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Dynamic Analysis for the Physically Correct Model of a Fractional-Order Buck-Boost Converter^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. DC–DC Buck-Boost Converter

- State 1: ${S}_{T}=\mathrm{ON}$ and ${S}_{D}=\mathrm{OFF}$, for $nT<t\le (n+{D}_{c})T$.
- State 2: ${S}_{T}=\mathrm{OFF}$ and ${S}_{D}=\mathrm{ON}$, for $(n+{D}_{c})T<t\le (n+1)T$.

#### 2.2. Fractional-Order DC–DC Buck-Boost Converter

- A state equation, where each state ${x}_{i}\left(t\right)$ is differentiated to a fractional-order ${\alpha}_{i}$, is given in the case of a generalized state–space model. All states ${x}_{i}\left(t\right)$ are differentiated to the same fractional order $\alpha $ for the commensurate case.
- An output equation depends on the internal states and the inputs, as in the integer case.

## 3. Stable-State Analysis

## 4. Numerical Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Equivalent circuits of the Buck-Boost converter in State 1 with ${S}_{T}=\mathrm{on}$ and ${S}_{D}=\mathrm{off}$. (

**a**) Equivalent circuit of the Buck-Boost converter in State 1 and ${S}_{T}=\mathrm{ON}$. (

**b**) Equivalent circuit of the Buck-Boost converter in State 1 and ${S}_{D}=\mathrm{OFF}$.

**Figure 3.**Equivalent circuit of the Buck-Boost converter in State 2 with ${S}_{T}=\mathrm{OFF}$ and ${S}_{D}=\mathrm{ON}$.

**Figure 4.**Visualization of $\alpha $ and $\beta $ influence the response of ${G}_{V}$ in stable-state. (

**a**) Relationship between ${G}_{v}$ and $\alpha $. (

**b**) Relationship between ${G}_{v}$ and $\beta $. (

**c**) Three-dimensional representation of $\alpha $ and $\beta $ variations.

**Figure 6.**Nondimensionalized voltage and current of the fractional-order Buck-Boost converter. (

**a**) Nondimensionalized inductor’s current. (

**b**) Nondimensionalized capacitor’s voltage.

**Figure 8.**Phase portrait of the fractional-order Buck-Boost converter at different values of $\alpha $: (

**a**) $\alpha =0.7$, (

**b**) $\alpha =0.8$, (

**c**) $\alpha =0.9$, (

**d**) $\alpha =1$.

**Table 1.**Parameters used for the simulations to obtain the Buck-Boost converter fractional model response in the stable state.

Parameter | Values |
---|---|

Stable–state time | ${t}_{s}=T=4$ ms |

Input voltage | ${V}_{i}=25$ V |

Inductor | $L=3$ mH |

Capacitor | $C=150$ $\mathsf{\mu}$F |

Load | $R=30\phantom{\rule{3.33333pt}{0ex}}\Omega $ |

Duty cycle | ${D}_{c}=0.6$ |

**Table 2.**Electrical characteristics of the fractional-order Buck-Boost Converter while varying $\alpha $.

Nondimensionalized | Real | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | Poles | $\widehat{\mathit{L}}$ | ${\widehat{\mathit{t}}}_{\mathbf{ss}}$ | ${\widehat{\mathit{v}}}_{\mathit{f}}$ | $\mathit{L}$ [v] | ${\mathit{t}}_{\mathit{ss}}$ [ms] | ${\mathit{v}}_{\mathit{f}}$ [v] |

$0.7$ | $-0.0391\pm 0.0165i$ | $1.7653$ | $20.420$ | $1.5003$ | $37.5075$ | $2.0420$ | $37.5075$ |

$0.8$ | $-0.0465\pm 0.0426i$ | $2.0369$ | $29.832$ | $1.5002$ | $50.9225$ | $2.9832$ | $37.5045$ |

$0.9$ | $-0.0457\pm 0.0725i$ | $2.4010$ | $48.5906$ | $1.5001$ | $60.0257$ | $4.8590$ | $37.5034$ |

1 | $-0.0375\pm 0.1029i$ | $2.8756$ | $318.6323$ | $1.5020$ | $71.8907$ | $31.8620$ | $37.5511$ |

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**MDPI and ACS Style**

Zambrano-Gutierrez, D.F.; Cruz-Duarte, J.M.; Valencia-Rivera, G.H.; Amaya, I.; Avina-Cervantes, J.G.
Dynamic Analysis for the Physically Correct Model of a Fractional-Order Buck-Boost Converter. *Comput. Sci. Math. Forum* **2022**, *4*, 2.
https://doi.org/10.3390/cmsf2022004002

**AMA Style**

Zambrano-Gutierrez DF, Cruz-Duarte JM, Valencia-Rivera GH, Amaya I, Avina-Cervantes JG.
Dynamic Analysis for the Physically Correct Model of a Fractional-Order Buck-Boost Converter. *Computer Sciences & Mathematics Forum*. 2022; 4(1):2.
https://doi.org/10.3390/cmsf2022004002

**Chicago/Turabian Style**

Zambrano-Gutierrez, Daniel F., Jorge M. Cruz-Duarte, Gerardo Humberto Valencia-Rivera, Ivan Amaya, and Juan Gabriel Avina-Cervantes.
2022. "Dynamic Analysis for the Physically Correct Model of a Fractional-Order Buck-Boost Converter" *Computer Sciences & Mathematics Forum* 4, no. 1: 2.
https://doi.org/10.3390/cmsf2022004002