Synchronization of Two Identical Improved Chaotic Colpitts Oscillators

Vu Van Yem and Nguyen Huu Long School Of Electronics and Telecommunications, Hanoi University Of Science and Technology

No.1 Dai Co Viet Road, Hanoi, Vietnam

Abstract -We design and simulate an improved wideband chaotic Colpitts oscillator based on micro-strip technology. The numerical and circuit simulation results are presented in order to verify the performance of the developed oscillator. In addition, we propose the synchronization model between two identical improved chaotic Colpitts oscillators coupled by a resistor. The chaotic synchronization, one of the most important requirements for chaotic communication systems, is validated by numerical and circuit simulation results. And synchronization thresholds are also estimated.

Keywords: Chaos, non-linear dynamic; improved chaotic Colpitts oscillator; identical synchronization; coupled oscillators; Wide band chaotic communication

I. INTRODUCTION Chaos-based communications have been received a lot of

attention of researchers since the discovery of property of self-synchronization of chaotic systems by Pecora and Carroll [1]. Chaotic signals have suitable characteristics for communications such as its wideband and ultra wide bandwidth, its stability to fading in multipath environment, its relative simply circuit, its possibility of control and synchronization as well as its possibility of signal generation at any frequency band, etc. These advantages promote the study of applying chaotic signal to modern communication systems, such as wireless communications, secure communications, spread spectrum communications and radar systems.

In chaotic wireless communication systems, a lot of researchers pay attention to some important problems such as chaotic signal generator and synchronization. Regarding to oscillators in chaos-based system, the Colpitts oscillator has been much more investigated thanks to its advantages. After it was reported that the standard Colpitts oscillator, with special settings of the circuit parameters can exhibit chaotic behavior [2], it has attracted great interests of researchers. They investigated this circuit at the low (kHz) frequencies, high (3–300 MHz) frequencies and ultra-high (300–1000 MHz) frequencies [3-6]. However, in [7] it is analyzed that the highest achievable value of the fundamental frequency in chaotic Colpitts oscillator is about 0.1fT where fT is the threshold frequency of the employed transistor. Due to this limitation, recently, several alternatives of this standard version of Colpitts oscillator have been suggested as a two-stage and an improved Colpitts oscillator [8-10] in order to achieve higher fundamental frequencies. In addition, another key issue in chaotic wireless communication is synchronization problem between transmitter and receiver that plays an important role

[11]. By chaotic synchronization, coherent correlation receivers have more advantages than non-coherent receivers in terms of noise performance and data rate [12].

In this paper, we are interested by two problems. At first, we develop an improved chaotic Colpitts circuit using Philips wideband BJT BFG425W with fT =25 GHz in both circuit and numerical simulation. We also exploit the parasitic effect of microwave transistors to make a simpler and more effective oscillator. Secondly, based on the developed Colpitts oscillator, the synchronization of two identical improved Colpitts oscillators by using a coupling resistor is proposed. In section II improved Colpitts circuit and its chaotic behavior are introduced followed by numerical simulation in session III. Session IV presents implementation of this oscillator on micro-strip technology. Section V describes mathematical model of a system consisting of two coupled improved chaotic Colpitts oscillators and identical synchronization notations. The numerical and circuit experiment results are shown in session VI. Finally, a brief conclusion is given in section VII.

II. IMPROVED CHAOTIC COLPITTS OSCILLATOR The configuration of an improved chaotic oscillator and a

standard version are shown in the Fig.1. We can see that they consist of a BJT as the gain element and a resonant network including an inductor and a pair of capacitors. The difference between conventional Colpitts oscillator and the improved version is that the inductor L is moved from the collector to the base of transistor where it is in series with resistor Rb.

Figure 1. Circuit diagrams of chaotic Colpitts oscillators

a. Standard version b. Improved version

The basic mechanism behind the improved configuration version is the diminishing of negative influence of capacitor CCB (zero-bias collector-base capacitance). In the standard version of Colpitts oscillator, capacitor CCB grounds the collector node and acts as a parasitic element destroying

978-1-4673-1909-6/12/$31.00 ©2012 IEEE

chaotic oscillations. In this novel version, L and Rb screen CCB from ground and reduce its negative influence.

The state equations for the improved chaotic Colpitts oscillator depicted in the Fig.1 are as:

BCB

LCB

CCBE

CBELbL

LCCC

BLCCC

ICCRI

CCRVVV

dtdVRC

VVIRdt

dIL

RIRIVVVdt

dVRC

IRRIVVVdt

dVRC

)1()1(

)1(

112101

2

02102

2

2101

1

β

β

++−++−−=

−−−=

+−−−=

+−+−−=

Where:

��

��

�

≤

>−=

*

**

,0

,

VV

VVr

VVI

BE

BEBE

B (2)

BC II β= VV 7,0* ≈ (3)

With r is the differential resistance of the forward-biased base-emitter junction, is the break-point voltage of its I–V characteristic (for silicon transistors � 0.7V) and � is the forward current gain of the device.

Let’s change the state equations in (1) into dimensionless ones by defining:

*1

VV

x C= , *VIy Lρ

= , *2

VVz C=

*VV

v BE= , τtt = ,

ρα bR

= ,CBC

C11+=γ .

1CL=ρ ,

1

2

CC

=ε , 1LC=τ (4)

ra ρ= ,

ρRb = ,

*0

VV

c = ,*0

VId ρ

=

The improved Colpitts oscillator can be described by a set of autonomous state-space equations as following:

)()()(1

)(1

)()1()(1

.

.

.

vFayzxbb

cv

dyzxbb

cz

vzyy

vFayzxbb

cx

βγγ

ε

α

β

+−++−=

−++−=

−−−=

+−++−=

�

�

�

�

(5)

Where F (v) is a non-linear function given by:

���

≤>−

=1,01,1

)(vvv

vF (6)

III. NUMERICAL SIMULATION We use Matlab for numerical simulations and in these

simulations, the constant c/b in (5) does not influent the over-all dynamics of the system so it can be omitted. For certain sets of parameters, the system (5) can exhibit chaotic behavior. For example, with the set of parameters: a = 0.8, b = 0.67, c = 21, d = 0.96, � = 0.87, � = 100, e = 1 and the initial conditions (x(0), y (0), z(0), v(0)) = (0.3, 0.1, -1, 0.7) and by using the fourth-order Runge–Kutta algorithm, we can obtain the projection of the chaotic trajectory in the plane [ x, z] ~ [ VC1, VC2] and the wave form of signal in the time domain as in the Fig.2.

(a)

-0.2 0 0.2 0.4 0.6-1.2

-1.1

-1

-0.9

-0.8

-0.7

x

z

(b)

Figure 2. Simulated waveform (a) and phase portrait (b) of chaotic oscillations from Eq. (5) in Matlab

To confirm chaos characteristic, we compute the Lyapunov exponents (LE) of (5) by Matlab. The result in Fig. 3 shows the LEs as a function of time. For more details, Table.1 provides several specific values of them.

0 50 100 150 200 250-80

-60

-40

-20

0

20Dynamics of Lyapunov exponents

Time

Lyap

unov

exp

onen

ts

Figure 3. Plot of the Lyapunov exponents as a function of time

(1)

*V*V

0 50 100 150

-0.2

0

0.2

0.4

0.6

t

x

TABLE I. SEVERAL VALUES OF LYAPUNOV EXPONENTS

Time (t)

Lyapunov exponent

�1 �2 �3 �4

25 0.083661 -0.22596 -1.64594 -30.8697

50 0.041822 -0.36921 -1.1113 -16.8178

75 0.027881 -0.42895 -0.92111 -12.1339

100 0.020911 -0.46841 -0.81641 -9.79188

125 0.016729 -0.4766 -0.76909 -8.38669

150 0.013941 -0.48579 -0.73381 -7.4499

175 0.011949 -0.49707 -0.70388 -6.78076

200 0.010455 -0.50508 -0.6819 -6.27891

225 0.009294 -0.50566 -0.67045 -5.88858

250 0.008364 -0.50979 -0.65762 -5.57631

We can see that there is a positive LE and the sum of the exponents is negative. These are evidences approving that there is the presence of chaos in the dynamical system (5).

IV. IMPLEMENTATION OF AN IMPROVED CHAOTIC COLPITTS OSCILLATOR

To implement this circuit, we use the transistor BFG425W of Philip with the threshold frequency of 25 GHz. Both circuit simulations and layout design were performed using ADS (Advanced Design System). Because the transistor works at high frequency, the parasitic effect of BJT will be equivalently modeled as in the Fig.4 with parameters that can be found in [13].

Similar to the fundamental frequency of conventional chaotic Colpitts [14], to increase the fundamental frequency of improved chaotic Colpitts circuit, value of inductor L in Fig.1 must be decreased. When the oscillation frequency of chaotic Colpitts is very high, the value of the inductor L in Fig.1 is very small, and the value of the packaging inductor L3 in Fig.4 becomes comparable to the value of L. Using this characteristic, we propose a modified circuit of the improved chaotic Colpitts oscillator, where inductor L in the originals circuit is eliminated and the parasitic inductance L3 is exploited. Simulation for the model of this circuit is established in ADS as shown in Fig.5.

L1R=

L2R=

C2

C1

L3R=

C3 1

2

3

CNum=2

PortENum=3

PortBNum=1

Figure 4. Package equivalent circuit of BFG425W

Vc1

Vc2

RR1

RR2

V_DCV2

V_DCV1

RRe

1

2

3

CC2

CC1

Figure 5. Simulation model in ADS

By using tuning function in ADS, following circuit parameters are chosen: V1=V2=10V, C1=C2=2 pF, Re=400 Ohm, R1=35 Ohm, R2=3.5 Ohm and the inductor L certainly equal to zero. The emitter current is I0=20mA and fundamental frequency (in regard to the parasitic components) f* is about 3.5 GHz. The waveforms of simulated signals VC1, VC2 are shown in Fig.6. It can be seen that the circuit goes into chaotic state by 7 nanoseconds. The chaotic attractor VC1 versus VC2 is shown in Fig.7. Spectrum distribution of signal VC2 is presented in Fig.8. These three graphs demonstrate randomness of signal in time domain, frequency domain and phase plane respectively.

Comparing the attractor in Fig.7 with that of numerical analysis of original improved chaotic Colpitts circuit in Fig.2, the attractor shapes are rather different. This is due to the parasitic components of the transistor when it works at high frequency. However, it is obvious that both oscillations are chaotic.

2 4 6 8 10 12 14 16 180 20

8.0

8.5

9.0

9.5

7.5

10.0

time, nsec

Vc1

Fig.6. Time-domain waveform of Vc1 and VC2

2 4 6 8 10 12 14 16 180 20

-1.0

-0.8

-0.6

-0.4

-0.2

-1.2

0.0

time, nsec

Vc2

1 2 3 4 5 6 7 8 9 10 110 12

-100

-80

-60

-40

-20

-120

0

freq, GHz

dB(V

C2)

Figure 7. Power spectrum of VC2 signal

8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6 9.87.8 10.0

-1.0

-0.8

-0.6

-0.4

-0.2

-1.2

0.0

Vc1

Vc2

Fig.8. Phase portraits- VC2 against collector VC1

Our proposal circuit is realized in the micro-strip design. We use substrate material Duroid with thickness H=1.59mm and dielectric constant Er=2.33. All the circuit components are in SMD (Surface Mount Device) package. The layout is shown in Fig.9. The phase portrait and the signal VC2 simulated in Figure 10 show that the designed oscillator can exhibit the chaotic oscillations.

Fig.9. Layout of the designed circuit

6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.05.5 10.5

-0.5

0.0

0.5

1.0

1.5

-1.0

2.0

Vc1

Vc2

(a)

60 70 80 9050 100

-0.5

0.0

0.5

1.0

1.5

-1.0

2.0

time, nsec

Vc2

, V

(b)

Fig.10. Phase portraits, the time-domain of signal Vc2 in the simulation with micro-strip line

V. COUPLED CHAOTIC COLPITTS OSCILLATORS Let us consider two identical Colpitts generators, G1 and

G2 with the transistor collectors coupled via linear resistor Rk Figure 11). By setting k=�/Rk is coupling coefficient, the overall system can be described by the set of eight differential equations

���������

�

���������

�

�

+−++−=

−−++−++−=

−−−=

−−+++−++−=

+−++−=

−−++−++−=

−−−=

−−+++−++−=

)()()(1

)()(1,

),()()1()(1

)()()(1

)()(1,

),()()1()(1

22222

22112222

2222

221122222

11111

11221111

1111

112211111

vFayzxbb

cv

zxzxkdyzxbb

cz

vzyy

zxzxkvFayzxbb

cx

vFayzxbb

cv

zxzxkdyzxbb

cz

vzyy

zxzxkvFayzxbb

cx

βγγ

ε

α

β

βγγ

ε

α

β

�

�

�

�

�

�

�

�

(7)

Fig.11.Coupled Oscillator

We also can synchronize the oscillators by coupling the emitters of the transistors. In this case the coupled system is given by:

���������

�

���������

�

�

+−++−=

−−++−++−=

−−+++−−−=

+−++−=

+−++−=

−−++−++−=

−−++−−−=

+−++−=

)()()(1

)()(1),(

),()1()(1

)()()(1

)()(1),(

),()1()(1

22222

22112222

22112222

22222

11111

11221111

11221111

11111

vFayzxbb

cv

zxzxkdyzxbb

cz

zxzxkvzyy

vFayzxbb

cx

vFayzxbb

cv

zxzxkdyzxbb

cz

zxzxkvzyy

vFayzxbb

cx

βγγ

ε

α

β

βγγ

ε

α

β

�

�

�

�

�

�

�

�

(8)

There are several notions of synchronization between chaotic systems such as: identical synchronization [15], generalized synchronization [8], [9] and phase synchronization [10], [11]. Here we use the strongest and most widely used notations: identical synchronization. In identical synchronization two systems are synchronized identically if:

0)()('lim =−

∞→txtx

t (9)

for any combination of initial states x (0) and x’ (0), with x and x’ are the state variables of the drive and response system.

VI. SIMULATION RESULTS

A. Numerical simulation We have solved numerically Eq. (7) and (8) using the

fourth-order Runge–Kutta algorithm. The simulation results show that there is a synchronization threshold value (kth). When the systems are not coupled (k=0) or the coupling coefficient is insufficient (k<kth) though the two generators G1 and G2 are fully identical ones, their oscillations do not coincide with each other. The estimated value of kth is 4.7 for coupled emitters and 0.47 for coupled collectors. Unsynchronized oscillators are illustrated in Figure 12 (k=0.1, coupled emitters). Different behavior of identical dynamical systems is caused by different initial conditions of the generators.

0 10 20 30 40 50 60 70 80 90 100-1.2

-1

-0.8

-0.6

-0.4

t

z1,

z2

z1

z2

(a)

0 20 40 60 80 100-0.5

0

0.5

t

z- z

2

(b)

Fig.12. Signals z in two oscillator (a) and the differential signal (b) when k=0.1 and coupled emitter

When the coupling is strong enough (k>kth) the oscillators “forget” their own initial conditions and after a short transient, approximate 10, they synchronize to each other. Meanwhile, the all different signals x1-x2, y1-y2, z1-z2 tend to zero (Figure 13.b). However, we note that each of the individual signals z1 and z2 remain chaotic as shown in Figure 13.a

0 10 20 30 40 50 60 70 80 90 100

-1.2

-1

-0.8

-0.6

-0.4

t

z1,

z2

z1

z2

(a)

0 5 10 15 20 25 30 35 40-0.4

-0.2

0

0.2

t

x1-x2

y1-y2z1-z2

(b)

Fig.13. Signals z in two oscillator (a) and the differential

signals (b) when coupled emitter and k=10

B. Circuit simulation The circuit simulation is done using ADS. Two emitter

coupled oscillators have been built using the following element values: L=0.10 nH, C1=C2=4.3nF (fundamental frequency is about 1 GHz), R=36�, V=10V, Rb=20�, Re=500�. Transistor Q is the BFG520 type bipolar junction transistors of Philips. With this set of parameters, oscillators are exhibited chaotic behavior and its chaotic attractor can be seen in Figure 14.

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-1.4 0.8

8.5

9.0

9.5

8.0

10.0

Vc2

Vc1

Fig.14. Simulated chaotic attractor in the phase plane

Figure 15 is Lissajous figure in the case of unsynchronized chaotic generators. It is a very complicated one, indicating random relation in phases and amplitudes of the oscillations.

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-1.4 0.8

-1.0

-0.5

0.0

0.5

-1.5

1.0

V2

V1

Fig.15. Emiter voltage from OSC1 versus emiter voltage from OSC2 for

unsynchronized case (Rk=300 �)

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-1.4 0.8

-1.0

-0.5

0.0

0.5

-1.5

1.0

V2

V1

Fig.16. Voltage from OSC1 versus emiter voltage from OSC2 for

synchronized case (Rk=200 �)

Meanwhile the fine diagonal in Figure 16 proves that the amplitudes and phases of the oscillations from the two generators do coincide with each other, and the oscillators can be considered as fully synchronized.

VII. CONCLUSION We demonstrate that the improved Colpitts oscillator with

BFG425W transistor can generate wideband chaotic oscillations effectively. The chaotic characteristic of the designed oscillator is well verified by calculating the Lyapunov exponent i.e. LE with a positive L.E and the sum if exponent is negative. We design and realize the layout of this circuit in micro-strip technology. The numerical and simulation results show that the oscillator generates chaotic signal in designed

microwave frequency band from 3 GHz to 10 GHz. In addition, we propose and demonstrate the ability of synchronization between two coupled improved chaotic Colpitts oscillators via a linear resistor. The synchronization threshold was also estimated. The chaotic technique appears as promising candidate for wideband wireless communications thanks to its simple structures.

ACKNOWLEDGMENT This work is carried out in the framework of the research

project supported by the Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) “Research and development of chaotic time pulse modulation methods for digital communication systems - Nghiên c�u phát tri�n các phng th�c �i�u ch th�i gian xung h�n lo�n cho h� th�ng thông tin s�”. The authors would like to thank the NAFOSTED for financial supporting this research.

REFERENCES [1] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems”,

Physical Review Letters, vol. 64, No. 8, pp. 821- 824, 1990. [2] M. P. Kennedy, “Chaos in the Colpitts oscillator”, IEEE Transactions on

Circuits and Systems I, vol. 41, No. 11, pp. 771-774 , 1994. [3] C.Wegener, M.P.Kennedy. “RF chaotic Colpitts oscillator”, in:

Proceedings of an international workshop on nonlinear dynamics of electronic systems NDES’95, Dublin Ireland;, p. 255–258, 1995.

[4] G. Mykolaitis, Tamasevicius. A, and S. Bumeliene, "‘Experimental demonstration of chaos from the Colpitts oscillator in the VHF and the UHF ranges’," Electron. Lett, vol. 40, pp. 91-92, 2004.

[5] A. Baziliauskas, A.T. Sevi, G. Mykolaitis, R. Krivickas, and E. Lindberg, "Chaotic Colpitts Oscillator for the Ultrahigh Frequency Range" Nonlinear Dynamics,vol. 46, pp. 159-165, 2006.

[6] S. Zhiguo, R. Lixin, and C. Kangsheng, "Simulation and experimental study of chaos generation in 1 microwave band using colpitts circuit," Journal of Electronics, vol. 23, pp. 433-436, 2006.

[7] S.Bumeliene, G. Mykolaitis, G. Lasiene, A. Cenys, and A. Tamasevicius, "Evaluation of high-speed bipolar transistors for application to chaotic colpitts oscillator," Ultrafast Phenomena Semiconduct, vol. 384/383, pp. 151-154, 2001.

[8] A. Tamasevicius, G. Mykolaitis, S. Bumelien, A. Cenys, A. N. Anagnostopoulos and E. Lindberg “Two-stage chaotic Colpitts oscillator” Electron. Lett, vol. 37, No. 9, pp. 549-551, 2001

[9] A. Tamasevicius, S. Bumeliene, and E. Lindberg, "Improved chaotic Colpitts oscillator for ultrahigh frequencies .," Electronics Letters, vol. 40, pp. 25-26, 2004.

[10] J.Y. Effa, B.Z. Essimbi, and J.M. Ngundam, "Commun Nonlinear Sci Numer Simulat Synchronization of Colpitts oscillators with different orders," Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 1598-1605, 2009.

[11] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems”, Physical Review Letters, vol. 64, No. 8, pp. 821- 824, 1990.

[12] M.P. Kennedy and L.O. Chua, "The Role of Synchronization in Digital Communications Using Chao - part II", IEEE Transactions on Circuits and Systems I, Vo1.45, No.11. pp.1129 -1 140, 1998.

[13] Philips Semiconductors. Data sheet BFG425W NPN 25 GHz wideband transistor. Mar. 11, 1998.

[14] S. Wei; S. Donglin; , "Microwave chaotic Colpitts circuit design", 8th International Symposium on Antennas, Propagation and EM Theory, 2008 (ISAPE 2008), pp.1127-1130, 2008.

[15] M. P. Kennedy, “Three steps to chaos part I: Evolution,” IEEE Trans. Circuits Syst. I, (Special Issue on Chaos in Nonlinear Electronic Cir- cuits—Part A: Tutorial and Reviews), vol. 40, pp. 640–656, Oct. 1993.