On the Calculation of Phenomena in the Input Device of a Klystron

I. I. Zilberman

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

General Equations

Let the alternating voltage that arises between the grids of the input resonator when a signal is applied be

\[ U_1 = U_{m1}\sin \omega t . \]

Then, if one adopts the assumptions customary for an input device, namely that it is possible to neglect the influence of the space charge and the edge effect, the equation of motion of the electrons may be written in the form

\[ \frac{dv}{dt}=\frac{e}{m}\frac{U_{m1}}{x_1}\sin \omega t , \tag{1} \]

where \(v\) is the electron velocity, \(\frac{e}{m}\) is the ratio of the numerical value of the electron charge to its mass, and \(x_1\) is the distance between the resonator grids.

Integration of this equation gives

\[ v-v_0=\frac{e}{m}\frac{U_{m1}}{\omega x_1}\left(\cos \omega t_0-\cos \omega t\right), \tag{2} \]

where \(t_0\) is the time of entry of the electrons into the resonator gap, \(v_0=\sqrt{2\frac{e}{m}U_0}\) is the velocity of the electrons at the entrance to the gap, and \(U_0\) is the constant potential of the input resonator with respect to the cathode.

If we denote by \(\Theta=\omega(t-t_0)\) the transit angle of an electron from the entrance to the gap to its cross section \(x\), by \(\zeta_1=\frac{\omega x_1}{v_0}\) the transit angle of the electrons between the grids in the absence of alternating voltage, and introduce the quantity \(\nu=\frac{U_{m1}}{2U_0}\), then the expression obtained makes it possible to determine the alternating component of the electron velocity as follows:

where \(\Theta_1\) and \(t_1\) are, respectively, the transit angle of the electrons between the resonator grids and the time at which the electrons leave the gap.

The function \(\Theta_1(t_1)\) cannot be obtained explicitly from this, but the inverse dependence is easily found; using it, the desired function is readily determined graphically,

\[ \omega t_1=\frac{\Theta_1}{2}+\operatorname{arctg} A(\Theta_1)+\arcsin \frac{2\zeta_1}{\nu}\,\frac{\zeta_1-\Theta_1}{\Theta_1^2\Phi(\Theta_1)} . \tag{8} \]

Fig. 1

The result obtained shows that, for given \(\zeta_1\) and \(\nu\), the possible values of the transit angle \(\Theta_1\) can lie only within the interval defined by the inequality

\[ -1\leq a\leq 1, \tag{9} \]

where

\[ a=\frac{2\zeta_1}{\nu}\,\frac{\zeta_1-\Theta_1}{\Theta_1^2\Phi(\Theta_1)} . \tag{9a} \]

The function \(\Theta_1^2\Phi(\Theta_1)\) decreases monotonically as \(\Theta_1\) decreases and, at \(\Theta_1=0\), also becomes zero. Therefore there are always finite values of \(\Theta_1\) satisfying the upper bound of inequality (9). Conversely, as will be shown below, for a given \(\zeta_1\) it is always possible to choose a sufficiently large \(\nu\) such that the equality \(a=-1\) will not be satisfied anywhere and the relation \(a>-1\) will always hold. The existence of such a relation indicates the possibility of arbitrarily large transit angles and physically corresponds to the ejection of electrons from the input gap.

\[ v_{\sim}=\frac{\nu v_0}{\zeta_1}(\cos \omega t_0-\cos \omega t) \tag{3} \]

or

\[ \frac{v_{\sim}}{v_0}=\frac{2\nu}{\zeta_1}\sin\frac{\Theta}{2}\sin\left(\omega t-\frac{\Theta}{2}\right). \tag{3a} \]

If \(eV\) is the energy of the electrons in electron-volts, then, correspondingly,

\[ V_{\sim}=V-U_0=\frac{mv^2}{2e}-\frac{mv_0^2}{2e} =U_0\frac{v_{\sim}}{v_0}\left(\frac{v_{\sim}}{v_0}+2\right). \tag{4} \]

and

\[ \frac{V_{\sim}}{U_0} =\frac{4\nu}{\zeta_1}\sin\frac{\Theta}{2}\sin\left(\omega t-\frac{\Theta}{2}\right) +\left[\frac{2\nu}{\zeta_1}\sin\frac{\Theta}{2}\sin\left(\omega t-\frac{\Theta}{2}\right)\right]^2 . \tag{4a} \]

From formula (3) it follows that

\[ \frac{dx}{dt}=v_0\left[1+\frac{\nu}{\zeta_1}(\cos\omega t_0-\cos\omega t)\right]. \tag{5} \]

Integrating this equation, we obtain¹

\[ \Theta=\zeta+\frac{\nu}{\zeta_1}(\sin\omega t-\sin\omega t_0-\Theta\cos\omega t_0), \tag{6} \]

where \(\zeta=\dfrac{\omega x}{v_0}\).

The expression obtained can, after certain transformations, be brought to the following form:

\[ \Theta=\zeta-\frac{\nu}{2\zeta_1}\Theta^2\Phi(\Theta) \sin\left[\omega t-\frac{\Theta}{2}-\operatorname{arctg}\Lambda(\Theta)\right], \tag{7} \]

where

\[ \Lambda(\Theta)=\frac{1}{\dfrac{\Theta}{2}}-\frac{1}{\operatorname{tg}\dfrac{\Theta}{2}}; \qquad \Phi(\Theta)=\frac{\sin\dfrac{\Theta}{2}}{\dfrac{\Theta}{2}} \sqrt{1+\Lambda^2(\Theta)}, \tag{7a} \]

and, for the latter function, only positive values are meant.

Graphs of the functions \(\Phi(\Theta)\) and \(\operatorname{arctg}\Lambda(\Theta)\) are given in Fig. 1. As applied to the exit from the gap, expression (7) is rewritten as follows:

\[ \Theta_1=\zeta_1-\frac{\nu}{2\zeta_1}\Theta_1^2\Phi(\Theta_1) \sin\left[\omega t_1-\frac{\Theta_1}{2}-\operatorname{arctg}\Lambda(\Theta_1)\right], \tag{7b} \]

\[ \Theta=\zeta-\frac{\nu}{\zeta_1}\left[(1-\cos\Theta)\sin\omega t_0+(\Theta-\sin\Theta)\cos\omega t_0\right] \tag{6a} \]

Then

\[ \frac{d\Theta_1}{d(\omega t_0)} = -\frac{\nu}{2\zeta_1}\,\Theta_1^{2}\Phi(\Theta_1)\, \frac{\sqrt{1-a^{2}}}{1+\dfrac{\tilde v}{v_0}} . \tag{16} \]

Consequently, if we restrict ourselves only to those values of the modulating voltage for which the total velocity of the electrons does not become zero and \(1+\dfrac{\tilde v}{v_0}>0\), then for \(a=\pm 1\) the derivative becomes zero and, thereby, the values of \(\Theta_1\) satisfying this condition will be, respectively, the minimum \((\Theta_{\min})\) and maximum \((\Theta_{\max})\) values of the transit angle.

Using (9a), it is easy to obtain the following expressions determining these values of the transit angle:

\[ \frac{2\zeta_1}{\nu}\,(\Theta_{\max}-\zeta_1) = \Theta_{\max}^{2}\Phi(\Theta_{\max}), \tag{17} \]

\[ \frac{2\zeta_1}{\nu}\,(\zeta_1-\Theta_{\min}) = \Theta_{\min}^{2}\Phi(\Theta_{\min}) . \tag{18} \]

Each of the equations obtained may formally have a number of solutions. However, in (17) the larger solutions should be excluded (since, moving from them toward smaller \(\Theta_1\), we inevitably enter the region \(|a|>1\), which according to (8) has no meaning), and in (18)—the negative solutions.

If the function \(\Theta_1^{2}\Phi(\Theta_1)\) is plotted, and then from the point \(\Theta_1=\zeta_1\) on the abscissa axis straight lines are drawn with slopes \(+\dfrac{2\zeta_1}{\nu}\) and \(-\dfrac{2\zeta_1}{\nu}\), then the intersections of these straight lines with the curve will determine, respectively, \(\Theta_{\max}\) and \(\Theta_{\min}\) for the chosen values of \(\zeta_1\) and \(\nu\).

When graphoanalytic methods are used, the relations obtained in the present paragraph make it possible to determine the velocity and energy of the electrons in the gap for given numerical \(\zeta_1\) and \(\nu\). For this purpose it is necessary, on the basis of expression (8), to plot the dependence of the transit angle on time and then, using it and relations (3) or (4), to determine graphically the time dependence of the velocity or energy of the electrons.

The analytic expressions corresponding to each specific case may then be obtained if the usual graphoanalytic methods of expansion into a harmonic series are used. In this case, construction of the dependence of the transit angle on time is substantially facilitated if one uses the simple methods indicated above for determining \(\Theta_{\max}\) and \(\Theta_{\min}\).

Other ways of calculating the phenomena in the input device of a klystron, associated with the use of graphical methods, may also be indicated [1, 2, 3].

Since in the drift space of the klystron the high-frequency field is absent, the electron velocity at the exit from the gap is princi-

The values of the amplitude of the alternating voltage that lead to electrons being thrown out of the gap, or in general to their reverse motion, will be specified specially below; for the present we shall restrict ourselves to smaller values of \(\nu\), for which the indicated phenomena do not occur.

Let us show that the values \(\Theta_1\) satisfying the condition \(\alpha = \pm 1\) will be, respectively, the minimum and maximum values of the transit angle for given \(\zeta_1\) and \(\nu\).

Making in expression (8) the substitution \(\omega t_1=\omega t_0+\Theta_1\) and then differentiating with respect to \(\omega t_0\), we obtain

\[ 1= \frac{\partial \Theta_1}{\partial(\omega t_0)} \left[ -\frac{1}{2} + \frac{\dfrac{d\Lambda(\Theta_1)}{d\Theta_1}}{1+\Lambda^2(\Theta_1)} + \frac{\dfrac{\partial \alpha}{\partial\Theta_1}}{\sqrt{1-\alpha^2}} \right]^{1)} \tag{10} \]

From (7a), taking into account the identity

\[ 2\frac{d\Lambda(\Theta_1)}{d\Theta_1}-\Lambda^2(\Theta_1)-1= \frac{4\Lambda(\Theta_1)}{\Theta_1}, \]

one can find

\[ \frac{d\Phi(\Theta_1)}{d\Theta_1} = -\frac{\left|\sin \dfrac{\Theta_1}{2}\right|}{\left(\dfrac{\Theta_1}{2}\right)^2} \frac{\Lambda^2(\Theta_1)}{\sqrt{1+\Lambda^2(\Theta_1)}} \tag{11} \]

and

\[ \frac{d}{d\Theta_1}\left[\Theta_1^2\Phi(\Theta_1)\right] = \frac{4\left|\sin \dfrac{\Theta_1}{2}\right|}{\sqrt{1+\Lambda^2(\Theta_1)}}, \tag{12} \]

whence

\[ \frac{\partial\alpha}{\partial\Theta_1} = -\frac{2\zeta_1}{\nu}\, \frac{1+\alpha\beta}{\Theta_1^2\Phi(\Theta_1)}, \tag{13} \]

where

\[ \beta= \frac{2\nu}{\zeta_1}\, \frac{\sin \dfrac{\Theta_1}{2}}{\sqrt{1+\Lambda^2(\Theta_1)}}. \tag{13a} \]

Using this last expression, we find

\[ \frac{\partial\Theta_1}{\partial(\omega t_0)} = -\frac{\nu}{2\zeta_1}\,\Theta_1^2\Phi(\Theta_1)\, \frac{\sqrt{1-\alpha^2}}{1+\alpha\beta+\beta\Lambda(\Theta_1)\sqrt{1-\alpha^2}}. \tag{14} \]

Eliminating the variable \(\omega t_1\) from (3) and (8), one may obtain the following dependence of the ratio of the components of the electron velocity on the transit angle:

\[ \frac{v_z}{v_0} = \beta\left[\Lambda(\Theta_1)\sqrt{1-\alpha^2}+\alpha\right], \tag{15} \]

where again both values of the root are meant.

\(^{1)}\) Here and below, both values of \(\sqrt{1-\alpha^2}\) are meant.

case when simultaneously with this \(\alpha=-1.^1\) This last case, when the conditions \(\dfrac{\tilde v}{v_0}=-1\) and \(\alpha=-1\) are simultaneously satisfied, is of special interest for the problem under consideration, and we shall dwell on it in greater detail.

In accordance with (14) and, taking into account that for \(\alpha=-1\), according to (9a),

\[ -\frac{\nu}{2\zeta_1}\Theta_1^2\Phi(\Theta_1)=\Theta_1-\zeta_1, \]

expression (20) can be transformed to the form

\[ \frac{\partial(\omega t_0)}{\partial\Theta_1} = -\frac{1}{\Theta_1-\zeta_1} \left[ \frac{1+\alpha\beta}{\sqrt{1-\alpha^2}} +\beta\Lambda(\Theta_1) \right]. \tag{21} \]

From formula (15) it follows that the simultaneous fulfillment of the conditions \(\dfrac{\tilde v}{v_0}=-1\) and \(\alpha=-1\) is possible only for \(\beta=+1\), as a consequence of which the ratio \(\dfrac{1+\alpha\beta}{\sqrt{1-\alpha^2}}\) in the case under consideration is also an indeterminate quantity.

The latter can be resolved in the usual way

\[ \left[ \frac{1+\alpha\beta}{\sqrt{1-\alpha^2}} \right]_{\substack{\alpha=-1\\ \beta=+1}} = \left[ -\frac{\beta\,\dfrac{\partial\alpha}{\partial\Theta_1} +\alpha\,\dfrac{\partial\beta}{\partial\Theta_1}} {\alpha\,\dfrac{\partial\alpha}{\partial\Theta_1}} \sqrt{1-\alpha^2} \right]_{\substack{\alpha=-1\\ \beta=+1}}. \]

According to expressions (13) and (9a),

\[ \left[ \frac{\partial\alpha}{\partial\Theta_1} \right]_{\substack{\alpha=-1\\ \beta=+1}} = -\frac{1}{\Theta_1-\zeta_1} [1+\alpha\beta]_{\substack{\alpha=-1\\ \beta=+1}} =0. \tag{22} \]

Differentiating expression (13a), after some transformations one can obtain

\[ \frac{\partial\beta}{\partial\Theta_1} = \beta\left\{ \frac{1}{\Theta_1} -\Lambda(\Theta_1) + \frac{2\Lambda^2(\Theta_1)} {\Theta_1[1+\Lambda^2(\Theta_1)]} \right\}, \tag{23} \]

whence it follows that, in the general case of arbitrary \(\zeta_1\), the quantity

\[ \left[ \frac{\partial\beta}{\partial\Theta_1} \right]_{\substack{\alpha=-1\\ \beta=+1}} \]

is different from zero.

Using (22) and taking into account that everywhere earlier both values of \(\sqrt{1-\alpha^2}\) were meant, we obtain

\[ \left[ \frac{1+\alpha\beta}{\sqrt{1-\alpha^2}} \right]_{\substack{\alpha=-1\\ \beta=+1}} = \pm\sqrt{\beta'_M\cdot(\Theta_1-\zeta_1)}, \tag{24} \]

\(^1\) According to relation (9a), values \(\alpha>0\) can occur only for accelerated electrons, and therefore the conditions \(\alpha=+1\) and \(\dfrac{\tilde v}{v_0}=-1\) are incompatible.

principally cannot be negative¹. Hence it follows that the relations obtained above will be valid only within the set of values \(\upsilon=\upsilon_M(\zeta_1)\) for which the smallest value of the electron velocity at the exit from the gap becomes zero. Therefore, before proceeding to a further investigation of the results obtained, it is necessary to determine these special values of the quantity \(\upsilon\).

Determination of the conditions corresponding to locking of the input gap

The input gap will be locked for some interval of time if, during this interval, electrons do not enter the drift space. The limiting case in which the gap will not be locked will obviously be that in which, at the exit from the gap, only the minimum value of the velocity becomes zero, i.e., when

\[ \upsilon_{\min}=0 \quad \text{or} \quad \left(\frac{\upsilon_{\sim}}{\upsilon_0}\right)_{\min}=-1 . \]

This latter condition is what we shall take as the basis for determining the function \(\upsilon_M(\zeta_1)\) mentioned above. Equation (3), as applied to the exit from the gap, may be written in the form

\[ \frac{\upsilon_{\sim}}{\upsilon_0} = \frac{\upsilon}{\zeta_1} \left[ \cos \omega t_0-\cos(\omega t_0+\Theta_1) \right], \]

whence

\[ \frac{\partial}{\partial \Theta_1} \left( \frac{\upsilon_{\sim}}{\upsilon_0} \right) = \frac{\upsilon}{\zeta_1} \left\{ \sin(\omega t_0+\Theta_1) + \left[ \sin(\omega t_0+\Theta_1)-\sin \omega t_0 \right] \frac{\partial(\omega t_0)}{\partial \Theta_1} \right\}. \tag{19} \]

Let us determine the value of the derivative \(\dfrac{\partial(\omega t_0)}{\partial \Theta_1}\) for the case under consideration, when

\[ \frac{\upsilon_{\sim}}{\upsilon_0}=-1. \]

It follows from expression (16) that

\[ \frac{\partial(\omega t_0)}{\partial \Theta_1} = -\frac{1}{\dfrac{\upsilon}{2\zeta_1}\,\Theta_1^2 \Phi(\Theta_1)} \, \frac{1+\dfrac{\upsilon_{\sim}}{\upsilon_0}}{\sqrt{1-\alpha^2}}, \tag{20} \]

and, consequently, if \(\dfrac{\upsilon_{\sim}}{\upsilon_0}=-1\), then \(\dfrac{\partial(\omega t_0)}{\partial \Theta_1}=0\) always; moreover,

II or I kind. The presence of such discontinuities while preserving the single-valuedness of the function is possible only in the case when, at a discontinuity of the II kind, the second derivative

\[ \frac{\partial^2 t_0}{\partial \Theta_1^2} \]

vanishes (i.e., the point where

\[ \frac{v_{\sim}}{v_0}=-1 \]

is an inflection point), and when, at a discontinuity of the I kind, for the first derivative the angle formed by the branches of the curve at the vertex of the break is less than \(\pi\). From the latter, in particular, it follows that if the values of the derivative in formula (26) have different signs, then the vertex of the break will correspond to the greatest value of the function \(\Theta_1(t_0)\), and the condition \(a=-1\) will remain the condition for a maximum also in the limiting case under consideration, when

\[ \frac{v_{\sim}}{v_0}=-1. \]

Fig. 2

Figures 2 and 3 give plots of the function \(\Theta_1(\omega t_0)\) for certain values of \(\zeta_1\) and \(\nu\), determined in such a way that, at the exit from the gap, only the minimum value of the electron velocity becomes zero. In this connection, the curve in Fig. 2 illustrates the latter case, when, for

\[ \frac{v_{\sim}}{v_0}=-1 \]

also \(a=-1\), and the derivative

\[ \frac{\partial \Theta_1}{\partial t_0} \]

has at this point such a discontinuity of the I kind that the value of the angle of flight for it proves to be the greatest. The curve in Fig. 3 corresponds to the case for which the derivative

\[ \frac{\partial \Theta_1}{\partial t_0} \]

at the point where

\[ \frac{v_{\sim}}{v_0}=-1 \]

undergoes a discontinuity of the II kind, while the second derivative

\[ \frac{\partial^2 t_0}{\partial \Theta_1^2} \]

accordingly vanishes.

where

\[ \beta'_m=\left[\frac{\partial \beta}{\partial \Theta_1}\right]_{\substack{\alpha=-1\\ \beta=+1}} =\frac{1}{\Theta_1}\Lambda(\Theta_1)+ \frac{2\Lambda^2(\Theta_1)}{\Theta_1[1+\Lambda^2(\Theta_1)]}. \tag{25}\!^{1)} \]

Then

\[ \left[\frac{\partial(\omega t_0)}{\partial \Theta_1}\right]_{\substack{\alpha=-1\\ \beta=+1}} = \frac{\Lambda(\Theta_1)\pm \sqrt{\beta'_m(\Theta_1-\zeta_1)}}{\Theta_1-\zeta_1}. \tag{26} \]

For \(\alpha\ne -1\), according to (8), one and the same value of \(\alpha\) corresponds to two values of the time of entry of the electrons \(t_0\) during one period. Therefore the two values of the derivative \(\dfrac{\partial(\omega t_0)}{\partial \Theta_1}\) obtained from formula (20) refer, respectively, to two different points of the function \(t_0(\Theta_1)\). However, to the value \(\alpha=-1\) there corresponds only one value of the entry time \(t_0\), and therefore both values of the derivative obtained in (26) refer to one and the same point of the curve \(t_0(\Theta_1)\).

Thus, in the case when \(\alpha=-1\) and \(\dfrac{v_{\sim}}{v_0}=-1\), the function \(t_0(\Theta_1)\) at the point where these conditions are satisfied has a kink, and its derivative has a discontinuity of the first kind. A physical interpretation of the result obtained may be given, and certain conclusions may be drawn from it.

If there is no return motion of electrons in the gap, then electrons that have entered the gap at a definite instant of time \(t_0\) will reach the cross section \(x\) after a perfectly definite time \(\vartheta=\dfrac{\Theta}{\omega}\). Thus, in the absence of return motion of electrons, the function \(\Theta(t_0)\) is in principle single-valued\(^{2)}\).

If, however, return motion of electrons takes place in the gap, then electrons that have entered the gap at a definite instant of time \(t_0\) will cross one and the same cross section of the gap two or several times, and, consequently, the function \(\Theta(t_0)\) will be multivalued.

It is quite natural that the branching points of the function \(\Theta(t_0)\) must lie where \(\dfrac{v_{\sim}}{v_0}=-1\), since this value of the ratio is the limit beyond which the return motion of electrons begins. In this case, since the function \(\Theta(t_0)\) is continuous, at its branching points the derivative \(\dfrac{\partial \Theta}{\partial t_0}\) must undergo a discontinuity of the second or first kind.

In the limiting case of interest to us, when at the exit from the gap only the minimum value of the electron velocity becomes zero, the function \(\Theta_1(t_0)\) will still remain single-valued, but its derivative \(\dfrac{\partial \Theta_1}{\partial t_0}\) at the point where \(\dfrac{v_{\sim}}{v_0}=-1\) will undergo a discontinuity

\(^{1)}\) Here and in formula (26) the values of \(\Theta_1\) satisfying the condition \(\alpha=-1\) are meant, i.e. the relation

\[ \frac{2\zeta_1}{\gamma}(\Theta_1-\zeta_1)=\Theta_1^2\Phi(\Theta_1). \]

\(^{2)}\) This does not, of course, exclude the possibility of multivaluedness of the inverse function \(t_0(\Theta)\). The latter, in particular, becomes multivalued when “overtaking” of electrons occurs, i.e. when, in one cross section, electrons with different entry times \(t_0\) coincide.

Let us now determine, within the intervals found, those regions where the conditions \(\alpha=-1\) and \(\beta=+1\) correspond to the greatest angle of flight of the electrons. For this, as indicated above, it is necessary that the values of the derivative in formula (26) have different signs. The latter will occur if

\[ |\Lambda(\Theta_1)|<\sqrt{\beta'_M(\Theta_1-\zeta_1)} \]

or

\[ \beta'_M>\frac{\Lambda^2(\Theta_1)}{\Theta_1-\zeta_1}. \tag{31} \]

Since in the case under consideration we must have \(\alpha\beta=-1\), it follows from formulas (9a) and (13a) that, when this condition is satisfied,

\[ \zeta_1=\frac{\Theta_1}{2}\,[1-\Lambda^2(\Theta_1)]. \tag{32} \]

Substituting into inequality (31) the expression for \(\beta'_M\) from (25) and \(\zeta_1\) from (32), and also taking into account that

\[ \Lambda(\Theta_1)=\frac{\Theta_1}{2}-\operatorname{tg}\frac{\Theta_1}{2}, \]

we obtain

\[ \Theta_1>\operatorname{tg}\frac{\Theta_1}{2}. \tag{33} \]

Fig. 4

The values of \(\Theta_1\) corresponding to the boundaries of inequality (33) and lying in the intervals indicated in (30a) will be

\[ \Theta_1=2.330;\; 9.208\ldots \tag{34} \]

and, consequently, the regions for which the conditions \(\alpha=-1;\ \beta=+1\) correspond to the greatest angles of flight will be

\[ 0<\Theta_1<2.330;\quad 2\pi<\Theta_1<9.208\ldots \tag{34a} \]

Returning to the initial problem of determining the conditions under which, at the exit from the gap, only the minimal value of the electron velocity is reduced to zero, we shall have in view, in accordance with the results obtained, two cases: \(\alpha\ne -1\) and \(\alpha=-1\). In the first case, as was established, at

\[ \frac{v_z}{v_0}=-1 \]

and

\[ \frac{\partial t_0}{\partial \Theta_1}=0. \]

In order that the value of the function \(\dfrac{v_z}{v_0}\) should at the same time be minimal, according to formula (19), it is necessary that

\[ \sin(\omega t_0+\Theta_1)=\sin \omega t_1=0 . \tag{35} \]

In accordance with this, one must have \(\cos \omega t_1=\pm 1\). However, substituting both these values into formula (3) and putting there \(\dfrac{v_{\sim}}{v_0}=-1\), it is easy to verify that the value \(\cos \omega t_1=+1\) cannot occur, since it leads to values of \(\cos \omega t_0\) greater than unity. Consequently, the final initial condition may be written in the form

\[ \omega t_0+\Theta_1=2\pi n \qquad (n=1,\,2,\,3\ldots). \tag{36} \]

The physical meaning of condition (36) is that the electrons leaving the gap at the end of the retarding half-period of the voltage between the grids of the input resonator can have the smallest velocity. However, while such a possibility raises no particular objections at large transit angles \(\zeta_1\), on the other hand it is well known that at small transit angles the electrons that pass through the gap near the middle of the retarding half-period are retarded most strongly, and it is precisely for these electrons that their total velocity at the exit from the gap may first become equal to zero. This leads one to suppose that condition (36), under which the value of the function \(\dfrac{v_{\sim}}{v_0}=-1\) is minimal, is not the only one. Therefore let us proceed to the investigation of the second case and show that the conditions \(\dfrac{v_{\sim}}{v_0}=-1\) and \(a=-1\), if their simultaneous fulfillment is possible, may also mean that the value of the ratio \(\dfrac{v_{\sim}}{v_0}=-1\) will be the smallest.

From expression (15) it follows that

\[ \frac{\partial}{\partial \Theta_1}\left(\frac{v_{\sim}}{v_0}\right) = \sqrt{1-a^2}\,\frac{\partial}{\partial \Theta_1}\,[\beta \Lambda(\Theta_1)] - \frac{a\beta\,\dfrac{\partial a}{\partial \Theta_1}\,\Lambda(\Theta_1)} {\sqrt{1-a^2}} + \]

\[ {}+a\frac{\partial \beta}{\partial \Theta_1} +\beta\frac{\partial a}{\partial \Theta_1}. \tag{37} \]

Since for \(\dfrac{v_{\sim}}{v_0}=-1\) and \(a=-1\) we also have \(\beta=+1\), then, taking (22) and (24) into account, we obtain

\[ \left[ \frac{\partial}{\partial \Theta_1}\left(\frac{v_{\sim}}{v_0}\right) \right]_{ \substack{ \dfrac{v_{\sim}}{v_0}=-1,\\ a=-1 }} = - \left[ \beta'_{\mu}\pm \Lambda(\Theta_1) \right] \sqrt{\frac{-\beta'_{\mu}}{\Theta_1-\zeta_1}} . \tag{38} \]

It follows from this that, when \(\dfrac{v_{\sim}}{v_0}=-1\) and \(a=-1\), the function \(\dfrac{v_{\sim}}{v_0}\) has a cusp, and therefore this value of the function may be the smallest. From Fig. 5, which illustrates precisely this case, it is seen that, for this, it is at least necessary that both values of the derivative at the cusp point be negative.

Since, in the range of values of the electron transit angle \(\Theta_1\) where simultaneous fulfillment of the conditions

\[ \frac{v_{\sim}}{v_0}=-1 \]

and \(\alpha=-1\) is possible, \(\beta'_M>0\), the circumstance indicated will always occur if inequality (31) is satisfied. Consequently, for \(\alpha=-1\) the value of the function

\[ \frac{v_{\sim}}{v_0}=-1 \]

may be the smallest if it pertains to electrons whose transit angle lies within the limits determined by inequality (34a).

If \(\zeta_1\) is such that the electron transit angle \(\Theta_1\) lies on the upper boundary of these regions (the equality sign in (31)), then one of the values of the derivative

\[ \frac{\partial}{\partial \Theta_1}\left(\frac{v_{\sim}}{v_0}\right) \]

and the corresponding value

\[ \frac{\partial t_0}{\partial \Theta_1} \]

vanish. Thus, in this particular case both conditions:

\[ \alpha=-1 \]

and

\[ \omega t_1=2\pi n, \]

under which the value of the ratio

\[ \frac{v_{\sim}}{v_0}=-1 \]

may be the smallest, are satisfied not only for the same values of \(\nu\), but also for the same electrons. Therefore, for the indicated values of \(\zeta_1\), not only must the values of \(\nu\) calculated from each condition separately coincide, but also the corresponding derivatives

\[ \frac{d\nu}{d\zeta_1}. \]

Fig. 5

Leaving the proof of the last proposition for later, we shall only indicate that the case is also possible when, for certain \(\zeta_1\), each of the conditions \(\alpha=-1\) and \(\omega t_1=2\pi n\) leads to one and the same value of \(\nu\), but for different electrons characterized, respectively, by different transit angles \(\Theta_1\).

These latter values of \(\zeta_1\) evidently correspond to points of simple intersection of the curves of the dependence of \(\nu\) on \(\zeta_1\), calculated from each condition separately. Let us denote the dependence of the quantity \(\nu\) on \(\zeta_1\), satisfying the conditions

\[ \frac{v_{\sim}}{v_0}=-1 \]

and \(\alpha=-1\), by \(\nu_1\), and the corresponding dependence for the conditions

\[ \frac{v_{\sim}}{v_0}=-1 \]

and

\[ \omega t_1=2\pi n \]

by \(\nu_2\). The sought set of values \(\nu_M(\zeta_1)\), for which the minimum value of the electron velocity at the exit from the gap becomes zero, is then obtained as the combination of the smaller values of both functions \(\nu_1\) and \(\nu_2\).

Putting in formulas (9a) and (13a) \(\alpha=-1\) and \(\beta=+1\) (which is equivalent to the condition \(\alpha=-1\) and \(\frac{v_{\sim}}{v_0}=-1\)), after some transformations we obtain the following expressions defining the function \(\nu_1\)

\[ \zeta_1=\frac{\Theta_1}{2}\left[1-\Lambda^2(\Theta_1)\right], \tag{32} \]

\[ \nu_1=\frac{\zeta_1}{2}\, \frac{\sqrt{1+\Lambda^2(\Theta_1)}}{\left|\sin \dfrac{\Theta_1}{2}\right|} =\frac{1}{2}\,\frac{1-\Lambda^4(\Theta_1)}{\Phi(\Theta_1)}. \tag{39} \]

Differentiating expression (32) with respect to \(\Theta_1\) and equating the derivative to zero, we obtain equation (29a). It follows from this that the maxima of the function

\[ \frac{\Theta_1}{2}\left[1-\Lambda^2(\Theta_1)\right] \]

lie at the values \(\Theta_1=2.732;\ 9.215\ldots\), corresponding to the inflection points of the curve \(\Theta_1^2\Phi(\Theta_1)\). As a result, the descending branches of the function \(\zeta_1=\dfrac{\Theta_1}{2}\left[1-\Lambda^2(\Theta_1)\right]\) correspond to the regions of convexity of the curve \(\Theta_1^2\Phi(\Theta_1)\), where the conditions \(\alpha=-1\) and \(\dfrac{v_{\sim}}{v_0}=-1\) have no meaning and therefore must be excluded from consideration. In exactly the same way, those portions of the ascending branches of this same function that lead to a multivalued inverse function\(^{1}\), as well as the regions of values of \(\Theta_1\) that lead to negative \(\zeta_1\), should be excluded from consideration.

Next let us find an analytic expression for the function \(\nu_2\). To this end we refer (1) and (6) to the exit from the gap and take there, in accordance with the definition of the function \(\nu_2\), \(\dfrac{v_{\sim}}{v_0}=-1\) and \(\omega t_1=2\pi n\).

Then

\[ \zeta_1=\frac{\Theta_1-\sin\Theta_1}{1-\cos\Theta_1}, \tag{40} \]

\[ \nu_2=\frac{\zeta_1}{1-\cos\Theta_1} =\frac{\Theta_1-\sin\Theta_1}{(1-\cos\Theta_1)^2}, \tag{41}^{2)} \]

which together determines the dependence of \(\nu_2\) on \(\zeta_1\).

To form the desired function \(\nu_M(\zeta_1)\), it now remains only to determine the regions of values of \(\zeta_1\) where \(\nu_2<\nu_1\), and in them to replace \(\nu_1\) by \(\nu_2\). As was already indicated, the lower boundaries of these regions lie where the conditions \(\alpha=-1\) and \(\omega t_1=2\pi n\) with \(\dfrac{v_{\sim}}{v_0}=-1\) are satisfied simultaneously for the same electrons. According to (33), this will occur if \(\operatorname{tg}\dfrac{\Theta_1}{2}=\Theta_1\), and in this case not only the functions \(\nu_1\) and \(\nu_2\) themselves, but also their derivatives, must coincide.

We shall now prove this more rigorously and at the same time determine numerically the boundaries of the regions under consideration. Since for \(\operatorname{tg}\dfrac{\Theta_1}{2}=\Theta_1\)

\(^{1)}\) The existence of these portions is possible only for values of the quantity \(\nu\) known to be larger than for the first of the corresponding portions lying at the same \(\zeta_1\).

\(^{2)}\) Eliminating \(\Theta_1\) from this, we obtain the dependence of \(\nu_2\) on \(\zeta_1\) in implicit form

\[ \arccos\left(1-\frac{\zeta_1}{2}\right) = \frac{\zeta_1}{\nu_2} \left[ \zeta_1-\sqrt{\frac{2\nu_2}{\zeta_1}-1} \right], \]

This dependence, however, is less convenient for calculations.

\[ \Lambda(\Theta_1)=\frac{1}{\Theta_1} \quad \text{and} \quad \sin \frac{\Theta_1}{2}=\frac{\Theta_1}{\sqrt{1+\Theta_1^{2}}}, \]
then from expressions (32) and (39) it follows that

\[ \zeta_1=\frac{\Theta_1^{2}-1}{2\Theta_1},\qquad \nu_1=\frac{\zeta_1}{2}\left[1+\frac{1}{\Theta_1^{2}}\right] =\frac{\Theta_1^{4}-1}{4\Theta_1^{3}}; \tag{42} \]

Upon substituting the numerical values corresponding to the condition

\[ \tan \frac{\Theta_1}{2}=\Theta_1, \]
this gives

\[ \zeta_1=0.950,\ 4.55,\ldots \qquad 2\nu_1=1.125,\ 4.604,\ldots \tag{43} \]

From (39) it follows that

\[ \frac{d\nu_1}{d\zeta_1} = \frac{\sqrt{1+\Lambda^{2}(\Theta_1)}}{2\sin \dfrac{\Theta_1}{2}} + \]

\[ + \frac{ \dfrac{\Lambda(\Theta_1)\dfrac{d\Lambda(\Theta_1)}{d\Theta_1}} {\sqrt{1+\Lambda^{2}(\Theta_1)}}\sin \dfrac{\Theta_1}{2} - \sqrt{1+\Lambda^{2}(\Theta_1)}\,\dfrac{\cos \dfrac{\Theta_1}{2}}{2} } {4\sin^{2}\dfrac{\Theta_1}{2}} \,\zeta_1 \left[ 1-\Lambda^{2}(\Theta_1)- \right. \]

\[ \left. -2\Theta_1\Lambda(\Theta_1)\frac{d\Lambda(\Theta_1)}{d\Theta_1} \right], \]

and since in the case under consideration

\[ 2\frac{d\Lambda(\Theta_1)}{d\Theta_1} = 1+\Lambda^{2}(\Theta_1)-\frac{4\Lambda(\Theta_1)}{\Theta_1} = 1-\frac{3}{\Theta_1^{2}} \]

and

\[ \cos \frac{\Theta_1}{2}=\frac{1}{\sqrt{1+\Theta_1^{2}}}, \]
then for the values of \(\zeta_1\) indicated in (43)

\[ \frac{d\nu_1}{d\zeta_1}=\frac{1}{\Theta_1^{2}}. \tag{44} \]

Taking further into account that, when \(\tan \dfrac{\Theta_1}{2}=\Theta_1\),

\[ \sin \Theta_1=\frac{2\Theta_1}{1+\Theta_1^{2}} \qquad \text{and} \qquad \cos \Theta_1=\frac{1-\Theta_1^{2}}{1+\Theta_1^{2}}, \]

From expressions (40) and (41), which determine the dependence of \(\nu_2\) on \(\zeta_1\), we obtain

\[ \zeta_1=\frac{\Theta_1^2-1}{2\Theta_1}, \qquad \nu_2=\frac{\zeta_1}{2}\left[1+\frac{1}{\Theta_1^2}\right] =\frac{\Theta_1^4-1}{4\Theta_1^3}. \tag{45a} \]

From the same expressions it follows that

\[ \frac{d\nu_2}{d\zeta_1} = \frac{1-\cos\Theta_1-2\zeta_1\sin\Theta_1} {(1-\cos\Theta_1)(1-\cos\Theta_1-\zeta_1\sin\Theta_1)} \]

and, if

\[ \tan\frac{\Theta_1}{2}=\Theta_1, \]

then

\[ \frac{d\nu_2}{d\zeta_1}=\frac{1}{\Theta_1^2}. \tag{45b} \]

Thus, for the values of \(\zeta_1\) indicated in (43), not only the functions \(\nu_1\) and \(\nu_2\) coincide, but also their derivatives, as was noted above.

The upper boundaries of the regions where \(\nu_2<\nu_1\) are determined as the points of intersection of the curves of the dependence of \(\nu_1\) and \(\nu_2\) on \(\zeta_1\), and correspond to the breaks in the sought function \(\nu_M(\zeta_1)\). In the range of transit angles \(\zeta_1\le 2\pi\) there is only one such point of intersection, lying at

\[ \zeta_1=3.321 \quad \text{and} \quad 2\nu_M(\zeta_1)=4.386. \tag{46} \]

In contrast to the values of \(\zeta_1\) indicated in (43), for which the coincidence of the functions \(\nu_1\) and \(\nu_2\) occurs at the same electron transit angles \(\Theta_1\), in the present case the coincidence of these functions occurs with respect to different values of \(\Theta_1\):

\[ \Theta_1=8.228 \]

—for dependences (32) and (39), which determine \(\nu_1\),

\[ \Theta_1=4.172 \]

—for dependences (40) and (41), which determine \(\nu_2\).

Fig. 6. Plot of

\[ \frac{U_m}{U_0}=2\nu_M(\zeta_1) \]

as a function of \(\zeta_1\). Marked points:

\[ A_1(0.950;\,1.125),\qquad A_2(3.324;\,4.386),\qquad A_3(4.550;\,4.604). \]

In other words, for the values \(\zeta_1\) and \(\nu\) indicated in (46), closure of the gap is attained at two separate instants of time and for electrons with different transit angles \(\Theta_1\).

Figure 6 gives the resulting plot of the function

\[ \frac{U_m}{U_0}=2\nu_m(\zeta_1) \]

for the range \(0 \le \zeta_1 \le 2\pi\); the values of the function, in accordance with the results obtained on the segments \(OA_1\) and \(A_2A_3\), are determined from (32) and (39), and in the remaining intervals from the relations (40) and (41).

A typical curve of the function \(\dfrac{v_{\sim}}{v_0}\), constructed as a function of \(\Theta_1\) for values of \(\nu\) and \(\zeta_1\) corresponding to points of the segment \(OA_1\) of the curve in Fig. 6, was given earlier in Fig. 5.

For \(\nu\) and \(\zeta_1\) corresponding to the point \(A_2\) in Fig. 6, the derivative at the point \(\dfrac{v_{\sim}}{v_0}=-1\) of the lower of the branches of the function \(\dfrac{v_{\sim}}{v_0}\) becomes zero; thereby both conditions \(\alpha=-1\) and \(\omega t_1=2\pi n\) are satisfied simultaneously.

For points of the initial part of the segment \(A_1A_2\) of the curve in Fig. 6, where \(\Theta_1<2\pi\) and fulfillment of the condition \(\dfrac{v_{\sim}}{v_0}=-1\) at \(\alpha=-1\) is impossible, the function \(\dfrac{v_{\sim}}{v_0}\) has only one minimum \(\dfrac{v_{\sim}}{v_0}=-1\) with a zero derivative. The typical dependence of \(\dfrac{v_{\sim}}{v_0}\) on \(\Theta_1\) corresponding to this case is shown in Fig. 7.

For points of the subsequent part of the same segment \(A_1A_2\), where \(\Theta_{1\max}>2\pi\), the curve of the dependence of \(\dfrac{v_{\sim}}{v_0}\) on \(\Theta_1\) acquires a more complex form, characterized by the appearance of a second minimum, smaller in absolute value. For values of \(\nu\) and \(\zeta_1\) corresponding to the point \(A_2\), the second minimum of the function \(\dfrac{v_{\sim}}{v_0}\) passes into a kink, whose vertex lies at the largest transit angle of the electrons \(\Theta_1\) and for which \(\dfrac{v_{\sim}}{v_0}=-1\); thus, in this case, closure of the input gap occurs twice during one period.

Fig. 7

The plot of the dependence of \(\dfrac{v_{\sim}}{v_0}\) on \(\Theta_1\), corresponding to the point \(A_2\) of the curve in Fig. 6, is given in Fig. 8. Beyond the point \(A_2\), the minimum of the function \(\dfrac{v_{\sim}}{v_0}\), rela-

corresponding to electrons with a smaller transit angle, decreases in absolute value.

Calculation of the Electron Velocity for the Case of Not Too Large Alternating Voltages \((U_{m1} \ll U_0)\)

In this region, which is of the greatest practical interest, it is possible to construct sufficiently accurate analytical expressions for the velocity and energy of the electrons in the input resonator.

If the transit angle of the electrons between the grids of the input resonator is represented in the form of the sum \(\Theta_1=\zeta_1+\varepsilon_1\), then for the variable component \(\varepsilon_1\), from expression (6a), we obtain

\[ \varepsilon_1 = -\frac{\nu}{\zeta_1} \left\{ [1-\cos(\zeta_1+\varepsilon_1)]\sin\omega t_0 + [(\zeta_1+\varepsilon_1)-\sin(\zeta_1+\varepsilon_1)]\cos\omega t_0 \right\} \tag{47} \]

\(\Theta_1\), or, after simple trigonometric transformations,

\[ \begin{aligned} \varepsilon_1 ={}& -\frac{\nu}{\zeta_1} \left[(1-\cos\zeta_1)\sin\omega t_0 + (\zeta_1-\sin\zeta_1)\cos\omega t_0\right] \\ &-\nu\varepsilon_1 \frac{\sin\dfrac{\zeta_1}{2}}{\dfrac{\zeta_1}{2}} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right) \\ &-\frac{\nu}{\zeta_1} \left[(1-\cos\varepsilon_1)\sin(\omega t_0+\zeta_1) + (\varepsilon_1-\sin\varepsilon_1)\cos(\omega t_0+\zeta_1)\right]. \end{aligned} \tag{48} \]

Fig. 8

Carrying out here the same transformations as in passing from (6a) to (7), we obtain

\[ \begin{aligned} \varepsilon_1 ={}& -\frac{\nu\zeta_1}{2}\Phi(\zeta_1) \sin\left[\omega t_0+\frac{\zeta_1}{2}-\operatorname{arctg}\Lambda(\zeta_1)\right] \\ &-\nu\varepsilon_1 \frac{\sin\dfrac{\zeta_1}{2}}{\dfrac{\zeta_1}{2}} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right) -\varepsilon_1^2\Phi(\varepsilon_1)\frac{\nu}{2\zeta_1} \sin\left[\omega t_0+\zeta_1+\frac{\varepsilon_1}{2}-\operatorname{arctg}\Lambda(\varepsilon_1)\right]. \end{aligned} \tag{49} \]

Hence, in particular, it follows that for \(\nu\ll 1\), \(\varepsilon_1\ll \zeta_1\). Therefore the result of the first approximation for the velocity and energy of the electrons at small \(\nu\) will be obtained if in (3) and (4) we take \(\Theta_1 \approx \zeta_1\). By neglecting

Neglecting quantities of small order higher than \(\nu\), this leads to the well-known expressions [4, 5, 6, 7]

\[ \frac{v_{\sim}}{v_0} = \nu\, \frac{\sin \dfrac{\zeta_1}{2}}{\dfrac{\zeta_1}{2}}\, \sin\left(\omega t_0+\frac{\zeta_1}{2}\right), \tag{50} \]

\[ \frac{V_{\sim}}{U_0} = 2\nu\, \frac{\sin \dfrac{\zeta_1}{2}}{\dfrac{\zeta_1}{2}}\, \sin\left(\omega t_0+\frac{\zeta_1}{2}\right). \tag{51} \]

The second approximation is obtained if in (49) the first term is retained and, correspondingly, one takes

\[ \Theta_1 = \zeta_1 - \frac{\nu\zeta_1}{2}\, \Phi(\zeta_1)\, \sin\left[\omega t_0+\frac{\zeta_1}{2}-\arctan \Lambda(\zeta_1)\right]. \tag{52} \]

Some authors [6] believe that the results of this approximation, as applied to the electron velocity, can be used without a large error up to \(\nu \leq 0.5\). However, as regards expression (52) itself, it means replacing the asymmetric curve \(\Theta_1(\omega t_0)\) by a sinusoid and is clearly unacceptable for large variable voltages, where the asymmetry of the half-periods of the function \(\Theta_1(\omega t_0)\) is quite significant. Therefore we shall investigate the third approximation, obtained by retaining the first two terms in (49), and shall show that it can be used over the entire region under consideration, \(\nu \leq 0.5\), both for calculating the function of the electron transit angle and for obtaining more accurate results for the electron velocity. It follows from formula (49) that the error of the approximation under consideration increases as \(\nu\) and \(\varepsilon_1\) increase. Therefore, as a criterion for estimating the error, we shall proceed from the maximum-in-time values of the variable component of the transit angle \(\varepsilon_m\), determined for \(\nu=0.5\) from equation (17).

The graph of the dependence of \(\varepsilon_m\) on \(\zeta_1\), calculated for \(\nu=0.5\), is given in Fig. 9.

Fig. 9

As is seen from Fig. 9, the quantity \(\varepsilon_m\), in turn, has a maximum; its existence is connected with the fact that, as \(\zeta_1\) increases, simultaneously with the increase in the time spent by the electrons in the gap, the action of the high-frequency voltage is also averaged. On this

In the same figure a plot is given of the function

\[ \frac{\varepsilon_m^2 \Phi(\varepsilon_m)}{4\zeta_1}, \]

corresponding to the largest values of the amplitude of the third term in (49). From a comparison of the curves it follows that, with the exception of the region of small \(\zeta_1\), where the influence of the electron transit time can in general be neglected, the last term in expression (49) is small in comparison with the sum of the other two. Omitting it from consideration, we then obtain from (49)

\[ \varepsilon_1 = -\frac{\nu\zeta_1}{2}\, \Phi(\zeta_1) \frac{ \sin\left[\omega t_0+\frac{\zeta_1}{2}-\operatorname{arctg}\Lambda(\zeta_1)\right] }{ 1+\nu\frac{\sin\frac{\zeta_1}{2}}{\frac{\zeta_1}{2}} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right) }. \tag{53} \]

Hence it follows that

\[ \Theta_1=\zeta_1- \frac{\nu\zeta_1}{2}\, \frac{ \frac{\sin\frac{\zeta_1}{2}}{\frac{\zeta_1}{2}} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right) - \frac{\sin\frac{\zeta_1}{2}-\frac{\zeta_1}{2}\cos\frac{\zeta_1}{2}} {\left(\frac{\zeta_1}{2}\right)^2} \cos\left(\omega t_0+\frac{\zeta_1}{2}\right) }{ 1+\nu\frac{\sin\frac{\zeta_1}{2}}{\frac{\zeta_1}{2}} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right) }. \tag{54} \]

A comparison of the values of the maximum transit angle \(\Theta_{\max}\), calculated by the formula obtained and determined graphically from (17), shows that the error of the proposed calculation is insignificant over the entire range under consideration \(\nu \leq 0.5,\ \zeta_1 \leq 2\pi\).

Since \(\Theta_1=\zeta_1+\varepsilon_1\), it follows from formula (3a) that

\[ \frac{v_{\sim}}{v_0} = \frac{2\nu}{\zeta_1} \sin\left(\frac{\zeta_1}{2}+\frac{\varepsilon_1}{2}\right) \sin\left(\omega t_0+\frac{\zeta_1}{2}+\frac{\varepsilon_1}{2}\right). \]

In accordance with the numerical estimate made earlier for the magnitude of \(\varepsilon_1\), one may, without introducing an appreciable error for the region considered, take

\[ \sin\frac{\varepsilon_1}{2}\simeq \frac{\varepsilon_1}{2} \quad\text{and}\quad \cos\frac{\varepsilon_1}{2}\simeq 1. \]

Then

\[ \frac{v_{\sim}}{v_0} = \nu\, \frac{\sin\frac{\zeta_1}{2}}{\frac{\zeta_1}{2}} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right) + \frac{\nu\varepsilon_1}{2}\sin(\omega t_0+\zeta_1) + \frac{\nu\varepsilon_1^2}{2\zeta_1} \cos\frac{\zeta_1}{2} \sin\left(\omega t_0+\frac{\zeta_1}{2}\right). \]

The last term in this expression is of the same order as the expression considered earlier, \(\dfrac{\varepsilon_1^2 \Phi(\varepsilon_1)}{4 \varepsilon_1}\), and may be omitted. Substituting \(\varepsilon_1\) from (54) into the resulting expression, after some transformations one can find

\[ \frac{v_{\sim}}{v_0} = \nu R(\zeta_1)\sin \omega t_c - \frac{\nu^2}{2}\, \frac{ Q(\zeta_1)\sin 2\omega t_c + R^2(\zeta_1)\sin^2 \omega t_c - L(\zeta_1) }{ 1+\nu R(\zeta_1)\sin \omega t_c }, \tag{55} \]

where

\[ R(\zeta_1)=\frac{\sin \dfrac{\zeta_1}{2}}{\dfrac{\zeta_1}{2}}, \qquad Q(\zeta_1)=\frac{1}{\zeta_1}\left(1-\frac{\sin \zeta_1}{\zeta_1}\right), \]

\[ L(\zeta_1) = \frac{\sin \dfrac{\zeta_1}{2}}{\left(\dfrac{\zeta_1}{2}\right)^2} \left( \sin \frac{\zeta_1}{2} - \frac{\zeta_1}{2}\cos \frac{\zeta_1}{2} \right), \qquad \omega t_c=\omega t_0+\frac{\zeta_1}{2}. \]

Fig. 10

The graphs of the functions \(R(\zeta_1)\), \(Q(\zeta_1)\), and \(L(\zeta_1)\) are given in Fig. 10. If \(eV_{\sim}\) is the variable component of the energy of electron motion, then, according to (4),

\[ \frac{V_{\sim}}{U_0} = 2\frac{v_{\sim}}{v_0} + \left(\frac{v_{\sim}}{v_0}\right)^2 . \]

Hence, neglecting the square of the last term in (55), as being smaller in magnitude than \(\nu^4\), we find

\[ \frac{V_{\sim}}{U_0} = 2\nu R(\zeta_1)\sin \omega t_c + \nu^2\, \frac{ Q(\zeta_1)\sin 2\omega t_c - L(\zeta_1) }{ 1+\nu R(\zeta_1)\sin \omega t_c }. \tag{56} \]

The formulas obtained, (55) and (56), may be used as initial formulas for calculating processes in the subsequent elements of the kly-

side in the case under consideration of large alternating voltages and large electron transit angles in the gap of the input resonator.

References

1. Philips H. B. and Ware L. A. Transit-Time Effect in Klystron Gaps, Proc. I. R. E., v. 35, October 1948.
2. Kompfner R. Velocity Modulating Grids, Wireless Eng., v. 19, April 1942.
3. R. Warnecke et P. Guénard. Les tubes électroniques à commande par modulation de vitesse, 1951.
4. Katsman Yu. A. Fundamentals of Radio-Tube Calculation, Gosenergoizdat, 1952.
5. Kovalenko V. F. Introduction to Microwave Electronics, Part 1, Sovetskoe Radio Publishing House, 1950.
6. Klystrons, translated from English, edited by E. D. Naumenko, Sovetskoe Radio Publishing House, 1952.
7. Electron Tubes, translated from English, edited by S. A. Obolensky, Sovetskoe Radio Publishing House, 1954.

1. For \(\upsilon>\upsilon_M(\zeta_1)\), some of the electrons simply will not reach the exit from the gap. If, however, one assigns values \(\upsilon>\upsilon_M(\zeta_1)\), then from formulas (8) and (3) electron velocities smaller than zero may be obtained. The latter is connected with the fact that, in deriving expressions (3) and (4), no condition was introduced indicating that, for \(x>x_1\), the high-frequency field is absent. In considering such a condition, however, there is no need if one restricts oneself, as was indicated, to values \(\upsilon \leqslant \upsilon_M(\zeta_1)\). ↩↩