205On the Synchronization of Oscillators
Coupled by One Negative Resistor
Hiroyuki Kanasugi t, Seiichirou Moro t, Yoshifumi Nishio $ and Shinsaku Mori tt Department of Electrical Engineering,$ Department of Electrical and ElectronicEngineering, Tokushima University
2-1 Minami-Jousanjima,
Tokushima, 770 JAPAN
Email: nishio@ee.tokushima-u.ac.jp

Keio University
3-14-1 Hiyoshi, Kohoku,Yokohama, 223 JAPAN
Tel: (+81) 45-563-1141 Ext. 3319
Email: kana@mori.elec.keio.ac .jp

ABSTRACT
There have been many investigations of the mutualsynchronization of oscillators. In this study we consider
N oscillators which have the same natural frequency, mutually coupled by one negative resistor. In this system,
according to the negative range of the coupling negative
resistor, we can observe various interesting synchronization phenomena. Because this system tends to minimize
the power consumption in the coupling negative resistor. Especially when i - v characteristics of the coupling
negative resistor are not third-power, but linear and negative, this system tends to be stable by producing thepower in the coupling one. The phase states are changed
accordeing to the negative range of the coupling negativeresistor.

Cnonlinear resistor(4INTRODUCTION
There have been many investigations of the mutual synchronization of oscillators( [I]-[7] and therein).
Kimura and Mano have confirmed that two oscillators
coupled by one positive resistor are synchronized at opposite phase, and that the system tends to minimize thepower consumption in the coupling positive resistor[l].
Endo et al have analyzed a large number of coupled vander Pol oscillators [2]-[4], and we have confirmed thatN-phase oscillation ( N = 213) can be stably excited for the case that the nonlinearity of each oscillator is strong [5][6]. In such systems various interesting synchronization phenomena are observed. Because
the system tends to minimize the power consumption in
the coupling positive resistor, i.e., the system tends tominimize the current through the coupling one, and thet ~ ) . For the case withsum of output voltage (21third-power nonlinear characteristics, all oscillators exhibit the oscillation. And N-phase oscillation can bestably excited, because of taking any phase differencesamong 4 = 2k?r/N(lc = 1 , 2 , . . . , N - 1). For the casethat N is not a prime number, N-phase oscillation seems
to be difficult t o observe and another types of synchronization are observed.
In this study, we investigate synchronization phenom---R
4.7kR

(b)

p(c)

Figure 1: Circuit Model. (a) Oscillators coupled by one negativeresistor. (b) Realization of nonlinear resistor. (c) Realization of
the coupling negative resistor.

ena observed from N oscillators with the same natural frequency, mutually coupled by one negative resistor.
This system tends to minimize the power consumption in
the coupling negative resistor, just as the positive resistor. Then various interesting synchronization phenomena
is observed. When i - v characteristics of the coupling
negative resistor are not third-power, but linear and negative, this system tends to be stable by producing thepower in the coupling one. i.e. , the system tends to bemaximum the current through the coupling one, and thesum of output voltage (21 ZN). As a result, when two,-Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:48:57 EDT from IEEE Xplore. Restrictions apply.

206

three or more oscillators are coupled, each oscillator ofthis system are synchronized at the in-phase, each other.
On the other hand, when i - v characteristics of the coupling negative resistor are third-power, this system tends
to be stable by producing the power in the negative rangeof the coupling one, and by consuming the power in thepositive range. As a result, according to the negative
range of the coupling one, the phase states is changed.
Especially when the negative range is very small, we canobserve N-phase oscillation.
We hope that our results can be applied to the designof some kinds of phase control systems in near future.

(a) N = 2(b) N = 2

CIRCUIT MODEL
The circuit model is shown in Figure 1. The circuit
equation is described as follows,

dihLdt(IC=V k - vr= 1,2,.’.,N)

where i, (vk) means v - i characteristics of nonlinear resistor, and v, means i - v characteristics of the coupling
negative resistor. In our system, these are approximatedby the following function respectively.
ir ( v h ) =-91Vk+

g3V:(91,93=(a) N = 3(b) N = 3(2)3N
Vr> 0)

L- T l C i j + T 3( c i j )

j=1

j=1

(TllT3>O)(3)

By changing the variables,t =m r ,(b) N = 4(1) is normalized as(b) N = 5N

3

Figure 2: (a) Numerical results for the case that N = 2 , 3 . (CI = 0.1, 0 = 0.1,y = 0,Awl = 0,A w = 0.001, A w =~ 0.002 ) (b)Experimental results for the case that N = 2
5. ( L = l h H ,N

C = 0.068pF,R = 4.70, horizontal scale: 50 psec/div., verticalscale: 1.0 V/div.

)

j=1( k = l , 2 , . . - , N)
where CY is the coupling factor and ,f?is the largeness ofnonlinearity. When ,B is small, the oscillation of each oscillator can be regarded as almost purely sinusoidal. And
7 is the negative range of the coupling negative resistor.
When 7 is small, the negative range is large. Especiallywhen 7 = 0, i - v characteristics of the coupling negative
resistor are not third-power, but linear and negative.

THEORETICAL ANALYSISThe synchronization phenomena in this system can beanalyzed theoretically by using averaging method. In thissystem, because nonlinear characteristics of all oscillators
is approximated by third-power, all oscillators exhibit theoscillation. At first, we consider the case that y = 0, N =2. When p is small, we can get the following equations

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207

about the amplitude ak and the phase e k .

N(7)( k = 1,2,...,N)In this study, the amplitudes has less effect for stabilityof steady states, so we only consider the equations on thephases. Then we can get the steady states of the phasedifference 00.

0o=2n?r(n=O,+l,f2,...)(8)

So we can see that x1 and x2 is synchronized at the inphase, each other for y = 0, N = 2. And we consider thecase that 7 = 0, N = 3 and y is very small, N = 2. Then
it is stable when the phase difference of each oscillator
is 0, just as 7 = 0, N = 2. So we can take in-phaseoscillations respectively.

990 992 994 996 998 1000z(b) N = 2(a) N = 22GO-22;io-22

9 0-2
990 992 994 996 998 1000

NUMERICAL CALCULATIONS ANDCIRCUIT EXPERIMENTS
In this section, we confirm the synchronization phenomena by numerical calculations and circuit experiments. For numerical calculations by using the RungeKutta-Gill method, in order to consider the difference
between the natural frequencies of real oscillators, (5) isrewritten as follows,

hJk= (k - 1) X

(k: = 1 , 2 , . ' . , N )On the other hand for circuit experiments, in order to
use the coupling negative resistor in linear and negative
range, we connect two diodes in parallel as shown in Figure l(b). Then each of oscillators exhibit the oscillation with the small amplitude, and the frequency of each
oscillator is almost 6.093kHz. The circuit as shown in
Figure l(c) is realized the coupling negative resistor. If
we change higher resistor than IkQ resistor, the negative
range of the coupling one is smaller.
Figure 2 shows the numerical results and the corresponding results of circuit experiments in the case thaty = 0, i.e. , i - v characteristics of the coupling negative resistor are not third-power, but linear and negative. From these results, we confirmed that in-phasesteady states are stably excited for all numbers of N .

7(a) N = 3(b) N = 3

Figure 3: (a) Numerical results for the caee that N = 2,3. (cy = 0.1,p = 0.1, -y = 0.001, Awl = 0,AW = 0.001, Aw3 = 0.002) (b)Experimental results for the case that N = 2,3.( L = lOmH,
C = 0.068pF, R = 42i-2, horizontal scale: 50 peec/div., verticalscale: 1.0 V/div. )Figure 3-5 shows the case that i - v characteristics ofthe coupling negative resistor are third-power. In Figure
3, because the negative range of the coupling negative resistor is much larger than the positive range, two or threesignals seem to be almost synchronized at the in-phase.
In Figure 4, because this system tends t o be stable byproducing the power in the negative range of the coupling one, and by consuming the power in the positive
range, two or three signals are synchronized at the different phase. In Figure 5, because the negative range ofthe coupling negative resistor is much smaller than thepositive range, two or three signals seem to be almost
synchronized at the N-phase. That is just like synchronization phenomena when N oscillators coupled by one
positive resistor.
From both of numerical calculations and circuit experiments we confirmed various interesting synchronization
phenomena according to the negative range of the coupling negative resistor.

CONCLUSION
In this study we have investigated N oscillators whichhave the same natural frequency, mutually coupled byAuthorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:48:57 EDT from IEEE Xplore. Restrictions apply.

208990 992 994 996 998 1000T(b) N = 2(a) N = 2(a) N = 2(b) N = 2(a) N = 3(b) N = 3900 920 940 950 980 10007(a) N = 3(h) N = 3Figure 4: (a) Numerical results for the case that N = 2 , 3 .
= 0.002 )a = 0.1,/3 = 0.1, = 0.1,Awl = 0, AWZ= 0.001, A w ~(b) Experimental results for the ca8e that N = 2 , 3 . ( L = 10mH,
C = 0.068pF, R = 203, horizontal scale: 50 psec/div., verticalscale: 1.0 V/div. )one negative resistor. In this system, according to thenegative range of the coupling negative resistor, we canobserve various interesting synchronization phenomena.
Because this system tends to minimize the power consumption in the coupling negative resistor. Especiallywhen i - U characteristics of the coupling negative resistor are not third-power, but linear and negative, thissystem tends to be stable by producing the power in thecoupling one. Because of the phase states is changed according to the negative range of the coupling negative
resistor, we hope that our results can he applied to thedesign of some kinds of phase control systems.
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H. Kimura and K. Mano, ”Some Properties of Mutually Synchronized Oscillators Coupled by Resistances,”
Tmns. ZEZCE, Vol. 48, No. 10, pp. 1647-1656 ( inJapanese )Figure 5: (a) Numerical results for the case that N = 2,3. (a = 0.1,/3 = 0.1, = 20, Awl = 0, AWZ= 0.001, AWS= 0.002 )(b) Experimental results for the case that N = 2,3. ( L = l O m H ,C = 0.068pF, R = 200, horizontal scale: 50 psec/div., verticalscale: 10 V/div. )Tmns. Circuits Syst., Vol CAS-23, No. 9. pp. 517-530,Sep. 1976.
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