# Behavioral Modeling of Memristor-Based Rectifier Bridge

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## Abstract

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## 1. Introduction

- Being an analog element, its resistance can take any values. This is a positive property in comparison with any binary element whose value can be either 0 or 1. Such variability of resistance is realized in one element, the memristor size is reduced to several nanometers and the response rate is reduced to nanoseconds.
- The memristor does not store a charge. This means that it is not prone to charge leaks, which must be dealt with when going to nanometer-scale microcircuits.
- The memristor is a non-volatile element, and data can be stored as long as the materials from which it is made exist.
- Memristors placed on crossing conductors (crossbars) can be used to form densely packed memory.
- Many memristor materials are compatible with complementary metal-oxide-semiconductor (CMOS) technology.

_{2}, NiO

_{2}, CuO, HfO

_{2}, etc. These memristors are called resistive ones due to the capability of resistance switching. In majority cases, other memristors are fabricated as spintronic, organic (polymeric) and ferroelectric ones [30]. However, the mentioned aspect has a negative effect that consists in the great variety of mathematical memristor models. These models describe different physical processes in memristors. As a result, we have to face the challenge of using mathematical memristor models, which essentially depend on the specifics of materials and technologies. These difficulties emerge on modeling memristor-based devices too.

## 2. Multi-Dimensional Split Polynomial as a Behavioral Nonlinear Model

- for all $\mathbf{A}\in {G}_{a}$, $t\in {G}_{t}$, there is an inequality$${S}_{\mathrm{sp}}\left(A,t\right)\ne 0;$$
- for any ${\mathbf{A}}_{\alpha}\ne {\mathbf{A}}_{\beta}$, ${\mathbf{A}}_{\alpha}\in {G}_{a}$, ${\mathbf{A}}_{\beta}\in {G}_{a}$, ${t}_{\alpha}\ne {t}_{\beta}$, ${t}_{\alpha}\in {G}_{t}$, ${t}_{\beta}\in {G}_{t}$, there is an inequality$${S}_{\mathrm{sp}}\left({A}_{\alpha},{t}_{\alpha}\right)\ne {S}_{\mathrm{sp}}\left({A}_{\beta},{t}_{\beta}\right).$$

## 3. Forming the Sets of Input and Output Signals of the Memristor-Based Rectifier Bridge

#### 3.1. The Yakopcic Model of a Memristor in LTspice

- Cx XSV 0 {1}
- .ic V(XSV) = xo

- .func F(V1,V2) = IF(eta*V1 >= 0, IF(V2 >= xp,exp(−alphap*(V2−xp))*wp(V2),1), IF(V2 <= (1−xn), exp(alphan*(V2+xn−1))*wn(V2),1))
- .func wp(V) = (xp−V)/(1−xp) + 1
- .func wn(V) = V/(1−xn)

- Gx 0 XSV value = {eta*F(V(P,N),V(XSV,0))*G(V(P,N))}
- Gm P N value = {IVRel(V(P,N),V(XSV,0))}

#### 3.2. Smoothing Output Signals of the Memristor-Based Rectifier Bridge

## 4. Results of the Rectifier Modeling Based on Multi-Dimensional Polynomials

- the uniform error$${\overline{\Delta}}_{k}({\overline{t}}_{n})={\overline{v}}_{\mathrm{out},\mathrm{sm},k}({\overline{t}}_{n})-{\overline{y}}_{k}({\overline{t}}_{n}),n=1,2,\dots ,2001,k=1,2,3;$$
- the maximum absolute error$$\mathrm{max}\left(\left|\overline{\Delta}({\overline{t}}_{n})\right|\right)=\underset{k=1,\hspace{0.17em}2,\hspace{0.17em}3}{\mathrm{max}}\left(\mathrm{max}\left(\left|{\overline{\Delta}}_{k}({\overline{t}}_{n})\right|\right)\right),$$$$\mathrm{max}\left(\left|{\overline{\Delta}}_{k}({\overline{t}}_{n})\right|\right)=\underset{{\overline{t}}_{n}\in \left(0,\hspace{0.17em}2\pi \right)}{\mathrm{max}}\left(\left|{\overline{v}}_{\mathrm{out},\mathrm{sm},k}({\overline{t}}_{n})-{\overline{y}}_{k}({\overline{t}}_{n})\right|\right),n=1,2,\dots ,2001;$$
- the root-mean-square error$${\epsilon}_{k}=\sqrt{\frac{1}{N-1}{\displaystyle \sum _{q=1}^{N-1}{\left(\hspace{0.17em}{\overline{v}}_{\mathrm{out},\mathrm{sm}}({\overline{t}}_{q})-{\overline{y}}_{k}({\overline{t}}_{q})\hspace{0.17em}\right)}^{\hspace{0.17em}2}}},N=2001,k=1,2,3.$$

## 5. Results of the Rectifier Modeling Based on Piecewise Multi-Dimensional Polynomials

## 6. Conclusions

- The form of the model and the method of its construction are universal, because they do not depend on the technology of memristor implementation.
- Since a polynomial is linear-in-parameters, these parameters are defined as globally optimal by solving the approximation problem.
- The splitting property allows the polynomial to be adapted to the assigned signal class, hence to construct a simpler model than other behavioral models.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**For the test input signals, the load voltages obtained sequentially after simulated in LTspice (

**a**), after smoothing by the low-pass filter (

**b**), after normalizing (

**c**).

**Figure 5.**For the 3rd trial input signal, the normalized output signals (the red curve is the load voltage of the circuit, the blue curve is the response of the split polynomial) and the uniform error (the black curve) in the case of smoothing by the low-pass filter (

**a**) and without smoothing (

**b**).

**Figure 6.**The block-scheme of dividing the input signal into sub-signals and the formation of the response of the split piecewise-polynomial model.

**Figure 7.**For the 3rd trial input signal, the normalized output signals (the red curve is the load voltage of the circuit, the blue curve is the response of the split piecewise-polynomial) and the uniform error (the black curve).

Signal Number | Amplitude and Frequency of Trial Input Signal | Errors | |
---|---|---|---|

$\mathbf{max}\left(\left|\overline{\mathbf{\Delta}}({\overline{\mathit{t}}}_{\mathit{n}})\right|\right)$ | $\mathit{\epsilon}$ | ||

1 | 12.5 V, 1.5 Hz | 1.5616 × 10^{−1} | 3.4918 × 10^{−3} |

2 | 14.5 V, 3.6 Hz | 2.1941 × 10^{−1} | 4.9062 × 10^{−3} |

3 | 15.5 V, 4.6 Hz | 1.7786 × 10^{−1} | 3.9771 × 10^{−3} |

Signal Number | Amplitude and Frequency of Trial Input Signal | Errors | |
---|---|---|---|

$\mathbf{max}\left(\left|\overline{\mathbf{\Delta}}({\overline{\mathit{t}}}_{\mathit{n}})\right|\right)$ | $\mathit{\epsilon}$ | ||

1 | 12.5 V, 1.5 Hz | 3.0720 × 10^{−2} | 6.8692 × 10^{−4} |

2 | 14.5 V, 3.6 Hz | 1.0967 × 10^{−1} | 2.4522 × 10^{−3} |

3 | 15.5 V, 4.6 Hz | 1.0967 × 10^{−1} | 9.6150 × 10^{−4} |

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**MDPI and ACS Style**

Solovyeva, E.; Schulze, S.; Harchuk, H.
Behavioral Modeling of Memristor-Based Rectifier Bridge. *Appl. Sci.* **2021**, *11*, 2908.
https://doi.org/10.3390/app11072908

**AMA Style**

Solovyeva E, Schulze S, Harchuk H.
Behavioral Modeling of Memristor-Based Rectifier Bridge. *Applied Sciences*. 2021; 11(7):2908.
https://doi.org/10.3390/app11072908

**Chicago/Turabian Style**

Solovyeva, Elena, Steffen Schulze, and Hanna Harchuk.
2021. "Behavioral Modeling of Memristor-Based Rectifier Bridge" *Applied Sciences* 11, no. 7: 2908.
https://doi.org/10.3390/app11072908