CN107145661A - A kind of circuit design method of real number index power memristor model - Google Patents

Abstract

A circuit design method of a real exponential power memristor model, based on a simple chaotic circuit that only includes a linear passive conductance, a linear passive capacitor and a nonlinear memristor, an exponential power memristive function is constructed Polynomial, the power of the exponent is a continuously variable positive real number. Numerical simulation is carried out on the real number exponent power memristor model system of the present invention, and the existence of the classical chaotic attractor of the system is verified. The corresponding circuit experiment simulation results show that the electronic device designed by the present invention satisfies the essential characteristics of the memristor, and the real number exponent power memristive function has more extensive and universal application value.

Description

A Circuit Design Method of Real Exponential Power Memristor Model

technical field

The invention belongs to the theoretical field of memristor circuits in nonlinear circuits and systems, and relates to the design and realization of the simplest chaotic system and memristor circuits.

Background technique

In 1971, Chinese-American scientist Leon O.Chua predicted based on the theory of electronics that in addition to resistors, capacitors, and inductance components, there is a fourth basic component of the circuit, namely the memristor. A memristor is a two-port device that connects the magnetic flux to the charge nonlinearly. In 2008, based on Chua's memristor model, the team of Stanley Williams of Hewlett-Packard Laboratory in the United States successfully developed a solid-state memristor using a double-layer titanium dioxide film, making Chua's theory physically realized. Since then, scholars at home and abroad have explored the basic properties, physical models, applications and fabrication of memristors from the perspective of mathematics and physics. Adhikari, Biolek and others proposed three essential characteristics of memristors, namely (i) when driven by a bipolar periodic signal, the device is a tight hysteresis loop tightened at the origin on the v-i plane, and the response is Periodic; (ii) starting from the critical frequency, the hysteresis sidelobe area decreases monotonously with the increase of excitation frequency; (iii) when the frequency approaches infinity, the tight hysteresis loop shrinks to a single-valued function.

At present, scientists at home and abroad analyze the dynamic characteristics and essential characteristics of memristors by looking for the equivalent circuit of the ideal memristor model, but there is little analysis of the associative memory ability of the general memristor model. In 2010, Muthuswamy and Chua realized the simplest chaotic circuit by using a linear passive inductor, a linear passive capacitor and a nonlinear memristor, that is, three basic circuit components, which we call the simplest chaotic system. On this basis, in 2014, Lin Teng et al. replaced the memristor function with a quartic polynomial function, which increased the complexity of the chaotic attractor of the system, and extended the integer order system to the fractional order system. For the general memristor model research, all focus on the case where the memristor function polynomial is a power of a specific integer exponent, but the research on variable real exponent power is seldom involved.

Contents of the invention

The purpose of the present invention is to propose a memristive function polynomial as a variable real exponent power model, construct the real exponent power memristive circuit, and analyze its feasibility and practicability.

The present invention designs a new memristor model on the basis of the simplest chaotic system. When the exponent power of the memristive function polynomial in the model is a variable positive integer, the simplest chaotic system can present chaotic behavior; the polynomial exponent The power is extended to positive real numbers, and the system can still present chaotic phenomena by adjusting the linear parameters. At the same time, the invention designs the circuit schematic diagram of the general memristor model, and verifies the tight hysteresis loop characteristic of the memristor.

The present invention is achieved through the following technical solutions.

The circuit design method of the real number exponent power memristive model described in the present invention comprises the following steps:

Step S01: Based on the simplest chaotic system, construct a general memristor model in which the polynomial exponent power of the memristive function is a variable parameter;

Step S02: Select the power of the polynomial exponent of the memristive function in step S01 as a positive integer to verify the chaotic characteristics of the simplest chaotic system;

Step S03: Design a circuit schematic diagram of a positive integer exponential power memristive model based on step S02, and verify the existence of the three essential features of the memristive element;

Step S04: Extend the positive integer in step S02 to a positive real number, and numerically calculate the chaotic characteristics of the simplest chaotic system based on the real number exponent power memristive model;

Step S05: Based on step S04, design a circuit schematic diagram of a general memristor model when the polynomial exponent power of the memristive function is a positive real number, and verify the three essential characteristics of the memristive element.

Further, the circuit design method of the real exponential power memristive model described in the present invention, its specific steps are as follows:

Step 1: Design the simplest chaotic system with exponential power.

The circuit diagram of the simplest chaotic system is shown in Figure 1, which includes three basic circuit elements, namely: a linear passive inductor, a linear passive capacitor and a nonlinear memristor. Its dynamic behavior is described as follows:

Among them, C is the capacitance value, L is the inductance value, R(z) is the resistance value of the memristive element, z is the state variable of the memristive element, i _C , i _L , i _M are respectively The current of the resistor, v _C , v _M are the voltages across the capacitor and the memristive element respectively. The present invention selects the memristive element model as:

Make x (t)=v _C (t), y (t)=i _L (t), because i _M (t)=-i _L (t) simultaneously, then the kinetic equation of the simplest chaotic system among the present invention correspondingly becomes:

In the formula, b ₁ , b ₂ , b ₃ , c ₁ , c ₂ , and c ₃ are all system parameters, and α is a variable exponent power parameter.

Step 2: Numerical simulation of the simplest chaotic system in which the polynomial of the memristive function is a power of an integer exponent.

In the present invention, the polynomial exponent α of the memristor function in the general memristor model is selected as a variable positive integer, the capacitance and inductance values are fixed, and the initial conditions of the system are set. By adjusting the linear parameters of the system, it is observed whether the system can generate chaotic attractors; At the same time, given the input signal, observe the volt-ampere characteristic curve of the integer exponential power memristor model, and verify whether it is an "8"-shaped tight hysteresis loop passing through the origin.

Using the definition method to calculate the Lyapunov exponent under the specific parameters of the system, it is theoretically proved whether the chaotic attractor of the system exists.

Step 3: Integer Exponential Power Memristor Circuit Schematic Design.

For the general memristor model of the integer exponent power in step 2, use the Multisim circuit simulation system to design the memristor circuit schematic diagram of the integer exponent power, and compare with the numerical calculation results in step 2 to verify the three essential characteristics of the memristor element existence.

Step 4: Numerical simulation of the simplest chaotic system in which the polynomial of the memristive function is a power of a real number exponent.

In order to make the memristor model in the present invention more general, the polynomial exponent power α of the memristive function is expanded from a positive integer to a positive real number, and the capacitance, inductance and initial conditions of the system remain unchanged. By adjusting the linear parameters of the system, it is observed that Whether the system can generate chaotic attractors; at the same time, given the input signal, observe the volt-ampere characteristic curve of the memristor model at this time, and verify whether it is an "8"-shaped tight hysteresis loop passing through the origin.

Also adopt the definition method to calculate the Lyapunov exponent under the specific parameters of the system, and theoretically prove whether the chaotic attractor of the system exists.

Step 5: Real number exponent power general memristor circuit schematic design.

For the general memristor model of real number exponent power in step 4, on the basis of the integer exponent power memristor circuit in step 3, a power operation module is added, wherein the power operation circuit consists of an integrated logarithmic operation circuit and an integrated exponential operation circuit combined. Arbitrary real exponential powers can be achieved by adjusting the values of the resistor-related components.

The present invention is characterized in that: the nonlinear memristor model in the simplest chaotic system is a general memristor model, and when the polynomial exponent power of the memristive function is a variable positive integer and a positive real number respectively, the system will generate classical chaotic attraction son. Simultaneously, the general memristor circuit schematic diagrams of integer exponent power and real number exponent power are designed, and the existence of three essential features of the memristor model of the present invention is verified.

Description of drawings

FIG. 1 is a circuit diagram of the simplest chaotic system including a memristive element according to the present invention.

Fig. 2 is the trajectory of each state variable of the real exponential power memristor model and the volt-ampere characteristic curve of the general memristor model when α=1 in the present invention. (a) is the xy variable trajectory, (b) is the xz variable trajectory, (c) is the yz variable trajectory, and (d) is the i _M -v _M variable trajectory.

Fig. 3 is the trajectory of each state variable of the real exponential power memristor model and the volt-ampere characteristic curve of the general memristor model when α=2 in the present invention. (a) is the xy variable trajectory, (b) is the xz variable trajectory, (c) is the yz variable trajectory, and (d) is the i _M -v _M variable trajectory.

Fig. 4 is the trajectory of each state variable of the real exponential power memristor model and the volt-ampere characteristic curve of the general memristor model when α=3 in the present invention. (a) is the xy variable trajectory, (b) is the xz variable trajectory, (c) is the yz variable trajectory, and (d) is the i _M -v _M variable trajectory.

FIG. 5 is a schematic diagram of a general memristor circuit of an integer exponential power of the present invention when α=1.

FIG. 6 is the volt-ampere characteristic curve of the general memristor when the input signal frequency f=1.7Hz when α=1 in the present invention.

Fig. 7 is the volt-ampere characteristic curve of the general memristor when the input signal frequency f=6.7Hz when α=1 in the present invention.

FIG. 8 is the volt-ampere characteristic curve of the general memristor when the input signal frequency f=45Hz when α=1 in the present invention.

Fig. 9 is the trajectory of each state variable of the simplest chaotic system (4) and the volt-ampere characteristic curve of the general memristor model when α=1.6 in the present invention. (a) is the xy variable trajectory, (b) is the xz variable trajectory, (c) is the yz variable trajectory, and (d) is the i _M -v _M variable trajectory.

Fig. 10, when α=3.3 of the present invention, the trajectory of each state variable of the simplest chaotic system (4) and the volt-ampere characteristic curve of the general memristor model. (a) is the xy variable trajectory, (b) is the xz variable trajectory, (c) is the yz variable trajectory, and (d) is the i _M -v _M variable trajectory.

Fig. 11 When α=3.8 in the present invention, the trajectory of each state variable of the simplest chaotic system (4) and the volt-ampere characteristic curve of the general memristor model. (a) is the xy variable trajectory, (b) is the xz variable trajectory, (c) is the yz variable trajectory, and (d) is the i _M -v _M variable trajectory.

Fig. 12 is a schematic diagram of a general memristor circuit of a power of real number exponent in the present invention when α=1.6.

Fig. 13 is a power analog operation circuit of the present invention.

Fig. 14 is the volt-ampere characteristic curve of the general memristor when the input signal frequency f=1.7Hz when α=1.6 in the present invention.

Fig. 15 is the volt-ampere characteristic curve of the general memristor when the input signal frequency f=6.7Hz when α=1.6 in the present invention.

Fig. 16 is the volt-ampere characteristic curve of the general memristor when the input signal frequency f=45Hz when α=1.6 in the present invention.

detailed description

The present invention will be described in further detail below in conjunction with the accompanying drawings.

Example 1. Numerical simulation of the simplest chaotic system when the polynomial power of the memristive function is a variable positive integer.

(1) Design of the simplest chaotic system with exponential power.

For the simplest chaotic system, the capacitance and inductance values are respectively selected as C=1, L=1, and the initial conditions are set as x(0)=0.1, y(0)=0.1, z(0)=-0.01, then the system ( 3) Correspondingly becomes:

(2): The power α of the polynomial exponent of the memristive function takes a positive integer.

When the memristive function polynomial power α=1, select the linear parameters b ₁ =-0.5, b ₂ =0.5, b ₃ =0.5, c ₁ =-1, c ₂ =-1.5, c ₃ =-3, then The phase diagram trajectories of the state variables of the system (4) are shown in Fig. 2(a), (b), and (c), respectively, all of which are classical chaotic attractors, and Fig. (d) depicts the general memristor volt-ampere The characteristic curve is a tight hysteresis loop in the shape of a backslash "8" passing through the origin, and the input signal is a sine wave with a frequency f=1.7Hz.

Using the definition method to calculate the Lyapunov exponents of the system are: LE ₁ = 0.3793, LE ₂ = -0.3638, LE ₃ = -1.4018. Since one of the LE values is greater than 0, and the sum of the three is less than 0, it is theoretically proved that the system (4 ) In Example 1 there is a classical chaotic attractor.

The present application has also completed when the memristive function polynomial exponent power α=2 and α=3, also select the linear parameters b ₁ =-0.5, b ₂ =0.5, b ₃ =0.5, c ₁ =-1, c ₂ = -1.5,c ₃ =-3, then the phase diagram trajectories of the state variables of the system (4) are shown in (a), (b) and (c) of Figure 3 and Figure 4 respectively, and they are all classical chaotic attraction Figure 3 and (d) of Figure 4 respectively depict the volt-ampere characteristic curve of the general memristor, which is a tight hysteresis loop in the shape of a backslash "8" passing through the origin, wherein the input signal is selected as frequency f= 1.7Hz sine wave.

(3) Integer exponent power general memristor circuit schematic design.

When α=1, the general memristor model of integer exponent power is shown in Figure 5, where U _0A , U _1A , and U _2A are operational amplifiers AD712JN, and A ₁ and A ₂ are analog multipliers. Choose R _s ＝10Ω, R _s1 ＝100kΩ, R _s2 ＝1kΩ, and set m＝-1000, then the voltage v ₀ converted from current i _M is expressed as:

Let R _f ＝100kΩ, R _b1 ＝R _b2 ＝R _b3 ＝200kΩ, then the memristive function expression is:

v _M ＝(-0.5+0.5z+0.5z) mi _M (6)

Set parameters C _f =10uF, R _c1 =100kΩ, R _c2 =66.7kΩ, R _c3 =33.3kΩ, then the internal state variable z of the memristor is expressed as:

The current source uses a sine wave with an amplitude of 10mA. In order to study the three essential characteristics of the memristor, experiments were carried out at frequencies equal to 1.7Hz and 6.7Hz. The voltage-to-current ratio of the current probe XCP1 was directly selected as 1V/mA=1000V/A, and the direction was selected as the opposite direction of the current source. , corresponding to m=-1000. At this time, the volt-ampere characteristic curves of the memristor are shown in Figures 6 and 7, all of which are tight hysteresis loops shrinking at the origin, satisfying the essential characteristic (i) of the memristor; and comparing Figures 6 and 7 at the same time It is found that the hysteresis sidelobe area of the memristor decreases monotonously with the increase of the frequency, satisfying the essential characteristic (ii) of the memristor. In order to verify the essential feature (iii) of the memristor, a frequency of 45 Hz (relatively tending to infinity) is selected for experiments. The circuit simulation is shown in Figure 8, which is approximately shrunk into a single-valued function.

Example 2. Numerical simulation of the simplest chaotic system when the polynomial power of the memristive function is a positive real number.

(1) Design of the simplest chaotic system with exponential power.

Referring to the step (1) in the implementation example 1, the design of the simplest chaotic system with exponential power is completed.

(2): The power α of the polynomial exponent of the memristive function is a real number.

When the memristive function polynomial exponent power α=1.6, select linear parameters b ₁ =-0.5, b ₂ =0.5, b ₃ =0.5, c ₁ =-1, c ₂ =-1.6, c ₃ =-3, then The phase diagram trajectories of the state variables of the system (4) are shown in Figure 9(a), (b), and (c) respectively, all of which are classical chaotic attractors, and Figure (d) depicts the general memristor volt-ampere The characteristic curve is a tight hysteresis loop in the shape of a backslash "8" passing through the origin, and the input signal is a sine wave with a frequency f=1.7Hz.

Using the definition method to calculate the Lyapunov exponents of the system are: LE ₁ = 0.3548, LE ₂ = -0.3426, LE ₃ = -1.4677. Since one of the LE values is greater than 0, and the sum of the three is less than 0, it is theoretically proved that the system (4 ) In Example 4 there is a classical chaotic attractor.

The present application has also completed the selection of linear parameters b ₁ =-0.5, b ₂ =0.5, b ₃ =0.5, c ₁ =-1, c ₂ =- 1.6,c ₃ =-3, then the trajectories of the phase diagrams of the state variables of the system (4) are shown in Figure 10 and Figure 11(a), (b) and (c) respectively, they are all classical chaotic attractors, Figure 10 and Figure 11(d) respectively depict the volt-ampere characteristic curve of a general memristor, which is a tight hysteresis loop in the shape of a backslash "8" passing through the origin, where the input signal is selected as a frequency f=1.7Hz sine wave. (3) Circuit design for powering real numbers.

For the general memristor model of the power of the real number exponent in (S4), the schematic diagram of the design circuit is shown in Figure 12, that is, a power operation module (Figure 13) is added to the far right of Figure 5, and it is used as an analog multiplier A ₂ an input terminal. The power operation circuit is composed of the integrated logarithmic operation circuit in the left box and the integrated exponential operation circuit in the right box, and its output voltage u ₀ can be expressed as:

If let I _R1 R ₁ ＝1, I _R2 R ₉ ＝1, but By adjusting the resistors R ₄ , R ₅ , R ₆ , and R ₇ , any real exponent power can be realized.

(4) Real number exponent power general memristor circuit schematic design.

In the general memristor circuit schematic diagram of a power of real number, continue to take α=1.6 as an example, and set the parameters of the circuit as follows: R ₁ =R ₂ =R ₃ =R ₈ =R ₉ =100kΩ; R _ref1 =R _ref2 = 1500kΩ; R ₄ =R ₅ =R ₇ =100kΩ, R ₆ =25kΩ; R _c2 =62.5kΩ, other parameter settings are the same as the general memristor circuit parameters of integer exponential power.

The measurement method is also the same as the measurement method in the implementation example 1 step (3), then the volt-ampere characteristic curve of the real number exponential power general memristor is depicted as Fig. 14, 15, 16, and it can be found by observation that it also satisfies the three requirements of the memristor essential features.

Claims (1)

1. A circuit design method of a real number exponent power memristive model, characterized in comprising the following steps: Step S01: Based on the simplest chaotic system, construct a general memristor model in which the polynomial exponent power of the memristive function is a variable parameter; Step S02: Select the power of the polynomial exponent of the memristive function in step S01 as a positive integer to verify the chaotic characteristics of the simplest chaotic system; Step S03: Design a circuit schematic diagram of a positive integer exponential power memristive model based on step S02, and verify the existence of the three essential features of the memristive element; Step S04: Extend the positive integer in step S02 to a positive real number, and numerically calculate the chaotic characteristics of the simplest chaotic system based on the real number exponent power memristive model; Step S05: Based on step S04, design a circuit schematic diagram of a general memristor model when the polynomial exponent power of the memristive function is a positive real number, and verify the three essential characteristics of the memristive element.