Application of Harmonic Oscillations of Electrons for Generating Ultra-High Frequencies

P. A. Borodovskii

(Presented by Professor A. B. Sapozhnikov, Doctor of Physical and Mathematical Sciences)

As is known [1], Barkhausen–Kurz generators have a low efficiency and a narrow range of operating frequencies. These shortcomings are due mainly to the large number of electrons striking the grid and to the impossibility of ensuring repeated interaction of the electrons, in a favorable phase, with the high-frequency field. The latter is caused by the fact that, when energy is delivered to the high-frequency field, the electron transit time decreases with each successive oscillation. Electrons undergoing their first oscillation require a lower frequency for optimum operation than electrons that have already undergone several oscillations. As a result, when the amplitude of the high-frequency voltage is sufficiently large, the electrons pass from a favorable phase into an unfavorable one within one or two cycles. This difficulty—the principal one for generators with a retarding field—is extremely significant and leads to the fact that, in practice, generation occurs only within a narrow range near the resonant frequency of the circuit.

In order for the time of the electron oscillations to be independent of the amplitude, the potential in the interelectrode space of the tube must vary parabolically. The striking of electrons on the electrode with a high positive potential can be eliminated by the use of a magnetic field. In other words, it is necessary to construct a system in which the electrons execute harmonic oscillations without striking the electrode with the accelerating potential.

The possibility of constructing such a system was first pointed out in [2]. In [3], brief information is given on the practical use of a similar system in tubes that have received the name strophotrons.

In the present work an analogous system for generating ultra-high-frequency oscillations is considered, and some experimental results are presented which are in satisfactory agreement with the conclusions of the approximate theory.

For an electron to execute harmonic oscillations along the \(x\)-axis, the potential along the \(x\)-axis must vary according to the law \(V \sim -x^2\). This occurs in a hyperbolic field

\[ V=K(y^2-x^2). \tag{1} \]

The required field configuration can be realized by means of electrodes having the form of hyperbolic cylinders [2], or by electrodes whose cross section is shown in Fig. 1.

Fig. 1.

Let us assume that electrode \(A\) has a high positive potential relative to electrode \(R\), from the point \(K\) of which electrons are emitted. Along the \(x\)-axis a uniform magnetic field \(B\) is applied, which prevents the electrons from reaching the accelerating electrode \(A\). The electric field along the \(z\)-axis is \(E_z=0\).

The equations of motion of the electrons, neglecting the effect of their space charge, for the given system have the following form:

\[ \frac{d^2x}{dt^2}=-2\frac{e}{m}Kx, \tag{2a} \]

\[ \frac{d^2y}{dt^2}=2\frac{e}{m}Ky-\frac{e}{m}B\frac{dz}{dt}, \tag{2b} \]

\[ \frac{d^2z}{dt^2}=\frac{e}{m}B\frac{dy}{dt}. \tag{2c} \]

Neglecting the initial thermal velocities of the electrons and taking, at \(t=0\),

\[ x=b,\quad y=b,\quad z=0, \tag{3} \]

we obtain the solution of equations (2a, b, c)

\[ x=b\cdot\cos\omega_0 t, \tag{4} \]

\[ y=-\frac{\omega_c^2}{\Omega^2}\,b\left(1-\frac{\omega_0^2}{\omega_c^2}\cos\Omega t\right), \tag{5} \]

\[ z=\frac{\omega_c}{\Omega}\,\frac{\omega_0^2}{\Omega^2}\,b(\Omega t-\sin\Omega t), \tag{6} \]

where

\[ \omega_0=\sqrt{2\frac{e}{m}K}, \tag{7} \]

\[ \omega_c=\frac{e}{m}B, \tag{8} \]

\[ \Omega=\sqrt{\omega_c^2-\omega_0^2}. \tag{9} \]

It follows from (4–6) that an electron, while executing harmonic oscillations along the \(x\)-axis with angular frequency \(\omega_0\), simultaneously moves along the system in the direction of the \(z\)-axis, describing a trajectory which in the \(yoz\) plane has the form of a cycloid.

The maximum displacement in the \(y\) direction is equal to:

\[ y_{\max}=\frac{\omega_c^2+\omega_0^2}{\Omega^2}\,b. \tag{10} \]

The displacement in the \(z\) direction corresponding to one period of motion \((\Omega t=2\pi)\) is:

\[ z_0=2\pi\,\frac{\omega_c\omega_0^2}{\Omega^3}\,b. \tag{11} \]

Thus, if a high-frequency field is produced between the electrodes, then, provided they are of sufficient length, it is possible to ensure multiple interaction of the electrons with the high-frequency field, which, as is known, is one of the conditions for obtaining high efficiency.

For an experimental verification of the possibility of using harmonic oscillations of electrons for the generation of ultrahigh frequencies, two tubes were made having the arrangement shown schematically in Fig. 2. The principle of operation of these tubes is analogous to the principle of operation of the strophotron [3] and is as follows. Electrons are emitted by the cathode \(K\) through a slit in the reflector \(R_1\) and are accelerated by the electrode \(A\), which has a positive potential \(V_a\) relative to the cathode. A uniform magnetic field \(B\), which prevents the electrons from striking the accelerating electrode \(A\), is directed perpendicular to the plane of symmetry of the electrodes. The reflecting electrodes \(R_1\) and \(R_2\), having a negative potential \(-V_R\), are connected to a two-wire line, which can be tuned by moving the short-circuiting bridge \(M\).

During their motion in the interelectrode space, the electrons will excite oscillations whose frequency will be equal to the frequency of the harmonic oscillations of the electrons. Under the action of the high-frequency field, “sorting” of the electrons emitted from the cathode takes place. Electrons that leave the cathode in an unfavorable phase will be accelerated by the high-frequency field and will strike the reflector \(R_2\). Electrons of a favorable phase, after leaving the interaction space, are collected by the collector \(C\), to which a potential approximately equal to \(1/2\,V_a\) is applied.

The frequency of the harmonic oscillations of the electrons is determined by the electrostatic field of the interelectrode space and does not depend on the magnitude of the magnetic field. The electrostatic field in the transverse section of the tube is chosen so that the potential distribution corresponds to expression (1).

Then the frequency of the electron oscillations is

\[ f_0=\frac{\omega_0}{2\pi}=\frac{1}{2\pi}\sqrt{2\frac{e}{m}K}. \tag{12} \]

The constant \(K\) can be determined from (1), assuming that for \(x=0\) and \(y=b+d_a\), \(V=V_a\). Substituting the expression obtained into (12), we have

\[ f_0=\frac{1}{2\pi}\sqrt{2\frac{e}{m}\cdot \frac{V_a}{b+d_a}}= \]

\[ =0.945\cdot 10^5\frac{\sqrt{V_a}}{b+d_a}. \tag{13} \]

Fig. 2. Schematic arrangement of the tube.
\(K\)—cathode, \(A\)—accelerating electrode, \(R_1, R_2\)—reflectors, \(C\)—collector, \(L\)—two-wire line, \(M\)—short-circuiting jumper. The tube is placed in a uniform magnetic field \(B\), whose direction is indicated by the arrow.

Since the electrodes \(R_1\) and \(R_2\) have a negative potential, the quantity \(b\) in formula (13) will be smaller than the geometrical dimension of the tube \(b_0\).

Integrating the expression

\[ -\,V_R=\int_{b}^{b_0}\frac{\partial u}{\partial x}\,dx, \]

one can find that

\[ b=b_0\sqrt{1-\left(\frac{b_0+d_a}{b_0}\right)^2\frac{V_R}{V_a}}. \tag{14} \]

The expressions obtained show that the oscillation frequency depends on the magnitude of the accelerating voltage \(V_a\) and on the potential of the reflectors, \(-V_R\). These dependences are shown graphically in Figs. 3a and 3b for tubes having dimensions \(b_0=4\) mm, \(d_a=1\) mm (tube No. 1) and \(b_0=7\) mm, \(d_a=3.5\) mm (tube No. 2).

As can be seen from the graphs presented, the measurement results are in quite satisfactory agreement with the calculated data according to formulas (13–14).

In order that the electrons not fall on the accelerating electrode \(A\), the magnitude of the magnetic field \(B\) must be not less than a certain “critical” value \(B_{\mathrm{cr}}\). The magnitude of the critical magnetic field can be found from the condition

\[ y_{\max}\leq b+d_a. \tag{15} \]

Substituting the expression for \(y_{\max}\) from (10) and using (7, 8, 9), we obtain

\[ B_{\mathrm{cr}}= \sqrt{\frac{2m}{e}}\sqrt{\frac{2b+d_a}{d_a}}\cdot \frac{\sqrt{V_a}}{b+d_a} = 3.37\cdot 10^{-6} \sqrt{\frac{2b+d_a}{d_a}}\cdot \frac{\sqrt{V_a}}{b+d_a} \left(\frac{\mathrm{Wb}}{m^2}\right). \tag{16} \]

As can be seen from Fig. 4, the data calculated from formula (16) agree well with the experimental data obtained for a tube having dimensions \(b_0=4\ \mathrm{mm}\), \(d_a=1\ \mathrm{mm}\), at \(V_R=0\).

Fig. 3a. Dependence of the generated frequency on the accelerating voltage.
— theoretical curve; x — experimental points.
A—tube No. 1, \(b_0=4\ \mathrm{mm}\), \(d_a=1\ \mathrm{mm}\), \(V_R=-50\ \mathrm{V}\)
B—tube No. 2, \(b_0=7\ \mathrm{mm}\), \(d_a=3.5\ \mathrm{mm}\), \(V_R=-75\ \mathrm{V}\)

Fig. 3b. Dependence of the generated frequency on the reflector potential.
— theoretical curve; x — experimental points.
A—tube No. 1, \(b_0=4\ \mathrm{mm}\), \(d_a=1\ \mathrm{mm}\), \(V_a=500\ \mathrm{V}\)
B—tube No. 2, \(b_0=7\ \mathrm{mm}\), \(d_a=3.5\ \mathrm{mm}\), \(V_a=850\ \mathrm{V}\)

It should be noted that the magnitude of the critical magnetic field can be found from purely energy considerations, without knowing the form of the electron trajectories.

According to the law of conservation of energy, one may write:

\[ \frac{1}{2}m\left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2\right]=eV, \]

if it is assumed that the electrons leave the cathode with zero velocity.

Fig. 4. Dependence of the magnitude of the critical magnetic field on the accelerating voltage for \(V_p=0\).
1 — calculated curve, 2 — experimental curve.

Taking into account that, for the critical conditions, at

\[ y=b+d_a,\quad x=0,\quad \frac{dy}{dt}=0,\quad V=V_a, \]

we obtain

\[ \frac{1}{2}m\left[\left(\frac{dx}{dt}\right)^2_{x=0}+\left(\frac{dz}{dx}\right)^2_{y=b+d_a}\right]=eV_a. \tag{17} \]

From equation (26)

\[ \frac{dz}{dt}=\omega_c(y-b). \tag{18} \]

The velocity component of an electron executing harmonic oscillations with amplitude \(b\) along the \(x\)-axis will obviously be equal to:

\[ \frac{dx}{dt}=-\omega_0 b\sin\omega_0 t. \tag{19} \]

Substituting (18) and (19) into expression (17), after the necessary transformations, we obtain

\[ B_{\mathrm{cr}}=\sqrt{2\frac{m}{e}}\sqrt{\frac{2b+d_a}{d_a}}\frac{\sqrt{V_a}}{b+d_a}. \]

On the basis of the investigations carried out, the following conclusions may be drawn:

a) The production and use of harmonic oscillations of electrons for generating ultrahigh frequencies is possible in tubes of simple design, not requiring great manufacturing precision.

b) Owing to electronic tuning, the operating frequency of the tube can vary over wide limits; in practice, the generator can cover a frequency range of 1:2 without mechanical retuning.

c) The loaded quality factor of the oscillatory system in generators of this type can be very low, which makes it possible to obtain an electronic-tuning band between the half-power points of the order of 10–15%.

d) The multiple interaction of the electrons with the high-frequency field makes it possible to obtain fairly high efficiencies (of the order of 20–30%).

The advantages indicated make it possible to speak of the advisability of developing tubes of this kind for use in radio communication links with frequency modulation, laboratory signal generators, and other microwave radio-engineering devices.

LITERATURE

1. V. I. Kalinin. Decimeter and Centimeter Waves, vol. I, Moscow, 1939.
2. J. Fogel and S. Braude. ZhETF, XVI, 187, 1946.
3. H. Alfven and D. Romell, Proc. IRE, 42, 1239, 1954.

ERRATA

Izv. TPI, vol. 95.