# Nonlinear Dynamics and Performance Analysis of a Buck Converter with Hysteresis Control

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## Abstract

**:**

## 1. Introduction

- The constant hysteresis control is applied to a nonlinear and fast dynamic power converter system.
- A mathematical model is derived and used to characterize the nonlinear dynamics induced by the hysteresis control.
- A cost-effective voltage control strategy is implemented using few analog electronic components.

## 2. Materials and Methods

#### 2.1. Model of the Buck Converter

#### 2.2. Buck Converter Voltage Control with Constant Hysteresis

#### 2.3. Physical Considerations

#### 2.4. System Parameters for Simulation

#### 2.5. Model Simulations and Numerical Methods

#### 2.6. Nonlinear Dynamics for the Zero Hysteresis Control

#### 2.6.1. Steady- and Transient-State Simulations

#### 2.6.2. Regulation for Different Voltage Set Points

#### 2.7. Nonlinear Dynamics for the Constant Hysteresis Control

## 3. Experimental Results

#### 3.1. Experimental Implementation of the Zero Hysteresis Control

#### Comparison of Experimental Observation with Simulations

#### 3.2. Steady- and Transient-State Operation

#### 3.2.1. Load Disturbance

#### 3.2.2. Chaos with the Zero Hysteresis Control

#### 3.3. Experimental Validation of the Constant Hysteresis Control

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Bisection Numerical Method

Algorithm A1 Bisection numerical Method |

1: $\mathit{t}\mathit{1}\leftarrow \left\{\mathit{Ti}\right\}$ Time interval initial value |

2: $\mathit{t}\mathit{2}\leftarrow \{\mathit{Tf}=\mathit{Ti}+\mathit{Ts}\}$ Time interval corresponds to $Ts$, the Time simulation step |

3: ${\mathit{V}}_{\mathit{ref}}\leftarrow \left\{\mathit{xc}\right\}$ Reference value to reach and find the time value (solution) |

4: $\u03f5\leftarrow \{\mathit{1}\mathit{E}-\mathit{15}\}$ Tolerance criteria for convergence |

5: $\mathit{error}\leftarrow \left\{\mathit{1}\right\}$ error as initial value |

6: while $\left|error\right|>\u03f5$ do |

7: $tm=(t1+t2)/2$ |

8: $x1=f\left(t1\right)$ |

9: $xm=f\left(tm\right)$ |

10: $error=xm-{V}_{ref}$ |

11: if $(x1-{V}_{ref})*(xm-{V}_{ref})<0$ then |

12: $t2=tm$ |

13: else |

14: $t1=tm$ |

15: end |

16: return $tr=tm$ |

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**Figure 2.**Topology of the equivalent circuits for a: (

**a**) continuous conduction mode, (

**b**) continuous conduction mode with the diode in forward bias, and (

**c**) discontinuous conduction mode with the diode in reverse bias.

**Figure 3.**Hysteresis function: (

**a**) hysteresis loop as a function of tracking error; (

**b**) state machine to describe the control transition as a function of error dynamics, hysteresis amplitude parameter and control state.

**Figure 4.**Schematics of the closed-loop control system for the Buck converter using a constant hysteresis control.

**Figure 5.**Application of the numerical method to solve an accuracy error on switching boundary: (

**a**) state trajectory without the numerical algorithm and (

**b**) state trajectory with the numerical algorithm.

**Figure 6.**Bisection numerical method results: one example to show evolution and convergence of numerical solution: (

**a**) boltage value convergence to ${v}_{C}=15$ and (

**b**) time solution evolution.

**Figure 7.**Hysteresis control applied to the Buck converter (

**a**) block diagram with the zero hysteresis control and (

**b**) switching state for the zero hysteresis control ($\xi \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0$ V).

**Figure 8.**Dynamic behavior in steady- and transient-state operations: (

**a**) phase diagram and (

**b**) temporal evolution.

**Figure 9.**Asymmetry in the phase plane for (

**a**) low reference voltage ${V}_{ref}=5$ V and (

**b**) high reference voltage ${V}_{ref}=25$ V.

**Figure 10.**Temporal dynamics of (

**a**) low reference voltage ${V}_{ref}=5$ V and (

**b**) high reference voltage ${V}_{ref}=25$ V.

**Figure 11.**Dynamics when considering symmetric vector fields with ${V}_{ref}=15$ V: (

**a**) phase diagram and (

**b**) temporal evolution of switching pattern for switch S.

**Figure 12.**Hysteresis control applied to the Buck converter (

**a**) block diagram with the constant hysteresis control and (

**b**) switching state for the constant hysteresis control ($\xi >0$).

**Figure 13.**$1T$-periodic orbits with the constant hysteresis control: (

**a**) $1T$-periodic orbit in CCM mode (${i}_{L}>0$), combining two solution pieces for $S=1$ and $S=0$; and (

**b**) $1T$-periodic orbit in DCM (${i}_{L}\ge 0$), combining three solution pieces for $S=1$, $S=0$ in CCM and for ${i}_{L}=0$.

**Figure 14.**Voltage control using the constant hysteresis control for $\xi =0.2$ V: (

**a**) phase portrait and (

**b**) time evolution.

**Figure 15.**Voltage control using the constant hysteresis control for $\xi =0.2$ V: (

**a**) phase portrait and (

**b**) time evolution.

**Figure 16.**Phase portrait varying the hysteresis control parameter $\xi $. Increasing parameter $\xi $ a

**Grazing bifurcation**appears to transform the CCM $1T$-periodic orbit into a $1T$-periodic orbit in DCM with ${i}_{L}\ge 0$.

**Figure 17.**Regulation indexes mean and variance when hysteresis parameter $\xi $ is varied: (

**a**) mean values and (

**b**) variance.

**Figure 21.**Numerical simulation vs. experimental results in the steady- and transient-state operations with $\xi =0$ V and ${v}_{c}=5$ V: (

**a**) experimental phase diagram, (

**b**) experimental temporal evolution, (

**c**) theoretical phase diagram, and (

**d**) theoretical temporal response.

**Figure 22.**Comparison of simulation and experiment of the zero hysteresis control when the load increases and decreases, and the reference voltage is ${V}_{ref}=14.53$ V for a (

**a**) simulated phase diagram, (

**b**) simulated temporal evolution, (

**c**) experimental phase diagram, and (

**d**) experimental temporal evolution.

**Figure 23.**Experimental chaotic dynamics with the zero hysteresis control due to existence of residual hysteresis created by physical limitations and internal parasitic effects of electronic components: (

**a**) DC voltage output and inductor currrent and (

**b**) AC signals showing the variable ripple on voltage and current.

**Figure 24.**Experimental dynamic behavior with residual zero hysteresis control $\xi \phantom{\rule{0.166667em}{0ex}}\simeq \phantom{\rule{0.166667em}{0ex}}0$ V, plotting (

**a**) phase diagram and (

**b**) temporal evolution.

**Figure 25.**Experimental validation of voltage control using the constant hysteresis control for $\xi =0.2$ V, converging to $1T$-periodic orbit in CCM: (

**a**) phase portrait, (

**b**) time evolution for simulation results, (

**c**) phase portrait, and (

**d**) time evolution for experimental results.

**Figure 26.**Experimental validation of voltage control using the constant hysteresis control for $\xi =1$ V, converging to $1T$-periodic orbit in DCM: (

**a**) phase portrait, (

**b**) time evolution for simulation results, (

**c**) phase portrait, and (

**d**) time evolution for experimental results.

**Figure 27.**Experimental validation of voltage control using the constant hysteresis control with low value, e.g., $\xi $ = 0.05 V, and required voltage is ${V}_{ref}=12\phantom{\rule{4pt}{0ex}}V$: (

**a**) phase portrait and (

**b**) time evolution.

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

${V}_{in}$ | Input voltage | 20 V |

${V}_{ref}$ | Reference voltage | 15 V |

R | Resistance | 22 $\phantom{\rule{0.166667em}{0ex}}\Omega $ |

C | Capacitance | 1000 $\mathsf{\mu}$f |

L | Inductance | 7 mH |

${X}_{o}$ | Initial conditions | 0 V; 0 A |

$\xi $ | Hysteresis band | 0 V |

$dt$ | Time discrete steps | $1\times 1{0}^{-6}$ s |

${t}_{o}$ | Initial simulation time | 0.0 s |

${t}_{f}$ | Final simulation time | 0.4 s |

**Table 2.**Parameters of the Buck converter with the zero-hysteresis control for the experimental test.

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

${V}_{in}$ | Input voltage | 30 V |

${V}_{ref}$ | Reference voltage | $--\phantom{\rule{4pt}{0ex}}V$ |

a | Constant of the sensor | 1/3 |

R | Load resistance | $--\Omega $ |

C | Capacitance | $299\phantom{\rule{4pt}{0ex}}\mathsf{\mu}$f |

L | Inductance | $10.6\phantom{\rule{4pt}{0ex}}$mH |

${R}_{inL}$ | Resistance of the inductor | $0.4$ $\Omega $ |

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**MDPI and ACS Style**

Hoyos Velasco, C.I.; Hoyos Velasco, F.E.; Candelo-Becerra, J.E.
Nonlinear Dynamics and Performance Analysis of a Buck Converter with Hysteresis Control. *Computation* **2021**, *9*, 112.
https://doi.org/10.3390/computation9100112

**AMA Style**

Hoyos Velasco CI, Hoyos Velasco FE, Candelo-Becerra JE.
Nonlinear Dynamics and Performance Analysis of a Buck Converter with Hysteresis Control. *Computation*. 2021; 9(10):112.
https://doi.org/10.3390/computation9100112

**Chicago/Turabian Style**

Hoyos Velasco, Carlos I., Fredy Edimer Hoyos Velasco, and John E. Candelo-Becerra.
2021. "Nonlinear Dynamics and Performance Analysis of a Buck Converter with Hysteresis Control" *Computation* 9, no. 10: 112.
https://doi.org/10.3390/computation9100112