RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Investigation of Simulation Frequencies and Theoretical Frequencies of Synchronization Statesby Using Electric Power
Ryoji Fukumasa1 , Masayuki Yamauchi1 and Yoshifumi Nishio2
1

Hiroshima Institute of Technology
2-1-1, Miyake, Saeki-ku, Hiroshima, 731–5193, Japan
Phone/FAX:+81-82-921-4248
E-mail: a112060@cc.it-hiroshima.ac.jp

Abstract
In this study, theoretical synchronization frequencies are calculated by using electric powers on coupled oscillator system.
This method is easily applied for complex and large systems
when stable synchronization states are clarified. The calculated theoretical frequencies are compared from frequencies
which are obtained by using simulations in changing a coupling parameter and a nonlinearity. We carry out that the theoretical frequencies can be applied for analyzing mechanismsof a phase-inversion waves or not.
1. Introduction
In this natural world, all kinds of oscillators are existing,their are moving mostly in cooperation. For example, a lotof fireflies which simultaneously flash are observed in southeastern Asia. In the sea, a lot of sardines simultaneously swim
and make a motion like a big fish. Especially, synchronization phenomena can be easily observed on the electric circuit
like coupled oscillators. Therefore, on the electric circuit,many researches of synchronization phenomena have been
carried out up to now. There are a ladder system, a ring system and 2D lattice system which are easy shape of coupledoscillators. On the ring system, in-phase synchronizations,anti-phase synchronizations and in-and-anti-phase synchronizations can be observed when the number of oscillators arean even number. The in-and-anti phase synchronization isa synchronization phenomenon which the in-phase synchronizations and the anti-phase synchronizations are alternatelyexisting.
Our previous study, the synchronization phenomena on theladder system, the ring system and the 2D lattice system,
which are constructed by using van der Pol oscillators andcoupling inductors, were investigated. We discovered a special wave motion which is continuously existing. We call thewave motion phase-inversion waves[1]. The phase-inversion
wave changes phase states between adjacent oscillators froman in-phase synchronization to an anti-phase synchronization
or from the anti-synchronization to the in-phase synchronization and propagates. The phase-inversion waves have some

2Tokushima University
2-1, Minamijyosanjima-cho, Tokushima, 770–8506, Japan
E-mail: nishio@ee.tokushima-u.ac.jp

characteristics. Frequencies of the in-phase synchronization,
the anti-phase synchronization and the in-and-anti-phase synchronization are needed for analyzing mechanisms of thecharacteristics. There are a method of using averaging as oneof method which the frequencies of the synchronization states
are obtained[2]. However, the averaging method is hard touse for a large system, complex shape system and so on.
In this study, a method, which finds a theoretical frequencyof each synchronization state, is developed by using reactive
power of an instantaneous electric power on the ring system.
It is already known that the ring system has the in-phase synchronization, the anti-phase synchronization and in-and-antiphase synchronization. The theoretical values of frequencies
are compared from the simulation results in changing circuit parameters. Finally, we check that the theoretical values
can be used for analyzing mechanisms of the phase-inversion
waves or can not.
2. Circuit Model
Our circuit model is shown in Fig. 1. Eight van der Pol
oscillators are coupled by inductor LC as a ring. An inductor
and a capacitor of each van der Pol oscillator are assumed as“L” and “C” respectively. Characteristic of nonlinear negative resistor of k-th oscillator is shown as f (vk ) in Eq. (1).
f (vk ) = −g1 vk + g3 vk3(1 ≤ k ≤ 8)

(1)

Circuit equations of this circuit are normalized by using Eq.
(2). The normalized equations are shown in Eq. (3).

- 447 -√
i=Cg1
xk , v =3Lg3L
α=, ε = g1
Lc√√√g1yk , t = LCτ,
3g3

Lg2, δ= 1 .
C
3g3

(2)

In this system, we can consider that a power effect is bestwhen a reactive power, which can be assumed as including instantaneous electric powers of L, C and LC , is zero. In otherwords, we can guess that this system is a steady state when
the reactive power is zero. In this study, angular frequenciesof the in-phase synchronizations, the anti-phase synchronizations and the in-and-anti-phase synchronizations are obtained
when the reactive powers are zero.
Prall = 0(9)Each phase θk is set as follows.
<Phase angles of the in-phase synchronization>(θk = 0)(10)<Phase angles of the in-and-anti-phase synchronization>()
θ1 = θ2 = θ5 = θ6 = 0(11)θ3 = θ4 = θ7 = θ8 = π

Figure 1: Circuit model.
dxk= yk ,dτdyk
1= −xk + α(xa − 2xk + xb ) + ε(yk − yk3 ).
dτ3(3)(If k = 1, a = N and b = 2.
If k = N, a = N − 1 and b = 1.
If 2 ≤ k ≤ N − 1, a = k − 1 and b = k + 1.)
Where α expresses a coupling parameter and ε shows nonlinearity of each oscillator. An instantaneous electric power of
each oscillators and an instantaneous electric power of eachinductor between adjacent oscillators are calculated by which
each oscillation wave shape is assumed as a sinusoidal wave.
Each current xk (1 ≤ k ≤ 8) is assumed as Eq. (4)xk = Xsin(ωτ + θk )yk = ωXsin(ωτ + θk )(4)<Instantaneous electric power of the indctor in eachoscillator>
δ(5)PLk = xk ykε<Instantaneous electric power of the capacitor in eachoscillator>
δ dyk
(6)
PCk = ykε dτ<Instantaneous electric power of the coupling indoctor>αδPLc(k+1,k) =(yk+1 − yk )(xk+1 − xk )ε<Phase angles of the anti-phase synchronization>()
θ1 = θ3 = θ5 = θ7 = 0
θ2 = θ4 = θ6 = θ8 = π(12)These angle frequencies are applied for Eq. (9), and weachieve Eqs. (13)–(15).
<Angle freqency of the in-phase synchronization>
ωin = 1(13)<Angle freqency of the in-and-anti-phase synchronization>√ωin−and−anti = 1 + 2α(14)<Angle freqency of the anti-phase synchronization>√ωanti = 1 + 4α(15)Each frequency of each phase state is shown in Table 1. An
arbitrary value of parameters α is applied for each equation
and a frequency of each phase state are obtained.

(7)

Table 1: Frequency of each synchronization states.
in-phase
in-and-anti-phaseanti-phase
f１2π

1 + 2α2π

1 + 4α2π

3. Comparison between theoretical frequencies and simA normalized equation of a total instantaneous reactive power
ulation frequencies
by using Eq. (2) are assumed as Eq. (8).
In this circuit, each oscillation frequency is calculated)
8 (whenthe synchronization state is the in-phase synchroniza∑δδ dykPrall =xk yk + yktion, the in-and-anti-phase synchronization or the anti-phase
εε dτ
k=1synchronization. Equations (13)-(15) depend on only cou7
plingparameter α. However, the frequencies of using numer∑αδ(8)+(yk+1 − yk )(xk+1 − xk )ical simulation depend on the coupling parameter α and theεk=1nonlinearity ε. The frequencies of the simulations are invesαδ
tigated while the coupling parameter α and the nonlinearity ε+ (y1 − y8 )(x1 − x8 )are changed. Calculation conditions are set as follows.
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5.0

6.0

4.0

α=0.1
α=0.3
α=0.5
α=0.7

4.0

Error Er [%]Error Er [%]

5.0

3.0
2.0

3.0
α=0.1 α=0.3

2.0
1.0

1.00.0

α=0.5

0.0

0.0

0.2

0.4

0.60.81.0

0.0

0.2

0.4

0.6

α=0.7

0.8

1.0

Nonlinearity εNonlinearity εFigure 2: Errors between fth and fsim of the in-phase syn- Figure 4: Errors between fth and fsim of the anti-phase synchronization.
chronization.
Table 2: Frequencies of the in-phase synchronization
5.0(Er<1%).
α=0.1

α=0.3

α=0.5

α=0.7

ε=0.400.990140.990160.990160.99017

theory1.00000

1.00000

1.00000

1.00000

error[%]

0.98%

0.98%

0.98%

0.98%

Error Er [%]

4.0
3.0
α=0.1

α=0.3

2.0

Table 3: Frequencies of the in-and-anti-phase synchronization (Er<1%).

1.0
α=0.5

0.0

0.2

0.4

0.6

α=0.1

α=0.3

α=0.5

α=0.7

ε=0.401.08641

1.25705

1.40715

1.54272

theory1.095441.26491

1.414211.54919

error[%]

0.82%

0.62%

0.49%

0.41%

α=0.7

0.0
0.8

1.0

Nonlinearity εFigure 3: Errors between fth and fsim of the in-and-antiphase synchronization.

Table 4: Frequancies of the anti-phase synchronization(Er<1%).

1. The coupling parameter α is changed from 0.00 to 1.00
every 0.05.
2. The nonlinearity ε is changed from 0.1 to 0.7 every 0.2.
The errors Er between the theoretical frequencies and thesimulation frequencies are obtained by using Eq. (16). The
frequency by using simulation is called fsim . The theoreticalfrequency is named fth .
fsim − fth
Er[%] =× 100(16)fth
When the synchronization state of the ring is the in-phase synchronization, relations between the errors Er and the parameters are shown in Fig. 2. The errors Er are shown in Fig. 3
when the synchronization state is the in-and-anti-phase synchronization. The errors Er are shown in Fig, 4 when thesynchronization state is the anti-phase synchronization.
Values of fth and fsim of which the errors Er are under 1%
are shown in Table 2 when the synchronization state of thesystem is the in-phase synchronization. When the synchronization state is the in-and-anti-phase synchronization, values

α=0.1

α=0.3

α=0.5

α=0.7

ε=0.45

1.172631.47473

1.72472

1.94281

theory1.183211.48323

1.73205

1.94935

error[%]

0.89%

0.71%

0.42%

0.33%of fth and fsim are shown in Table 3. When the synchronization state is the anti-phase synchronization, values of fth andfsim are shown in Table 4.
4. Instantaneous Freqencies of Phase-Inversion waves
We check that the frequency equations and value can beused or can not be used in analysis of mechanism of phaseinversion waves. An phase-inversion wave, which changes
phase states between adjacent oscillators from the in-phasesynchronization to the anti-phase synchronization, propagates with another phase-inversion wave, which changes thephase states from the anti-phase synchronization to the inphase synchronization.
Observation conditions of the phase-inversion waves are configured as follows.

- 449 -0.190

Instantaneous Frequency

ε0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.185

anti-phase synchronization

0.180
0.175
0.170
0.165
0.160
in-phase synchronization

0.155
600 650 700 750 800 850 900 950 1000

Time

α

Figure 6: Itinerancies of the instantaneous freqencies by two
Figure 5: Domains which the phase-inversion waves are ob- phase-inversion waves.
served.
1. α is changed from 0.00 to 0.80 every 0.05.

A domains, which the phase-inversion waves are observed,
are shown in Fig. 5. The phase-inversion waves can beobserved in the colored domain. We investigate the phaseinversion waves when α and ε are set 0.1 and 0.2 respectively(see X-mark in Fig. 5). Instantaneous frequencies
and phase differences of when the phase-inversion waves
are propagating are shown in Figs. 6 and 7. The phaseinversion wave, which change from the anti-phase synchronization to the in-phase synchronization propagate immediately after the phase-inversion wave, which change from thein-phase synchronization to the anti-phase synchronization,propagates(see around 700τ in Fig. 7). Therefore, the instantaneous frequencies start to decrease before arriving at
the anti-phase synchronization frequency(see around 700τ inFigs. 6 and 7). The instantaneous frequencies arrive at thein-phase synchronization frequency and become to a stablestate(see around 770τ in Figs. 6 and 7). In this Fig. 6, we
can think that our theoretical value can be used for analysing
mechanism of the phase-inversion wave because the instantaneous frequencies arrive at near the in-phase synchronization
frequency and does not arrive at the anti-phase synchronization frequency.

5. Conclusions

Phase Difference[degree]

02. ε is changed from 0.00 to 0.80 every 0.05.

-20-40-60-80-100-120-140600 650 700 750 800 850 900 950 1000

TimeFigure 7: Itinerancies of the phase differences by two phaseinversion waves.

Acknowledgment
This work was supported by JSPS KAKENHI Grant Number
25730152.
References[1] Masayuki Yamauchi, Masahiro Wada, Yoshifumi Nishio
and Akio Ushida, “Wave Propagation Phenomena ofPhase States in Oscillators Coupled by Inductors as aLadder,” IEICE Trans. Fundamentals, vol.E82-A, no.11,
pp.2592-2598, Nov. 1999.

In this study, we developed the method which frequencies
are obtained by using reactive powers. The errors between
theoretical frequencies and simulation frequencies were in- [2] Tetsuro Endo and Sinsaku Mori, “Analysis of a Ring of avestigated in changing the coupling parameters and the nonLarge Number of Mutually coupled van der Pol Oscillalinearities. Furthermore, we considered whether the theorettors,” IEEE Trans. on Circuits And Systems, vol. cas-25,
ical frequencies could be used for analyzing the mechanisms
no. 1, pp.7-18, Jan. 1978.
of the phase-inversion waves.

- 450 -