CHAPTER 4

EFFECT OF NONLINEARITY IN SINUSOIDAL

OSCILLATORS

The development ofchaotic oscillators and their 5ynchronization is an active

.field ofresearch in communication technology. Tl1'O nonlinear Wien-bridge oscillator

configurations based on current feedback operational ampltfier as the active element

generating chaotic oscillations are presented in the first part of this chapter. The

proposed structures are based on the simplest possible abstract model of RC

sinusoidal oscillators that satisfy the basic Barkhausen criteriafor oscillation and the

frequency ofoscillation. The chaotic signals generatedfrom both circuits persist for a

wide and continuous range ofparameter values and in different frequency bands that

are easily adjusted by simple parameter scaling. The numerical and simulation results

obtained confirm the chaotic behaviour ofthe modtfied circuits.

Recent investigations show that wireless communication 5ystems are a very

promising application area of chaotic dynamics. The Colpitts oscillator is one of the

chaotic oscillators that have been extensively used as a sinusoidal waveform source

because of its low harmonic content. The latter part of this chapter deals with the

numerical and simulation study C?t' Colpitts oscillator for its chaotic dynamics. A rich

variety ofbtfurcation phenomena is observed in the circuit .fhr dttferent values of the

control parameter. Simulation study is also carried out/hI' this oscillator byappZving

a sinusoidal signal to ihe base of the transistor. thus providing the scope for

understanding the dynamics (~{a driven oscillator.

77

4.1 Introduction

Since its introduction, Chua's circuit has been the main source for studying

chaos in electronic circuits. Extensive studies of this circuit's performance have been

introduced in the literature, notably in references [Chua et aI., 1993; Chua, 1994;

Pivka et aI., 1994]. After the introduction of chaotic behaviour in Colpitts oscillator

by Kennedy [1994], there has been an increasing interest in investigating the possible

dynamical behaviour of the classical sinusoidal oscillators [Senani & Gupta, 1998;

Elwakil & Kennedy, 1998, 1999]. This is because chaotic signals have shown to be

very useful in a large number of applications, including secure communications,

signal encryption, lnedical field, neural net\vorks and even music generation [Cuomo

& Oppenheim, 1993; Xie & Leung, 2005]. Due to the increase in such chaotic signal

applications, an increasing demand on electronic chaotic signal generators that are

simple, easy to construct, easy to tune and easy to operate in different frequency

bands have elnerged [Kilic, 2003; Hanias & Tombras, 2006]. Most of the proposed

chaotic circuits are modifications of the LC oscillators, which are inconvenient to

build for audio frequencies. The use of inductors n1akes then1 bulkier, costlier and

difficult for integration. The lnain thrust of this research area has been to introduce

new chaotic oscillator circuits and to further concentrate on studying the nonlinear

dynamics responsible for generation of chaos in these circuits. Due to easiness in

implementation, Wien-bridge based chaotic oscillators have been designed and

studied in the literature [Morgul, 1995b; Natnajunas & Tamasevicius, 1995 & 1996;

Elwakil & Soliman, 1997]. Section 4.2 deals with detailed analyses of two Wien­

bridge oscillator configurations that are modified for chaos using diode-inductor

combination and Chua's diode [Gopakumar et. aI., 2010]. Results obtained by

nUlnerical analysis and Multisiln sinlltIation are also presented.

It is required that the generated signal has a voltage s\ving that persists for a

wide and continuous range of parameter values and that periodic and chaotic

waveforn1s can be sustained. The Colpitts oscillator is one of the chaotic oscillators

which have been extensively used as a sinusoidal waveforn1 source because of its h.)\v

harn10nic content and rich dynan1ical behaviour. The Colpitts oscillator is COn11110nly

designed to generate periodic oscillations [Sedra & Sll1ith, 1998]. However, vvith

special settings of the circuit paraIlleters and certain circuitry lnodifications, the

78

bipolar junction transistor (BJT) based Colpitts oscillator [Kennedy, 1994; Wegener

& Kennedy, 1995] is found to exhibit a rich dynamical behaviour. The dynamics of a

single stage Colpitts oscillator for different values of control parameter and the

performance of a driven oscillator for varying values of signal amplitudes are

investigated. The results obtained by numerical analysis and PSpice simulation are

discussed in section 4.3.

4.2 Inducing Chaos in Wien-bridge Oscillator

This section describes two Wien-bridge oscillator configurations based on

current feedback operational amplifier (CFOA) as the active element that are

modified for chaos using diode-inductor combination and Chua's diode. The use of

CFOA [Soliman, 1996] as the active building block makes the circuit more flexible

and versatile than conventional voltage operational anlplifier (VOA) based circuit. It

provides a constant closed loop bandwidth at required gain and dynamic range. The

circuits presented in this section have the following important features:

(i) absence of inductor,

(ii) large output voltage swing,

(iii) wide range of chaotic band and

(iv) the ability to function as sinusoidal oscillator even when nonlinearity is present.

4.2.1 Chaotic Wien-bridge structure using a diode-inductor composite

An oscillator circuit in which a balanced bridge is used as the feedback

network is the Wien-bridge oscillator [Clarke, 1953] shown in Fig. 4.1. The active

elen1ent is a voltage tnode operational mnplifier (VOA) as shown in Fig. 4.l(a) or a

current feedback operational an1plifier (eFOA) as in Fig. 4.1 (b). In both cases, the op

an1p has a very large voltage gain, negligible output resistance and very high input

resistance. Assutning further that gain is constant over the range of frequencies of the

operation 0 rthe circuit, loop gain is given by

79

loop gain = - Kp

\vhere K is the closed loop gain and f3 is the feedback factor.

(4.1 )

Two auxiliary voltages VI and V2 are indicated in Fig. 4.1(a) such that Vi =

V2 -V I. It can be shown that [Millman & Halkias, 1972] impedances ZI and Z2 have

the same phase angle at a frequency of

f = 1 / (2rcRC) (4.2)

(by assuming R 1 = R2 = Rand C 1 = C2 = C). If a null is desired, then RA and Rs must

be chosen such that Vi = o. It gives Rs = 2RA• In the present case, 'when the bridge is

used as the feedback net\vork for the oscillator, the magnitude of loop gain of Eqn.

(4.1) must equal unity and must have zero phase. This is accomplished by taking the

ratio RA / (RA+Rs) less than 1/3 or gain K greater than 3 for the circuit in Fig. 4.1 (a).

Gain K is given by the relation Rs/RA for the CFOA based Wien-bridge oscillator

sho\\'11 in Fig. 4.1 (b). With C 1 = C2 = C, the state equations for the Wien-bridge

oscillator shown in Fig. 4.1 (b) are given by:

(4.3)

It is possible to realize a chaotic oscillator that fulfils a set of circuit specific

constraints. For this, a sinusoidal circuit that meets the desired requirements in telms

of passive element structure, tunability, sensitivity and active building block is

designed first. Folmulae defining the necessary condition for oscillation and the

frequency of oscillation are then derived. Based on the design and the structure of the

oscillator, a suitable position is selected to insert a simple nonlinear element or a

composite. If the designed sinusoidal oscillator is of order less than three, an

additional energy storage elelnent should also be added in a suitable position. In the

first design procedure, a diode-inductor con1posite has been introduced as the

nonlinear elelnent.

80

V2

C16 Va

R1 Vcl

Z1RA . RB

-::-

V1·

(a)

R.2

Z2 VC2

.p

R1 C·1

Z1

Fig. 4.1 Wien-bridge oscillator (a) VOA based and (b) CFOA based.

D

I--7

I )o

Fig. 4.2 A diode inductor composite.

81

Fig. 4.2 shows a diode-inductor (D-L) composite, containing a parallel

combination of diode and an inductor. The diode gets forward and reverse biased

according to the voltage developed across the inductor. This voltage appears across

the diffusion capacitance CD of the diode. The circuit can then be described by the

following equations:

L I L(4.4)

where IL is the inductor current, I is the composite current, IDis the diode current and

VD is the diode voltage. For modifying the basic Wien-bridge oscillator circuit for

chaos, this composite module should be connected in series with one of the resistors

of the feedback network as shown in Fig. 4.3. The composite current I then becomes)

(4.5)

Clearly, grounding the composite (Vp = 0 orVQ = 0) is preferable. The resultant

circuit modified for chaos by inserting the D-L composite is shown in Fig. 4.4. The

modified oscillator is then a fourth order chaotic oscillator. The state equations given

in (4.3) modify as given below:

(4.6)

For the choice of R1

quantities:

R2 R and by introducing the following dimensionless

82

p

c

Q

Fig. 4.3 Use of D-L composite in a Wien-bridge arm.

R2 C2N---4

7 VC1

6 Va

VC2

C1 5

Fig. 4.4 Wien~bridge oscillator modified for chaos using D-L composite.

83

the state space representation of the chaotic oscillator changes as shown below.

dx =(K-2)x-y+wdr

dy =(K-l)x-ydrdz-=pwdrdw-=x-z-w-K1dr

(4.7)

(4.8)

Although the systelTI described by Eqn. (4.8) is a fourth order system, it is

effectively equivalent to a three dimensional one as the value of internal capacitor Co

is much smaller compared to the other two capacitors in the circuit. The phase space

trajectory obtained by numerically integrating Eqn. (4.8) using Runge-Kutta foulih

order algorithm with K = 3.3, E = 0.01, K, = 50 and ~ = 0.1 is plotted in Fig. 4.5. The

generation of chaos in the circuit is a result of linking sinusoidal oscillators to simple

to

e't·,.5' .

. (Q

Fig. 4.5 Numerical simulation results of Eqn. (4.8) in X-V projection.

84

nonlinear devices and is associated only with the nonlinear characteristics of these

devices and not with any amplitude control mechanism of the sinusoidal oscillator.

The same characteristics responsible for chaotic generation guarantee sustained

oscillation.

4.2.2 Wien-bridge based realization of Chua's circuit

In the second modification, a nonlinear resistor is used as the nonlinear element.

The modified circuit is shown in Fig. 4.6, which consists of a VOA-based nonlinear

resistor namely the Chua diode [Elwakil & Kennedy, 2000] and the CFOA based

Wien-bridge oscillator. In the circuit diagram, Wien-bridge oscillator comprises of op

amp A, resistors RI, R2, RA and RB and capacitors Cland C2• Chua's diode contains

op amps Al and A2, resistors R3 - R9, and a capacitor C3. The modified circuit can be

described by the following state equations:

3

2

Fig. 4.6 Wien-bridge oscillator modified for chaos using Chua's diode.

85

(4.9)

,vhere g(VR) is the current in the voltage-controlled nonlinear resistor which is

modelled as follows:

(section 2.2 contain details).

(4.10)

For the choice of R1 = R2 = R; C1 = C2 = C and by introducing the following

dimensionless quantities:

the Eqns. (4.9) and (4.10) modify to dimensionless form as follows:

dx-=(K-2)x+ydr

dy =(I-K-&/"K)x-(I+&/")y+&/"zdr

dz& -=& Kx+& y-& z-g(z)

C dr r r r

where

g(z) = K tz+[lz+II-lz- 11]K2

86

(4.11 )

(4.12)

4.2.3 Simulation results

The circuit shown in Fig. 4.4 is siluulated by Multisim for component values

R) = R2=1 kQ, RA = 2.2 kQ, RB = 7.2 kQ, L=10 mH and C)= C2= 1 nF. AD844 is the

current feedback op amp with supply voltage of ±12V and IN914 is the diode used.

The simulation results are shown in Fig. 4.7. Fig. 4.7(a) shows the time domain

response ofvoltage across C2 and 4.7(b) that of C I. The Fourier spectrum of voltage

across C I is shown in Fig. 4.7(c). The wideband nature of this waveform establishes

the chaotic nature of the circuit. The phase spectra VC2 VS VC), VL VS Yo, and VC2 VS

VL are shown respectively in figures (d), (e) and (t).

The circuit shown in Fig. 4.6 is simulated by ~v1ultisim for component values

RI= R2= R7 = R9 = 220 Q, R3= 1.5 kn, RA=1 kn, RB = 3.5 kn, ~= 3.3 ill, R5 = Rg =

22 kl1 and R6 = 2.2 kQ and capacitors C) = C2 = 50 and C3 = 1.5 nF. AD844 is the

current feedback op amp A and AI and A2 are AD712 voltage op amps with supply

voltage of ± 12V. Multisin1 simulation results obtained for the circuit are depicted in

Fig. 4.8. The time don1ain response of voltage across C3 is shown in Fig. 4.8(a) and its

Fourier transform is shown in Fig. 4.8(b). The phase spectra showing the variations of

VC2 VS VCI and VC2 vs VC3 are shown in figures 4.8(c) and (d) respectively.

From the results it can be concluded that the Wien-bridge oscillator behaves

as siluple divergently oscillating part and that only the nonlinearity of the coupling

diode or the Chua diode controls the amplitude. In other words, the oscillator plays

the role of expanding while the nonlinear part that of folding. These are known as the

essence of generating chaos.

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88

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(b) Fourier spectrum of V C3: Phase spectrum showing the variation of (c) VC2 VS VCI and

(d) VC2 VS VC3.

(a) (b)

Fig. 4.9 (a) BJT Colpitts oscillator and (b) its small signal model.

89

4.3 Chaotic Dynalnics of Colpitts Oscillator and its Control

In this section, the use of nonlinear analysis as a tool for designing oscillators

to obtain either nearly sinusoidal oscillations or chaotic behaviour is presented

[Maggio et aI., 1998: 1999]. The investigation is carried out by numerical and PSpice

simulation on a modified version of the Colpitts oscillator described in standard

textbooks. Though it is silnple, all the lnajor features of chaos can be understood

analytically. One of the advantages of the Colpitts oscillator is that its frequency can

vary from a few Hz to a few GHz, depending on the circuit components chosen.

Another advantage is that the active device possesses an intrinsic nonlinearity (the

current-voltage relation is exponential), which can easily give rise to regular

(periodic) oscillations or chaotic behaviour depending upon the choice of parameters.

In this oscillator, a rich variety of nonlinear phenomena is obtained, i.e., successive

qualitative and quantitative changes in the nature of oscillations at critical value of the

control parameter [Feo & Maggio, 2000~ Feo et aI., 2000]. Simulation study is also

can'ied out for this oscillator by applying a sinusoidal signal at the base of the

transistor, thus providing the scope for understanding the dynamics of a driven

oscillator.

4.3.1 Circuit model of Colpitts oscillator

Fig. 4.9(a) shows the circuit diagram of a Colpitts oscillator. It consists of

three frequency detennining elements~ inductor L and a pair of capacitors Ct and C2.

The transistor Q is biased in its active region by l11eans of V cc, VEE and RE, which

plays the role of both an active device and a nonlinear elenlent. The resistor R

controls the quality of the LC tank circuit. The circuit is therefore a third order

autonomous systenl containing just one nonlinear elenlent, a two segment piecewise

linear resistor, as shown in the equivalent circuit of Fig. 4.9(b). The key elel11ent in

the transistor nl0del is this nonlinear resistor (modelling the base-emitter junction)

which is responsible for lnost of the chaotic phenonll~na discussed here. The neglected

elements have only a seal ing erICct on the ObSCl ved behaviour. In other words, luore

complete models do not affect the qualitative dynanlics of the circuit.

90

If it is assumed that the BJT is a purely resistive element, then the circuit may

be described by the following set of three autonomous differential equations:

(4.13)

L I L = Vcc - I L R - Vel - VC 2

where VCl and VC2 are voltages across capacitors C1 and C2 & IL is the current

through the inductor L. The transistor operates in two regions, forward active and cut­

off Hence, the transistor can be modelled as a two segment piecewise linear voltage

controlled resistor N R and a linear current controlled current source as shown in Fig.

4.9(b). Thus,

Is =0

= [VBE - VBE(cut-in)] / RON

if VBE :s VBE(cut-in)

if VBE ~ VBE(cut-in), and (4.14)

where VBE(cut-in ) is the cut-in voltage of the BJT( = 0.7 V), RoN is the small signal

on-resistance of the base-emitter junction and Bis the forward current gain of the

device. The resonant frequency of the oscillator is given by the expression

(4.15)

The current equation of a BJT is

(4.16)

where Is is the scale current, VBE is the base en1itter voltage. VT is the volt-equivalent

of temperature given by VI = kT/q ('1' denotes the absolute temperature, k the

Boltzmann constant and q thc charge of the electron) and Ie = a Ir:- In the circuit

sho\\'I1, VI3E = -VC2. It is a gcneral practice to write the equations in dimensionless

91

form. By scaling voltages by Vr , time by l/coo, where COo is the natural frequency of

oscillation of the LC tank circuit, and using the Eqn. (4.16), the dimensionless form of

Eqn. (4.13) can be written in the form

where

dx = Q K[z - ar (e-.l' -1)]dT

dy = Q(I- K) [z + (1-a)r(e-Y -1)- p(y + VEE)]dT ~.

dz =_(x+ y+z)dT Q

(4.17)

(4.18)

The numerical solution of Eqn. (4.17) is computed using fourth order Runge-Kutta

algorithm with parameter values K = 0.5, y = 1.85 x 1O-ll, Q = 1.8, and p = 0.1. Figure

4.10 shows the scaled voltage across the capacitor C2. The phase portrait of the

chaotic attractor plotted with scaled emitter voltage (y) versus scaled collector voltage

(x+y) is depicted in Fig. 4.11.

-50

1000 1050 1100 1150T

Fig. 4.10 Time domain plot of the scaled emitter voltage.

92

50

BCI.) 0bf)c;:t+-l.....

-50Q

>~CI.)

~ -100'0-1

ECI.)

-150~CI.).....c;:t

-200urn

-250-40 -20 0 20 40

Scaled collector voltage (x+Y)

Fig. 4.11 Phase portrait obtained by plotting the

scaled emitter voltage vs scaled collector voltage.

4.3.2 Simulation results

The circuit 5ho\\'n in Fig. 4.9(a) is simulated using PSpice with component

values R = 35 Q, L = 100 IlH, C I = C2 = 50 nF and with supply voltages = ±5V. The

transistor chosen is 2N2222. Emitter resistance RE is the control parameter. The actual

sinusoidal signal expected from the oscillator is presented in Fig. 4.12(a) which is

obtained for RE = 120 Q. When the resistance value is increased gradually, chaotic

behaviour of the circuit dominates. The emitter voltage, its Fourier transform and the

phase pOlirait are shown respectively in figures 4. I2(b), (c) and (d) for RE = 370 n.Evidently chaotic oscillations in Fig. 4. 12 are not frequency lilnited. However, in

practical designs, real characteristics of the BJT should be taken into account. Finite

values of current gain and threshold frequency of the transistor govern the frequency

of operation of the circuit. For very higher values of RE, the circuit exhibit different

steady state behaviour, as illustrated in Fig. 4.12(e).

93

4.3.3 Effect of sinusoidal excitation on the driven oscillator

Simulation study is also carried out for the circuit by applying a sinusoidal

signal at the input of the transistor as shown in Fig. 4.9(a). The frequency of the

sinusoidal signal is kept at 100 kHz, the resonant frequency of the circuit and the

value of emitter resistance RE is fixed at 370 n. For signal amplitudes in the range 0 :sYin ~ 450 mV, the circuit exhibits the same chaotic behaviour as the one without the

external signal. This is illustrated by the emitter waveform plotted for Yin = 350 mV

shown in Fig. 4.13(a). For higher signal muplitudes, nonlinearity is under control and

characteristics of the circuit vary. The elnitter voltage, its Fourier transform and the

phase portrait are shown respectively in figures 4.l3(b), (c), and (d) for Yin = 750

mV. It has been shown that [Murali & Lakshlnanan, 1992] near bifurcation points of

dynamical systems, the application of external periodic forcing at some resonant

frequencies can cause an amplification of the periodic signal and a shift in bifurcation

point, thus stabilizing the periodic state. As a result, at large signal amplitudes, say in

the range 1.6 V~ Yin ~ 6 V, the chaotic nature of the circuit is under control and a

pure sinusoidal signal is obtained. A typical waveform observed for Yin = 2 V is'

shown in Fig. 4.13(e).

94

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Fig. 4.12 Emitter voltage of the Colpitts oscillator with (a) RE =1200 and(b) RE =3700:

(c) Fourier transform of emitter voltage with RE = 3700, (d) phase spectrum showing emitter

voltage vs collector voltage with RE =3700 and (e) emitter voltage with RE =1000 O.

1V 1V

0.6V

i OV

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i~ -2V

-3V

-1 0 2 3 4 5 6 7Ve, volt

(d)

·6V 0 10 20 30 40 50 60 70

TIme, IJs(e)

Fig. 4.13 Emitter voltage of a driven Colpitts oscillator for f =100 kHz and RE =370 0, with

(a) Yin =350 mV and (b) Yin =750 mY: (c) Fourier transform of emitter voltage with Yin =750 mY, (d) phase spectrum showing emitter voltage vs collector voltage with Yin =2 V and

(e) emitter voltage with Yin =2 V.

95

4.4 Conclusions

A family of minin1um component oscillators that employ current feedback op

amp as the active building block has been modified for chaos using a simple

asymmetrical linearity introduced by D-L composite and Chua's diode. The resulting

RC chaotic generators are found to be extremely simple, easy to tune and offer a

buffered output voltage with large swing. Since one of the proposed circuits does not

contain an inductor, it may be lTIOre suitable for integrated circuit applications. The

chaotic signals generated from both circuits persist for a wide and continuous range of

parameter values and in different frequency bands that are easily adjusted by simple

parameter scaling. The numerical and simulation results presented in the first part of

this chapter confirm the chaotic behaviour of the modified circuits. Since the circuit

has more parameters which could be used for chaos tuning, it may be useful for

various practical applications.

In the second part of this chapter, numerical and PSpice simulation results for

a model of the Colpitts oscillator are assessed. A rich variety of bifurcation

phenomena is observed in the circuit for different values of the control parameter. The

qualitative behaviour exhibited by the oscillator turns out to be fairly insensitive with

respect to most of the simplifying assumptions. The effect of a sinusoidal driving

signal in the dynamics of the oscillator is also investigated. The circuit can be easily

implemented in a laboratory and can be studied for wide range of periodic oscillations

and chaotic behaviour.

96