Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment
Characteristic regimes of Kijashko-Pikovsky-Rabinovich (KPR) noise generator were investigated on the basis of analytic theory and numerical simulation. Previously unknown regimes were found out. Modified circuit of KPR noise generator using the operational amplifiers was developed. Signal waveforms...
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irk-123456789-1109882017-01-08T03:03:43Z Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment Anisimov, I.O. Schur, A.V. Siversky, T.V. Нелинейные процессы Characteristic regimes of Kijashko-Pikovsky-Rabinovich (KPR) noise generator were investigated on the basis of analytic theory and numerical simulation. Previously unknown regimes were found out. Modified circuit of KPR noise generator using the operational amplifiers was developed. Signal waveforms were measured. Several regimes of KPR noise generator predicted by simulation were experimentally verified. Especially the relaxation regimes for large gain factors of the operational amplifier were obtained. 2003 Article Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment / I.O. Anisimov, A.V. Schur, T.V. Siversky // Вопросы атомной науки и техники. — 2003. — № 4. — С. 85-87. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/110988 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы Нелинейные процессы Anisimov, I.O. Schur, A.V. Siversky, T.V. Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment Вопросы атомной науки и техники |
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Characteristic regimes of Kijashko-Pikovsky-Rabinovich (KPR) noise generator were investigated on the basis of analytic theory and numerical simulation. Previously unknown regimes were found out. Modified circuit of KPR noise generator using the operational amplifiers was developed. Signal waveforms were measured. Several regimes of KPR noise generator predicted by simulation were experimentally verified. Especially the relaxation regimes for large gain factors of the operational amplifier were obtained. |
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Anisimov, I.O. Schur, A.V. Siversky, T.V. |
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Anisimov, I.O. Schur, A.V. Siversky, T.V. |
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Anisimov, I.O. |
| title |
Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment |
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Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment |
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Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment |
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Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment |
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Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment |
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kijashko-pikovsky-rabinovich noise generator: computer simulation and experiment |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2003 |
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Нелинейные процессы |
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Kijashko-Pikovsky-Rabinovich noise generator: computer simulation and experiment / I.O. Anisimov, A.V. Schur, T.V. Siversky // Вопросы атомной науки и техники. — 2003. — № 4. — С. 85-87. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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KIJASHKO-PIKOVSKY-RABINOVICH NOISE GENERATOR:
COMPUTER SIMULATION AND EXPERIMENT
I.O. Anisimov, A.V. Schur, T.V. Siversky
Taras Shevchenko National University of Kyiv, Radiophysics Faculty, Kyiv, Ukraine, ioa@u-
niv.kiev.ua
Characteristic regimes of Kijashko-Pikovsky-Rabinovich (KPR) noise generator were investigated on the basis
of analytic theory and numerical simulation. Previously unknown regimes were found out. Modified circuit of KPR
noise generator using the operational amplifiers was developed. Signal waveforms were measured. Several regimes
of KPR noise generator predicted by simulation were experimentally verified. Especially the relaxation regimes for
large gain factors of the operational amplifier were obtained.
PACS: 52.35.Mw
1. INTRODUCTION
Kijashko-Pikovsky-Rabinovich (KPR) noise genera-
tor (fig.1) is one of the simplest systems that can
demonstrate stochastic dynamics. Consequently this
system can be widely used for the preliminarily ac-
quaintance with such systems [1-2].
One of the stochastic regimes of KPR noise genera-
tor was experimentally investigated in [3]. The simple
analytical theory that describes this regime and its statis-
tics was proposed in [4]. Only the tunnel diode non-lin-
earity was taken into account in this theory.
Modified set of differential equations taking into ac-
count the non-linearity of an active amplifier element
and piecewise-linear approximation of the tunnel diode
volt-ampere characteristic (VAC) was proposed for the
description of the KPR noise generator in [5]. This set
was solved numerically. Two new regimes of the
stochastic oscillations and two relaxation regimes were
found out. Corresponding diagram of the regimes de-
pending on two driving parameters was obtained.
Further investigations estimated several bifurcation
parameters of the KPR generator, i.e. described the do-
main boundaries of the KPR generator characteristic
regimes.
Regimes predicted in [5-6] were not observed in the
experiment [3] because it was impossible to vary the pa-
rameters of the original circuit over a wide range. Of
course, computer simulation had no such limitation.
Results of the experimental investigation of the
modified KPR noise generator based on the operational
amplifier are described in this article. Experimental ver-
ification of several regimes predicted in [5-6] is ob-
tained.
2. MODIFIED THEORY OF KPR NOISE
GENERATOR
The electrical circuit of KPR noise generator is giv-
en on fig. 1. This circuit is described by the following
set of equations [5-6]:
( )
( )
−=
=
−−−=
.
;
;2
ντνε
τ
νγτ
diidd
iddu
uiuddi
(1)
Here
v=V/ ∞U ; u=U/ ∞U ; i= ρ I/ ∞U ; τ =tω; ε=C1/C;
Fig.1. Electrical circuit of KPR noise
generator
Fig. 2 VAC of tunnel diode
Fig. 3 Surfaces of the slow motion on the
phase portrait of KPR generator
γ =K2/ 2U ∞ –R/ρ; v1=V1/ ∞U ; rA=RA/ρ; rB=RB/ρ; ω
2=1/LC; ρ2=L/C; 2U ∞ =K2/ωMS, (2)
values I, U, V, R, L, M, C are shown on fig.1, C1 is the
junction capacitance of tunnel diode, S is the slope of
the triode transfer characteristic approximated by cubic
polynomial
Ia = SU – SU3/3K2,
id(v) is VAC of tunnel diode approximated by the piece-
wise-linear dependence
( ) ( ) ( )
( )
+≥
+<≤−
<
=
.2,
;2,2
;,
1
111
1
BABB
BABA
A
d
rrrr
rrrr
r
i
ννν
ννννν
ννν
ν (3)
For γ < rA the amplitude condition for self-excitation
is not satisfied. So solution of the equations’ set (1) is
quasi-linear damped oscillation. When γ exceeds the
critical value rA (i.e. the amplitude condition for self-ex-
citation satisfied) Andronov-Hopf bifurcation takes
place and thus quasi-harmonic continuous oscillations
are installed.
With the further increase of γ when the magnitude of
the steady oscillations (for the linear resistor used in-
stead of the tunnel diode) exceeds imax, i.e. for
2(γ-rA)1/2> v1/rA, the signal waveform qualitatively
changes and stochastic regime appears. When the cur-
rent in oscillatory circuit exceeds imax the jump from
low-resistance increasing brunch A to the high-resis-
tance one B occurs (fig.2-3). So the ohmic resistance
abruptly increases (voltage drop across the tunnel diode
increases while the current through diode does not
change). It results to the aperiodic damping in the oscil-
latory circuit. After the current magnitude decreases to
the value imin, the jump from brunch B to brunch A takes
place. The new package of oscillations with new ampli-
tude and phase is generated, because the probability to
get to the previous phase trajectory vanishes [3-4]. This
regime can be referred as monomodal one because it
corresponds to the monomodal mapping.
With the further increase of γ the oscillations’ incre-
ment grows. So the representative point during one rota-
tion can perform several couples of jumps between the
surfaces A and B (fig 3.). Thus the new stochastic
regime appears. It differs from the one described above.
It can be referred as multimodal regime because it cor-
responds to the multimodal mapping [5].
For very large γ the oscillations’ voltage magnitude
becomes much more then the non-monotonic interval of
the tunnel diode VAC. So this interval becomes inessen-
tial and system generates the relaxation-type oscillations
similarly to the Van-der-Pole generator.
3. RESULTS OF KPR GENERATOR NU-
MERICAL SIMULATION
Numerical solution of the set (1) allows us to inves-
tigate the behavior of the KPR noise generator depend-
ing on the driving parameters γ and v1. The appropriate
diagram of the characteristic regimes based on the anal-
ysis of the obtained signals’ waveforms was built (fig.
4.). These regimes can be pointed out: 1 – damped oscil-
lations; 2 – quasi-harmonic oscillations; 3 – monomodal
stochastic regime; 4 – multimodal stochastic regime; 6,
7 – relaxation regimes (fig. 5 а).
Fig. 5 b demonstrates another type of the multimodal
stochastic regime (regime 5). Its representative point
performs one and only one couple of jumps between the
surfaces of slow motion A and B. Quasi-harmonic oscil-
lations occur for this regime. The oscillations’ magni-
tude varies randomly from one period to another.
Fig. 4. Diagram of KPR generator regimes in (γ, v1)
coordinates 1 – damped oscillations; 2 – quasi-har-
monic oscillations; 3,4,5 stochastic regimes; 6,7 –
relaxation regimes
a
b
Fig.5. Relaxation (a) and multimodal stochastic
(b) oscillation regimes
The hysteresis i.e. the signal waveform dependence
on the initial conditions was noticed for the driving pa-
rameters corresponding to the boundaries between the
different regimes.
4. ELECTRICAL SCHEMATIC DIAGRAM
OF THE MODIFIED KPR NOISE
GENERATOR
It was already noticed that many parameters (gain
factor, non-linearity of amplifier, Q-factor, etc) of the
KPR noise generator given on fig. 1 can not be varied
over a wide range. The modified electrical schematic di-
agram of the KPR noise generator (fig. 6) is proposed.
Operational amplifier is used in this circuit instead of
radio tube. For convenient data processing ADC and
computer were used. So the oscillation frequency and
it’s harmonics should not exceed the upper frequency
limit of available ADC (24 kHz). Under this circum-
stances high Q-factor values are unreachable because of
the high resistance of the inductance coil at low fre-
quencies. That’s why equivalent gyrator circuit was
used instead of the inductance coil. Consequently the
value of Q-factor of about 103 could be reached. The os-
cillatory circuit was moved from the feedback circuit to
the load circuit. The modified KPR noise generator al-
lows varying many parameters (gain factor, nonlinear
characteristics of feedback loop, the current over tunnel
diode, and Q-factor) in the wide range.
Fig 6. Modified circuit of the KPR noise generator.
R1, R3, R4, R5=560Ω, R6=1.5МΩ, R7=10kΩ,
R8=47kΩ, R2=1.5kΩ, C1–C2=1mkF, DA1-DA3 –
LM101A
The KPR noise generator is based on the gyrator
circuit that consists of the following elements: DA2,
DA3, R1, C2, R3, R4, and R5. It has the parallel
connection with the tunnel diode VD1 and the capacitor
C1. These elements form the high-Q oscillatory circuit
for small signals because the resistive component of
impedance of this circuit depends only on the value of
R2. A source of the feedback signal of the oscillatory
circuit is resistor R1. Then it is amplified by DA1. The
feedback resistor R8 allows varying the gain factor and
non-linearity. The output of DA1 is connected with the
potentiometer-type voltage divider R7. It linearly
decreases the generator feedback voltage. Resistor R6
completes the feedback loop. It varies the current over
the load oscillatory circuit. The signal waveforms were
taken using the dual-channel ADC Philips UDA1361T
(CNR=88 dB, THD+N=83 dB, sampling rate is
48 kHz). Waveforms taken from resistor R2 represent
current in the oscillatory circuit.
5. EXPERIMENTAL INVESTIGATION OF
THE MODIFIED KPR NOISE GENERATOR
For the low feedback factors the damping quasi-har-
monic oscillations were observed.
Some increase of the feedback factor caused the
generator self-excitation. The magnitude of obtained os-
cillations (voltage on oscillatory circuit) was about 1V
less then the power supply voltage (15 V).
Further increase of the feedback factor caused the
stochastic regime appearance (described in [3-4]). One
could observe the bursts of up to 50 oscillations due to
the high Q-factor.
Further increase of the feedback factor resulted to
the establishment of the periodic relaxation-type oscilla-
tions. The signal waveform and spectrum for this
regime are given on fig. 7. The first harmonic is 141±
1 Hz.
In the narrow band of parameters one of the multi-
modal stochastic regimes similar to the regime 5 (see
fig. 4 and fig. 5b) was observed. The corresponding
waveform and it’s spectrum are presented on fig. 8.
a
b
Fig 7. Relaxation oscillation regime: waveform of cur-
rent in oscillation circuit (а) and it’s spectrum (b)
a
b
Fig 8. Multimodal stochastic regime: waveform of cur-
rent in oscillation circuit (а) and it’s spectrum (b)
6. CONCLUSION
Modified circuit of the KPR noise generator giving
the possibility to vary several parameters in the wide
range is proposed. It’s behavior was investigated experi-
mentally. Some regimes predicted by the modified theo-
ry of KPR noise generator [5-6] were observed, i.e. one
of the multimodal stochastic regimes and relaxation-
type oscillations. Bifurcation chain caused by the gain
factor varying predicted by this theory was also ob-
served.
Some difference between the results of experiment
and theory can be explained by taking into account the
difference between the experimentally investigated
scheme and one being simulated.
REFERENCIES
1. M.I. Rabinovich, D.S. Trubetskov. Introduction to
the theory of waves and oscillations. Moscow:
"Nauka", 1985. (In Russian).
2. I.O. Anisimov. Oscillations and waves. Kyiv:
"Academpress", 2003. (In Ukrainian).
3. S.V.Kiyashko, A.S.Pikovsky, M.I.Rabinovich.
Self-generator of the radio band with the stochastic
behavior // Radiotechnika i electronika. 1980, v. 25,
№2, p. 336-343. (In Russian).
4. A.S. Pikovsky, Stochastic oscillations in dissipative
systems // Physica. 1981, v. 2D, p.8-24.
5. I.O. Anisimov, A.M. Didovyk, T.V. Siversky.
Modified theory of the Kiyashko-Pikovsky-Rabi-
novich noise generator // Bulletin of Kyiv Universi-
ty. Physics and Mathematics. 2000, issue 2, p. 367-
374. (In Ukrainian).
6. I.O. Anisimov, T.V. Siversky. Bifurcations in the
Kiyashko-Pikovsky-Rabinovich noise generator //
Bulletin of Kyiv University. Radio physics and elec-
tronics. 2002, issue 4, p. 12-19. (In Ukrainian).
Taras Shevchenko National University of Kyiv, Radiophysics Faculty, Kyiv, Ukraine, ioa@univ.kiev.ua
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