- Email: [email protected]
A family of Colpitts-like chaotic oscillators
~) PERGAMON
Journal of The Journal of the Franklin Institute 336 (1999) 687-700
Franklin Institute
A family of Colpitts-like chaotic oscillators A.S. Elwakil,M.P. Kennedy* Department of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, Ireland Received 11 June 1998; accepted 2 September 1998
Abstract A family of chaotic oscillators with qualitative dynamics similar to the chaotic Colpitts oscillator is introduced. The oscillators use a single current feedback op amp, configured as a noninverting voltage-controlled voltage source, as the active building block, and a nonlinear element with an antisymmetrical current-voltage characteristic. A procedure for obtaining the chaotic oscillators by modifying simple harmonic oscillators is demonstrated. These chaos generators are suitable for high-frequency operation; device parasitics have negligible effect. Experimental results, PSpice circuit simulations and numerical simulations of the derived mathematical models are included. © 1999 The Franklin Institute. Published by Elsevier Science Ltd.
Keywords: Chaos; Oscillators; Chaotic oscillators
1. Introduction Since the introduction of the chaotic Colpitts oscillator [1, 23, an area of research aiming at investigating the possible chaotic nature of conventional sinusoidal oscillators has been opened. Basically, it is required to investigate whether a sinusoidal oscillator circuit can by nature behave chaotically under specific conditions, as is the case with the classical Colpitts oscillator, or whether it can be modified to behave chaotically as in [3-7]. In this work, a family of chaotic oscillators that require a single coil and two capacitors and that display behaviour which is qualitatively similar to that of the chaotic Colpitts oscillator is introduced. By contrast with the chaotic Colpitts oscillator, where the active element providing gain is itself the nonlinear element (a bipolar transistor operating in both the forward active and the cut-off regions), the proposed oscillators
* Corresponding author. Fax: 00 353 1283 0921; E-mail: [email protected] 0016-0032/99/$20.00 + 0.00 © 1999 The Franklin Institute. Published by Elsevier Science Ltd. PII: S0016-0032(98)00046-5
A.S. Elwala'l, M,P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
688
rely on a current feedback op amp (CFOA) employed as a linear noninverting voltage-controlled voltage source (VCVS) to provide the necessary gain, while the nonlinearity is introduced by a junction field effect transistor (JFET) operating as a two-terminal voltage-controlled resistor. This decoupling of the active and nonlinear functions facilitates circuit design and tuning and permits the oscillator to inherit the most attractive features from both the active and the nonlinear elements. In particular, two achievements are emphasized: (1) The possibility of designing a chaotic oscillator by intentionally modifying a sinusoidal oscillator circuit is confirmed. (2) The possibility of obtaining desirable features in the chaotic oscillator through inheritance from the original sinusoidal oscillator with appropriate oscillator design and choice of the active and nonlinear elements. Such features may include suitability for high-frequency operation, insensitivity to parasitics and component variations, low-voltage low-power operation, possibility of certain elements being grounded or floating and providing for current or voltage outputs. In Section 2, a brief discussion of the CFOA and its major sources of nonideality is given while in Section 3, a family of three sinusoidal oscillators that are to be modified for chaos is presented. Section 4 describes the modification procedure. The new family of chaotic oscillators is then presented in Section 5. Experimental results, PSpice simulations and numerical simulations of the derived mathematical models are shown.
2. The current feedback op amp
The current feedback op amp (CFOA), also known as the transimpedance op amp, is a versatile four terminal active building block which is recognized for its excellent performance in high speed and high slew rate analog signal processing [-8]. The CFOA is considered to be a cascade of a second generation current conveyor [9] and a voltage buffer. The CFOA is described by the following matrix equation:
Iy It Vo
=
{ilOO[ 0
0
0
Vy
0
0
0
V~
0
1 0
(1)
Io
In a practical implementation of a CFOA, the X terminal (inverting input terminal) is characterized by a very low input impedance (Rx) while the Y terminal (noninverting input terminal) has a practically infinite input impedance. The two outputs C and O exhibit a very high (Re) and a very low (Ro) output impedance respectively. For the commercial AD844 [10] the values of Rx, Rc and Ro are approximately 65 ~, 2 M ~ and 15 f~. A parasitic capacitance (Cc) of about 5 pF also appears at the C terminal, as shown in the macromodel of the device in Fig. 1 [-8]. The CFOA does not suffer from the finite gain-bandwidth product of conventional voltage op amps (VOAs) and can
A.S. Elwakil, M.P, Kennedy~Journalof the Franklin Institute 336 (1999) 687-700
+o
C
~>
689
0
Av=+ 1
I
Rx
Rc T Cc
I Fig. 1. Macromodel of the CFOA device.
operate without external negative feedback. The capability of a CFOA to provide both voltage and current outputs is particularly useful and results in circuits that generally require fewer components. Many analog signal processing blocks can be realized using the CFOA [11]. A novel CMOS CFOA was recently introduced in [12].
3. A family of simple harmonic oscillators
Figure 2 represents a family of simple harmonic oscillators employing a single CFOA as a linear noninverting voltage-controlled voltage source (VCVS) [13]. With equal capacitor values and neglecting the parasitic elements Rx, Re, Ro and Cc, the state equations of this family are given as
l a,,
l?c2J = C kaza
1
a22_J LVc2J"
The parameter set Jail, [~2(R~--l)
(2)
alz, a z b a223
for the three circuits is given respectively as
1 -1,1(R4
for the circuit in Fig. 2a, l(R4 1~--1 l(R4 )R,--1 R2 k,R3 - J' R 2 ' R2 R3 - 1 + R1R3 , R2 for the circuit in Fig. 2b and ~ L ( R4 LR2 \ R 3 --
1
1 RI'
for the circuit in Fig. 2c.
R4 R2R3 +
11 '
'
1] R1
690
A.S. Elwakil, M.P. Kennedy/Journal of the Franklin Institute 336 (1999) 687 700
Vcc
C2
R2 AD844A/AD
R3
(a)
_
-Vcc
_
R4
._
Vcc
C2
R2 AD844NAD
(b)
_
_
=
=
Vcc
(c)
=
-
=
=
F i g . 2. T h e f a m i l y o f s i n u s o i d a l o s c i l l a t o r s .
The condition for oscillation (ax 1 + a22 thus found to be
R4
Rz
K=R33 =2+
R-~
= 0)
for the oscillators of Fig. 2a and b is
(2a)
A.S. Elwakil, A~P. Kennedy~Journal of the Franklin Institute 336 (1999) 687 700
691
and for the oscillator of Fig. 2c it is given by K=I
+2R-z R1
(2b)
which are both satisfied by choosing R2 = R1 = R and R4 = 3R3. The ratio K = 3 is the necessary loop gain provided by the CFOA, The radian frequency of oscillation is then found to be
1 ~o
-
RC
(2c)
"
It is worth noting that the circuit of Fig. 2c is related to that of Fig. 2b by the R C - C R transformation [14].
4. Modification for chaos In order to modify the family of oscillators for chaos, there are two necessary requirements. First, an additional energy storage element (inductor or capacitor) should be added to the oscillator structure to allow for a third-order autonomous system to exist. Second, some form of nonlinearity must be introduced. Targeting a structure similar to the chaotic Colpitts oscillator [1], an inductor and a voltage-controlled nonlinear resistor with an antisymmetrical current-voltage characteristic are selected. This type of nonlinearity can be simply realized by a J F E T operating as a two-terminal device. A suitable position to add the inductor in the three harmonic oscillators is in series with R1. This location allows the inductor to be grounded and causes minimal changes to the basic oscillator architecture. Since JFETs are well known for their good performance as voltage-controlled resistors [15], a J F E T might then replace resistor Rg. The current-voltage characteristic of this device can be modelled in a piecewise-linear form as
l ~'V~s, Is = ~ (vp,
V~s/> Vp, Vos < vp,
(3)
where Rj is the J F E T small signal resistance at the operating point (approx. 750 fl for the J2N4338), Vp is the J F E T pinch-off voltage (approx. - 0 . 7 V) and V~s is the gate-to-source voltage. Equations (2a), (2b) and (2c) are used to estimate the required values of circuit components and the centre frequency of the generated chaotic spectrum. The modification process is semi-systematic in the sense that the choices of the position and value of the inductor and the nonlinear element are subject to the designer's intuition. In the following section, the above process is applied to the family of harmonic oscillators resulting in a family of Colpitts-like chaotic oscillators.
692
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687 700
5. The family of chaotic oscillators
5.1. The first chaotic oscillator configuration Figure 3a represents a chaotic oscillator resulting from the modification of the harmonic oscillator in Fig. 2a. Guided by Eq. (2a), the value of R1 should be taken
Vc2 +1
J2N4338
AD84
Vcc
I
-
R1 Cl
.~
Vcl
-Vcc
1
(a) 2 • 5V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
t I I
2.ov"
.
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•
, I
,
b I
a.
ov~
.
I I I I I I I I I J I
-0.0V J
.
.
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.
-I.OV+---
.
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-I.2V [] V ( C 2 : I ) (b)
.
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2. OV
I .'O V
O.OV V(C2:2) V(CI:2)
Fig. 3. (a) and (b) The first chaotic oscillator and the corresponding PSpice Trajectories obtained by numerical integration of Eq. (5).
Vcl -
Vc2
trajectory. (c) and (d)
A.S. Elwakil, M.P. Kennedy~Journal o f the Franklin Institute 336 (1999) 687 700
693
-4 >.-6
-10 -12
-12
-10
-8
-6
-4
(c)
-2
0
X
-6.
-15
(d)
-10
Y
-5
X Fig. 3. (continued).
2
6
8
694
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
around the same value of the J F E T small signal resistance while the gain K should then be varied around the nominal value K = 3. The equal capacitors are arbitrarily chosen to define the frequency band of interest which can be estimated from Eq. (2c). Fig. 3b shows a PSpice simulation of the Vcl - Vc2 phase space trajectory obtained with R~ = 750 f~, Cl = C2 = 50 pF, L --- 50 pH, R 3 = 1 kf~, R4 = 2.85 kf2 and using + 9 V supplies. It can be seen that the parasitic input resistance Rx of the C F O A appears in series w i t h R 3 and can thus be accounted for. The very small CFOA output resistance Ro has very little effect on the circuit's behaviour; it simply adds in series with the J F E T resistance. However, in order to neglect the effect of the parasitic capacitance Cc, especially at high frequencies, it is recommended to make C1 and C2 much larger than Cc. The low gain (K = 3) provided by the C F O A further reduces its effect. The values of the gain resistors (R 3 and R4) should be sufficiently large to minimize the current supplied by the C F O A to the respective terminals. Neglecting the parasitics, the circuit can be described by the following set of equations: Cll/c I = Ij - IL, C2 I;'c2 = 1j,
(4a)
LIL = Vcl -- R I l L ,
where 1L is the inductor current and Ij is the nonlinear J F E T current modelled as 1 ~'(K - 1)Vcl - Vc2, (K - 1)Vc, - Vc2 >/ Vp, I, = ~jj ( Vp,
(4b)
(K -- 1)VC1 -- Vc2 < Vp.
Setting Vcl X=
Vc2
Vp '
Y=
RIlL
Vp'
Z
- -
2
--Vp
t '
~
--
RxC1
R1 ,
~=
Rjj
R2C1 ,
fl
--
L
and with C1 = Cz = C, the dimensionless state-space representation of equation set (4) becomes
=
a(K
1)
--a
O
0
-fl
+
,
(5a)
if(K--1)X-Y~ 1, if(K--1)X--Y>I.
(5b)
where a=c~ a=0
and and
b=0 b=c~
Numerical integration of equation set (5) was carried out using a fourth-order Runge-Kutta algorithm with 0.005 time step taking a = 1,2, fl = 0.45 and K = 2.37. The X - Y phase-space trajectory is plotted in Fig. 3c along with the switching plane: (K -- 1)X - Y = 1. For clarity, a three-dimensional view of the attractor is shown in Fig. 3d.
A.S. Elwakil, M.P. Kennedy~Journal o f the Franklin Institute 336 (1999) 687-700
695
Fig. 4. (a) Experimental Vcl - VL trajectory: X-axis: 0.1 V/div; Y-axis: 0.1 V/div. (b) Spectrum of the CFOA output voltage.
From Eq. (5) it is seen that the circuit has a single real equilibrium point at the origin. The eigenvalues of this saddle focus are -0.71424 and 0.35412 +j0.79413. Although JFETs suffer from a large manufacturing spread and are rather temperature sensitive, they offer a simple means of implementing voltage-controlled resistors and operate well at high frequencies. The chaotic oscillator of Fig. 3a has been experimentally constructed using the following component values: R~ = 950fl, C1 = C2 = 82 pF, L = 100 ~tH, R3 = 1 kfl and R~ -- 5 kfl pot. for tuning. Figure 4a represents the observed Vcl - VL trajectory where VL is the voltage developed across the inductor. The frequency spectrum of the C F O A output voltage is shown in Fig. 4b starting at 200 kHz and extending up to 5 MHz. The centre
A.S. Elwala'L M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
696
frequency of the limit cycle was measured at 1.35 MHz. As the C F O A gain is gradually increased, the centre frequency decreases until it reaches 1.188 M H z where a period doubling route to chaos starts. The mechanism for chaos generation is similar to that of the Colpitts oscillator [1, 2].
5.2. The second chaotic oscillator configuration Figure 5a shows a chaotic oscillator based on the harmonic oscillator of Fig, 2b. A typical V c l - Vcz trajectory is shown in Fig. 5b obtained using PSpice with
J2N4338AD844~ Vcc
Vc2 +11 -
"
k
It. ~
R1
.
(a) 5. OV
/
.
.
R3
R4
.
.................................................................
4. OV:
2"OV]iIIi
. . . . . . .
'4/ i i i i I i
.
ov4
J I I
:!
. .
.
.
, .
I
-I.OV+
...........
r ...........
-0.5V -0.0V D V(C2:I)V(C2:2) (b)
r ...........
0.5V
r ...........
1.0V
r ...........
r ....
1.5V
2.0V
V(CI:2)
Fig. 5. (a) and (b) The second chaos generator and the corresponding PSpice Vc, (c) Attractor obtained by solving equation set (6).
--
Vc2
trajectory.
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
697
o-i N-2~-
i
-4--
,
-4
5
O
-5
-10
-15
-20
.
-25
(c)
-30
-'10 -8
-12
X
y Fig. 5. (continued).
R1 = 750 D, C 1 = C 2 ~-- 1 nF, L = 1 m H , R 3 = 10 kf~ and R 4 = 24.5 kl'~. With the same settings as for the first chaotic oscillator, the circuit model is given as 2
r a ( K - 1) = |a(K
L/~K
1)
- a --a
-/~
°l[ l L!] 1
Y -4-
-/~
z
(6)
where a and b are defined as in Eq. (5b). Numerical integration of equation set (6) was carried out with ~ = l, fl = 0.45 and K = 2.45. The resulting attractor is plotted in Fig. 5c. This oscillator also has a single equilibrium point at the origin. The following eigenvalues are calculated for the above parameter values: - 0 . 6 5 9 5 , 0.3297 -t-j0.7574, 0, - 0 . 2 2 5 +j0.63196,
a = ~, a = 0.
5.3. The third chaotic oscillator configuration In a similar manner, the h a r m o n i c oscillator of Fig. 2c is modified resulting in the chaotic oscillator of Fig. 6a. With the same settings as for the first and second chaotic
698
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
L
Vcc
J2N4338
+lIVe2.
[
AD844~
02
R1
(a)
-
.
.
N
-2-~' -J'0.50 -0.5 ~ -1 (b)
~ -1.5
_2"-~-'-~---..,~_~> -2,5 -3
Y
°~t"j -2 -3 X
Fig. 6. (a) The third chaotic oscillator. (b) Attractor obtained by solving equation set (7). oscillators, the circuit is described by the following state-space representation:
I!l
=
0
,,
0
where a and b are as given by
+
Eq. (5b).
[!] ,
(7)
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
699
T h e result of n u m e r i c a l l y i n t e g r a t i n g e q u a t i o n set (7) is p l o t t e d in Fig. 6b for = 0.9,/~ = 1.3 a n d K = 2.6. T h e circuit also has a single e q u i l i b r i u m p o i n t at the origin a n d the following eigenvalues are calculated: - 0 . 7 9 5 1 , 0.46755 + j 1 . 1 1 9 3 ,
a = ~,
0, - 0 . 6 5 _ j l . 4 7 5 6 ,
a = 0.
Circuit s i m u l a t i o n s similar to t h o s e of the s e c o n d c h a o t i c o s c i l l a t o r can be achieved with R1 = 800 f2, C1 = C2 = 1 n F , L = 0.5 m H , R3 : 10 kf2 a n d R4 : 25.7 kfL
6. Conclusion A family of c h a o t i c oscillators with q u a l i t a t i v e d y n a m i c s similar to those of the c h a o t i c C o l p i t t s o s c i l l a t o r has been designed. T h e p r o c e d u r e for o b t a i n i n g these oscillators s t a r t i n g from s i n u s o i d a l o s c i l l a t o r circuits is d e m o n s t r a t e d . T h e p r o p o s e d c h a o s g e n e r a t o r s are suitable for high-frequency o p e r a t i o n because a low gain c u r r e n t f e e d b a c k o p a m p is used as the active element. T h e b e h a v i o u r of the family m e m b e r s is c a p t u r e d by m a t h e m a t i c a l m o d e l s t h a t are i n d e p e n d e n t of a n y device parasitics. Hence, it is e v i d e n t t h a t m o r e c h a o t i c oscillators with similar features can be designed with a l t e r n a t i v e r e a l i z a t i o n s of the s a m e C F O A function.
Acknowledgement T h e A D 8 4 4 C F O A s used in these e x p e r i m e n t s were p r o v i d e d by A n a l o g Devices.
References [1] M. P. Kennedy, Chaos in the Colpitts oscillator, 1EEE Trans. Circuits and Systems - I 41 (1994) 77l 774. [2] M.P. Kennedy, On the relationship between the chaotic Colpitts oscillator and Chua's oscillator, IEEE Trans. Circuits and Systems - I 42 (1995) 376-379. [3] A. Namajunas, A. Tamasevicius, Modified Wien-bridge oscillator for chaos, Electron. Lett, 31 (1995) 335 336. [4] O. Morgul, Wien bridge based RC chaos generator, Electron. Len. 31 (1995) 2058 2059 [5] A. Namajunas, A. Tamasevicius, Simple RC chaotic oscillator, Electron. Lett. 32 (1996) 945-946. [-6] A.S. Elwakil, A.M. Soliman, A family of Wien-type oscillators modified for chaos, Int. J. Circuit Theory Appl. 25 (1997) 561-579. [7] A.S. Elwakil, A.M. Soliman, Two twin-T based op amp oscillators modified for chaos, J. Franklin Inst. 335B (1988) 771 787. [8] C. Toumazou, J. Lidgey, A. Payne, Emerging Techniques for High Frequency BJT Amplifier Design: A Current Mode Perspective, Parchment Press, Oxford, 1994. [-9] A.S. Sedra, K.C. Smith, A second generation current conveyor and its applications, IEEE Trans. Circuits and Systems I 17 (1970) 132 134. [10] Analog Devices, Amplifier Reference Manual, 1992.
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A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
[11] A.M. Soliman, Applications of the current feedback operational amplifiers, Analog Integrated Circuits Signal Process 14 (1996) 265-302. 1-12] A. Assi, M. Sawan, J. Zhu, An offset compensated and high gain CMOS current feedback op amp, IEEE Trans. Circuits and Systems - 1 45 (1998) 8540. [13] A.M. Soliman, A.S. Elwakil, A new generalized oscillator, Electron. Engng 70 (1998) 22 29. [14] A. Budak, Passive and Active Network Analysis and Synthesis, Houghton Mifflin, Boston, MA, 1974. [15] M. Hribsek, R.W. Newcomb, VCO controlled by one variable resistor, IEEE Trans. Circuits and Systems - I 23 (1976) 16(~169.
Journal of The Journal of the Franklin Institute 336 (1999) 687-700
Franklin Institute
A family of Colpitts-like chaotic oscillators A.S. Elwakil,M.P. Kennedy* Department of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, Ireland Received 11 June 1998; accepted 2 September 1998
Abstract A family of chaotic oscillators with qualitative dynamics similar to the chaotic Colpitts oscillator is introduced. The oscillators use a single current feedback op amp, configured as a noninverting voltage-controlled voltage source, as the active building block, and a nonlinear element with an antisymmetrical current-voltage characteristic. A procedure for obtaining the chaotic oscillators by modifying simple harmonic oscillators is demonstrated. These chaos generators are suitable for high-frequency operation; device parasitics have negligible effect. Experimental results, PSpice circuit simulations and numerical simulations of the derived mathematical models are included. © 1999 The Franklin Institute. Published by Elsevier Science Ltd.
Keywords: Chaos; Oscillators; Chaotic oscillators
1. Introduction Since the introduction of the chaotic Colpitts oscillator [1, 23, an area of research aiming at investigating the possible chaotic nature of conventional sinusoidal oscillators has been opened. Basically, it is required to investigate whether a sinusoidal oscillator circuit can by nature behave chaotically under specific conditions, as is the case with the classical Colpitts oscillator, or whether it can be modified to behave chaotically as in [3-7]. In this work, a family of chaotic oscillators that require a single coil and two capacitors and that display behaviour which is qualitatively similar to that of the chaotic Colpitts oscillator is introduced. By contrast with the chaotic Colpitts oscillator, where the active element providing gain is itself the nonlinear element (a bipolar transistor operating in both the forward active and the cut-off regions), the proposed oscillators
* Corresponding author. Fax: 00 353 1283 0921; E-mail: [email protected] 0016-0032/99/$20.00 + 0.00 © 1999 The Franklin Institute. Published by Elsevier Science Ltd. PII: S0016-0032(98)00046-5
A.S. Elwala'l, M,P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
688
rely on a current feedback op amp (CFOA) employed as a linear noninverting voltage-controlled voltage source (VCVS) to provide the necessary gain, while the nonlinearity is introduced by a junction field effect transistor (JFET) operating as a two-terminal voltage-controlled resistor. This decoupling of the active and nonlinear functions facilitates circuit design and tuning and permits the oscillator to inherit the most attractive features from both the active and the nonlinear elements. In particular, two achievements are emphasized: (1) The possibility of designing a chaotic oscillator by intentionally modifying a sinusoidal oscillator circuit is confirmed. (2) The possibility of obtaining desirable features in the chaotic oscillator through inheritance from the original sinusoidal oscillator with appropriate oscillator design and choice of the active and nonlinear elements. Such features may include suitability for high-frequency operation, insensitivity to parasitics and component variations, low-voltage low-power operation, possibility of certain elements being grounded or floating and providing for current or voltage outputs. In Section 2, a brief discussion of the CFOA and its major sources of nonideality is given while in Section 3, a family of three sinusoidal oscillators that are to be modified for chaos is presented. Section 4 describes the modification procedure. The new family of chaotic oscillators is then presented in Section 5. Experimental results, PSpice simulations and numerical simulations of the derived mathematical models are shown.
2. The current feedback op amp
The current feedback op amp (CFOA), also known as the transimpedance op amp, is a versatile four terminal active building block which is recognized for its excellent performance in high speed and high slew rate analog signal processing [-8]. The CFOA is considered to be a cascade of a second generation current conveyor [9] and a voltage buffer. The CFOA is described by the following matrix equation:
Iy It Vo
=
{ilOO[ 0
0
0
Vy
0
0
0
V~
0
1 0
(1)
Io
In a practical implementation of a CFOA, the X terminal (inverting input terminal) is characterized by a very low input impedance (Rx) while the Y terminal (noninverting input terminal) has a practically infinite input impedance. The two outputs C and O exhibit a very high (Re) and a very low (Ro) output impedance respectively. For the commercial AD844 [10] the values of Rx, Rc and Ro are approximately 65 ~, 2 M ~ and 15 f~. A parasitic capacitance (Cc) of about 5 pF also appears at the C terminal, as shown in the macromodel of the device in Fig. 1 [-8]. The CFOA does not suffer from the finite gain-bandwidth product of conventional voltage op amps (VOAs) and can
A.S. Elwakil, M.P, Kennedy~Journalof the Franklin Institute 336 (1999) 687-700
+o
C
~>
689
0
Av=+ 1
I
Rx
Rc T Cc
I Fig. 1. Macromodel of the CFOA device.
operate without external negative feedback. The capability of a CFOA to provide both voltage and current outputs is particularly useful and results in circuits that generally require fewer components. Many analog signal processing blocks can be realized using the CFOA [11]. A novel CMOS CFOA was recently introduced in [12].
3. A family of simple harmonic oscillators
Figure 2 represents a family of simple harmonic oscillators employing a single CFOA as a linear noninverting voltage-controlled voltage source (VCVS) [13]. With equal capacitor values and neglecting the parasitic elements Rx, Re, Ro and Cc, the state equations of this family are given as
l a,,
l?c2J = C kaza
1
a22_J LVc2J"
The parameter set Jail, [~2(R~--l)
(2)
alz, a z b a223
for the three circuits is given respectively as
1 -1,1(R4
for the circuit in Fig. 2a, l(R4 1~--1 l(R4 )R,--1 R2 k,R3 - J' R 2 ' R2 R3 - 1 + R1R3 , R2 for the circuit in Fig. 2b and ~ L ( R4 LR2 \ R 3 --
1
1 RI'
for the circuit in Fig. 2c.
R4 R2R3 +
11 '
'
1] R1
690
A.S. Elwakil, M.P. Kennedy/Journal of the Franklin Institute 336 (1999) 687 700
Vcc
C2
R2 AD844A/AD
R3
(a)
_
-Vcc
_
R4
._
Vcc
C2
R2 AD844NAD
(b)
_
_
=
=
Vcc
(c)
=
-
=
=
F i g . 2. T h e f a m i l y o f s i n u s o i d a l o s c i l l a t o r s .
The condition for oscillation (ax 1 + a22 thus found to be
R4
Rz
K=R33 =2+
R-~
= 0)
for the oscillators of Fig. 2a and b is
(2a)
A.S. Elwakil, A~P. Kennedy~Journal of the Franklin Institute 336 (1999) 687 700
691
and for the oscillator of Fig. 2c it is given by K=I
+2R-z R1
(2b)
which are both satisfied by choosing R2 = R1 = R and R4 = 3R3. The ratio K = 3 is the necessary loop gain provided by the CFOA, The radian frequency of oscillation is then found to be
1 ~o
-
RC
(2c)
"
It is worth noting that the circuit of Fig. 2c is related to that of Fig. 2b by the R C - C R transformation [14].
4. Modification for chaos In order to modify the family of oscillators for chaos, there are two necessary requirements. First, an additional energy storage element (inductor or capacitor) should be added to the oscillator structure to allow for a third-order autonomous system to exist. Second, some form of nonlinearity must be introduced. Targeting a structure similar to the chaotic Colpitts oscillator [1], an inductor and a voltage-controlled nonlinear resistor with an antisymmetrical current-voltage characteristic are selected. This type of nonlinearity can be simply realized by a J F E T operating as a two-terminal device. A suitable position to add the inductor in the three harmonic oscillators is in series with R1. This location allows the inductor to be grounded and causes minimal changes to the basic oscillator architecture. Since JFETs are well known for their good performance as voltage-controlled resistors [15], a J F E T might then replace resistor Rg. The current-voltage characteristic of this device can be modelled in a piecewise-linear form as
l ~'V~s, Is = ~ (vp,
V~s/> Vp, Vos < vp,
(3)
where Rj is the J F E T small signal resistance at the operating point (approx. 750 fl for the J2N4338), Vp is the J F E T pinch-off voltage (approx. - 0 . 7 V) and V~s is the gate-to-source voltage. Equations (2a), (2b) and (2c) are used to estimate the required values of circuit components and the centre frequency of the generated chaotic spectrum. The modification process is semi-systematic in the sense that the choices of the position and value of the inductor and the nonlinear element are subject to the designer's intuition. In the following section, the above process is applied to the family of harmonic oscillators resulting in a family of Colpitts-like chaotic oscillators.
692
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687 700
5. The family of chaotic oscillators
5.1. The first chaotic oscillator configuration Figure 3a represents a chaotic oscillator resulting from the modification of the harmonic oscillator in Fig. 2a. Guided by Eq. (2a), the value of R1 should be taken
Vc2 +1
J2N4338
AD84
Vcc
I
-
R1 Cl
.~
Vcl
-Vcc
1
(a) 2 • 5V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
t I I
2.ov"
.
.
.
.
.
.
.
.
.
.
.
•
, I
,
b I
a.
ov~
.
I I I I I I I I I J I
-0.0V J
.
.
.
.
.
-I.OV+---
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
-I.2V [] V ( C 2 : I ) (b)
.
•
.
.
.
.
.
.
-I
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
.
.
.
.
.
.
2. OV
I .'O V
O.OV V(C2:2) V(CI:2)
Fig. 3. (a) and (b) The first chaotic oscillator and the corresponding PSpice Trajectories obtained by numerical integration of Eq. (5).
Vcl -
Vc2
trajectory. (c) and (d)
A.S. Elwakil, M.P. Kennedy~Journal o f the Franklin Institute 336 (1999) 687 700
693
-4 >.-6
-10 -12
-12
-10
-8
-6
-4
(c)
-2
0
X
-6.
-15
(d)
-10
Y
-5
X Fig. 3. (continued).
2
6
8
694
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
around the same value of the J F E T small signal resistance while the gain K should then be varied around the nominal value K = 3. The equal capacitors are arbitrarily chosen to define the frequency band of interest which can be estimated from Eq. (2c). Fig. 3b shows a PSpice simulation of the Vcl - Vc2 phase space trajectory obtained with R~ = 750 f~, Cl = C2 = 50 pF, L --- 50 pH, R 3 = 1 kf~, R4 = 2.85 kf2 and using + 9 V supplies. It can be seen that the parasitic input resistance Rx of the C F O A appears in series w i t h R 3 and can thus be accounted for. The very small CFOA output resistance Ro has very little effect on the circuit's behaviour; it simply adds in series with the J F E T resistance. However, in order to neglect the effect of the parasitic capacitance Cc, especially at high frequencies, it is recommended to make C1 and C2 much larger than Cc. The low gain (K = 3) provided by the C F O A further reduces its effect. The values of the gain resistors (R 3 and R4) should be sufficiently large to minimize the current supplied by the C F O A to the respective terminals. Neglecting the parasitics, the circuit can be described by the following set of equations: Cll/c I = Ij - IL, C2 I;'c2 = 1j,
(4a)
LIL = Vcl -- R I l L ,
where 1L is the inductor current and Ij is the nonlinear J F E T current modelled as 1 ~'(K - 1)Vcl - Vc2, (K - 1)Vc, - Vc2 >/ Vp, I, = ~jj ( Vp,
(4b)
(K -- 1)VC1 -- Vc2 < Vp.
Setting Vcl X=
Vc2
Vp '
Y=
RIlL
Vp'
Z
- -
2
--Vp
t '
~
--
RxC1
R1 ,
~=
Rjj
R2C1 ,
fl
--
L
and with C1 = Cz = C, the dimensionless state-space representation of equation set (4) becomes
=
a(K
1)
--a
O
0
-fl
+
,
(5a)
if(K--1)X-Y~ 1, if(K--1)X--Y>I.
(5b)
where a=c~ a=0
and and
b=0 b=c~
Numerical integration of equation set (5) was carried out using a fourth-order Runge-Kutta algorithm with 0.005 time step taking a = 1,2, fl = 0.45 and K = 2.37. The X - Y phase-space trajectory is plotted in Fig. 3c along with the switching plane: (K -- 1)X - Y = 1. For clarity, a three-dimensional view of the attractor is shown in Fig. 3d.
A.S. Elwakil, M.P. Kennedy~Journal o f the Franklin Institute 336 (1999) 687-700
695
Fig. 4. (a) Experimental Vcl - VL trajectory: X-axis: 0.1 V/div; Y-axis: 0.1 V/div. (b) Spectrum of the CFOA output voltage.
From Eq. (5) it is seen that the circuit has a single real equilibrium point at the origin. The eigenvalues of this saddle focus are -0.71424 and 0.35412 +j0.79413. Although JFETs suffer from a large manufacturing spread and are rather temperature sensitive, they offer a simple means of implementing voltage-controlled resistors and operate well at high frequencies. The chaotic oscillator of Fig. 3a has been experimentally constructed using the following component values: R~ = 950fl, C1 = C2 = 82 pF, L = 100 ~tH, R3 = 1 kfl and R~ -- 5 kfl pot. for tuning. Figure 4a represents the observed Vcl - VL trajectory where VL is the voltage developed across the inductor. The frequency spectrum of the C F O A output voltage is shown in Fig. 4b starting at 200 kHz and extending up to 5 MHz. The centre
A.S. Elwala'L M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
696
frequency of the limit cycle was measured at 1.35 MHz. As the C F O A gain is gradually increased, the centre frequency decreases until it reaches 1.188 M H z where a period doubling route to chaos starts. The mechanism for chaos generation is similar to that of the Colpitts oscillator [1, 2].
5.2. The second chaotic oscillator configuration Figure 5a shows a chaotic oscillator based on the harmonic oscillator of Fig, 2b. A typical V c l - Vcz trajectory is shown in Fig. 5b obtained using PSpice with
J2N4338AD844~ Vcc
Vc2 +11 -
"
k
It. ~
R1
.
(a) 5. OV
/
.
.
R3
R4
.
.................................................................
4. OV:
2"OV]iIIi
. . . . . . .
'4/ i i i i I i
.
ov4
J I I
:!
. .
.
.
, .
I
-I.OV+
...........
r ...........
-0.5V -0.0V D V(C2:I)V(C2:2) (b)
r ...........
0.5V
r ...........
1.0V
r ...........
r ....
1.5V
2.0V
V(CI:2)
Fig. 5. (a) and (b) The second chaos generator and the corresponding PSpice Vc, (c) Attractor obtained by solving equation set (6).
--
Vc2
trajectory.
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
697
o-i N-2~-
i
-4--
,
-4
5
O
-5
-10
-15
-20
.
-25
(c)
-30
-'10 -8
-12
X
y Fig. 5. (continued).
R1 = 750 D, C 1 = C 2 ~-- 1 nF, L = 1 m H , R 3 = 10 kf~ and R 4 = 24.5 kl'~. With the same settings as for the first chaotic oscillator, the circuit model is given as 2
r a ( K - 1) = |a(K
L/~K
1)
- a --a
-/~
°l[ l L!] 1
Y -4-
-/~
z
(6)
where a and b are defined as in Eq. (5b). Numerical integration of equation set (6) was carried out with ~ = l, fl = 0.45 and K = 2.45. The resulting attractor is plotted in Fig. 5c. This oscillator also has a single equilibrium point at the origin. The following eigenvalues are calculated for the above parameter values: - 0 . 6 5 9 5 , 0.3297 -t-j0.7574, 0, - 0 . 2 2 5 +j0.63196,
a = ~, a = 0.
5.3. The third chaotic oscillator configuration In a similar manner, the h a r m o n i c oscillator of Fig. 2c is modified resulting in the chaotic oscillator of Fig. 6a. With the same settings as for the first and second chaotic
698
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
L
Vcc
J2N4338
+lIVe2.
[
AD844~
02
R1
(a)
-
.
.
N
-2-~' -J'0.50 -0.5 ~ -1 (b)
~ -1.5
_2"-~-'-~---..,~_~> -2,5 -3
Y
°~t"j -2 -3 X
Fig. 6. (a) The third chaotic oscillator. (b) Attractor obtained by solving equation set (7). oscillators, the circuit is described by the following state-space representation:
I!l
=
0
,,
0
where a and b are as given by
+
Eq. (5b).
[!] ,
(7)
A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
699
T h e result of n u m e r i c a l l y i n t e g r a t i n g e q u a t i o n set (7) is p l o t t e d in Fig. 6b for = 0.9,/~ = 1.3 a n d K = 2.6. T h e circuit also has a single e q u i l i b r i u m p o i n t at the origin a n d the following eigenvalues are calculated: - 0 . 7 9 5 1 , 0.46755 + j 1 . 1 1 9 3 ,
a = ~,
0, - 0 . 6 5 _ j l . 4 7 5 6 ,
a = 0.
Circuit s i m u l a t i o n s similar to t h o s e of the s e c o n d c h a o t i c o s c i l l a t o r can be achieved with R1 = 800 f2, C1 = C2 = 1 n F , L = 0.5 m H , R3 : 10 kf2 a n d R4 : 25.7 kfL
6. Conclusion A family of c h a o t i c oscillators with q u a l i t a t i v e d y n a m i c s similar to those of the c h a o t i c C o l p i t t s o s c i l l a t o r has been designed. T h e p r o c e d u r e for o b t a i n i n g these oscillators s t a r t i n g from s i n u s o i d a l o s c i l l a t o r circuits is d e m o n s t r a t e d . T h e p r o p o s e d c h a o s g e n e r a t o r s are suitable for high-frequency o p e r a t i o n because a low gain c u r r e n t f e e d b a c k o p a m p is used as the active element. T h e b e h a v i o u r of the family m e m b e r s is c a p t u r e d by m a t h e m a t i c a l m o d e l s t h a t are i n d e p e n d e n t of a n y device parasitics. Hence, it is e v i d e n t t h a t m o r e c h a o t i c oscillators with similar features can be designed with a l t e r n a t i v e r e a l i z a t i o n s of the s a m e C F O A function.
Acknowledgement T h e A D 8 4 4 C F O A s used in these e x p e r i m e n t s were p r o v i d e d by A n a l o g Devices.
References [1] M. P. Kennedy, Chaos in the Colpitts oscillator, 1EEE Trans. Circuits and Systems - I 41 (1994) 77l 774. [2] M.P. Kennedy, On the relationship between the chaotic Colpitts oscillator and Chua's oscillator, IEEE Trans. Circuits and Systems - I 42 (1995) 376-379. [3] A. Namajunas, A. Tamasevicius, Modified Wien-bridge oscillator for chaos, Electron. Lett, 31 (1995) 335 336. [4] O. Morgul, Wien bridge based RC chaos generator, Electron. Len. 31 (1995) 2058 2059 [5] A. Namajunas, A. Tamasevicius, Simple RC chaotic oscillator, Electron. Lett. 32 (1996) 945-946. [-6] A.S. Elwakil, A.M. Soliman, A family of Wien-type oscillators modified for chaos, Int. J. Circuit Theory Appl. 25 (1997) 561-579. [7] A.S. Elwakil, A.M. Soliman, Two twin-T based op amp oscillators modified for chaos, J. Franklin Inst. 335B (1988) 771 787. [8] C. Toumazou, J. Lidgey, A. Payne, Emerging Techniques for High Frequency BJT Amplifier Design: A Current Mode Perspective, Parchment Press, Oxford, 1994. [-9] A.S. Sedra, K.C. Smith, A second generation current conveyor and its applications, IEEE Trans. Circuits and Systems I 17 (1970) 132 134. [10] Analog Devices, Amplifier Reference Manual, 1992.
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A.S. Elwakil, M.P. Kennedy~Journal of the Franklin Institute 336 (1999) 687-700
[11] A.M. Soliman, Applications of the current feedback operational amplifiers, Analog Integrated Circuits Signal Process 14 (1996) 265-302. 1-12] A. Assi, M. Sawan, J. Zhu, An offset compensated and high gain CMOS current feedback op amp, IEEE Trans. Circuits and Systems - 1 45 (1998) 8540. [13] A.M. Soliman, A.S. Elwakil, A new generalized oscillator, Electron. Engng 70 (1998) 22 29. [14] A. Budak, Passive and Active Network Analysis and Synthesis, Houghton Mifflin, Boston, MA, 1974. [15] M. Hribsek, R.W. Newcomb, VCO controlled by one variable resistor, IEEE Trans. Circuits and Systems - I 23 (1976) 16(~169.