# A Maximum Power Transfer Tracking Method for WPT Systems with Coupling Coefficient Identification Considering Two-Value Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coupling Coefficient Identification and Maximum Power Transfer Tracking

#### 2.1. System Topology

_{b}is the load resistance. R

_{p}and R

_{s}are the internal resistance of L

_{p}and L

_{s}, respectively. L

_{p}, C

_{p}constitute the primary series resonant circuit, while L

_{s}, C

_{s}constitute the secondary series resonant circuit. M is the mutual inductance, and satisfies $M=k\sqrt{{L}_{p}{L}_{s}}$ (k represents the coupling coefficient). S

_{1}~S

_{4}constitute the full-bridge inverter, while D

_{1}~D

_{4}constitute the rectifier. S

_{b}, D

_{b}, L

_{b}, C

_{b}constitute the Buck converter. R

_{bin}and R

_{rin}are the equivalent input impedances of the Buck converter and rectifier circuit, respectively. i

_{p}and i

_{s}are the resonant currents. V

_{s}, V

_{r}, U

_{b}are the input voltage of the resonant circuit, the rectifier and the Buck converter respectively. To reduce EMI, assuming the operating frequency f of the inverter is same with the nature frequency of the resonant circuit, i.e., $f=\frac{1}{2\pi \sqrt{{L}_{p}{C}_{p}}}=\frac{1}{2\pi \sqrt{{L}_{s}{C}_{s}}}$.

#### 2.2. Identification of the Coupling Coefficient

_{bin}and R

_{b}is:

_{b}can be derived by $\frac{{U}_{o}}{{I}_{o}}$.

_{s}is continuous. Figure 2 is presented to show a case of continuous current. i

_{s}is continuous with the following condition [25]:

_{p}is:

_{s}is the RMS value of V

_{s}satisfying ${U}_{s}=\frac{2\sqrt{2}}{\pi}E$. Z

_{ref}is the reflected impedance, satisfies ${Z}_{ref}=\frac{{\omega}^{2}{M}^{2}}{{R}_{rin}+{R}_{s}}$.

_{r}is:

_{o}can be calculated as:

_{o}can be derived as:

_{p}, L

_{s}, R

_{p}, and R

_{s}are the known parameters. k can be calculated from (8) with the measured DC parameters U

_{o}and I

_{o}. However, Equation (8) shows that derivation of k may have two values based on one instance of U

_{o}. Therefore, at least two instances of output voltage are needed to identify the actual k. In the next part, this two-value issue during the identification will be solved.

#### 2.3. Dealing with the Two-Value Issue When Identifing the Coupling Coefficient

_{o}can be derived. By calculating Equation (8), four solutions of coupling coefficient can be obtained. Then compare the four solutions if there is a difference between two solutions is less than the tolerance error e

_{k}. The actual coupling coefficient can be determined by calculating the mean value of these two solutions. Otherwise, we will increase the duty cycle and restart the identification process. It should be noted that we just need the identification of the coupling coefficient once before the maximum power tracking process when there is no relative movement between the primary and secondary coils. When the coupling coefficient is identified, the maximum power tracking can be achieved through impedance matching and this will be introduced in the following section.

#### 2.4. Maximum Power Transfer Tracking

_{p}= 0; X

_{s}= 0), the output power P

_{L}can be derived as:

_{s}is the RMS value of i

_{s}, U

_{s}is the RMS value of V

_{s}.

_{p}is the RMS value of i

_{p}.

_{opt}can be derived:

_{b}is detected during the whole charging process. If R

_{b}is varied, the optimum duty cycle d derived by Equation (14) will fed to the Buck converter.

## 3. Simulation Analysis

#### 3.1. System Parameters and the Control Structure

_{b}is the frequency of the Buck converter.

#### 3.2. Analysis of the Coupling Coefficient Identification and Maximum Power Transfer Tracking

_{s}is continuous. According to Equation (3) and the parameters shown in Table 1, we can derive that when d is larger than 0.3, the current is continuous. Figure 7 shows the identification accuracies of the coupling coefficient when d varies from 0.2 to 0.9. The reference k is selected as 0.15 while R

_{b}= 10 Ω.

_{s}is continuous), the identification accuracies are all larger than 95%, while the identification accuracies are below 90% when i

_{s}is discontinuous. Therefore, only the continuous current case is taken into consideration in this paper.

_{b}is set at 10 Ω, d

_{1}= 0.5, d

_{2}= 0.6, e

_{k}= 0.0027. The identification accuracy is over 94% at all of the coupling coefficient conditions. These high accuracy identification results indicate that the identification method is feasible.

_{b}changes from 10 to 30 Ω, and two coupling coefficient cases are considered (the given coupling coefficients are 0.0448 and 0.0811, while the identified coupling coefficients are 0.0474 and 0.0839). The theoretical maximum power can be derived from (12), which are shown in the top of Figure 8a,b with dot line. The middle of the figure indicates the practical output power while the load R

_{b}variation is shown in the bottom of the figure. Both the two figures (a) and (b) share the similar features, due to the identification error of k and the non-ideal of the semiconductor switch and diode, the practical output power is slightly lower than the theoretical value. The steady state of the practical output power (P

_{o}) is almost the same at different load condition, this can prove that the maximum power tracking is achieved. At the changing instant of load R

_{b}, the practical output power is pulsatile due to the switch noise, and it cannot be totally eliminated.

## 4. Experimental Analysis

#### 4.1. Experimental Setup

#### 4.2. Experimental Results

_{o}, as shown in Equation (8), so we need to motivate the system twice by changing d as shown in Figure 3. When there is difference lower than e

_{k}between two solutions, k can be calculated by averaging these two solutions. e

_{k}should be selected as a small value to ensure the high identification accuracy, in this paper we choose e

_{k}to be 0.0027 (i.e., the mutual inductance is 1 µH). One thing about the selecting of d is that, as can be seen from Figure 7, the identification accuracy is relatively higher when d is larger until d increase to a certain value. Therefore, when k cannot be identified, we just increase d as shown in Figure 3.

_{1}and d

_{2}are selected as 0.5 and 0.6. k

_{1}and k

_{3}has the difference lower than e

_{k}, so the identification of k can be determined, and the accuracies of the two conditions are higher than 93%.

_{b}at 10, 20, and 30 Ω. ARM chip can detect the load variation, and then the tracking duty cycle d can be calculated by Equation (14). The maximum power tracking results under the two coupling conditions are shown in Figure 11, where the values in the blue rectangular boxes indicate the measured output power P

_{o}, I

_{o}, and U

_{o}under different load and coupling conditions.

_{o}and current I

_{o}. By comparing Figure 11a,b and Figure 8a,b the experimental output power is slightly smaller than the simulation results because of the identification error of k and the losses in the experiment system. When load R

_{b}changes, the tracked maximum output power is almost the same at the steady state, this can prove that the maximum power tracking is achieved.

_{b}is about 10 ms in steady state. After the identification of k, the maximum power transfer tracking can be done through impedance matching. As for the changing of load R

_{b}, the gap time between the two variable loads just need to be larger than the identification time. The proposed identification method shown in Figure 3 is also suitable for the dynamical identification of the coupling coefficient, and the variation of the coupling coefficient should be slower enough for the coupling coefficient identification.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**Maximum power tracking when load R

_{b}changes with different k: (

**a**) k = 0.0811; (

**b**) k = 0.0448.

**Figure 9.**Simulation analysis of system efficiencies under maximum power transfer tracking (MPTT) condition.

**Figure 11.**Experimental MPTT results when load R

_{b}changes with different k: (

**a**) k = 0.0811; (

**b**) k = 0.0448.

Items | Parameter | Value | Parameter | Value |
---|---|---|---|---|

Resonant circuits | L_{p} | 365.96 µH | L_{s} | 363.68 µH |

R_{p} | 0.83 Ω | R_{s} | 0.51 Ω | |

C_{p} | 34.25 nF | C_{s} | 34.30 nF | |

Litz-wire coils | Diameter | 14 cm | Number of turns | 25 |

Frequencies | f | 45 kHz | f_{b} | 100 kHz |

Buck converter | L_{b} | 120 µH | C_{b} | 470 µF |

Input source | E | 10 V |

Reference k | Identified k | Accuracy |
---|---|---|

0.0448 | 0.0474 | 94.16% |

0.0811 | 0.0839 | 96.59% |

0.0930 | 0.0962 | 96.67% |

0.1500 | 0.1531 | 97.93% |

0.2000 | 0.2037 | 98.15% |

0.2500 | 0.2546 | 98.16% |

Separation Distance | Reference k | Motivating d_{1} | Motivating d_{2} | Identified k | Accuracy |
---|---|---|---|---|---|

6 cm | 0.0811 | d_{1} = 0.5 k _{1} = 0.0877; k_{2} = 0.0183 | d_{2} = 0.6 k _{3} = 0.0854; k_{4} = 0.0098 | 0.0866 | 93.2% |

9 cm | 0.0448 | d_{1} = 0.5 k _{1} = 0.0484; k_{2} = 0.0332 | d_{2} = 0.6 k _{3} = 0.0464; k_{4} = 0.0207 | 0.0474 | 94.2% |

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**MDPI and ACS Style**

Dai, X.; Li, X.; Li, Y.; Deng, P.; Tang, C.
A Maximum Power Transfer Tracking Method for WPT Systems with Coupling Coefficient Identification Considering Two-Value Problem. *Energies* **2017**, *10*, 1665.
https://doi.org/10.3390/en10101665

**AMA Style**

Dai X, Li X, Li Y, Deng P, Tang C.
A Maximum Power Transfer Tracking Method for WPT Systems with Coupling Coefficient Identification Considering Two-Value Problem. *Energies*. 2017; 10(10):1665.
https://doi.org/10.3390/en10101665

**Chicago/Turabian Style**

Dai, Xin, Xiaofei Li, Yanling Li, Pengqi Deng, and Chunsen Tang.
2017. "A Maximum Power Transfer Tracking Method for WPT Systems with Coupling Coefficient Identification Considering Two-Value Problem" *Energies* 10, no. 10: 1665.
https://doi.org/10.3390/en10101665