On Electrical and Mechanical Ferroresonance

A. E. Salomonovich

1. Introduction

The theory of oscillations stands apart among other branches of physics by criteria different from those according to which physics is divided into mechanics, acoustics, optics, the theory of electricity and magnetism, etc. Whereas the latter division is based on grouping phenomena according to the unity of their physical nature, the theory of oscillations groups together physical phenomena that are identical in the form of the laws governing them.

In every field of science, in the words of Acad. L. I. Mandelstam[^1], “it is desirable to single out those guiding points of view which allow us to unite an entire class of problems.” From this point of view, the joint consideration of a whole series of oscillatory phenomena from mechanics, optics, the theory of electromagnetism, etc., proves highly fruitful. Under such consideration, results obtained in one field—for example, in mechanics—may serve not only to enrich the theory of oscillations itself, but also to advance another field—for example, radio engineering. Therefore, one of the guiding ideas of the modern theory of oscillations is the idea of the “oscillatory mutual assistance of various fields of physics and technology.”[^2]

A vivid and most typical example illustrating such an “oscillatory approach” to phenomena from different fields of physics and technology, having the same form of regularities, is provided by resonance phenomena (see, for example[^3]).

The concept of resonance has received an extraordinarily profound deepening and broadening in the theory of nonlinear oscillations, whose development over the last twenty years has been due chiefly to Soviet scientists and, to a large extent, to L. I. Mandelstam, N. D. Papaleksi, and the school of Soviet physicists they created.[^4],[^5]

We encounter one of the distinctive varieties of resonance phenomena, in particular, in studying the behavior of a mechanical or electrical nonlinear system under the action of an external periodic force. In the case where, in this system left to itself, there can occur (in the absence of damping) periodic ...

processes; their frequency, unlike in the linear case, depends not only on the properties of the system, but also on the amplitudes of the oscillations. As we shall see below, resonance phenomena of this kind, known in electrical engineering under the name of ferroresonance, also occur in other fields of physics and technology; it therefore seems expedient to consider them alongside ferroresonance proper.

Our survey is devoted to such a joint consideration of phenomena which, in the indicated sense, are identical with ferroresonance.

2. FORCED OSCILLATIONS IN A CIRCUIT WITH AN IRON CORE

The problem of ferroresonance in the proper sense of the word is posed by considering forced oscillations of the current in an electric circuit consisting of a capacitor, an ohmic resistance, and a self-induction coil inside which an iron core is placed (Fig. 1). Assuming that an external e.m.f. acts on such a circuit, varying with time according to a sinusoidal law (as is often the case in practice), we solve the problem by determining how the current in the circuit depends on time. In doing so, the dependence of the current strength on the parameters of the external force—its frequency and amplitude—as well as on the parameters of the circuit, must be revealed.

Fig. 1.

Kirchhoff’s equation for the circuit shown in Fig. 1 is obtained in the following form:

\[ R\dot q+\frac{q}{C}=E_0\sin pt-\frac{d\Phi}{dt}, \tag{1} \]

where \(q\) is the charge on the capacitor plates, \(C\) is the capacitance, \(R\) is the resistance, \(p\) is the frequency of the external e.m.f., and \(\Phi\) is the magnetic flux through the self-induction coil.

If the current flowing in the circuit is sufficiently small, i.e. there is practically no magnetization of the core by a direct current, and hysteresis phenomena in the core may be neglected, then the magnetic flux crossing the coil may be considered proportional to the current flowing through it, i.e. one may take

\[ \Phi(\dot q)=L\dot q \quad\text{and}\quad \frac{d\Phi(\dot q)}{dt}=L\ddot q \]

where \(L\) is the coefficient of self-induction.

ON ELECTRICAL AND MECHANICAL FERRORESONANCE

In this case Kirchhoff’s equation (1) is linear:

\[ L\ddot q + R\dot q + \frac{q}{C} = E_0 \sin pt \tag{2} \]

and, accordingly, the system described by this equation—the electric circuit—is called linear. The solution of such a linear problem shows that a sinusoidal current flows in the circuit, whose period coincides with the period of the external e.m.f., and whose amplitude is proportional to the amplitude of the external force. The coefficient of proportionality depends on the parameters of the circuit \((L, C, R)\) and on the frequency of the e.m.f. As for the phase of the current, it differs from the phase of the external e.m.f. by an amount that also depends on the parameters of the circuit and on the frequency of the source.

The curve showing the change in the amplitude of the current in the circuit when the frequency of the external e.m.f. is varied—the so-called resonance curve—is given in Fig. 2. When the frequency of the source coincides with the frequency \(\omega_1\), characteristic for the given circuit, the amplitude of the oscillations increases sharply. If the resistance of the circuit becomes ever smaller, the amplitude of the oscillations at the resonance frequency increases without bound, while the resonance frequency itself merges with the frequency of the natural oscillations of the circuit occurring in it when resistance is neglected, i.e. with

\[ \omega_0 = \frac{1}{\sqrt{LC}}. \]

We obtain the classical linear resonance. It is characteristic of linear systems, in which the frequency of the natural oscillations (for \(R = 0\)) is a constant quantity, independent of the amplitude of the oscillations occurring in the system.

Fig. 2.

When the self-induction coil of the circuit is free of an iron core, the law of proportionality between magnetic flux and current is well satisfied within the limits of practical interest. However, in the presence of a core the situation changes. The current flowing through the coil forms inside it a magnetic field which, in turn, produces a flux of magnetic induction (magnetic flux)

\[ \Phi = BS = \mu HS, \]

where \(S\) is the cross-sectional area of the coil, and \(\mu\) is the magnetic permeability of the core (we consider it closed, and neglect leakage). So long as \(\mu = \mathrm{const}\) and does not depend on the magnetic field, the flux is proportional to the field \(H\), and hence also to the current.

Experience shows, however, that \(\mu\) changes with a change in the magnetic field. One of the curves, obtained experimentally\(^6\) for a core in which hysteresis is negligibly small, is shown in Fig. 3. With an appropriate choice of coefficients, the course of the ex-

experimental curves of the type shown in Fig. 3 can be expressed analytically[^7] with sufficient accuracy by the formula

\[ \Phi(\dot q)=A\dot q+B\operatorname{arctg}(a\dot q); \tag{3} \]

where \(A, B, a\) are constants.

With such a dependence of the magnetic flux on the current, proportionality is violated, and the equation becomes nonlinear. Indeed, after differentiating (3) and substituting in (1), we obtain

\[ \left(A+\frac{Ba}{1+(a\dot q)^2}\right)\ddot q+R\dot q+\frac{q}{C}=E_0\sin pt. \tag{4} \]

Instead of the constant coefficient of self-induction \(L\), we now have a factor depending on the current. Figure 4 shows how \(L(\dot q)\) now changes as \(\dot q\) changes. Owing to the nonlinearity of the system described by equation (4), one cannot expect that the results obtained in analyzing such a system will coincide with the results of investigating a linear system—with \(L=\mathrm{const}\).

Fig. 3.

Fig. 4.

In particular, we can no longer assert that, when a sinusoidal e.m.f. acts on our circuit, the current in the circuit will remain sinusoidal. On the contrary, the appearance of harmonics should be expected. The “amplitude” of the current may also prove not to be proportional to the amplitude of the e.m.f., and the resonance curve may lose its symmetric form when the frequency of the applied action is varied on both sides of resonance.

This can be expected because in our nonlinear system the frequency of the free undamped oscillations (for \(R=0\)) depends on the amplitude of the oscillations. The system has ceased to be isochronous. Anharmonic oscillations occur in it, called, rather unfortunately, pseudoharmonic. In our case the “mean coefficient of self-induction” decreases with increasing current amplitudes (see Fig. 4), and consequently the “mean natural frequency” of the circuit increases with the growth of the oscillations. The circuit becomes more “rapid.”

Near resonance, the amplitudes of the oscillations increase sharply with a change in the frequency of the excitation. As resonance is approached, the circuit, owing to the change in its “natural frequency,” either “deviates” from resonance or “moves toward it,” depending on whether we approach resonance from the side of higher or lower frequencies.

These preliminary considerations, which are qualitative in character, will be confirmed by the results of the investigation of the nonlinear equation of our problem obtained above.

3. ESTIMATE OF THE SMALLNESS OF THE PARAMETERS

In order to proceed to a quantitative consideration of the problem of ferroresonance, it is necessary to make certain assumptions about the smallness of the parameters entering into equation (4). Bearing in mind that in what follows we shall apply, for the solution of the nonlinear equation, a method suitable in the case of systems differing little from linear and conservative ones, we shall assume first of all that the nonlinearity of the function expressing the dependence of the magnetic flux on the current is small.

However, before comparing the individual terms of equations (3) and (4), it is necessary to make the coefficients independent of units of measurement (to make them of the same dimensionality). Otherwise, when comparing quantities of different dimensions with one another, we shall obtain different results depending on what units of measurement we use.^8 But in order that all the coefficients have the same dimensionality, it is necessary to make dimensionless the variable quantities appearing in (3) and (4).

Therefore, as the variable we shall take not the charge on the plates of the capacitor, but its ratio to the greatest charge accumulated during the oscillations: \(z=\dfrac{q}{q_0}\). We shall also measure time in dimensionless units, putting \(pt=\tau\). The period of the external e.m.f. will then be equal to unity.

After the indicated changes, equation (4) becomes

\[ \left[A+\frac{Baq_0p^2}{1+(aq_0p\dot z)^2}\right]\ddot z+q_0pR\dot z+\frac{q_0}{C}z=E_0\sin\tau. \tag{5} \]

Now we can formulate the requirement of smallness of the nonlinearity. It has the form:

\[ \chi=a^2 i_0^2\ll 1 \tag{6} \]

and means that the square of the product of the greatest current flowing in the circuit by the coefficient of nonlinearity is much less than unity.

It is precisely this case that we shall consider in what follows. It should be pointed out that the majority of authors who have studied the phenomenon of ferroreso-

... (6, 9, and others), essentially are limited precisely to this case.
When condition (6) is fulfilled,

\[ \frac{1}{1+xz^{2}} \simeq 1-xz^{2}, \]

and, after simple transformations, we obtain instead of (5):

\[ \ddot z+\frac{\omega^{2}}{p^{2}}z= \frac{E'}{L'}(1+l\dot z^{2})\sin\tau -\frac{R'}{pL'}\dot z -\frac{R'l}{pL'}\dot z^{3} -\frac{\omega^{2}l}{p^{2}}\dot z^{2}z, \tag{7} \]

where the following notation has been introduced:

\[ q_{0}(A+B\sigma)=L';\qquad Rq_{0}=R';\qquad \frac{q_{0}}{C}=\frac{1}{C'};\qquad \frac{Bax}{L'}=l;\qquad \frac{E_{0}}{p^{2}}=E'; \]

\[ \frac{1}{L'C'}=\omega^{2}. \]

Considering the circuit near resonance, we shall further put

\[ \frac{\omega^{2}-p^{2}}{p^{2}}=a_{0}. \]

Here \(a_{0}\) is a small relative “detuning” between the frequency of the action and the “natural frequency” of the circuit, coinciding with the frequency of its free oscillations in the absence of damping and nonlinearity. For \(a_{0}>0\) we have an action with a frequency smaller than the “natural” one, and for \(a_{0}<0\)—with a larger one.

Since the resonance phenomenon becomes distinct only when the resistance is decreased, we consider the case when the damping in the circuit is small:

\[ \frac{R'}{pL'}=\mu \ll 1. \]

Near resonance, the amplitude of the oscillations, for small damping, considerably exceeds the amplitude of the action, and it is natural to assume that

\[ \frac{E'}{L'}=A_{0}\ll 1. \]

We shall assume that all these small quantities, namely \(l\), \(a_{0}\), \(A_{0}\), and \(\mu\), are of the same order, i.e., that the ratio of the first three of them to \(\mu\) is of order unity. Such an assumption makes it possible to obtain a solution of the problem with an approximation sufficient for many practical cases and, in any case, to give a complete qualitative picture of the characteristic features that occur in ferroresonance.

We replace equation (7) by an approximate one in which terms of order of smallness no higher than \(\mu\) are retained:

\[ \ddot z+z=\mu\left[A'\sin\tau-az-\dot z-\frac{l}{\mu}\dot z^{2}z\right], \tag{8} \]

where

\[ A'=\frac{A}{\mu}\qquad\text{and}\qquad a=\frac{a_{0}}{\mu}. \]

In order to simplify the equation obtained somewhat, put \(z=Cz_{1}\) and choose \(C\) so as to make equal to unity the multiplier

at \(z^{2}\chi\). Owing to the homogeneity of the remaining terms of the equation, we finally obtain

\[ \ddot z_{1}+z_{1}=\mu\left[A''\sin\tau-az_{1}-\dot z_{1}-z_{1}^{2}z_{1}\right], \tag{9} \]

where \(A''=\dfrac{A'}{C}\) is the amplitude of the external force, \(C^{2}=\dfrac{\mu}{l}\), \(\mu\) is a small parameter, and \(a\) is the detuning. In what follows we omit the subscript at \(z\).

Thus we have obtained the equation of the ferroresonance problem in “canonical” form.

4. THE PROBLEM OF THE VIBRATION GALVANOMETER

Before proceeding to the solution of the problem formulated in the preceding paragraphs, let us consider the problem of the theory of the vibration galvanometer. A work by Appleton\(^{10}\) is devoted to the study of such a galvanometer. The errors made by him were corrected in an article by A. G. Lyubina,\(^{11}\) who examined the whole question exhaustively, using the methods of the nonlinear theory of oscillations.

The equation of motion of the moving system of a vibration galvanometer measuring a sinusoidal current whose amplitude is \(i_{0}\) has the form

\[ I\frac{d^{2}\vartheta}{dt^{2}}+b\frac{d\vartheta}{dt}+M(\vartheta)=G_{0}\sin pt, \tag{10} \]

where \(\vartheta\) is the angle of deflection of the moving system, and the constants \(I\), \(b\), \(G\) characterize the mechanical and electrical properties of the galvanometer. \(M(\vartheta)\)—the restoring moment of the galvanometer—only at very small angles may be considered proportional to the angle of deflection from the equilibrium position. At larger angles \(\vartheta\), as measurements show, the relation

\[ M(\vartheta)=k\vartheta-\gamma\vartheta^{3} = k\vartheta(1-\lambda\vartheta^{2})=k_{1}(\vartheta)\vartheta \tag{11} \]

is valid.

Fig. 5.

With increasing deflection the elasticity of the suspension decreases (Fig. 5). The “natural frequency” of the system decreases. The system becomes more “soft.” When the frequency of the measured current approaches the “natural frequency” of the galvanometer, phenomena analogous to ferroresonance should occur, and the asymmetry of the resonance curves in the two cases should be directed to opposite sides, since the “natural frequency” changes differently with increasing amplitude of oscillations.

Using expression (11) and repeating, with obvious modifications, the course of reasoning of § 2, we obtain the equation of the vibration galvanometer near resonance in the “canonical” form*):

\[ \ddot z+z=\mu\left[A\sin\tau-\dot z-az+z^3\right], \tag{12} \]

where \(A\) is the amplitude of the external force, \(\mu\) is a small parameter proportional to the damping of the system, and \(a\) is the detuning.

Equation (12) is very similar to (9), obtained for ferroresonance. It differs only in the nonlinear term: \(+z^3\) instead of \(-z^2z\).

A whole series of problems on forced oscillations in systems with nonlinear elasticity \(^{12-16}\) are reduced to a similar equation, under the corresponding assumptions on the smallness of the parameters. Here the case is possible of a characteristic of the elastic force shown in Fig. 5, as well as the opposite case, when the stiffness increases with the growth of the amplitude of oscillations (Fig. 6). In this latter case the sign of the nonlinear term in (12) is changed to the opposite. In the more general case considered in 1943 by B. V. Bulgakov \(^{17}\), the elastic force is taken equal to

\[ f(x)=m(\omega^2+\mu)x-U'(x), \]

where the first term gives the linear approximation, and \(U'\), representing the derivative of some function \(U\), is the nonlinear correction. The solution of such a problem is quite analogous to the solution of A. G. Lyubina (the van der Pol method, investigation of the equations by the Poincaré method). For not too large values of \(x\), the function \(U'(x)\) can usually be represented in the form of the first two or three terms of a power series, which reduces this case to the one considered above.

Fig. 6.

5. CIRCUIT WITH A SEIGNETTE-ELECTRIC CAPACITOR

Investigations by I. V. Kurchatov and Kobeko showed \(^{18}\) that Rochelle salt possesses electrical properties analogous to the magnetic properties of iron. Owing to the peculiarities of the polarization of Rochelle salt, the charge on the plates of a capacitor filled with it is not proportional to the applied voltage. The dielectric permittivity (like \(\mu\) of iron) ceases to be constant (Fig. 7). It changes together with the field, and the capacitance of the capacitor turns out to be

*) Eppleton’s error consisted in the fact that, without estimating the order of smallness of the various terms of equation (10), he obtained a superfluous condition of stability of the solution “in the small.” Stability “in the large” was investigated only by A. G. Lyubina (see below).

depending on the applied potential difference. In this case we call the capacitance

\[ C(q)=\frac{q}{V}, \]

where \(q\) is the charge on the plates of the capacitor, and \(V\) is the potential difference caused by this charge.

Figure 8 shows the experimentally obtained dependence, determined in this way, of the capacitance of a capacitor on the charge on its plates.

Fig. 7.

Fig. 8.

Fig. 9.

Neglecting losses in the Rochelle salt, we obtain the equation for the change of the charge on the plates of a ferroelectric capacitor included in a circuit (Fig. 9) together with a sinusoidal source of e.m.f.:

\[ L\ddot q+R\dot q+\frac{q}{C(q)}=E_0\sin pt. \tag{13} \]

For moderate \(q\), the curve of Fig. 8 may be satisfactorily expressed analytically in the form

\[ C(q)=\frac{C_0}{1+C_1q^2}, \tag{14} \]

where \(C_0\) and \(C_1\) are constants. After simple transformations, analogous to those carried out in §§ 3 and 4, we obtain the simplified equation in “canonical” form:

\[ \ddot z+z=\mu\left[A\sin\tau-\dot z-az-z^3\right]. \tag{15} \]

As was to be expected, equation (15) is identical to the equation of the problem of a nonlinear spring whose stiffness increases with the amplitude of the oscillations.

6. RESONANCE PHENOMENA IN THE MOTION OF A RELATIVISTIC PARTICLE IN A CYCLOTRON

In order to exhaust the class of “ferroresonance” problems of mechanics and electrodynamics (for systems with one degree of freedom), let us turn to the problem of the motion of a charged particle under the action of constant magnetic and alternating electric fields—a case occurring in the cyclotron.

A particle of mass \(m\) and charge \(e\), under the action of a constant magnetic field \(\mathbf H\), perpendicular to its velocity at some initial ...

at a given instant of time, performs circular motion governed by the equation \(m\dot{\mathbf v}=\dfrac{e}{c}[\mathbf v\mathbf H]\), where \(c\) is the speed of light. Written in Cartesian coordinates, this equation splits into two:

\[ (a)\quad m\ddot x=\frac{eH}{c}\dot y \quad \text{and} \quad (b)\quad m\ddot y=-\frac{eH}{c}\dot x . \tag{16} \]

After integration we obtain, up to an inessential constant,

\[ \ddot x+\omega_0^2 x=0, \]

where \(\omega_0=\dfrac{eH}{cm}\) is the angular frequency of rotation, independent of the radius of the orbit, which is determined by the initial conditions. Under the action of an alternating electric field applied to the gap (Fig. 10), coinciding in frequency with \(\omega_0\) and in phase with the particle, the particle’s velocity increases and the radius of its orbit increases. However, the energy losses due to radiation and collisions with other particles, which we have not taken into account, may lead to the establishment of a stationary radius of the orbit.

Fig. 10.

As A. A. Andronov and G. S. Gorelik pointed out\(^{19}\), even in the case of a particle of constant mass there occurs not the usual linear resonance, but a nonlinear resonance, since the electric field acts only in the gap between the dees and, consequently, the periodic external force depends on the coordinate, and moreover nonlinearly. Without considering this problem in its general form (for the case of an infinitely narrow gap this was done in the cited work of Andronov and Gorelik), we shall restrict ourselves to the case when the energy imparted by the electric field is small in comparison with the energy stored by the particle. In this case the force acting from the electric field may be regarded approximately as independent of the coordinate of the particle, and, introducing damping proportional to the velocity, we obtain the equation of motion of the particle along the \(x\)-axis near resonance in the form

\[ \ddot x+\omega_0 x+h\dot x=E_0\sin pt. \tag{17} \]

In doing so, of course, we lose the possibility of considering phenomena due to the indicated dependence of the force on the coordinate (resonance at \(\omega_0=\frac{p}{2}, \frac{p}{3}, \ldots\)).

If the velocity of the particle is so great that it becomes necessary to take into account the relativistic change of mass, then we obtain an equation characteristic of ferroresonance. Indeed, putting

\[ m=m_0\left(1-\frac{\dot{x}^2}{c^2}\right)^{-1/2}\simeq m_0(1+\alpha \dot{x}^2), \]

where \(m_0\) is the rest mass, and, regarding damping, detuning, and the amplitude of the acting force as quantities of order \(\mu\), we obtain the equation in “canonical” form

\[ \ddot{z}+z=\mu\,[A\sin\tau-\dot{z}-az+\dot{z}^{\,2}z]. \tag{18} \]

7. SOLUTION OF THE FERRORESONANCE PROBLEM

Let us write again in “canonical” form the four equations obtained above for problems analogous to the ferroresonance problem:

\[ \begin{aligned} \mathrm{I}\quad &\ddot{z}+z=\mu\,[A\sin\tau-\dot{z}-az-z^2z] &&\text{(ferroresonance)}, &&(9)\\ \mathrm{II}\quad &\ddot{z}+z=\mu\,[A\sin\tau-\dot{z}-az+z^3] &&\text{(vibr. galvan.)}, &&(12)\\ \mathrm{III}\quad &\ddot{z}+z=\mu\,[A\sin\tau-\dot{z}-az-z^3] &&\text{(seignettelectric)}, &&(15)\\ \mathrm{IV}\quad &\ddot{z}+z=\mu\,[A\sin\tau-\dot{z}-az+\dot{z}^{\,2}z] &&\text{(relativ. part.)}. &&(18) \end{aligned} \]

We shall solve the first of these, following, in the main, the work of Lyubina\(^{11}\), who considered equation (12).

According to the method of Mandelstam and Papaleksi\(^{20}\), we pass from equation (9) to the corresponding van der Pol equations.

In the phase plane, i.e. in the plane \(z,\dot{z}\), let us take a coordinate system \(x,y\), the origin of which coincides with the origin of the system \(z,\dot{z}\) (Fig. 11), and let us rotate it clockwise with angular velocity equal to 1 (we measure time in units of \(\tau\)).

The formulas for passing from the coordinates \(z,\dot{z}\) to the coordinates \(x,y\) will be

\[ z=x\cos\tau+y\sin\tau, \]

\[ \dot{z}=-x\sin\tau+y\cos\tau. \tag{19} \]

Fig. 11.

For \(\mu=0\), the oscillations of the harmonic oscillator correspond to the motion of the representative point in the plane \(z,\dot{z}\) along a circle having its center at the origin, with angular velocity 1. Hence,

for \(\mu=0\) the representing point is immobile relative to the rotating plane \(x,y\), and each point of this plane for \(\mu=0\) corresponds to a state of equilibrium.

For \(\mu \ne 0\), \(x=x(\tau)\) and \(y=y(\tau)\). Substitute in (9) \(z,\dot z\), and \(\ddot z\), computed with the aid of (19), and seek \(x(\tau)\) and \(y(\tau)\) satisfying this equation.

We obtain
\[ -\dot x\sin\tau+\dot y\cos\tau =\mu\{A\sin\tau-\dot z-az-\dot z^{\,2}z\}, \]
\[ \dot x\cos\tau+\dot y\sin\tau=0, \tag{20} \]
whence
\[ \begin{aligned} \dot x&=-\mu[A\sin\tau-\dot z-az-\dot z^{\,2}z]\sin\tau,\\ \dot y&=\mu[A\sin\tau-\dot z-az-\dot z^{\,2}z]\cos\tau. \end{aligned} \tag{21} \]
Expanding the right-hand sides in Fourier series in \(\cos\tau\) and \(\sin\tau\), we obtain
\[ \dot x=-\mu\left[\frac{\varphi_0(x,y)}{2}+\varphi_2(x,y)\cos2\tau+\overline{\varphi}_2(x,y)\sin2\tau+\ldots\right], \]
\[ \dot y=\mu\left[\frac{\psi_0(x,y)}{2}+\psi_2(x,y)\cos2\tau+\overline{\psi}_2(x,y)\sin2\tau+\ldots\right]. \tag{22} \]
Discarding the terms with \(\cos2\tau,\ldots,\sin2\tau,\ldots\), we obtain the auxiliary or “shortened” van der Pol equations with the discarded “oscillating” terms (see p. 435):
\[ \begin{aligned} \frac{dx}{d\tau} &=\frac{\mu}{2}P(x,y) =-\frac{\mu}{2}\left\{A+x-y\left[a+\frac14(x^2+y^2)\right]\right\},\\ \frac{dy}{d\tau} &=\frac{\mu}{2}Q(x,y) =-\frac{\mu}{2}\left\{y+x\left[a+\frac14(x^2+y^2)\right]\right\}. \end{aligned} \tag{23} \]
To study the motion described by the shortened equations (23), let us first consider the singular points on the phase plane \(x,y\), corresponding to states of equilibrium in the system of coordinates \(x,y\) and, consequently, to periodic oscillations with the frequency of the external force.

To each singular point \(x_0,y_0\) there corresponds the periodic solution
\[ z=x_0\cos\tau+y_0\sin\tau. \]
The investigation of the singular points will thus make it possible to find the amplitudes of the possible periodic motions and to decide the question of their stability. We find the coordinates of the singular points by solving simultaneously the equations
\[ P(x,y)=0,\qquad Q(x,y)=0. \tag{24} \]
We obtain
\[ x_0=-\frac{\rho}{A},\qquad y_0=\frac{\rho}{A}\left(a+\frac{\rho}{4}\right), \]
where \(\rho=x_0^2+y_0^2\) is determined by the equation
\[ \rho\left(a+\frac{\rho}{4}\right)^2+\rho=A^2. \tag{25} \]
For \(A=\mathrm{const}\), equation (25) determines in the plane \(a,\rho\) a resonance curve. For \(a=\mathrm{const}\) this same equation gives the “characteristic”

of the circuit, i.e., the curve of the dependence of the amplitude of the voltage across the capacitor on the amplitude of the external e.m.f. (in the plane \(A^2,\rho\)). Rewriting (25) in the form

\[ a=-\frac{\rho}{4}\pm \sqrt{\frac{A^2}{\rho}-1}, \]

it is easy to construct the resonance curve in the plane \(a,\rho\) (Fig. 12).

We see that the resonance curve differs from the curve of linear resonance (Fig. 2) by its asymmetry, which is the more noticeable the greater the amplitude of the external e.m.f. For sufficiently large \(A^2\), one of the branches of the curve becomes multivalued, which is a characteristic feature of ferroresonance.

The investigation of equations II, III, and IV leads, naturally, to analogous expressions. The solution for case III is identical, up to coefficients, with the solution of case I, while cases II and IV differ from the case considered by us, apart from this, by the sign at \(\frac{\rho}{4}\). Consequently, the resonance curves in these cases are inclined not toward high frequencies, but toward low frequencies. Physically this result is quite understandable. In cases I and III the “natural frequency” of the system increases with amplitude—the system becomes more “rigid” (in case I due to a decrease in inductance, in case III due to a decrease in capacitance). In cases II and IV, however, with increasing amplitude the “natural frequency” of the system decreases—the system becomes more “soft” (in case II as a consequence of a decrease in the stiffness of the suspension, in case IV as a consequence of an increase in the mass of the particle). What exactly caused the change in the natural frequency of the system—the “elastic” or the “inertial” term—does not play an essential role in the resonance properties of the system.

Fig. 12.

8. INVESTIGATION OF STABILITY. HYSTERESIS OF RESONANCE CURVES

Having found the amplitudes of the possible periodic motions with the frequency of the external force, let us determine the character of the stability of these motions. Only stable periodic motions are actually realizable. The stability of the periodic solutions \(z=x_0\cos\tau+y_0\sin\tau\) is determined by the stability of the singular points specified by system (24), i.e., by the stability of the equilibrium positions in the plane \(x,y\). The equilibrium is sta-

Lyapunov-stable if the characteristic equation

\[ \left| \begin{array}{cc} P'_x(x_0,y_0)-S, & P'_y(x_0,y_0)\\ Q'_x(x_0,y_0), & Q'_y(x_0,y_0)-S \end{array} \right|=0 \tag{26} \]

for the system of linear equations of Lyapunov’s first approximation

\[ \left. \begin{aligned} \frac{d\xi}{d\tau} &= \frac{\mu}{2}\,[P'_x(x_0,y_0)\xi+P'_y(x_0,y_0)\eta],\\ \frac{d\eta}{d\tau} &= \frac{\mu}{2}\,[Q'_x(x_0,y_0)\xi+Q'_y(x_0,y_0)\eta] \end{aligned} \right\} \]

(\(\xi=x-x_0,\ \eta=y-y_0\) are small deviations from the equilibrium position corresponding to the singular point with coordinates \(x_0,y_0\)) has roots whose real parts are negative. If at least one of the roots of equation (26) has a positive real part, the equilibrium position is unstable. Indeed, in the first case the deviations decay, while in the second they grow. Calculations for our case give the following characteristic equation:

\[ S^2+2S+\frac{3\rho^2}{16}+a\rho+a^2+1=0. \tag{27} \]

The roots of this equation

\[ S_{1,2}=-1\pm\sqrt{1-\left(\frac{3\rho^2}{16}+a\rho+a^2+1\right)} \]

have negative real parts when

\[ \frac{3\rho^2}{16}+a\rho+a^2+1>0, \]

and one of the roots has a positive real part when

\[ \frac{3\rho^2}{16}+a\rho+a^2+1<0. \]

Consequently, the hyperbola

\[ \frac{3\rho^2}{16}+a\rho+a^2+1=0 \]

with asymptotes

\[ a=-\frac{\rho}{4} \quad\text{and}\quad a=-\frac{3\rho}{4} \]

divides the \(a,\rho\)-plane into regions of stable and unstable (saddle-type) singular points, and hence also into regions of stable and unstable forced oscillations. In turn, the straight lines

\[ a=-\frac{\rho}{4} \quad\text{and}\quad a=-\frac{3\rho}{4} \]

inside the stability region separate the region of stable foci

\[ \left(\frac{3\rho^2}{16}+a\rho+a^2>0\right) \]

from the region of stable nodes

\[ \left(\frac{3\rho^2}{16}+a\rho+a^2<0\right). \]

In the \(a,\rho\)-plane we obtain the picture shown in Fig. 13.

For

\[ A^2=\frac{13}{9}\sqrt{3} \]

the resonance curve touches the boundary of the instability region, and for

\[ A^2>\frac{13}{9}\sqrt{3} \]

the curves become multivalued. On curves intersecting the hyperbola (Fig. 13), to certain values of

ON ELECTRICAL AND MECHANICAL FERRORESONANCE

to detunings \(a\) there correspond three amplitudes of the forced oscillations. In this case the mean value of the amplitude proves to be unstable.

The study of the general picture of the integral curves in the phase plane \(x, y\) shows, however, that when \(A^2 < \dfrac{13}{9}\sqrt{3}\), for any \(a\) there is a single singular point (a stable focus or node). Since limiting cycles are impossible in our system and infinity is unstable (both assertions can be proved), the picture in the phase plane in this case has the form shown in Fig. 14.

Fig. 13.

For \(A^2 > \dfrac{13}{9}\sqrt{3}\) the picture in the phase plane changes substantially depending on the magnitude of the detuning. If the abscissas of the points of intersection of the resonance curve with the hyperbola (Fig. 13) are denoted by \(a_1\) and \(a_2\) \((a_1 < a_2)\), then for \(a < a_1\) or \(a > a_2\) the qualitative picture of the phase plane does not change (Fig. 14): there is one stable singular point, and hence a forced oscillation is possible with only one definite amplitude. If, however, \(a_1 < a < a_2\), then there exist three singular points in the plane—two stable ones (a focus or a node) and one unstable one (a saddle). The picture in the phase plane has the form shown in Fig. 15*).

Fig. 14.

In this case the plane is divided by the integral curves going to the saddle (separatrices) into two spiral-shaped “domains of attraction” of the two stable singular points. A representative point which, owing to the initial conditions, has fallen into one or the other domain will move toward one of the nodes. In the circuit, depending—

* A study of the phase plane for this case, carried out by I. S. Zhukova, eliminated an inaccuracy in the placement of the singular points in the phase plane admitted in [1], and showed the transition to the conservative case (see below).

depending on the initial conditions, forced oscillations of two different amplitudes may be established.

When the detuning is varied, when \(a\) becomes equal to \(a_1\) or \(a_2\), the unstable singular point merges with one of the stable ones and “infects” it with its instability. The representative point, having been in it, passes over into another stable singular point. Turning to the resonance curve, it is easy to understand how the amplitude of oscillations will change when the detuning is varied. If the frequency of the e.m.f. approaches the “natural frequency” of the circuit from the side of higher frequencies \((a < 0)\), the amplitude of the forced oscillations increases up to the point \(a_1\) (Fig. 16), where a jump-like change of amplitude occurs. With a further decrease of frequency the amplitude decreases smoothly. Let us note that the point at which the frequency of the external e.m.f. coincides with the “natural frequency” of the circuit is in no way remarkable. With the reverse increase of frequency the amplitude grows smoothly up to the frequency at which a secondary jump occurs, this time downward, after which the amplitude continues to decrease smoothly.

Fig. 15.

Fig. 16.

Such a “hysteretic” character of the resonance curves in ferroresonance has repeatedly been observed experimentally in a number of the cases enumerated above (I, II, III) for a sufficiently large amplitude of the external force. As an example, Fig. 17 gives a curve from the work of Martinsen\(^9\). Figure 18 shows an oscillogram of the current in the circuit at the moment of the jump. The deviation of the curve from a sinusoid illustrates the appearance of harmonics owing to the nonlinearity of the oscillatory system, for which the principle of superposition no longer applies. The method considered by us makes it possible to calculate easily only the fundamental harmonic of the forced oscillation [we have discarded the “oscillating” terms in equation (22)].

A rigorous analysis shows that the original equation (9) has stable periodic solutions, all the closer to the solutions of the shortened equations obtained by us the smaller \(\mu\) is (see, for example\(^8\),

p. 455); the process of establishing stationary oscillations occurs in such a way that, for sufficiently small \(\mu\), the obtained solution of the shortened equations differs from the solution of the unshortened equation, satisfying the same initial conditions, by an arbitrarily small amount over an arbitrarily large interval of time.\(^{20}\)

Fig. 17.

To take into account the influence of the ohmic resistance of the circuit on its ferroresonance properties, it is sufficient to take the small damping of the circuit equal not to \(\mu\), but to \(h\rho\), where \(h \sim 1\). In this case the equation of the resonance curve (25) is transformed into

\[ \rho\left(\alpha+\frac{\rho}{4}\right)^2+h^2\rho=A^2, \tag{25a} \]

and the characteristic equation (27) into

\[ S^2+2hS+\frac{3\rho^2}{16}+\alpha\rho+a^2+h^2=0, \tag{27a} \]

which leads to the expression for the boundary of the region of instability

\[ \frac{3\rho^2}{16}+\alpha\rho+a^2=-h^2. \]

Fig. 18.

With increasing damping of the circuit, the asymmetry of the resonance curves is smoothed out, and the appearance of an unstable regime occurs at larger amplitudes of the external force.

With decreasing damping, the spiral-shaped region on the phase plane (Fig. 15) narrows and, in the limit as \(h=0\), the picture becomes the one obtained by B. V. Bulgakov for the case of forced oscillations in a nonlinear conservative system considered by him and shown in Fig. 19.

Fig. 19.

The closed curves on the phase plane of Fig. 19 correspond in this case to beats between forced and free oscillations. The latter do not decay because of the absence of resistance in the circuit. Depending on the initial conditions, the beats occur about one of the two possible values of the amplitude of the forced oscillations, to which there correspond on the phase plane \(x,y\) special points of center type. The point represented is then located either in the hatched region or outside it.

9. FERRORESONANCE IN THE PRESENCE OF BIAS MAGNETIZATION

It is of interest to consider the phenomenon of ferroresonance in the case where the core is biased by a direct current.

In Fig. 20 a diagram of a circuit with bias magnetization is shown. In contrast to expression (3), the magnetic flux is given by the formula

\[ \Phi(q)=Aq+B\operatorname{arctg}(aq+bI), \tag{28} \]

where \(I\) is the strength of the direct bias current. With such a dependence of the flux on the current,

\[ \frac{d\Phi}{dt} = \left[ A+\frac{Ba}{1+(a\dot q+bI)^2} \right]\ddot q = L_1(\dot q)\ddot q, \tag{29} \]

and the matter reduces to shifting the curve \(L(\dot q)\) (Fig. 4) to the right of the ordinate axis by an amount proportional to \(I\) (Fig. 21). Since we are considering the fundamental harmonic of the forced periodic oscillation of the current and are not interested in higher-order harmonics, for our investigation only the symmetric part of this curve is essential (⁸, p. 502), i.e. the curve

\[ \frac{L_1(\dot q)+L_1(-\dot q)}{2}, \]

shown in Fig. 22. Indeed, all the odd terms of the expansion of \(L_1(\dot q)\)

in the power series drop out of the shortened van der Pol equations and do not leave an imprint on the final result for the fundamental harmonic. The symmetric part of the curve \(L_1(\dot q)\), however, is in our case approximated with sufficient

Fig. 20.

Fig. 21.

degree of accuracy by the expression

\[ L_1^*(\dot q)=L'(1+l'\dot q^2-m'\dot q^4). \]

Substituting it into (29)*) and repeating all the arguments of § 3, we obtain, after discarding terms of order of smallness higher than \(\mu\),

\[ \ddot z+z=\mu\left[A\sin x-\dot z-az+\dot z^{\,2}z-m\dot z^{\,4}z\right], \tag{30} \]

where \(m\sim 1\). For \(m=0\) we obtain case IV.

This equation is characteristic of all cases of “asymmetric nonlinearity.” A similar equation is obtained for the problem of a ferroelectric capacitor with constant “subelectricization” (see, for example, \({}^{8}\), p. 133), or the problem of nonlinear elasticity having a quadratic term in the expansion of the force with respect to displacement (Fig. 23). Systems with nonlinearities of this kind behave in such a way that, as the amplitudes of the oscillations increase, their “mean frequency” first decreases and then increases again (a core with bias mag-

Fig. 22.

Fig. 23.

*) Instead of the original circuit we consider another one, for which the “coefficient of self-induction” is equal to \(L_1^*(\dot q)\), knowing in advance that the final results for both circuits coincide.

nancing), or, conversely, first increases and then decreases (the spring in Fig. 23). We may, consequently, expect that the resonance curves in these cases will exhibit asymmetry twice—at small amplitudes in one direction, at large amplitudes in the other. The regions of instability may also split, with jumps appearing on both sides of the resonance. Investigation by means of the van der Pol equations confirms these assumptions.

For the amplitudes of the periodic motion we obtain the relation

\[ \rho\left(a-\frac{\rho}{4}+\frac{m\rho^2}{16}\right)^2+\rho=A^2, \tag{31} \]

where, as before, \(\rho=x_0^2+y_0^2\). For \(A^2=\mathrm{const}\) we obtain the resonance curve

\[ a=\frac{\rho}{4}-\frac{m\rho^2}{16}\pm \sqrt{\frac{A^2}{\rho}-1}. \tag{32} \]

Fig. 24.

In the plane \(a,\rho\) (Fig. 24) two such curves for different \(A^2\) are shown by heavy lines. At small amplitudes \(A^2\) the curves are skewed toward the low frequencies; then there appears (for a certain value of the parameter \(m\)) a region of instability. With a further increase of \(A^2\) the instability is eliminated, but a skewness of the resonance curves toward the high frequencies appears. For sufficiently large \(A^2\) a region of instability again appears, this time without disappearing.

Investigation of the phase plane, carried out by the method indicated above, shows that in this case the characteristic equation has the form

\[ S^2+2S+\left(a_1-\frac{\rho}{4}+\frac{m\rho^2}{16}\right) \left(a-\frac{3\rho}{4}+\frac{5m\rho^2}{16}\right)+1=0, \tag{33} \]

and, hence, the boundary of the region of instability is represented in the plane \(a,\rho\) by the fourth-order curve

\[ \left(a-\frac{\rho}{4}+\frac{m\rho^2}{16}\right) \left(a-\frac{3\rho}{4}+\frac{5m\rho^2}{16}\right)+1=0. \tag{34} \]

The boundary between the regions of stable foci and stable nodes is

\[ \left(a-\frac{\rho}{4}+\frac{m\rho^2}{16}\right) \left(a-\frac{3\rho}{4}+\frac{5m\rho^2}{16}\right)=0. \tag{35} \]

represents two parabolas intersecting at points with coordinates \(a_1=0,\ \rho_1=0\) and \(a_2=\frac{1}{4m},\ \rho_2=-\frac{2}{m}\). The coordinates of the points of the curve (34) at which the tangent to this curve is horizontal are found as the simultaneous solution of equation (34) and of the equation of the curve \(\frac{d\rho}{da}=0\), which leads to the quadratic equation in \(\rho\)

\[ a-\frac{\rho}{2}+\frac{3m\rho^2}{16}=0, \tag{36} \]

i.e., to the equation of a parabola intersecting the parabolas (35) at their common points. The three real positive roots \(\rho_1,\rho_2,\rho_3\) (having physical meaning) of the equation

\[ \rho^2\left(1-\frac{m\rho}{2}\right)^2-16=0, \tag{37} \]

obtained as a result of eliminating \(a\) from (34) and (35), give the required ordinates

\[ \rho_1=\frac{1}{m}+\sqrt{\frac{1}{m^2}+\frac{8}{m}}; \qquad \rho_{2,3}=\frac{1}{m}\pm\sqrt{\frac{1}{m^2}-\frac{8}{m}}, \]

the latter two only for \(m<\frac{1}{8}\). In a similar way one can find the ordinates of the points of the curve (34) at which the tangents to the curve are vertical. The results of the analysis show that for small \(m\) \(\left(m<\frac{1}{8}\right)\) we have two separate regions of instability—one closed, and the other with infinite branches.

Fig. 25.

For large \(m\) \(\left(m>\frac{1}{8}\right)\), the first region contracts to a point, while the second, by its vertex, approaches the axis of abscissas. As \(m\to0\) the second region recedes to infinity, while the first, expanding, passes (for \(m=0\)) into a hyperbolic one with asymptotes \(a=-\frac{\rho}{4}\) and \(a=\frac{3\rho}{4}\).

In the plane \(a,\rho\), for not too large \(m\) we obtain the picture shown in Fig. 25. In Fig. 26 are given experimentally recorded ferroresonance curves in the presence of a const-

current.^6 Their shape agrees well with the curves obtained as a result of our consideration. In analogous problems the whole picture either does not change qualitatively at all, or else rotates about the ordinate axis, depending on how the “mean frequency” of the system changes with increasing amplitude of the oscillations.

Fig. 26.

10. FERRORESONANT STABILIZERS

In conclusion let us consider the physical principles of operation of ferroresonant voltage stabilizers, which have become widely used.

A ferroresonant stabilizer is a device whose principal element is a circuit with a choke operating near saturation of the core. Depending on the method of connecting the circuit (according to the voltage-resonance scheme or according to the current-resonance scheme), two different stabilizer designs are obtained. The first, the so-called Kéinat circuit, described by V. V. Kovalevskaya,^21 is shown in Fig. 27. The second, using current resonance, was considered in works by E. V. Sazanov^22,23 and is shown in Fig. 28.

Fig. 27.

Fig. 28.

To clarify the principle of operation of ferroresonant stabilizers of these types, let us turn to formula (25) of § 7, which expresses the dependence of the square of the voltage amplitude on the circuit capacitor (Fig. 1) on the square of the amplitude of the external e.m.f. and on the detuning (in relative units).

In contrast to linear resonance, for which $\rho$ is always proportional to $A^2$ (at constant detuning), in the case of ferroresonance

ON ELECTRICAL AND MECHANICAL FERRORESONANCE

with increasing \(A^2\), the resonance curves shift toward higher frequencies, and \(\rho\) (for \(a=\mathrm{const}\)) increases very little. The greater the negative detuning, the more noticeable this feature is.

However, for a sufficiently large fixed \(a<0\), as \(A^2\) decreases we enter the region of instability, and this sets a limit to the use of ferroresonance for stabilization. The whole picture is especially clear when considering curve (25) in the plane \(A^2,\rho\) at constant detuning \(a\).

For various fixed detunings, curves (25) give a family of circuit characteristics (Fig. 29). In the circuit of Fig. 27, the input voltage \(U_1\), which is to be stabilized, plays the role of the external e.m.f. of the series circuit. The voltage \(U_2\) taken from the circuit is composed of the difference between the voltage taken from the choke and a small part (10–20%) of the voltage taken from the capacitor (the transformer \(T\) is step-down). Thus, \(U_2\), in a first approximation, is proportional to the current in the circuit, i.e. proportional to \(\sqrt{\rho}\).

Fig. 29.

For a definite detuning (for example, \(a_3\), Fig. 29), the output voltage \((\simeq \sqrt{\rho})\), beginning with a certain minimum input voltage \((\simeq A_1)\), called “critical,” depends very little on changes of \(A_1\), i.e. on changes of \(U_1\). For example, when \(U_1\) fluctuates by 10–20%, \(U_2\) in some circuits varies within 0.2–0.5%. A small increase of \(U_2\) is compensated by means of the transformer \(T\). Lowering the input voltage to a value below the critical one leads to instability of the stabilizer and to voltage jumps: at \(A=A_2\), the output voltage drops abruptly from \(\sqrt{\rho_1}\) to \(\sqrt{\rho_2}\).

The stabilizer can operate stably only at input voltages greater than the critical voltage. As the load increases, this critical voltage rises and, at a certain value of it, becomes greater than the nominal voltage to be stabilized.

In the operation of the circuit of Fig. 28, the inclusion of a choke with an air gap is essential; in it the self-inductance may be regarded as constant. In the working part of the characteristic (Fig. 29), an increase of the total current through the ferroresonant circuit almost does not lead to an increase of the voltage falling across it, which is equivalent to a reduc-

...to a decrease in the resistance of the circuit as the input voltage increases. The excess input voltage therefore drops across the choke with an air gap. The additional winding of this choke plays a role analogous to that of the transformer \(T\) in the circuit of Fig. 27.

A serious disadvantage of ferroresonant stabilizers is the strong dependence of their characteristics on the frequency of the stabilized voltage. Therefore their use in networks with considerable frequency fluctuations (low-power alternators) is inadvisable. It must also be taken into account that the waveform of the voltage taken from the ferroresonant circuit differs greatly from a sinusoidal one, owing to the appearance of harmonics. In some cases the amplitude of the third harmonic reaches 35%1. To improve the waveform of the stabilized voltage, when this is necessary, the harmonics have to be filtered out by connecting circuits tuned to the frequency of these harmonics.

A number of voltage-stabilizer circuits based on the use of ferroresonance have been described in a considerable number of works (24–29, etc.).

For stabilization purposes, circuits without capacitors are also used; these constitute the case of degenerate nonlinear systems. Stabilizers are also possible in which the nonlinear element of the circuit is not a saturated choke, but a capacitor with a ferroelectric dielectric (see § 5).

CITED LITERATURE

1. L. I. Mandelstam, UFN, 13, issue 2, 1933.
2. A. A. Andronov, Izv. AN, physical series, 9, Nos. 1–2, 1945.
3. G. S. Gorelik, ibid.
4. L. I. Mandelstam, N. D. Papaleksi et al., New Investigations of Nonlinear Oscillations, Svyazizdat, 1936.
5. N. D. Papaleksi, “Evolution of the Concept of Resonance,” UFN, 31, 447 (1947); “Nonlinear Oscillations,” jubilee collection of the Academy of Sciences of the USSR dedicated to the 30th anniversary of the Great October Socialist Revolution.
6. H. Schunke u. L. Zenneck, Jahrbuch d. drahtl. Telegr., 19, 170, 1922.
7. L. Dreyfuß, Arch. f. Electrotechnik, 2, 343, 1913.
8. A. A. Andronov and S. E. Khaikin, Theory of Oscillations, GTTI, 1937.
9. O. Martienssen, Phys. Zeits., 11, No. 10, 448, 1910.
10. E. Appleton, Phyl. Mag. [6], 47, 609, 1924.
11. A. G. Lyubina, ZhETF, issue 8, 1934.
12. G. Duffing, Erzwungene Schwingungen bei veränderl. Eigenfrequenz und ihre techn. Bedeutung, Braunschweig, 1918.
13. J. Horn, Arch. f. Math. u. Phys., 28, 1920.
14. Rüdenberg, Zeits. f. angew. Math. u. Mech., 3, issue 6, 1923.
15. J. P. Den Hartog and S. J. Mikina, Trans. ASME, 54, 1932.
16. J. P. Den Hartog, J. of the Frankl. Inst., 216, 1933.
17. B. V. Bulgakov, Applied Mathematics and Mechanics, 7, No. 1, 1943.
18. I. V. Kurchatov, Ferroelectrics. Series “Problems of Recent Physics,” GTTI, 1933.

1. A. A. Andronov and G. S. Gorelik, DAN, 49, 664, 1945.
2. L. I. Mandelstam and N. D. Papaleksi, ZhETF, 4, issue 2, 1934.
3. V. V. Kovalevskaya, Izv. Elektroprom. Sl. Toka, 8—9, 63, 1938.
4. E. V. Sazanov, Izv. Elektroprom. Sl. Toka, 12, 30, 1937.
5. E. V. Sazanov, Izv. Elektroprom. Sl. Toka, 1, 44, 1939.
6. G. Keinath, Die Technik electrischer Messgeräte, vol. 2, 1928.
7. A. Sonlier, Rev. Gén. El., No. 4, 196, 1928.
8. I. Peskis, Radiofront, No. 2, 1934.
9. R. Greiner, EFZ, 18, 489, 1936.
10. Way, Electronics, 7, 14, 1937.
11. A. Shpigler, Tekhnika svyazi, No. 1, 44, 1937.

1. Reference number as printed in the source. ↩