ILLIN I
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
PRODUCTION NOTE
University of Illinois at
Urbana-Champaign Library
Large-scale Digitization Project, 2007.
UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION
Bulletin Series No. 400
ALMOST SINUSOIDAL OSCILLATIONS IN NONLINEAR SYSTEMS
Part II : Synchronization
JOHANNES S. SCHAFFNER
Formerly Research Assistant Professor
of Electrical Engineering
Pulblished bi the UniecrsitY of Illinois, Urbana
UNIVERS'TY
300--4-52--49621 O PRLINOIS
PR YESS I'
CONTENTS
I. INTRODUCTION 5
II. OUTLINE OF THE METHOD 7
1. Definition of Terms 7
2. Linearization of the Oscillator 8
3. Equilibrium and Stability 9
[II. ORDINARY SYNCHRONIZATION 12
4. Internal 1:1 Synchronization 12
5. External 1:1 Synchronization 16
IV. SUBHARMONIC SYNCHRONIZATION 18
6. Oscillations in the Absence of Synchronism 18
7. Synchronized Oscillations 19
I. INTRODUCTION 5
8.- Particular Examples of Subharmonic
Synchronization 24
9. Bandwidth of Synchronization 27
V. EXPERIMENTAL VERIFICATION 28
APPENDIX: SELECTED BIBLIOGRAPHY 32
FIGURES
1.1 Tuned-Plate Oscillator with External Sinusoidal
Synchronizing Voltage 5
1.2 Entrainment of Frequency 6
2.1 General Oscillatory Circuits 7
2.2 Equivalent Linearized Oscillatory Circuits 10
3.1 Equivalent Linearized Circuit for Internal 1:1 Synchronization 13
3.2 Regions of Stability and Variation of X as a Function of p4
for p = 1, q = 1 14
3.3 Region of Stability and Variation of X as a Function of
6tyAw for p = 1, q = 1 15
4.1 Range of pp for Stable Synchronous Oscillations (X02 = 1) 23
4.2 Regions of Stability and Variation of X as a Function of pp
for p = 1, q = 3 25
4.3 Region of Stability and Variation of X as a Function of pP
for p = 3, q 1 26
5.1 Circuit Used for Experimental Verification of the
Theoretical Results 28
5.2 Region of Stability for p = 1, q = 3 (XA2 = 1) 29
5.3 Region of Stability for p = 3, q = 1 (Xo2 = 1) 30
5.4 Bandwidth of Svnrhrnnization. (Avi')}. as a Funrtiom of
5 4 Bandwidth...... , as a .. .... .. .
Q, Q/, for External 1:1 Synchronization
I. INTRODUCTION
In the absence of the external voltage vs the circuit of
Fig. 1.1 is a tuned-plate oscillator. It consists essentially
of a resonant circuit and an electron tube. This electron tube
has a double function. It feeds power into the resonant circuit
in order to compensate for the losses in the passive elements and
because of its nonlinearity, it limits the amplitude of oscilla-
tion. The frequency of oscillation in the first approximation
is Wo = 1/vLC.
1. INTR
In th(
Fig. 1.o1
of a resc
has a dou
in order
because o
tion. Tl-
is oo = 1
If the
varied 01
Lation d(
1. 2. If
becomes ;
integers,
^o to ( 3Ce
+ K
2C 30
Necessary and sufficient conditions for this are that
V +-- w >0
3CG 3C G
V W +0
av -a v ýd
(2.13)
(2.14)
Systems satisfying Eqs. 2.11 and Ineqs. 2.14 will produce stable
steady-state oscillations
V 3G,
e +K
2C 3V
a 3C
2C 'V
III. ORDINARY SYNCHRONIZATION
4. Internal I:1 Synchronization
In Fig. 2.la, the voltage Vs is inserted in the resonant
circuit in series with the inductance L. If the voltage v, = v
cos ast synchronizes the oscillation, then the voltage v across
the nonlinear element is
v = V cos(6 t+t) (3.1)
In this Chapter the subscript s of os is dropped, since for 1:1
synchronization w = w.s
It is assumed that the voltage v and the current i passinf
through the nonlinear network are related by
i = av + ftv2 + yv3 (3.2)
where a < 0, y > 0. This is a good approximation for most
oscillators. It can then be shown that the equivalent impedanco
of Fig. 2.2a consists of a conductance Ge only (Fig. 3.1)
Ge = a + y2 (3.3)
If no synchronizing voltage is present (vs 0), the conditiol
for steady-state oscillation is
G + Ge * 0
or 3
G + a +- yV2 = 0 (3.4)
The value of V satisfying this equation is
V0 4 G+a (3.5)
Vo2 3 7
The discussion in this section is limited to oscillators foi
which Vo2 > 0.
In Fig. 3.1 the current passing through the inductance L is
1
iL =- [V sin(wt+O) - Vsin wt] (3.6)
The same current would flow through the network consisting of L.
G1 and C1 due to the voltage v = V sin (wt + 0) alone. The value
for G1 and C1 are
Theri
state or
where h
The cone
After ri
Fig. 3.1. Equivalent Linearized Circuit for Internal 1:1 Synchronization
Theri
state or
where h
The cone
After ri
Bul. 400. NONLINEAR SYSTEMS 13
wCV
G 8 sin 0
C, = C cos (3.7
V
Therefore in the presence of a synchronizing voltage steady-
state oscillations occur if
C + G1 + Ge = 0
Aw Cl = 0 (3.8)
w 2C
where Aw = 1//LC - c. In terms of V and 4, these equations are
3 2 V
G + a + -yV2 + wC-- sin 0 = 0
4 V
(3.9)
Aco Vs
- - cos = 0
w 2V
The conditions for stability are
D3 C 3 V
V-(G, +G ) + - yY2 " 2C 8 sin
3V a q 2 V
3
y(2V2 - V 2) > 0
(3.10)
ac, a ac a
-1 (G- + G) - - (Ge + G1
3 2 V
= --yCV sin 0 + wC - > 0
2 V3
After rearranging the terms, the conditions for stability are
2V2 > V02
(3.11)
2wC V
sin 0 <-- -
3y V3
13
(3.7)
e steady-
(3.8)
ions are
(3.9)
(3.10)
2
-> 0
ty are
(3.11)
C
ILLINOIS ENGINEERING EXPERIMENT STATION
This last condition is satisfied automatically for sin 4 < 0. Foi
sin 4 > 0, it can be simplified to
(3.12)
The equations for stability and equilibrium can be simplifiec
somewhat through the introduction of dimensionless parameters
V V, G+a
X=- Xs =-V' A -31)
X - - X = ---A
V s V 2,-C
Fig. 3.2. Regions of Stability and Variation
of po for p = I, q = I
of X as a Function
Making these substitutions they become, for equilibrium,
S1X
A(X2-1) + - - sin 0 = 0
2 X
- cos (P = 0
wa 2r
and for stability
(3.14)
s'n2 < 1 Vo
sin2 (< -1)
2 V2
(3.13)
Bul. 400. NONLINEAR SYSTEMS
X2 >
2
and either
sin p < 0
or if sin ( > 0
1 1
sin2p < - )
2 X2
The phase angle p may be eliminated from these equations so
that those for equilibrium reduce to
2 ý -4 2 (3.16)
a[A2d 2(2e f2 + X (-)y to
and those for stability to
. (2 - 2 - 1) 0
(-T) + (x2 - l)(3X2 - 1) > 0
Fig. 3.3. Region of Stability
NVAU for
-A)
A J
and Variation of X as a Function of
p = I, q = I
(3.17)
ILLINOIS ENGINEERING EXPERIMENT STATION
The regions of stability may be represented in a plane with X
Aw
as the ordinate and either W or A as the abscissa. (Figs. 3.2
and 3.3). These figures also indicate the variation of X .with T-
and 4, respectively. As stated before, these figures have been
discussed extensively in the literature.
5. External 1:1 Synchronization
In Fig. 2.1b, it can be assumed that the voltage v across the
resonant circuit is approximately sinusoidal; that is
v1 i ' V cos(wt+<) (3.18)
If the oscillator is synchronized by the external voltage, the
frequency of oscillation is w, and as in Section 4 the subscript
of w, is dropped.
The voltage across the nonlinear element is
v = V cos(wt+p) + V8 cos Wt
- (V + V, cos c) cos(wt+4) + V. sin P sin(wt+g) (3.19)
It is again assumed that the current i and the voltage v are
related by
i = av + +V2 + yv3 (3.2)
where a < 0, y > 0. Using Eqs. 3.19 and 3.2, the current i can
be expressed as a function of time
i = I1 cos(wt+c) + 12 sin(ot+0)
+ components at frequency 3w (3.20)
The circuit of Fig. 2.1b can then be replaced by that of Fig.
2.2b where
1 12
Ge -, C, = - (2.7)
V WV
The values of Ge and Ce are
V. 3
G, = (1 +-- cos 4) [a +-y (V2 + 2VV, cos q + V,2)]
V 4
(3.21)
Ce -" V sin q [a + -4(V2 + 2VVs cos p + V 2)]
WV 4
If V, << V, these equations can be simplified considerably to
Ge a +- yV2
4
(3.22)
V 3 V
C= -1- sin p (a +--V2) = -(-3 sin G2
V 4 wV
Bul. 400. NONLINEAR SYSTEMS
As before the conditions for steady-state oscillations are
G + G =0
AW. C (3.23)
e 0
S-2C
where Aw = 1//LC - w. In terms of V and 4, Eqs. 3.23 are
3
G + a + - V2 = 0
Ao Vs (3.24)
- 2OV sin Q Ge = 0
The conditions for stability are
'Ge 3Ce
V"JF + W -W > 0
3G 3C, 3Ge 3Ce (3.25)
3v 3¢- 3- 3 V > 0
3G
Since e = 0, these inequalities are satisfied if
3G, a3C
-V > 0, 7 > 0 (3.26
or if
y > 0 cos 0 > 0 (3.27)
The permissible variation in P is, therefore,
7T 77
- < ^ < 2
The variation in frequency corresponding to this is
Ao VW G V 1
o V C Vo
where Q = cC/G is the quality factor of the resonant circuit.
Thus the maximum variation in frequency is proportional to the
ratio of amplitudes of the external synchronizing voltage to the
internal voltage. Consequently if the external voltage increases,
the bandwidth of synchronization increases likewise. In addition
the maximum frequency variation is inversely proportional to the
Q of the circuit. Therefore for small Q, the frequency can be
entrained over a large band; but for high Q, the bandwidth of
synchronization is relatively small.(7)
IV. SUBHARMONIC SYNCHRONIZATION
For external subharmonic synchronization the voltage v1 across
the resonant circuit of Fig. 2.1 is again assumed to be approxi-
mately sinusoidal
v1 ^ V cos(Wt+k) (3.17)
where w is approximately 1/ILC. For synchronization, w and the
frequency ws of the external voltage v, are related by Eq. 1.1.
The voltage across the nonlinear network is-then
v = V cos(at+() + Vs cos wst (4.1)
It is assumed that this voltage and the current i passing
through the nonlinear network are related by a rapidly convergent
power series
S= av + /3v2 + yv3 + Sv4 +... (2.1)
If this expression is combined with Eq. 4.1, then the current
can be expressed as a function time
I = I1 cos(wt+4) + 12 sin(wt+0)
+ components at frequencies other than w (2.6)
The circuit of Fig. 2.lb can then be replaced by that of
Fig. 2.2b where again
Ge = 1 C = 12 (2.7)
V e wV
Methods for calculating Ge and Ce have already been discussed in
University of Illinois Engineering Experiment Station Bulletin
395, Chapter II.(15)
6. Oscillations in the Absence of Synchronism
For some frequencies the external voltage will not synchronize
the oscillator. No relation p) = qw8 will then be preserved over
any appreciable length of time and it can be assumed that W/w, is
irrational. The equivalent impedances for )/w, irrational have
been calculated in Bulletin 395, Chapter II.15) They are
Gc = a + - y(V2+2V,) + .
S(4.2)
Ce = 0
Bul. 400. NONLINEAR SYSTEMS
The prime indicates that no synchronism is present. To faci-
litate further calculations, it is assumed that neglecting all
terms of Ge other than the first two will not lead to appreciable
errors. This is true for most oscillators.
Steady-state oscillations are possible if a < 0, y > 0 and
G + G' = 0 (4.3)
or 3
G + a + y(V2 + 2V,) = 0 (4.4)
The value of V that satisfies this equation is
v2 4 G+a
Vo2 = - 7 - 2V,2 (4.5)
Vo2 has a physical significance only if it is positive, otherwise
it is just a parameter satisfying Eq. 4.5. If Vo2 > 0, then the
oscillations of the system are called "free" oscillations; if
V02 < 0, "forced" oscillations.
It is desirable that Eq. 2.1 be changed to a form correspond-
ing to Eq. 11.1 of Bulletin 395.(15)
aO
i = Io axx (4.6)
1
where the ax and x are dimensionless. This can be done by
defining
v a 3 y
V I \V \ 2=V 28 3 7
So2 V 1- 2- o -Vo 13 (4.7)
Similarly, dimensionless variables corresponding to V and vs
are defined as
V V V I
XX ' -- X G0 = (4.8)
for free oscillations Xo2 = 1; for forced oscillations X2 - 1.
An expression that will be helpful in the next section is
3 2)
G + Ge = G + a +-(V2 +2,2) (4.9)
Combining this with Eq. 4.5 and making the substitutions of
Eq. 4.8
3
G + G'e -4a 3G(X2- Xo2) (4.10)
7. Synchronized Oscillations
Locking phenomena occur if the two frequencies c and ws are
related by an equation
(1.1)
pw = qoW,
ILLINOIS ENGINEERING EXPERIMENT STATION
where p and q are small integers. The special case p = q is not
considered in this chapter.
Some of the combination frequencies (nw+±mn) are now identical
with w; as a consequence I1 and 12 and therefore Ge and Ce may
contain additional terms. In fact these terms are necessary for
the mechanism of synchronization. A double-prime is used to in-
dicate such additional terms. For synchronization the equivalent
impedances are
G = G' + C" Ce = Ce (4.11)
It is assumed that the series Xaxx converges rapidly. It is
1
then sufficient to consider only the first term that contributes
to G" and C", As shown in Bulletin 395, Eq. 10.19,(15) this term
is
1oa xx = p+q-l (4.12)
and the corresponding G" and C" are (Bulletin 395, Eqs. 11.4 and
11.5)
akGo h
G" -i (q) xp-2 Xs cos P(
ahC G (.4.13)
C" = 2-l(q) Xp-2 X q sin cp
These can be simplified by introducing a coefficient
aX X
77 = q^r(q)
G" and C" are then
e e
G' = 7Go XP-2Xsq cos pq
C (4.14)
C" -r-Xp-2 Xq sin p0
Some values for 77 are presented in Table I. Equations 4.14
correspond to a first-order approximation. As stated above, none
of the terms Ioakxx where X = (p+q-l) will contribute to G' and
C'e. This does not hold if approximations higher than the first
are considered.
For example, for 5w = w, (p = 5, q = 1) the contribution of
Io a3x towards G" and C" is for a second-order approximation
27 G 2
G" a 2 -G X3Xs cos 5
27 G 2 (4.15)
Ce = 128 " X3X, sin 50
snC
Bul. 400. NONLINEAR SYSTEMS
TABLE I
Values of )
p q
1 2 3 4 5
1 ... 1/2 a2 1/4 a3 1/8 a4 1/16 a5
2 a2 .. 1/2 a4 ... 3/16 a6
3 3/4 a3 3/4 a4 ... 15/32 a6 21/64 a7
4 1/2 a4 ... 5/8 a6 ... 7/16 a8
5 5/16 a5 15/32 a6 35/64 a7 35/64 a8 ...
The contribution due to a first-order approximation would be zero
since X = 3 is smaller than (p+q-1) = 5. In most cases the terms
of Eq. 4.15 are small compared with those corresponding to Ioa5x5
and a first-order approximation, particularly if the circuit
has a large Q (G/wC small). They can therefore be neglected.
As a result of Eqs. 4.14 the total equivalent impedances are
then
Ge = Ge + G" = Go [al + a3(X2+2X2) + rXP-2Xsq cos pc
G (4.16)
Ce = C" = - -- XP-2 X sin p
As shown in Chapter II, steady-state oscillations can occur if
G+Ge =0
A C, (2.11)
o 2C
where
Ao) 1 q
--r= / O
In terms of X and 4, these equations are
X2 - X 2 + --_ " X2 cos pc = 0 (4.17)
Aco G
-- + 7 G - XP-2Xq sin pb = 0 (4.18)
w 2wCo
If the phase angle is eliminated, then these two equations
combine into
3 2 =o £] 2
[r a(X2-Xo )] + [2-L 2 = [77XP-2X q] (4.19)
Equations 4.18 and 4.19 permit a representation of X as a func-
tion of either pc or Aw/w. For the purpose of this bulletin,
p is the more adequate independent variable.
ILLINOIS ENGINEERING EXPERIMENT STATION
The conditions for stable steady-state oscillations are
3G aC
X - + " - > 0
(4.20)
aC eG Ce aG
e +e e > 0
The first of these two inequalities is, in terms of X and c,
X3G 3C
X-e + a)w c
BX 34
ax 4€
= Go a3X2 + ?(p-2)oX-2Xq Cos pc - 7pXp-2Xs Cos p4]
= Go [-a X2 - 27,X-2X, cos p4]
23 -7
= Go[a3X2 +ta3(X2-Xo2)] > 0 (4.21)
Since a3 > 0, this inequality is satisfied if
X2 > X 2 (4.22)
2 o
For forced oscillations (X02 = -1), Ineq. 4.22 is satisfied
automatically.
The second part of Ineq. 4.20 is
WCe aGe + 3Ce 'Ge
3 3X ~X 3
G 2 3
O.- {[77Pp-2X cos pq] [--aX2 + (p-2)X^2Xq cos p4]
+ (TpXp-2Xs sin pf] [7(p-2)Xp-2X"q sin p4]}
G 2 3
- -pXp-'1 q9[-a3 cos p - 2(p-2)XP-4 Xs] > 0 (4.23)
co 2
This inequality is satisfied if
7 cos p0 < - 2 p- XP-4X q (4.24)
3 a
For Ineq. 4.24 three different cases of subharmonic synchron-
ization have to be distinguished: p = 1, p = 2 and p > 2.
a. p = 1
If p - 1, then the right-hand side of Ineq. 4.24 is positive.
The range of the phase angle 4 for which the oscillation is
stable is therefore larger than 7T. For positive 7 it extends
Bul. 400. NONLINEAR SYSTEMS
approximately from 7/2 to 37/2 and for negative - from -7/2 to
77/2 (Fig. 4.1). Equation 4.17 shows that for Xo2 = -1 (forced
oscillation) the possible variation of 4 for which X is real is
smaller than 77. Inequality 4.24 is therefore automatically
satisfied. For Xo2 = 1 (free oscillations), 4 may however vary
over a much wider range than 7 as indicated in Fig. 4.1, and
therefore Ineq. 4.24 must be considered. The critical regions
are shaded in Fig. 4.1. In these regions 7 cos 4 is positive.
Using Eq. 4.17, Ineq. 4.24 can be simplified to:
cos2 - <1 X 2
2 X2
(4.25)
Fig. 4.2 shows the region of possible steady-state oscillations
defined by Ineqs. 4.22 and 4.25 for ] > 0. The regions for q < 0
can easily be obtained by a phase shift of 1800.
6. p = 2
For p = 2, Ineq. 4.24 reduces to
7 cos 20 < 0
(4.26)
The total variation of (24) is therefore 77 for Xo2 = 1 and
smaller for XJ2 = -1 (Fig. 4.1).
c. p > 2
If p > 2, then the right-hand side of Ineq. 4.24 is negative.
Therefore the variation of the phase angle for which the oscilla-
tions are stable is smaller than 77 (Fig. 4.1). Using Eq. 4.17,
Ineq. 4.24 transforms to
cos2 p4 > 2 -
2 X2
Fig. 4.1. Range of p4 for Stable Synchronous Oscillations (Xo2 = I)
(4.27)
0
L_?
I 0
4.!
-77
-.77 -
31r
n7 -
2
~
ILLINOIS ENGINEERING EXPERIMENT STATION
Equation 4.17 shows that for this variation and for Xo2 = 1,
X > 1 and therefore Ineq. 4.22 need not be considered. The
region of possible steady-state oscillations is indicated ir
Fig. 4.3 for p = 3.
For Xo2 = -1, Ineq. 4.27 can be satisfied for p = 3 only
(Fig. 4.3b). For p ' 4, Ineq. 4.27 cannot possibly be satisfied.
Therefore, for a first-order approximation no forced subharmonic
oscillations are possible for p =4. This, of course, would notbe
true if more than the first two terms of Eq. 4.2 were considered.
8. Particular Examples of Subharmonic Synchronization
The discussion in this section deals with three examples of
subharmonic synchronization:
o = 3w ,, 2w = Ws, 3w = W.
a. 0 = 3w8 (p = 1, q = 3)
I
For this example 7 from Table I is -a3. Eq. 4.17 is then
1 X3
X2 X2 cos
V 3X
The regions for stability for Xo2 = 1 are indicated in Fig. 4.2a.
For Xo2 = -1, all steady-state oscillations are stable as shown
in Fig. 4.2b. Fig. 4.2 also shows the variation of X with pc for
X. = constant. The variations of the phase angle are
-77/2 < A4 < T/2 (X2 = 1)
0 < A0 < 77 (Xo2 = -1)
b. 20 = w, (p = 2, q = 1)
From Table I 7 is a2 and Eq. 4.17 is
X2 = Xo2 _4 a2X cos 20
3 a3
As shown in Fig. 4.1, the range of 20 is ~T for Xo2 = 1. For
Xo2 = -1, all steady-state oscillations are stable.
c. 3w = w, (p = 3, q = 1)
For this case 77 is-a3. Eq. 4.17 is then
X2 = Xo2 - X Xs cos 34
For Xo2 = 1 the region of stability is shoun in the shaded area
of Fig. 4.3a; for Xo2 = -1, in Fig. 4.3b. The variation of X
Bul. 400. NONLINEAR SYSTEMS 25
0
4-
H-
0
C)
Cf
0
0
41
C'
L-
0
C
0
L.
U-
26 ILLINOIS ENGINEERING EXPERIMENT STATION
0
0
0,
cn
'U
C4
C
4-
0
C)
·-
Bul. 400. NONLINEAR SYSTEMS
with pp is also shown in Fig. 4.3. As can be seen from these
figures, the total change in phase is
7/2 < A3 < 7 (Xo2 = 1)
0 < A3 < 7T/2 (Xo2 = -1)
9. The Bandwidth of Synchronization
So far in this bulletin the limits of stable synchronization
have been expressed in terms of phase. For practical purposes it
is, however, more important to know the maximum permissible de-
viation in frequency, or the bandwidth of synchronization (ao/w)o
It is of course possible to find precise expressions, but in most
cases an approximate value is satisfactory. Such an approximate
value can be found if it is assumed that X = 1 and that the phase
angle varies by an amount 7 either from -7/2 to 7/2 or from
7T/2 to 37/2. The bandwidth of synchronization is then (from
Eq. 4.14)
Aw G aG a
( q-) - x - ( ) X q (4.28)
o 2- wC
This equation shows that the bandwidth of synchronization be-
comes rapidly smaller as k = p+q-1 increases.
V. EXPERIMENTAL VERIFICATION
In the discussion of the theory some questionable approxima-
tions have been made. It was therefore necessary to investigate
how well the theoretical results are confirmed by the experiment.
The circuit used in the experiment is shown in Fig. 5.1.
In this circuit, designed by Dr. Giuseppe Francini, a con-
ventional transitron circuit served as the nonlinear element.
The operating point of the tube was adjusted by the potentio-
meters R1 and R2 and slight changes in the current-voltage
characteristic were made by changing tubes. The synchronizing
voltage was supplied by a Hewlett-Packard oscillator. The ex-
perimental results were obtained using resonant circuits tuned
to a frequency of 6 kc and having a Q of about 35.
Synchronization was observed with the aid of Lissajou figures
on the screen of an oscilloscope. The voltages V and V, were
measured directly across the oscillatory circuit and R .
It was to be expected that the critical part of the theory
was that dealing with stability rather than equilibrium. The
conditions of stability for cases (3a = wS) and (a = 3ws) were
checked in this experiment. The results of the experiment are
shown in Figs. 5.2 and 5.3. It can be seen from these results
that the correspondence between theory and experiment is reason-
ably good.
Equation 3.28 was also verified by experiment. Good corres-
pondence was obtained for ratios of V,/V as high as 0.15. The
experimental results for V,/V = 0.1 are shown in Fig. 5.4. Gen-
erally, the maximum variation of the phase angle was somewhat
larger than ?; therefore, the measured values of (Aw/w)o are
somewhat too large.
Bul. 400. NONLINEAR SYSTEMS
---------___-
Fig. 5.1. Circuit Used for Experimental Verification of the
Theoretical Results
Fig. 5.2. Region of Stability for p=I, q = 3 (Xo2 = I)
- -
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 5.3. Region of Stability for p = 3, q = I (X2 = I)
Bul. 400. NONLINEAR SYSTEMS
06"
OE 03 a4 US 0h6 07
1/Q
Fig. 5.4. Bandwidth of Synchronization, (A&/,w)o, as a Function of
Q, G/cC, for External I:1 Synchronization
7
0
z
o
Theoret/ca/ Curve
0 Epementa/ Points
nz'zzz/
_ _° z _ _ _ _
n-
o
/
^ ^^ "-
APPENDIX: SELECTED BIBLIOGRAPHY
1. N. Minorsky, "Introduction to Nonlinear Mechanics," J. W. Edwprds,
Ann Arbor, 1947.
2. J. J. Stoker, "Nonlinear Vibrations," Interscience Publishers Inc.,
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