# The Coupled Reactance-Less Memristor Based Relaxation Oscillators for Binary Oscillator Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Behavior of Reactance-Less Memristor Based Oscillator

#### 2.1. Operating Principles of Oscillator

_{out}. The current generator Im, which is included in the negative feedback network of the comparator, converts the binary output signal (“0” and “1”) into negative current and positive current through the memristor (−I, +I), respectively.

_{in}that is proportional to the input signal V

_{in}. The proportionality coefficient k affects the coupling strength between the connected MBOs.

_{out}is equal to the logical “0” for −V

_{M}< v < V

_{m}and is equal to the logical “1” otherwise, |V

_{M}|> |V

_{m}|. Here V

_{m}and −V

_{M}are the threshold voltage values. The reference voltages V

_{m}and V

_{M}set the initial minimum and maximum threshold voltages, respectively. The initial threshold voltage values are set by negative Vn and positive Vp voltages on the adder in accordance with Figure 1. Here, we have Vn = −V

_{M}, Vp = V

_{m}.

_{in}and V

_{in}. The input current is added to the current Im(V

_{out}). The input voltage V

_{in}is subtracted from the specified values V

_{m}and V

_{M}. The current input I

_{in}is the conventional input for reactance-less MBO.

_{out}. When connecting a memristor with an anode to a current generator Im (Figure 1), the equation

_{on}and maximal R

_{off}resistances. The range of variation R is further narrowed in the considered circuit (Figure 1) under input signal V

_{in}(t) due to the maximal V

_{M}and minimal V

_{m}threshold voltages of comparator:

#### 2.2. Two Control Types in MBOs

_{in}affects the rate of change in the resistance of the memristor, increasing the speed at the same signs of the input current and the generator current Im, and decreasing the speed otherwise.

_{in}is applied to the comparator to change its thresholds. It determines the range of change in the memristor resistance. In this case, the threshold control principle [32] is applied.

_{in}or V

_{in}) lead to different responses of the MBO to the shape of the input signal.

## 3. Features of Coupled Reactance-Less Memristor Based Oscillators

- -
- During the action of the high output level (logical “1”) of the transmitting MBO, both comparator thresholds of the receiving MBO decrease; after the completion of the action of the high output level of the transmitting MBO, the comparator thresholds of the receiving MBO are restored to their original values. The low output level (logical “0”) of the transmitting MBO does not impact on the comparator thresholds of the receiving MBO;
- -
- Threshold changes are small enough to provide the condition of oscillations receiving MBO;
- -
- Input potential signal does not impact the amount of current flowing through the memristor.

_{1}and R

_{2}, as well as signs of their derivatives dR

_{1}/dt and dR

_{2}/dt .Their behavior is shown on the phase plane R

_{1}and R

_{2}where the trajectories of the representing point are the straight lines which are parallel or perpendicular to the main diagonal of the quadrant (Figure 6). Thus, one of the four trajectories defined by the signs of the derivative dR/dt can pass through each point of the phase plane.

_{M}, R

_{M}) and $\left({R}_{m}-r,{R}_{m}-r\right)$ lying on the main diagonal passing through these points. This square has the region of stationary trajectories of periodic motion of the system. The specific stationary trajectory characterizes the state of coupled MBOs. The region of stationary trajectories is bounded by straight lines that are parallel to the main diagonal. They cross a straight line perpendicular to the main diagonal and spaced from the vertex by the distance $r/\sqrt{2}$. This area is bounded by dotted lines in Figure 6. The stable trajectories themselves are straight parallel to the main diagonal. They correspond to synchronous oscillations. The trajectories on the main diagonal correspond to oscillations of MBO1 and MBO2 of equal amplitude.

## 4. Example of Application of Coupled MBOs in Oscillatory Networks

_{0}is included in the network. This oscillator generates some averaged (centered) frequency and operates under the control of the averaged value of the current generator Im

_{0.}

_{j}.The coupling of the oscillators is assumed to be weak enough to maintain the independence of the specified frequencies f

_{i}.

_{1}= 100 uA, Im

_{2}= 200 uA. Then, the reference oscillator can be specified by the average of the current value: Im

_{0}= 150 uA.

_{m}set in accordance with “correct” image (Table 2). In this case, zero excitations fall on current inputs of oscillator circuits, since the real input signal is determined in this approach by the difference between stored and incoming signals. For this reason, we can see the same reference frequencies for all the oscillators. In Figure 9F, the triangular character of the memristor resistance change R (t) for the first oscillator is shown.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The time diagram for control by current pulse: input current pulse; character of the sawtooth change of memristor resistance R under current control; waveforms of voltage changes at the output of the oscillator circuit V(out).

**Figure 4.**The computed waveforms of the memristor-based oscillator with controlled threshold parameters.

**Figure 5.**The potential control with a narrow input signal with impact (

**a**) and without impact (

**b**) on output pulse train.

**Figure 8.**The output waveforms of the coupled oscillators according to encoded frequencies. The following set of oscillator frequencies corresponds to considered example: f

_{0}= 25.2975, f

_{1}= 11.2486, f

_{2}= 44.6236, f

_{3}= 11.2486, f

_{4}= 44.6236 Hz.

**Figure 9.**(

**a**–

**f**) The computed waveforms for five coupled oscillators for the “correct” image; (

**f**)–the triangular character of change memristor resistance R (t) for the first oscillator.

**Figure 10.**(

**a**–

**g**) The computed waveforms for five coupled oscillators; (

**f**,

**g**) the triangular character of change memristor resistances R (t) for the first and second oscillator.

**Figure 11.**(

**a**–

**g**) The computed waveforms for five coupled oscillators; (

**f**,

**g**) the triangular character of change memristor resistances R (t) for the third and fourth oscillator.

Parameter | Description | Value |
---|---|---|

Ron | Resistance in ON State, [kOhm] | 1 |

Roff | Resistance in OFF State, [kOhm] | 10 |

Rinit | Initial resistance at t = 0, [kOhm] | 4 |

uv | Migration coefficient, [m^{2} s^{−1} V^{−1}] | 10^{−14} |

D | Width of the thin film, [nm] | 10 |

p | Parameter of the window function | 10 |

White Im_{1} = 100 uA |

Black Im_{2} = 200 uA |

White Im_{3} = 100 uA |

Black Im_{4}= 200 uA |

Gray I_{M1} = 120 uA |

Gray I_{M2} = 180 uA |

White I_{M3} = 100 uA |

Black I_{M4} = 200 uA |

White I_{M1} = 100 uA |

Black I_{M2} = 200 uA |

Gray I_{M3} = 145 uA |

Gray I_{M4} = 155 uA |

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**MDPI and ACS Style**

Rakitin, V.; Rusakov, S.; Ulyanov, S. The Coupled Reactance-Less Memristor Based Relaxation Oscillators for Binary Oscillator Networks. *Micromachines* **2023**, *14*, 365.
https://doi.org/10.3390/mi14020365

**AMA Style**

Rakitin V, Rusakov S, Ulyanov S. The Coupled Reactance-Less Memristor Based Relaxation Oscillators for Binary Oscillator Networks. *Micromachines*. 2023; 14(2):365.
https://doi.org/10.3390/mi14020365

**Chicago/Turabian Style**

Rakitin, Vladimir, Sergey Rusakov, and Sergey Ulyanov. 2023. "The Coupled Reactance-Less Memristor Based Relaxation Oscillators for Binary Oscillator Networks" *Micromachines* 14, no. 2: 365.
https://doi.org/10.3390/mi14020365