**4. Behavior of coupled memristor based oscillators with inverting connections**

#### **4.1 Operating principles**

The connection types of the coupled memristor based oscillators (MBO) and the values of the coupling strengths between them impact significantly on the character of their behavior.

The schematic of coupled MBOs with inverting connections and binary output signals is shown in **Figure 10**. The circuit of this oscillator element contains [49] two identical oscillators MBO1 and MBO2. If the direct signal *V*<sup>1</sup> applied to input MBO2 then MBO1 receives an inverted signal (�*V*2Þ from MBO2 output. The circuit contains also output phase detector *F V*ð Þ 1,*V*<sup>2</sup> and input adder. The phase detector performs logical function over the binary outputs MBO1 and MBO2. The input adder provides the receipt of both the control analog signal *VC* and the inverted signal (�*V*2Þ at MBO1 input.

The state of the considered oscillator system can be specified by the variables *R*<sup>1</sup> and *R*<sup>2</sup> and time derivatives *dR*1*=dt* and *dR*2*=dt*. The modules of the rates of change of memristor resistances *R*<sup>1</sup> and *R*<sup>2</sup> are the same for identical MBOs but signs of these rates may differ.

The detailed analysis of behavior of this system using phase plane for variables *R*<sup>1</sup> and *R*<sup>2</sup> is given in [49]. The feature of phase portrait for coupled MBOs with inverting connections is related with the derivatives of variables *R*<sup>1</sup> and *R*<sup>2</sup> that take the values �1. Due to this feature the trajectories of the image point are inclined straight lines with angle of � π/4 relative to the coordinate axes.

The ratio of coupling coefficients with opposite signs *r*<sup>1</sup> ¼ �*k*1*V*2ð Þ*t =I* and *r*<sup>2</sup> ¼ *k*2*V*1ð Þ*t =I* significantly affects character of behavior. The cases of equal values ( rj j <sup>1</sup> ¼ r2) and different values (j j *r*<sup>1</sup> 6¼ *r*2) are discussed in [49].

Introducing the additional notations *r* and *rM* two possible versions can be considered for different values of coupling coefficients: j j *r*<sup>1</sup> ¼ *rm* <*r* ¼ *r*2, j j *r*<sup>1</sup> ¼ *rM* >*r* ¼ *r*<sup>2</sup> . First case with the dominance of direct positive coupling corresponds to antiphase oscillations and second case with dominance of inverting negative coupling corresponds to in-phase oscillations.

The period of antiphase oscillations equals to *T* ¼ 2 ð Þ *RM* � *Rm* � *rm =γI* and the period of in-phase oscillations is *T* ¼ 2 ð Þ *RM* � *Rm* � *r =γI*.

The external control signal impacts on the phase trajectory of the system. Consider then the case with j j *r*<sup>1</sup> ¼ *rm*.

The range of varying MBO1 threshold voltage is shifted due to applying the control signal *VC*. The additional shift in the threshold resistances *rC* ¼ *VC=I* is generated by control signal *VC* . Due to action of the *VC* signal the following active restrictions determine the interval of varying memristor resistances of MBO1 and MBO2 circuits.

$$R\_m + r - r\_C \le R\_1(t) \le R\_M + r - r\_C \text{ at } dR\_2/dt < \mathbf{0} \tag{11}$$

$$R\_m - r\_C \le R\_1(t) \le R\_M - r\_C \text{ at } dR\_2/dt > 0 \tag{12}$$

$$R\_m - r \le R\_2(t) \le R\_M - r \text{ at } dR\_1/dt < 0 \tag{13}$$

$$R\_m \le R\_2(t) \le R\_M \text{ at } dR\_1/dt > 0 \tag{14}$$

**Figure 11** illustrates such shift at the phase portrait of the system with control signal. As follows from formulas Eqs. (11) and (12), the threshold resistances for *R*<sup>1</sup> are decreased (**Figure 11**) but the threshold resistances for *R*<sup>2</sup> remained unchanged.

**Figure 10.** *The system of coupled MBOs with inverting connections.*

*Functional Capabilities of Coupled Memristor-Based Reactance-Less Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97808*

**Figure 11.** *Phase plane of system of coupled MBOs with inverting connections under external excitation.*

For this reason, the parallel shift of trajectories at the phase portrait (**Figure 11**) corresponds to impact of external control signal.

Let the initial stable trajectory of the system before an external excitation correspond to line segment (*ab)* and after an external excitation the displaced trajectory corresponds to line segment (*gh*). When a constant control signal is applied for sufficiently long time, the transition to the trajectory (*gh*) is inevitable. It is caused by change in the sign of *dR*2*=dt* and reducing the threshold resistance to *Rm* þ *r* � *rC* when the point *b* reaches the border at *R*1ðÞ¼ *t Rm* þ *rm*. **Figure 11** illustrates the movement of image point from *b* to *c*, then to *d*, until it falls on the trajectory (*gh).*

The difference Δ ¼ *r*<sup>2</sup> � j j *r*<sup>1</sup> determines the width of the stability area and impacts on the speed of the transition process to new trajectory. It should be expected that due to the piecewise linear character of transients the return time will be proportional to the input amplitude in certain range of amplitude variation of control signal.

### **4.2 Simulation example**

The results of the behavior simulation of the coupled MBOs with inverting connections are given below for case of short input signal.

The computed waveforms for the oscillator system with phase detector NOR are shown in **Figure 12**. In this case stable antiphase oscillations are observed in the system under the absence of an external signal.

The following values of coupling factors were selected: *r*<sup>2</sup> ¼ 0*:*1,*rm* ¼ 0*:*09. These values of coupling factors mean that positive coupling in connected MBOs is stronger than negative coupling. The dimensionless parameters and variables are used below and the dimensionless time is also applied.

Let a starting point of system state be the stable trajectory with antiphase oscillations. The initial values of resistances *RM* ¼ 0*:*8, *Rm* ¼ 0*:*4 are selected. The output signal *VS* ¼ 0 corresponds to the stable trajectories of initial state (**Figure 12**). The positive pulse with amplitude of 0.05 and duration of 0.1 arrives at time t = 0.9. It causes the delay of switching of MBO1. The series of four output pulses is generated (**Figure 12**) while the system is in an excited state and the antiphase is violated.

The difference Δ in the coupling factors significantly impacts on the speed of transition to a stable trajectory. If this value is small the transient process can be

**Figure 12.**

*Example of simulating the timing diagram of the generation of output pulse series in the coupled MBOs with inverting connections.*

significantly delayed. At fixed values of the coupling strengths the input amplitude and time of arrival of the input pulse determine the time of transition to a stable trajectory. Due to this property the conversion of the input amplitude to the duration of the transition process can be performed.
