1
Improved Implementation of Sprott’s Chaotic
Oscillators Based on Current-Feedback Op Amps
Banlue Srisuchinwong and Chun-Hung Liou
Sirindhorn International Institute of Technology, Thammasat University, Bangkadi Campus,
131 Moo 5, Tiwanont Road, Muang, PaThum Tani, 12000, Thailand, banlue@siit.tu.ac.th
Abstract- Improved implementation of Sprott’s chaotic
oscillators based on current-feedback op amps (CFOAs) is
proposed. A Sprott’s jerk function and four different types of
nonlinear components are implemented using the attractively
high-frequency features of the CFOAs operating in both voltage
and current modes. Four trajectories of the chaotic attractors are
demonstrated. The chaotic spectrums are easily scaled and
extended to higher frequencies by a factor of 277 to 380
I.
In this paper, high frequency implementation of Sprott’s
chaotic oscillators is presented using current-feedback op
amps (CFOAs). The CFOAs are currently recognized as
versatile alternatives to the traditional op amps for their
excellent performance in bandwidth and slew rates [9].
II. CIRCUIT IMPLEMENTATION
INTRODUCTION
Over the past two decades there has been increasing interest
in the study of chaotic oscillators [1-3]. Chaotic oscillators are
useful tools not only for investigation of nonlinear phenomena,
bifurcation and chaos, but also for a variety of applications
such as synchronizations, control [4] and chaos-based
communications systems [5]. Chua’s circuit [3] is one of the
best-known chaotic circuits but is difficult to scale to arbitrary
frequencies because of the inductor with its frequencydependent resistive losses [6] although inductorless versions
of Chua’s circuit have been possible [7]. Three reactive
components (capacitors or inductors) and a nonlinear
component are typically required for chaos systems with
continuous flows so that the Kirchhoff representation of the
circuit contains three first-order ordinary differential equations
(ODEs).
Recently, Sprott [6] has alternatively proposed chaotic
oscillators based on a single third-order ODE in a simple form
of d3x/dt3 = F (d2x/dt2, dx/dt, x) called a “jerk function” (time
derivative of acceleration). One of the Sprott’s jerk functions
in a general form is [6]
"3x
"2x
"x
$
$
# G (x )
A
3
2
"t
"t
"t
(1)
where G(x) is a nonlinear function and A = 0.6. The most
straightforward implementation involves three successive
active integrators to generate d2x/dt2, dx/dt and x from d3x/dt3
coupled with a nonlinear element G(x) and feeds it back to
d3x/dt3. Although Sprott’s chaotic oscillators [6] based on (1)
can be easily constructed using operational amplifiers (op
amps) and be easily scaled to different frequencies, the
operation has been somewhat delicate and the circuit has
exhibited hysteresis because of the finite gain-bandwidth
product and slew rates of the op amps [6, 8]. This problem has
been circumvented by operating at a lower frequency of
around 1.59 kHz [6, 8].
Figure 1. High frequency implementation of Sprott’s chaotic oscillator using
CFOAs and each of G(x) shown in Figure. 2.
Fig. 1, shows the high frequency implementation of the
Sprott’s chaotic oscillator based on CFOAs. The nonlinear
components G(x) [6] can be implemented using CFOAs as
shown in each of the system in Figs. 2(a) to 2(d) where related
parameters are also listed. In Fig. 1, the CFOA U0 forms an
integrator U0 where the zero-dB crossing (ZdB) frequency 0
= 1/!0 and !0 = R0C0. The CFOA U2 forms another integrator
U2 where the ZdB frequency 2 = 1/!2 and !2 = R2C2. An
expected integrator between U0 and U1 is replaced by a
simpler passive RC filter. A routine analysis in Fig. 1 reveals
that the resulting jerk function at node N1 is
"3x
"2x
"x
$
# G ( x)
$ K1
K
2
"t
"t 3
"t 2
! ! ! R
K3 # 0 X 2 1
R X R2
K3
K2 #
K1 #
! 0! 2 *
R2
R
((1 $ 1
) RX
!0
R2
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'
%%
&
(2)
2
Figure 2. Nonlinear components G(x) using CFOAs.
Figure 3. Chaotic attractors produced by Fig.1 using each of the
nonlinear components G(x) shown in Fig. 2.
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where !1 = R1C1, RX = (R0R2)/( R0+R2), and !X = RXC1. As (1)
= (2), therefore K1 = 1, K2 = A = 0.6 and K3 = 1. For
simplicity, let R2 = R1 = R0 = RT. Consequently, C2 = A/(3RT),
C1 = 3/(ART) and C0 = R2/R0 = 1. Table I summarizes the
calculated values of the capacitors for RT = 1 + and also, with
slight modification or scaling, the practical values of the
capacitors for RT = 1 k+.
(OAs). It can be seen from Table II that the proposed
implementation using CFOAs enables higher fS by a factor of
277 to 380.
TABLE I
Calculated and practical values of resistors and capacitors.
Components
(Fig. 1)
Calculated
Values
C0
C1
C2
R0
R1
R2
1
3/A
A/3
1
1
1
III.
Practical Values
G(x) in Figs.
2(a), (b), (d)
0.01 ,F
0.50 ,F
0.02 ,F
1 k+
1 k+
1 k+
G(x) in
Fig. 2(c)
0.5 ,F
25.0 ,F
1.00 ,F
1 k+
1 k+
1 k+
Figure 4. An example of the output voltage waveform and the
chaotic spectrum (dBm) centered around 1.21 MHz indicated
in Table II.
SIMULATION RESULTS
The performances of the circuits shown in Figs. 1 and 2
have been simulated through Pspice. Models AD844/AD and
AD-845/AD are CFOAs with and without current feedback
terminal, respectively. By using the practical values shown in
Table I, the simulated trajectories of the chaotic attractors in
the x-(dx/dt) plane are shown in Figs. 3(a)-3(d) for each of the
nonlinear components G(x) shown in Figs. 2(a)-2(d),
respectively. In Table I, the practical values of capacitors C0,
C1, C2 may be scaled down until the simulated chaotic
attractors shown in Fig. 3 corrupt at which point Table II
records the minimum values of such capacitors for each G(x)
shown in Fig. 2 whilst maintaining the value of R0 = R1 = R2 =
1 k+. Consequently, the operating frequency fS where the
chaotic spectrum is centered will be maximum as shown in
Table II.
Table II
Minimum values of capacitors and the corresponding
maximum operating frequencies fS.
G(x)
Types
C2
(F)
C1
(F)
C0
(F)
fS
(Hz)
Fig.
2(a)
OAs
CFOAs
0.02 u
0.05 n
0.50 u
1.25 n
0.01 u
25.0 p
1.49 k
544 k
Fig.
2(b)
Fig.
2(c)
Fig.
2(d)
OAs
CFOAs
OAs
CFOAs
OAs
CFOA
2.00 n
5.00 p
1.00 u
2.50 n
0.01 u
25.0 p
50.0 n
125.0 p
25.0 u
62.5 n
0.25 u
625.0 p
1.00 n
2.50 p
0.50 u
1.25 n
5.00 n
12.5 p
14.6 k
4.05 M
24.49
9.33 k
3.81 k
1.21 M
f s (CFOAs)
f s (OAs)
IV.
High frequency implementation of Sprott’s chaotic
oscillators has been presented using CFOAs. The Sprott’s jerk
function and four different types of nonlinear components
have been implemented using CFOAs. Four trajectories of the
chaotic attractors have been illustrated. The operating
frequencies are easily scaled and extended by a factor of 277
to 380.
ACKNOWLEDGMENTS
Authors are grateful to Mr Wimol San-Um who brings the
topics of chaotic oscillators to the authors’ attentions.
REFERENCES
[1]
[2]
[3]
365
[4]
277
[5]
380
317
As an example in Table II where G(x) is shown in Fig. 2(d),
Fig. 4 shows the corresponding output voltage waveform x
and the chaotic frequency spectrum (dBm) centered around fS
= 1.21 MHz. The latter is obtained through the fast fourier
transform of x. For purposes of comparisons, Table II also
includes the minimum values of the capacitors and the
resulting frequency fS for the similar cases using op amps
CONCLUSIONS
[6]
[7]
[8]
[9]
Special issue on chaos in nonlinear electronic circuits, Part A : Tutorials
and Reviews, IEEE Transactions on Circuits and Systems 40(10), 1993.
Delgado-Restituto, M. and Rodriguez-Vazquez, A. (2002), “Integrated
Chaos Generators”, Proceedings of the IEEE, vol. 90, No. 5, May, 2002,
pp. 747-767.
Chen, G. and Ueta T. ed., Chaos in Circuits and Systems, World
Scientific, Singapore, 2002.
Special issue on chaos synchronization, control, and applications, IEEE
Transactions on Circuits and Systems 44(10), 1997.
Mandal, S. and Banerjee, S., “Analysis and CMOS Implementation of a
Chaos-Based Communication System,” IEEE Trans. Circuits and
Systems – Part I 51(9), Sep. 2004, pp. 1708-1722.
Sprott, J.C. “A New Class of Chaotic Circuits,”, Physics Letters A, 266,
2000, pp. 19-23.
Morgul, O. “Inductorless realization of chua oscillator,” Electron. Lett.
31(17),1995,pp.1403-1404.
Sprott, J.C. “Simple Chaotic Systems and Circuits,” Am. J. Phys. 68(8),
Auguest, 2000, pp. 758-763.
Elwakel, A.S. and Kennedy, M.P. “Improved Implementation of Chua’s
Chaotic Oscillator Using Current-Feedback Op Amp,” IEEE Trans.
Circuits and Systems, Part I, 47(1), 2000, pp. 76-79.
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Whereas, previous CCCII-based MSO scheme is only
suitable for bipolar technology and not guarantee
generate the sinusoidal at the high frequency. Therefore,
the proposed MSO has more flexibility for IC
implementation and dealing with the applications than the
previous CCCII-based MSO.
I2
" C2 C1
#
I1 1! s$C2 g m %
Based on Fig. 2, the generated circuit for realizing an MSO
is shown in Fig. 3. Assuming that all OTA in Fig. 3 are
identical, by using (1) the loop again can be expressed as
II. CIRCUIT DESCRIPTIONS
Fig. 1 shows the symbol of the multiple-output OTA. An
ideal OTA is a finite bandwidth voltage-controlled current
source, with an infinite input and output impedance. The
output currents of the ideal multiple-output OTA is given by
I o # " g m (V! V )
(1)
where gm is the transconductance gain, Io the output current,
V+ and V- is the non-inverting and inverting input voltages,
respectively. For the case of OTA implemented with MOS
transistors operating in saturation, the transconductance (gm)
is proportional to (Iabc)1/2 and it implemented with bipolar
transistors, the gm is directly proportional to Iabc.
I abc
V+
+
$C2 C1 % (&
L$s % # ))
$C2 g m % &'
1
s
!
*
Io
+
$C2 C1 % (&
))
#1
$C2 g m % &' S # j,o
s
1
!
*
N
$1 ! j,o $C1
g m %%N ! ( 1) N !1 $C2 C1 %N # 0
CO
-I 2
,o #
gm
+. (
tan ) &
C2
*n'
C2
-2
C1
C2
Fig. 2. Basic building block of the proposed multiphase oscillator.
(OTA)N
(OTA)2
-Io2
+Io2
C1
C2
-IoN
+IoN
gm
C1
C2
(8)
and
fo #
C1
(7)
From (6) and (7), the FO and CO of a three-phase sinusoidal
oscillator (N=3) can be given as
+I 2
gm
(6)
C2 +
+ . ((
- ))1 ! tan 2 ) & &&
C1 *
* n ''
C1
-Io1
+Io1
(5)
12
I1
gm
(4)
or
Fig. 1. Circuit symbol of the multiple-output OTA.
(OTA)1
(3)
According to the Barkhausen criterion, the condition for the
proposed circuit to provide sinusoidal oscillation of
frequency is
FO
-I o
gm
N
By expanding (5), it is show that (5) would have a solution
only if the value of N is odd (N-3). By equating the
imaginary and real parts to zero, respectively, the frequency
of oscillation (FO) and the condition of oscillation (CO) can
be expressed as
gm
V-
(2)
C2
Fig. 3. Generalized circuit for realizing multiphase oscillator.
Fig. 2 shows the basic block of the proposed multiphase
oscillator circuit. It consists of a multiple-output OTA and
two grounded capacitors. The transfer functions between the
output and input terminal of the circuit in Fig. 2 can be
given by
3g m
2.C2
(9)
The frequency of oscillation and the condition of oscillation
for realizing N-phase sinusoidal oscillator, equal in
amplitude and equally spaced in phase, are summarized in
Table. From Fig. 3, the use of multiple output-OTA
provides an inverted of the output current. Thus, there are
2n=6, 10, 14 even-phase available output currents.
Therefore, the MSO circuit of Fig. 3 can generate both oddnumber and even-number of phase by a single circuit. From
(8) and (9), the frequency of oscillation and the condition of
oscillation can be orthogonally controllable. The oscillation
condition can be adjusted the capacitor C1 and the frequency
condition can be tuned by electronically the transconductance gm through the bias current. The high frequency
oscillation can be obtained without the effect of OTA
bandwidth in term of oscillation condition. Since the output
impedance of the OTA is very high, the MSO current
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outputs can be directly connected to the next stage without
the using additional current followers. The active and
passive sensitivities of proposed MSO circuit have low,
approximately -1 to 1.
CONDITION AND FREQUENCY OF OSCILLATION OF MULTIPHASE
SINUSOIDAL OSCILLATOR.
Number of phase
(N)
Condition of
oscillation
Frequency of oscillation
(,o)
3
5
7
9
C2=2C1
C2=1.237C1
C2=1.11C1
C2=1.063C1
1.732gm/C2
0.728gm/C2
0.482gm/C2
0.364gm/C2
Fig. 5. The simulated output waveform of three phase oscillator of Fig. 3.
III. SIMULATION RESULT
To the theoretical analysis of the proposed multiphase
sinusoidal oscillator, a CMOS design example has been
simulated through PSPICE simulation program. The
PSPICE model parameters for NMOS and PMOS transistor
are standard 0.5/m CMOS process of MOSIS. The
multiple-output plus/minus OTA schematic is modified
from well know single-ended OTA structure [19]-[20], as
shown in Fig. 4. It consists of a source-coupled pair with
identical MOS devices (M1-M2) operating in the saturation
region [19] where the output current is replicated using the
current mirrors. The MOS transistors aspect ratios are:
20/m/2/m for M1, M2; 40/m/2/m for M3-M6, M9-M10,
M13-M14, M17-M18; and 46/m/2/m for M7-M8, M11M12, M15-M16, M19-M20. The power supply is VDD=-VSS
=2.5V. Fig. 5 presents the simulation results of proposed
MSO circuit with C1=10pF, C2=20.12pF, Iabc=250/A
(gm=0.371mS) for N=3 where C2 was designed to be larger
than 2 times of C1 to ensure the oscillation will start. Fig. 6
shows sinusoidal waveform for six phases.
Fig. 6. Simulated output waveform of six phase oscillator of Fig. 3.
8
M13
M9
M5
M3
M4
V-I o
-I o
M6
V+
M1
M2
M10
M14
+I o
M18
+Io
Iabc
Oscillation Frequency, MHz
7
M17
6
5
4
3
Simulated
Theoretical
2
M7
M8
1
0
M11
100
200
300
400
500
600
Bias Current, /A
M12
Fig. 7. Variation of the oscillation frequency with the bias current.
M15
M19
M16
M20
Fig. 4. CMOS multiple-output OTA implementation used in simulation.
Fig. 7 presents the simulation results of oscillation
frequency of Fig. 3 by varying the value of the bias current
Iabc (i.e. 50/A to 500/A or equal gm from 0.1928mS to
0.48mS) with C1=10pF and C2=20.12pF. The relationship
between C2 and oscillation frequency is shown in Fig. 8,
which C1 is varied by C2 on (8). It shows that the proposed
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MSO can be generated the frequency high up to 10MHz
according with theoretical. The circuit be generated the
frequency higher than 10MHz but the simulation results not
confirm with theory, this error cause from the parasitic
capacitor of OTA. Finally, Fig. 9 shows the output currents
against varied bias current Iabc.
[1]
[2]
[3]
30
[4]
25
Oscillation Frequency, MHz
REFERENCES
[5]
20
[6]
15
Simulated
10
[7]
Theoretical
5
[8]
0
[9]
0
20
40
60
80
100
120
C2, pF
[10]
Fig. 8. Variation of the oscillation frequency with capacitor C2.
[11]
500
[12]
450
Output Current, /Ap-p
400
[13]
350
[14]
300
250
[15]
200
150
[16]
100
[17]
50
0
100
200
300
400
500
600
Bias Current, /A
[18]
Fig. 9. Output currents of proposed multiphase sinusoidal oscillator again
varying bias current with C1=10pF and C2=20.12pF.
[19]
IV. CONCLUSIONS
In this paper, a new electronically tunable MSO circuit
has been presented. The proposed MSO circuit has a simple
configuration which uses a multiple-output OTA and two
grounded capacitors per section. The MSO circuit can be
configured to provide an odd-number of equal-amplitude
equally special in-phase output current. The frequency and
condition of oscillation are independent controlled. The
proposed MSO enjoys simple structure, an electronically
tunable and suitable for IC implementation as both CMOS
and bipolar technologies. Simulation results, which confirm
the theoretical analysis, are obtained.
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