View of ANALYSIS OF SYNCHRONIZATION IN COUPLED COLPITT OSCILLATORS OPERATING UNDER THE CHAOTIC CONFIGURATION

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Vol.03, Issue 09, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

ANALYSIS OF SYNCHRONIZATION IN COUPLED COLPITT OSCILLATORS OPERATING UNDER THE CHAOTIC CONFIGURATION

1Babita Saxena Lecturer SIRT Bhopal

2Shalini Pushpadh Scholar In SIRTS Bhopal

Abstract :. In this work we present a detail investigation of the effect of synchronization of emitter coupled and collector coupled colpits oscillator with different values of coupling coefficient in driver response configuration system (unidirectional coupling between two identical systems). This research work focuses on the performance of emitter coupled and collector coupled chaotic colpitts oscillators by numerical investigation of settling time of synchronization in MATLAB. Synchronization. Results of investigations have been plotted as the plot of RMS of Error , Standard Deviation of Error , settling time versus amplitude for the maximum positive deviation and max negative deviation. These graphs shows that emitter coupled system has much smoother error graph for coupling coefficient and have lower positive deviation while Collector coupled systems have lower negative deviation.

These plots also shows that Colpits oscillators are settle down early in emitter coupling as compare to collector coupled oscillator.

1 INTRODUCTION

Chaos signal is a noise type random signal having a comparable long unpredictable nature . It is generally expressed mathematically by a sensitivity to initial conditions—where the equations of system dynamical condition takes it is difficult to predict from the initial point. This random nature of chaotic systems advised early is very useful in some type of secure communications. Noise like and ultra wide bandwidth waveform are very beneficial for number of applications such as secure communication , medical application and RADAR.

Fundamental components for secure chaos communication system are Chaotic Signals, oscillators which generates chaos waveform , Masking Circuit, Nonlinear Filter, Synchronizing method, and Encryption/Decryption. For practical realization of chaos based communications system two chaotic oscillators are required one act as a transmitter (or master) and another as receiver (or slave).

Fig 1 Chaotic Communication System

1.1 Complete Synchronization

Complete synchronisation of non-linear systems i.e colpitts oscillator , now consider two fully diffusively coupled syatem

We assume that H(0) = 0 so that the synchronization subspace x1 = x2 is invariant for all coupling strengths α.Meaning that for any synchronized initial condition the entire solution remains synchronized: as in the synchronized state the diffusive coupling term vanishes, the dynamics is identical to that of the uncoupled system (with α = 0). Consequently, the coupling has no influence on the synchronized motion. In particular, it could be the case that the synchronised motion is chaotic, if the uncoupled systems exhibit such behavior.

We aim to show that if the coupling is sufficiently strong, the system Equation will synchronize x1(t) − x2(t) → 0 as t

→∞. We consider H = I (the identity matrix) then the term reads as

αH(x2 − x1) = α(x2 − x1).

The aim is to identify sufficient conditions for the coupling parameter α to guarantee that locally near z = 0 we have limit→∞ z(t) = 0. Another possibility is that we use certain sets of variables to drive a subsystem. For appropriate choices, we can observe synchronization . We illustrate this scheme in the Lorenz

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system where x-component can be driving signal of another identical system. The master is fully replaced to the x variable in the slave

˙x = σ(y − x)

y˙ = x(ρ − z) − y y˙s = x(ρ − zs) − ys z˙ = −βz + xy z˙s = −βzs + xys

where (x, y, z) are the states of the master system and (ys, zs) are the states of the slave system. In order to check the behaviour of the trajectories, we track the simultaneous variation of the trajectories by y(t) = y(t) − ys(t) and )z(t) = z(t) − zs(t).

The model and the nonlinear analysis of Colpitts oscillators which have been presented in earlier has been taken here as it is [39-41] .

The mathematical model for three-point (transistor)scheme is described as follows.

Here VC1 - voltage at C1 capacitor, VC2 – voltage at C2 capacitor, I L - current through L inductor, f - nonlinear characteristics of the transistor, αF - ratio of collector current to emitter current.

Note that αF =1 for most transistors.

Let αF =1, i.e. neglect base current. For our purposes, this approximation is rather fair. It is usual to describe transistor characteristics as follows:

IE = Is exp( ).

The collector current IK is proportional to the emitter current: IK=αIE. The nonlinear current-voltage characteristic of

the emitter-base (EB) junction can be approximated by two linear segments:

Here r is the small signal ON resistance of the EB junction and U* is the break- point voltage (U*≈0.7V).

Assuming for simplicity that the forward current gain α≈1, that is IK ≈IE and in troducing the following dimensionless variables and parameters [39-41]

we come to the set of differential equations convenient for numerical integration

Here

For certain sets of the circuit parameters, i.e. the coefficients in Eq. (1) has been simulated in matlab and this system exhibits chaotic oscillations which has been shown in next chapter

1.2 Synchronization of identical oscillators

Now we consider two identical Colpitts generators, G1 and G2 with the transistor collectors coupled via linear resistor Rk

Figure 2 Coupled generators Introducing the coupling coefficient k=ρ/Rk the overall system can be

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described by the set of six differential equations: 7

An alternative way to synchronize the oscillators is to couple the emitters of the transistors. In this case the coupled system is given by Equation 8..

These diifrential Equations are numerically investigated in MATLAB for . synchronization threshold estimated numerically in Matlab which is kth ≈0.5 (coupled collectors). Unsynchronized oscillators are illustrated in Fig. 10 (t<150) [39].

Different behavior of identical dynamical systems is caused by different initial conditions of the generators. When the generators are coupled to each other first by collector at (t=150) and the coupling is strong enough (k>kth) the oscillators ― forget‖ their own initial conditions and after a short transient (t=150…160) they synchronize to each other: (x1+z1) ⇒ (x2+z2). Meanwhile, the difference signal (x1+z1)−(x2+z2) tends to zero (bottom trace in Fig. 4). We note, however, that each of the individual signals (x1+z1) and (x2+z2) remain chaotic as evident from the top trace in Fig. 4 at t>150. 100 150 200

Similar synchronization results are obtained from above equations in the case of coupled emitters as well. The threshold value of the coupling coefficient kth≈0.14.

The experimental circuit (Fig.11) slightly differs from the one shown in Fig.

8. The emitter current I0 is set by means of the base divider R1-R2 and the series emitter resistor Re. The capacitor C0 grounds the base with respect to the ac

signals, thus ensures the common-base configuration of the transistor. The oscillators G1 and G2 have been built using the following element values:

L=0.86mH, C1=C2=470nF (f*≈11kHz), R=36Ω, V0=12V, C0=47μF, R1=R2=3kΩ, Re=510Ω. The Q are the 2N3904 type bipolar junction transistors.

Investigate of synchronization of chaotic Colpitts oscillators by emitter coupling and collector coupling by means of integration of above simplified (piecewise-linear) differential equations has been implemented in MATLAB .

1.3 Simulation and Results

At first we plotted chaotic waveform for single chaotic colpittts oscillator which was formulated in dynamical system discussed above simulated in MATLAB and results for simulation is following waveform

Figure 3 Plot of resulting waveform of Colpitts oscillator

2 RESULTS OF SIMULATION OF COLLECTOR COUPLED CHAOTIC OSCILATOR

Then similar chaotic oscillators are couple linearly and directly via collector as described section 3.2 chapter 3 .Coupling is done for different coupling coefficient in following manner . Result of these variations are plotted in matlab . Combined Amplitude of chaotic signal versus settling time has been plotted .

Figure 4 Synchronization of colpitts oscillator coupled through collector using coupling factor 0.8, (synchronization starts at time = 100).

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Figure 5 Synchronization of colpitts oscillator coupled through emitter using coupling factor 0.8, (synchronization starts at time = 100).

Figure 6 plot for the settling time values listed in table 1 and 2, clearly shows that emitter coupled systems can synchronize at much lower coupling factor (at 0.4) than the collector coupled systems (at 0.6).

Figure 7 coupling coefficient versus maximum positive deviation plot for the maximum positive deviation

Figure 8 plot for the max negative deviation . The figure depicts that emitter coupled systems have lower positive deviation while Collector coupled systems have lower negative deviation.

Figure 9 Lissajous waveform for synchronized colpitts oscillator

Figure 10 Lissarious waveforms for uncorrelated and non –synchronized system.

Figure 11 Impact of coupling factor on RMS value of Error

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Figure 12 Impact of coupling factor on Standard Deviation of Error

3 CONCLUTION & FUTURE SCOPE Though the two chaos generators G1 and G2 are fully identical ones, the dynamic equations of scillations of both oscillators x1(t) and x2(t), also y1 and y2, as well as z1 and z2 do not coincide with each other until the systems are not coupled (k=0) or the coupling coefficient is insufficient (k<kth). Figure 2 shows the plot of simple chaotic waveform of a single generator .

The figure 4 plots shows coupling both oscillators confirms that both coupling techniques provide similar results.

Another verification of synchronization can be done using Lissajous waveform the waveforms at left shows non-synchronization condition as the states shows no correlation, however after synchronization an stable correlation can be seen in the figure at the right.

Plot for the max negative deviation and negative . These figure depicts that emitter coupled systems have lower positive deviation while Collector coupled systems have lower negative deviation.

We have analyzed synchronization of emitter couple and collector coupled oscillators using different coupling cofficient by linear difference method . The result are shown in fig . 3, 4, 5, 6, 8 and 9. Fig 8 shows synchronization condition of two oscillators. If two are not synchronized wave will be lissajous.

The plot of settling time in fig. 5 shows how time varies according coupling cofficient for emitter coupled (Blue) and collector coupled (Red). We can see in figure that oscillators are settle down early in emitter coupling which is 0.4 and after at 0.6 collector coupled oscillator are settle down . So we conclude that emitter coupled oscillator are better

in terms of settling time as compare to collector coupled.

4 FUTURE SCOPE

The application of chaos theory is very vast field . One of its application in secure communication successfully acomplshised by the use of synchronization in two chaotic colpitts oscillator . However there are other schemes also possible for secure chaotic communication . Here synchronisation has been analysed for emitter and collector coupling which is linear and bidirectional . There is a scope for other schemes for synchronizing coupled colpitts oscillator which would have less synchronizing error and less settling time

The application of chaos in digital

communication needed less

synchronization methods . Chaos theory can be used in various field also as in medical , cinema etc .

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