213

Synchronization Phenomena in RC Oscillators
Coupled by One Resistor
Seiichiro Morot, Yoshifumi NishioS and Shinsaku Moritt Dept. of Electrical Engineering, Keio University
3-14-1 Hiyoshi, Kouhoku, Yokohama, 223 Japan
Tel: +81-45-563-1141 Email: moro@mori.elec.keio.ac.jp$ Dept. of Electrical and Electronic Engineering, University of Tokushima
2-1 Minami-Josanjima, Tokushima, 770 Japan

ABSTRACT
There have been many investigations of the mutual synchronization of oscillators ([1]-[6] and therein). Especially, we have analyzed N van der Pol type LC oscillators coupled by one resistor, and confirmed that thissystem can take a huge number of steady states. In
this study, we analyze N Wien-bridge oscillators withthe same natural frequency mutually coupled by one resistor by both of computer calculations and circuit experiments. Because there is no inductor in this system,
it is suitable for VLSI implementation. Moreover, thesystem have much more phase states than that of thesystem with van der Pol type LC oscillators when N isnot so large. So we can utilize this system as a structuralelement of cellular neural networks.

INTRODUCTION
There have been many investigations of the mutual synchronization of oscillators ([1]-[6] and therein). Endoand we have analyzed the systems of large numberof coupled van der Pol oscillators[l]. Kimura et al.
have confirmed that two oscillators coupled by resistors
are synchronized a t opposite phase[2] and three oscillators coupled by resistors are synchronized at 3-phase[3].
Moreover, we have investigated the synchronization phenomena in N oscillators coupled by one resistor[4][6].
When the nonlinearity is the third-power characteristic, N-phase oscillations are stably excited and the system has (N - l)! phase states[4]. When the nonlinearity is the fifth-power characteristic, not only N but also
N - 1, N - 2,. ,2-phase oscillations are stably excitedand the system has much more phase states than that ofsystem with the third-power nonlinear characteristics[b].
Because of the coupling structure and extremely large
number of steady states of the systems [4] and [6], thesesystems could be used as a structural element of cellularneural networks[5].

Fig. 1: Circuit model.

In this study, we investigate the synchronization phenomena in the RC Wien-bridge oscillators coupled by
one resistor with both of computer calculations and circuit experiments. Because the Wien-bridge oscillator
doesn’t include inductors and consists of only capacitors, resistors and op-amp but doesn’t include inductors, the system is suitable for VLSI implementation.
And in this system, we can see different type of synchronization from N-phase oscillation and we can get much
more phase states than that of the system of van der
Pol oscillators when N is not so large. So this system ismuch more suitable for a structural element of cellularneural networks than the system of van der Pol type LCoscillators.

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SYNCHRONIZATION PHENOMENA IN
WIEN-BRIDGE OSCILLATORS' SYSTEM
Recently, we have reported synchronization phenomena
in N van der Pol oscillators with the same natural frequency coupled by one resistor. In such systems, various
synchronization phenomena can be stably observed, because they tends to minimize the current through thecoupling resistor.
When the nonlinear characteristics are the thirdpower, we can see N-phase oscillations. When we takethe waveform observed in one oscillator as reference signal, the other oscillators can take any phase differencesamong f#Jk = 2kn/N(k = 1 , 2 , . - . , N- 1). Therefore,
this system can take ( N - l ) ! phase states. Becauseof there coupling structure and extremely large numberof steady states, the systems described above would besuitable for an extremely large memory or a structuralelement of cellular neural networks.
But because the systems of van der Pol oscillators include inductors, these systems are not available for VLSI
implementation. So we need to get many steady states
on RG oscillators coupled by one resistor. We show an
example of such systems as follows.
The circuit model is shown in Fig. 1. N Wien bridge
oscillators with the same natural frequencies are coupledby one resistor. Op-amps.
- have nonlinear gain characteristics approximately as follows.
3= Qlvk3 - g 3 V k 3t4

1-2190

192

194

196

198

200z

So circuit equations are described as follows,Fig. 2: Numerical results for the case of N = 4 ( a =0 . 0 1 , ~= 1.5).

By changing variables,t = RCT,(2) is normalized as

In equations (4), a is coupling factor and E is thestrength of nonlinearity. Because the condition for oscillation of the Wien-bridge oscillators is g1 > 3, thestrength of nonlinearity E must be larger than 0.
Next, we show an examples of numerical results with
Runge-Kutta-Gill method and corresponding circuits experimental results for the case of N = 4 , 5 (Figs. 2-5).
On computer calculations, we have to consider the differences among the natural frequencies of real oscillators. On circuit experiments, we take Ri = 4.7kO andRf = 14.7kR.
From these results, we can see the different type ofsynchronization from N-phase oscillation. In these figures, we can see only 3-phase oscillation. In figs. 2 and3, oscillators 1, 2 and 4 synchronize at almost 3-phasebut oscillator 3 synchronize with oscillator 1 at in-phase.
Similarly, in figs. 4 and 5, oscillators 1, 2 and 5 synchronize at almost 3-phase, but oscillators 1 and 3 synchro-Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:49:21 EDT from IEEE Xplore. Restrictions apply.

215

w-1Fig. 3: Experimental results for the case of N = 4(R= 100kR,C = 0.0015pF, T = 2.0kfl, horizontal scale:O.Smsec/div., vertical scale: 5V/div.)
nize at in-phase and oscillators 2 and 4 synchronize atin-phase.
Because the type of synchronization is different fromthe case with van der Pol oscillators, the number of phasestates must be different from previous case. In this case,
we can summarize the way of synchronization that 3 ofN oscillators synchronize at 3-phase and the other oscillators synchronize at in-phase with the oscillator which
synchronize at 3-phase. So when the number of oscillators are N,we can consider the number of the phasestates PN as follows.

From this equation, we can see this system have much
more phase states than that of the system with van der
Pol type LC oscillators. For comparison, we show thenumber of the phase states of this system and the systems with van der Pol type oscillators in Table 1.
When N is large (about over lo), the number of thephase states PN becomes smaller than that of the system with van der Pol oscillators. But on cellular neuralnetworks, the number of connections should be small.
So when we use these oscillator system as a structuralelement of cellular neural networks, the number of coupled oscillators should be small. Therefore this system isvery suitable for a structural element of cellular neuralnetworks.

-2'190

''

192

.

'194.

z'

196.

"198'200Fig. 4: Numerical results for the case of N = 5 (a =0 . 0 1 , ~= 1.5).

CONCLUSIONSIn this study, we have investigated the synchronization
phenomena of the Wien-bridge oscillators coupled by one
resistor by both of computer calculations and circuit
experiments. In this system, we can see only 3-phaseoscillations while N-phase oscillations in the system ofvan der Pol oscillators. Because there is no inductor in
Wien-bridge oscillators, this system is suitable for VLSI
implementation. Moreover, we can get much more phasestates than that of the system with van der Pol type oscillators.
In article [5] and [6], we show that the system of theoscillators coupled by one resistor would be utilized asa structural element of cellular neural networks becauseof their coupling structure and extremely large numberof phase states. Moreover, the system has no inductor.
So this system is much more suitable for a structural

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216

N

with Wien-bridge oscillators

R2”

611

with LC oscillators
Third-power I Fifth-power251080

45

33,776,028

13169
I

I

479,001,600,792,712,283

REFERENCES[l] T. Endo and S. Mori, “Mode Analysis of a Ringof a Large Number of Mutually Coupled van derPol Oscillators,” IEEE Trans. Circuits Syst., Vol.
CAS-25, NO. 1, pp. 7-18, 1978.
[2] H. Kimura and K. Mano, “Some Properties of Mutually Synchronized Oscillators Coupled by Resistances,” Trans. IECE, Vol. 48, No. 10, pp. 16471656, 1965 (in Japanese).

(31 H. Kimura and K. Mano, “Three-phase Oscillators
by Resistive Coupling,” IECE Technical Report ofNonlinear Theory, 64.2-20, 1965 (in Japanese).

- .t .

--n--r---rlr..--, ,, , , , , ,j__..

r-._.._._[4] Y. Nishio and S. Mori, “Mutually Coupled Oscillators with an Extreme Large Nuniber of SteadyStates,” Proc. of ISCAS ’92, pp. 819-822, 1992.
[5] Y. Nishio, S. Mori and A. Ushida, “On Coupled Oscillators Networks -for the Cellular NeuralNetwork-,” Proc. of ISCAS ’93, pp. 2327-2330,
1993.

Fig. 5: Experimental results for the case of N = 5(R = 100kR, C = 0.0015pF, T = 2.0kR, horizontal scale:
O.Zmsec/div., vertical scale: 5V/div.)[6] S. Moro, Y. Nishio and S. Mori, “Coupled Oscillators with a Huge Number of Steady States -for a
Structural Element of a Cellular Neural Network,” Proc of ECCTD ’93, pp 27-32, 1993.

element of new type of VLSI chip for cellular neural networks.

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