induced by individual electrons. It is easy to show that it is equal to zero if in the cathode—anode space the electron stream is not modulated in density. In this case the currents induced by electrons moving in the cathode—grid space are completely compensated by the currents induced by electrons moving in the grid—anode space.

The latter occurs in operation at long and medium waves, when the transit time of the electrons in the device is much smaller than the period of oscillation; the electron stream in the tube changes in phase with the voltage on the grid, modulation of the electron stream in density (i.e., the dependence of the density of the electron stream on the coordinate \(x\), measured from the cathode) is absent, and the total current induced on the grid is equal to zero.

In operation at meter and decimeter radio wavelengths, the time of flight of the electrons through the device becomes not only comparable with the period of oscillation, but may even exceed it. This will lead to the electron stream no longer being determined by only one-

instantaneous value of the voltage on the grid. The density of the electron current will now depend on \(x\), and the total current induced in the grid will be different from zero. It contains, generally speaking, both a component shifted in phase by \(\dfrac{\pi}{2}\) and a component coinciding in phase with the grid voltage. This latter component is responsible for the creation of \(R_{\text{in}}\): \(R_{\text{in}}=\dfrac{V}{I_{\text{ind}}}\), where \(V\) is the amplitude of the voltage applied to the grid and \(I_{\text{ind}}\) is the component of the induced current that coincides in phase with the voltage on the grid.

In the case of microwaves, specific requirements are also imposed on oscillatory circuits. Ordinary circuits, containing lumped self-inductance and capacitance, become very small in size, and no appreciable power can be obtained from them. The damping of a circuit, given by the expression \(\left(\dfrac{R}{\sqrt{L/C}}\right)\), increases when \(L\) and \(C\) are decreased, owing to the decrease of the ratio \(\dfrac{L}{C}\). An increase in the damping of the oscillatory circuit leads to a decrease in the resonant voltage that can be taken from this circuit; i.e., in the final analysis, a decrease in the size of the oscillatory circuit leads to a deterioration in the gain of the stage.

Finally, let us note that an amplifier containing many stages is very easily self-excited through parasitic couplings, which are extremely difficult to eliminate.

In what directions should methods of microwave amplification be improved?

One such possible direction is a return to the use of triodes. However, these triodes must have very small dimensions, so that the distance between the electrodes will amount to fractions of a millimeter. Obviously, with such tube dimensions the influence of the electron transit time will be very small. As circuits in these special triodes there are used so-called endovibrators, which are closed metallic cavities inside which very intense oscillations of the electromagnetic field are possible. The resonant wavelength of an endovibrator depends on the type of oscillation, the shape of the endovibrator, and its dimensions. The largest among the resonant wavelengths usually has the order of the dimensions of the endovibrator, i.e. lies in the region of microwaves. The damping of oscillations in such circuits is very small.

Endovibrators are also used in klystrons—electron-beam devices intended for the amplification and generation of microwaves.

A schematic representation of a two-cavity klystron is given in Fig. 2. It contains a cathode \((K)\), an accelerating electrode \((A)\), endovi-

vibrator-modulator, the drift space \((BC)\), the endovibrator-receiver and, finally, the electron collector \((D)\). The signal to be amplified is fed along the cable line \(E\), through a coupling loop, to the endovibrator-modulator. The electron stream passing through the grids of the first endovibrator is velocity-modulated by the signal being amplified.

In the drift space this velocity modulation is converted into density modulation of the electron stream. It can be arranged so that the density-modulated electron stream will excite, in the second endovibrator, oscillations more powerful than those that were applied to the first resonator. These amplified oscillations can be taken from the endovibrator-receiver by means of a coupling loop.

Fig. 2. Two-circuit klystron in amplifier mode.

A serious drawback of the klystron as an amplifier is its very narrow pass band, associated with the small damping of the endovibrators and, consequently, with the “sharp” resonance curve.

Finally, one should note one more fundamental shortcoming inherent in all devices of the klystron type. In these electron-beam devices the ratio of the interaction time of the electrons with the retarding electric field in the endovibrator-receiver to the total residence time of the electrons in the device is very small. (This is due to the fact that the space between the grids of the endovibrator is small in comparison with the space in which the drift of the electrons takes place.) The indicated circumstances seriously limit the field of applicability of the klystron as an amplifier.

In recent years, as an amplifier at microwaves, the “traveling-wave tube” has begun to be used; it possesses truly remarkable operating characteristics both with respect to the gain and with respect to the bandwidth. These characteristics are as follows[^3]: the power gain is 200, and the pass band at a carrier frequency of the amplified signal of \(3600\) megacycles (which corresponds approximately to a wavelength \(\lambda = 10\) cm) is about \(800\) megacycles.

2. DESIGN OF THE TRAVELING-WAVE TUBE

In the traveling-wave tube, use is made of the interaction of a beam of electrons with an electromagnetic wave having a longitudinal component of the electric field. The principle of operation of the tube is as follows (see Fig. 3). In one and the same direction there are passed an electron beam, accelerated by a constant potential, and an electromagnetic

wave propagating along a special device that reduces its phase velocity by a factor of 10–15 in comparison with the velocity of propagation of a wave in empty space. Usually such a device is a conducting helix; the electron beam passes along the axis of this helix. The interaction of the electrons with the electromagnetic wave continues during the entire time of motion of the electrons inside the helix, which has a considerable length (of the order of 30–40 cm). Therefore

Fig. 3. Diagram of a traveling-wave amplifier. In Fig. (a) are shown the heated cathode, the accelerating electrode, the helix that slows the wave, and the electron collector. The signal input and output are carried out by means of waveguides. In drawing (b) the helix and the electron beam are shown on an enlarged scale.

the effectiveness of such interaction for small signals can be very substantial, and thus the gain coefficient can be sharply increased in comparison with ordinary tubes and klystrons. The absence of grids (which are present in klystrons) makes it possible to avoid unnecessary losses in the electron beam. Finally, the absence of any resonant circuits in the device makes it possible to expect a wide pass band.

In Fig. 4 is given a photograph of a traveling-wave tube whose operating characteristics were given above. In the elongated part of the tube, which, as can be seen from the photograph presented, is 14 inches (i.e., about 40 cm) long, there is placed a helix that slows the electromagnetic wave.

Shown in Fig. 5 is the same tube, with a coil (solenoid) placed over the elongated part of the tube. The magnetic field of this solenoid focuses (compresses toward the axis) the electron beam moving along the axis of the helix.

Fig. 4. External view of the traveling-wave tube.

Fig. 5. Traveling-wave tube together with the focusing coil placed over it. The cross sections of the waveguides through which the signal is supplied and taken off are visible.

The figure also shows the rectangular cross sections of the input and output waveguides, through which microwave signals are supplied and taken off.

In the following paragraph we shall set forth a brief theory of wave propagation along a conducting helix. As has already been noted, such a helix is used in a traveling-wave tube for slowing down an electromagnetic wave.

3. THEORY OF THE HELIX\(^{9,15}\)

Consider an infinite helix made of an ideal conductor, having a circular cross-section of radius \(a\) (the section is taken perpendicular to the axis of the helix, along which the \(z\)-axis is directed) and with the turns inclined to the said section by an angle \(\alpha\) (see Fig. 6). The position of any point of the helix may be defined by specifying a single parameter—the arc length \(s\), measured from some arbitrary point. This parameter is expressed by the formula:

\[ s=\widehat{AP}=\frac{a}{\cos\alpha}\,\varphi, \]

where \(A\) is the point from which the count begins, \(P\) is the current point, and \(\varphi\) is the azimuthal angle corresponding to the point \(P\).

Fig. 6.

Let us assume that along the helix there propagates a current wave of the form:

\[ I=I_0\cdot e^{i(\omega t-ks)}, \]

where \(\omega\) is the angular frequency, \(k=\frac{\omega}{c}\) is the wave number, and \(c\) is the velocity of light in vacuum. In what follows we shall take all fields to vary harmonically in time. The vector potential on the axis of the helix, produced by the current propagating along the wire of the helix, will be:

\[ \mathbf{A}(M)=\int_{-\infty}^{+\infty} I(s)\,\frac{e^{-ikr}}{r}\,ds, \]

where \(M\) is a point on the axis of the helix, and \(r\) is the distance from the current point \(P\) to the point \(M\), at which the vector potential is evaluated (see Fig. 6).

Replacing \(I(s)\) by \(I_0 e^{-iks+i\omega t}\), we have:

\[ \mathbf{A}(M)=I_0 e^{i\omega t}\int_{-\infty}^{+\infty}\frac{e^{-ik(s+r)}}{r}\,ds. \]

It is obvious that \(r\) is expressed in terms of the basic parameters of the helix in the form

\[ r=\sqrt{a^2+\left(a\varphi\tan\alpha-z\right)^2}. \]

If we now make the change of variables

\[ (a\varphi \operatorname{tg}\alpha - z)=\eta,\qquad \text{so that}\quad s=\frac{1}{\sin\alpha}(z+\eta), \]

then the expression for the projection of the vector potential onto the axis of the helix takes the form:

\[ A_z(M)=I_0 e^{-i\frac{k}{\sin\alpha}z} e^{i\omega t} \int_{-\infty}^{+\infty} \frac{e^{-ik\left(r+\frac{\eta}{\sin\alpha}\right)}}{r}\,d\eta, \]

where

\[ r=\sqrt{a^2+\eta^2}. \]

Already from this expression it is clear that, along the axis of the system, the wave of the vector potential propagates with phase velocity

\[ v_{\text{phas}}=\frac{\omega}{k/\sin\alpha}=c\sin\alpha, \]

i.e. (for \(\alpha\ne 0\)) with a velocity \(\sin\alpha\) times smaller than the velocity of propagation of a wave in free space. As is known, the electric field can be obtained from the expression for the vector potential by simple differentiation. Consequently, the electric and magnetic fields determined by this vector potential also propagate with phase velocity \(c\sin\alpha\). Hence it is clear that the helix can be used as a slowing-down device for an electromagnetic wave.

Fig. 7. Longitudinal field on the axis as a function of frequency.

Let us list, without proof, the most important properties of the fields inside the helix, which can be obtained by studying the above expression for the vector potential.

In Fig. 7 the amplitude of the component of the electric-field intensity \(E_z\) on the axis of the helix is presented as a function of the frequency

\[ f=\frac{\omega}{2\pi}. \]

What is essential for us is that this component of the field intensity decreases with increasing frequency at sufficiently large \(\omega\). This can be interpreted in the following way. With increasing frequency, the capacitive conductance of the turns of the helix, equal to \(C\omega\) (where \(C\) is the capacitance), increases; therefore the field is concentrated in the immediate vicinity of the conductor, and the field on the axis of the helix correspondingly falls.

The fall of \(E_z\) to zero as \(\omega\to 0\) is explained by the general decrease of the fields as \(\omega\) decreases, since in the limit at \(\omega=0\) all alternating fields are absent.

Fig. 8 depicts the distribution of \(E\) (i.e. the amplitude of the electric field) near the helix as a function of \(r\) and \(z\) at a fixed instant of time \(t\). From this figure it is seen that the field is concentrated,

mainly near the helix and is considerably weakened on the axis of the system.

In the simplest theory of the helix, which we have set forth, the questions connected with the dispersion of electromagnetic waves as they propagate along the helix (i.e., the questions connected with the dependence of the phase velocity of the propagating waves on frequency) are completely left out.

At the very beginning of the theory being presented it was assumed that the phase velocity of wave propagation along the helix is, for all frequencies,

\[ C=\frac{1}{\sqrt{\varepsilon\mu}}, \]

where \(\varepsilon\) and \(\mu\) are the dielectric and magnetic permeabilities of the medium surrounding the helix.

Fig. 8. Field of the helix \(E\) as a function of the coordinates \(z\) and \(r\).

In reality, as was shown in the work of Lojakov and Ol’derogge \(^{13}\), this assumption is valid only for very high frequencies \(\omega \to \infty\) (or, correspondingly, for small wavelengths \(\lambda\), when, for example, \(\lambda\) exceeds the diameter of the helix by only a few times). For all other frequencies (or, correspondingly, for all wavelengths \(\lambda\) considerably exceeding the diameter of the helix) the velocity of propagation of the electromagnetic wave along a turn of the helix will have the form \(c \cdot f(\omega)\), where \(f(\omega)\) is a dispersion factor, greater than unity, increasing as the frequency \(\omega\) decreases.

In \(^{15}\) the general case of the propagation of electromagnetic waves along a helix placed inside a waveguide was considered, and questions relating to the dispersion of electromagnetic waves were studied in detail.

4. ELEMENTARY EXPOSITION OF THE PRINCIPLES OF AMPLIFICATION IN A TRAVELING-WAVE TUBE

Before presenting the theory of amplification in analytical form, it is useful to give a qualitative picture of the phenomena occurring in such tubes. We have seen that along an infinite helix in the absence of electron-

of the beam, an electromagnetic wave may propagate both in the positive and in the negative direction of the \(z\)-axis, possessing a phase velocity smaller than the velocity of light \((v_{\mathrm{ph}}=c\cdot\sin\alpha)\).

For brevity we shall call the first wave the forward wave, and the second the backward wave. As will be shown below (see § 5), in the presence of a weak electron beam, in which the electrons move, for example, in the direction of positive \(z\), the backward wave will change its phase velocity only slightly, while the forward wave will split into three waves propagating in the same positive direction as the beam.

The first of these three waves will decay with distance, the second will have a constant amplitude, and the third will grow. The phase velocity of the decaying and growing waves will be somewhat less than the mean velocity of the electrons in the beam \(u_0\), which is assumed to be equal to the phase velocity of the wave in the spiral in the absence of the electron beam.

The effect of the interaction of the growing wave with an electron beam overtaking this wave may be compared with the action of wind producing waves on the surface of the sea. Without claiming accuracy, and emphasizing that this is only an analogy, one may say that the electronic wind drives the electromagnetic wave along. Let us explain here why an electron beam overtaking an electromagnetic wave of increasing amplitude will, on the average, transfer its kinetic energy to the wave. (The growth of the amplitude of the wave occurs at the expense of this energy.)

Imagine that an electron moves somewhat faster than the growing electromagnetic wave. Let the longitudinal component of the electric field strength \(E_z\) (the \(z\)-axis is directed along the motion of the beam) have the form:

\[ e^{\gamma z+i(\omega t-kz)}, \tag{a} \]

where \(\gamma\) is the growth coefficient, \(k\) is the wave number, and \(\omega\) is the angular frequency. Consider a coordinate system moving together with the phase of the wave, i.e. put \(\omega t-kz=-kz'\); then expression (a) will take the form:

\[ e^{\frac{\gamma}{k}(\omega t+kz')}e^{-ikz'}. \]

It is clear that for a fixed point of the wave, i.e. \(z'=\mathrm{const}\), we have an increase of the wave amplitude in time.

Let us consider an initially homogeneous beam of electrons moving with a velocity slightly exceeding the velocity of the electromagnetic wave (see Fig. 9). Suppose that, during the motion of an electron over the segment from \(A\) to \(B\), it is accelerated. Then over the segment \(BC\) the electron will be decelerated, and its deceleration will be greater than the acceleration on \(AB\), since during the time of its motion from \(A\) to \(B\) the amplitude

(and consequently the curvature of the barrier which the electron overcomes) has increased. Hence it is clear that the electrons will be grouped on the “ascending” portions of the potential wave. Consequently, on the average there will be more slow electrons than fast ones. This loss of kinetic energy of the electrons goes into increasing the energy of the electromagnetic wave. We note that for a wave of constant amplitude the energy effect is absent on the average. Conversely, if one considers the interaction of a damped electromagnetic wave with an electron beam moving faster than this wave, one may arrive at the conclusion that there is additional damping of such a wave.

Fig. 9. Motion of electrons relative to the potential wave. The amplitude of the potential increases. Electrons concentrate on the ascending (decelerating) slopes of the potential wave.

5. MATHEMATICAL THEORY OF THE TRAVELING-WAVE TUBE[^5][^6][^7][^8][^11][^12][^13]

All that was discussed in the preceding section can be obtained analytically.

A rigorous and complete theory of the traveling-wave tube has not yet been given. At present several approximate methods of solving the problem have been proposed, proceeding from definite physical assumptions. These different methods give results that agree qualitatively with one another and are fairly well confirmed by experimental data.

The first of these methods may be called the method of successive approximations. It consists of the following.[^5] As the zeroth approximation for the field in the helix–beam system, one takes an electromagnetic wave propagating along the system as if there were no electron beam in it. Next, one considers the motion of the electrons and the modulation of the electron beam in density and velocity in the field of this wave. The convection current calculated in this way induces charges on the turns of the helix. These charges generate an electromagnetic wave, which in turn again acts on the electron beam, and so on. The successive approximations calculated by this method converge rather rapidly. The resulting field is the limit of these successive approximations.

The second method[^12][^13] consists in reducing the problem of the traveling-wave amplifier to an eigenvalue problem.

The problem is solved under reasonable physical assumptions (the beam has the form of a circular cylinder13 or of a cylindrical surface12, radial currents in the system are absent, and the signal at the input to the system is sufficiently small that the product of the variable components of the electron density and their velocity may be neglected). The solution is given in successive stages:

1) The equation of motion of the electrons is integrated under the assumption that the field in the helix, in the presence of electrons, has the form \(e^{i\omega t-\Gamma z}\), where \(\Gamma\) is the propagation constant to be determined. This integration makes it possible to find the convection current \(q\) as a function of \(\Gamma\) and \(\omega\), i.e. \(q = F(\Gamma,\omega)\).

2) The solutions of Maxwell’s equations are found separately for region \(a\) (the space inside the beam), region \(b\) (the space enclosed between the beam and the helix), and region \(c\) (outside the helix). These solutions also assume for the fields a dependence of the form \(e^{i\omega z-\Gamma z}\); in addition, they use the results of stage 1.

3) The solutions of Maxwell’s equations are matched at the boundaries of the regions \(ab\) and \(bc\). In doing so the helix is approximated by a cylindrical surface possessing conductivity only in the direction of the turns of the helix.

The matching conditions give a transcendental equation for \(\Gamma\); by solving it one can compute a series of eigenvalues for \(\Gamma\), i.e. determine the types of oscillatory processes possible in the beam–helix system16–18.

The method described is a development of works in which the simpler system electron beam—ideal waveguide was considered6–18.

The third method6,7,8,11, which will be presented in more detail than the first two and which, in the main, gives the same results, consists (like the preceding one) in the joint solution of the field equations and the equations of motion of the electrons. In coupling the solutions, however, it is assumed that the problem has a one-dimensional character. This question will be set forth on the basis of works6,7,8. The solution of the problem will be given in separate stages.

A. Determination of the Convection Current as a Function of the Field

A system is considered consisting of a helix whose axis is directed along the \(z\)-axis. At the input of this system there is supplied a beam of electrons having one and the same prescribed velocity \(u_0\), equal to the phase velocity of propagation of an electromagnetic wave along the helix in the absence of the electron beam. The electromagnetic wave, possessing a longitudinal component of the electric field, will modulate the electron beam in density and velocity; in turn, the electron beam will act upon the field of the electromagnetic wave. Our first task will be the calculation of the modulation of the electron

of the beam as a function of the resulting electric field in the beam–helix system.

We shall assume that all quantities (i.e., the current, fields, and electron velocities) vary proportionally to the factor

\[ e^{-\Gamma z+i\omega t}, \]

i.e., a steady harmonic process is considered, whose character (growth or decay as a function of the coordinate) is determined by the values of the quantity \(\Gamma\), which have not yet been determined.

Let us suppose that there is no radial motion of the electrons. This assumption is not fundamental, since the introduction of a sufficiently strong longitudinal constant magnetic field ensures focusing of the beam. Let us also assume that near the helix, where the electron beam is located, the fields do not depend on the radius.

Next, let us put the total velocity of an electron \(u\) in the form

\[ u=u_0+v, \]

where \(u_0\) is the velocity with which the electron enters the wave, and \(v\) is the variable component of the velocity, which is a small quantity, so that the condition \(|v|\ll u_0\) is satisfied at all times (the small-signal condition).

The equation of motion of the electron has the form

\[ \frac{dv}{dt}=-\eta E, \tag{1} \]

where

\[ \eta=\frac{e}{m}, \qquad \text{and} \qquad \frac{dv}{dt}=\frac{\partial v}{\partial z}\frac{dz}{dt}+\frac{\partial v}{\partial t}. \]

Since

\[ \frac{dz}{dt}=u, \qquad \frac{\partial}{\partial z}\sim -\Gamma, \qquad \frac{\partial}{\partial t}\sim i\omega, \]

we obtain from (1):

\[ -\Gamma vu+i\omega v=-\eta E. \]

In the last equality, evidently, one may replace \(u\) by \(u_0\), since in doing so terms of order \(v^2\) will be discarded. Thus we obtain the equation for \(v\) in the form

\[ -\Gamma vu_0+i\omega v=-\eta E. \]

From this relation, for the variable component of the electron velocity along the \(z\)-axis, one obtains the expression

\[ v=\frac{-\eta \dfrac{E}{u_0}}{-\Gamma+\beta i}, \tag{2} \]

where

\[ \beta=\frac{\omega}{u_0}. \]

Now one can calculate the variable component of the convection current \(q\), using the continuity equation for the charge

\[ \frac{dq}{dz}=-\frac{d\rho}{dt}, \]

where \(\rho\) is the variable component of the charge density. This equation gives

\[ -\Gamma q=-i\omega \rho . \tag{3} \]

The convection current \(q+q_0\), where \(q_0\) is the constant component, is represented in the form

\[ q+q_0=(\rho_0+\rho)(u_0+v)\simeq \rho_0u_0+u_0\rho+v\rho_0. \]

Here quantities with a zero subscript refer to the constant components of the corresponding quantities, and in the last equality the term \(v\rho\), which is assumed small, has been discarded. This term will indeed be small in comparison with the preceding terms when \(\rho/\rho_0 \ll 1\), \(v/u_0 \ll 1\), i.e. within the limits of linear effects. Evidently,

\[ q=u_0\rho+v\rho_0. \tag{4} \]

Using relations (2) and (3), we obtain from (4)

\[ q=i\beta\,\frac{q_0E}{2V_0}\,\frac{1}{(i\beta-\Gamma)^2}, \tag{5} \]

where

\[ q_0=-\rho_0u_0,\qquad u_0^2=2\eta V_0. \]

Formula (5) expresses the amplitude of the variable component of the convection current in terms of the amplitude of the field \(E\), the input current \(q_0\), the quantity \(\beta=\dfrac{\omega}{u_0}\), and the wave number \(\Gamma\). Expression (5) is an integral of the equation of motion, obtained under the assumption that the modulation of the beam is small, both in velocity \((v)\) and in density \((\rho)\).

B. Determination of the amplitude of the electric field \(E\) as a function of the convection current \(q\)

In the absence of an electron beam, along the helix (enclosed within a waveguide) there can propagate an infinite, but discrete, set of waves, called the normal waves of the given system\({}^{14}\). An individual normal wave has the form:

\[ E_n e^{-\Gamma_n z+i\omega t}. \]

Any wave field can be represented as a superposition of normal waves. Therefore, a wave traveling through the system in the direction of positive \(z\) can be represented in the form:

\[ \overrightarrow{E}=\sum_{n=0}^{\infty} E_n^{\rightarrow} e^{-\Gamma_n z}, \]

where \(E_n^{\rightarrow}\) does not depend on \(z\). A wave traveling in the opposite direction is represented in the form

\[ E^{\leftarrow}=\sum_{n=0}^{\infty} E_n^{\leftarrow} e^{\Gamma_n z}. \]

In order to find the total radiation field at a certain point, it is necessary to sum the radiation from all current elements. Using the principle of superposition, we shall consider the radiation belonging to a separate normal wave, and then sum over all normal waves.

The power passing in the direction of positive values of \(z\) through any cross section of the spiral perpendicular to the \(z\)-axis can be written in the form:

\[ P_n^{\rightarrow}=E_n^{\rightarrow} E_n^{*\rightarrow}\cdot \frac{\psi_n^{*}}{2}, \tag{6} \]

where \(\psi_n^{*}\) is a normalization factor, numerically equal to twice the value of the power that has passed through the given plane in one direction when the field amplitude is equal to unity. For waves propagating in the opposite direction, we have analogously:

\[ P_n^{\leftarrow}=E_n^{\leftarrow} E_n^{*\leftarrow}\frac{\psi_n^{*}}{2}. \tag{7} \]

For definiteness, let us consider the plane \(z=0\) and suppose that on the axis of the system at the point \(z=0\) there is a current element \(q\), directed along the \(z\)-axis*).

If the length of the current element is \(l\), then the average power of interaction of this current with the normal wave \(E_n^{\rightarrow}\) will be:

\[ P_n=-\frac{1}{2} ql E_n^{*\rightarrow} \tag{8} \]

(the asterisk, as usual, denotes complex conjugation; let us also recall that \(\frac{1}{2} qE_n^{*\rightarrow}\) represents, in the absence of losses,

*) Everywhere below we shall take the cross section of the electron beam to be unity, so that the current density coincides with the total current.

...the time average of the quantity \(q E_n^{\to}\). Similarly, for the oppositely directed wave we have:

\[ P_n=-\frac{1}{2} q l E_n^{*\leftarrow}. \tag{9} \]

Each of these two powers may be radiated in both directions, i.e.,

\[ P_n^{\to}=-\frac{1}{4} q l E_n^{*\to}, \tag{10} \]

\[ P_n^{\leftarrow}=-\frac{1}{4} q l E_n^{*\leftarrow}. \tag{11} \]

Comparing (6), (7) with (10), (11), we have:

\[ P_n^{\to}=E_n^{\to} E_n^{*\to}\frac{\psi_n^*}{2} =-\frac{1}{4} q l E_n^{*\to}, \]

\[ P_n^{\leftarrow}=E_n^{\leftarrow} E_n^{*\leftarrow}\frac{\psi_n^*}{2} =-\frac{1}{4} q l E_n^{*\leftarrow}. \]

Whence

\[ E_n^{\to}=-\frac{1}{2}\frac{q l}{\psi_n^*}, \qquad E_n^{\leftarrow}=-\frac{1}{2}\frac{q l}{\psi_n^*}. \]

Let the current distribution \(q(\zeta)\) be given. A current element generates (radiates) a field, the component of which belonging to the type of normal wave \(E_n\), is equal to:

\[ -\frac{1}{2}\frac{q(\zeta)\,d\zeta}{\psi_n^*}. \]

Let us find the field at some point \(z\ne \zeta\). At this point the field is composed of the fields generated by current elements located both to the left and to the right of the point \(z\).

For the former we have:

\[ dE^{\to}=-q(\zeta)\sum_{n=0}^{\infty}\frac{1}{2\psi_n^*}e^{-\Gamma_n(z-\zeta)}\,d\zeta, \tag{12} \]

where the sign \((-)\) before \(\Gamma_n z\) means that the radiated waves propagate in the direction of increasing \(z\). The sum corresponds to the superposition of all normal waves. Similarly, from points located to the right of the point \(z\), we obtain the field:

\[ dE^{\leftarrow}=-q(\zeta)\sum_{n=0}^{\infty} e^{\Gamma_n(z-\zeta)}\frac{1}{2\psi_n^*}\,d\zeta. \tag{13} \]

We are interested in solutions of the form

\[ q(\zeta)=I e^{-\Gamma \zeta}, \]

where \(I\) is the amplitude.

From (12) and (13) we have, respectively,

\[ E^{\to}=-\frac{I}{2}\sum_{n=0}^{\infty}\frac{e^{-\Gamma_n z}}{\psi_n^*}\int_{-\infty}^{z} e^{-\Gamma \zeta+\Gamma_n \zeta}\,d\zeta, \tag{14} \]

\[ E^{\leftarrow}=-\frac{I}{2}\sum_{n=0}^{\infty}\frac{e^{+\Gamma_n z}}{\psi_n^*}\int_{z}^{\infty} e^{-\Gamma \zeta-\Gamma_n \zeta}\,d\zeta, \tag{15} \]

or, carrying out the integration,

\[ E^{\to}=-\frac{I}{2}\sum_{n=0}^{\infty}\frac{e^{-\Gamma_n z}}{\psi_n^*}\cdot \frac{1}{\Gamma_n-\Gamma}\,e^{+(\Gamma_n-\Gamma)\zeta}\Big|_{-\infty}^{z}, \tag{16} \]

\[ E^{\leftarrow}=+\frac{I}{2}\sum_{n=0}^{\infty}\frac{e^{\Gamma_n z}}{\psi_n^*}\cdot \frac{1}{\Gamma_n+\Gamma}\,e^{-(\Gamma+\Gamma_n)\zeta}\Big|_{z}^{\infty}. \tag{17} \]

In order to ensure convergence of the integrals, one must take

\[ \operatorname{Re}(\Gamma_n-\Gamma)>0 \quad \text{and} \quad \operatorname{Re}(\Gamma+\Gamma_n)>0, \]

then:

\[ e^{(\Gamma_n-\Gamma)\zeta}\Big|_{-\infty}^{z}=e^{(\Gamma_n-\Gamma)z}, \qquad e^{-(\Gamma+\Gamma_n)\zeta}\Big|_{z}^{\infty}=-e^{-(\Gamma+\Gamma_n)z}. \]

Taking into account the equalities just written, we have:

\[ E^{\to}=\frac{I}{2}\sum_{n=0}^{\infty}\frac{e^{-\Gamma_n z}}{\psi_n^*}\frac{1}{\Gamma-\Gamma_n}e^{(\Gamma_n-\Gamma)z} =\frac{I}{2}e^{-\Gamma z}\sum_{n=0}^{\infty}\frac{1}{\psi_n^*(\Gamma-\Gamma_n)}, \tag{18a} \]

\[ E^{\leftarrow}=-\frac{I}{2}\sum_{n=0}^{\infty}\frac{e^{\Gamma_n z}}{\psi_n^*}\frac{1}{\Gamma_n+\Gamma}e^{-(\Gamma+\Gamma_n)z} =-\frac{I}{2}e^{-\Gamma z}\sum_{n=0}^{\infty}\frac{1}{\psi_n^*}\frac{1}{\Gamma+\Gamma_n}. \tag{18b} \]

To find the total field, one must add (18) and (19). Thus we have:

\[ E=\frac{I}{2}e^{-\Gamma z}\sum_{n=0}^{\infty}\frac{1}{\psi_n^*} \left(\frac{1}{\Gamma-\Gamma_n}-\frac{1}{\Gamma+\Gamma_n}\right), \tag{19} \]

or

\[ E=I e^{-\Gamma z}\sum_{n=0}^{\infty}\frac{1}{\psi_n^*}\frac{\Gamma_n}{\Gamma^2-\Gamma_n^2}. \]

In order to obtain the correct expression for the total field in the space occupied by the electron beam, it is necessary to po-

so that to the field given by (19) one must add \(E_1\), equal to \((q/ - i\omega\varepsilon)\), so that for any \(z\) we have \(q + \dfrac{d}{dt}\varepsilon E_1 = 0\). This means that in the electron beam the convection current \(q\) is compensated by the displacement current of the additional field \(E_1\), which is created by the space charges of the beam.

Taking the foregoing into account, we have for the total field in the beam

\[ E = I e^{-\Gamma z} \left[ \sum_{n=0}^{\infty} \frac{1}{\psi_n}\, \frac{\Gamma_n}{\Gamma^2-\Gamma_n^2} + \frac{i}{\omega\varepsilon} \right]. \tag{20} \]

Thus the second part of the problem has been solved: the electric field in the beam, at the point \(z\), produced by the current \(q = I e^{-\Gamma z}\), has been found.

B. Simultaneous solution of the equations of motion of the electrons and of the field equations

It remains for us to solve equations (5) and (20) simultaneously. Equation (5) is the equation of motion of the electron beam (more precisely, of the electron) in a certain specified electric field. Equation (20) determines the electric field created by the distributed current. Substituting (20) into (5), we obtain an equation for \(\Gamma\), into which the amplitudes \(E\) and \(q\) do not enter.

\[ 1 = i\beta \frac{q_0}{2V_0}\, \frac{1}{(i\beta-\Gamma)^2} \left[ \sum_{n=0}^{\infty} \frac{\Gamma_n}{\psi_n(\Gamma^2-\Gamma_n^2)} + \frac{i}{\omega\varepsilon} \right]. \tag{21} \]

G. Investigation of the equation for \(\Gamma\)

The problem consists in determining from this equation with an infinite number of terms the possible values of the quantity \(\Gamma\), which represents the propagation constant of the oscillatory process. In the general case it is difficult to find the roots of this equation. Let us recall that in the absence of the electron beam an infinite number of normal waves propagate along the helix, only one of which (with propagation constant \(\Gamma_0\)) is nonattenuating. We shall restrict ourselves to finding roots which are close in value to \(\Gamma_0\). The meaning of such a consideration is that we assume the perturbation introduced by the electron beam into the conditions of wave propagation in the system to be small.

Assuming the closeness of \(\Gamma\) to \(\Gamma_0\), we can obviously retain from the entire infinite sum only the principal term, corresponding to the value \(n=0\), i.e., reduce equation (21) to the form

\[ 1 = i\beta \frac{q_0}{2V_0}\, \frac{1}{(i\beta-\Gamma)^2} \cdot \frac{\Gamma_0}{\psi_0(\Gamma^2-\Gamma_0^2)}. \tag{22} \]

Naturally, the approximation made limits the applicability of formula (22); such a treatment is not suitable for electron beams of considerable density, which introduce large perturbations into the values of the fields. In other words, the discarded sum is connected with the influence of space charges, which we shall not take into account.

It is easy to see that (22) is an equation of the fourth degree with respect to \(\Gamma\); hence it follows that, generally speaking, it must have four different roots. This means that, within the framework of the assumption of weak interaction between the beam and the field, as a result of this interaction we obtain, instead of the two waves (direct and reverse) which existed in the system in the absence of the electron beam, four waves. Let us consider the character of these waves.

Let us rewrite equation (22) in the form

\[ (\Gamma^2-\Gamma_0^2)=\frac{i^2\beta^3 V_0 C^3}{(i\beta-\Gamma)^2}, \tag{23} \]

where

\[ C^3=-\frac{q_0}{4V_0\beta^2\Phi_0^*} =\frac{q_0}{8V_0}\frac{EE^*}{\beta^2 P_0}, \]

and

\[ \beta=\frac{\omega}{u_0},\qquad \Gamma_0=i\beta. \]

The quantity \(P_0\) is the time-average value of the flux of the Poynting vector associated with the wave \(\Gamma_0\) in the helix in the absence of the electron beam. If the heat losses in the helix are small, then the quantity \(P_0\) may be regarded as real. Therefore the parameter \(C^3\) may also be regarded as real.

To investigate the deviation of \(\Gamma\) from the unperturbed value

\[ \Gamma_0=+i\beta \]

we put:

\[ -\Gamma=-i\beta+\delta,\qquad \text{i.e.}\quad \Gamma=i\beta-\delta, \]

where \(\delta\) is small. Then

\[ \Gamma^2 \simeq -\beta^2-2i\beta\delta \]

(neglecting terms with \(\delta^2\));

\[ i\beta-\Gamma=\delta;\qquad \Gamma^2-\Gamma_0^2=-2i\beta\delta. \]

Substituting the values of these differences into (23), we obtain:

\[ \delta^3=-i\beta^3 C^3, \]

whence

\[ \delta=C\beta\sqrt[3]{-i}, \]

or

\[ \delta_1=(0.866-i0.5)\beta C, \tag{a} \]

\[ \delta_2=(-0.866-i0.5)\beta C, \tag{b} \]

\[ \delta_3=i\beta C. \tag{c} \]

Taking the solution in the form

\[ e^{-\Gamma z+i\omega t}=e^{-i\beta z+\delta z+i\omega t}, \]

we see that the waves corresponding to \(\delta_1,\ \delta_2\) and \(\delta_3\) differ in the character of their propagation.

Wave (a) is a wave moving in the same direction as the electron beam, with a phase velocity somewhat smaller than the velocity of motion of the beam \(u_0\).

The latter becomes clear if the solution is written in the form

\[ e^{0.866\,\beta C z}e^{-i\beta(1+0.5C)z+i\omega t}; \]

the phase velocity of this wave is equal to

\[ v_{\mathrm{phase}}=\frac{\omega}{\beta(1+0.5C)} =\frac{u_0}{1+0.5C}. \]

The amplitude of this wave grows exponentially. This wave, in accordance with the mechanism set forth above, is responsible for the amplification of the signal. Of waves (b) and (c) we shall say briefly: the first of them is a damped one, its phase velocity is also less than \(u_0\); wave (c) propagates without damping with a phase velocity somewhat greater than \(u_0\). These waves are not connected with signal amplification.

Considering next waves close to the counter-propagating undamped wave, i.e. seeking solutions \(\Gamma'\) in the form \(-\Gamma=i\beta+\delta\), it is easy to see that we obtain a solution of the form

\[ e^{i\omega t+i\beta z\left(1-\frac{1}{4}C^3\right)}. \]

This wave travels toward the electron beam. Its amplitude is unchanged; moreover, \(C\ll 1\), so that in general this wave is very close to the counter-propagating unperturbed wave and likewise does not participate in signal amplification.

Using the expression for the growing wave (a), we calculate the signal amplification coefficient given by a system of length \(L\):

\[ k=\frac{E(L)}{E_0}=\frac{1}{3}e^{0.866\,\beta C L}. \tag{24} \]

The coefficient \(1/3\) has the following meaning: at the beginning of the system the signal energy is divided equally among three forward waves, the amplitude of each of them being equal to \(\frac{1}{3}E_0\). Thus, the amplification grows exponentially with the length of the tube.

The solution found does not give the experimentally observed decrease of the amplification coefficient at very small, as well as very large, frequencies. The required character of the dependence of the amplification coefficient on frequency is obtained, however, if one takes into account that \(E_z\) on the axis tends to zero as \(\omega\to 0\) and \(\omega\to\infty\) (see Fig. 7). Further, expression (24) does not take into account the damping of the normal waves associated with thermal losses

valid. It likewise does not take into account the influence on the amplification of the space charge, i.e., it does not take into account the influence of nonpropagating types of oscillations. The theory set forth is a “small-signal theory,” when the products of the variable parts of the charge velocity and density may be neglected. This condition, for the tube described above, is satisfied \(^{8}\) at a full input current of \(\sim 1\) mA up to power gains of 200 times.

In the theory presented, the case was considered in which the initial velocity of the electrons at the entrance into the wave \(u_0\) is equal to the phase velocity of propagation of the wave through the system in the absence of the electron beam, \(v_{\mathrm{phas}}\). In the case where \(u_0 \ne v_{\mathrm{phas}}\), the picture of the interaction of the helix fields and the electron beam becomes more complicated. The amplification coefficient depends on the ratio \(\dfrac{u_0}{v_{\mathrm{phas}}}\). Its optimum value is attained at

\[ \frac{u_0}{v_{\mathrm{phas}}}=1, \]

decreasing on both sides of this value. When \(u_0 > u_2\) (where \(u_2\) is a certain critical velocity), amplification disappears altogether.

If one takes into account the fields produced by the space charge in the beam, the picture of the interaction of the beam and the field becomes still more complicated \(^{7,13}\).

Signal amplification will occur only within a certain interval of electron velocities at the entrance, \(u_1 < u_0 < u_2\); moreover, as the current in the beam is increased, both the interval itself and the value \(u_0\) at which the amplification is maximal shift into the region of higher velocities.

If \(u_0 \ll u_1\) or \(u_0 \gg u_2\), i.e., if \(u_0\) lies far from the range of velocities where amplification is possible, then one of the three waves will have a propagation constant very close to

\[ \frac{\omega}{v_{\mathrm{phas}}}, \]

whereas the other two will have propagation constants very close to

\[ \frac{\omega}{u_0}. \]

In other words, one of the waves will then be close to the unperturbed helix wave, while the other two will be closely connected with the electron beam. For the electron-beam–waveguide system corresponding to the case \(u_0 \ll u_1\), these three waves have long been known \(^{6-18}\).

Furthermore, a detailed analysis shows that the ratio of the amplitudes and phases of the three waves into which, in the presence of the beam, the direct wave is split depends also on the ratio of the velocities \(u_0/v_{\mathrm{phas}}\) \(^{7,13}\), so that the assumption made above,

\[ E_1=\frac{E_0}{3}, \]

is valid only in a special case.

Let us note that, according to the data of theory \(^{13}\), a traveling-wave tube may be used as a microwave generator.

Conclusion

The brief survey presented here has given only the basic information relating to the application of a new type of amplifier—the traveling-wave tube. In conclusion, three points should be emphasized.

First, the traveling-wave tube has a large gain (of the order of 200 in power) and a wide pass band (of the order of 800 mc/s). In this it compares favorably with other microwave amplifiers currently in use.

Second, the theoretical treatment of the processes taking place in the traveling-wave tube, to the extent that it is presented in the literature available to us, is far from complete.

Third, in the nature of its operation the traveling-wave tube is an inverse linear resonant accelerator, in which the traveling electromagnetic wave interacts with the electron beam and transfers its energy to it.

It may be expected that, in time, the traveling-wave tube will enter into everyday use in centimeter-band radio engineering.

References Cited

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