GENERATION AND APPLICATION OF ULTRASHORT UNDAMPED ELECTRIC WAVES

H. E. Hollmann*

Contents

Introduction

I. Oscillations in circuits with feedback with a positive anode
    a) The feedback limit
    b) Inversion of ultradynamic characteristics

II. The retarding-field method
    a) Purely electronic oscillations
    b) The electron generator as a coupled system
    c) Theory of space-charge oscillations
    d) Short waves
    e) Inversion oscillations (Inversionsschwingungen)
    f) Influence of the gas
    g) Special tubes and oscillatory circuits

III. General behavior of a discharge gap at high frequencies

IV. Magnetron
    a) Cylindrical anode in a magnetic field
    b) Magnetron with a split anode (split magnetron)

V. Practical applications of microwaves
    a) Reception
    b) Quasi-optical propagation

Literature

Introduction

A review of the present state of methods for generating ultrashort waves by means of electron tubes has already been given several times^2,3. Although since then, in the sense of obtaining ultrashort waves or microwaves with higher frequencies, no particularly noteworthy successes have been achieved, experimental studies and theoretical ideas about the operation of the electron tube have broadened and deepened appreciably, making it possible to construct decimeter transmitters with practically acceptable powers and to make fruitful use of decimeter communication.

* Hochfrequenztechnik und Elektroakustik 44, 37, 1934. Translated by D. R. Kanaskov.

At the present time, microwaves are entering the first stage of their practical application, and it therefore makes sense to supplement the articles published earlier with a survey of the work that has appeared since then, and thereby to broaden the topic by including questions of reception and quasi-optical propagation.

1. Oscillations in Feedback Circuits with a Positive Anode

a) The Feedback Limit

The generation of ultrashort undamped waves in feedback circuits, ordinarily used for longer waves, i.e., in circuits in which, unlike Barkhausen–Kurz circuits with a retarding field, the grid controls the anode current, encounters, as is known, difficulties as the elements determining the frequency of the oscillatory circuit become smaller and smaller.

According to the view that has prevailed until now, there are two causes that prevent an arbitrary increase in frequency: first, the ever-increasing difficulty of matching the rising frequency to the still more rapidly decreasing apparent resistance of the anode circuit, and second, the need to maintain high anode voltages in order to preserve a definite electron transit time.

According to the experimental data available up to the present, the limit determined by these causes is reached at wavelengths of about 60 cm. It is quite evident that the effects mentioned must make themselves felt even before this wavelength is reached, in the form of deviations from the usual behavior of longer waves. In this connection we shall point to the precise experiments of Maslennikov and Smirnov5, who, in investigating waves from 1–20 m, observed separate regions of oscillation whose behavior, when the anode voltage was varied, changed noticeably with frequency. These results have a certain similarity to those obtained in studying the retarding-field method, and they show that even with ordinary feedback in the meter-wave region the electron transit time already plays a noticeable role.

Particularly striking in this respect is the fact that, as the wavelength is shortened, the intensity maximum shifts toward higher voltages.

In this case, if in the measurements one confines oneself to the rectilinear part of the characteristic and neglects its curvature, the relation holds

\[ \lambda^{2} E_g = \text{const}, \]

analogous to the Barkhausen–Kurz dependence

\[ \lambda^{2} E_a = \text{const}. \]

At very high anode voltages, from 600 to 700 V, and \(E_g = 0\), waves of 10 to 15 cm wavelength arise. With their emerg-

…by the renewal the grid current changes its sign, as though the grid itself emitted electrons or captured positive ions. In exactly the same way as in special arrangements with a retarding field, these oscillations can be brought into resonance with a circuit made, according to Fig. 1, of the helical grid and its holder; however, the mechanism of excitation of these oscillations has not yet been theoretically clarified.

A precise consideration of the causes of the appearance of a limiting wavelength in circuits with feedback led Krebel6 to the conclusion that the oscillations must arise again when the transit time of the electron becomes equal to a whole period. In ordinary tubes, however, clean results cannot be expected, because the resonant system, owing to the difference in the capacitances of the grid and the anode with respect to the cathode, is asymmetrical, and as a result of this the alternating voltages on the electrodes are not exactly opposite in phase.

Fig. 1. Resonant circuit consisting of a grid and holder.

Proceeding from these considerations, a special arrangement was constructed which, with the smallest interelectrode capacitances, ensures perfect symmetry of the oscillations in the Lecher system.

In this way the boundary for excitation of oscillations with feedback was shifted to waves of length 31 cm. If the calculated electron transit time \(\vartheta\) is compared with the measured period of oscillation \(T\) for various wavelengths, it turns out that there are two transit times satisfying the conditions for excitation of oscillations, namely, one slightly greater than a half-period of the oscillation, and the other slightly less than \(\frac{3}{4}T\).

The reason for this is the alternating voltages induced on the filament, which change the phase relations between the grid and anode voltages. This has an especially strong effect when the amplitude of the oscillation on the filament reaches the magnitude of the grid amplitude, because then one can no longer neglect the reverse effect of the alternating voltages at the anode on the anode current.

A theoretical consideration of the question on the basis of these assumptions does indeed give two electron transit times \(\vartheta\), namely

\[ \frac{3}{4}T > \vartheta > \frac{5}{8}T . \]

By an extremely simple method, Thomson7 shifted the boundary of the region of application of electron tubes toward shorter waves. He not only reduced the dimensions of the external resonant system, but at the same time brought the dimensions of the parts of the tube itself down to one tenth of the previously used values. True, this produced distances between the electrodes of \(1/10\) mm, which can be used only with flat electrodes and heated cathodes.

At 115 V anode voltage and 3 mA anode current it is possible to obtain a wave of 30 cm; for a wave of 40 cm the current and voltage

can be reduced to \(-0.5\) mA and 45 V. What energy transformations are possible in a system with such small electrodes will be shown by the future.

b) Inversion of ultradynamic characteristics

The starting point of Krebel’s theory, namely, the assumption of the equality of \(\vartheta\) and \(T\), leads directly to the theory of inversion of ultradynamic characteristics, given in exhaustive form by Sakhanek\(^8\), generalized and experimentally verified by Golman\(^9\).

The theory of inversion oscillations, which has turned out, incidentally, as will be shown further on, to be especially fruitful in considering the braking-field method, proceeds from the fact that the finite velocity of the electrons causes, at very high frequencies, a change in the form of the characteristics. This change is connected with the fact that the anode current lags behind the control voltage by an angle \(\varphi\), which is determined by the relation \(\varphi=\omega\vartheta\), where \(\vartheta\) is the electron transit time, and \(\omega\) is the angular frequency of the control voltage.

Fig. 2. Idealized ultradynamic characteristic (after Golman).

The ultradynamic characteristics of some tube at high frequency, taken, for example, with the aid of a Braun tube synchronized in turn with the high frequency by eliminating the internal phase shift in it, may, when the transit time or the frequency is varied, be regarded as the superposition of two angular functions shifted in phase, as shown in Fig. 2.

If the static characteristic at \(\varphi=0\) is rising, then with a gradual increase of \(\varphi\) it assumes the form of an ellipse. At \(\varphi=\pi\) it again becomes a straight line, but with the opposite slope. With further increase of the phase shift it is again deformed into an ellipse and, finally, at \(\varphi=2\pi\) it once more assumes its usual rising form, after which the process may be repeated again any number of times.

At phase angles

\[ \varphi=(2n+1)\pi,\quad (n=1,\,2,\,3,\ldots) \]

the ultradynamic characteristic becomes falling, while at all angles \(\varphi=2n\pi\) the reverse inversion takes place, and the characteristics become rising again.

In reality the relations are not so simple, since for the current flowing in the external circuit of the tube the convection current is essential, and not the displacement current.

For a theoretical consideration of the question one may replace the control-

... a triode-diode, controlled by a space charge, with inertialess emission control, if the alternating voltages at the anode are neglected in comparison with the constant one. The theory of such a discharge gap with periodic emission gives, for purely rising or purely falling ultradynamic characteristics—which in this case correspond to an oscillatory maximum or minimum—the phase relation:

\[ \tg \vartheta=\varphi . \]

According to this relation, one can no longer assume that \(\omega\vartheta\) increases simply proportionally to \(n\pi\); rather, the individual ordinal numbers \(n\) give the following angles:

\(n=\)	0	1	2	3	4
\(\omega\vartheta=\)	0	1.41	2.45	3.46	4.48

It further turns out that the alternating anode current likewise does not remain unchanged, as is assumed in the schematic diagram of Fig. 2, but decreases as the phase shift increases. Physically this must be understood as follows: the maximum of the volume charge present between the electrodes is always compensated by its minimum, so that in the limiting case, when \(n=\infty\), all minima and maxima mutually cancel out and no alternating current at all enters the external circuit. As a result, ultradynamic characteristics with increasing inversions should have a more gentle course. Further, it is known that the onset of oscillations in a self-excited circuit depends not only on the feedback, which is an element of the circuit, but also on the characteristic of the controlled flow. With a rising characteristic, the output and control voltages must be opposite, while with a falling one they must be identical in phase.

In a circuit with feedback, at phase angles of \(2n\pi\), oscillations can arise again when the ultradynamic characteristics undergo reverse inversion. In the intermediate regions, however, at phase angles of \((2n-1)\pi\), the circuit operates as a purely watt resistance.

In self-excited circuits with a positive anode, it has still not been possible experimentally to obtain, in a periodic sequence, inversion oscillations corresponding to the representations given above, although the short-wave oscillations obtained by Smirnov are, in essence, nothing other than inversion oscillations. On the contrary, with the aid of the arrangement shown in Fig. 3, one can experimentally demonstrate a sequence of several inversion regions.

The circuit consists of two parts, namely: a short-wave generator \(II\), with a retarding field, which excites the self-excited circuit \(I\) connected with it. Depending on whether the excitation of this measuring circuit is negative in the inversion regions or positive in the regions of reverse inversion, the energy will be

or be given back to generator II, or be expended on the excitation of oscillations in generator I. Consequently, if the ultradynamic phase shift is varied, for example, by means of the anode voltage, while the frequency of the exciter is kept constant, then, correspondingly to the oscillations of generator II, the anode current \(I_{a2}\) oscillates about its mean value in the way shown in Fig. 4 by the dotted curve.

Fig. 3. Circuit for measuring excitation (after Gollmann).

At the zero point for \(\Delta I_{a2}\), the mean ultradynamic steepness is also equal to zero, and the tube can neither absorb energy nor reduce the damping. For the practical production of oscillations in the regions of inversion, excitation is evidently no longer sufficient unless the resonance conditions are improved by special devices.

II. Method of the retarding field

In contrast to circuits with tubes having a positive anode, for which, owing to their limited applicability for obtaining decimeter waves, there are only isolated investigations, Barkhausen’s retarding-field method is studied in a considerable number of the works reviewed here.

Fig. 4. Regions of excitation, ×—tube with retarding field, — — — feedback.

The enormous number of concomitant factors here, together with the impossibility of classifying them exactly, makes the verification of often contradictory theoretical considerations extremely difficult. It is possible only in broad outline, omitting individual details, to set forth in a certain order the principal points of view, by reviewing the present state of research on this question.

a) Purely electronic oscillations

Usually Barkhausen’s theory proceeds from the idea of free electrons oscillating between the electrodes, whose wavelength, according to the well-known relation \(\lambda^{2} E_{\sigma} = \mathrm{const}\), depends

depend solely on the potential of the electrodes. In most cases a Lecher system is connected to the grid and anode, which, owing to the alternating voltages on the electrodes, acts on the oscillating space charge and produces complex frequency jumps and pulling phenomena. In the opinion of most authors, such resonant voltages on the electrodes are necessary for phase-coherent and synchronous control of the electronic oscillations. In this case, however, the frequency is pulled or stabilized by the resonant frequencies of the external circuit, especially when the external system is tuned to resonance.

Alongside this, the hypothesis has recently received confirmation from various quarters that, besides these forced oscillations, so-called “pure” Barkhausen oscillations arise.

Thus, for example, Morita observed electronic oscillations arising independently of the external circuit. He points, however, to a special selection of the dimensions of the parts of his tube. In particular, if the frequent grid is replaced by a sparser one, then no difference in the static characteristics is obtained. Further, the ratio of the anode radius to the grid radius \(\frac{r_a}{r_g}\) must be equal to 2.5. If the value of this ratio deviates strongly from 2.5, then the oscillations become unstable, and at \(\frac{r_a}{r_g}=3\), contrary to Barkhausen’s formula, the waves become longer as the grid voltage is increased. If, by appropriate measures, the resonant effect of the external circuit or of the electrodes of the tube itself is eliminated, then purely Barkhausen oscillations also arise. Such elimination of interfering concomitant oscillations may occur, for example, with a short capacitive shunt inside the tube. Kalinin \(^{12}\) constructed the grid in the form of a freely oscillating dipole with its own wavelength from 4 to 6 cm, lying far beyond the range of wavelengths investigated by him. In such an “aperiodic” tube, at a small positive anode potential he obtained oscillations depending only on the voltage on the grid, with an energy of 0.1 W. Comparing the long-wave oscillations observed by Widinton, caused by gas ions, with Barkhausen electronic oscillations, Orgel \(^{13}\) arrived at the inevitable conclusion that Barkhausen oscillations can arise independently of the external circuit, since in the region of waves of 500 m the electrode circuits play no role whatever, and there are no other resonant circuits in the arrangements.

b) The electron generator as a coupled system

According to Barkhausen’s conceptions, each electron is considered separately. In reality, however, electrons flying out of the filament at different times must create a phase-coherent oscillatory motion, i.e., must form a dense oscillating cloud of space charge, if the tube is subjected to

to a constant external action. To interpret such synchronization of individual oscillations, most investigators assume that alternating voltages of appreciable magnitude are induced on the electrodes; these, however, may be small at the moment when oscillations begin. Especially large alternating voltages are superposed on the electrode potentials when the electrodes themselves, or together with the Lecher system connected to them, are in resonance with the electronic oscillations.

In earlier works, the appearance in these cases of frequency jumps accompanied by a considerable increase in energy was ascribed to an oscillatory change of the frequency itself, the alternating voltages at a definite phase position causing an increase of the electronic frequency up to the resonant one (“frequency feedback”).

At present various investigators regard these phenomena of frequency jumps with their various wavelength regions as waves

Fig. 5. Electronic oscillations as the primary circuit (after King).

of coupling of an excited coupled system. In this, the role of the primary circuit is played by an oscillating space charge, while the connected Lecher system acts as a secondary circuit, as is clearly shown according to King14 in Fig. 5.

The oscillating space charge is replaced by an ordinary generator with feedback, to which, to the right of the bridge \(B\), the Lecher system is connected. In the electronic generator, the coupling between the space charge serving as the primary circuit and the tunable circuit is effected by alternating fields common to both systems between the grid and the anode. In the feedback generator, however, this coupling is effected during radiation through the common impedance of bridge \(B\). When bridge \(B'\) is shifted, the frequency curves of the feedback generator are obtained similar to the curves already known for the electronic generator. It should, however, be noted that the comparison must not be too strict, because the feedback action of the secondary circuit on the electronic oscillations is a phenomenon of a substantially different kind than the interaction between two coupled circuits.

Strutt15, 16 also considers the wave jumps that appear when the Lecher system connected to the lamp circuit is detuned as phenomena of pulling in coupled systems, occurring under weak excitation of the lamp with a retarding field and disappearing under strong excitation. Wundt17 likewise explains the simultaneous dependence of the wavelength both on the regime and on the external resonant system by the effect of coupling between the external Lecher system and the electronic oscillations coupled with it through the electrodes of the lamp. However, he does not give an exact theoretical formulation for the interaction of the voltages in this case.

To verify his theory, given for plane electrodes, Wundt constructed a special tube with plate-like electrodes. Since in this case everything depends very strongly on the homogeneity of the field, so that in ordinary tubes with plane electrodes and with one or several filaments no oscillations arise, Wundt uses, instead of the tube proposed by Romanova1, 29 with a plane cathode heated by electron bombardment, a tube in which 14 parallel-connected oxide filaments are placed opposite a plane grid.

For the sake of simplifying the theoretical calculations he connected a Lecher system between the anode and the cathode, since in principle the experimental results are the same as when it is connected in the usual way between the anode and the grid.

Fig. 6. Theoretical regions of oscillation (according to Wundt).

Starting from the differential equation of the oscillating space charge and the known equations for the energy of a Lecher system, short-circuited at one end and loaded at the other by the capacitance of the tube \(C_0\), Wundt derived the following relation for the established wave of coupling:

\[ \operatorname{tg}\frac{2\pi l}{\lambda} = \frac{\lambda}{2\pi c C_0 Z} \left[ 1- \frac{\left(\dfrac{\lambda_0}{\lambda_1}\right)^2} {1-\left(\dfrac{\lambda_0}{\lambda}\right)^2} \right], \]

where \(\lambda_0\) is the purely Barkhausen electron wave arising independently of the externally variable voltages, \(\lambda_1\) is the wave arising from the induced charges on the electrodes, and \(Z\) is the wave impedance of the wire. The quantity \(\left(\dfrac{\lambda_0}{\lambda_1}\right)^2\), obviously, is very important for the coupling between the two systems and determines the deviation of the coupling frequency from the natural frequencies of the constituent circuits. For

\[ \frac{\lambda_0}{\lambda_1}=0 \]

the above equation becomes the following:

\[ \operatorname{tg}\frac{2\pi l}{\lambda} = \frac{\lambda}{\mathrm{const}}, \]

which, as is known, gives the expression for the natural wave of the external circuit in the absence of the action of electron oscillations. The presence of the tangent function in the equation gives an infinite number of branches of the curve, differing by \(\dfrac{\lambda}{2}\); however, the course of the overtones coincides in principle with the course of the fundamental oscillation.

In Fig. 6 the dashed line gives the curve of the dependence of the natural co-

oscillations of the Lecher system on the length of the wire, and next the wavelengths of coupling for \(\left(\dfrac{\lambda_0}{\lambda_1}\right)^2 = 0.19\), likewise as a function of the length of the wire. Considering, in connection with this, the dotted course of the damping decrement \(\alpha_0, \alpha_1, \alpha_2 \ldots\) for different overtones, one can trace how, with the continuous increase of \(l\), the fundamental frequency is excited as soon as \(\alpha_0\) becomes sufficiently small. However, the damping decrement \(\alpha_1\) of the first overtone gradually decreases and becomes less than \(\alpha_0\). At this moment the coupling frequency jumps from the curve for \(\lambda_0\) to the curve for \(\lambda_1\), etc. The course of the energy change, with its periodic maximum, also finds its explanation in the corresponding minimum for \(\alpha\). The calculations of Elder\(^{18}\), who derived the same dependences on the basis of energy relations and showed in particular how the power of the direct current supplied to the tube is converted into the energy of oscillation, are also in agreement with the theory presented. The oscillatory power delivered by the tube is given by the relation

\[ L = - \frac{U \beta i a_0}{4\pi}, \]

where \(U\) is the amplitude of the voltage between cathode and anode, \(i\) is the emission current in the tube in the absence of oscillations, the reduced value of which in the presence of oscillations is obtained by multiplication by \(\beta\), where \(\beta\) is the permeability of the grid, i.e. a quantity proportional to the ratio of the area of the grid openings to its total surface, and \(a_0 = \omega t\) is the “lifetime” of the electron up to the moment it reaches the grid. On the other hand, the power required to maintain, in the external system, oscillations with amplitude \(U\) is equal to \(CU^2 \alpha\), so that one may write

\[ \frac{U \beta i a}{4\pi} \geq CU^2 \alpha \]

or

\[ \frac{\beta i a_0}{4\pi CU} \geq \alpha. \]

The left-hand side does not depend on the wavelength and, in the Hund diagram, can be represented by a straight line parallel to the abscissa axis. Depending on the position of the straight lines, separate regions of oscillations overlap. The oscillations then become unstable and give “pulling loops”; in other words, the frequency jumps upon lengthening the Lecher system occur in different places than upon shortening it, which is in agreement with experiment.

According to Elder, the maximum oscillatory power has the magnitude

\[ L_{\max} = 0.13 \beta k i E_g. \]

In view of the fact that \(\beta < 1\), it constitutes only a few percent of the supplied direct-current power.

Morita\(^{11}\) investigated, on a large number of lamps, how the excitability and intensity of oscillations in a lamp with a retarding

field and by the external system with variable tuning on the conditions and, in particular, on the dimensions of the tube. Generally speaking, oscillations are excited even before the grid current reaches saturation, but in the saturation region they are most stable. The anode current is not a reliable measure of the oscillatory energy, although it does have a greater value the more strongly \(I_a\) oscillates when the external system is detuned.

To clarify the role of the anode diameter, various tubes were studied with a constant grid radius of \(2\) mm, but with different ratios

\[ r=\frac{r_a}{r_g}. \]

Consideration of the different oscillation regions at various grid voltages, given in Fig. 7, shows that, although at large \(r\) considerable oscillatory currents occur, nevertheless the course of the changes in wavelength is irregular, and, most importantly, very high grid voltages are required for short waves. At \(r=2\) no oscillations at all are obtained.

Fig. 7. Oscillation regions for different values of the ratio of the anode radius to the grid radius (after Morita).

In addition to the anode diameter, the pitch of the grid helix \(p\) has a noticeable influence on the intensity of the oscillations. If the pitch of the helix is large, then the mean velocity of the electrons passing through the grid is less than for small \(p\), and, correspondingly, the wavelength increases, with a simultaneous shift of the oscillation maximum toward higher voltages. With a decrease in the pitch of the helix, the fraction of the current passing through the grid decreases, and hence so does the oscillation energy. At excessively large pitches, owing to distortion of the field, the synchronism of the oscillations of the individual electrons is disturbed.

If tubes with different helix pitches are constructed by using different grid wires and the dimensions of the electrodes are chosen so that the grid permeability \(\beta\), and consequently the static characteristics as well, remain approximately constant, it turns out that fine grids with small pitches are considerably more advantageous, as was to be expected from considerations of field homogeneity. With a proportional reduction of all the dimensions of the tube, the frequency increases with a simultaneous decrease in energy.

To realize plane electrodes satisfying the theoretical requirements, Morita constructed a tube with a cathode consisting of 16 parallel-connected filaments forming a cylindrical surface. In contrast to previous observations, the frequency jumps in this tube are symmetrical, and it is impossible to distinguish “Barkhausen” oscillations from “Gill–Morell” oscillations. Evidently, this is a particularly simple coupling effect, which is all the more probable since electronic oscillations also take place in the absence of an external circuit. If the distance between the wires is increased, then the coupling becomes weaker, and the phenomena of frequency jumps gradually disappear.

Of the various ways of connecting the Lecher system between the anode, grid, and cathode, the strongest oscillations and the sharpest resonance curves are obtained by the usual connection of it between the anode and cathode, opposite in phase.

In the following paragraph a detailed theory of the coupling effect will be given, with simultaneous consideration of the oscillation of the space charge as such.

c) Theory of space-charge oscillations

In all the preceding arguments, the oscillating space charge was assumed to be given. It could be set into oscillation either under the action of static potentials on the electrodes, or it could be entrained or synchronized by an external oscillatory system connected into the tube. At the same time, the question already emphasized by us remained open: in what way is that ordering of the motion of electrons created which is necessary to produce their in-phase and synchronous oscillatory motion?

In this direction Möller^20 develops his theory of “anode and phase sorting” (“transit-time oscillations”), in which, for quantitative calculations, it is assumed that between the anode and cathode there exist, even if small at first, variable voltages which play a noticeable role in both these processes. In “anode sorting,” first described by Kaptsov, the electrons that approach the anode during a certain part of the period, when the anode is positive, are completely captured by the anode and are thus removed from the subsequent process. The remaining electrons return to the grid and sustain the oscillations. “Anode sorting,” however, is not always sufficient to explain Barkhausen oscillations, for it is known that in the Barkhausen generator oscillations are excited even at a negative anode potential, when, generally speaking, electrons cannot reach it and be sorted by it. These oscillations, it is true, continue until the amplitude of the voltage on the anode becomes greater than the negative anode voltage, when anode sorting begins again. One can construct such a sorting mechanism which is suitable also for the case when oscil-

tions also arise with a negative anode, and consequently also for application to purely Barkhausen oscillations. It must be borne in mind here that the transit time of electrons excited when the anode-voltage oscillations are still weak is, owing to the greater “height of fall,” greater than the transit time of electrons reflected by the retarding field. This difference in transit times \(d\vartheta\) is assumed proportional to the difference of amplitudes \(dx\). However, as a result of this difference in transit time, the electrons with increased amplitudes later return from the cathode, i.e., their second oscillation comes too late, and the “sorting” already takes place in another phase. The whole process is shown in Fig. 8, where at the top the electrons, in accordance with their uniform emission from the cathode, are uniformly distributed along the time axis. Electrons with large amplitudes are slowed down before the anode, and, for simplicity, it is assumed that \(d\vartheta = dx\), and later return to the grid. As will be seen further, the space charge becomes denser after each oscillation the more strongly, the more rapidly \(dx\) decreases, whereas under other conditions the space charge becomes rarefied. Theoretically the relations here are as they would be if the anode current oscillated as indicated in Fig. 8, drawing 3.

Fig. 8. Phase sorting (after Möller).

If the electrodes are blocked at high frequencies by ideal chokes, then the oscillating space charge creates a “no-load voltage” proportional to the output voltage

\[ U_a = U \cos \omega t. \]

The coefficient of proportionality, which bears the name “excitation coefficient,” has, for “phase sorting,” the form

\[ D_{ph}=\frac{\pi^4 I_0 x_0}{2\omega E_g}\left(\frac{1}{1-\beta}-\Delta U\right), \]

and for “anode sorting”:

\[ D_a=\frac{i\pi^3 I_0 x_0}{\omega e_h}\left(\frac{1}{1-\beta}-\Delta U\right), \]

where \(I_0\) is the density of the emission current, and \(\beta\) is the permeability of the grid. Consideration of these results leads to the following experimentally confirmed conclusions. \(D\) increases with the increase of the current through the tube (\(I_0\)); hence the given tube must be given as strong a filament heating as possible. In “phase sorting” \(E_g\) stands in the denominator, and in “anode sorting” the filament voltage \(e_h\) stands in the denominator. Consequently, small tubes with weak emission can be excited only with a positive anode voltage, owing to the considerably stronger anode sorting in this case. Fur-

Further, the “excitation coefficient” is proportional to the distance between the electrodes, or more precisely to the electronic amplitude \(x_0\). If the electron path \(x_0\) is shortened by increasing, for example, the negative anode voltage, so that the electrons are forced to return to the grid earlier, then \(D\) decreases and the oscillations cease.

With regard to the energy of the oscillations, it should be noted that the anode current, like the grid current, in an ordinary feedback oscillator is a harmful loss. Between the cessation of oscillations at negative \(E_a\) and \(E_a = 0\) there lies a maximum of intensity.

If a Lecher system with complex impedance \(R\) is connected into the tube, then the relations are

\[ D \gtrless 1 - \frac{1}{\omega^2 LC} \]

or

\[ D \gtrless C_1 \left(\frac{1}{C} + \frac{1}{C_1}\right) \]

depending on whether the Lecher system represents a capacitive or an inductive impedance. The most powerful oscillations arise between \(L=\infty\) and the resonant value \(\frac{1}{\omega^2 C}\).

Fig. 9. Regions of oscillation of a tube with a retarding field.

But even when \(D > 1\), for very large negative values of \(R\), weak oscillations arise. In this case regions of oscillation are obtained which, taking into account the phase shift produced by the damping resistance, are represented in Fig. 9b.

Experimentally, these regions were observed by Tank and Schiltknecht. The course of the changes in wavelength of the excited phase oscillations, given in Fig. 9c, is also in agreement with the theory presented above.

In connection with the theory presented above, Möller and Gills subjected the character of the excitation of oscillations to a careful examination, because, for example, for the reception of ultrashort waves a soft excitation is desirable.

Considering the steepness of the amplitude curve as a function of the “excitation coefficient,” they came to the conclusion that the width of the region of jumps is greatest for intermediate excitation coefficients and that it decreases both with an increase and with a decrease of \(D\). With

at very small excitation coefficients the frequency-jump phenomena completely disappear. Helmholtz[^21] accurately investigated phase sorting and calculated the complete excitation coefficient from the sum of all oscillations, all half-periods, taking into account the fact that the density of the oscillating space charge at each passage through the grid decreases by the factor \(\beta\), and for cylindrical tubes obtained the expression

\[ D_{\text{calc}}=\frac{\pi^4}{2}\,\frac{I_0 x_0}{2\omega E_g}\,\frac{1}{2(1-\beta)} . \]

If one works at the point of “excitation” of the oscillation, one obtains the formula

\[ D_{\text{meas}}=1-\frac{1}{\omega C R}, \]

according to which \(D\) can be found from experimentally determined quantities and compared with that calculated by Möller’s formula.

For simplifying the measurements, the resonance curve plotted on a quadratic scale is extrapolated along a straight line to the voltage equal to zero. Experiment gives agreement with the formula for \(D_{\text{calc}}\), derived from the above considerations, to an accuracy of \(2\%\), and, consequently, the process is described by this corrected formula more accurately than by Möller’s formula.

Fig. 10. Electron paths (after Sirs).

The equations of motion given by Kaptzov for explaining anode sorting of electrons emitted from the filament at different times did not yield a general solution and were solved by Kaptzov empirically, by means of far-reaching simplifications. They were solved by graphical integration by Sirs[^22], who on this basis calculated the frequency of the free electrons by the well-known Scheibe formula.

In this way it was possible to take into account the influence of space charge, the initial velocities of the electrons, and the amplitude of the alternating voltage. In Fig. 10 several electron paths are drawn for \(E_a=0\) and for an alternating voltage, opposite in phase, between grid and anode, with an amplitude of \(10\text{ V}\). In the figure one can see how the alternating field deforms the electron paths, which for the case of absence of the alternating field are represented by the dashed curve.

The example given confirms Kaptzov’s theory, according to which all electrons emitted at phase angles from \(90\) to \(270^\circ\) cannot form any oscillating group and therefore are sorted out. Gill[^23] also investigated a circuit with a retarding field with a Lecher system between the grid and anode. In a tube with an anode diameter of \(25\text{ mm}\) and with a grid voltage

from 24 to 157 V, he excited overtones with maximum energy at wavelengths of 5.75, 4.00, 3.04, and 2.48 m. In these regions certain regularities characteristic of the retarding-field method were observed. If, at the corresponding intensity maximum of the various overtones, one adjusts not only the grid voltage but also the heating current, then the general Barkhausen relation \(\lambda^{2}E_g=\mathrm{const}\) proves to hold. But these overtones arise also at constant grid voltage and with variation of the heating current alone; moreover, for each \(E_g\) the dependence \(\lambda^{2} i_g=\mathrm{const}\) proves applicable. For any value of \(E_g\) and \(i_g\) the condition holds

\[ \frac{\lambda^{2} i_g}{\sqrt{E_g}}=\mathrm{const}, \]

which may be regarded as a generalization of the simple Barkhausen formula.

Theoretical considerations proceed from the assumption that oscillations are maintained only in the case when the space between the anode and the grid is saturated, but not the space between the grid and the cathode, for only in this case can the electrons flying through the grid control the current coming from the cathode. From the fact of periodic rarefaction and densification of the space charge in front of the anode and the associated oscillations of the potential distribution, it follows that oscillations can arise only when the period is equal to \(2\vartheta\), where \(\vartheta\) is the time of flight of the electron along the path \(d\) from the grid to the turning point at the anode. Since this time of flight under saturation is equal to \(\frac{3}{2} d \sqrt{\frac{2m}{eE_g}}\), it follows that the wavelength is

\[ \lambda=\frac{30d}{\sqrt{E_g(\text{volts})}} \]

or

\[ \lambda^{2}E_g=900d^{2}, \]

which agrees well with the measurements in the above-mentioned wavelength region, a result that differs greatly from the value given by Barkhausen[^11]. The first oscillation arises when the fraction of the current passing through the grid becomes sufficient to saturate the space between the anode and the grid. If a larger number of electrons passes through the grid, then, according to the formula derived above, the wavelength is shortened until the ensuing saturation of the space between the cathode and the grid sets a limit to this shortening of the wavelength. In tubes used in practice the two values are often very close, so that in most cases only one wave arises, as was also obtained in the earlier measurements of Gill and Morrell. In accordance with other theories, it follows from the considerations given above that the space between the grid and the anode must be greater than the space between the grid and the cathode if the tube is to operate by the retarding-field method.

Morita, in the work already mentioned, considers oscillations of the spatial charge density outside the grid as the cause of the occurrence of electron oscillations at grid potentials close to saturation. If there is a space charge \(Q\) in the anode space, then the number of electrons passing through the grid decreases, so that in the stationary state the grid current assumes the value \(I_g\). If, however, under the influence of some impulse \(Q\) suddenly increases, then the occurrence of oscillation is possible if the grid passes many electrons; in the opposite case, when the grid captures electrons, the oscillations die out. This condition is formulated so that the occurrence of oscillations depends on whether

\[ \frac{dI_g}{dQ} \lessgtr 0. \]

However, the condition

\[ \frac{dI_g}{dQ}<0 \]

is fulfilled only at a grid current that has not reached saturation, because only under this condition can the space charge that has penetrated, by virtue of its kinetic energy, into the anode space decrease as a result of the growth of the grid voltage owing to the increase of the grid current \(I_g\). If a sudden change in the space charge causes voltage oscillations \(dE_g\) on the grid, and if \(q_1\) is an oscillation of the space charge on one square centimeter of the electrode surfaces, then we have the following differential equation for the oscillations of the space charge:

\[ \left[2.7\pi\left(\frac{m}{e}\right)\frac{d^3}{E_g}\right]\left(\frac{2}{\pi}\right)\frac{d^2 q_1}{dt^2} + \left[2.7\pi\left(\frac{m}{e}\right)\frac{d^3}{E_g}\right]\frac{dI_g}{dQ}\frac{dq_1}{dt} + 8d=0. \]

For negative

\[ \frac{dI_g}{dQ} \]

oscillations are obtained from this with angular frequency

\[ \omega=\frac{8}{\sqrt{\,5.4\left(\frac{m}{e}\right)\frac{d^2}{E_g}\,}} \]

or with wavelength

\[ \lambda=\frac{895}{\sqrt{E_g}}\,4d, \]

—a formula coinciding with the Barkhausen formula, with numerical factor 1000. If the external Lecher system is replaced by a series connection of self-inductance \(L\) and capacitance \(C\), which is certainly permissible in the resonance case of interest to us, and if \(C_0\) is the grid–anode capacitance, then instead of the equation given above, describing processes only in the tube itself, we obtain the differential equation:

\[ \frac{d^4 q_1}{dt^4} + (\omega_1^2+\omega_2^2)\frac{d^2 q_1}{dt^2} + \omega_1\omega_2\left(1-\frac{C'}{2C_0}\right)q_1 =0, \]

where \(C'\) is a combination of \(C\) and \(C_0=\frac{CC_0}{C}+C_0\). This equation corresponds to the general theory of coupled circuits and represents the change in wavelength at resonance \((\omega_1=\omega_2)\) as a function of the coupling

\(\dfrac{C'}{2C_0}\). Although the general theory of coupled systems also covers jumps of frequencies, the intensity oscillations near the resonance position, however, are not in agreement with the simple theory of coupled systems, and find their explanation only in special coupling conditions under which the external inductive circuit sustains, while the capacitive one damps, the oscillations of the space charge.

For the practical calculation of a tube with a braking field for a specified wavelength, the considerations given lead to the formula

\[ \lambda=\frac{195 d_a^{2/3}}{\left(\dfrac{I_s}{l_a}\right)^{1/3}}, \]

if the ratio of the diameters \(\dfrac{d_a}{d_g}=2.5\). According to this, for a wave of length \(50\ \mathrm{cm}\), \(I_s\) must be equal to \(20\ \mathrm{mA}\), the anode length \(l_a=2\ \mathrm{cm}\), and the diameter \(=6.8\ \mathrm{mm}\).

Rostany \(^{24}\) proceeds from the known fact that gas ions affect the capacitance of a capacitor with a rarefied gas as dielectric. The current passing through a capacitor of surface \(F\) and distance between the plates \(d\), under an alternating voltage \(V\), is equal to

\[ I=j\frac{ne^2F}{md\omega}\,V=j\frac{1}{L\omega}\,V, \]

where

\[ L=\frac{md}{ne^2F}, \]

and \(m\), \(e\), and \(n\) are the mass, charge, and concentration of the ions.

In addition to this inductive component of the current, a displacement current also passes through the capacitor. If \(L\) and \(C\) are connected in parallel, they form an independent oscillatory system with a natural frequency determined by Thomson’s formula. From the expression

\[ C=\frac{F}{4\pi d} \]

and the expression for \(L\), we then obtain the resonance frequency

\[ \omega=\sqrt{\frac{4\pi ne^2}{m}}, \]

which, obviously, is completely independent of the geometrical dimensions of the capacitor and is determined only by the gas pressure and the gas ions.

These ideas may be transferred to an electron oscillator, if the ions are replaced by oscillating electrons. Although the conditions at individual points of the electron oscillator differ greatly from the conditions in a gas capacitor, it is nevertheless assumed that, under certain limiting assumptions, the density determined by the...

because of the spatial charge density, the natural frequency of the capacitor with the electron gas can be excited only when it coincides with the natural frequency of the electronic oscillations. Under this assumption one can find the wavelength as a function of the number of electrons passing per unit time, i.e. as a function of the current through the tube:

\[ \lambda=\frac{3.35}{\sqrt{n}}\,10^6. \]

If, instead of \(n\), one substitutes the total number of electrons \(N=nv\), where \(v\) is the volume of the space between the grid and the anode, then

\[ \lambda \sqrt{N}=3.35\cdot 10^6\sqrt{v}, \]

a result confirmed on various tubes. From this there further follows the relation

\[ \lambda^2 I=\mathrm{const}, \]

which had already been derived by another method by Gill\(^3\). If, as is the case in most practical instances, there is in addition an external resonant circuit, which is excited by the oscillations of the space charge and is tuned to maximum intensity, then in all there are three different oscillatory systems, namely: 1) electronic oscillations depending on the static potentials on the electrodes; 2) the electronic capacitor with its inductive component, depending on the current density; and, finally, 3) the external circuit. It turns out, however, that for the resonance case, when the transit time of the electron \(\vartheta\) from the grid to the anode is equal to the oscillation period \(T\), or to a number which is a multiple of it, i.e. \(\vartheta=kT\) \((k=1,2,3\ldots)\), the electronic oscillations do not diminish the oscillation frequency. Since \(\vartheta\) varies inversely proportional to \(\sqrt{E_g}\), then in the case when \(\vartheta=T\), the relation is obtained

\[ \lambda^2 E_g=\mathrm{const}. \]

On the other hand, the general relation

\[ \vartheta=kT, \]

from which one obtains

\[ \lambda^2 E_g=\frac{\mathrm{const}}{k^3}, \]

indicates Potapenko’s dwarf waves and inversion oscillations. The latter will be considered more precisely later.

Moullin\(^ {26}\) observed, with a gradual increase of the filament heating, several successive regions of oscillations with different wavelengths, which, upon a sharp jump of the anode current, mutually destroyed one another. Since the oscillations occur without any external oscillatory system, and since their frequency does not change when it is switched in, a theory was developed according to which here

what is involved is the natural oscillations of the electrodes, whose excitation is also connected with the capacitance of the electrodes themselves, which depends on the density of the electron gas. With an increase in the emission current both the tuning of this resonant system and the electronic frequency change, so that the increase in wavelength with increasing emission current, which contradicts all other observations, can be explained by this.

Gerber \(^{27}\) used tubes with a filamentary anode and cathode, which, in the manner indicated in Fig. 11, were connected into a Lecher system, the displacement of the bridges \(B_1\) and \(B_2\) taking place in such a sequence that the tubes were always located at a voltage antinode.

Fig. 11. Generator with two filaments (after Gerber).

Although in this case the oscillations in question are not purely Barkhausen oscillations, tubes with filamentary electrodes exhibit all the characteristic properties of three-electrode tubes with a retarding field, so that here one may speak of analogous Barkhausen oscillations of the space charge. Thus, for example, here the Barkhausen relation \(\lambda^2 E_a = \mathrm{const}\) holds in the voltage region up to 700 V. However, beginning from this point, the value of the wavelength changes abruptly to some other value \(\lambda_2\), and at the corresponding voltages a third wave \(\lambda_3\) can also be observed.

Fig. 12. Oscillation regions of a generator with two filaments.

In contrast to ordinary tubes with a retarding field, which generate only near saturation, the oscillation regions in filamentary tubes extend over the entire space-charge curve. In this case the longer waves lie in the saturation region, and their length decreases with increasing heating. The shorter waves lie in that part of the \(I_a—E_a\) characteristic which is determined by the space charge and does not depend on the heating.

Typical tuning curves of a filamentary tube are given in Fig. 12. Shown here are the wavelength \(\lambda\), the anode current \(I_a\), and the amplitude of oscillations

$E$ as a function of the length of the wire $l$ between the bridges $B_1$ and $B_2$. Here the frequency jumps already known from the Barkhausen generator appear very clearly. These frequency jumps are accompanied both by changes in the anode current and intensity, and by quenching phenomena. However, the phenomena here proceed in the diametrically opposite direction; namely, the phenomena that occur in the Barkhausen generator when the Lecher system is lengthened occur here precisely when it is shortened.

From the theory of Tank and Schiltknecht, Gerber drew the conclusion that the space charges of a filament lamp oscillate only in the inductive phase, and that the feedback between the oscillating space charge and the Lecher system must be capacitive. However, the simple theory of coupled systems cannot explain the striking difference in the intensity of the two coupling waves. The cause of electron oscillations in filament lamps may be regarded as the periodic traversal by electrons of the closed curves shown in Fig. 13; moreover, owing to symmetry, both directions of traversal are possible simultaneously.

Fig. 13. Closed (a) and unclosed (b) paths of electrons in a lamp with two filaments.

Along such paths, with a double direction of circulation, standing waves of space charge arise, which can excite various anharmonic overtones $\lambda_1$, $\lambda_2$, $\lambda_3\ldots$ All electrons falling onto the unclosed paths shown in Fig. 13b cannot oscillate synchronously and are excluded from the process.

Experimental confirmation of this idea of two different directions of circulation can be obtained by placing the lamp in a weak magnetic field. This magnetic field lengthens or shortens the transit time of the two electron groups, depending on their direction of circulation, so that the frequency is split into two separate frequencies—of the electrons circulating to the right and to the left. Thus, for example, in a magnetic field of $2.5$ C there is a splitting of a wave of $172$ cm by about $\pm 4.5$ cm, which, in order of magnitude, agrees with the classical formula for the Zeeman effect.

d) Dwarf Waves

Up to now we have considered chiefly Barkhausen oscillations, which obey, to a rough approximation, the relation

$$ \lambda^2 E_g = 10^6 d_a^2 = C_0. $$

As has been shown by many investigators, there can be obtained

regions of oscillations with much smaller numerical factors, i.e., with shorter waves.

Potapenko showed for the first time that the frequency of these oscillations of higher order is a quantity that is a multiple of the fundamental Barkhausen frequency \((n = 1, 2, 3\ldots)\), or that the general constant

\[ \lambda_2^n E_g = C_n \]

decreases by factors of 4, 9, 16, etc. These dwarf waves were subsequently investigated by Potapenko in a new, thoroughly prepared work. The circuit of the apparatus used in this work is shown in Fig. 14. It consists of a symmetrical circuit with tunable anode and grid circuits, in which the second tube is replaced by a capacitor with a capacitance of 0.1 to 0.2 cm. This avoids the irregularities that arise when two tubes are used and that smooth out the fine structure of the oscillation characteristics. By varying \(C\), one can produce various regions of dwarf waves, depending on whether \(C\) is located at a node or at an antinode. From the amplitudes of the oscillation and of the corresponding anode potential, determined from the point at which the anode current appears, the energy of the oscillations in one tube was determined. It proved to be equal to 0.08 W at \(\lambda = 10\ \mathrm{cm}\) and 0.2 W at

Fig. 14. Circuit of Potapenko’s generator.

Fig. 15. Complete operating diagram.

1 fundamental waves
2 dwarf waves of the first order
3 “ ” second “
4 “ ” third “
5 “ ” fourth “

\(\lambda = 60\ \mathrm{cm}\). Since the regions of oscillations and the wavelengths depend on the grid potential and, on the other hand, on the tuning, i.e., on the length of the wire of the Lecher system, the \(E_g\)—\(L\) characteristics were combined into spatial “operating diagrams.” Fig. 15 gives the complete “operating diagram” of the Russian R-5 tube, which is especially rich in higher-frequency regions of oscillations.

If \(\lambda_0\) is determined from the above equation, then for the dwarf waves the following relations hold:

\[ \lambda_1 \]
is equal to the dwarf wave of the 1st order, equal to \(\dfrac{\lambda_0}{2}\);

\(\lambda_2\) is equal to the dwarf wave of the 2nd order, equal to \(\dfrac{\lambda_0}{3}\);

\(\lambda_3\) is equal to the dwarf wave of the 3rd order, equal to \(\dfrac{\lambda_0}{4}\).

The experimentally found values of the constant \(C_n\) do not enter into the theoretically assumed integral ratios, which may be attributed to the reverse influence of the alternating voltages induced on the electrodes upon the motion of the electrons. External tuning has an effect only insofar as various regions of oscillation are either damped as a result of energy transfer or, conversely, are excited at resonance. The difference in the number of oscillation regions in different tubes must be ascribed to the difference in the flight time of the electrons in both directions between the electrodes, because symmetry of the transit time is a necessary condition for exciting dwarf waves. Connected with this, according to Weinberg\(^{29}\), is the necessity for perfect coaxiality of the electrode arrangement, since an asymmetry of the filament by \(0.5\ \mathrm{mm}\) already produces a 20% difference in the electron flight time, which in the region of the 2nd order increases to 80%. This also explains the fact that, in tubes ordinarily available for sale, relatively few regions of dwarf waves can be excited. At most, with \(E_g = 330\ \mathrm{V}\), a wave of \(9.5\ \mathrm{cm}\) can be obtained as a dwarf wave of the 4th order.

Fig. 16. Simplified diagram of electron motion.

To interpret the fact of excitation of regions of higher order, it is necessary to obtain a diagram of the motion of electrons and the corresponding resonant voltages on the electrodes.

In Figs. 16a and 16b two such fundamental diagrams are given, which differ from one another in that the resonant period \(T\) of the oscillatory circuit lying between the anode and the grid is, in one case, equal to the electron period \(\tau\), while in the other case it is only one third of it.

It is evident from the figure that an electron leaving the cathode at the moment \(a\), under the influence of the positive grid alternating voltage, experiences a greater accelerating action in the space between the grid and the cathode than the retarding action in the grid–anode space, so that it gives up its excess energy to the anode. Conversely, for an electron leaving at the moment \(b\), the conditions are exactly the opposite, since its retardation in the anode space is greater than its acceleration before the grid, so that it turns back before the anode and returns-

falls on the grid. Of these two groups, only the electrons flying out at the moment \(b\) deliver energy to the anode circuit; the electrons of group \(a\), however, constitute a purely wattless resistance.

The same considerations are also applicable to Fig. 16b; here only, in accordance with the higher frequency, the groups \(a, a', a''\ldots\) and \(b, b', b''\ldots\) follow one another much more rapidly in time. To obtain a more precise picture of the motion of the electrons, graphs of the motion were calculated not only for the fundamental oscillation \(T=\tau\), but also for the harmonic waves \(T=\dfrac{\tau}{n}\). The calculations were made on the basis of Kaptsov’s theory, taking into account the variable fields at the electrodes.

Fig. 17. Motion of electrons in an alternating field.

Two such graphs of electron motion for \(T=\dfrac{\tau}{2}\) and \(T=\dfrac{\tau}{3}\) are given in Figs. 17a and 17b.

It is clear that the phase position of the electrons which supply the energy of oscillations is very different in the two cases. Thus, for example, in Fig. 17a only those electrons which leave the cathode at the moment \(\omega t = 270^\circ\) have a turning point before the anode. In Fig. 17b, however, the corresponding phase angle is \(180^\circ\). Further, it may be seen that the alternating voltages on the grid have a decisive significance for the distribution of the electrons into groups. Although, according to Fig. 17, those electrons which appear at the moments \(\omega t=0\) and \(270^\circ\) reach the anode at equal voltage amplitudes, nevertheless only the latter belong to the group returning back, because they receive a stronger acceleration in the space between the grid and the anode.

The great importance which the alternating voltages on the electrodes have for the entire mechanism of oscillations makes it necessary to investigate the influence of the filament current, which primarily determines the energy of the oscillations and, consequently, also the magnitude of the alternating voltages. As follows already from Rostagni’s theory,^2 the Barkhausen relation is

\[ \lambda_n^2 E_\sigma = C_n = \frac{C_0}{n^2}. \]

can also be extended to the region of dwarf oscillations; however, as has already been said above, this is done experimentally only approximately, and, generally speaking, the deviations are the smaller the weaker the heating and, consequently, the intensity of the oscillations. Moreover, the behavior of dwarf waves is essentially different from that of ordinary Barkhausen oscillations. In ordinary Barkhausen oscillations the wavelength decreases, as is known, with increasing heating, whereas for dwarf waves, on the contrary, the wavelength becomes larger under the same conditions. If the anode current \(I_a\) is taken as a measure of the intensity of the oscillations, then, according to Fig. 18, one can represent the various constants as functions of \(I_a\).

By extrapolation one can find their values corresponding to \(I_a=0\), at which the alternating voltages also vanish. It turns out that the ratio of the constants obtained in this way agrees very well with the theoretically expected value \(n^2\). It follows from Fig. 18 that the alternating voltages affect the frequency of the electronic oscillations in the same way as the constant voltages on the electrodes. If we assume that the amplitude of the alternating voltage \(E_e=kI_a\), where \(k\) is a proportionality factor, then the generalized Barkhausen relation takes the form

\[ \lambda_n^2(E_g \pm nE_e)=\frac{C_0}{n^2}. \]

Fig. 18. Constant \(C\) as a function of the emission current.

Here the plus sign refers to ordinary waves, and the minus sign to dwarf waves. The coefficient \(n\) takes into account all possible kinds of influence of alternating voltages on the region of dwarf waves. The Barkhausen relation, corrected in this way, describes well the experimentally observed dependence between wavelength and heating current, provided that the corresponding alternating voltage \(E_e\) is determined from the anode potential at the point where the anode current \(I_a\) arises. Whereas Potapenko worked preferably at an anode potential equal to zero, Krebel \(^{30}\), as well as Uda and Makami \(^{31}\), investigated the course of the intensity of dwarf waves characterized by the natural oscillations of the grid with its holder (as in Fig. 1), and found a clearly expressed maximum at a negative anode potential and at a correspondingly reduced grid voltage. The anode current, however, decreases all the time with increasing \(E_a\) toward negative potentials and thus can in no case be a measure of the energy of the oscillations.

Kollenbusch \(^{32}\) used various tubes with a spiral grid, short-circuited either inside the spiral itself or outside the anode. In addition to waves resonant with this grid, waves of a still lower order were excited, whose length was determined by other

electrodes and the leads connected with them. In several tubes with \(r_a/r_g = 4\), one more energy maximum was obtained at positive \(E_a\), at which the current \(I_a\) decreased with the onset of oscillations. Precise measurements of the dependence of the regions of oscillation on the potentials of the anode and grid in many tubes gave about seven voltage regions for one and the same wave. Fig. 19 gives four such regions for \(\lambda = 12.6\ \mathrm{cm}\).

Closed curves bound that region of grid and anode voltage within which, at a definite emission current, oscillations of this natural frequency are excited. As the emission increases the regions become larger; as the emission decreases the regions become smaller and narrower. They narrow until, at a certain emission, they contract into a “working point” with energy always close to zero. The reason for the existence of such a voltage region obviously lies in the fact that in any oscillatory system there are several excited motions of electrons with transit times in the anode–grid space that differ from one another by an integer number times the period of oscillation.

Fig. 19. Regions of oscillation (after Kohlenschuss).

e) Inversion oscillations

The phase shifts caused by the finite transit time of the electron must, of course, affect the tube with a retarding field, just as they affect the ordinary triode. This means that the ultradynamic characteristics must invert when the phase angle is equal to \(\omega \vartheta = n\pi\), and assume a cyclic character in the intermediate regions. Sachanek\(^{8}\) calculated the excitation of a tube with a retarding field and plane electrodes, proceeding from energy considerations. For the case in which the anode has a potential equal to the mean potential of the filament, and alternating voltages are applied to the anode and grid, the energy of the oscillations is equal to

\[ E = \frac{1}{2} i_0 e_0 \left[ \frac{\sin^2 \frac{\omega \vartheta}{2}}{\frac{\omega \vartheta}{2}} - \frac{3}{2}\frac{\sin \omega \vartheta}{\omega \vartheta} - \frac{1}{2} \right]. \]

Since the expression in brackets can never be positive, in this special case oscillations cannot be excited

at a positive potential on the electrodes. Conversely, with cylindrical electrodes the oscillation energy may, depending on \(\dfrac{r_a}{r_g}\), increase, especially when the anode potential is negative relative to the cathode.

Theoretical consideration gives, in this case,

\[ E=-\frac{e_0 i_0}{\pi}\left[2\,\frac{\sin \omega \vartheta}{\omega \vartheta} -\frac{\sin^2 \dfrac{\omega \vartheta'}{2}}{\left(\dfrac{\omega \vartheta'}{2}\right)^2}\right]\cos \omega t_1, \]

where \(\vartheta'\) is the duration of the return time to the grid, and \(t_1\) is the time during which the electron charge reaches the anode. Since \(\dfrac{T}{4}<t_1<\dfrac{3T}{4}\), \(\cos \omega t_1\) is always negative, and, consequently, the energy periodically also passes through a positive and negative maximum with plane electrodes. The more closely \(e_{gk}\) and \(e_{ak}\)—the grid–cathode potential and the grid–anode potential—approach one another, the greater \(\cos \omega t_1\) becomes, and the energy increases. Consequently, the excitation of oscillations is associated with the occurrence of an anode current, which must diminish as the oscillations become established.

Fig. 20. Ultradynamic characteristic for the case of a retarding field (according to Hollmann).

Hollmann\(^{9}\) extends the region of inversions of the characteristics, since he sees the difference between a tube with a retarding field and an ordinary triode controlled by space charge in the interchange of the roles of the electrodes, so that in the former the grid plays the role of the anode, while the anode, which here plays the role of the retarding electrode, is transformed in the triode into the controlling electrode. In this case the control is effected no longer electrostatically, but by the distribution of the currents. If from this point of view one considers the static characteristics of a tube with a retarding field, given in Fig. 20 in the form of non-closed curves, it may be noted that the characteristic of the tube with a retarding field \([i_b=f(e_b)]\) has the form of an ordinary characteristic with a saturation region and with the limiting value \(\beta i_s\), if \(i_s\) denotes the emission current.

Assuming that the grid–cathode space is saturated, i.e., that the grid potential exceeds the saturation voltage, we have \(d i_g=-d i_b\), i.e. the characteristic of the tube with a retarding field is the mirror reflection of the grid characteristic \(i_g=f(e_b)\). If the space near the cathode possesses clearly expressed saturation and if the electrodes are exactly coaxial, then the characteristics of the tube with a retarding field over a wide region do not depend on the grid voltage, provided only that it does not become lower

saturation voltage. It follows from this that a distinguishing feature of a tube with a retarding field is that it can operate as a generator with an infinitely large saturation resistance, at any external resistance \(R_g\), without practically noticeable permeability phenomena.

If, by analogy with the usual amplification coefficient, the “transformation ratio” (Übersetzungsverhältnis) is expressed by the ratio of the output and input voltages, we have

\[ U=\frac{e_g}{e_b}=-\frac{i_g R_g}{i_b R_{ib}}=\frac{R_g}{R_{ib}}, \]

where \(R_{ib}\) is the internal resistance of the tube, i.e.

\[ R_{ib}=\frac{d e_b}{d i_b}. \]

Thus the entire circuit of the tube with a retarding field represents the ratio of two resistances: a “resistance transformer,” which lowers an external resistance \(R_g\) of any magnitude to the value \(R_{ib}\). At the same time it is clear that, because of \(R_{ib}\), the tube constitutes a noticeable load for the controlling voltage, and that in this case control can no longer take place without loss of power, as in ordinary triodes. The power required for control can be taken from the tube itself by means of a special self-excitation circuit, provided only that the delivered power \(i_g^{2}R_g\) is considerably greater than the controlling power \(i_b^{2}R_{ib}\), or if \(R_g \gg R_{ib}\).

The theory of inversions developed earlier for ordinary tubes can also be extended to tubes with a retarding field, since at high frequencies there is a phase shift corresponding to the transit time of the electrons that turn around at the retarding electrode and return to the grid. In view of the presence of a considerable space charge between the grid and the retarding electrode, produced by the current to this electrode and by the current of electrons returning to the grid, a surface of zero potential is formed immediately in front of the controlling electrode, which can play the role of a virtual cathode. Depending on the magnitude of the retarding potential \(e_b\), a greater or lesser current \(i_b\) goes to the retarding electrode, while the remaining fraction of the current returns to the grid. Since the distance between the surface of zero potential and the retarding electrode is extremely small, the time lag of the current \(i_b\) relative to the retarding potential \(e_b\) is also so small that it cannot have an effect at the frequencies considered here (we shall return to this later when considering questions of reception of ultrashort waves).

Since the retarding potential controls, by means of the current to the retarding electrode as well as the reverse current to the grid, it also affects—in practice without inertia—the emission from the virtual cathode, so that tubes with a retarding field of the type introduced earlier are in fact equivalent to a diode with inertia-free

emission control. Especially significant is the fact that for ultradynamic phase mixing inside a tube with a retarding field the same distance \(d\) is involved between the control and trapping electrodes as in an ordinary triode; only the direction of electron motion here is the reverse. Thus the assumption becomes possible that the relations here are the same as in a tube controlled by a space charge, while the course of the ultradynamic characteristics is the same as in Fig. 20.

Considering now the feedback of a certain oscillatory circuit and a tube with a retarding field, one can see that, owing to the descending course of the characteristic, the input and output voltages are no longer opposite in phase, as in ordinary tubes, but are in the same phase. This determines, however, that the external feedback \(K\) must also be positive in order that oscillations can be excited.

In contrast to this, with an ordinary circuit with negative feedback, excitation in tubes with a retarding field is negative, and the circuit operates as a wattless resistance. If an ordinary triode with positive transconductance and with output and input voltages opposite in phase is denoted by \(-R\), and a tube with a retarding field with its descending characteristic by \(+R\), and the external feedback by \(\pm K\), then the sign of excitation can be obtained from the formula

\[ \operatorname{sgn} A = \operatorname{sgn} R \cdot K \cdot (-1)^n . \]

This very simple dependence shows that the ultradynamic regions of excitation of any self-oscillating circuit operating once with an ordinary triode and another time with a tube with a retarding field must be shifted in phase by \(180^\circ\). If a tube with a retarding field is included in an ordinary three-point circuit with external negative feedback, then the excitation is negative at low frequencies in the region determined by the static characteristics, and only under ultradynamic inversion does the feedback operate with the correct phase. From this, generally speaking, it follows that the frequency regions of inversion oscillations of a tube with a retarding field lie precisely between the regions of oscillations of the very same circuit, but with ordinary tubes.

With the aid of the apparatus already described in the first part, one can measure the ultradynamic excitation (as in Fig. 3) and unambiguously verify these theoretical assumptions. For this purpose the excitation of measuring circuit I is again observed, with the only difference that the potentials of the electrodes of tube \(R_1\) are interchanged, i.e., the tube operates as a generator with a retarding field. The result is given in Fig. 4 in the form of a continuous curve, on which one can see a series of periodically following maxima of positive and negative excitation. Comparison of this curve with the previously obtained excitation curve under ordinary feedback gives the result that for each region of positive excitation in the tube with a retarding field

there is always a corresponding negative region of excitation of the ordinary tube, and conversely. In this we neglect the inaccuracies that occur at low grid potentials.

Accordingly, ultrashort waves excited by means of a tube with a retarding field must be regarded as oscillations caused by feedback, the feedback opposite in phase being compensated in this case by the inversion of the ultradynamic characteristics. Of course, the question remains open as to why excitation by means of a tube with a retarding field is considerably stronger than in the inversion regions of an ordinary tube. In considering the oscillation regions practically obtained by means of a generator with a retarding field, one must proceed from the fact that an apparatus like that shown in Fig. 21 contains a whole series of different resonant systems—for example, the electrodes themselves, or these together with their holders, or with an external Lecher system. Depending on the capacitive or inductive ratios of all these natural frequencies, the feedback coefficient in different regions of wavelengths may be positive or negative. At the same time, all these different systems can be excited only in the regions of ultradynamic inversion, if the above excitation condition is fulfilled.

Fig. 21. Generator with a retarding field, having several degrees of freedom.

Fig. 22 gives the frequency spectrum, obtained with an apparatus similar to that shown in Fig. 21, in which the wavelengths of the individual regions are almost independent of \(E_g\) and can therefore be regarded as resonant oscillations of some system. On the diagram are plotted the curves corresponding to the formula

\[ n^2 \lambda^2 E_g = \text{const} \]

Fig. 22. Frequency spectrum of a generator with a retarding field.

curves on which the centers of the oscillation regions should correspond to even and odd ordinal numbers. The place in the diagram—that is, the voltages at which inversion oscillations appear in practice—depends exclusively on the accidental resonances that may occur in the tube. More

more detailed than those provided by the inversion formula, the theoretical explanations for various regions of oscillations are not justified by experiment.

Since the inversion theory does not, generally speaking, assume any electronic oscillations or coupling effects, and since all phenomena—such as, for example, the jump in the currents \(i_b\) and \(i_g\) to the control electrode that is characteristic for the establishment of oscillations—are determined from comparison with an ordinary generator, the question arises of how to reconcile this point of view with other theories and, in particular, with experiment. In this respect it should be pointed out that purely Barkhausen electronic oscillations with wavelength \(\lambda_B\) arise only in tubes with an exceptionally dense grid. The coarser the grid, the greater the deviation from this wavelength toward oscillations synchronized by the external system, until, finally, there remains only one Gill–Morell region with a wavelength of almost \(0.6\,\lambda_B\) (see, for example, Fig. 20). In this case, consequently, the primary excited system is entirely absent, and therefore there can be no question here of coupling waves. What remains is only the inversion theory, which gives the correct frequency relation for this case, because for the first region of inversion we obtain

\[ \omega \vartheta = \pi \]

and

\[ \lambda = \frac{500\,da}{\sqrt{E_g}} = 0.5\,\lambda_B . \]

f) Influence of the Gas

A survey of electronic oscillations would be incomplete without considering an effect whose significance some investigators regard as decisive for the oscillations—namely, the action of residual gases in the tube.

Rindfleisch \(^{33}\) distinguishes here between direct and indirect influence; the direct influence causes a change in the transit time of the electron and of the regions of oscillation, while the indirect influence manifests itself in a shift of the resonance with a rigidly coupled external system. The influence of the gas is especially clearly seen in the grid-voltage characteristics shown in Fig. 23, taken with a tuned external circuit. It can be firmly established that the maximum value of the oscillation energy obtained when the grid voltage is varied, with increasing content—

Fig. 23. Intensity and wavelength as functions of grid voltage at various gas pressures (after Rindfleisch).

the gas content decreases, while the point of origin of the oscillation (Einsatzpunkt) and the intensity maximum shift parallel to the static characteristics toward lower grid voltages. The higher the grid voltage, the more clearly this weakening of the energy appears, so that, for example, the sharply expressed maximum occurring in vacuum at \(E_g = 250V\) almost completely disappears as the pressure increases. The oscillations continue to be maintained even with a glowing discharge, and at high pressures the previously negative anode current becomes positive, i.e., becomes an ion current. The wavelength becomes several percent greater as the gas content increases; however, an exact accounting for this, owing to the influence of the external circuit, is very difficult. Depending on the position of the operating point—for example, on whether it lies before saturation or after it—one or another region of oscillations, in the presence of gas, may either be excited or quenched. From this it can be seen how easily one may be misled regarding the necessity of the presence of gas for the excitation of some definite wavelength if, beforehand, one does not form a complete picture of the oscillation regions in the tube by means of a corresponding investigation of all the conditions of operation of the tube in vacuum.

The cause of the decrease in the energy of the oscillations is the change in the potential distribution and collisions of electrons with gas ions. Under all conditions it must here be taken into account that ionized atoms are attracted to the anode and to the cathode and recombine there. Since in both places the electrons still, or already, do not have sufficient energy for ionization, the collisions are purely elastic. Such elastic impacts can scatter, in synchronism with the high frequency, the oscillating electron cloud, and thereby reduce the amplitude of the oscillation.

Kalinin \(^{34}\) compares the regions of oscillation of a vacuum tube and of a tube with gas, in which no regularity in the appearance of oscillation regions can be established. The Barkhausen product \(\lambda^{2}E_g\) assumes a different value in each region; however, as is evident from Fig. 24, within each region it is a linear function of the grid voltage. Accordingly, the Barkhausen constant may be expressed as follows:

\[ \lambda^{2}E_g = aE_g + b, \]

where the quantities \(a\) and \(b\) characterize the individual oscillation regions. This formula is a generalization of the usual Barkhausen formula, for, by putting \(a = 0\), we obtain the condition for purely Barkhausen oscillations, while for \(b = 0\) Zhil-Morelev oscillations arise with a wavelength independent of \(E_g\). What is striking in Fig. 24 is the presence of three kinds of oscillation regions, namely:

1. Regions with \(a > 0\) and \(b \ne 0\), which occur only in vacuum tubes.

2. Regions with \(a > 0\) and \(b \ne 0\), i.e., with a constant wavelength.

1. Regions with \(a < 0\) and \(b = 0\), which occur only in tubes containing gas.

It also follows from these observations that the residual gas in the tube has a very strong effect on electronic oscillations. Gotton and Bove \(^{35}\) also found that individual regions of oscillation in an apparatus with a retarding field are connected in very diverse ways with the gas pressure. Likewise, Morita \(^{11}\) firmly established that the character of the regions of oscillation changes very strongly in comparison with high vacuum if the pressure rises to \(10^{-3}\)—\(10^{-4}\) mm Hg. At still higher pressures oscillations do not arise at all. The static characteristics have the same course as in Fig. 23.

Fig. 24. Barkhausen’s constant as a function of grid voltage (after Kalinin).

In some tubes an increase of energy with increasing pressure was observed; however, it has not been firmly established whether this result, which contradicts the preceding measurements, is not the product of some extraneous influence. When the tube is overloaded too strongly, occluded gases are liberated, and when the voltage or filament-current characteristics are taken, “loops” appear if one does not wait for the temperatures to equalize. In general, the tuning curves of a poorly evacuated tube have a very complicated form. Especially sharply expressed in the presence of residual gas are hysteresis loops, which far exceed the normally expected dimensions in coupled circuits.

Finally, Gill \(^{23}\) points to the action of positive ions, which neutralize the negative space charge and thus influence the oscillations. Ionescu \(^{36}\) attempts in general to reduce oscillations in a tube with a retarding field to the natural oscillations of the ionized residual gas, and his calculation of the Barkhausen constant contains, in addition to the dimensions of the tube, also the ratio of the number of positive ions to the number of electrons. Ro-

Stani[^37] showed, however, that such a number of ions should produce an ionic current to the anode which has never yet been observed experimentally.

In conclusion it must be said in general that, although the residual gas exerts a noticeable effect on electron oscillations in certain regions, this effect cannot be used for the excitation of oscillations.

g) Special tubes and oscillatory circuits

To increase the energy and frequency of the oscillations, the most varied tubes and oscillatory circuits are included in the circuit. Almost all investigators use cylindrical symmetrical electrodes and tungsten cathodes. Incidentally, according to Giacomini[^38], for circuits with a retarding field oxide cathodes with indirect heating are also used.

Fig. 25. Grid oscillatory circuit (after Gossel).

With weak excitation of dwarf waves (owing to the decrease of the inversion current with increasing ordinal number), it is very important that the resonant system in the tube be, as far as possible, without losses. It has been shown in various ways that, for this purpose, the grid must occupy a special (particular) position. To create a definite symmetry, Gossel[^39], according to Fig. 25, replaces the arc short-circuiting the grid spiral by another spiral. The excitation of this arrangement can be considerably improved if this second spiral is also located between the anode and the cathode. The form of the resulting, thus, push-pull device is shown in the same Fig. 25.

Fig. 26. Freely oscillating grid (after Hollmann).

Fig. 27. Tubes for waves of 10 cm and 7 cm (after Kohl).

Hollmann constructed the grid in the form of a freely hanging turn (Fig. 26). The current is fed to its middle, so that it can play the role of an oscillating dipole. In the ninth inversion region he obtained,

i.e., for \(\varphi = 9\pi\), oscillations with a wavelength of 13 cm at a grid voltage of only 128 V. The dimensions to which one must go in order to obtain the shortest possible waves are shown by the short-wave Kohl lamps\(^2\) for \(\lambda\) of 10 and 7 cm, shown in Fig. 27, in which, instead of a spiral, only the two ends of a short-circuited turn are introduced into the space between the anode and the cathode. There are various possibilities for increasing the intensity. Thus, for example, Hollmann\(^9\) describes the arrangement shown in Fig. 28, where a lamp with a grid and anode is connected in a Lecher system short-circuited at both ends.

Fig. 28. Generator with a retarding field, having 1 degree of freedom (after Hollmann).

In contrast to the circuit shown in Fig. 21, this installation can oscillate only with one degree of freedom, namely, with a node of oscillation on the capacitor bridge and a voltage antinode at the electrodes. Such a lamp with a grid allowing greater dissipation, manufactured by the Telefunken company, gives, at a wave of 50 cm, a high-frequency power of up to 7 W. The possibility of theoretically any increase in energy is provided by the parallel connection of several generators. In practice, in most cases, one confines oneself to a push-pull circuit. A particularly large “excitability” is exhibited by the push-pull lamp with two filamentary anodes on both sides of the filament, proposed by Gerber\(^27\). In this lamp, even at an anode current of \(5 \cdot 10^{-5}\) A, the first oscillations are established. Morita estimates the oscillation power of a push-pull arrangement at 38 mW, as against 10 mW for one separate lamp. Marconi\(^41\) worked with a push-pull radiator with 3 tunable circuits placed between both grids, anodes, and cathodes. Koczanowski\(^42\) investigated the push-pull arrangement shown in Fig. 29, with 2 American UX 852 tubes, and observed, in contrast to the usual circuit with a Lecher system between grid and anode, a shortening of the wavelength from 90 to 70 cm at an output power of 5 W. Finally, one must mention one more method of increasing the power, namely, the replacement of the solid anode cylinder by a grid or spiral, which, in turn surrounded by a metal sheet playing the role of an anode, represents nothing other than a two-grid tube. The explanation of the increase occurring in this case

Fig. 29. Asinphase generator (after Koczanowski).

energy, consists in the fact that the internal-anode resistance, or “braking resistance,” is, for oscillations, a loss which is reduced when a braking grid is used. In addition, the smaller capacitance of the electrodes also plays a role.

Morita¹¹ investigated the influence of the permeability of such a grid-like anode on the energy of the oscillations. The results, given in Fig. 30, show that what occurs here is not only an increase in intensity, but also a widening of the oscillation region; moreover, here again, a coarse grid has a smaller effect. The wavelength in this case increases somewhat, which may be ascribed to the fact that the electrons pass through the braking grid and only after that return back. In contrast to a solid cylindrical anode, the maximum energy of a grid-like anode occurs at weak positive anode potentials.

Fig. 30. Influence of a grid anode on the intensity of oscillations (after Morita).

Sahanek⁴³ studied in detail the electronic oscillations in a two-grid tube. It is easy to see that an electron stream passing by the positive grid is partly turned back before the second braking grid, partly passes through this grid, and only before the anode turns back. Depending on whether, in the same phase or in the opposite phase, this second return stream is with the first return stream, there is either an increase or a decrease in the energy of the oscillation. These relations, consequently, depend entirely on the voltage between the braking grid and the anode. For an in-phase superposition of both return currents the theory gives the condition

\[ E_{ag_2} = 399 \frac{d}{\lambda} \ \text{and}\ 164 \frac{d}{\lambda}, \]

whereas at \(232 \frac{d}{\lambda}\) compensation of the streams occurs (here \(d\) is the distance between the braking grid and the anode in millimeters).

III. GENERAL BEHAVIOR OF THE DISCHARGE GAP AT HIGH FREQUENCIES

Earlier the behavior of an ordinary triode, controlled by a space charge, at high frequencies was already considered. This

behavior received especially clear expression in ultradynamic characteristics. In this, neglecting the alternating voltages at the anode, the tube was, for convenience of consideration, replaced by a diode with inertia-free emission control. However, on the other hand, a diode with a constantly emitting source of electrons, or, more generally, any discharge phenomenon in a gas or in a vacuum, must give an analogous effect if the electric or also the magnetic field changes in time so rapidly that the motion of the charge carriers can no longer occur without inertia. In the simplest case, let there be a charge \(e\) with mass \(m\) in an electric field oscillating with angular frequency \(\omega\). The current caused by this charge in the external circuit is, very approximately, expressed as follows:

\[ i_a=-\frac{e^2}{m d^2}\,\frac{V_0}{\omega}\cos \omega t=-\frac{V_0}{\omega L}\cos \omega t, \]

where \(m\dfrac{d^2}{e^2}\) is the “self-inductance” of the discharge gap, which is manifested only at very high frequencies and differs from ordinary self-inductance in that it is an accumulator of mechanical, not magnetic, energy.

If the charge carriers are, in addition, accelerated by a constant field, then the relations here are fundamentally similar to those that occur under periodic emission control, and the difference consists only in the fact that here, instead of the current density \(j\), the electron velocity \(v\) oscillates. The relation for the space charge

\[ \rho=\frac{i}{v} \]

makes it possible to express the velocity oscillations through a space charge oscillating in space and in time.

Thus, for a diode with a periodically varying anode potential as well, ultradynamic phase displacement and inversion regions with a falling characteristic and with alternately positive and negative internal resistance are obtained. The relations here become very greatly complicated if one attempts to take into account the influence of the space charge.

Sakhanek \(^{8}\) applied his theoretical considerations, in the work already repeatedly cited here, to a simple diode and obtained for the energy of oscillations the expression

\[ E=\frac{e_n i_0}{2}\left[\frac{\sin^2 \dfrac{\omega \vartheta}{2}}{\left(\dfrac{\omega \vartheta}{2}\right)^2}-2\,\frac{\sin \omega \vartheta}{\omega \vartheta}\right]. \]

For small values of \(\omega\vartheta\) the energy of the oscillations, in accordance with the positive internal resistance, is negative. With increasing frequency or transit time, for values of \(\omega\vartheta\) between \(\dfrac{2\pi}{2.7}\) and \(2\pi\), the first inversion region arises, in which

the energy of the oscillations is positive; up to \(\frac{2\pi}{0.65}\) extends the second inversion region, in which the lamp operates as a purely watt resistance, etc. Since the absolute value of the expression in parentheses decreases with increasing \(\omega\), in regions of higher order one must expect considerably lower oscillation intensities.

Bengam\(^{44}\) investigated the dielectric constant \(\varepsilon\) and the resistance \(R\) to the electron current, and gave the course of these quantities as a function of the phase angle, shown in Fig. 31. The fact that the dielectric constant of the diode at low frequencies is less than unity is experimentally confirmed; for high frequencies, however, no experimental material is available. The internal resistance, on the contrary, with increasing phase shift assumes the form of curve \(b\) and is alternately positive and negative. In this case the negative maximum coincides with the potential region of dwarf waves.

Fig. 31. Dielectric constant and internal resistance of a discharge gap filled with electron gas (after Bengam).

Witt\(^{45}\) solves the problem for the saturation region, neglecting the space charge, and obtains for the region of excitation of oscillations the relation

\[ \frac{4n-1}{\frac{\pi}{2}} < \omega\vartheta < \frac{4n+1}{\frac{\pi}{2}}, \]

whereas the centers of the oscillation regions are determined by the formula

\[ \lambda_n^2 E_g = \frac{2c^2 d^2 m}{n^2 e}, \]

in which we again recognize the generalized Barkhausen formula and the regularity established for inversion oscillations.

A detailed theoretical treatment of the problem is given by Möller\(^{46}\), who assumes that at every point of space there are electrons of only one velocity. He applies his theory in two special cases. First, in the case when the electrons appear in space in the form of a current of constant density and move with constant velocity, and, secondly, in the case when near

of the electron source there is any number of them with zero velocity. Müller’s investigation gives the result that the influence of the electron stream may be regarded as equivalent to a series or parallel connection of a capacitance with a resistance. In the region of space charge the equivalent resistance may become negative for certain frequencies, as was already explained by means of Fig. 31b. It is especially interesting whether this negative resistance, if sufficiently large, can extinguish the high frequency corresponding to some oscillatory circuit. For this purpose it is necessary that the loss angle of the entire external oscillatory arrangement be smaller than the negative phase-shift angle of the tube. As an example it was shown that this is possible for \(\omega \vartheta = 2.37\pi\).

However, the practical production of short-wave oscillations solely by inversion in a diode is not so simple experimentally. This, of course, does not apply to diodes with a grid or wire anode.

Sahanek\(^{47}\) considers the cause of the insufficient excitation to be the unfavorable distribution of potential in an ordinary diode with an axial filament cathode. In these diodes saturation already sets in at such low voltages that inversion within the region determined by the space charge, at the frequencies that occur here, becomes impossible. It is necessary, therefore, to shift the saturation region toward higher voltages. Sahanek achieves this by reversing the arrangement of the electrodes, surrounding the axial anode with an externally situated cathode. An easily excited arrangement with diodes, constructed on this principle, is shown in Fig. 32.

Fig. 32. Diode arrangement (after Sahanek).

On both sides of the tungsten anode, 0.5 mm in diameter, there is a V-shaped filament \(V\), which is protected on the outside by two segments. At one end of the anode there is a metal cylinder \(A\) of the same length as the anode, serving for electrical balancing, so that the anode as a whole constitutes an oscillating dipole. The saturation boundary lies at 2500 V, but already at 1000 V an entire wave spectrum of intense oscillations arises, corresponding to the overtones of the whole anode system. The shortest wave, of 5 cm, is obtained in the ninth-order region.

Mac Petrie\(^{48}\) places, outside the anode wire, another 4 filaments; however, he explains the occurrence of oscillations by the oscillatory motion of electrons around the anode.

In conclusion of this section we shall point out some data of Sahanek\(^{49}\) on the production of electric waves of less than 1 m by means of an arc at atmospheric pressure. According to the considerations given earlier, the arc may also be regarded as one of

forms of discharge, and therefore in its inversion regions it can readily excite resonating electrodes.

IV. Magnetron Oscillations

The magnetron is a special form of discharge gap in which electrons are deflected by a magnetic field from the direction of the electric lines of force and return to the cathode along circular paths. In this case the same conditions must be fulfilled for the occurrence of electron oscillations as in a tube with a retarding field; only here the electric retarding field is replaced by a magnetic field, and the accelerating and retarding fields are not located in separate regions between the electrodes, but mutually penetrate one another.

a) Cylindrical Diode in a Magnetic Field

In order to investigate the influence, described by Slutskin and Steinberg, of the inclination of the magnetic lines of force relative to the axis of the tube, Ranin\(^ {50}\) took static magnetic characteristics of a tube inclined relative to the magnetic field and found on the volt-ampere characteristic a falling region of negative resistances, in which oscillations with wavelengths up to 3 m could be excited. It turned out that, for the occurrence and extension of the region of negative resistances, both the ratio of the length of the anode to its radius and the angle of inclination of the magnetic field are very essential.

Fig. 33. Spiral paths of electrons in an inclined magnetic field (according to Holmann).

According to the theoretical investigation of these phenomena given by Holmann\(^ {51}\), the change of sign of the internal resistance of the tube is explained by the cycloidal form of the spiral paths of the electrons, whose axes coincide with the direction of the magnetic lines of force and whose screw thread cuts into the edge of the cylindrical anode. Qualitatively the process is clearly explained by Fig. 33, which gives the projection of such spiral paths between plane electrodes onto the plane of the drawing, perpendicular to the surfaces of the electrodes. As the voltage is decreased, the anode current falls to zero at 630 V and then arises again. Practically, of course, the current does not disappear completely; nevertheless the theory gives a simple explanation for the abnormal behavior of a magnetron inclined to the magnetic field. By heavily loading the tube, one can confirm the idea of spiral paths of electrons, na-

observing the helical motion of the heated spots arising on the anode cylinder when such operating conditions of the tube as the voltage or the magnetic field are changed.

It turns out that the electronic oscillations in the magnetron arise independently of statically negative resistances. At the same time, because of the longer transit time of the electron along spiral paths, one must assume that their inertia no longer favors the production of waves shorter than one meter.

Okabe[^52] observed in a magnetron generator, consisting of a cylindrical diode with an included Lecher system, two kinds of oscillations, known as coupling waves of the electronic generator. In one case the oscillations did not depend on the external tuning; in the other case, on the contrary, they depended on the resonant frequency. In contrast to the observations cited earlier, the wavelength of the latter region is greater than the wavelength of the first region, which reaches \(2.8\ \mathrm{cm}\) in a tube with an anode diameter of \(3\ \mathrm{mm}\) and at a voltage of \(1500\ \mathrm{V}\). In individual cases both oscillations with different wavelengths were observed simultaneously[^53], as was also observed in the generator with a retarding field.

Kollenbusch[^32] showed that a three-electrode tube can operate as a magnetron if the grid acts as the anode and if, in addition, the grid spiral is in resonance with its holder. Whereas this grid circuit, under purely electric excitation, can oscillate only antiphase, as a magnetron it can also be excited in phase, which gives almost half-waves.

For investigating the magnetic permeability and the velocity of propagation of electric waves along wires, Hogg[^54] considers a Lecher system, one end of which is closed by a reflecting bridge and the other is excited by a “diode-magnetron.” The tuning of this Lecher system is given by the relation

\[ S = s + \frac{\lambda}{360}\arccos \sqrt{\frac{i}{I}}, \]

where \(S\) is the position of the bridge, \(s\) is the position of the detector circuit inductively coupled to a galvanometer measuring the current \(i\), and \(I\) is the maximum swing between two successive maximum and minimum. The tuning curve is a straight line in the coordinate system

\[ S = f\!\left(\arccos \sqrt{\frac{i}{I}}\right). \]

b) Magnetron with a Split Anode

A noticeable increase in the intensity of the oscillations and an expansion of the oscillation regions can be achieved, as is known from Okabe, by splitting the anode and dividing it into two or more segments.

MacArthur and Spitzer^55 investigated the static characteristics of such a magnetron with a split anode, treating it as a dynatron. Energy measurements gave an efficiency of 6% for a 75-centimeter wave and 35% for a 5-meter wave at full powers of 5 and 35 W.

As with the cylindrical “diode magnetron,” the magnetron with a split anode can be substantially improved if the electrodes are given a small angle of inclination to the magnetic field. Thus, according to Kilgore,^56,57 the maximum energy of an obliquely positioned magnetron occurs at very small anode currents. Fig. 35 gives curves showing the influence of rotating the magnetron relative to the magnetic field. Curves A and B show, however, that with a coaxial magnetic field oscillations do not arise, and that they arise only when it is rotated through a positive or negative angle.

Fig. 34. Magnetron with a split anode.

Fig. 35. Oscillation intensity at different angles of inclination of the tube relative to the magnetic field (after Kilgore).

Small deviations of the two branches of the curves from one another may be attributed to inhomogeneities of the magnetic field and to asymmetry in the arrangement of the electrodes. On curve C the effect of inclination is not so clearly expressed. Generally speaking, the optimum inclination amounts to only a few degrees; however, depending on the anode voltage and current, it reaches as much as 14°. The oscillatory energy of such a magnetron with a split anode, at the optimum inclination, is 7 W for a wave of 42 cm. The efficiency is 8%, not counting the power expended to produce the magnetic field.

A theoretical consideration of the conditions in a magnetron with a split anode is, of course, impossible, since the nonuniform distribution of the field between the segments is unknown, being caused by the corresponding arrangement of the space charges. An approximate solution is given by Dellinger,^58 who assumes the presence of a very small rectangular voltage between the segments and, as the condition for excitation of oscillations, obtains the relation

\[ 1 - 2p - 2(1 - p)\frac{\theta}{\pi} > 0, \]

where \(p\) is a known percentage of the total number of electrons whose radial distances from the cathode already exceed some critical value at the moment when they leave the sector of the anodic segment with high positive potential, and \(\theta\) is the angle through which the electrons are deflected by the magnetic field during their full time of flight.

If the potential distribution can be described by the expression

\[ E_a \left(\frac{r}{r_a}\right)^n, \]

then the above-mentioned critical radius is

\[ r' = r_a \left(\frac{1}{2}\right)^{\frac{1}{n}}. \]

Since in the inequality it is assumed that the alternating voltage may be neglected, it gives no indication of the amplitudes of oscillation of practical interest to us, although the known effect of the space charge is taken into account by means of \(p\) and \(\theta\).

Megaw \(^{59}\) observed in a magnetron with a split anode, as did Okabe in an ordinary magnetron, two kinds of oscillations. One is excited owing to the negative resistance of the tube in the resonant system placed between the anode segments, and the other is a purely electronic oscillation with a frequency which, analogously to the diode, is determined by the relation

\[ \lambda = \frac{\mathrm{const}}{H}, \]

the anode segments oscillating in phase.

In order to obtain electronic oscillations of the greatest possible intensity, it is necessary to match as accurately as possible the following quantities: the length \(l\) of the Lecher system connected between the anode segments, the anode voltage, and the current and intensity of the magnetic field. It should be noted here that the wavelength increases linearly with increasing \(l\). The theoretical dependence \(\lambda H = \mathrm{const}\) and \(\frac{E_a}{H} = \mathrm{const}\) is well confirmed experimentally.

Sometimes, when the operating conditions are varied, two adjacent maxima of oscillation intensity appear, which, as is known, is characteristic of coupled waves. The reason for this may be considered to be the presence of a concentrated capacitance between the two anode leads, so that the internal electrode system—namely, the anode segments with their holders (up to the glass wall itself), of length \(l_1\)—constitutes the primary circuit, while the external part of the conductor of length \(l_2\) is the secondary circuit. From this we find that the anode segments themselves must be regarded as a resonant system with a natural wavelength approximately \(25 \sqrt{d_a}\). The nature of the oscillations in the segment is such that a nodal line passes through the middle of each segment, while voltage antinodes arise at the edges. If the magnetron is rotated relative to the magnetic field, then at an angle of inclination of about \(8^\circ\) a fivefold increase in the high-frequency current is obtained; moreover, this increase depends strongly on the axial length of the magnetron. The difference between negative and positive angles of inclination

may be attributed to the voltage drop along the filament. By giving the electrodes a special shape and, moreover, by adjusting the electrodes inside the tube, one can obtain an output power of 1.5 W at a wavelength of 24 cm. The wavelength is calculated from the formula

\[ \lambda=\frac{920\,d_a}{\sqrt{E_a}}, \]

analogous to Barkhausen’s. If the maximum permissible load of the anode is taken to be \(5\ \mathrm{W/cm^2}\) and the optimum emission is allowed for, then the shortest wave to which the magnetron can still be tuned will be

\[ \lambda_{\min}=40\,d_a^{\frac{5}{5}}\ \mathrm{cm}. \]

This gives, for example, for an anode diameter of 3 mm a wave of length 20 cm.

Excitation of oscillations by means of negative resistances occurs when the magnetic field passes through a certain critical value. By taking static characteristics it can be shown that indeed, at a certain magnitude of the magnetic-field strength, the current falling on the negative anode segment may be greater than the current going to the positive segment. The energy of oscillations excited in this way falls very rapidly as soon as the magnitude of the period of oscillation begins to approach the magnitude of the electron transit time. The point from which this rapid fall of energy begins corresponds to four times the value of the electron wave. The intensity of these two-stroke oscillations reaches, at a 50-cm wave, from 1.5 to 2 W.

In concluding this section, let us mention the investigation of Okabe\(^{60}\), who obtained oscillations in a horizontal magnetron without a heated cathode. In doing so he again obtained two forms of oscillations differing in their properties, namely, one with a wavelength of several hundred meters (as in the oscillations obtained by Biddington), determined only by the data of the tube and obtained both in an ordinary magnetron and in a magnetron with a split anode, and another with a wavelength of several meters, dependent on the excitation of an external resonant system and arising only in a magnetron with a split anode.

V. Practical Applications of Microwaves

a) Reception

Owing to the continuous increase in power and the improvement of methods of investigation, decimeter waves, in addition to effective demonstration experiments, have opened up broad new possibilities for physical research. A detailed consideration of individual problems in this direction is not the purpose of the present review, since

development is still at its very beginning. Here only a few remarks will be given concerning the methods of measurement so far available and the aims set by investigators.

The methods of investigation here differ depending on whether one observes the velocity of propagation of the waves within or along the medium under study \(^{54, 62–65}\), their reflection \(^{66}\), or their refraction. In particular, the phenomenon of anomalous dispersion makes it possible to draw important conclusions about the magnitude and shape of molecules, about the forces acting between them, and their arrangement.

Microwaves open up a broad field for their application in transmitting messages over a distance, not only in the sense of expanding the range of waves that can be used for this purpose. Their properties make it very easy to produce directed radiation with their aid, and make it possible—an extremely important matter for television—to transmit modulated high frequencies by their means. However, the quasi-optical propagation of microwaves limits the field of their application to communication over short distances. The same reasons that hinder the production of decimeter waves also make it difficult to use electronic tubes as detectors and amplifiers.

Although the previously described tubes with minimal distances between electrodes \(^{7}\) make it possible to amplify high frequency with the aid of decimeter waves, nevertheless in all the works available up to the present they deal exclusively with detection. This is because the ordinary triode does not operate at all in a circuit with plate and grid rectification; this becomes quite understandable if the effective path of the electron is regarded as a diode, in which inversions of the ultradynamic characteristics and a strong decrease of the inversion current arise. This phenomenon is especially sharply expressed in grid detection, where, although the distance between the electrodes is very small, the accelerating fields are very weak.

That at very high frequencies a change in the detecting action takes place is shown by the measurements of Müller and Tank \(^{68}\), the results of which are given in Fig. 36, where, it is true, no comparison of amplitude values is given, but the rectified current \(\Delta I\) is given for different points of the static characteristic. Depending on the circumstances, at shorter wavelengths more favorable conditions for rectification may occur than at longer ones; moreover, the behavior of the upper and lower bends of the characteristic must be different in the two cases.

A simple diode, at least in its usual form, with a coaxial cathode, operates only as a detector and therefore causes strong damping of the receiving system. To eliminate the damping from the receiving system when oscillations are excited, it is recommended to use, for reception, a tube with a retarding field, as was done after Barkhausen in many works. Thus, for example, Okabe \(^{69}\) employs as a receiver a diode-grid arrangement, in which an antenna situated outside the bulb and capacitively coupled acts on the electron streams in the tube. With a suitable choice of condi—

clear maxima of sensitivity are obtained as a consequence of resonance in the motion of the electrons. However, a three-electrode tube with an antenna connected into the anode proves better than an arrangement with a diode^70. In addition, various receivers were subjected to the action of the corresponding magnetic field.

Uda^73 improved the tuning of the receiver with a retarding field according to the Lecher system by providing it with a capacitor bridge of variable capacitance. Changing the latter acted as a displacement of the bridge.

Fig. 36. Detector characteristic of a diode at various frequencies (after Müller and Tank).

Pistor^74 distinguishes, as in an ordinary regenerative receiver, 4 different possibilities of reception.

1. Ordinary audion reception with reduced damping owing to feedback.
2. Audion reception of the first kind, when, with critical feedback, oscillations arise only upon excitation at large distances.
3. Heterodyne reception.
4. Audion reception of the second kind, when entrainment of the weakly oscillating receiver takes place.

Pistor worked with a two-grid tube and a resonant circuit placed between the space-charge grid and the anode. He regards his receiver as a coupled system with the oscillating space charge as the primary circuit and with the external circuit as the secondary circuit. In order that the coupling between the tube and the external circuit could be varied, they were connected to one another by a wire of variable length, whereby it was possible to obtain the soft excitation favorable for reception. For large received fields, audion reception of the second kind proved more suitable; for weak fields from the transmitter, audion reception of the first kind; whereas heterodyne reception, owing to the instability of the wave, proved impossible. Variable feedback, obtained by modulating the voltage on the space-charge grid with the aid of an auxiliary high frequency, gave an insignificant increase in sensitivity, although the characteristic noise was also produced. With a decrease in the length of the auxiliary

of the wave at 1000 m the sensitivity of the receiver even decreased, which must be ascribed to an excessively short build-up time between the modulation periods. Self-oscillation of the feedback, owing to the falling volt-ampere characteristic of the generator with a retarding field, acts less favorably because of the difficulty of tuning. In all the works cited hitherto it has simply been assumed, following Barkhausen, without a detailed concrete consideration of the demodulation process itself, that the tube with a retarding field is a rectifier also at high frequencies. It is assumed only that the receiver with a retarding field has the greatest sensitivity when it is directly set to its natural frequency.

Karara^75 recently gave an exhaustive explanation of the demodulation effect in a tube with a retarding field. He compared the rectified currents flowing in the circuit of the retarding electrode with the static characteristic of a tube with a retarding field \(i_b=f(e_b)\) (Fig. 20), assuming that the disturbing action of the electron oscillations is absent. The characteristic of a tube with a retarding field can be represented by the relation

\[ i_b = C e_b^n, \]

and the rectifying action is proportional to its curvature, i.e.,

\[ \frac{d^2 i_b}{d e_b^2} = n(n-1) C e_b^{\,n-2}. \]

For \(n=2\) the rectifying action obviously does not depend on \(e_b\), whereas for \(n \gtrless 2\) it increases or decreases with \(e_b\). Experimentally, \(n\) can be determined as \(5/2\). Hence, on the basis of the known equation of the diode characteristic, one can determine the ratio \(\dfrac{r_a}{r_k}\), i.e. the ratio of the anode radius to the filament radius and, consequently, \(r_k\) itself. The calculation performed gives practically coincident values of \(r_a\) and \(r_k\). This, however, means that \(r_k\) is the radius of a virtual cathode located immediately in front of the anode. A tube with a retarding field therefore acts as a diode with very small distances between the electrodes, which technically cannot be realized at all, and with an internal phase shift which, in the frequency region under consideration, can be neglected. In a later work Karara^74 compares the rectifying action of a diode with a suitably selected antenna with the action of a tube with a retarding field and comes to the conclusion that the diode can give only half the rectified current. This was confirmed by comparative measurements at longer wavelengths. Hollmann^75 experimentally showed that the rectified current in the circuit of a tube with a retarding field is practically proportional to the curvature of the characteristic, as is shown in Fig. 37; in accordance with this it changes its sign in the region of the upper and lower curvatures of the characteristic. Since the internal resistance

operating in the saturation region of a tube with a retarding field is equal, on the grid side, to infinity, while on the side of the retarding electrode it has a value of the order of \(1000\,\Omega\), the grid load of an audion with a retarding field, owing to the output resistance \(R_g\), proves less advantageous than the Barkhausen connection in the circuit of the retarding electrode. In order to avoid an unnecessarily high voltage drop across \(R_g\), a choke coil should be preferred to a purely ohmic resistance.

An extremely sharp setting of the operating point in the lower region of curvature of the characteristic can be softened if the retarding electrode is loaded with a high-ohmic biasing resistance; moreover, the positive grid voltage can serve as the biasing potential. This biasing resistance must, of course, be shunted by a very large capacitance, for in this way the rectified current will flow in the circuit of the retarding electrode and can be transmitted into the grid circuit. On this principle is built the circuit of an audion receiver with a retarding field shown in Fig. 38. In it there is retained the method, known from the transmitter with a retarding field and possessing certain advantages, of connecting the tube in the voltage antinode of a Lecher system short-circuited at both ends.

Fig. 37. Detector action of a tube with a retarding field when operating on microwaves.

Fig. 38. Circuit of a receiver with a tube with a retarding magnetic field (after Hollmann).

When the grid voltage is varied, a series of periodically successive receiving maxima appears, which satisfy the inversion relation for a standing wave

\[ n^2 E_g = \mathrm{const}. \]

With sufficient transmitter energy these maxima show up sharply also in the current through the retarding electrode, according to the observations of Gill and Donaldson \(^{78}\), in the first region of first order at a wavelength of 10 m. Since the operating points for soft excitation and for optimal rectification, generally speaking, do not coincide, the interference-free superposition of oscillations for purposes of demodulation and the creation of undamped oscillations in a tube with a retarding field is impossible. This difficulty, however, can be overcome if both functions are assigned to two tubes connected to each other by a common lead, or to a double system (Doppelsystem), in which the electrodes separated by a capacitance, with respect to the high-

at the frequency constitute a single whole, whereas the connection to the power source is such that one system serves for rectification, and the other for obtaining undamped oscillations.

In order to obtain a receiver which, over a sufficiently wide range, can be tuned by a single grid voltage, without special tunable external circuits, Hollmann ^77 built an audion receiver with a retarding field, using a two-cycle tube (Fig. 39). As can be seen from the figure, the retarding electrode, as in the magnetron with a split anode, is divided into two segments \(B_1\) and \(B_2\) and placed in the antinode of the current of the receiving dipole \(AA_1\). The current in the circuit of the retarding electrode passes through the chokes \(D\) and \(D'\), and the operating point is automatically set by means of \(W\), while the grid has the load \(R_g\). Owing to the nonuniform distribution of the field between the two segments, a transition of current occurs from the negative to the positive segment; moreover, because of the inertia of the electrons, this static negative resistance can invert.

Fig. 39. Circuit of an asiphase audion with a retarding field.

It is possible, therefore, to regard the tube as a complex resistance, the imaginary and real parts of which change with variation of the grid potential, so that the behavior of the dipole over a wide range is determined only by the grid voltage. The elimination of damping almost up to the natural frequency can be effected by regulating the filament heating and by shifting the operating point by means of \(W\). In order, with the generally high sensitivity of the receiver, to eliminate interference of any kind, the entire receiver must be placed in a closed spiral, which allows only correctly polarized centimeter waves to pass to the dipole, while it is a shield against all other fields.

Finally, Giacomini ^86 investigated a magnetron used as a receiver, in which the anode current decreases when extraneous oscillations appear. If the filament heating and the voltages are constant, and only the magnetic-field intensity is increased, then on the volt-ampere curve in the falling portion several unevenly distributed reception maxima appear.

b) Quasi-optical propagation of microwaves.

In all experiments in which the maximum range and the properties of quasi-optical propagation of decimeter waves or microwaves were established, in most cases work was carried out with an intense beam of transmitted radiation, obtained either by means of a parabolic mirror or by a reflector system or

by a combination of several transmitters with a directional antenna. Particularly simple seem to be the all-metal parabolic mirrors that were used in the transmission experiments from Dover to Calais1 at a wavelength of 18 cm.

Marconi3 used a mirror consisting of several reflectors, in whose focal line there were 4 dipoles fed by the same number of generators operating in parallel. Generally speaking, all the directional systems so far developed for long waves, a brief survey of which has recently2 been given, can, with a reduction of their dimensions, be applied to microwaves.

Whereas for long waves the chief consideration is, as it were, surface radiation, in the so-called short waves propagation depends on volume radiation; with regard to ultrashort microwaves and the phenomena connected with their propagation, one is inclined to take as a basis the optical phenomena corresponding to visible light. If the transmitter and receiver are at distances \(H\) and \(h\) from the surface of the earth, then theoretically the range of action is

\[ d=\sqrt{2RH}+\sqrt{2Rh}, \]

or, if for the radius of the earth \(R\) the value 6370 km is taken, then

\[ d_{\text{km}}=3.55\left(\sqrt{H_m}+\sqrt{h_m}\right). \]

Fig. 40. Propagation of microwaves in various layers of the atmosphere.

Let us note that this formula does not take into account the absorption of that part of the radiation which is directed tangentially to the surface of the earth.

The notion of a sharp boundary of the range of action because of the shadow created by the horizon cannot be strict. It is much more acceptable that for microwaves laws similar to those for light waves are at work. According to these ideas, a ray, passing on its way at various heights above the surface of the earth through layers of air of different temperature, pressure, and humidity, undergoes deviation.

Approximately one can make a calculation by assuming that the path of the ray is, as shown in Fig. 40, a circumference of radius \(mR\), where \(m\) is a factor taking into account the condition of the lower layers of the atmosphere. Then the range of action is

\[ d_{\text{km}}=\sqrt{2H_m R_{\text{km}}\frac{m}{m-1}}+\sqrt{2h_m R_{\text{km}}\frac{m}{m-1}}. \]

Whereas in the region of meter waves this theory had been confirmed in various ways, recently Marconi3 showed at a 57-cm wave that its range of action considerably exceeds the optical horizon. In these experiments the transmitter was at a height of 750 m above sea level, and the receiver was moved away on a vessel

from the transmitter. At first the reception was sufficiently distinct up to 10 km beyond the optical horizon, i.e., up to 107 km; then, however, it deteriorated sharply, so that at 150 km only very weak signals were being received. Between 161 and 182 km the strength of the receiving field suddenly increased almost to its value at the limit of optical visibility, only then to decrease to zero at 203 km. The receiver was then placed at a height of 340 m; in that case a distance of 269 km was covered, although the optical visibility did not reach even half this value.

According to Pession[^85], the explanation of the fact that meter waves propagate far beyond the limits of the optical horizon lies in scattering phenomena, which obey the formulas of Eckersley and Watt. For decimeter waves this assumption is not suitable, because it presupposes excessively large temperature gradients in the lower layers of the atmosphere, which is especially strange if one takes into account the elevated location of the transmitter and receiver in Marconi’s experiments. Here, consequently, the issue can only be refraction in the very lowest layers of the atmosphere.

The propagation of waves along the earth’s surface is complicated not only by the different absorptive capacity of different portions of the earth’s surface, but also by disturbances penetrating into the path of the radiation and by diffraction phenomena caused by the form of individual portions of the earth’s surface. Experiments in this direction exist only for waves down to 1.3 m in length; the behavior of microwaves has not yet been clarified.

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