Research on Chaotic Synchronization Circuit Based on Analog Inductance

1 Chaos and Chaos Synchronization Principle

Chaos refers to the random-like output produced by a deterministic system. The so-called deterministic circuit means that the parameters and inputs of the circuit are deterministic values, and there are no random factors. The so-called uncertain, random-like output means that the output of a circuit is neither periodic nor quasi-periodic. Neither tends to infinity nor to stationary, but presents an output that never repeats within a region. Generally speaking, chaos synchronization belongs to the category of chaos control. To date, several types of chaos synchronization have been found, one of which is the synchronization scheme proposed by Pceora and Carrolll. In this scheme, there is a drive-to-drive relationship in the circuit, where the drive circuit can be divided into stable and unstable parts, in which the stable part copies a response, and then the response system and the drive system are coupled with the drive signal, so that the corresponding system and the drive system can be synchronized.

With the development of the research on non-linear circuits, there have been many reports on the actual circuits that produce chaos to study the mechanism of chaos generation. Chaotic phenomena exist widely in non-linear circuits. The typical circuit which has been deeply studied is Cai's circuit. Cai's circuit is shown in Fig. 1 (a). Nonlinearity in the circuit is introduced by a piecewise linear negative resistance. The volt-ampere characteristics of the nonlinear resistance are shown in Figure 1 (b).

When the parameters of the circuit satisfy certain conditions, it will produce a self-excited oscillatory attractor which becomes a double-scroll volume. Figure 2 shows a Cai's chaotic synchronization circuit.

2 analog inductor

Three commonly used analog inductor circuits are introduced in this paper: the Rioden inductor circuit, the lossless analog inductor circuit and the low loss analog inductor circuit.

2.1 Rioden inductor circuit

The circuit (Fig. 3) consists of two integrated operational amplifiers, four resistors and one capacitor. Since the op-amp is considered to be an ideal integrated op-amp, the open-loop differential-mode voltage amplification factor Aod < # and the current I+ < < # flowing into both inputs 0, U+ < U_.

The op amps implement the same proportional operation circuit, so they can be obtained:

Therefore, the Rioden circuit can be equivalent to an analog inductance of L=R2C.

2.2 New Lossless Analog Inductor Circuit

The circuit (Fig. 4) consists of an operational amplifier, four resistors and two capacitors. Ui is the input signal and Uo is the output signal. From the characteristics of ideal operational amplifiers, it can be concluded that:

Therefore, the circuit can be equivalent to an analog inductance of L=2R2C.

2.3 Low Loss Analog Inductor Circuit

The circuit (Fig. 5) consists of an operational amplifier, four resistors and one capacitor. Ui is the input signal and Uo is the output signal.

Based on the characteristics of the ideal operational amplifier, the following can be listed:

It can be concluded that this circuit can be equivalent to a series combination of a resistor and an L=R1R2C.

3 Simulation Research

The inductances in Figure 2 are replaced by the three simulated inductances mentioned above. The time domain waveforms, chaotic attractors and output voltage spectrum of the three circuits are observed when the 18mH analogue inductance is used instead of the actual inductance.

Application of 3.1 Rioden Circuit in Chaotic Synchronization Circuit

If R=1k and capacitance C=18nF are chosen for the parameters of the Rioden circuit described earlier, the Rioden circuit can be equivalent to L=R2C=(103)2 × 18 × 10-9=18mH inductance.

The time domain waveform, chaotic attractor and voltage spectrum obtained by applying them to a chaotic synchronous circuit are shown in Figure 6.

3.2 Application of New Lossless Analog Inductor Circuit in Chaotic Synchronization Circuit

If the parameters in the lossless analog inductor circuit described earlier are R=1k, C=9nF, the new analog inductor circuit can be equivalent to an L=2R 2 C=2 × (103)2 × 9 × 10-9=18mH inductance.

The time-domain waveform, chaotic attractor and voltage spectrum obtained by applying them to a chaotic synchronous circuit are shown in Figure 7.

3.3 Low Loss Analog Inductor Circuit Applied to Chaotic Synchronization Circuit

If the parameters in the low-loss analog inductor circuit described above are R4=R2=0.05k, R1=R3=4k, and C=90nF, the low-loss analog inductor circuit can be equivalent to an L=R1R2C=4 × 103 × 0.05 × 103 × 90 × 10-9=18mH inductance.

The time domain waveform, chaotic attractor and voltage spectrum obtained by applying them to a chaotic synchronous circuit are shown in figure 8.

4 Conclusion

By applying the Rioden circuit, the new lossless analog inductance circuit and the low loss analog inductance circuit to the chaotic synchronous circuit for simulation, it is found that the analog inductance can replace the actual inductance without affecting the chaotic characteristics of the chaotic circuit. The chaotic circuit based on the analog inductance has not only the spectrum characteristics of white noise, but also the autocorrelation function is close to that of the chaotic synchro δ The behavior of the function provides the conditions for the development of chaotic oscillators with integrated functions. The simulation results show that the chaotic circuit based on analog inductance not only does not affect the characteristics of the chaotic circuit itself, but also has the characteristics of small size and easy integration. Therefore, he has a very wide application prospects.