1. INTRODUCTION
One of the most significant trends of the twentieth century is the
departmentalization of science. The broad field of human experience
has been divided into innumerable sections and subsections, each mas-
tered only by a small group of specialists. One of the consequences of
this division of science is that structural similarities between different
fields are often overlooked. The recognition of these similarities can
frequently be a great source of inspiration; a source closed to the
specialist who fails to develop any interest outside his own field.
A typical example of such a similarity is the occurence of oscillatory
phenomena in such diversified fields as electronics, aerodynamics, eco-
nomics, biology, etc. In recent years a uniform theory of oscillations
applicable to all of these fields has been developed. However, the main
applications of this theory at present are in electrical and aeronautical
engineering. Mathematically, oscillatory phenomena lead to nonlinear
differential equations. Because of this, the theory of oscillations is often
called "nonlinear mechanics."
Most important for the study of an oscillatory system are the steady-
state oscillations. They may be stable or unstable, depending on whether
the oscillator will or will not return to its original state after being
subjected to a small disturbance (noise, etc.). An oscillator may have
several possible steady-state oscillations. In this bulletin particular
emphasis is placed on the nonsteady-state or transient oscillations.
These terms, nonsteady-state and transient, are used synonomously.
The study of these transient oscillations is often important for the
complete understanding of the circuit behavior. For example, in a
system having several possible steady-state oscillations, the transient
study predicts which one of these will be reached from a given set of
initial conditions. It also determines the manner in which oscillations
will build up and describes the behavior of the system if a circuit para-
meter is changed abruptly. Generally, a study of the transient oscilla-
tions results in a more complete understanding of the oscillatory system.
One important problem to be overcome by the investigator is that
of representing the transient oscillations. Analytical representation is
not practicable, since, even if an approximate formula could be found,
it would be so complicated that it could not be interpreted easily. The
ILLINOIS ENGINEERING EXPERIMENT STATION
most common graphical method of representation makes use of a plane
with time as the abscissa and the dependent variable as the ordinate.
This method is, however, limited to the cases in which the differential
equation can be reduced to
dt
dt = f (x,t).
This reduction is possible for only a few oscillators.
Autonomous systems with one degree of freedom can often be
described by the differential equation:
d2x / dx dx ( dx
dt2 + f (x' dt) dt ± g z'xdi = 0 (1.1)
The corresponding oscillations can be represented in a plane with x as
the abscissa and dx/dt as the ordinate, the "phase-plane." Such a repre-
sentation is shown in Chapter II.
No general method for the graphical representation of oscillations
in more complicated systems is available. If it can be assumed, however,
that the oscillations are almost sinusoidal, then a large class of oscillators
can be represented in some sort of a phase plane. In this new plane, an
oscillation with a certain amplitude, phase angle, etc., will correspond
to a single point.
II. AUTONOMOUS OSCILLATIONS WITH ONE
DEGREE OF FREEDOM
The study of the autonomous oscillators with one degree of freedom
takes a rather special place in nonlinear mechanics insofar as these
oscillators can be treated conclusively and in all generality by topological
methods. Equation 1.1, corresponding to autonomous oscillators with
one degree of freedom, can be transformed into two simultaneous dif-
ferential equations of the first order by the substitution dx/dt = y
dy
dt = -f (x,y) y - g (x,y) x
dx
dt y (2.1)
The time can be eliminated from these two equations by dividing
one by the other:
dy = -f (x,y) - g (x,y) X (2.2)
dx y
The solutions of this equation can be represented by curves in the x-y
plane, which can be found by graphical construction (method of iso-
clines). If the system is at rest, then x and y are constant in time;
that is,
dx dy
dt dt
and the differential, dy/dx, at such a point, is indeterminate (singular
point).
The x-y plane may also contain a number of closed curves, or "limit
cycles," corresponding to periodic solutions of Eq. 1.1. The singular
points and the limit cycles determine the structure of the x-y plane.
The state of the oscillator is completely described at any one time by a
point in the x-y plane. If the system is not disturbed, then this "repre-
sentative" point will move along a curve defined by Eq. 2.2.
Typical oscillatory systems that can be treated by this method are,
for example, the ordinary triode oscillator, the Prony-brake and the
multivibrator. It is possible but impractical to treat more complicated
systems by this method because spaces with three or more dimensions
are required to describe them.
III. ALMOST SINUSOIDAL OSCILLATIONS
A large group of oscillatory phenomena can be represented in a plane
entirely different from that discussed in Chapter II provided that it
can be assumed that the oscillations are almost sinusoidal. In electrical
circuits, for example, oscillations are nearly sinusoidal if the quality
factors (Q's) of the resonant circuits are high. Then if it is assumed
that the oscillations are purely sinusoidal, the circuits can be described
completely by a few parameters; for example, by the amplitude of
oscillation, the phase angle relative to an external sinusoidal voltage,
etc. For steady-state oscillations, these parameters remain constant;
for transient oscillations, they will change. A representation in a plane
is possible if the system can be described by two parameters only, say
X and Y. When one of these parameters is a phase angle, the plane
lies on the surface of a cylinder so that the lines, 0 = 0 and = 27r, are
identical. Many oscillatory phenomena can be described by two para-
meters and can, hence, be represented in this plane.
The rate of change of X and Y is determined by the state of the
system and, hence, depends on X and Y only.
dX
dt --= (X,Y)
dY
dt = A (X,Y) (3.1)
Equations 3.1 can be obtained by various analytical and experimental
methods. The method used in this bulletin is that of equivalent lineariza-
tion. The mathematical formulation of this method has been described
extensively in Buls. 395 and 400, and hence is not presented here.
The time can be eliminated from Eqs. 3.1 by dividing one by the
other.
dX _ f (X, Y)
dY f (X,Y)
The solutions of this differential equation can be represented as
curves in the X-Y plane. If the system is not disturbed, then the
"representative point," corresponding to an oscillation with parameters
X and Y, will move along one of these curves. For steady-state oscilla-
tions, X and Y remain constant in time.
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
dX dY
- d 0 (3.3)
dt dt
This corresponds to a singular point in the X-Y plane. The symbols,
X0 and Yo, are used for the parameters of the steady-state oscillation.
A singular point and the corresponding steady-state oscillation are
stable if the oscillation and the corresponding point will return to their
original position after a small disturbance. In order to discuss the
behavior of the system in the neighborhood of the steady-state oscilla-
tions, it is necessary to expand Eq. 3.1 around Xo and Yo. Letting
SX = X - X0
SY = Y - Y0
then the first terms of the expansion are
d (6X) f af a y
t - X x + SY +...
dt aX aY
d(aY) f 2X (3f4)
d()tX + 0-Y + ... (3.4)
dt aX aY
It is assumed that SX and 5Y are small and therefore that the higher
terms of this expansion can be neglected. Equation 3.4 is a set of two
simultaneous linear differential equations which can be solved by the
usual methods. The variables 6X and 6Y will approach zero from any
initial conditions in the neighborhood of the steady-state oscillations
provided that both the roots Ki and K2 of Eq. 3.5 have negative real parts.
af1 _ af1
9X f aY
ax =0
af2 af2
a- -a K (3.5)
9X aY
Necessary and sufficient conditions for this to exist are that:
9f + - < 0
ax ah
af af2 f f2 > 0 (3.6)
aX aY aY aX
Like the limit cycles of the x-y plane described in Chapter II, the
X-Y plane may contain closed curves which are called "drift curves."
Physically, these drift curves correspond to oscillations with periodically
varying amplitude.
IV. PARAMETRIC EXCITATION OF NONLINEAR SYSTEMS
Oscillations in resonant circuits can be generated in several different
ways; for example, by an external sinusoidal force or by a negative
resistance. Another possibility is the excitation of the circuit by means
of the periodic variation of a circuit parameter. This excitation is
particularly strong if the capacitance or inductance of the circuit is
varied with a frequency approximately twice that of the resonant
frequency of the circuit. The amplitude of the oscillations so generated
will build up to a value determined by the nonlinearities of the system.
Assume for example that in the circuit of Fig. 4.1, the distance d
between the plates of the capacitance C is varied according to
d = do + di cos 2wt (4.1)
where di<<do and co is close to the resonant frequency of the circuit.
Then if the resistance R is sufficiently small, oscillations with a fre-
quency w will build up. As is shown later, they may reach a stationary
value if either R or L is a function of the current i (nonlinear resistance
or inductance).
In the first part of this chapter, it is assumed that R and L are
constant; R= Ro, L= Lo. The capacitance C corresponding to Eq. 4.1 is:
C __Co
1 + 7 cos 2wt
where 7= di/do<<1.
The circuit of Fig. 4.1 can be transformed to an "equivalent linear
circuit" by the method of equivalent linearization described in Buls.
395 and 400. The first step is to split the capacitance C into two parts,
C1 and Cio, as shown in Fig. 4.2.
)2
Cio = W Co (4.2)
1 - - +7 cos 2wt
5002
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
where co'o= 1/LOCo is the resonant frequency of the system. It was
assumed that w--wo; therefore Cio is very large.
For a large C10 and a small R0, the current i is approximately
i I cos (Wt + 4)
The amplitude I and the phase 4 will of course change, but over a few
cycles they will remain practically constant. The voltage v across Ro
and Co1 is (Fig. 4.2):
v = idt + iRo
1 - w/wo02 - -y/2 cos 204 sin (wt + -)
oCo
+ I( R + --C sin 20) cos (wt + ()
+1 2I - sin (3cot + 0)
Due to the component of v at frequency 3w a small current at this
frequency will flow through Lo and C1. Since this component is very
small, however, it can be neglected. As is shown, the components at
frequency w will cause a change of I and 04.
d
C06
Fig. 4.1. General Oscillatory Circuit Fig. 4.2. Oscillatory Circuit with Variable
with Excitation by Variable Capacitance Capacitance Excitation. R and L Constant
If all components of the current except those at frequency w are
neglected, then the circuit of Fig. 4.2 can be replaced by that of Fig.
4.3 where:
R = Ro + sin 240
2wCo
C = Co(4.3)
1 - wC2/o2 - 7/2 cos 204 (4.3)
The circuit of Fig. 4.3 is linear and can be discussed by the method
of linear analysis.
ILLINOIS ENGINEERING EXPERIMENT STATION
2 Fig. 4.3. Equivalent Linearized
Circuit of Fig. 4.2
Ce
The energy stored in the oscillator, LOP/2, will remain approxi-
mately constant over one cycle. Over a longer period of time, it will
change slowly since part of it is dissipated in the resistance Re.
d (L-OP/2) - 2 R
dt 2
or
dl I (Ro + sin 2) (4.4)
dY 2L, 2Co0
Similarly, the phase angle will change slowly.
C A dt L, C,
SC, + C,
or
de 1 Co
dt 2 0 C,
_A A - ^ cos 2A (4.5)
where
A0 == COO - W
The time can be eliminated from these two equations by dividing one
by the other:
de Ac - (7/4)co cos 20
dl - (I/2Lo)[Ro + 2 sin 20]
z - cos 2p (4.6)
-I (p +sin 2)
where
z 4--
2RiooCo
p-z is proportional to the "detuning" of the resonant circuit.
z is proportional to the "detuning" of the resonant circuit.
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
The trajectories in the I-0 plane corresponding to Eq. 4.6 permit
graphical representation of the transient solutions. They can be ob-
tained either by the method of isoclines or by direct integration. Typical
groups of trajectories are shown in Fig. 4.4.
Steady-state oscillations exist if dl/dt = dk/dt = 0. This corresponds
to singular points in the I-0 plane. The stability of these singular points
and of the corresponding steady-state oscillations can be determined
from the behavior of the trajectories in their neighborhood. It can be
seen that they are unstable, because after a small disturbance of k the
amplitude I will increase or decrease indefinitely.
In order to obtain stable steady-state oscillations in the circuit of
Fig. 4.1, it is necessary that the circuit contain a nonlinear inductance
or a nonlinear resistance.
( ( (
0 (C) z- O, p = (Small Resistance)
0 (b) z-2,, p=O
(a z= , p=-2 (Large Resistance)
Fig. 4.4. Trajectories
77
7T
a.
(a) z=0,, o=0
p
I
rf
*
*
,_.(_
I
ILLINOIS ENGINEERING EXPERIMENT STATION
Consider first the case where the inductance L of Fig. 4.1 is a func-
tion of the current i passing through it:
L = Lo (1 + ei2) e2 « 1
The oscillatory curcuit corresponding to Fig. 4.1 with a nonlinear in-
ductance is shown in Fig. 4.5. It is again assumed that
i= I cos (wt + 4)
The voltage v is then:
v = d(LoEi) + Roi + 1 fidt
dt CIO
3 I 1 - W2/W02 - 7/2 cos 2 sin (wt
= I --eLoPI + - -C sin (wt + )
S4 wCoc
+ I (Ro + 2y cos 24 cos (wt + 4)
+ terms at frequency 3w.
The terms at the frequency 3w are again neglected. The equivalent
conductance and capacitance are:
Re = Ro + 2 _ sin 24
Co
C, = c 3 (4.7)
1 - W2/w0o cos 24 - T E2
The differential equation corresponding to Eq. 4.6 is:
de A0w - (7/4)w cos 24 - 8 eJ2w
dl - (I/2Lo)[Ro +(7/2wCo)sin 24]
z - cos 24 - aI2
I (p + sin 24)
where
z= 4--
o07
2RowOCo
-y
p=-
3 e
2 ly
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
Fiz Fig. 4.5. Oscillatory Circuit with
Variable Capacitance Excitation.
Inductance Nonlinear
For a steady-state oscillation it is necessary that
dl d4_ 0
dt dt ~
z - al2 - cos 24 = 0
p + sin 24p = 0
The symbols I0 and 0o are used for the
Eqs. 4.8. For each set of z, a, p there
solutions:
102 = z - 1 - p2
102 + / 1 - p2
a
(4.8)
values of I and 0 that satisfy
are at most two independent
(cos 20 > 0)
(cos 24 < 0)
Only one of these solutions is stable, however.
The characteristic equation corresponding to Eqs. 4.4 and 4.5 is
Io ORe
2Lo aI K
+ Co 0 (1
S2 I Ce
Io OR,
2Lo 04
WCo 0 (1
2 61 \ C -/
Necessary and sufficient conditions for stability are that
Io OR WCo a( 1
2Lo 1 2 04 C,
l 0 \ C_ OR7 0 (1 ) >
01T ao4C /T ao4?8 C'
or, for I 10,
=0
77
0
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 4.6. Region of Stable Oscillations
for a > 0
I
Fig. 4.7. Trajectories for p = 0, z = 2 (Large Detuning)
0
17
Fig. 4.8. Trajectories for p = '/2, z = 2 (Large Detuning)
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
or, in terms of 0 that
sin 24 < 0
a cos 2a < 0
The first of these inequalities is satisfied if p>0 (Eq. 4.8). The
second inequality shows that, of the two solutions for l0, only one is
stable. For a>0 the solution with cos 204<0 is stable; for a<0 the
solution with cos 24 > 0 is stable. The regions of stable oscillations can
be represented in the p-z plane (Fig. 4.6). Additional boundaries of the
region correspond to: 12 > 0 and p = -sin 20 < 1. The region does
not depend on the magnitude of a as long as a remains finite. If p is
small, then the inductance may arbitrarily be increased or decreased
depending on the sign of a. This result, however, is due to the many
simplifications made for this calculation; for example, with regard to
the dependence of the inductance, L, on the current, i. In any physical
system, stable oscillations will occur only within a finite range of varia-
tion of the inductance. Figures 4.7 and 4.8 show two sets of trajectories
corresponding to systems with nonlinear inductance. For Fig. 4.7 the
system is assumed to contain no damping (p=0) and large detuning
(z =2). The singularity is of the center-type and all trajectories are
also drift curves. The case p=O cannot occur in a physical system,
however, since a small amount of damping is necessarily contained in
any system. Figure 4.8 corresponds to small damping (p = Y2) and large
detuning (z = 2). The stable singularity is a focal point which, however,
cannot be reached by the system from rest (I= 0) unless the detuning
is considerably smaller. This corresponds to hard self-excitation of the
ordinary oscillator.
Stable steady-state oscillations may also exist if the circuit contains
a resistance that is a function of the current i passing through it.
(See Fig. 4.9.)
R = Ro (1 + gi')
R0 and ju may be positive or negative. If Ro<0, u,>0, and -y=0 (C is
constant in time), then the circuit of Fig. 4.1 will oscillate with an
amplitude
4
J2 = - (4.10)
3A
which result is used later.
It can be shown (Fig. 4.10) that the equivalent capacitance and
resistance are
Co
C = 1 - W1/wo - (7/2 cos)20
R, = Ro ( +- + -sin 2
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 4.9. Oscillatory Circuit with Variable Fig. 4.10. Equivalent Line
Capacitance Excitation. Resistance Nonlinear Circuit of Fig. 4.9
and that the differential equation corresponding to Eq. 4.6 is
d _ z - cos 24
dl I (p (1 + m2) + sin 2)
where
z=4WY
2RowCo
The conditions for steady state oscillations are
The conditions for steady state oscillations are
z - cos 24 = 0
p (1 + 3%AI) + sin 20 = 0
arized
(4.11)
(4.12)
and the conditions for stability
sin 24 < 0
pn > 0
Figure 4.11 shows the region in the p-z plane where stable oscilla-
tions are possible. Of these stable oscillations two different cases must
be distinguished - p <0 and p > 0.
Fig. 4.11. Region of Stable Oscillations
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
0<p<1
The amplitude of oscillation I can be found from Eqs. 4.12. The
dependence of the amplitude on detuning is shown in Fig. 4.12. Outside
the "region of synchronization" the amplitude of oscillation is zero. A
group of trajectories for z =12, p = , p = 1 is shown in Fig. 4.13.
p<0
The dependence of the amplitude on the detuning for this case is
shown in Fig. 4.12b. Inside the region of synchronization there is little
difference between this and case a, but outside the region the amplitude
does not drop to zero. This is due to the fact that even without para-
metric excitation the circuit will oscillate.
z
(a) p.>0 (b) p <0
Fig. 4.12. Dependence of the Amplitude I on Detuning
((K I
Fig. 4.13. Trajectories for Circuit Nonlinear Resistance, z -= /2,, P = 3, P = I
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 4.14. Trajectories for z = 1.05, p = -'/2, it = -1
Fig. 4.15. Oscillations Corresponding to Drift Curve Shown in Fig. 4.14
Figure 4.14 shows the trajectories for z = 1.05. Because this is out-
side the "region of synchronization," no singular points exist. There
exists, however, one (and only one) drift curve shown dotted in Fig.
4.13. Figure 4.15 shows the oscillation corresponding to that curve.
This figure shows a good agreement with some experimental results
published by W. L. Barrow.'"
*Parenthesized superscripts refer to correspondingly numbered entries in the Bibliography.
V. SYNCHRONIZATION
In this chapter the oscillatory circuit of Fig. 5.1 is discussed. This
circuit is the equivalent of an ordinary plate-tuned oscillator with an
external sinusoidal synchronizing voltage vs. When v, is zero, the circuit
will oscillate with a frequency,
1
0 V'LC
The frequency of oscillation w for v, 0- is not necessarily wo. As the
frequency of the external voltage w, is varied and becomes approxi-
mately equal to 00, w will suddenly change from wo to so. This phenomena
is called locking, synchronization, or entrainment of frequency. It occurs
not only if , a-- but also if ws/ýwo=p/q, where p and q are "small
integers." This phenomenon is called subharmonic synchronization and
has been discussed in detail in Bul. 400.
If it is assumed that v, = V, cos w,t, then the voltage vi between points
A and B is approximately sinusoidal, particularly if the losses in the
resonant circuits are small
vi - V cos (wt + )
It was shown in Bul. 400 that the circuit of Fig. 5.1 can then be
reduced to that of Fig. 5.2 in which the non-linear element is replaced
by an "equivalent network." The differential equations for V and q6
for this circuit are
dV V [G + G (5.1)
dt 2C
d - 1 C (5.2)
dt 2 -
where Aw = wo - Ws
These equations determine the transient behavior of the oscillator.
Dividing Eq. 5.2 by Eq. 5.1 eliminates the time
do_ _ 2XwC - c0oC, (5.3)
dV v (G + Ge)
This equation permits representation in a plane with V and 4 as
coordinates, the V-4 plane. Steady-state oscillations (d V/dt = de/dt = 0)
correspond to singular points. Besides singular points, the V-0 plane
ILLINOIS ENGINEERING EXPERIMENT STATION
C
Fig. 5.7. Equivalent Circuit of a Tuned-Plate Oscillator with External Synchronizing Voltage
QGe Ce L 6 C
Fig. 5.2. Equivalent Linearized Circuit of Fig. 5.1
may also contain drift curves. They occur particularly if an oscillator
is close to synchronization.""
As an example, the special case w = 3w, is presented here. The fre-
quency of the oscillator is three times that of the external voltage
(frequency-multiplication). It is assumed that the current i of Fig. 5.1
can be expressed by a third degree polynomial of the voltage v
i = av + 3v2 + 7v3
The equivalent conductance and capacitance for w=3o, and a third
degree polynomial were calculated in Bul. 400. They are:
3 ly Vý3
G = a + 7(V2 + 2V,2) + - cos
Ce = V sin 0 (5.4)
4co V
Using these two expressions, Eq. 5.3 may be written as
d zV2 +(V8/V)sin (5.5)
dV 3V (V - Vo2) + V3 cos
where
8wC Aco
V,2 4 (G + a) _ 2V
3-y
Introducing the dimensionless parameters X= V/Vo and X,/Vo, Eq.
5.5 can be simplified to
d Xz Xz + 3 sinX
dX X [3X (X2 - 1) + X,3 cos (5.6)
The solutions of this differential equation can be represented as
trajectories in the X-4 plane. The singular points in this plane are
stable if they fall inside the region of stability in Fig. 5.3. Figure 5.4
Bul. 421. OSCILLATIONS IN NONLINEAR SYSTEMS
Fig. 5.3. Regions of Stability and Variation of X
as a Function of p for p = 3, q = 1
oS 10 v
Amplitude
Fig. 5.4. Trajectories with Stable Singular Point
r
ILLINOIS ENGINEERING EXPERIMENT STATION
shows the family of trajectories in the X-4 plane corresponding to
a stable singular point (point A). This point is a focus. Figure 5.5
shows the trajectories in the X-0 plane for an unstable singular point
(point B). This figure also shows a drift curve corresponding to "near
synchronization."
Amphliude
Fig. 5.5. Trajectories with Unstable Singular Point
VI. SIMULTANEOUS OSCILLATIONS
One of the most controversial and also one of the more interesting
phases of nonlinear mechanics is that of simultaneous oscillations in
circuits with two degrees of freedom. In recent years a number of
papers dealing with this problem have been written and it has been
proven both theoretically and experimentally that simultaneous oscilla-
tions are possible. (171 19)
For simultaneous oscillations, the voltage between points A and B
of Fig. 6.1 is, approximately,
vi = V1 cos (wit + 0i)
and the voltage between B and C is
v2 = V2 cos (Wt + 42) (6.1)
where 1 # C2. A further distinction between wi and W2 gives rise to the
necessity for distinguishing between synchronous and asynchronous
simultaneous oscillations. For synchronous oscillations the frequencies
Wi and 02 are related by PWi = qW2, where p and q are small integers,
and comes about as a result of an effect of mutual synchronization.
When the sum (p+q) is large, the synchronization becomes so weak
that it can no longer counteract the continuous disturbances caused by
line fluctuations, noise, etc. For asynchronous oscillations no relation
P01 = qgc2 is preserved over an appreciable length of time. Mathemati-
cally it is assumed in this case that the ratio Wi/w2 is constant but irra-
tional. Asynchronous simultaneous oscillations were discussed briefly
in Bul. 395.
The oscillations in a circuit with two degrees of freedom can be
represented in a four-dimensional phase space with (vi, dvi/dt, v2, dv2/dt)
as coordinates. For a successful treatment, however, it is imperative
that the oscillations of any system be represented in a plane or at most
in a three-dimensional space. The four dimensions of the present case
can be reduced to three if it can be assumed that vi and v2 are sinusoidal.
The oscillations can then be represented in a space having as coordinates
Vi, V2 and, for synchronous oscillations, the angle = p - q42 which
is defined as the phase angle between vi and v2. An oscillation which
has amplitudes Vi, V2 and phase angle 4 corresponds then to a point
in this space.
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 6.1. Oscillator with Two Degrees of Freedom
The voltage v of Fig. 6.1 is
v = v1 + v2 = V1 cos (wit + 01) + V2 COS (w2t + 02)
This voltage applied to the nonlinear element will produce a current
i = In cos (wit + 01) + 112 sin (wit + 0i)
+ 121 cos (W2t + 02) + 122 sin (w2t + 02)
+ terms at frequencies other than w1 or W2. (6.2)
As shown in Bul. 395, Chapter I, the circuit of Fig. 6.1 can then be
replaced by the two circuits of Fig. 6.2 where
Gie = TIll. CI 112
V,1 WVl (6.3)
G2e = 121 Ce - = 122
V2 W2 V2
where the equivalent conductances and capacitances are functions of
V1, V2, and 4.
If p and q are small integers, then the expression 4 = pfl -q02 is
determined independent of a shift of the time axis. However, if either
p or q is irrational, 4 can still be adjusted arbitrarily close to any value
Fig. 6.2. frequent Circuit of Fig. 6.1
Fig. 6.2. Equivalent Circuit of Fig. 6.1
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
by a suitable shift of the time axis. Since this shift is a purely mathe-
matical operation and cannot have any influence on the oscillator, it
is clear that the equivalent impedances for asynchronous oscillations
cannot be functions of 4.
The transient and steady-state behavior of the system can be
calculated from the circuits of Fig. 6.2. The conductances of Fig. 6.2
will damp the oscillations and change the amplitudes. The capacitances
C1 and C2 will change the frequencies, wi and 02, and cause a shift of
the phase angle, 0.
The corresponding differential equations are for asynchronous oscil-
lations
dVi Vi
dV = - (G( + G-e)
dt 2C1 1
dV V (G2 + G2) (6.5)
dt 2C2
and for synchronous oscillations
dV1 V ±
d - V- (G, + Gie) (6.6)
dt 2C1
dV2- V2 (G + G2e) (6.7)
dt 2C2
d A-P , C (6.8)
dt 2 C1
where
C6 = C1- C2
By dividing Eqs. 6.5 it is possible to eliminate the time between them,
giving
dVi _ V1C2 (G + G) (6.9)
dV2 V2C1 (G2 + G2)
This differential equation can be solved by the methods of isoclines
in a plane with Vi and V2 as coordinates. A point in this plane corre-
sponds to an oscillation with the amplitudes, Vi and V2. If the system
is not disturbed, then the representative point will move along a tra-
jectory. Figures 6.3 and- 6.4 show typical groups of trajectories. As
shown in Bul. 395, stable simultaneous steady-state oscillations are not
possible if the current and voltage across the nonlinear element are
related by
(6.10)
i = av + fv2 + yva
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 6.3. Trajectories for Asynchronous Oscillator and i = av + jav2 + Yv'
Fig. 6.4. Trajectories for Asynchronous Oscillator and i = av + Gv' + Yv' + Bv + ev
Bul. 421. OSCILLATIONS IN NONLINEAR SYSTEMS
The trajectories shown in Fig. 6.3 correspond to those obtained from
such a polynomial. The point A shown in this figure is one that would
ordinarily correspond to simultaneous oscillations, but it is a saddle
point and is therefore unstable. On the other hand, if the current-
voltage relationship is a fifth-degree polynomial and if certain other
conditions are satisfied, then stable simultaneous oscillations are possi-
ble. Such a condition is represented by point B in Fig. 6.4. This figure
also shows that as a rule asychronous simultaneous oscillations are not
self-starting.
In general the representation of the synchronous oscillations in the
Vi-V2-4 space is not practicable. For certain conditions, however,
Eq. 6.8 can be reduced to
dt= f (V1, V2) sin 0 (6.11)
where f(Vi,V2) >0 for VI >0, V2 >0. All trajectories will then approach
the 4 = v plane and a trajectory starting in this plane will remain in it.
Figure 6.5 shows a typical group of trajectories in the 0= ~ plane.
The simultaneous oscillations corresponding to point A in this figure
are stable and self-starting.
Fig. 6.5. Trajectories for Synchronous Oscillations (eW = 3W2)
VII. THE LIMITATION OF THE AMPLITUDE OF
OSCILLATIONS BY LAMPS
The amplitude of oscillations in an oscillator containing only linear
elements will either decrease or increase indefinitely. It can reach a
stationary value other than zero only if the oscillator contains a non-
linearity of some sort, in most cases an electron tube. The electron tube
has a double function: it delivers a-c power to the passive elements
and limits the amplitude of oscillation.
In the oscillator discussed here, these functions have been separated.
The electron tube delivers the a-c power, but the amplitude is limited
by a lamp inserted in the resonant circuit. The temperature of this
lamp depends on the power dissipated in it and, hence, on the amplitude
of oscillation. The resistance of the lamp varies as a function of fila-
ment temperature.
For small amplitudes of oscillation more power will be generated in
the electron tube than is dissipated in the lamp, and as a consequence
the amplitude will increase. Due to this increase, the temperature
of the lamp, and also its resistance, will become greater. At a certain am-
plitude the resistance of the lamp will be so great that the power gener-
ated in the electron tube will be equal to that dissipated in the lamp,
and steady-state conditions for amplitude are achieved.""4
Fig. 7.1. Oscillator
An idealized oscillator using a lamp to limit the amplitude of oscilla-
tion is shown in Fig. 7.1. Actual oscillation using lamps to limit their
amplitudes are usually more complex."', 16)
The variation in resistance of a conductor with the temperature is
RT, = RTr (1 + aTi) (7.1)
Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS
where a is a constant depending on the material; RT, the resistance at
the ambient temperature To; and Ti= T- To the difference between
the temperature of the conductor and the ambient temperature.
It would be difficult to treat the circuit of Fig. 7.1 in full generality;
therefore a number of simplifying assumptions must be made. It is
assumed that no grid current flows and that only the linear portion of
the triode characteristic is used. The plate-current is then a function
of the plate voltage only:
ip = gm 9. + -- V
= gm-v = - GovP (7.2)
where
(M 1)
Go = gm L-
is a constant with the dimensions of a conductance. Equation 7.2 shows
that the triode of Fig. 7.1 can be replaced by the equivalent negative
conductance -Go. (Fig. 7.2) The lamp has been replaced in this figure
by a resistance RT,
The circuit of Fig. 7.2 can be further simplified. Since the frequency
of oscillation w differs by only a very small amount from 1/ VLC ,
the conductance -Go can be replaced by a resistance -Ro in series
with the inductance L (Fig. 7.3).
L
Ro = -- Go
The total resistance of this resonant circuit is
R = -Ro +- RT
= -Ro+RT (1 +aTi)
= - R1 (1 -/3T) (7.3)
where
Ri = Ro - RT
R,
It is assumed that the quality factor Q of the circuit is high. The
current i can then be approximated over a few cycles by
i = I cos ct
(7.4)
34 ILLINOIS ENGINEERING EXPERIMENT STATION
0
R(T7)
Fig. 7.2. Oscillator with Triode Replaced by Negative Conductance
The amplitude of current I will vary slowly and, for a short period of
time, it can be assumed constant.
The total energy stored in the circuit of Fig. 7.3 is LP/2. During
one cycle, it changes twice from electromagnetic to electrostatic energy
and back again, but the total amount remains approximately constant.
A slow change is caused, however, since some of the energy is dissipated
(or generated) in the resistance R.
L dP R1
Sd = 2 (1 - ) (7.5)
2 dt 2
A similar differential equation can be found for the thermal energy
mcT, of the lamp
dT1 12
mc d = -KT1 - RT, (7.6)
where m is the mass of the heat element, c is the specific heat of the
element, and K is the constant of thermal conductivity. For steady-
state oscillations it is necessary that both I and Ti remain constant; i.e.,
dI dT
--- --0
dt dt
The values of I and T1 corresponding to steady-state oscillations are
2K 1
1o2 = f- ; T1 = 1 (7.7)
These oscillations may be stable or unstable depending on whether they
will or will not resume their original amplitude after being subjected to
a small disturbance. This can be determined by calculating and examin-
ing the conditions for stable steady-state oscillations from Eqs. 7.5 and
7.6. Such an examination shows that these oscillations are stable.
Fig. 7.3. Equivalent Circuit of Fig. 7.2
Bul. 421. OSCILLATIONS IN NONLINEAR SYSTEMS
The behavior of the nonsteady-state oscillations can be determined
also from Eqs. 7.5 and 7.6. Dividing one of these equations by the
other eliminates time as a variable and makes possible the representation
of the transient oscillations in a plane with P and T1 as the coordinates
12
dT L -- T, + 2K [R0 - R, (1 - 3 - Ti)]
dTi L 2K
d - R1T 12(1 - T) (7.8)
where T= K/mc is the thermal time constant of the lamp.
This differential equation cannot be solved in closed form. An ap-
proximate solution, however, can be found by the method of isoclines.
In Fig. 7.4, a typical solution is represented by a group of trajectories
in a plane with the amplitude of oscillation as the abscissae and the
temperature as the ordinates, the I2-T1 plane.
At a given time the oscillator will have an amplitude, I, and the
lamp a temperature, Ti. The point in the P-Ti plane defined by these
two values will be called the "representative point" since it represents
the state of the oscillator. As temperature and amplitude vary, the
representative point will move along a line or "trajectory" in the
P-Ti plane. For example, if the oscillator starts from rest (point 0
in Fig. 7.4), it will move toward the point B along the dotted line.
T,
1
0
Amph/>ude
Fig. 7.4. Trajectories Describing the Transient Behavior of the Oscillator
ILLINOIS ENGINEERING EXPERIMENT STATION
The point B corresponds to steady-state oscillations with amplitude,
Io, and temperature, Ti0.
The steady-state oscillations corresponding to point B will be dis-
turbed constantly because of noise, variations in circuit parameters,
and change of the ambient temperature; but it can be seen that the
representative point, after having been displaced from B, will always
return to its original position. Consequently the steady-state oscillations
corresponding to point B are stable.
By a similar consideration it can be seen that point A is unstable
since the representative point will never return to this point after it
has been given a small displacement.
By assuming the current i to be
i = I cos ct
it was possible to describe the oscillator by the equations 7.5 and 7.6.
If no assumption regarding the current i is made, however, it is neces-
sary to use three differential equations of the first order. For example,
the. circuit of Fig. 7.3 leads to
di
w= y
dy 1 1.
dt L C(1-T1)y-
dTt = T + m1 (Ro - Rl (1 - fT1))i2 (7.9)
where i is the current through the resonant circuit.
Similar to the 12-Ti plane it is possible to set up a i-y-Ti space.
The solutions of Eqs. 7.9 could then be represented by a group of
trajectories in this space.
VIII. CONCLUSIONS
Several oscillatory systems have been discussed in this bulletin with
special emphasis on nonsteady-state or "transient" oscillations. For
autonomous systems with one degree of freedom and any degree of
nonlinearity, a complete representation of the transient oscillations is
possible in the phase plane. For more complicated systems, this method
of representation is either not possible or impracticable since it leads
to spaces with four or more dimensions.
A representation in a plane (which is no longer the phase plane) is,
however, still possible in some important cases if it can be assumed
that the oscillations are almost sinusoidal. As has been shown, the
transient oscillations can then often be described by two first order
differential equations:
dX
dt = fA (X, Y)
dY = f2 (X,Y) (8.1)
where X and Y may be an amplitude, phase angle, temperature, etc.
These equations do not give a complete description of the oscillations,
however, since they neglect the higher harmonics. This does not lead
to appreciable errors if the losses in the resonant circuits are small
(high Q) since then the amplitudes of the higher harmonics are also
small. Even for low Q circuits the results calculated from Eq. 8.1 are
confirmed surprisingly well by experiments.
The general procedure is to eliminate from Eqs. 8.1 by division:
dX f- (X, Y)
dY f (X,Y)
The solutions of this differential equation can then be represented as a
group of trajectories in the X-Y plane, in which an oscillation with
parameters X and Y corresponds to a point lying on one of the tra-
jectories in this plane. If the system is not disturbed, then this point
will move along the trajectories defined by Eq. 8.2. Steady-state oscil-
lations correspond to singular points of this equation. These oscillations
38 ILLINOIS ENGINEERING EXPERIMENT STATION
are stable if after being subjected to a small disturbance the parameters
X and Y will return to this point. Besides the singular points, the X-Y
plane may also contain closed trajectories or drift curves.
Four different oscillatory phenomena were treated by this method:
parametric excitation, synchronization, simultaneous oscillations and
amplitude limitation by means of lamps. The method can be applied
to a number of other problems, such as oscillations in conservative
systems with several degrees of freedom, self-modulating oscillators, and
the motor-boating effect occurring in audio oscillators.
The results obtained in Chapter V and VI were confirmed experi-
mentally. The degree of correspondence between theory and experiment
was good.