On the Question of Space Charge in a Klystron

V. P. Panov

Klystrons at the present time have become widely used as low-power generators of ultrahigh frequencies. However, with the aid of klystrons, large powers can also be obtained. For this purpose one should use the so-called straight-through klystrons, since they possess a higher efficiency than reflex klystrons. Since the obtaining of large powers requires the use of large current densities as well, the question of the influence of the space charge created by the electrons on the processes in a klystron is of interest.

The fundamental process in a klystron is the bunching process, which consists in the formation, from a uniform electron stream, of bunches (groups) of electrons as a result of the electrons acquiring different velocities in the input resonator. On the basis of the assumptions of one-dimensionality of the problem, small amplitude of the alternating voltage in the input resonator, and small electron velocity in comparison with the velocity of light, two theories of bunching have been developed. In the so-called kinematic theory the influence of space charge is not taken into account \([1; 2; 3]\). In the other theory \([2; 3; 4]\), the influence of space charge on bunching is taken into account. Both theories are constructed in analogous form, and the formulas of these theories differ only in the expression for the bunching parameter entering the formulas, which determines the degree of concentration of the electron charges. According to the kinematic theory, the bunching parameter is determined as follows:

\[ r_k = \gamma_1 \xi, \tag{1} \]

where

\(\gamma_1\) is the velocity-modulation coefficient, \(\xi = \dfrac{\omega x}{v_0}\) is the transit angle in the drift space, \(x\) is the distance in the drift space to the point under consideration, \(v_0\) is the constant electron velocity, and \(\omega\) is the frequency.

According to the theory that takes into account the influence of space charge, the bunching parameter has the following expression \([4]\):

\[ r = \frac{\gamma_1}{a}\sin a\xi = \gamma_1 \xi \frac{\sin a\xi}{a\xi}, \tag{2} \]

where

\[ a = 0.0975\,\lambda \sqrt{\frac{i_0}{\sqrt{U_0}}}, \]

\(\lambda\) is the wavelength, \(i_0\) is the current density, and \(U_0\) is the constant accelerating voltage.

It is not difficult to see that as \(a\xi \to 0\), \(r \to r_k\), i.e., the influence of space charge may be neglected. In accordance with what has been said, it is customary to distinguish two cases: 1) when the influence of space charge may be neg-

neglected, 2) when this influence must be taken into account. However, the analysis given below shows that it is more correct to distinguish not two, but three cases of the influence of the space charge.

Let us denote the quantity \(\dfrac{\gamma_1}{a}\) entering into formula (2) by

\[ r_e=\frac{\gamma_1}{0.0975\lambda\sqrt{\dfrac{i_0}{U_0}}}. \tag{3} \]

This quantity is conveniently taken as a parameter determining the influence of the space charge on bunching. From expression (3) it is seen that the influence of the space charge will be the stronger, the smaller the quantity \(r_e\).

Depending on the numerical value of \(r_e\), three different pictures of bunching may occur, and, correspondingly, one may distinguish three cases of the influence of the space charge.

Case I: \(r_e \gg 1\)\(^{1}\). The influence of the space charge will be negligibly small. The picture of bunching may be represented by the well-known diagram shown in Fig. 1, I. In this figure the graphs of motion are plotted for three electrons which have acquired different velocities in the input resonator; along the abscissa axis the time \(t\) is plotted, and along the ordinate axis the distance \(x\). We note that in this case the graphs of motion of the electrons will be straight lines, which means constancy of the electron velocity in the drift space.

Fig. 1

Let us next draw attention to one feature of the motion of electrons in the drift space. At first (up to the point \(a_1\)) the motion of electrons in the drift space will be homogeneous, i.e., in a given cross section at a definite instant of time there will be electrons having the same velocity. After the point \(a_1\), the electrons will overtake one another; in this process the motion of the electrons becomes nonuniform, i.e., in one cross section and at one and the same instant of time there will be electrons which left the input resonator at different instants of time \(t_1\), and, consequently, which have different velocities. In this connection the function \(t_1=\varphi(t)\) becomes multivalued in the nonuniform motion of the electrons. As is known, the arrival time \(t\) of the electrons at the point under consideration and the time \(t_1\) at which the electrons leave the input resonator are related by the following dependence [1, formula 282]:

\[ t=t_1+\frac{x}{v_0}-\frac{r}{\omega}\sin\omega t_1. \]

\(^{1}\) A more exact boundary for this case is established below.

Differentiating with respect to \(t_1\), we shall have:

\[ \frac{dt}{dt_1}=1-r\cos\omega t_1 . \]

It is evident from this that \(t=\varphi_1(t_1)\) is monotonic for \(r<1\); for \(r>1\) the function \(t=\varphi_1(t_1)\) ceases to be monotonic, since the derivative can change sign. For \(r>1\) the inverse function, \(t_1=\varphi(t)\), becomes multivalued; consequently, overtaking occurs for \(r>1\).

II case: \(r_e<1\). Since in this case the bunching parameter \(r\) always remains less than unity, overtaking cannot occur in this case, no matter how much we increase the length of the drift space \(x\). This is explained by the very strong influence of the space charge. The kinetic energy of the variable velocities of the electrons will be expended in overcoming the forces of electrostatic repulsion. The absolute magnitude of the variable velocities will gradually decrease, and in some transverse section the electrons will have identical velocities. Further concentration and overtaking will be impossible, since their cause—the variable velocities—disappears. In this section the kinetic energy of the variable velocities passes into the potential energy of the forces of electrostatic repulsion. In subsequent motion, under the action of the repulsive forces, variable velocities will be imparted to the electrons (with the electrons accelerated in the input resonator becoming retarded and, conversely, the retarded ones becoming accelerated). In this process the potential energy of the repulsive forces will again pass into the kinetic energy of the variable velocities. In accordance with what has been said, in Fig. 1, II the graphs of electron motion are shown in the form of nonintersecting curved lines.

III case: \(r_e>1\). For values of \(r_e\) not very large compared with unity, the influence of the space charge will be significant; however, the pattern of bunching will differ qualitatively from that shown in Fig. 1, II, since overtaking of electrons will take place. The bunching pattern for this case is shown in Fig. 1, III; here, for comparison, the graphs of electron motion when the influence of the space charge is absent are also plotted with dashed lines.

As is seen from this figure, case III differs from case I in that overtaking (point \(a_3\)) occurs somewhat farther from the input resonator. As \(r_e\) increases, the overtaking point \(a_3\) approaches the overtaking point \(a_1\) (when the influence of the space charge is absent).

The influence of the space charge may be considered negligible if the transit angle at which overtaking occurs, \(\zeta\), calculated with allowance for the space charge, exceeds by no more than \(3\%\) the transit angle \(\zeta_k\) at which overtaking occurs in the case of complete absence of the influence of the space charge (i.e., \(\zeta_k\) is determined from the kinematic theory). When the space charge is taken into account, overtaking will occur for

\[ r=\nu_1'\zeta\frac{\sin a\zeta}{a\zeta}=1. \]

In the case of absence of the influence of the space charge, overtaking occurs for

\[ r_k=\nu_1'\zeta_k=1. \]

Let \(\zeta=1.03\zeta_k\); then

\[ \nu_1'\zeta=1.03\nu_1'\zeta_k=1.03. \tag{4} \]

In this case

\[ \frac{\sin a\zeta}{a\zeta}=\frac{r}{\nu_1'\zeta}=\frac{1}{1.03}=0.971, \]

whence

\[ a''=0.42. \tag{5} \]

Dividing (4) by (5), we obtain:

\[ r_e=\frac{\nu_1}{a}=2.45\simeq 2.5. \]

We shall conventionally take the obtained value \(r_e=2.5\) as the boundary between regions I and III.

In Fig. 2, in the coordinate system \(i_0\)—\(U_0\), the curves \(r_e=1\) and \(r_e=2.5\) are plotted, with \(\lambda=10\ \mathrm{cm}\) and \(\nu_1=0.1\) adopted. These curves divide the entire set of values into three regions corresponding to the three cases of the influence of space charge considered above. These regions are designated by the corresponding Roman numerals.

Fig. 2

The curve \(r_e=1\) in Fig. 2 determines the minimum current density at which bunching does not occur. We shall call this current density critical. The critical current density may be determined from the following formula

\[ i_{0\mathrm{cr}}=\frac{\nu_1^2\sqrt{U_0}}{0.0095\,\lambda^2}. \tag{6} \]

Naturally, for calculations in region I one may use the kinematic theory, whereas for calculations in regions II and III it is necessary to use a theory that takes space charge into account.

Application of the theory that takes space charge into account to region III for electron motion after bunching leads to very unexpected results. According to the kinematic theory, after bunching, as is known [5], two current peaks are formed, which, as \(x\) increases, diverge in opposite directions. It would seem that the influence of space charge should lead to a more rapid divergence of the current peaks after bunching because of the forces of mutual repulsion. But the theory leads precisely to

opposite result. As \(x\) increases, the bunching parameter

\[ r=\frac{v_1}{a}\sin az \]

will at first increase, until \(az\) reaches \(\frac{\pi}{2}\), and then will decrease. Therefore the two current peaks that arise for \(r>1\) will at first move apart, and then, as \(r\) decreases, again merge into one peak¹. Construction of the graphs of electron motion leads to a similar result: after the crossing of the electrons at the point \(a_3\) (Fig. I, III), according to the theory there should be observed a second crossing at the point \(a'_3\). It then becomes unclear what forces cause the electrons to approach one another and again overtake one another. It is evident that such a character of the motion cannot be explained by the influence of electrons belonging to neighboring periods, since for values of \(r_e\) only slightly exceeding unity the electrons will separate after the overtaking point by very small distances. In this connection one may doubt the correctness of the result obtained from the theory.

The disagreement of the result obtained with physical ideas is probably explained by the fact that we were not entitled to apply the theory analyzed here, which takes account of space charge, to the nonuniform motion of electrons (it should be noted that such an application occurs in many works). The point is that the various existing derivations of formula (2) usually proceed from the equations of charge conservation (7) and of total current (8):

\[ I_0\Delta t_1=I\Delta t, \tag{7a} \]

\[ j=\rho v+\varepsilon_0\frac{dE}{dt}, \tag{8a} \]

where \(I\) is the current through some transverse section in the drift space, \(\Delta t\) is the interval of time during which a certain space charge passes through the selected section, \(I_0\) is the current at the beginning of the drift space, \(\Delta t_1\) is the interval of time during which the electrons making up the charge under consideration pass through the beginning of the drift space, \(\varepsilon_0\dfrac{dE}{dt}\) is the displacement-current density, \(\rho v\) is the electron-current density, and \(j\) is the total-current density.

However, the form of equations (7a) and (8a) given above is valid only for uniform motion of electrons. For nonuniform motion these equations must be written in the following form:

\[ I_0\sum \Delta t_i=I\Delta t, \tag{7b} \]

\[ j=\sum \rho_i v_i+\varepsilon_0\frac{dE}{dt}, \tag{8b} \]

where \(\Delta t_i\) is one of the intervals of time during which the electrons making up the space charge under consideration pass through the beginning of the drift space, the summation being carried out over all such intervals; \(\rho_i\) is the density of the space charge made up of electrons with velocity \(v_i\).

Since formula (2) was obtained from equations (7a) and (8a), which are valid only for uniform motion of electrons, extending this formula to nonuniform motion of electrons is unjustified. Thus, one may say that a theory taking space charge into account has not been developed for the case of nonuniform motion.

The undeveloped state of the theory leads to the fact that, as yet, there is no basis for calculating klystrons operating in region III after overtaking. Nor can the question be resolved of the maximum electronic efficiency in this case, \(\eta_{e\,max}=I_1(r_e)\), where \(I_1\) is a Bessel function of the first order. Formal application of the theory taking space charge into account gives that, for \(1<r_e<1.84\), \(\eta_{e\,max}<0.58\). If, however, one assumes that the absolute values of the alternating velocities will not decrease after overtaking (or, what is the same, that the peaks will not converge again), then, for values of \(r_e\) close to unity, one may expect \(\eta_{e\,max}=0.58\).

We have already noted that in region II there are mutual transformations into one another of kinetic energy and potential energy of the forces of electrostatic repulsion. In this case, oscillations of the velocity and density of the electrons are also observed. In region III only some part of the kinetic energy of the alternating velocities is transformed into potential energy, while in region I this part of the energy may be neglected¹. Therefore region II may be called the region of oscillations in the electron stream [4].

Conclusions

1. The influence of space charge on bunching can be characterized by the parameter \(r_e\) (3).

2. For \(r_e<1\), overtaking of electrons is impossible. It is proposed to call the minimum current density at which overtaking is absent the critical density (6).

3. Three cases of the influence of space charge may be distinguished:

1) \(r_e>2.5\) (negligible),

2) \(r_e<1\) (very strong, as a result of which overtaking is absent),

3) \(1<r_e<2.5\) (significant, but overtaking may occur).

1. A theory taking into account the influence of space charge becomes unjustified for region III in the case of nonuniform motion of electrons.

References

1. Vlasov V. F. Electron-vacuum devices, Svyazizdat, 1949.

2. Savel’ev V. Ya. ZhTF, vol. 10, p. 1365, 1940.

3. Hamilton D. R., Knipp J. K. and Kuper J. B. H. Klystrons and microwave triodes, Newyork Mc Graw Hill Book Co, 1948.

4. Katsman Yu. A. Equations of oscillations of uniform electron streams, ZhTF, vol. XXII, no. 9, 1952.

5. Handbook of Radio Engineering, ed. by Smirenin, Gosenergoizdat, p. 418, 1950.

1. From this point of view, operating modes in region I and partly in region III may be regarded as modes in which the kinetic energy of the alternating velocities remains not fully utilized. ↩↩