Mandelstam-Papalexi - Report on Recent Research on Nonlinear Oscillations - 1935.pdf

Mandelstam-Papalexi - Report on Recent Research on Nonlinear Oscillations - 1935.pdf
81
Original Title: “Expose Des Recherches Recentes Sur Les Oscillations Non Lineaires”
Published In: Technical Physics of the USSR, Leningrad; Volume 2, Number 2-3, pp 81 – 134, 1935.
Original English Translation: Techtran Corporation, NASA TT F-12,678 (59 pages). November 1969
Revised, Edited and Enhanced Translation: © Philip Pesavento, November 2005
REPORT ON RECENT RESEARCH ON
NONLINEAR OSCILLATIONS1
L. Mandelstam, N. Papalexi, A. Andronov, S. Chaikin and A. Witt
Introduction
Since Dr. Balthazar Van der Pol, in his capacity as the President of the Physics Section of
the International Congress of the ISRU, has invited us to provide a summary of recent
research on nonlinear oscillations performed at the Physics Institute of Moscow
University, at the Central Leningrad Radio Laboratory, at the Electrophysics Institute of
Leningrad and at Gorki University, and since a large part of our work in this important
field is closely related to the basic research and experiments of Dr. Van der Pol, it gives
us great pleasure to submit this report. We will only be able to discuss a limited portion
of this subject. So, we will be reporting on some general concepts that guided us during
our research, and also on some of our more interesting experimental results as well.
Until rather recently the predominant theory of oscillations dealt with so-called linear
systems (small perturbations of mechanical systems having a finite number of degrees of
freedom, electronic circuits, and classical problems with boundary conditions). At the
present time more interest is being focused on nonlinear systems in the various fields of
pure and applied science (mechanics, acoustics, biology2 and above all, since the
invention of the electron tube, radio engineering).
The systems presently being used in radio engineering for transmitting and receiving are
essentially nonlinear, and this is by no means an accidental condition. It is sufficient to
merely examine – let us take the simplest example – a triode oscillator, in order to see
that an autonomous linear system, i.e. a device in which the current and voltage are
governed by linear differential equations in which time does not explicitly enter,3 cannot
have the properties that are possessed by, or need to be possessed by, a radio transmitter.
(We shall confine ourselves to autonomous systems in order to exclude those whose
oscillations are transmitted from outside the system. Naturally, when a system receives
oscillations from an external source the question immediately arises as to how
1
2
3
This report was delivered to the Radiophysics Section of the General Assembly of the International
Scientific Radio Union (ISRU), London 12-18 September 1934.
B. van der Pol, J. van der Mark; Philosophical Magazine, 1928: “The heartbeat considered as a relaxation
oscillation and an electrical model of the heart”.
We define as autonomous any system whose differential equations do not contain any explicit reference
to time.
82
oscillations from this source are produced. In order to discuss the problem of the
emission of oscillations, systems functioning without external electromotive forces
should be observed). Indeed the basic property of linear systems is that the amplitude is
not intrinsic to the system, but depends entirely on the initial conditions. Now the
distinguishing characteristic of modern oscillating devices is that, independent of the
initial conditions, an oscillatory operating mode is established whose amplitude and
frequency are completely defined. This is why modern radio engineering has had to call
upon physical concepts and a mathematical approach that are able to cope with nonlinear
systems. The great diversity of phenomena that are revealed in nonlinear systems makes
them extremely interesting from a purely physical standpoint. It is their diversity as well
as their flexibility which has enabled the broad applications that these systems have
received over the course of the last few years.
Since the study of nonlinear differential equations is much more difficult and complicated
than that of linear equations, the tendency from the beginning has naturally been to
“linearize” the problems, i.e. to treat the essentially nonlinear problems from a linear
viewpoint. It cannot be denied that, in order to clear up some aspects of the known
phenomena, such a method (linearization) can sometimes have its utility. However, since
it is still incomplete, artificial, and requires complementary ad-hoc hypotheses, this
method of linearization often leads to errors. One of these errors, which is still
encountered quite often, is referred to below4.
After nonlinear systems had completely dominated the field of practical applications,
they began to reveal phenomena absolutely foreign to linear systems, and inspired the
search for a mathematical approach that could cope with these new phenomena. Very
soon, publications appeared which purposely began from a nonlinear viewpoint. These
are, primarily, the remarkable works of Van der Pol to which we will have occasion to
refer several times. As has been stated, the results that they contain are of fundamental
importance to the entire field in which we are involved. However, and this was
reasonable, the first works had the production of tangible results as their goal rather than
to develop a general and rigorous theory. Thus, for example, the existence of periodic
solutions was accepted as an assumption. Series were often used whose convergence was
questionable. Nevertheless, and let us emphasize this, the results obtained were good.
After using these methods (which originally owed their existence to the works of Van der
Pol) for some time, a great quantity of valuable results were produced, and it was natural
to pass on to more general points of view and look for a mathematical approach that
could cope with nonlinear problems. It is in this direction that a portion of our work has
been executed.
4
See page 87.
83
It turned out that the mathematical approach that could deal with nonlinear oscillation
problems had been in existence for a long time. On the one hand, it was contained in the
famous works of Henri Poincaré [1], [2], and on the other hand, in the remarkable
investigations of Liapunov [3].
The relationship existing between the works of Poincaré carried forward by Birkhoff [4],
as well as those of Liapunov and our present physical problems was indicated out by one
of us [5]. Three things should be pointed out. First of all, the qualitative theory of
differential equations, developed earlier by Poincaré, has turned out to be very effective
for the qualitative discussion of physical phenomena taking place in systems presently
used in radio engineering. Nevertheless, neither the physicist nor, with greater reason, the
engineer can be happy with a qualitative analysis. The later works of Poincaré supplied
an approach that allowed the treatment of our problems on a quantitative basis. Finally,
the works of Liapunov permit applying questions of stability to the mathematical
discussion.
The first part of this report summarizes these mathematical theories and shows how we
can apply them to our present problems. We shall virtually disregard questions of
stability. The second part of this report discusses the theoretical and experimental aspects
of some concrete questions. These partially involve problems for which we have
perfected and supplied results, previously obtained by other authors, with a strict
mathematical foundation. In addition, we shall examine resonance phenomena of the n-th
degree and shall concisely report certain experiments and theoretical considerations on
the so-called parametric excitation phenomena.
We will conclude by making several observations on the role of statistics in oscillatory
phenomena.
Section 1: Geometric Presentation of the Motion of an Oscillatory System; The
Phase Plane
There is no question but that the mathematical methods concerned here are noticeably
more complicated and difficult than those used to study linear systems. This arises from
the very nature of the physical problems, which are far from being simple. Also, there is
no doubt that the characteristic features of this approach will prove capable of dealing
with nonlinear systems on a theoretical as well as a practical basis. It is our position that a
mathematical approach can only be charged with being cumbersome and
overcomplicated when it leads to a result after a long succession of operations in which
each operation, taken separately, has no physical interpretation. Now, this is not at all the
case in the geometric approach connected with the name of Poincaré. Here, each
geometrical component possesses a direct physical meaning. This is why this geometrical
approach, though complicated, is far from being an obstacle. It actually simplifies the
description and understanding of the physical phenomena involved.
84
This well-founded method, which consists of showing the operation of an oscillating
system by using a geometric figure, has been used in science for quite some time. The
idea is, essentially, as follows. In order to characterize the state of a system with N
degrees of freedom it is necessary to provide 2N numbers (N coordinates and N
velocities). These 2N numbers can be considered as specifying the position of a point in
space with 2N dimensions. To each point of this space there corresponds a certain state (a
certain “phase”) of the system. This is why this space is called “extension in phase”. In
the case of a system with one degree of freedom, this space has two dimensions. In the
simplest case it is a plane.
Let us take the simplest example of a harmonic oscillator. Its equation has the form:
..
x + ωo2 x
= 0
(1)
or
.
y = − ωo2 x
.
x = y





(2)
These equations, likewise, describe an electrical circuit that has capacitance and selfinduction (but no resistance) if, for example, x represents the charge on the condenser.
We will describe the behavior of the oscillator on a plane related to the rectangular axes
.
x , x (the voltage-current plane). This will be the phase plane. Each new state of the
system corresponds to a new “representative” point on the phase plane. A succession of
states of the system corresponds to a movement of the representative point on the phase
plane, or phase trajectory.
Planck has familiarized physicists with the phase trajectories of the harmonic oscillator.
They form a family of ellipses, each enclosing the other and having a common center as
.
the origin. The equation x = y indicates that the representative point is moving in a
clockwise direction. The origin can be considered as an ellipse that has degenerated into a
point.
In the case of equations (2), the point x = 0, y = 0 is a “singular point”, since at this point
dy/dx = 0/0 and the direction of the phase trajectories is indeterminate. According to the
description acceptable in mathematics, a singular point surrounded by a family of ellipses
may be termed the center. It is clear that in the instance of our subject, singular points are
of special interest. In order for dy/dx = 0/0, i.e. a singular point, it is sufficient that dy/dt
= 0 and dx/dt = 0. However, if the current and voltage in a simple circuit are
simultaneously equal to zero, the system is in equilibrium. Therefore, all the states of
equilibrium of the system under study are represented by the singular points of the
differential equations (2).
Each ellipse and each closed trajectory represents a periodic phenomenon corresponding
to certain suitably selected initial conditions. The origin represents a stable state of
equilibrium in the sense that any small disturbance remains small.
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In the case under consideration, it is clear that, regardless of the initial conditions, the
system describes a periodic motion, except in the situation where the initial conditions
correspond to the coordinate origin.
In the same manner, we present the phase plane for a damped oscillator that is governed
by the equation:
..
.
x + 2 h x + ωo2 x
= 0
(3)





(4)
or
.
y = 2 h y − ωo2 x
.
x = y
We shall assume, first of all, that h2 < (ω0) 2 . Since x is the voltage at the condenser
terminals, these equations describe an oscillating circuit that has resistance. All the
integral curves are spirals that are
wound asymptotically with the origin of the coordinates (Figure 1). Each one of the
spirals represents a damped oscillation corresponding to suitable initial conditions. Like
the preceding example, the origin represents a state of equilibrium and is found to be a
singular point, although of a new type. The singular points that are used as asymptotic
points with a family of spirals are called the focus. In our case, the singular point is stable
and is therefore called a stable focus.
Giving consideration to the phase plane of a system governed by equations (3) or (4), but
whose damping is substantial enough for h2 > (ω0) 2 to be true, we shall see parabolic
curves passing through the origin substituted for the spirals (Figure 2). As in the
proceeding example, the origin represents a state of equilibrium. It is a singular point of
the type called a node. Integral curves reflect the aperiodic movement of the system
towards the state of equilibrium. Therefore, we have a stable node.
In addition to the center, the focus, and the node, there also exists an important type of
singular point for us: this is the saddle (Figure 3). The saddle represents, for example, the
upper equilibrium position of a pendulum.
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The behavior of integral curves in the vicinity of the saddle shows that the system always
ends by diverging. A saddle is, therefore, always unstable.
In the case of a triode oscillator whose oscillating circuit is inserted between the filament
and the grid, the voltage v at the condenser terminals satisfies the equation:
..
.
Lv + Rv +
1
v =
C
M
C
 .
f  v , v




(4)
or
dv
dt
di
dt
=
i
C
= − ωo2 v + f (v , i )
The form of the function f (v, i) is provided by the plate characteristic of the tube (for
simplicity’s sake, we disregard the grid current). The mathematical discussion of this
equation on the phase plane gives us the following picture. In the case of a soft mode and
a weak excitation (M small), the integral curves are spirals which uncoil in a closed
curve. Some approach one another from the outside, coming from infinity. Others
approach one another from the inside, unwinding beginning from the point of origin
(Figure 4, 16). It is easy to establish the relationship between the essential lines of the
geometrical figure and those of the physical system. The origin of the coordinates still
represents an equilibrium state; it is an unstable focus. In the circuit we will observe the
appearance of oscillations whose amplitude will gradually increase, even if the values of
voltage and current differ very little from zero (as will be the case, for example, if the
initial perturbation i0,v0 is produced by random fluctuations). After some time, the
increase will slow down and then stop, and we shall see the setting up of a stationary
oscillatory mode of operation, which may be depicted on the phase plane by a closed
curve. If the initial conditions correspond to a point located outside of the closed curve,
the circuit will oscillate with decreasing amplitude until a stationary mode of operation is
established.
87
The closed curves, on which the integral curves are wound, or from where they are
unwound, are the limiting cycles of Poincaré. This mathematical concept has a very
simple physical interpretation: the limiting cycles depict periodic stationary modes of
operation. In the same way as with singular points, the limiting cycles can be stable (if all
of the neighboring integral curves come closer) or unstable (if all of the neighboring
integral curves diverge). It is clear that only the stable limiting cycles represent the actual
periodic motion of a physical system.
If a strong excitation M is given in the same case as the previous “soft” mode of
operation, there will still exist a limiting cycle with which the other integral curves will
be wound from the inside and from the outside since the property was suitably selected.
Nevertheless, in the vicinity of an equilibrium state the behavior of the system will be
essentially different. The integral curves will diverge from the singular point in an
aperiodic rather than an oscillatory manner, and the singular point will be an unstable
node. The presence of the cycle shows that, no matter what the initial conditions may be,
there will definitely be a well-defined periodic “amplitude” phenomenon. This periodic
phenomenon will, again, be independent of the initial conditions. However, the transitory
phenomenon assumes a different characteristic than in the case of a small excitation. It
should be sufficient to study the beginning of the transitory phenomenon of a linear
idealization in the case where the initial perturbation is assumed to be small.
Nevertheless, in the case of an unstable node, the absence of oscillatory phenomena in a
linear system does not permit one to conclude anything concerning the absence of
periodic oscillatory phenomena out at a distance from the state of equilibrium, i.e. - out at
some point where the system cannot be considered as linear.
The case of the node enclosed in a limiting stable cycle is the most striking example of
the lack of capability of linear methods to decide on the existence of periodic phenomena
in a self-exciting system. Furthermore, if this circumstance is disregarded, it is possible to
commit a serious error, as has been done by several authors……
The singular points and the limiting cycles constitute the geometric components
characterizing, to a certain degree, stationary movements in the systems. According to
Poincaré, the knowledge of these components is enough to judge the properties of all
other movements. The coexistence of these components is, likewise, controlled by
general topological laws. Also, if the properties of one of these components is known, it
is often possible to deduce the existence of the others. If, for example, far from the point
of origin, all the integral curves converge towards the origin, with the latter being an
unstable focus or node, and, provided that there are no other singular points, there exists
at least one stable limiting cycle. If there exist several limiting cycles enclosing one
another and between which there are no singular points, then there is an alternation of
stable and unstable cycles. From the point of view of “Analysis Situs” (topology), these
statements are almost syllogisms. Nevertheless, the physical phenomena corresponding to
these geometric properties are far from being trivial. The qualitative theory of Poincaré is
so valuable because it permits the formation of an overview of the physical phenomenon
by a relatively simple analysis.
88
The following is a very simple example. Although without practical significance, it
clearly illustrates what has been said. Let us assume that the characteristic of the tube has
the shape that can be seen in Figure
5. Under what conditions will the
oscillator have stationary (periodic)
oscillations? In order to know the
behavior of the system at infinity, it
is clearly possible to assume that the
operational point is at the apex of the
angle formed by the two rectilinear
parts of the characteristic. Depending
on the slope of the inclined part, two
cases can be seen: either the cycle at
infinity will be stable, i.e., all the
integral curves will go towards
infinity (if the slope exceeds a
certain critical value), or else the
cycle at infinity will be unstable (if
the slope is less than this critical
value). It can easily be seen that it is
only possible to have one finite limited cycle for finite values of the slope. If the
operational point is located in the horizontal part of the characteristic, then the origin is a
stable point of equilibrium and, consequently, owing to general topological laws, there
can be no limiting cycle. Therefore, it is not possible to have oscillations. If the
operational point is located on the inclined portion of the characteristic, three cases can
occur. If the singular point is stable, there are no oscillations. If the slope increases, the
singular point becomes unstable. If the cycle at infinity is likewise unstable, the
oscillations certainly continue to exist. However, when the slope increases beyond a
certain critical value, the cycle at infinity becomes stable and oscillations again become
impossible.
Section 2: Analytical Methods for the Study of Nonlinear Systems
The general qualitative theory of differential equations, partially discussed above, is still
in the developmental stage. It allows analysis, however incomplete, in the case of two
and possibly three autonomous equations (incomplete analysis in this last case), provided
that the second terms are either polynomials of not too high a degree (third, fifth), or
functions that can be geometrically characterized with sufficient simplicity. However, the
radio engineer cannot be content with a qualitative study of the problem. He requires a
quantitative study that, alone, can be used as a basis for practical calculations. On the
other hand, the radio engineer can accept a quantitative theory with only an approximate
basis provided that it satisfactorily takes into account the cases that are important for
practical purposes.
89
From this point, there is a clear necessity to prepare approximate methods for the study of
nonlinear systems. These methods should, of course, take into account what these
systems have in the way of specifics. One approximate quantitative method which can
deal with the analysis of nonlinear systems is the one involving coefficients with slow
variation, or, as we shall term it, the Van der Pol method. Although this method has
actually been used for quite some time in celestial mechanics, it was Van der Pol who
was the first to systematically apply it to problems of radio engineering. He produced a
series of basic results concerning forced synchronization, “resistance”, etc. [6], [7].
But it was only recently that this method was supported on a mathematical basis.
Additionally, there still remains a certain amount of uncertainty in the very method of its
application. The chief difficulty, in this respect, was obviated [8] in a way which we shall
explain using the equation:
..
x
+
x
 . 
= µ fx , x , t




(5)
in which the second term is a periodic function of t of period 2π, and µ is a “small
parameter” on which the degree of approximation will depend, as we shall see. It is
possible to reduce the equation of a regenerative receiver, etc to this form. According to
Van der Pol, one should pose that:
x = u cos t + v sin t
(6)
u and v being functions of time t with “slow variation,” i.e. whose derivatives are small
with respect to u and v, and whose second derivatives are small with respect to the
primary derivatives. Introducing the hypotheses involved in expressions (6) and (5), and
disregarding all the terms of higher degrees as well as the harmonics, we now obtain the
approximate equations of Van der Pol:
du
= a o (u , v),
dτ
dv
= b o ( u , v)
dτ
in which τ = µt and ao(u,v), bo(u,v) are functions of u and v.
Let us view the problem from another point of view. Let us substitute two new variables,
u and v for the variable x, u and v being defined as follows:
x
= u cos t
+
v sin t
Substituting two variables u and v for a single variable x enables us to impose upon them
the supplementary condition:
.
u cos t
+
.
v sin t
= 0
and replace equation (5), so that the variable x may be checked from the equations:

µ f (u sin t − v cos t , u cos t + v sin t ) cos t 

.

v = µ f (u sin t − v cos t , u cos t + v sin t ) sin t . 
.
u =
(7)
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Replacing the second terms by their averages, we again arrive at the “truncated” or Van
der Pol equations. By producing them in the same way, it is possible to clearly state an
approximation problem: it is a matter of establishing when and how much (as a function
of the value of µ) the solutions of the truncated equations are themselves close to those of
the exact equations (7). This is purely a mathematical question that has been studied by P.
Fatou [9] in a memorandum, which only came to our attention after our investigations
concerning the Van der Pol method.
Our works, as well as the mathematical results of Fatou that apply to our present
problems, while reporting the conditions and ranges that the truncated equations of Van
der Pol can take into account, can also, with sufficient approximation, deal with transitory
phenomena. In addition, the results of Fatou confirm that when µ is sufficiently small,
each equilibrium position of the truncated system corresponds to a periodic solution of
the exact system, and that if this position of equilibrium is stable the periodic solution is
likewise stable. Therefore, the question relating to the mathematical basis of the Van der
Pol method has been clarified. It is possible to hope that this method will likewise be
justified for more complicated cases.
Let us explain again the advantages of the Van der Pol method. In the case of an
autonomous system with one degree of freedom, the Van der Pol equations can be
reduced to a single equation, which may be easily solved by quadrature. In the case of
non-autonomous systems with one degree of freedom – the case which has just been
reported above – the Van der Pol equations are autonomous, and consequently, soluble
by the methods of Poincaré [29]. In the case of more complicated systems, for example
those with two degrees of freedom (autonomous or subject to external effects), the Van
der Pol equations are systems of autonomous equations of the first degree – two
equations in the most simple cases – which can be processed by the methods of Poincaré
[10]5. The Van der Pol method therefore allows replacing a system of nonlinear equations
by another, more simple one. It is possible to use the Van der Pol equations successfully,
which has been done in a study of the concepts of extension of phase, singular points,
limiting cycles, and the theory of bifurcations (Section 3). We shall see below that by
applying the Poincaré methods to the approximate equations of Van der Pol, it is possible
to produce some new results that are of physical interest.
We have seen that the Van der Pol method provides a solid basis for dealing with the
simplest cases with which we are concerned. But it only provides a “zero order
approximation”. For some questions arising in radio technology this would be sufficient.
But there are other questions that require further approximations: these mainly concern
questions about the correctness of frequency, the latter only appearing in many problems
as a second approximation.
5
There are 3 van der Pol equations for a self-oscillating system with two weakly coupled circuits. This case
was discussed by Mayer at Gorki.
91
A theory that allows improvement of precision and calculation is therefore necessary.
Unfortunately, such a theory only exists in the case of purely periodic phenomena. This is
the “small parameter method” to which we likewise are indebted to Poincaré. This
method allowed the latter to scientifically demonstrate the existence of periodic solutions
of a very general character for the three-body problem in celestial mechanics. In
substance, this method consists of the following. Let us assume that when the parameter
µ = 0, our system exhibits certain periodic motions. A search is then made for the motion
existing when µ ≠ 0 in the form of an ordinate series according to the powers of µ, in
which the zero approximation is one of the solutions corresponding to µ = 0. If, with µ
being zero, the system produces a family of periodic solutions, it includes a discontinuous
system of periodic solutions close to those existing when µ ≠ 0 and which should be
determined. This method is especially convenient when used with the zero
approximation; this system is linear and conservative6.
We have applied this method to a whole series of self-exciting problems [12], [13], [14],
[15].
Section 3: Variation in One Parameter; Stability on a Large Scale
In order to study the important subject of the transformations undergone by the phase
plane in the case of variation of one parameter, we should consult Poincaré once again.
Poincaré was led, in his famous theory of the equilibrium of rotating fluid masses, to state
and brilliantly solve questions relative to the development of equilibrium states of a
conservative system in the case of the variation of one parameter. The concept created by
Poincaré concerning the bifurcation value of the parameter can be generalized and
applied to the problems in which we are involved. One value of the parameter λ = λ0 is
called ordinary if there exists a finite quantity ε (ε > 0) such as for |λ – λ0| < ε. The
integral curves on the phase plane have the same qualitative appearance and show
bifurcation in the contrary case.
In the general case, the theory of the development undergone by the qualitative
appearance of the phase plane, in the case of variation of the parameter, is quite
complicated and insufficiently perfected. However, in the case of approximately
sinusoidal oscillations, the theory is simplified in the extreme and returns to the theory of
Poincaré for to the equilibrium states of a conservative system. It is sufficient to replace
the coordinates of the states of equilibrium by the squares of the amplitudes of stationary
motion (limiting cycles and singular points) [16]. Without explaining the Poincaré theory,
we will provide an example of its application.
Let us take the two principal types of excitation, the “soft” excitation and the “abrupt”
excitation. We shall concern ourselves with the oscillator of Figure 6 and shall select the
coefficient of mutual induction as a parameter.
6
Pontryagin has provided a general method for the case in which, with zero approximation, the system is
Hamiltonian.
92
Soft Excitation: Let λ = λ1 be the value of this parameter corresponding to the excitation
(Figure 7). In the case of λ < λ1, the only stable stationary state is the state of equilibrium
depicted by one focus (Figure 8). Regardless of the initial position of the representative
point, at the end of a certain length of time it will be found in the vicinity of this focus.
When λ = λ1 is a bifurcation value of the parameter: the focus loses its stability at the
same time that it generates a small stable limiting cycle (Figure 9) on which the
representative point begins to rotate. In the parlance of physics, we say that the oscillator
has been excited.
With λ increasing the radius of the limiting cycle becomes larger (Figure 10), and with λ
decreasing all the phenomena are reproduced in the reverse direction: the limiting cycle is
reduced to one point, and the oscillations cease. On the physical diagram I2, (λ being the
amplitude of the current), we obtain a “soft” transition from the state of equilibrium to the
periodic motion and vice versa: the amplitude of the oscillations varies in a continuous
manner (Figure 11).
Abrupt Excitation: If, in the case of small values for λ, the system is found to be in the
proximity of the state of equilibrium, it remains there until λ assumes the value λ = λ1
(Figure 12, 13, 14, 15). The creation of two twin limiting cycles -- one stable and the
other unstable -- at the instant in which λ = λ0 does not disturb our representative point
since it leaves the stability of the equilibrium state intact.
93
In the range λ0 to λ1, the unstable cycle becomes smaller. Then when λ = λ1, it disappears
and dissipates, so to say, the singular point by its instability. At this instant, the
representative point, following the integral curves, rejoins the stable limiting cycle whose
amplitude has gradually increased from the instant at which λ = λ0 (Figure 16).
Causing the parameter to vary in reverse, we observe that, on “return”, the oscillator
takes a different path than when “going”. Indeed, the representative point will stay with
the limiting cycle until the instant at which λ = λ0. At this instant, the two cycles are
merged, compelling the representative point towards the state of equilibrium. The fact
that the latter becomes stable when λ = λ1,
94
produces no effect on the movement of the representative point since, at the instant in
which λ = λ1, the characteristic of the cycle used as its path does not change.
The diagram I2, λ (Figure 17) emphasizes a discontinuous (“abrupt”) variation of the
amplitude, a variation which, owing to its irreversibility, recalls hysteresis phenomena.
This phenomenon of abrupt excitation, which is quite interesting for the radio engineer,
finds its natural and proper interpretation in the language of singular points, limiting
cycles, and the bifurcation values of the parameter.
For this reason, it can be seen immediately that
when λ0 < λ < λ1, the representative point can be
“thrown” from one stationary mode of operation
into another by a sufficiently strong impulse. This is
a claim that we suspect can be demonstrated
intelligibly by quasi-linear theory.
At this point, we should like to discuss a concept
that would have no significance in the case of a
conservative system, but which, in the case of a
nonconservative system, has a great deal of interest.
Let a path be stable. On the phase plane we can mark off a region containing the initial
position of the representative point from which the latter, ultimately (when t J +∞)
rejoins this path. This region is called the “large scale region of stability” or “region of
attraction” of the stationary movement under consideration. Figure 18 and 19 represent
two examples.
The phase plane of Figure 18 has three singular points: two stable nodes and one saddle.
The range of attraction of node A is in the right semiplane, whereas that of node B is in
the left semiplane. Let us now consider (Figure 19) a stable singular point surrounded by
an unstable cycle, which is, in turn, contained within a stable cycle. The range of stability
on a large scale of the position of equilibrium is the portion of the plane contained within
the unstable cycle. The remainder of the plane is the range of attraction of the stable
cycle.
95
Without providing more complicated examples, let us point out that the sharing of the
extension in phase by the fields of attraction comes up against some obstacles, even in the
case of one degree of freedom. It appears that considerations of probability must be taken
into consideration here.
Let us note, in conclusion, that the concept of stability on a large scale imparts the true
physical interpretation to the unstable limiting cycles and separating curves. These are the
boundary motions similar to what is called, in geography, the tidal line: according to
which, at the initial instant, the representative point is placed on one side or the other of
these components and takes its direction towards differing destinations.
Section 4: Autonomous Systems
At the forefront of the many nonlinear problems is the one of autoexcitation, i.e. those
oscillations created by the oscillating system itself without the participation of any
external forces varying with time and at the expense of a constant source of energy (for
example a storage battery). The qualitative questions and, partially, the quantitative
questions that are concerned directly with the study of an autonomous nonlinear system,
can be solved by methods that we have described above. We have already shown
examples of the application of these methods to the problems relating to soft and abrupt
excitation, and to transient phenomena. As we have said, quantitative methods of
approximation allow us to find the amplitude of autoexcitations and the correct frequency
in the case of almost sinusoidal oscillations. Permit us to comment that, at least insofar as
only the zero-order approximation is concerned, in such problems the methods placed
previously known results solidly upon a mathematical basis. In this way, it is possible to
precisely demonstrate the existence of periodic solutions in certain cases, and establish
their stability. We estimate that these demonstrations of existence have a high degree of
usefulness, and the following is why.
When we set up any problem of physics as a differential equation, we are always forced
to simplify it. We do not write the equation of the problem that is given to us, but rather
that of a simplified and idealized problem. Now, how can we be certain that we have not
disregarded any of the essential features of the real problem? The situation changes if we
have demonstrated the existence of a periodic solution that has been verified by
experiment. This demonstration is an argument to maintain that we have not omitted
essential features (assuming as the latter those that make available the production of
stationary oscillations). But these are only indirect data. The same will not be true if it
can be demonstrated that the differential equations do not have a periodic solution,
whereas the system that they claim to describe does possess such solutions. Most
certainly we have then disregarded some essential feature, and we must attempt to
recapture it.
96
Practical experience recognizes cases where experimental investigations demonstrate the
existence of a solution, which has suggested the means for correcting omissions of this
type and placing further discussion on the right track.
The following is one of the most elementary examples. Everyone knows how many
technical manuals – chiefly among the older ones – describe the theory of the bell or
electric buzzer. The armature, in the state of equilibrium, closes the circuit of the
electromagnet. When a battery is placed in the circuit the electromagnet attracts the
armature, the current is interrupted, the magnet loses the force of attraction, the spring
pulls the armature back into its original position, and as the old expression goes – the
game continues. If this reasoning is translated into differential equations, it is possible to
show very easily that they do not permit the game to continue and that they allow only a
periodic solution. Some essential point has therefore been disregarded. In reality, the
theory of buzzers is less simple than it appears at first glance. Self-induction is necessary
for oscillations to be possible at all. Mr. Leontovitch [17] has successfully discussed the
problem of the buzzer and has clearly shown that not only is self-induction necessary for
the existence of the phenomenon, but that it becomes a factor in the period of the
oscillations, this period differing from that of the tuning fork or armature return spring.
Other examples could be quoted showing the actual utility of experimental investigations
devoted to questions of the existence of solutions.
Radio engineering often reveals conditions in which oscillators are practically
sinusoidal7. However, during the last two years, and to a great extent owing again to the
works of Van der Pol, interest has grown concerning systems executing oscillations
which greatly differ from a sinusoidal shape and may be termed “relaxation oscillations”.
The characteristic of these oscillations is, essentially, a function of the resistance or
parameters equivalent to it. This implies that an equation of the type:
..
Lq +
q
C
.
.
 .
= f  q , q  − R q = ϕ(q , q)




(8)
.
describes the autoexcitation system. The function ϕ(q , q) is not limited to small values,
as in the case of “almost sinusoidal” oscillations, but, as Van der Pol has pointed out, it
may take on substantial values. Since, it is assumed here that we are concerned with
periodic solutions, this case is entirely within the purview of the qualitative theory of
Poincaré. The singular unstable point is a node, and the periodic solution corresponds to a
limiting cycle.
According to the simplest hypothesis, when the characteristic of the tube can be modeled
as a cubic parabola, the equation can be described in the form:
.
v − ε (1 − v 2 ) v +
7
v = 0
There is one very simple mechanical system that allows production of autoexcitations that are almost
sinusoidal. This is the Froude pendulum, which has been studied by Strelkov [18].
(9)
97
.
whereupon, assuming v = y , it becomes:
dy
dv
− ε (1 − v 2 ) +
v
y
= 0.
(10)
If ε << 1 we have oscillations which are almost sinusoidal (“Thomsonian” oscillations).
However, the qualitative theory is applied to the general case no matter what the value of
ε may be. When ε << 1, the singular point is a focus, but when ε >> 1 it is a node; in both
cases there is a limiting cycle.
Nevertheless, in quantitative studies of relaxation oscillations it is possible to proceed in
another way. It is possible to idealize the problem by setting L = 0 and replacing the
equation of the second order (9) by the equation of the first order:
.
v =
v
ε (1 − v 2 )
(11)
which is easily integrated. Clearly, this equation does not allow for a periodic solution. At
the close of a finite time duration, the velocity (or the electric current, or its derivative),
represented by v, becomes infinite. After having idealized the problem in this way, in
order to take the physical phenomenon into account in an approximate manner, a new
condition is required to be introduced, which in our case assumes that at a certain instant
the current undergoes a discontinuity whereas the voltage at the condenser terminals
remains constant. This assumption or “condition of discontinuity” is physically justified
by the fact that the energy cannot vary discontinuously. It is possible to provide it with
another form by explicitly requiring conservation of energy. This “discontinuous” theory,
together with the condition of discontinuity, permits evidence of a “discontinuous“
periodic motion and finding its amplitude as well as its period.
Without being identical, this mode of treating relaxation oscillations, which is applicable
to electrical and mechanical systems, is analogous to methods employed in mechanics to
analyze elastic collisions. It is assumed that at the moment of collision the velocity
changes discontinuously. The conservation of energy and momentum allows a velocity
decrease after the collision from that existing beforehand. In principle, this method
excludes the possibility of studying what is happening during the extremely short
duration of the collision. The results that it gives are often sufficient since the collision is
quite brief. However, if we wish to follow the phenomenon of the collision itself, the
problem becomes extremely complicated. It is enough to merely recall the investigations
of Hertz. Likewise, in our theory of relaxation oscillations, we can simplify the
mathematical description of the problem by idealizing it. As a consequence we do not see
how the system can “leap” from one state to the other.
We have applied this method to the study of electrical systems with one degree of
freedom in which self-induction plays a secondary role [19], as well as to mechanical
autoexcitation systems with a small mass and a high degree of damping [20].
98
In this case it should be noted that by taking the “parasitic” self-induction into account,
nothing is obtained which is of physical interest. There still remains the parasitic
capacitance of conductors etc. to be considered. Now, it is impossible to take into
consideration all of the parasitic parameters. Our idealization has the advantage that it
allows us to study relaxation systems that are relatively complicated, such as the
multivibrator of Abraham-Bloch, a system with two degrees of freedom. This system has
already been studied by Van der Pol [21],
but with one essential restriction: he
assumed the phenomenon to be
completely symmetrical, and rejected the
consideration that there were transitory
phenomena following an initial
asymmetrical state. This restriction
allowed him to produce an equation of the
second order. Nevertheless, in the general
case, he would have obtained two
equations of the second order, and this
would have greatly complicated the problem. For the general case, our idealization
provides two equations of the first order, which can easily be studied by the methods
described above [22]. Thus, we have been able to study not only the stationary mode of
operation (by calculating the amplitude and the period) but also the transitory
asymmetrical phenomena in the multivibrator of Abraham-Bloch. These results were
experimentally verified.
The discontinuous theory is not merely
applicable to the case in which one of the
parameters is small. Even in a circuit with
capacitance and self-induction there are cases
that can occur where, in certain regions, the
velocity becomes so great that it is possible to
replace the very rapid variation of state of the
system by a discontinuity and to determine the
final result of the motion in this region by using
the “condition of discontinuity”. Removing, in
this way, those regions where quantitative research is most difficult, we can solve (with a
precision satisfactory for all practical purposes) a whole series of questions that involve
oscillations whose form is essentially nonsinusoidal. It is possible to apply this method,
for example, to the configuration depicted in Figure 20.
The parasitic parameters again give rise to the following observation. By studying an
oscillatory system, we always idealize it, disregarding certain parameters (for example,
the self-induction of a condenser or the capacitance of a coil). Nevertheless, we often
obtain good results. The cause for this rests in the rather extensive properties of the
mathematical approach, concerning which we will not go into detail. However, there are
cases where, by disregarding certain parameters, no matter how small, we qualitatively
change the overall picture of the phenomena. Let us consider the system shown in Figure
21.
99
The circuit is formed by a battery E, a resistance R, a capacitance C, and an electric arc.
By using the characteristic of the arc, we produce three stationary values for the current
using the customary graphical method. Analyzing the stability of these values by wellknown methods, we find that two of them, A and B, are stable (Figure 22).
However, if we introduce an arbitrarily small selfinduction L (Figure 23), the state of equilibrium
becomes unstable. And so, in reality, it does not
exist.
It is easy to see how cases of this kind can occur. If
the order of the differential equation from which we
draw conclusions concerning stability or instability
(in the case of equilibrium this will be an equation
of the n-th linear order with constant coefficients)
does not rise as a result of the introduction of the
parasitic parameter, the latter, if it is sufficiently small, is not able to change anything. On
the other hand, if this parameter appears in the equations and increases their order, it can
render equilibrium states (that would be considered stable by disregarding it) unstable.
The physical meaning of this result is clear. In composing a system of equations of the nth order, forming a system with n dimensions, we only assume initial conditions. And
when we raise the order of the differential equations by taking into account the parasitic
parameter, we allow by this a greater diversity of initial conditions. It is then possible that
among the initial states newly allowed, the conditions are right for the system to diverge
from the state of equilibrium. Therefore, a certain caution is necessary in idealized
models.
Some words are in order concerning self-exciting
systems with distributed parameters that play an
important role in radio engineering and in
mechanics (oscillators containing antennas or
Lecher wires, tubes whose grid makes up an
oscillating system resulting in very high frequency
waves (which were studied by Grechowa),
telegraphic wires emitting a sound owing to the
effect of wind, vibrating airplane wings, bowed
musical instruments, organ pipes etc.). There still does not exist a precise mathematical
theory for these phenomena. Nevertheless, without speaking rigorously, it is rather easy
to create a possible theory for some of them by analogy with those produced under rather
precise conditions - those systems existing with a finite number of degrees of freedom.
This theory allows the calculation of amplitudes, the solution of questions of stability,
etc. [23]. However, since this theory does not have a strict mathematical basis, it is
necessary to use its results with care. It takes into account most characteristic phenomena
occurring in distributed self-exciting systems. It anticipates that oscillatory modes of
100
operation at different frequencies can be established under the same operating conditions.
(The same is true in the self-exciting systems having a finite number of degrees of
freedom). The production of such-and-such an operating mode is a function of the initial
conditions or the history of the system. These phenomena have been produced and
studied experimentally by Bendrikov and Brailo at Moscow, as well as by Gaponov at
Gorki. It is possible to cause the disappearance of a particular oscillating wavelength in a
Lecher wire oscillator by touching them with a finger. The system then begins to oscillate
at another wavelength. Under certain conditions, when the finger is taken away the
system will not return to the original wavelength, but continues oscillating at the new
wavelength. This phenomenon can likewise be produced by capturing the energy using a
resonant circuit. Apparently, similar phenomena occur in bowed musical instruments.
Strelkov has performed similar experiments with a string vibrating under the effect of a
jet of water or air. These quite simple experiments allow the observation of phenomena
that are characteristic for distributed self-exciting systems.
Given that, under the same operating conditions different oscillating modes can occur, the
question may be asked as to which of them is produced when the system is triggered.
This question is often within the capacity of the theory of probabilities. It has not yet
been solved theoretically for distributed systems. The statistical phenomena are easy to
observe experimentally in the Lecher wire oscillator. They also take place in organ pipes.
Section 5: Effects of an External Force on a Self-Exciting System
One characteristic property of self-exciting systems, which is quite important for the
entire realm of radio engineering, is the appearance of the phenomenon of forced or
automatic synchronization, or frequency locking. This phenomenon, previously noted by
Huygens for clocks hanging from the same wall, was first observed in radio engineering
by G. Moeller [24] and Vincent [25]. It gave rise to many experimental and theoretical
investigations, among which special mention should be made of those by Van der Pol [6].
As is well known, the simplest feature of this phenomenon consists in the following.
When an external force of frequency ω acts on a self-exciting system of frequency ω0, no
beats are observed, just as would be the case in an undamped linear system when the
difference frequency, ω – ω0, is sufficiently small. The system is automatically
synchronized to the frequency of the external force.
A similar phenomenon occurs in a system subjected to the effect of a force which is not
periodic, but only quasiperiodic (which can be depicted by a sum of terms of
incommensurable frequencies).
101
The locking likewise occurs, as observed by Van der Pol and Van der Mark [26] in
relaxation oscillator systems, and as seen by Koga [27] in ordinary oscillators when one
of the frequencies of the external force is close to a multiple of the frequency of the
system. When the out-of-tune condition exceeds a certain value, which can be termed a
locking limit, the appearance of beats may be ascertained. Since the force is sinusoidal,
when the misalignment or out-of-tune condition far exceeds the locking limit, it can be
broadly stated that two oscillations exist in the system, one in response to the frequency
of the external force, and the other characteristic of the system.
Nevertheless, if the frequency misalignment only slightly exceeds the locking limit, this
last frequency is shifted towards the frequency impressed from the outside, and, as we
shall see, the whole phenomenon becomes complicated.
The theoretical study of the locking phenomenon consists in searching for stationary
solutions of the differential equation:
..
.
y + ωo2 y = µ f ( y , y) + Σ A s cos (ωs t + δ s )
(12)
In the simplest case in which an electromotive force acts upon a system in a soft mode of
operation with almost sinusoidal oscillations, the “curves of amplitude” have, as is
known, the appearance shown in Figure 24. The portions of the curves labeled 4-7-12, 58-11, etc. belong to the synchronization mode. The parts 1-4-12-15 belong to the beat
mode. Theoretically, these curves have a symmetrical appearance, but, most often,
experience shows asymmetrical curves (for example those of Figure 25). This
deformation is probably owing to the presence of a grid current. This is what appears to
be confirmed by the recent results of Bakoulov (Moscow).
In his classical study, Van der Pol had studied a self-exciting system with a soft mode,
modeling the characteristic of the tube as a cubic parabola, and he was able to detail the
principal characteristics of the phenomena of synchronization and frequency shift.
Nevertheless, there are still several questions remaining. In synchronization phenomena,
as assumed by Ollendorff [28], it still remains to be explained whether there exists a
threshold for the amplitude of the applied electromotive force (EMF).
102
Employing the “truncated” equations that Van der Pol set up for this problem:
dx
dt
dy
dt

− a y + x (1 − r 2 ) 


= A + a x + y (1 − r 2 ) 

=
(13)
[where a = 2(ω0 - ω)/α is the tuning
misalignment, ω0 is the frequency of the
self-excitations, A = (Bω0)/(αa0). B is
the amplitude of the force used, α is a
constant depending on the two
parameters, a0 is the amplitude of the
self-oscillations, τ = at/2 and finally t
represents time], it was possible to show
that, in the topographical analysis of Poincaré, a threshold did not exist [29]. This has
also been established experimentally [30].
Fig. 27. The integral curves of the differential equations.
dx
dt
dy
dt
=
− y + x (1 − r 2 )
= 0.303 + 0.3 x + y (1 − r 2 )
for the initial conditions τ = 0 , x o = 1 , y o = 0 , A =
8
27
Consequently, in the weak signal case it was easy to quantitatively demonstrate that the
normalized width of the synchronization band is a function of the ratio of the amplitude
of the signal to that of the self-oscillations (ω0 – ω)/ ω.
103
This is what has permitted the application to weak signals of the method of field intensity
measurement by the width of the synchronization band, as suggested by Appleton [31],
[32]. A similar method, based on the phenomenon of synchronization of acoustic selfexciters, has been used to measure the
intensity of sound [33], [34]. The
phenomena of acoustic locking leads
to an interesting problem that we are
preparing to study, which is that of the
automatic synchronization of
woodwind and bowed orchestra
instruments.
Let us return to the analysis of
equation (13). If the square of the amplitude A2 is plotted on the ordinate, and detuning,
a, is plotted on the abscissa, Figure 26 is produced. The resonance curves are those of
Van der Pol, but our figure shows the fields corresponding to the various types of
transitory phenomena. One might be interested in what is happening when an
electromotive force is applied to the oscillator and it becomes active.
It is also possible to observe what occurs when the oscillator is triggered after the
electromotive force has been applied. Let us examine the first case, which is physically
the most interesting. The theoretical study of transient phenomena consists of discussing
the nonstationary solutions of the equations for amplitudes (13), i.e. in following the
amplitude fluctuations, which are components x and y as a function of time. These results
are summarized in the diagram of Figure 26. In order to know how the stationary
oscillations are set up, with the variance and amplitude of the applied EMF being given,
it is necessary to get the resonance curve corresponding to this EMF, and, on the latter,
the data corresponding to the given variance. If this point is located in the domain of a
stable
104
node, the establishment of the mode is accomplished aperiodically: the coefficients with
slow variation in the Van der Pol solution tend aperiodically toward constant values. If
this point is in the domain of a stable focus, the phenomenon is oscillating. Finally, if the
point falls within the range of instability, then there are no stable periodic solutions.
These results have been experimentally verified in the works of Riazine [35] who
made oscillographs of various types of transient phenomena. He calculated the solutions
of equation (13) by methods of numerical integration and confirmed the theoretical
results by low frequency oscillograms. The calculated curves of aperiodic transitions are
shown in Figure 27, those of oscillating transitions are shown in Figure 28, and the
respective oscillograms are given in Figures 29 and 30. A pure sinusoid appears before
the application of the signal, but the oscillograms do not show it.
The same method of numerical integration
was used to study what occurs, to a minor
degree, outside of the synchronization
band. In this case, the nonstationary
solution of equation (13) on the x, y plane,
does not tend toward a singular point, as in
the domain of entrainment, but is coiled
around a limiting cycle. The curves
showing current as a function of time and
limiting cycle are depicted in Figures 31
and 32. The theoretical results agree closely
with the experimental findings (Figure 33).
With regard to beats, the theoretical curves
and oscillograms harmoniously indicate
that their amplitude increases at a
perceptibly greater rate than it decreases.
In order to establish the spectral
composition of the beats, Riazine performed a harmonic analysis of the curves shown in
Figure 31. The spectrum so obtained (Figure 34) shows that, in the vicinity of the
synchronization region, the application of an electromotive force causes the appearance
of a self-oscillation in the spectrum, that has equidistant combinations of frequencies
clearly delineated. Oscillograms were produced for beats in the vicinity of the
105
synchronization bands for ratios of frequencies characteristic of the applied frequency,
which were approximately equal to 1:2, 1:3, 1:4 and 1:5 (Figures 35, 36, 37, 38). It
became obvious from these oscillograms that we have a pulsation of amplitude in the
region of the limit of forced synchronization, just as in the case of the 1:1 ratio. The
envelope of the beats always shows practically the same characteristic:
a rapid increase and then a slow decrease of amplitude. Let us again emphasize that in the
vicinity of the synchronization band, the oscillations cannot be expressed by the linear
superposition of two sinusoidal terms, as has been done up until now. There are at least
three oscillations of amplitude that are almost equal.
The study, likewise, was concerned with the
phenomena that occur in the case of abrupt
modes (H. Sekerska [36]). In this case the
so-called domain of potential selfoscillations has a special interest. In this
region the resonance curves end in a peak,
recalling those that we produced in the
resonance phenomena of the n-th class
(Section 6). There still exist synchronization
phenomena in the frequency combinations
given by several electromotive forces. As an
example, we shall give a special case that
we observed some time ago:
synchronization “in the middle”. The
oscillator is tuned approximately to the
frequency (ω1 + ω2)/2, with ω1 and ω2 being
the frequencies of the two electromotive
forces. Synchronization phenomena are then clearly observed, and especially so when
there exists a simple relationship between the frequency of the oscillator and those of the
electromotive forces. It is understood that this phenomenon is important for reception of a
signal without a carrier (DSB). In reality, when the carrier frequency is produced at the
transmitter site, for practical purposes it is quite difficult to arrive exactly at the center of
the sideband frequencies, this being, moreover, absolutely necessary. The phenomenon
that has just been described lends its assistance: the frequency of the oscillator is
automatically located in the middle of the sideband frequencies.
106
For this purpose, it is sufficient for the oscillator to only be approximately tuned. Similar
phenomena occur when the oscillator is tuned to the frequency (ω1 + ω2)/4, and to other
combinations of frequencies and their submultiples. The phenomenon of “in the center”
synchronization has been studied theoretically by Goldstein and Petrossian8.
8
This publication is in preparation.
107
Section 6: Resonance Phenomena of the n-th Class
We shall devote a special paragraph to the phenomena that can be termed “resonance of
the n-th class”. The mathematical theory of these phenomena is based on the general
results provided in the well known works of Poincaré [2] without any relationship to the
physical applications concerning us. Poincaré shows that in nonlinear systems there can
be periodic oscillations whose period is a multiple of that of the applied force (“periodic
solutions of the second class”). It is useful for the discussion that follows to call any nonself-exctiting system that becomes self-oscillating if the regenerative feedback is
sufficiently increased, or if the value of any parameter is suitably modified, a potentially
self-exciting system.
In Section 5 we referred to the phenomenon of forced or automatic synchronization that
occurs in self-exciting systems subjected to the effect of a sinusoidal force that has a
frequency close to the eigen-frequency. Koga [27] as well as Van der Pol and Van der
Mark [26] have observed similar phenomena in self-exciting systems subjected to the
effect of a force whose frequency is a multiple of the eigen-frequency. As for potentially
self-exciting systems, in this case it is observed that when the frequency ω of the applied
EMF is equal, or approximately equal, to a multiple of the eigen-frequency, the mode of
operation being suitably selected, there is a special phenomenon of synchronous
excitation. As long as its eigen-period is not close to a multiple of that of the applied
force, a potential self-exciting system is the source of very weak “forced” oscillations.
Nevertheless, when the system’s eigen-frequency is sufficiently close to (ω/n), n being an
integer, there can appear intense oscillations of frequency exactly equal to (ω/n).9 This is
the phenomenon of resonance of the n-th class [15], [37].
There still exists, in potentially self-exciting systems, another phenomenon which should
be mentioned here. The resonance of the n-th class requires a well-defined mode of
operation of the tube. If, starting from this mode of operation, the regeneration is
increased, quite slightly in order that the system does not become self-exciting, under
certain conditions there may be seen to appear, intense oscillations almost identical to the
system’s own oscillations, no matter what the period of the EMF may be. To these
intense oscillations very weak “forced” oscillations are added in such a way that the
system’s own phenomenon is almost periodic. It can then be termed asynchronous
excitation [39], [40], [41].
In the simplest case of n = 2, if the EMF is E = E0 sin ωt and acts upon an oscillating
circuit interposed between the plate and the filament of a tube whose characteristic
9
The capability for exciting a potential self-exciting system on a frequency equal to that of half of that of
the EMF has been likewise reported by Groszkowski.
108
can be represented by a polynomial of the third degree, then the equation of the
phenomenon (the grid current being assumed nonexistent) can be described in the form:
We assume that γ0 < 0. Using suitable transformations and notations:
The equation can be rephrased in the form:
and the regeneration factor
Applying the Van der Pol method to equation (16), by the transformation:
we produce the system of truncated equations:
109
where z = u2 + v2 is the square of the instantaneous amplitude.
These equations may be easily solved if ψ = u/v and z [42] are selected as variables.
Their initial values will be designated by ψ0 and z0. Taking:
we obtain the solution in the following form:
Equations (22) and (23) approximately describe what occurs beginning from any initial
conditions whatsoever as well as the case when the system is not excited (k < 0) and also
when it has self-exciting (k > 0). If q = 0, ε = 0, p = 0, i.e. if there is no applied force, we
resort to the Van der Pol solution for autonomous systems [6].
It is evident from equation (23) that z only tends toward a constant value zst different
from zero when p is real and M + 2p > 0, i.e. when
q2
>
ξ2
β2
(24)
and
k +
γ1 q 2
+
2
q2 −
ξ2
β2
>
0;
these are the existence conditions for constant solutions in the case of u and v. When
these conditions are satisfied, periodic oscillations whose period is double that of the
(25)
110
EMF will establish themselves in the system (c.f. Equation 18). These oscillations with
double period only appear in a limited frequency interval that is characteristic of the
system and, by its appearance, the phenomenon calls to mind certain resonance
phenomena. The term “resonance of the second class” (and more generally of the n-th
class) recalls that this theory is closely connected to the existence of periodic solutions of
the second class of Poincaré [2].
Given that γ1 < 0, condition (25) cannot be
satisfied in the case of a potentially selfoscillating system (k < 0) except when the
value of q is included within a certain interval
qmin < q < qmax. We state that there is a
threshold and a ceiling for the value of EMF that is capable of exciting a potentially selfoscillating system with double period oscillations. In the case of self-oscillating systems,
condition (25) is satisfied, no matter how small q may be. There is, therefore, no
threshold for the automatic synchronization of a self-oscillating system with a period that
is double that of the applied EMF.
The square of the amplitude of the stationary oscillations, with a period double that of the
EMF, which have been excited by resonance of the second class, is provided by the
formula:
z st
=
4
γ1

2
 k + γ1 q +

2

q2 −

ξ2 
.
β 2 
(26)
According to this formula, the stationary amplitude is a function of the detuning
parameter, ξ, quite apart from the case of ordinary resonance. The curves which provide z
as a function of ξ -- they can be termed resonance curves of the second class – are shown
in Figure 39 (theoretical) and in Figure 40 (experimental). Formula (26) likewise
provides the stationary amplitude as a function of the value of the applied EMF. This
function (“the characteristic amplitude”) is depicted in Figure 41 (theoretical) and in
Figure 42 (experimental). Note that the “excitation band” (i.e. the frequency interval in
which the second class resonance occurs) is equal to zero in the case of q = qmin, and
becomes wider at first as q increases, then decreases and again drops to zero in the case
of q = qmax .
111
The increase of oscillations up to the stationary amplitude has quite another characteristic
than in the case of ordinary resonance as shown in Figure 43. The latter depicts the
variation of amplitude as a function of time, beginning from the instant at which the
external force becomes a factor in the resonance of second class (curve 2) and in ordinary
resonance (curve 1).
Let us note the similarity between curve 1 and the curve expressing the increase in
amplitude of self-oscillating phenomena. This similarity is not accidental. In ordinary
resonance, the excitation of oscillations takes place no matter what the initial conditions
may be and can start more particularly beginning from absolute
equilibrium (i = 0, di/dt = 0). In resonance of the second class
pulses, whether they are very small or not, are necessary to
cause the system to deviate from the initial state z0 = 0. Under
the effect of an external force, the position of equilibrium of a
potentially self-oscillating system that satisfies resonance
conditions of the second class becomes an unstable focus
enclosed in a stable limiting cycle.
The special characteristic of the curve of oscillation growth, in resonance of the second
class, can be used advantageously for practical ends (c.f. below)
When the system is in an abrupt mode of operation, resonance of the second class shows
certain special features. Thus, at the boundaries of the regions of excitation, “resistance”
phenomena can be observed owing to the partial superposition of different ranges of
dynamic stability. If the characteristic of the tube is expressed by a fifth degree
polynomial, which was done successfully by Appleton and Van der Pol [43] as well as
112
other authors [44], then, by applying the methods of Poincaré (c.f. Section 2) to the case
of self-oscillations, it is possible to provide an approximate theory of the phenomena.
Figure 44 gives resonance curves of the abrupt mode of operation calculated in this
manner. These agree closely with experimental results (Figure 45).
When an experimental study is made of
the resonance of the second class in
systems with abrupt excitations, certain
precautions should be taken in order to
avoid the phenomena of asynchronous
excitation as defined above. In one of our
laboratories (Central Radio Laboratory), it
has been shown theoretically [39] and
confirmed experimentally, by E.
Roubtchinski [45]), that asynchronous
excitation is only possible if the mode of
operation is abrupt, and if the values of the
regenerative feedback and the amplitude
of the EMF are each included within a
specific range.
Figure 46 shows the modes of operation corresponding to the different values of the
feedback factor k. The region to the right of zero is that of spontaneous excitation. The
region of resistance is found between zero and A (0 > k > γ2/(8|ξ1|)). Assuming the circuit
is suitably tuned and that the EMF has a suitable value, it is possible to produce
resonance of the second class to the left of A. In addition, phenomena of asynchronous
excitation can occur in the shaded portion between A and B (--γ21/(8|ξ1|) > k > -γ21/(6|ξ1|)). In order to produce resonance of the second class, in the pure state, it is
113
necessary to work to the left of B. This is a very important matter for receiving stations
making use of this phenomenon.
In an abrupt mode of operation, the transitory phenomena in resonance of the second
class have been studied by A. Melikian [46]. His experiments show that the phase is set
up much more quickly than the amplitude. In the case of a soft mode of operation, theory
gives the same result, provided that customary numerical values of the parameters are
used.
Assuming this condition to be true, as a hypothesis in his calculations, Melikian obtained
a relatively simple set of theoretical formulae for the abrupt mode of operation. Figure 47
shows one of the theoretical curves of growth of oscillations. On these pages, we have
reproduced some oscillograms produced by Melikian. The one shown in Figure 48, which
has been produced using an electronic oscillograph with an incandescent cathode
requiring the synchronous repetition of the phenomenon, shows the effect of a
rectangularly shaped signal. In Figures 49 and 50, the oscillograms shown illustrate the
increase and decrease of oscillations in a triode transmitter, and in a potentially selfoscillating system, under the effect of a force of double the frequency compared to its
own frequency. They were taken using an oscillograph with a cold cathode and with an
internal photographic device which allowed the recording of a single event.
114
The theory developed for resonance of the n-th class likewise allows for the analysis of
automatic synchronization phenomena of self-oscillating systems at a frequency equal to
a submultiple of the frequency used [37]. Just as in ordinary locking (c.f. Section 5) there
is no threshold for the EMF and the decrease of the latter only causes the synchronization
band to become narrower. Theoretical and experimental data show that, above a certain
limit, the amplitude of the “self-oscillations” decreases when the amplitude of the force
increases, and, beginning at a specific value of the latter, becomes equal to zero. The selfexcitations are damped by the EMF, and only forced oscillations having the frequency of
the EMF remain in the system.
Theoretical researches conducted by the Central Radio Laboratory have shown that it is
impossible to produce oscillations whose resonance is of the third class in a potentially
self-oscillating system with one degree of
freedom and a soft mode of operation.
Experimental data show (Tschikhatchov) that
oscillations synchronized in a period three
times that of the EMF or resulting from
asynchronous excitation can be “driven” to the
left of B (Figure 46). In potentially selfoscillating systems with two or more degrees of
freedom, it is possible to excite oscillations
whose resonance is of a higher order.
Tschikhatchov produced resonances
phenomena of the fourth class in the
installation described in Figure 51.
Resonance phenomena of the n-th class have a
certain relationship with the excitation of
oscillations by periodic variation of a system’s
parameters. This is the “parametric excitation”
to be discussed in the following section (Section 7). Consequently, in reality, we can
interpret the excitation by resonance of the n-th class in a purely qualitative manner. The
external force acting on a potentially self-oscillating system causes “forced” oscillations
to appear, which have the same period as the force. By reproducing the reasoning used to
analyze the stability of motion by the methods of Poincaré and Liapounov, we can
115
consider our system to be nonlinear and, in the vicinity of the forced oscillations, to be a
linear system whose parameters are functions of the forced solution q sin nt. The
properties of this linear system, with parameters varying periodically, are well known
(Section 7). If it is located in one of the regions of instability, the forced oscillation will
be unstable and the system will perform increasing oscillations whose frequency will be a
submultiple of that of the force. This method of reasoning offers certain advantages, and
it is often useful to consider the resonance of the n-th class as a sort of parametric
excitation. In order to distinguish the parametric phenomena in their true sense, which
occur when the parameters of an electrical or mechanical system are caused to vary
indirectly with those of resonance of the n-th class, we shall call the first ones
heteroparametric and the second ones self-parametric.
The phenomena that have just been described possess encouraging properties for certain
advantages in their practical application. Experience has shown that resonance of the
second class can be used successfully for frequency reduction as well as the generation of
very high amplifications in cases where the frequency should remain very stable, as, for
example, in the case of transmitters with independent excitation, and above all in the case
of reception.
The resonance curves of the second class (with abrupt edges, the existence of a threshold,
and a ceiling for the applied EMF) give rise to new methods for selective reception.
Nevertheless, of course, it should not be forgotten that with receivers as well as with
high-speed operating automatic transceivers, we are not only dealing with stationary
phenomena, but with increasing and decreasing phenomena which, as we have seen
(Figure 43), are basically different from those occurring in the case of ordinary
resonance.
Experiments performed in 1930 and 1931 showed that, under practical conditions, a
device with a resonance of the second class has applications in a radio receiver. The
device was used as a selective filter (“self-parametric filter”) and gave excellent results.
Figure 52 shows a photograph of two simultaneous recordings of signals emitted by radio
station WCI (wavelength 16,317 meters) made on 4 February 1931 at the radio receiving
exchange of Boutovo near Moscow. Track A comes from a radio receiver furnished with
116
a self-parametric filter. Track B comes from a radio receiver that has a crystal filter. The
difference is obvious.
Prolonged tests on a radio receiver with a self-parametric filter were performed at
Sagaredjo, near Tiflis, in a region that is subject to intense atmospheric disturbances, and
these have shown that this filter is very effective in distinguishing an extended harmonic
signal from a static crash.
This property of systems with resonances of the second class is explained by the
characteristics of the law according to which oscillations increase. In a linear filter, a
brief impulse (with respect to time duration) that
is sufficiently strong can cause oscillations whose
amplitude is comparable or even greater than the
desired signal. However, owing to the peculiar
features of the curve of growth for a selfparametric filter, it only gives small oscillations.
In this way, the auto-parametric (or selfparametric) filter practically suppresses
atmospheric disturbances that have the form of
short impulses, even though they may attain a
considerable value. This insensitivity to
atmospheric impulses remains when they are
superimposed on a signal. However, very strong
atmospheric discharges have the effect of “segmenting” the signal (as was observed by I.
Borouchko and N. Weissbein in 1931). In recordings on tape there are certain signs (dots
or dashes) that show discontinuities (1, 3 on Figure 53).
A. Melikian studied this last phenomenon in one of our laboratories (the Central Radio
Laboratory). Using the oscillographic method he very skillful examined in detail the
simultaneous effect of a signal plus short pulses (a train of damped oscillations) in a selfparametric system. He showed that “segmenting” (breaking up) occurs when the impulse
arrives at an instant for which the double period oscillations, due to the signal, are already
almost established (Figure 54). However, if the pulse arrives at a time when the double
period oscillations are still weak, it accelerates their growth.
117
Figures 55 and 56 allow comparison of the oscillograms for growth and decay of a selfparametric system (Figure 55) with those of a linear system (Figure 56).
Section 7: Parametric Excitation
The phenomena incited in nonlinear circuits by an external stimulus (Section 6) are
closely related to the excitation of oscillations by periodic variation of the parameters of
an oscillating system. This effect, which can be called parametric excitation (for short),
has been known to physicists for quite some time. (Melde [47], Rayleigh [48]) The great
importance that it has in radio engineering is likewise known. However, although the
possibility of parametric excitation of electrical oscillations has been known for a long
time (Rayleigh [49], Poincaré [50], Brillouin [51] and later Van der Pol [52]), it is only in
the last few years that the full value of this phenomenon was realized, and its systematic
study undertaken. We should like to mention the experiments of Heegner [53] and
Guenther-Winter [54] concerning the excitation of electrical oscillations at acoustic
frequencies by alternately magnetizing the iron core of a self-induction coil, as well as
the experiments of Guenther-Winter [55] and I. Wantanabe, T. Saito, and K. Kaito [56]
on the excitation of electrical oscillations by mechanical periodic variation of the selfinduction or capacitance of an electrical oscillating system.
We have also performed experiments on the parametric excitation of electric oscillations
by mechanical periodic variation of the self-induction [57] of a circuit, but using very
different devices from those used by Guenther-Winter and Wantanabe. In addition, we
produced the parametric excitation of electric oscillations by periodic variation of the
capacitance of a circuit10 [58].
10
W.L. Barrow (Proceedings of the Institute of Radio Engineers, Volume 22, p. 210, 1934) wrongly
assumed that his experiments showed the capability for the parametric excitation of an oscillating circuit
by periodic variation of its capacitance. He caused the variation of not only the capacitance of an
oscillating circuit, but also the ohmic resistance of a shunt containing a condenser. Now, the variation of
a positive resistance can be carried out (and is carried out) without expenditure of energy. Therefore, by
118
Theoretically, the authors mentioned are not limited to the use of a linear differential
equation with periodical coefficients, which provides the excitation conditions but can
say nothing as to the capability and characteristic of a stationary mode of operation. Now,
this question is no less important than the preceding one. This is why, in order to be
complete, we began with a general overview of the theory of parametric excitation, a
theory that should be supported by a nonlinear differential equation.
It is easy to show by energy considerations that it is possible to excite oscillations in the
latter by causing the capacitance of a circuit to vary suitably. Let us assume that, at an
initial instant in time, t = 0, when the current is equal to zero and the condenser possesses
a charge q, we reduce its capacitance by a small quantity ∆C. Having done this, we
supply the work (∆C/2C2) q2. Let us then allow the condenser to discharge, and at the
instant t = T/4 (T being the period of the circuit) when all the energy is magnetic and the
charge on the condenser is zero, let us restore its capacitance to its initial value. We do
this with no expenditure of work. At the instant t = T/2 the current is reduced to zero and
the condenser carries a charge that is greater or smaller than q, depending on whether the
energy supplied to the system at reduced capacitance is bigger or smaller than the energy
dissipated. At the instant t = T/2, the cycle of variation of capacitance is complete. Let us
state this differently. The oscillations will gradually increase, no matter how small the
initial charge (on the capacitor) provided that the following condition is fulfilled:
∆C
2 C2
q2
1
π
Ri
2
ω
>
,
m >
ε
2
in which
ε
πR
ωL
=
ε being the mean logarithmic decrement of the system and
m
=
∆C
2C
=
C max − C min
C max + C min
being the modulation index of the varying parameter.
The initial charge q is always present, even in the absence of outside disturbed inductions
(electric field lines, atmospheric discharges), owing to statistical fluctuations.
this principle alone, the device does not allow supplying to the circuit (by mechanical work) the energy
that is necessary for the excitation and maintenance of oscillations. There is no doubt that Barrow
observed in his experiments not only the parametric excitation of oscillations by periodic variation of the
capacitance, but also phenomena owing to the presence of an electronic tube and from regenerative
feedback.
119
By causing a periodic variation of the circuit capacitance (by some mechanical process)
at a frequency double that of the circuit’s own frequency, we can thereby excite electrical
oscillations without using any EMF. A similar reasoning is applicable in the case of
mechanical variation of the circuit’s self-induction.
This abridged discussion is enough to show that, in order to produce parametric
excitation, two conditions should be satisfied:
1. The frequency of variation of the parameter should be suitably selected (in our
example it is double the frequency of the circuit).
2. In the case of a given mean logarithmic decrement, the modulation index of
the parameter should be sufficiently high.11
A more complete study of the initiation of oscillations in phenomena of parametric
excitation leads, as is known, to the discussion of “unstable” solutions of linear
differential equations with periodic coefficients. If for example, the capacitance varies
according to the law:
1
C
=
1
( 1 + m cos ν t )
Co
We have for q = ∫ i d t the equation
L
d2 q
dq
1
+ R
+
dt
dt
C
(1
+ m cos ν t ) q = 0
(28)
which leads to the form
..
x + λ 2 ( 1 + m1 cos 2 τ ) x
= 0
(29)
(the equation of Mathieu), by taking
R
t
2
q = xe L ,
−
ωo2 =
1
L Co
,
τ=
νt
2
2δ =
R
L
2 (ωo2 − δ 2 )
,
λ2 =
,
m1 = m
ν2
ωo2
ωo2 − δ 2
Equation (29) was discussed from a mathematical viewpoint by Mathieu, Hill, Poincaré,
etc. It was also discussed, with respect to our present problem, by Rayleigh, then by
Andronov and Leontovitch [59] and by Van der Pol and Strutt [60]. It is known that
equations of the same type appear in a great number of problems of celestial mechanics,
optics, elasticity, acoustics, etc. The general solution of equation (29) is in the form:
x
11
= C1 e h τ χ o (τ) + C 2 e − h τ χ (− τ)
In the case of sinusoidal variation of capacitance the condition m > ε/2 is replaced by m > 2ε/π.
120
χ(τ) being a periodic function. In order for there to be parametric excitation, it is
necessary for h to have a real part which has an absolute value that is greater than δ. This
condition between the parameters λ and m defines the “unstable regions” of equation
(28). They are located in the vicinity of 2ω1/ν = n, n being an integer. Their boundaries
can be calculated by the approximate method of Rayleigh [61]. Thus, in the case of the
first region of instability (n = 1), we have, approximately, with terms of order m2:
1+
m2
− 4 ϑ2
4
≥
2 ω1
ν
≥
1−
m2
− 4 ϑ2
4
(30)
In order to find the second region (n = 2), terms on the order of m4 should be taken into
account. It follows that:
4+
2 2
m +
3
≥
4+
m 4 − 64 ϑ 2
2 2
m −
3
≥
2 ω1
ν
≥
(31)
m 4 − 64 ϑ 2
The width of the regions of instability decreases as mn.
As is shown by relations (4), (5) in order for the initiation to be possible, it is necessary
that, in the case of n = 1:
m > 4ϑ
(32)
and in the case of n = 2:
m > 2 2ϑ
(33)
The modulation index required for initiation is therefore greater, the decrement remaining
the same, in the case of n = 2, than for n = 1. Initiation becomes still more difficult in the
case n = 3, 4, etc. This is why the case of n = 1 is, for practical purposes, the most
interesting. It is the only one that we shall discuss here.
Once conditions (30) or (31) were satisfied, if the linear equation (29) was exact for any
values of q, the amplitude of the oscillations would increase without bound. Therefore, in
order for a system with periodically varying parameters to reach a stationary mode of
operation and become a generator of alternating current, it is necessary that it conform to
a nonlinear differential equation12. In this case, the linear equation (2) is only valid
(approximately) in the case of sufficiently small amplitudes. It allows only the setting up
of conditions (30), (31) which should confirm the parameters for which there was
initiation.
As will be seen, our experiments confirm this manner of regarding the phenomenon. In
order to obtain a permanent mode of operation, it is necessary to introduce into the circuit
12
The problem of the frequency modulation of a triode oscillator, also a nonlinear one, has been studied by
S. Rytov in an article carried in the journal Technical Physics USSR.
121
nonlinear components such as an iron cored coil, incandescent lamps, etc.; in the first
case, the equation of the problem is:
.
.
d ϕ (q )
1 + m cos 2 ω t
+ Rq +
q = 0
dt
Co
(34)
.
with the nonlinear dependence of the flux on the current, ϕ ( q ) , being given in the form
of a polynomial, for example. The mathematical theory of the phenomena includes the
search for periodic solutions of equation (34) and the discussion of their stability, in
addition to investigating the condition in which the state of equilibrium becomes unstable
(condition of initiation).
If the nonlinear part of the self-induction is small with respect to its linear part and if, in
addition, m is small, it is possible to apply the methods of Section 3 to this equation. In
the simplest hypothesis, in which
.
.
.
.
.
ϕ ( q ) = A + Lo q + β q 2 + γ q 3
(35)
by setting
1
,
Lo Co
ω o2 =
τ = ωt ,
ω 2 − ω o2
ξ =
ω2
χ =
q
qo
it follows that
..
.
χ + χ = µ f (χ , χ , τ)
whose solutions approximately satisfy the “truncated” equations
.
u
.
= −
v =
1  m

 (1 − ξ ) + 2 ϑ u
2  2

γ  

+  ξ + 1 z  v
4  

γ1  
1  m


z  u
 (1 − ξ ) + 2 ϑ v +  ξ +
2  2
4  


which can be treated by the methods of Section 2. The stationary solution is:
γ1
z = − ξ + Sign γ1
4
m 2 (1 − ξ) 2
4
− 4 δ2
(36)
Whence the condition:
m (1 − ξ) > 4 δ
(37)
which is practically identical to equation (32).
The curves of Figures (37) and (38), which can be called heteroparametric resonance
curves (c.f. Section 5), provide z (the square of the amplitude) as a function of the
“detuning” ξ. They differ essentially from the ordinary resonance curves and resonance
curves of the second class. As shown by Figure 57, whereas:
ξ < −
m2
(1 − ξ) 2
4
− 4 δ2
(36)
(where γ1 < 0)
122
no oscillations are observed. When:
ξ1
= −
m2
(1 − ξ) 2
4
− 4 δ2
the oscillations begin, starting with small amplitudes and, z increasing, are gradually
amplified. z increases linearly until, when the detuning assumes the value
ξ1
m2
(1 − ξ) 2
4
>
− 4 δ2
,
the oscillations stop abruptly.
As shown by equation (10), the zero-order approximation theory is only limited on one
side by the detuning ξ. Stable, finite amplitudes exist outside of the range of values of ξ
in which the conditions of initiation are fulfilled. In other words, the parametric
oscillations, once excited, can be “driven” into the regions where the equilibrium is
stable. When ξ varies in a reverse direction, the oscillations appear when ξ = +ξ1, then
decrease and vanish when ξ = −ξ1. Resistance, therefore, appears on only one side. In
order to calculate the extent of the circuit resistance as well as to solve several other
questions, µ will have to be used and the harmonics taken into consideration. Figure 58
shows that when γ1 > 0 the phenomenon is reversed: z increases whereas ξ decreases, and
the circuit resistance appears for positive values of ξ = +ξ1.
These two cases were observed experimentally by W. Lazarew [66]. V. Gouliaev and V.
Nigouline [62] and have shown that the same results are produced by expressing the flux,
as Dreyfus [63] and Zenneck [64] have proposed, by the function:
.
Φ(q )
=
.
.
Φ o arc tg ( k q ) + L 2 q .
123
We set the production of the effect of parametric excitation and the verification of the
theory described above as a goal in our
experiments.
In the first experiments we cause the
self-induction of a circuit to vary by
using the device shown in Figures 59,
60, 61. The variable self-induction was
composed of seven pairs of flat coils
fastened face to face with two parallel
discs on the periphery of two perimeters.
A serrated metallic disc was able to
rotate within the empty space fixed
between the coils. The teeth, likewise,
seven in number, were cut in such a
fashion as to synchronously occupy and
vacate the region simultaneous with the
field of the coils. As the disc spins, the
self induction of the circuit decreases
when the teeth enter the field of the
coils, and then increases when they leave
it. By using a disc made from duraluminum, we were able to achieve a peripheral
velocity of 220m/sec, and in this way attain a
considerable rate of variation of the parameter (1700-2000 per/sec). The coils were
supplied with iron cores divided in such a way as to concentrate the field and increase
self-induction. The apparatus permitted production of the parametric excitation of rather
powerful oscillations in the circuit shown in Figure 62, which has no current or voltage
source. By tuning this circuit to a frequency approximately equal to ω/2, ω being the
frequency of variation of the self-induction, the presence of oscillations whose frequency
is exactly equal to ω/2 may be ascertained. The amplitude swiftly increased until the
124
apparatus ruptured, either in the condenser or in the conductors of the circuit. In our
experiments, the voltage went as high as 12,000-15,000V. As required by theory, it was
necessary to introduce a nonlinear component into the system in order to
produce a stationary mode of operation. In the first experiments this was a bank of 100watt incandescent lamps connected into the oscillating circuit.
More detailed experiments were
carried out in our laboratory at the
Institute of Electrophysics of
Leningrad by W. Lazarew [66] with
an apparatus providing a greater
modulation index (40% instead of
14% as in the first experiments) and
a greater power (as high as 4kW).
The device causing the self-induction
to vary is depicted in Figures 63 and
64. The duraluminum rotor had eight
teeth. The self-induction varied at a rate of approximately 1900 per/sec, which provided
oscillations of approximately 950 per/sec. A stationary mode of operation was produced
using nonlinear self-induction, either in the iron cores of the coils of the stator or of a coil
with a special iron core possessing an auxiliary winding for direct current magnetization.
By causing the intensity of the latter to vary, it is possible to shift the operating point of
the iron on the curve of magnetic induction and modify the coefficients β and γ of
formulae (35), (36).
125
By measuring the maximum damping occurring with the initiation of oscillations, we find
that experiment agrees very satisfactorily with formula (4), as shown by the table below.
The experimental curves of
“heteroparametric” resonance clearly have the
features indicated below (Figures 65 and 66).
They rise and fall according to whether the
permanent mode of operation occurs owing to
the induction coils themselves or to a special
coil magnetized by a continuous current.
Figures 67 and 68 are oscillograms of the
stationary current. Figure 69 is an oscillogram
of the transient regime of operation.
A circuit diagram is provided in Figure 70 for
the excitation of electrical oscillations by the
126
periodic mechanical variation of the capacitance of a device. The oscillation circuit is
formed by condenser C, whose capacitance varies periodically, shunted with an oil
condenser C (serving to tune the circuit) and a self-induction coil L (several sections of
the secondary of a nonferrous core inductor). Condenser C (in Figure 71) includes
two systems of armatures, one stationary (stator) and the other rotating (rotor). The stator
is made up of 26 square aluminum plates, each one having 14 radial grooves arranged
symmetrically. The rotor is an assembly of 25 circular aluminum plates that are
perforated in the same manner as those of the stator and actuated by a direct current
motor with a maximum rotational speed of 4000 rpm. When the motor performs at n
revolutions per second, the capacitance varies with a frequency of 14n per/sec.
127
Six 220V neon tubes in series and a Hartmann-Braun static voltmeter of 1200V allowed
observation of the presence of oscillations and the evaluation of their intensity. Neon
tubes were used to limit the growth of the oscillations.
Since the rotor revolves at a fixed rate,
there is a range of values of c in which
the voltmeter fluctuates and the neon
tubes light up. This range corresponds
to the frequencies characteristic of the
circuit in the vicinity of n/2. Checking
the frequency of oscillations by means
of a tuning fork, we were able to
ascertain that it is constant within the whole range of excitation and equal to 7n (n being
measured with a tachometer).
If the neon tubes are removed, the system becomes linear, and it is possible to predict that
the oscillations will increase until the apparatus is ruptured. This is what actually occurs.
The voltage, which the neon tubes kept between 600-700V, increased in their absence
until a spark was produced between the armatures of the condenser (at between 20003000V). The frequency at which the spark occurs decreased proportionally as the
frequency characteristic of the circuit departs from the value of 7n. This observation is
likewise supported by theoretical considerations. According to the latter, the increase in
oscillations is reduced proportionally as one approaches the boundaries of the region of
instability of linear equation (29).
In these experiments the modulation index of the capacitance was 0.175.
The experimental curves of Figure 72 show the amplitude of the voltage excited as a
function of detuning and circuit damping. When the latter is increased, the range of
parametric excitations is reduced. Its measured width is certainly in agreement with that
provided by theoretical considerations.
128
In conclusion, permit us to add that H. Sekerska (Institute of Physics, Moscow) has
provided the details of a new process allowing
production of the Melde [47] phenomenon, i.e.
the parametric excitation of a vibrating string13.
A variable weight suspended by a metal wire
permits tuning the normal modes of this latter to
different frequencies. The wire completes the
circuit of an alternating current at 50 per/sec.
Consequently, the temperature and, therefore,
the tension of the wire are caused to vary
periodically at the rate of 100 per/sec. Provided
the current strength is sufficient, when one of the
normal modes of the wire is approximately
tuned to a frequency of 50 per/sec, the
parametric excitation of this mode is observed.
Section 8: Forced Oscillations of a System with Periodic Parameters - Parametric
Coupling
As we have seen in Section 7, if a system with periodically varying parameters is found
in an unstable region, the stationary state cannot be described by a linear equation. But, if
it is located in a stable region, the stationary mode differs from equilibrium only owing to
the effect of a periodic or quasi-periodic external force. Since the oscillations are small
enough, the phenomenon can then be described by an inhomogeneous linear equation
with periodic coefficients which, in the simplest case, is in the form:
..
.
q + 2 δ q + ρ (t ) q = f (t )
(38)
where δ is a damping coefficient, ρ(t) a periodic function, and f(t) a periodic or quasiperiodic function. Equation (38) represents a generalization of the well known
phenomena of resonances produced by the action of a periodic, or quasi-periodic, force
f(t) on a linear system with constant parameters (harmonic resonator). G. Gorielik14
subjected these phenomena of generalized resonance to a detailed theoretical study based
on a few considerations of principles that were provided by one of our group [57].
The phenomena, which have their source in harmonic resonators, endow a special
physical importance to sinusoidal functions and the harmonic analysis of an arbitrary
function. It is the language of sinusoidal functions that is used by theory to treat
resonance phenomena in systems with constant parameters.
13
14
The publication is in process.
The article is in this issue of Technical Physics of the USSR.
129
However, this language ceases to be able to handle systems with periodically varying
parameters. The function f(t) should satisfy certain conditions in order for there to be
resonance. The form of forced oscillations, which are expressed for a resonator with
periodic parameters, involves selectivity principles using new periodic or quasi-periodic
functions. These may be considered as generalizations of the sine and cosine functions,
and are determined by the intrinsic properties of the resonator.
In linear systems with periodic parameters, resonance phenomena, in the absence of an
external force and damping, present a character that changes, depending upon the regime
of operation of the resonator, i.e. whether the ideal system described by the equation
..
q + ρ (t ) q = 0
is in a stable region, at the boundary with an unstable region, or in an unstable region. In
the first case, the forced oscillations of resonance are proportional to 1/δ just as in the
case of a harmonic resonator. In the second case, there are two kinds of resonance: a
“strong” resonance in which the forced oscillations are proportional to 1/δ2 and a “weak”
resonance where they are proportional to 1/δ. If f(t) = E cos(ωt + ϕ), it has been found
that by varying the phase ϕ, it is possible to pass out of the weak resonance regime. In the
third case the resonance becomes more pronounced as the modulation index increases; if
the force is sinusoidal the nature of the phenomena is likewise a function of its phase.
The theory of resonators with periodically varying parameters takes into account certain
phenomena that have some similarity to those occurring in regenerative receivers. In the
latter, the coupling between the grid circuit and the plate circuit “regenerates”. This
allows partial restoration of the energy dissipated by the forced oscillations at the expense
of the plate battery. The theory of “regeneration” can be made using a linear equation
with constant coefficients by disregarding the nonlinear terms of the tube characteristic.
The regeneration decreases the coefficient of the dissipating term.
A “regeneration effect” is produced in a similar manner. In other words, by means of
utilizing a local source, it is possible to partially compensate for the losses of energy in a
circuit performing forced oscillations if one of its parameters is caused to vary at a
suitable frequency. The phenomenon is especially advantageous if the frequency
characteristic of the circuit, that of the EMF, and that of the variation of the parameter are
all in the ratio of 1:1:2. The modulation index plays a role analogous to that of the
coupling coefficient in a regenerative receiver. An essential difference between
conventional regeneration and this “parametric regeneration” is that this one depends
primarily on the phase that the parameter variation has relative to the EMF.
130
The effect of “parametric regeneration” was observed and studied at the Central Radio
Laboratory by Divilkovski and Rytov, as well as by Roubtchinski15.
The work of our laboratories has revealed new effects of parametric coupling between
oscillating systems. They differ essentially from the well-known phenomena that take
place in linear coupled systems.
Let us take, for example, a mass suspended by a string with one fixed point. This is the
elastic pendulum studied by G. Gorelik and a member of our group [67] regarding a
question in optics16. When the mass oscillated vertically, the length of the pendulum
underwent a periodic variation. If the frequency of the elastic oscillations is double that
of the angular oscillations, there will be parametric excitation of the latter by the former.
(Whence the term “parametric coupling”.) With respect to those phenomena discussed in
Section 7, the phenomenon has this difference: the variation of the parameter itself is a
function of the oscillation that it excites. In effect, the angular oscillation causes the
appearance of a centrifugal force at the frequency of the elastic oscillations, and
consequently it reacts on the latter by ordinary resonance. The naturally autonomous
coupling is expressed in the differential equations of the system by nonlinear terms.
Parametric coupling can likewise be seen in self-exciting systems with two degrees of
freedom: for example that of Figure 51, which was studied by Tourbovitsch17. Since the
operating point is selected in such a manner that the polynomial expressing the
characteristic of the tube has a term of the second degree, which is clearly indicated, the
equations of the system will be in the form:
..
x + ω2 x
..
. .
= 2 β x y + ...
y + 4 ω2 y =
.
β y 2 + ...
(We have written only the most significant terms.) It can be seen that oscillation y, at
frequency 2ω, causes a variation of circuit “resistance” at the critical frequency ω
(parametric action) and that, in return, oscillation x generates energy at frequency 2ω,
which reacts by resonance on oscillation y.
A. Tscharakhtschian18 studied the action of a sinusoidal force on two circuits with
parametric coupling forming a “parametric transformer”; in this system, the variation of
the current in the primary circuit causes the induction coil of the secondary circuit to vary
by modifying the magnetization of the iron-core coils. This allowed the production of
parametric excitation phenomena.
15
The publications are in preparation
This model allows providing a standard qualitative table of certain anomalies of combinative diffusion
(Raman effect) with CO2 molecules, which Fermi (Zeitschrift für Physik, Vol. 71, 1931, p. 250) treated
suitably by quantum mechanical methods.
17
The publication is in preparation.
18
The publication is in preparation.
16
131
Section 9: The Role of Statistics in Dynamic Systems
We shall conclude with a few words on some questions whose theoretical and
experimental study has just begun at our laboratories19, and which relate the theory of
oscillations to statistical theories.
Even in the simplest case of oscillation initiation in a triode oscillator, the role of
statistics can be clearly seen in the behavior of the system [69], [70]. Even without a
deviation from the normal, if, at the initial instant, the system is found in a state of
equilibrium, it will always diverge owing to random pulses [71] (produced, for example
by random fluctuations). Now, the time for the system to arrive at a stationary state is a
function of the magnitude of the initial perturbation (this, of course, concerns the time
necessary for the state of the system to arrive at a value differing from the stationary state
by a given value). However, in a triode oscillator, the oscillating circuit levels the pulses
to a “mean”. This is why their influence is always shown by the formation of small
oscillations characteristic of the circuit, whose magnitude is a function of the spectral
intensity of the pulses.
This leveling of the pulses will not have time to be carried out if the system is very “fast”
one. The instantaneous values of the current and voltage will then be random. The effect
of various initial conditions will be directly observed in systems that have a small
variation of initial conditions in the establishment of such and such a final state. This is
the case, for example, for rocker-relays (Kipp relays) which possess a saddle (at the
origin of the coordinates) and two stable nodes (one on the right and the other on the left
of the saddle). It is possible to produce a relay having two stable nodes located on the
phase plane, to the right and to the left of saddle O and symmetrical with respect to the
latter. Let us assume that when the relay is triggered, the representative point is located at
O. It is clear that, if the initial perturbations are distributed according to the laws of
chance, deviations to the right and to the left will likewise be probable and if,
consequently, the system is triggered without applying an external pulse, it will travel
either towards the right node or towards the left node, according to a statistical law. This
statistical law will be altered if steady pulses are used. By comparing the effect of steady
pulses with those of random pulses, it is possible to evaluate the magnitude of the latter.
Experiments of this kind were performed in one of our laboratories. The amplified
current fluctuations of a vacuum tube were used as a source of random pulses. The
magnitude of the fluctuations determined in this way agrees satisfactorily with the wellknown theoretical and experimental findings of Schottky.
19
The experiments are still in progress.
132
It is possible to raise another question concerning the transition of a system from one
state to another owing to the effect of random pulses. This problem was treated
theoretically using the Fokker equation. In particular, L. Pontryagin mathematically
calculated the expected duration of the transition from one state to the other. Using the
results obtained it is possible to compute the duration of the transition from one stationary
state to another, thus allowing discussion of the mean duration that the system remained
in such-and-such a stationary state. Naturally, this is a function of the amplitude of the
random pulses. We were able to experimentally ascertain the existence of these
“spontaneous” transitions from one stationary state to the other.
By observing how long the system remained in such-and-such a stationary state, it is
possible to determine, the magnitude of the random pulses by using some plausible
supplementary postulates. Note that, in principle, the existence of the random pulses
limits the precision with which it is possible to attribute a definite period to an oscillatory
phenomenon.
Section 10: Closing Remarks on Oscillations
Our desire was to provide a short overview of some research carried out during the last
few years in the laboratory of the Institute of Physics of the University of Moscow, the
Central Radio Laboratory (Leningrad), the Laboratory of Nonlinear Oscillations of the
Institute of Electrophysics (Leningrad) and the University of Gorki. In order to not
encumber our report, we have omitted a whole series of questions relating, for example,
to systems with several degrees of freedom20. And, within the topics that we have
discussed, we have had to confine ourselves to the most essential topics. In this way a
great number of interesting details have often been sacrificed. References are provided
below. Soon a number of works, so far published only in Russian, will appear in other
languages, too.
References
1.
2.
3.
4.
5.
6.
7.
8.
20
Poincaré, Henri, “On the Curves Defined by a Differential Equation,” Oeuvres 1928,
Volume 1.
Poincaré, Henri, ”New Methods of Celestial Mechanics”.
Liapounov, A., “The General Problem of Stability of Movement,” Charkow 1892, Annals
of the Faculty of Sciences of Toulouse, 1907.
Birkhoff, “Dynamical Systems,” New York, 1927.
Andronov, A., “The Limiting Cycles of Poincaré and the Theory of Self-Maintained
Oscillations,” Comptes-Rendus, Volume 189, p. 559, 1927.
Van der Pol, B., “Forced Oscillations in a Circuit with Nonlinear Resistance,”
Philosophical Magazine, Volume 7, Number 13, pp. 3-65, January 1927.
Van der Pol, B., “On Oscillation Hysteresis in a Triode Generator with Two Degrees of
Freedom,” Ibid, Volume 6, Number 43, p. 700, 1922.
Mandelstam, L., Papalexi, N., “Concerning the Basis of a Method for the Approximate
Solution of Differential Equations,” Zhurnal Eksperim. i Tekhnicheskoi Fiziki, Volume 4 p.
2, 1934; Technical Physics in the USSR, Volume 1, Number 4, pg 415, 1934.
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133
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with Two Coupled Oscillating Circuits,” Zhur. Tekhnicheskoi Fiziki, Vol. IV, p. 158, 1934.
Pontryagin, L., “Concerning Self-Excitation Systems Resembling Hamilton Systems,”
Zhurnal Eksper. i Teoretich. Fiziki, Volume 4, #9, 1934, and Volume 6, # 25, 1934.
Andronov, A, Witt, A., “On the Mathematical Theory of Cooling,” Zhurnal Prikladnoi
Fiziki, Volume VII, Number 4, pp. 3-20, 1930.
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Degrees of Freedom,” Zhurnal Tekhnicheskoi Fiziki, Volume IV, Number 1, 1934.
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Strelkov, S., “The Froud Pendulum,” Zhur. Tekhnicheskoi Fiziki, Volume III, p. 563, 1933.
Chaikin, S., “Continuous and ‘Discontinuous’ Oscillations,” Zhurnal Prikladnoi Fiziki,
Volume 7, p. 6, 1930.
Chaikin, S., Kaidanowsky, N., “Mechanical Relaxation Oscillations,” Zhurnal
Teknicheskoi Fiziki, Volume 3, p. 1, 1933.
Van der Pol, B., “Relaxation Oscillations,” Philosophical Magazine, Number 43, pp. 978992, 1926; Z. f. Hochfrequenztechn. Number 28, pg 176, 1926; Z. f. Hochfrequenztechn.,
Number 29, p. 114, 1927.
Andronov, A., and Witt, A., “Discontinuous Periodic Movements and Theory of
Multivibrators of Abraham and Bloch,” Bulletin of the Academy of Sciences of the USSR,
Volume 189, 1930.
Witt, A., “Self-Excitations in Continuously Distributed Systems,” Phys. Z. der SowjetunionVolume 5, p. 777, 1934; Zhurnal Tekhnicheskoi Fiziki, Volume 1, 1934.
Moeller, G., “Concerning Disturbance Free Direct Current Reception with the Oscillating
Detector,” Jahrb. D. drahtl. Telegr. U. Telef. (Yearbook of Radio Telegraphy and
Telephony) Volume 17, p. 256, 1921.
Vincent, J., H., Phys. Soc. Proc., p. 84, February 1920.
Van der Pol, B., and Van der Mark, “Frequency Demultiplication,” Nature, p. 363, 10
September 1927.
Koga, Proceedings of the Institute of Radio Engineers, Volume 15, p. 679, 1927.
Ollendorff, “Forced Oscillations in Induced Systems” Arch. f. Elektrotechnik, Volume 16,
p. 280, 1926.
Andronov, A., and Witt, A., “On the Theory of Drift of Van der Pol,” Arch. f.
Elektrotechnik, Volume 24, p. 99, 1930.
Chaikin, S., and Witt, A., “Drift in the Case of Small Amplitudes,” Zhurnal Tekhnicheskoi
Fiziki, Volume 1, Number 5, p. 428, 1931.
Appleton, Proc. Cambr. Soc. Volume 23, p. 231, 1923.
Chaikin, Maisels, and Witt, “Direct method of Measurement of Field Intensities,” Bulletin
of Industrial Electrotechnology, Number 3, 1933.
Chaikin and Theodortschik, K., “Acoustical Drift,” Zhurnal Tekhnicheskoi Fiziki, Volume
II, Number 1, p. 111, 1932.
Chaikin, S., “Quantitative Investigations of Acoustical Drift Phenomena,” ENT, Volume 9,
Number 10, 1932.
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Groszkowsky, Proceeding of the Institute of Radio Engineers, Volume 18, p. 1960, 1930.
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Zhurnal Tekhnicheskoi Fiziki, Volume IV, Number 1, 1934.
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IV, Number 1, 1934.
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of Resonance Phenomena of the Second Class,” Zhurnal Tekhnicheskoi Fiziki, Volume IV,
Number 1, 1934.
Appleton and Van der Pol, Philosophical Magazine, Volume 43, p. 177, 1922
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Triode Oscillators,” Philosophical Mag., Volume 6, Numbers 45, and 267, p. 401, 1923.
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Japan, Number 506, Volume 53, p. 21, 1933.
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Volume IV, p. 1141, 1933.
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Telegraphy and Telephony) Number 19, p. 170, 1922.
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Barrow, W., L., “On the Oscillations of a Circuit Having Periodically Varying
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Fiziki, Volume 3, Number 3, 1933; Phys. Z. der Sowjetunion, Volume 6, p. 1, 1934.
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