# An Improved Estimation Method of Mutual Inductance Angle for a Two-Dimensional Wireless Power Transfer System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 2D WPT System

#### 2.1. Structure of Coils

#### 2.2. Coupled Circuit Model

_{α}= L

_{β}= L and $X=\omega L-1/\omega C$, respectively, where ω denotes the frequency of the input voltages. The mutual inductances between in Tx and Rx coils are represented as ${M}_{\alpha r}$ and ${M}_{\beta r}$ for the α- and β-axis, respectively. The mutual inductance between the α- and β-axis is assumed as zero because two Tx coils are orthogonal, as mentioned.

#### 2.3. Power Transmission Efficiency (PTE) and Mutual Inductance Angle

_{αm}and I

_{βm}, are defined as

## 3. Estimation of Mutual Inductance Angle

#### 3.1. Structure Od Proposed Estimator

_{r}, X = 0, thus the q-axis voltage equation can be simply rewritten as

_{p}and T denote the proportional gain and time constant, respectively.

#### 3.2. Design of Proposed Estimator

_{r}of the Rx circuit is selected as 20 nF for matching to the resonant frequency.

_{m}means the measured signals of the voltage and current modulated with cosωt. The carrier frequency is given as 530 kHz, which is the same as the frequency of the input voltage and current. The second-order low-pass filter with a cut-off frequency of 10 kHz is used to extract the envelope of the voltages and currents of the Tx coils, which is a trade-off between the filtering performance and control loop dynamics.

## 4. Simulation and Experiments

_{αr}and M

_{βr}can be measured using the open and short circuit test of the Tx and Rx as shown in Figure 10, and the mutual inductance angles are calculated using (8).

_{p}= 1.0 and T = 0.002, respectively. It is shown in this figure that the settling time is 1.2 ms for the given gain.

_{α}and I

_{β}for the input terminals in the α- and β-axis are calculated using the estimated mutual inductance angle $\theta $ as

_{d}= 3 A and I

_{q}= 0 for the experiment. The load resistance is given as R

_{L}= 10 Ω. Figure 15 shows the experimental results for the 2D WPT system when the physical angle is Φ = 30° and d = 0 cm between the Tx and Rx coils. Figure 14a,b show the input and output currents when $\theta =\varphi =30\xb0$ and $\theta =\widehat{\gamma}=26.79\xb0$, respectively. The output powers of both cases are calculated as 8.56 W and 9.32 W, respectively, where the PTEs are 80.2% and 81.4%, respectively. Figure 16 shows the results for Φ = 30° and d = 2 cm. The output powers and PTEs are 11.2 W and 12.4 W, and 83.7% and 84.9% for $\theta =\varphi =30\xb0$ and $\theta =\widehat{\gamma}=36.49\xb0$, respectively. The output power and PTE of the 2D WPT system for various physical angles are shown in Figure 17. It is noted in these results that the power transferred to the Rx coil and PTE is improved when the estimated mutual inductance angle is applied instead of the physical angle.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ng, W.M.; Zhang, C.; Lin, D.; Hui, S.Y. Two- and Three-Dimensional Omnidirectional Wireless Power Transfer. IEEE Trans. Power Electron.
**2014**, 29, 4470–4474. [Google Scholar] [CrossRef] [Green Version] - Zhao, J.; Huang, X.; Wang, W. Wireless Power Transfer with Two-Dimensional Resonators. IEEE Trans. Magn.
**2014**, 50, 4002804. [Google Scholar] [CrossRef] - Chabalko, M.J.; Sample, A.P. Three-Dimensional Charging via Multimode Resonant Cavity Enabled Wireless Power Transfer. IEEE Trans. Power Electron.
**2015**, 30, 6163–6173. [Google Scholar] [CrossRef] - Lin, D.; Hui, S.Y.; Zhang, C. Omni-directional wireless power transfer systems using discrete magnetic field vector control. In Proceedings of the 2015 IEEE Energy Conversion Congress and Exposition (ECCE), Montreal, QC, Canada, 20–24 September 2015; pp. 3203–3208. [Google Scholar]
- Lin, D.; Zhang, C.; Hui, S.Y. Mathematical Analysis of Omnidirectional Wireless Power Transfer—Part-I: Two-Dimensional Systems. IEEE Trans. Power Electron.
**2017**, 32, 625–633. [Google Scholar] [CrossRef] - Lin, D.; Zhang, C.; Hui, S.Y. Mathematical Analysis of Omnidirectional Wireless Power Transfer—Part-II: Three-Dimensional Systems. IEEE Trans. Power Electron.
**2017**, 32, 613–624. [Google Scholar] [CrossRef] - Guo, T.; Seol, W.K.; Chung, S.K. Estimation of Mutual Inductance Angle for 2-D Wireless Transfer System. In Proceedings of the 2017 Korea Institute of Power Electronics Fall Conference, Gyeongbuk, Korea, 4–6 July 2017; pp. 49–50. [Google Scholar]
- Sasaki, M.; Yamamoto, M. Exciting voltage control for transfer efficiency maximization for multiple wireless power transfer systems. In Proceedings of the 2017 IEEE Energy Conversion Congress and Exposition (ECCE), Cincinnati, OH, USA, 1–5 October 2017; pp. 5523–5528. [Google Scholar]
- Seol, W.K.; Chung, S.K. Current Vector Control of Wireless Power Transfer System with 2D Transmitting Coils. Electron. Lett.
**2018**, 54, 91–92. [Google Scholar] [CrossRef] - Su, M.; Liu, Z.; Zhu, Q.; Hu, A.P. Study of Maximum Power Delivery to Movable Device in Omnidirectional Wireless Power Transfer System. IEEE Access
**2018**, 6, 76153–76164. [Google Scholar] [CrossRef] - Sergkei, K.; Lombard, P.; Semet, V.; Allard, B.; Moguedet, M.; Cabrera, M. Omni-directional Inductive Wireless Power Transfer with 3D MID inductor. In Proceedings of the 2019 IEEE Wireless Power Transfer Conference (WPTC), London, UK, 18–21 June 2019; pp. 154–157. [Google Scholar]
- Li, J.; Yang, Y.; Yan, H.; Liu, C.; Dong, L.; Wang, G. Quasi-Omnidirectional Wireless Power Transfer for a Sensor System. IEEE Sens. J.
**2020**, 20, 6148–6159. [Google Scholar] [CrossRef] - Chung, S.K. A Phase Tracking System for Three Phase Utility Interface Inverters. IEEE Trans. Power Electron.
**2000**, 15, 431–437. [Google Scholar] [CrossRef] [Green Version] - Boyes, G. Synchro and Resolver Conversion; Analog Devices Inc.: Norwood, MA, USA, 1980. [Google Scholar]
- Gardner, F.M. Phaselock Techniques, 3rd ed.; Wiely: Hoboken, NJ, USA, 2005. [Google Scholar]

**Figure 10.**The short and open circuit test for the mutual inductance measurement between the α-axis and RX coils. (

**a**) Short circuit test and (

**b**) open circuit test.

**Figure 12.**Steady-state response of proposed estimator (d = 0 cm). (

**a**) Φ = 15°, (

**b**) Φ = 30°, (

**c**) Φ = 45°.

**Figure 13.**Steady-state response of proposed estimator (d = 2 cm). (

**a**) Φ = 15°, (

**b**) Φ = 30°, (

**c**) Φ = 45°.

**Figure 14.**Experimental results for proposed estimator. (

**a**) Measured and estimated mutual inductance, (

**b**) estimation error.

**Figure 15.**Experimental results for the 2D WPT system using the proposed estimator (d = 0 cm). (

**a**) Input and output currents for $\theta =\varphi =30\xb0$. (

**b**) Input and output currents for $\theta =\widehat{\gamma}=26.79\xb0$.

**Figure 16.**Experimental results for the 2D WPT system using the proposed estimator (d = 2 cm). (

**a**) Input and output currents for $\theta =\varphi =30\xb0$. (

**b**) Input and output currents for $\theta =\widehat{\gamma}=36.49\xb0$.

**Figure 17.**Output power and power transmission efficiency (PTE) of the 2D WPT system for various physical angles. (

**a**) Output power (

**b**) Power transmission efficiency.

Item | Value | Item | Value |
---|---|---|---|

a | 10 cm | b | 4 cm |

R | 0.42 Ω | Rr | 0.26 Ω |

L | 9 uH | Lr | 4.4 uH |

C | 10 nF | Cr | 20 nF |

f | 530 kHz | R_{L} | 10 Ω |

φ (deg) | d = 0 cm | d = 2 cm | ||||
---|---|---|---|---|---|---|

M_{αr} (uH) | M_{βr} (uH) | γ (deg) | M_{αr} (uH) | M_{βr} (uH) | γ (deg) | |

0 | 1.777 | 0.003 | 0.09 | 1.085 | 0.004 | 0.21 |

15 | 1.602 | 0.327 | 11.54 | 1.179 | 0.517 | 23.68 |

30 | 1.308 | 0.655 | 26.60 | 1.267 | 0.925 | 36.13 |

45 | 1.008 | 1.008 | 45 | 1.407 | 1.407 | 45 |

60 | 0.655 | 1.308 | 63.40 | 0.925 | 1.267 | 53.87 |

75 | 0.327 | 1.602 | 78.46 | 0.517 | 1.179 | 66.32 |

90 | 0.003 | 1.777 | 89.90 | 0.004 | 1.085 | 89.79 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, S.; Lee, J.; Kwon, J.; Chung, S.-K.
An Improved Estimation Method of Mutual Inductance Angle for a Two-Dimensional Wireless Power Transfer System. *Symmetry* **2021**, *13*, 448.
https://doi.org/10.3390/sym13030448

**AMA Style**

Lee S, Lee J, Kwon J, Chung S-K.
An Improved Estimation Method of Mutual Inductance Angle for a Two-Dimensional Wireless Power Transfer System. *Symmetry*. 2021; 13(3):448.
https://doi.org/10.3390/sym13030448

**Chicago/Turabian Style**

Lee, Sangyong, Jeonho Lee, Jongkyum Kwon, and Se-Kyo Chung.
2021. "An Improved Estimation Method of Mutual Inductance Angle for a Two-Dimensional Wireless Power Transfer System" *Symmetry* 13, no. 3: 448.
https://doi.org/10.3390/sym13030448