平成23年度 電気関係学会四国支部連合大会

21-2

Synchronization Phenomena
in Cross-coupled Two Colpitts Oscillators
Kosuke MATSUMURA, Takahiro NAGAI, Yoko UWATE and Yoshifumi NISHIO(Tokushima University)
11
1. Introduction
ρ y22 −y21ẋ21 = −α1 x21 +

α3α3


Synchronization phenomena are observed everywhere in11γ
natures, such as vibration of a pendulum, the lumines ẋ22 = −α3 ε x22 + 2 (|x21 − γ | − |x21 + γ |)cence phenomena of a firefly, and so on. Also, synchronization with electric circuit is eﬀective in analyzing thenatural phenomena. It is easy to observe the synchronization phenomena of coupled oscillators by various elements
and techniques.
In this study, we observe synchronization phenomena incross-coupled two Colpitts oscillators by using computer
simulations and circuit experiments.

R2+-R1

R2R3C1

i21v21

C2

L1
L2

√√

We show the our circuit model in Fig.1. In this study,
we investigate the case where two Colpitts oscillators arecross-coupled via inductors.

vin+v22vout1αr
We use the following normalizations:√
γ = (R2 /R1 R3 ) L/C1 ,αr = (1/r)

2. Circuit Model

R11+y21 +ρ y12α3α3ẏ21 = α3 (x21 − ε x22 ) ẏ22 = α3 (ε x12 − x21 ) − 1 ρ y22

α3 = (1/R3 )(3)√

L/C1 , α1 = (1/R1 )L/C1 ,L/C1 , ε = C1 /C2 , ρ = L1 /L2 .

3. Synchronization Phenomena
Figure 3 is the circuit experiment result, (horizontal : [10 V/div], vertical : [10 V/div], C1 = 1 nF, C2 = 10 nF,
R1 = 10 kΩ,
R2 = 100 kΩ,
R3 = 1 kΩ,
L1 = 10 mH,
L2 = 5 mH). Figure 4 is the computer simulation result
by the normalized Eq. (2) and (3). Attractor on ẋ11 ẋ21 plane, γ = 31.62, α1 = 0.3162, α3 = 3.162, αr = 316.2,
ε = 0.10, ρ = 2.0.

L2i22vout

i12rr

v21

V

L1
C1

i11

v11

C2

v12v11vin

R30R2
R1--V+

Fig 3: Experiment result. Fig 4: Simulation result.
Fig 1: Circuit model.

Fig 2: OP-Amp model.

The operational amplifier (OP-Amp) currently used forour circuit model is not an ideal linear function, and weassume that the OP-Amp model is taken as the piecewise
linear function as shown in Fig. 2.
Therefore, input-output characteristics of an OP-Ampis expressed with Eq. (1).
vout =R2
R1
R1(|vin −
V | − |vin +V |)
2R1
R2
R2(1)

The normalized circuit equations of our circuit model
are given as follows:11ρ y12 −y11ẋ11 = −α1 x11 −αα3311γ ẋ12 = −α3 ε x12 + 2 (|x11 − γ | − |x11 + γ |)11+y11 −ρ y22
α3α3ẏ=α(x−εx) 1131112
 ẏ12 = α3 (x11 − ε x22 ) − 1 ρ y12αrPowered by TCPDF (www.tcpdf.org)

We show the in-phase synchronization in Fig.2 andFig.3. When we compare two Lissajous figures, phase
diﬀerence of computer simulation result is 0.04◦ , but Lissajous figure of circuit experiment result is diﬀerent. Asthe cause, we assume that two oscillators in our circuit
experiment does not have a perfectly same elements.
Furthermore, we summarize synchronization phenomena in circuit experiment by change of L2 in Table 1. We
observe in-phase state when L2 is less than 50mH, asynchronous state when L2 is more than 50mH.

Tab. 1: Synchronization state by change of L2 .
L2 [mH]
5〜50
more than 50(2)synchronization statein-phaseasynchronous

4. Conclusions
In this study, we have observed synchronization phenomena in the proposed circuit. In the future, we would
like to investigate what kind of phenomena it becomes
about the case of four oscillators are coupled.