2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Investigation of Strange Synchronization Phenomena
in Coupled Wien-Bridge Oscillators
Koichi Matsumoto† , Yoko Uwate† , Yoshifumi Nishio† and Seiichiro Moro‡†Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan‡Dept. of Electrical and Electronics Engineering, University of Fukui
3-9-1 Bunkyo, Fukui 910-8507, Japan
E-mail: †{koichi, uwate, nishio}@ee.tokushima-u.ac.jp, ‡moro@ppc8100.fuee.fukui-u.ac.jp
The Wien-Bridge oscillator is shown in Fig.1. This cirAbstract – The synchronization phenomena are therepresentative phenomena in the nonlinear science and can cuit consists of two capacitors, four resistors and one OP
be often observed in nature world. Coupled oscillatory amp.
The parameters for the circuit experiments are chosen
systems have attracted a great deal of attentions in variousfields. In particular, synchronization in such systems is as follows. C = 15nF, R=10kΩ, Ri = 4.7kΩ, R f = 14.7kΩ.
very important phenomenon and many researches haveRC
been reported. In this study, we investigate the interesting
synchronization phenomena with 143 degree phase difference in detail, as varying for example the nonlinearity of＋
the oscillator. The results would be a first step to makeclear the mechanism of the generation of the 143 degrees−synchronization.
CR
Rf
1. Introduction
Ri
Coupled oscillatory systems have attracted a great dealof attentions in various fields. In particular, synchronization in such systems is very important phenomenon
and many researches have been reported. In our paststudies [1][2], we have reported that a certain class of coupled systems of the Wien-Bridge oscillators synchronizes
with 0 or 120 degrees.
In [3], we have investigated synchronization phenomena
in coupled Wien-Bridge oscillators by both circuit experiments and computer simulations using SPICE. We observed the synchronization state with 143 degrees phasedifference. The phase difference of 143 degrees is very
strange, because in-phase, anti-phase, or N-phase synchronization is typical.
In this study, we investigate the strange but interesting
synchronization phenomenon with 143 degrees phase difference in detail, as varying for example the nonlinearityof the oscillator. The results would be a first step to makeclear the mechanism of the generation of the 143 degrees
synchronization. In Sec.2, some results in our past studies
are introduced. In Sec.3, the circuit model in this study andthe circuit equations are given. In Sec.4, simulation results
are shown. In Sec.5, we conclude our study.

Fig.1 Wien-Bridge oscillator.
Figure 2 shows one of the coupled systems of the WienBridge oscillators reported in our past studies. In thissystem, N Wien-Bridge oscillators are coupled as a ring.
We observed the synchronization state with 143 degrees
phase difference for the ring by both circuit experiments
and SPICE simulations. Typical examples of the results
for the case of N=2 are shown in Figs. 3 and 4. This
strange synchronization coexists with the in-phase synchronization.
v

C − 12+C2 ＋

v11L−＋C
2 v11R−＋＋−− C＋v

R −13

Riv

C − 22+R

Rfr +v- r1rR＋v21L

v21R−＋ C＋2R

v23−

Rf− 2Ri+v- r22. Coupled Wien-Bridge Oscillators

Fig.2 Ring of coupled Wien-Bridge oscillators (N=2).

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of the 143 degrees synchronization. The circuit model isshown in Fig. 5. In this model, two Wien-Bridge oscillators are coupled by one resistor r.

v12R+−v
C − 22+ RC

i11+

v13+ R-

Rf++CC +-(a)i21v v21− R +v
−1123−

Ri

Ri+vr rir-

RfFig.5 Coupled Wien-Bridge oscillators.
The circuit equations are given as follows.

dvk1
dv
1= C k 2 − vk 3
dt
Rdtdvk 2 1
C= g1vk 3 − g 3vk33 − vk 2 − vk 3 (1)dt
Rdv ⎞ 1⎛ dv
C ⎜ 11 + 21 ⎟ = (vk 3 − vk1 )dt ⎠ r⎝ dt
C((b)
Fig.3 Strange synchronization with 143 degrees phasedifference (circuit experiment) . Two phase states coexist.

)(k = 1,2) ．
By changing variables and parameters,

t = RCτ ,xk =g1 − 3
vk 1 , y k =3g 3zk =g1 − 3rvk 3 , α = , ε = g1 − 3 ，
3g 3
R(a)

g1 − 3
vk 2 ,
3g 3(2)the circuit equations are normalized as follows.

x& k = y& k − z k⎛z k3 ⎞y& k = ε ⎜⎜ z k − ⎟⎟ − yk + 2 z k3⎠⎝z k = xk + α (x&1 − x&2 )(b)
Fig.4 Strange synchronization with 143 degrees phasedifference (SPICE simulation) . Two phase states coexist.

(k = 1,2) ．

3. Circuit ModelIn this study, we consider a simple coupled system of
two Wien-Bridge oscillators to investigate the generation(3)

In equation (3), α is the coupling factor and ε is thestrength of nonlinearity.

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We investigate synchronization phenomena whenε ischanged by the computer simulation using the RungeKutta method. The relation between ε and φ is shown inFig. 6. φ is the phase difference between the two oscillators. We set the coupling parameter as α = 0.01.
180i11i21ir

160(a )

140120φv21(b) ε=3.0.

(d )100(e)

8060(b )40(f )

v11(c )

v2120

i11

0012345ε

678910i21ir

Fig. 6 The relation between nonlinearity and phasedifference.

(c) ε=9.0.
This figure shows that only the anti-phase synchronization state (φ=180 ) can be observed for ε < 2.3.
Namely, all of the initial states converge to the anti-phasesynchronization. The voltage and the current waveforms
for this case are shown in Fig. 7(a). Because the nonlinearity of the oscillators is weak, the voltage waveforms
are similar to sine waves.
For ε> 2.3, we can observe the coexistence of the inphase synchronization and the other strange synchronization whose phase difference varies with ε. From Fig.
6, we can see that the phase difference of the strange
synchronization state decreases from 180 to around 135 as
ε increases. However, for ε>5.0, the phase difference
increases toward 180 again.
The voltage and the current waveforms for the in-phasesynchronization are shown in Figs. 7(b) and (c). We canobserve that the shapes of the waveforms are different
because of the difference of the nonlinearity.
The voltage and the current waveforms for the strange
synchronization are shown in Figs. 7(d)-(f). We consider
that the shape of the current through the coupling resistor
ir plays an important role to decide the phase difference of
this strange synchronization.

v11v21i11i21ir(d) ε=3.0.

v11v21i11i21ir(e) ε=5.0.

v11v11v21v21i11i11i21i21irir(f) ε=9.0.

(a) ε=1.0.

Fig.7 The voltage and the current waveforms.

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6. Conclusions
5. Effect of Coupling Strength
Next, we investigate how the graph of the phase difference of the strange synchronization changes according to
the change of the coupling strength α . The relation
between ε and φ for different coupling strengths (α =0.005 and 0.001) are shown in Figs. 8 (a) and (b).
We can obtain an interesting change of the graph.
However, at the moment we cannot explain the reason ofthe change.
180160140

φ12010080604020In this study, we investigate the strange synchronization
phenomena with 143 degrees phase difference in detail, asvarying for example the nonlinearity of the oscillator. Theresults would be a first step to make clear of the mechanism of the generation of the 143 degrees synchronization.
Reference[1] S. Moro, Y. Nishio and S. Mori, "Synchronization
Phenomena in RC Oscillators Coupled by One
Resistor," IEICE Transactions on Fundamentals, vol.
E78-A, no. 10, pp. 1435-1439, Oct. 1995.
[2] S. Moro and T. Matsumoto, "Various Kinds ofCoupled Networks with Wien-Bridge Oscillators,"
Proc. of NOLTA’00, pp. 547-550, Sep. 2000.
[3] K. Matsumoto, Y. Uwate, Y. Nishio and S. Moro,"Synchronization Phenomena in a Ring of Coupled
Wien-Bridge Oscillators," Proc. of NDES’05, p. O28,
Sep. 2005.

0012345ε67891078910(a) α=0.005.
180160140

φ120100806040200012345ε6(b) α=0.001.
Fig. 8 Effect of Coupling Strength.

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