# Li-Ion Battery Charging with a Buck-Boost Power Converter for a Solar Powered Battery Management System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lithium Battery Model

_{6}, LiBF

_{4}, or LiClO

_{4}. The negative electrode is connected to the negative terminal of the cell and usually contains graphite. The positive electrode is connected to the positive terminal of the cell and is a metal oxide or a blend of several metal oxides such as Li

_{x}Mn

_{2}O

_{4}and Li

_{x}CoO

_{2}. The separator is a solid or liquid solution with a high concentration of lithium ions. It is an electrical insulator that prevents electrons from flowing between negative and positive electrodes, but the electrolyte allows ions to pass through it. In the discharging process, the lithium ions at the surface of the solid active material of the porous negative electrode undergo an electrochemical reaction, transferring the ions to the solution and the electrons to the negative terminal. The positive ions travel through the electrolyte solution to the positive electrode where they react with, diffuse toward, and are inserted into the metal oxide solid particles. Ions and electrons reverse traveling direction in the charging process.

## 3. Synchronous Buck-Boost Converter

_{S}. Thus, the output voltage can be higher than the input supply voltage. When supply voltage is close to the desired load voltage, the converter is set to buck-boost operation. In this mode, transistors ${Q}_{1}$ and ${Q}_{3}$ work as a group and ${Q}_{2}$ and ${Q}_{4}$ work as another group. To charge the inductor, switches ${Q}_{1}$ and ${Q}_{3}$ are closed and ${Q}_{2}$ and ${Q}_{4}$ are open. Transistors ${Q}_{2}$ and ${Q}_{4}$ are closed and ${Q}_{1}$ and ${Q}_{3}$ are opened to engage the inductor discharge cycle. In this mode, average load voltage ${V}_{\text{out}}$ equals d/(1 − d)V

_{S}. This implies that output voltage can be changed to more or less than the supplied voltage. Table 1 shows a summary of power converter operation.

Buck mode | Buck-Boost mode | Boost mode | ||
---|---|---|---|---|

${Q}_{1}$ | Inductor charge | ON | ON | ON |

Inductor discharge | OFF | OFF | ON | |

${Q}_{2}$ | Inductor charge | OFF | OFF | OFF |

Inductor discharge | ON | ON | OFF | |

${Q}_{3}$ | Inductor charge | OFF | ON | ON |

Inductor discharge | OFF | OFF | OFF | |

${Q}_{4}$ | Inductor charge | ON | OFF | OFF |

Inductor discharge | ON | ON | ON | |

Average load voltage | ${V}_{\text{out}}=d{V}_{\text{S}}$ | ${V}_{\text{out}}=\frac{d}{1-d}{V}_{\text{S}}$ | ${V}_{\text{out}}=\frac{1}{1-d}{V}_{\text{S}}$ |

## 4. Battery Charging with a Buck-Boost Power Converter

_{Qi}represents the corresponding switch on resistance of the MOSFET. For simplicity, resistances ${R}_{CS}+{R}_{B}$ are replaced by a single resistor with resistance ${R}_{S}$. In this model, switch ${Q}_{3}$ is off and ${Q}_{4}$ is on. Buck operation is achieved by controlling switches ${Q}_{1}$ and ${Q}_{2}$ to charge and discharge the inductor and then transfer energy to the battery. Figure 7(a) shows the equivalent circuit for the inductor charging cycle and Figure 7(b) shows the inductor discharging equivalent circuit. If the duty cycle for inductor charging is $d$ and ${d}^{\prime}=1-d$ during inductor discharge, the averaged dynamic system for battery charging in buck mode operation is:

**Figure 10.**Battery charging for buck-boost mode operation. (

**a**) Inductor charge, (

**b**) inductor discharge.

_{L}= 50 mW, R

_{C}= 5 mW and R

_{O}= 7 mW, Figure 11 shows the stability margins of the system from duty cycle variation $\tilde{d}$ to output voltage variation ${\tilde{V}}_{\text{out}}$ with respect to supply voltage variation. The figure indicates that the phase margins are inadequate.

_{1}= 10 kW, R

_{2}= 668 W, R

_{3}= 630 W, C

_{1}= 1.05 nF, C

_{2}= 161 nF and C

_{3}= 10.1 nF. Table 2 shows a summary of stability margins. The results show that the phase margins improve significantly. The results for 1-A and 0.1-A charge current are also investigated. The stability margin will be improved when the load current decreases. Figure 14, Figure 15 and Figure 16 are the examples of the bode plots for boost, buck-boost, and buck modes respectively. The bode plots examples clearly show the improvement of the phase margin for the compensated system.

Mode | Stability margin | Without compensator | With compensator |
---|---|---|---|

Buck mode $11.8\text{V}{V}_{\text{S}}\le 30\text{V}$ | Gain margin (dB) | $\infty $ | $\infty $ |

Phase margin (deg) | 31.77–38.22 | 78.05–113.65 | |

Buck-Boost mode $10.8\text{V}{V}_{\text{S}}\le 11.8\text{V}$ | Gain margin (dB) | 36.89 | 31.77–31.95 |

Phase margin (deg) | 31.65–32.35 | 106.82–111.27 | |

Boost mode $8\text{V}{V}_{\text{S}}\le 10.8\text{V}$ | Gain margin (dB) | 36.89 | 31.06–31.77 |

Phase margin (deg) | 33.70–36.27 | 118.65–134.11 |

## 5. Electric Circuit Simulation

_{p}and θ

_{n}(θ

_{p0%}= 0.895, θ

_{p100%}= 0.305, θ

_{n0%}= 0.126 and θ

_{n100%}= 0.676 were selected). Table 3 lists the parameters used in the simulation. Constant current or constant voltage feedback determined the battery charging mode. The duty cycle controller determined the proper duty ratio for the power switches. In this simulation, $d=\overline{d}+\tilde{d}$ determined the duty ratio and the type III compensator described in the previous section determined variation $\tilde{d}$. Mean value $\overline{d}$ was defined using the following equations:

**Buck**and

**Boost**in (53) and (54) are Boolean signals and ${\overline{V}}_{\text{out}}$ and ${\overline{I}}_{\text{BATT}}$ are desired regulated voltage and current, respectively. Equations (53) and (54) are formulated using Equations (35) to (37) by setting $d{i}_{L}/dt=0$ and $d{v}_{C}/dt=0$ and assuming that R

_{L}= 0, R

_{C}= 0 and R

_{Q}= 0.

Parameter | Negative electrode | Separator | Positive electrode |
---|---|---|---|

Thickness, ${L}^{-},{L}^{\text{sep}},{L}^{+}$ (cm) | 50 × 10^{−4} | 25.4 × 10^{−4} | 36.4 × 10^{−4} |

Particle radius, ${R}_{\text{s}}$ (cm) | 1 × 10^{−4} | - | 1 × 10^{−4} |

Active material volume fraction, ${\epsilon}_{\text{s}}$ | 0.580 | - | 0.500 |

Electrolyte phase volume fraction, ${\epsilon}_{\text{e}}$ | 0.332 | 0.5 | 0.330 |

Solid phase conductivity, $\sigma ({\text{\Omega}}^{-1}{\text{cm}}^{-1})$ | 1.0 | - | 0.1 |

${\text{Li}}^{+}$ Transference number, ${t}_{+}^{0}$ | 0.363 | ||

Electrolyte phase ionic conductivity, $\kappa ({\text{\Omega}}^{-1}{\text{cm}}^{-1})$ | $\kappa =15.8{c}_{\text{e}}\text{exp}[0.85{(1000{c}_{\text{e}})}^{1.4}$ | ||

Electrolyte phase diffusion coefficient, ${D}_{\text{e}}({\text{cm}}^{2}{\text{s}}^{-1})$ | 2.6 × 10^{−6} | ||

Solid phase diffusion coefficient, ${D}_{\text{s}}({\text{cm}}^{2}{\text{s}}^{-1})$ | 2.0 × 10^{−12} | - | 3.7 × 10^{−12} |

Maximum solid phase concentration, ${c}_{\text{s},\text{max}}(\text{mol}\cdot {\text{cm}}^{-3})$ | 16.1 × 10^{−3} | - | 23.9 × 10^{−3} |

Exchange current density, ${i}_{0}(\text{A}\cdot {\text{cm}}^{2})$ | 3.6 × 10^{−3} | 2.6 × 10^{−3} | |

Average electrolyte concentration, ${\overline{c}}_{\text{e}}(\text{mol}\cdot {\text{cm}}^{-3})$ | 1.2 × 10^{−3} | ||

Charge-transfer coefficients, ${\alpha}_{a},{\alpha}_{c}$ | 0.5, 0.5 | - | 0.5, 0.5 |

Active surface area per electrode unit volume, $a({\text{cm}}^{-1})$ | ${a}_{\text{s,n}}=3{\epsilon}_{\text{e}}/{R}_{\text{s}}$ | - | ${a}_{\text{s,n}}=3{\epsilon}_{\text{e}}/{R}_{\text{s}}$ |

Utilization ratio at 0% SOC, ${\theta}_{\text{n}0\%},{\theta}_{\text{p}0\%}$ | 0.126 | - | 0.895 |

Utilization ratio at 100% SOC, ${\theta}_{\text{n}100\%},{\theta}_{\text{p}100\%}$ | 0.676 | - | 0.305 |

Electrode plate area, A $({\text{cm}}^{2})$ | 10452 | - | 10452 |

Current collector contact resistance, ${R}_{\text{f}}(\text{\Omega}{\text{cm}}^{2})$ | 20 | ||

Equilibrium potential, Negative electrode | $\begin{array}{c}{U}_{\text{n}}({\theta}_{\text{n}})=8.0029+5.0647{\theta}_{\text{n}}-12.578{\theta}_{\text{n}}^{0.5}\\ -8.6322\times {10}^{-4}{\theta}_{\text{n}}^{-1}+2.176\times {10}^{-5}{\theta}_{\text{n}}^{3/2}\\ -0.46016\text{exp}[15.0(0.06-{\theta}_{\text{n}}]\\ -0.55364\text{exp}[-2.4326({\theta}_{\text{n}}-0.92)]\end{array}$ | ||

Equilibrium potential, Positive electrode | $\begin{array}{c}{U}_{\text{p}}({\theta}_{\text{p}})=85.681{\theta}_{\text{p}}^{6}-357.70{\theta}_{\text{p}}^{5}+613.89{\theta}_{\text{p}}^{4}\\ -555.65{\theta}_{\text{p}}^{3}+281.06{\theta}_{\text{p}}^{2}-76.648{\theta}_{\text{p}}\\ +13.1983-0.30987\text{exp}(5.657{\theta}_{\text{p}}^{115})\end{array}$ |

## 7. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Shiau, J.-K.; Ma, C.-W.
Li-Ion Battery Charging with a Buck-Boost Power Converter for a Solar Powered Battery Management System. *Energies* **2013**, *6*, 1669-1699.
https://doi.org/10.3390/en6031669

**AMA Style**

Shiau J-K, Ma C-W.
Li-Ion Battery Charging with a Buck-Boost Power Converter for a Solar Powered Battery Management System. *Energies*. 2013; 6(3):1669-1699.
https://doi.org/10.3390/en6031669

**Chicago/Turabian Style**

Shiau, Jaw-Kuen, and Chien-Wei Ma.
2013. "Li-Ion Battery Charging with a Buck-Boost Power Converter for a Solar Powered Battery Management System" *Energies* 6, no. 3: 1669-1699.
https://doi.org/10.3390/en6031669