On Bunching in a Cylindrical Klystron with a Radially Diverging Electron Beam

V. P. Panov

(Presented by the scientific seminar of the Radio Engineering Faculty)

In the development of high-power klystrons, the question naturally arises of ways to increase the input power, which is determined by the product of the total current of the electron beam \(I_0\) and the constant accelerating electron voltage \(U_0\). The most expedient course is to increase the voltage, since this reduces the influence of the space charge, and at the same time makes it possible to increase the current density in the beam.

However, it should be noted that the use of very high voltages (for example, above 400 kilovolts) is undesirable, chiefly because at such voltages the increase in the energy of the electrons in the input resonator of the klystron will occur mainly through an increase in the mass of the electron, and not in its velocity; consequently, the application of the principle of operation of the klystron—velocity control—becomes ineffective.

Another way of increasing the input power is also possible: increasing the total current \(I_0\) by increasing the cross-sectional area of the electron stream. The latter can be accomplished, for example, in a klystron whose design is schematically shown in Fig. 1. In the figure the designations are: 1—the gap of the input resonator, 2—the gap of the output resonator, 3—the collector, 4—the cylindrical cathode. The electron stream has the form of a disk; the electrons diverge radially from the center, as shown in the figure by arrows. The arrangement of the electrodes of such a klystron is the same as in tubes of cylindrical construction; therefore such klystrons, in contrast to ordinary ones, may be called “cylindrical.” Various designs of such klystrons have been proposed by N. D. Devyatkov [6], Yu. A. Katsman [1], V. F. Kovalenko [2], and others.

In the present work an analysis is given of bunching in a cylindrical klystron, taking into account the influence of space charge. The theory of bunching for the cylindrical case can be constructed in the same way as for the ordinary case [4]. The difference will consist only in one of the initial equations—the equation of conservation of charge—which for our case will have the form:

\[ i_{\sim}=i_0\left[\left(\frac{dt_1}{dt}\right)_x-1\right]\frac{x_1}{x}, \]

where \(x_1\) is the mean radius of the input gap (Fig. 1),

Fig. 1

\(x\) — radius of the cross section under consideration of the electron beam,

\(i_0\) — density of the electron current at the input gap \(^{1)}\),

\(i_{\sim}\) — variable component of the electron-current density in the section \(x\),

\(dt_1\) and \(dt\) — time intervals during which the electron group under consideration passes through the corresponding cross sections.

Carrying out mathematical transformations analogous to those given in [4], we obtain an equation for determining the bunching parameter in the cylindrical case, \(r_u\),

\[ \frac{d^2 r_u}{dx^2}+\frac{\Lambda}{x}\,r_u=0, \tag{1} \]

where

\[ \Lambda=\frac{e i_0 x_1}{\varepsilon_0 m v_0^3}, \tag{2} \]

\(e\) and \(m\) are the charge and mass of the electron, \(\varepsilon_0=8.85\cdot10^{-12}\ \dfrac{\text{farad}}{\text{meter}}\),

\(v_0\) is the electron velocity determined by the voltage \(U_0\). Equation (1) has a solution [5, p. 238] expressed in terms of Bessel functions. The final expression for \(r_u\) is obtained in the form

\(^{1)}\) In what follows, when speaking of the current density in a cylindrical klystron, we shall always refer this density to the cross section in the middle of the input gap.

\[ r_u=\nu_1\pi\frac{\omega}{v_0}\sqrt{x x_1}\left[ \mathrm{N}_1(2\sqrt{\Lambda x})J_1(2\sqrt{\Lambda x_1})- \mathrm{N}_1(2\sqrt{\Lambda x_1})\cdot J_1(2\sqrt{\Lambda x}) \right], \tag{3} \]

where \(J_1\) and \(\mathrm{N}_1\) denote Bessel functions of the first order, of the first and second kinds, respectively;
\(\nu_1\) is a coefficient characterizing the efficiency of imparting variable velocities in the input resonator;
\(\omega\) is the operating frequency.

It is interesting to note that at small current densities (when \(\Lambda\to 0\), and the known expressions for Bessel functions at small arguments may be applied), the expression for the bunching parameter for the cylindrical case (3) passes into the usual expression for the bunching parameter according to the kinematic theory,

\[ r_k=\nu_1\frac{\omega(x-x_1)}{v_0}, \]

as was to be expected from physical considerations.

By analogy with klystrons of the usual design, one may introduce the concept of the critical current density (3), as the minimum current density at which electron overtaking does not occur. In view of the complexity of the expression for the bunching parameter \(r_u\), determining, for given \(x_1\), \(\omega\), and \(v_0\), the admissible critical current density \(i_{0\mathrm{cr}}\) and the length of the drift space \(x-x_1\) presents certain difficulties. Below we shall describe how the above quantities can be determined.

The condition for electron overtaking in our case remains the same, that is, \(r_u=1\), which may be written in the following form:

\[ \frac{1}{\pi\nu_1\frac{\omega}{v_0}\sqrt{x_1}} = \sqrt{x}\left[ \mathrm{N}_1(2\sqrt{\Lambda x})J_1(2\sqrt{\Lambda x_1})- \mathrm{N}_1(2\sqrt{\Lambda x_1})J_1(2\sqrt{\Lambda x}) \right]. \tag{4} \]

Let us denote

\[ \frac{1}{\pi\nu_1\frac{\omega}{v_0}\sqrt{x_1}}=A \tag{5} \]

\[ \sqrt{x}\left[ \mathrm{N}_1(2\sqrt{\Lambda x})\cdot J_1(2\sqrt{\Lambda x_1}) - \mathrm{N}_1(2\sqrt{\Lambda x_1})J_1(2\sqrt{\Lambda x}) \right]=B. \tag{6} \]

If one plots the dependence of the quantities \(A\) and \(B\) on \(x\), then \(A\) will be represented by a straight line parallel to the abscissa axis, while \(B\) by a certain curve having a maximum.

Such a construction has been carried out in Fig. 2 for the following specific values: \(x_1=2\ \mathrm{cm}\), \(U_0=2500\ \mathrm{V}\), \(\lambda=10\ \mathrm{cm}\), \(\nu_1=0.1\). In the figure three curves \(B_1\), \(B_2\), \(B_3\) are plotted, corresponding to the following current densities:

\[ B_1-1.1\ \mathrm{a/cm^2},\qquad B_2-0.7\ \mathrm{a/cm^2},\qquad B_3-0.88\ \mathrm{a/cm^2}. \]

At the points of intersection of the straight line \(A\) and the curves \(B\) we have a solution of equation (4). Obviously there may be three fundamentally different cases, which are illustrated by Fig. 2:

Fig. 2

1. \(A\) and \(B_1\) do not intersect.
2. \(A\) and \(B_2\) intersect at two points.
3. \(A\) and \(B_3\) intersect at only one point.

It is obvious that in case 1 there is no electron overtaking, and the corresponding current density \(i_0 = 1.1\ \mathrm{a/cm^2}\) is greater than the critical one. In case 2, overtaking occurs for \(x > 4.1\ \mathrm{cm}\), and the corresponding current density is less than the critical one. Case 3 corresponds to the maximum bunching of the electrons at a distance \(x = 4.7\ \mathrm{cm}\) from the klystron axis. The current density in this case is equal to the critical one.

Thus, of all possible solutions of equation (4), only this last one will correspond to the critical current density. To find this solution it is necessary to determine the maximum of the curve \(B_3\). It should be noted that the maxima of the curves \(B_1\) and \(B_2\) also correspond to critical current densities, but for other values of \(v_0\) (or of other quantities). It is therefore expedient to construct the curve \(C\) (shown in Fig. 2 by the dashed line), which is the locus of the maxima of the family of curves \(B\). To determine the points of the curve \(C\), we differentiate the expression for \(B\) and set it equal to zero. The result obtained may be written in the form of the equation:

\[ \frac{J_0\left(2\sqrt{\Lambda x}\right)} {N_0\left(2\sqrt{\Lambda x}\right)} = \frac{J_1\left(2\sqrt{\Lambda x_1}\right)} {N_1\left(2\sqrt{\Lambda x_1}\right)}, \]

whose solution is written in the form

\[ 2\sqrt{\Lambda x}=\frac{0.5}{k-1}\,\pi(1+\alpha), \tag{7} \]

where

\[ k=\frac{2\sqrt{\Lambda x_1}}{2\sqrt{\Lambda x}}=\sqrt{\frac{x_1}{x}}, \]

and \(1+\alpha\) is determined from Fig. 111 [5] as a function of \(k\).

The calculation of the critical current densities and of the radii of the output slit \(x\) may be carried out as follows. We assign a definite value of \(x_1\) and a series of values of \(x\). For each \(x\) we determine the quantity \(k=\sqrt{x_1/x}\) and \(2\sqrt{\Lambda x}\) from formula (7). After this it is not difficult to determine \(2\sqrt{\Lambda x_1}\), \(\Lambda\), and \(B\) from formula (6). The value of \(B\) obtained is the maximum for one of the curves of the family \(B\), and therefore can be used to determine \(i_{\mathrm{cr}}\) and the voltage \(U_0\) corresponding to this critical current density. Indeed, from formula (5), assigning arbitrary values of \(v_1\) and \(\lambda\), one can determine \(v_0\), and consequently also \(U_0\):

\[ U_0=x_1\left(\frac{A v_1 10^4}{\Lambda}\right)^2, \]

and from (2)

\[ i_{\mathrm{cr}}=\frac{1.05\cdot 10^{-5}\Lambda U_0^{3/2}}{x_1}, \]

Fig. 3

Thus, for each \(x\) one can determine \(U_0\) and \(i_{\mathrm{cr}}\). From the calculations carried out in this way, Fig. 3 gives curves of the critical current densities as functions of \(U_0\) for different radii of the input slit \(x_1\). For comparison, the curve for a klystron of ordinary design is also given here

\[ (x_1=\infty). \]

The curves presented can readily be understood from simple physical considerations. It follows from the curves that the observed increase in the critical current density together with an increase in the cross-sectional area of the electron stream makes it possible to obtain, in klystrons of cylindrical design, considerably larger values of beam current than in ordinary klystrons.

It should be noted that the limitation on increasing the cross-sectional area of the electron stream is determined by the possibility of exciting parasitic oscillations of the TE type in the klystron resonators.

The method described can also be applied to determine the radius of the output aperture in the case where the current density exceeds the critical value (which corresponds to region II according to the classification given in [3]). In this case one carries out a solution, analogous to that given above, of an equation differing from (4) only in that in the numerator of the left-hand side there stands a number less than unity. This number determines the maximum possible electronic efficiency of the klystron. For small current densities (i.e., in region I [3]) the calculation can be performed, as shown above, according to the usual kinematic theory.

If the current density is less than the critical value, but the influence of space charge is nevertheless significant (which corresponds to region III), then, as for an ordinary klystron [3], the theory ceases to be valid after overtaking, and therefore a reliable calculation can be made only up to the overtaking of the electrons.

REFERENCES

1. Katsman Yu. A. A device for the generation and amplification of electron oscillations. Author’s certificate No. 60958 of May 8, 1940.

2. Kovalenko V. F. The history of the development of electron-beam tubes for microwaves. Izvestiya Akademii nauk SSSR, Seriya fizicheskaya, vol. 4, No. 3, 1940.

3. Panov V. P. On the question of space charge in a klystron. Izvestiya TPI, vol. 73, 1952.

4. Klystrons (translated from English). Sovetskoe radio, pp. 114–117, Moscow, 1952.

5. Jahnke E. and Emde F. Tables of Functions. GITTL, 1949.

6. Devyatkov N. D., Sliozberg M. L., Daniltsev E. N. Author’s certificate No. 61000 with priority of November 11, 1940.