# Optimal Solutions for Underwater Capacitive Power Transfer

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- 1.
- It extends the analysis proposed in [14] to include three possible solutions, namely, the one that maximizes the efficiency, the one that maximizes the transferred power, and the one that realizes power matching.
- 2.
- It considers the dielectric losses of seawater, unlike the previous analysis for lossless medium in [17].
- 3.
- It investigates the under seawater CPT system behavior at the 0.3 to 1 MHz frequency range and separation distance up to 300 mm.

## 2. System Analysis

#### 2.1. Passive Linear Reciprocal System

- (a)
- Maximum efficiency: determine the value of the source and the load admittance that achieve maximum efficiency.
- (b)
- Maximum power: determine the value of the source and the load admittance that achieve maximum power transfer to the load.
- (c)
- Conjugate-image: determine the values of the admittance that realize the principle of power matching.

#### 2.2. Maximum Efficiency Solution

#### 2.3. Maximum Power Solution

#### 2.4. Conjugate-Image Solution

## 3. Calculated and Measured Results

#### 3.1. Calculated Results

#### 3.2. Measurement Results

## 4. Discussion

#### 4.1. The Parameters $\psi $ and $\chi $

#### 4.2. The Frequency Effect

#### 4.3. The Distance Effect

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

WPT | wireless power transfer |

IPT | inductive power transfer |

CPT | capacitive power transfer |

EMI | electromagnetic interference |

## References

- Ahmad, A.; Alam, M.S.; Chabaan, R. A Comprehensive Review of Wireless Charging Technologies for Electric Vehicles. IEEE Trans. Transp. Electrif.
**2018**, 4, 38–63. [Google Scholar] [CrossRef] - Covic, G.A.; Boys, J.T. Modern Trends in Inductive Power Transfer for Transportation Applications. IEEE J. Emerg. Sel. Top. Power Electron.
**2013**, 1, 28–41. [Google Scholar] [CrossRef] - Lu, F.; Zhang, H.; Mi, C. A Review on the Recent Development of Capacitive Wireless Power Transfer Technology. Energies
**2017**, 10, 1752. [Google Scholar] [CrossRef] [Green Version] - Mei, Y.; Wu, J.; He, X. Common Mode Noise Analysis for Inductive Power Transfer System Based on Distributed Stray Capacitance Model. IEEE Trans. Power Electron.
**2021**, 37, 1132–1145. [Google Scholar] [CrossRef] - Zhang, H.; Lu, F.; Hofmann, H.; Liu, W.; Mi, C.C. A Four-Plate Compact Capacitive Coupler Design and LCL-Compensated Topology for Capacitive Power Transfer in Electric Vehicle Charging Application. IEEE Trans. Power Electron.
**2016**, 31, 8541–8551. [Google Scholar] [CrossRef] - Lu, F.; Zhang, H.; Mi, C. A Two-Plate Capacitive Wireless Power Transfer System for Electric Vehicle Charging Applications. IEEE Trans. Power Electron.
**2018**, 33, 964–969. [Google Scholar] [CrossRef] - Lu, F.; Zhang, H.; Hofmann, H.; Mi, C.C. A Double-Sided LC-Compensation Circuit for Loosely Coupled Capacitive Power Transfer. IEEE Trans. Power Electron.
**2018**, 33, 1633–1643. [Google Scholar] [CrossRef] - Urano, M.; Ata, K.; Takahashi, A. Study on underwater wireless power transfer via electric coupling with a submerged electrode. In Proceedings of the IMFEDK 2017—2017 International Meeting for Future of Electron Devices, Kansai, Kyoto, Japan, 29–30 June 2017; pp. 36–37. [Google Scholar] [CrossRef]
- Tamura, M.; Naka, Y.; Murai, K.; Nakata, T. Design of a Capacitive Wireless Power Transfer System for Operation in Fresh Water. IEEE Trans. Microw. Theory Tech.
**2018**, 66, 5873–5884. [Google Scholar] [CrossRef] - Tamura, M.; Naka, Y.; Murai, K. Design of capacitive coupler in underwater wireless power transfer focusing on kQ product. IEICE Trans. Electron.
**2018**, E101C, 759–766. [Google Scholar] [CrossRef] - Tamura, M.; Murai, K.; Matsumoto, M. Design of Conductive Coupler for Underwater Wireless Power and Data Transfer. IEEE Trans. Microw. Theory Tech.
**2021**, 69, 1161–1175. [Google Scholar] [CrossRef] - Zhang, H.; Lu, F. Insulated Coupler Structure Design for the Long-Distance Freshwater Capacitive Power Transfer. IEEE Trans. Ind. Inform.
**2020**, 16, 5191–5201. [Google Scholar] [CrossRef] - Mahdi, H.; Hoff, B.; Ostrem, T. Evaluation of Capacitive Power Transfer for Small Vessels Charging Applications. IEEE Int. Symp. Ind. Electron.
**2020**, 2020, 1605–1610. [Google Scholar] [CrossRef] - Mahdi, H.; Hoff, B.; Østrem, T. Maximum Available Power of Undersea Capacitive Coupling in a Wireless Power Transfer System. In Proceedings of the 2021 IEEE Wireless Power Transfer Conference (WPTC), San Diego, CA, USA, 1–4 June 2021; pp. 1–4. [Google Scholar] [CrossRef]
- Lecluyse, C.; Minnaert, B.; Kleemann, M. A Review of the Current State of Technology of Capacitive Wireless Power Transfer. Energies
**2021**, 14, 5862. [Google Scholar] [CrossRef] - Ohira, T. Extended k-Q product formulas for capacitive- and inductive-coupling wireless power transfer schemes. IEICE Electron. Express
**2014**, 11, 20140147. [Google Scholar] [CrossRef] [Green Version] - Dionigi, M.; Mongiardo, M.; Monti, G.; Perfetti, R. Modelling of wireless power transfer links based on capacitive coupling. Int. J. Numer. Model. Electron. Netw. Devices Fields
**2017**, 30, e2187. [Google Scholar] [CrossRef] - Roberts, S. Conjugate-Image Impedances. Proc. IRE
**1946**, 34, 198p–204p. [Google Scholar] [CrossRef] - Sverdrup, K.A.; Duxbury, A.B.; Duxbury, A. An Introduction to the World’s Oceans; McGraw-Hill Higher Education: New York, NY, USA, 2005; pp. 149–150. [Google Scholar]
- Griffiths, D. Introduction to Electrodynamics, 3rd ed.; Pearson: London, UK, 2008. [Google Scholar]
- Jackson, J. Classical Electrodynamics, 3rd ed.; John Wiley & Sons: Hoboken, NJ, USA, 1999. [Google Scholar]

**Figure 2.**A general representation of CPT system: (

**a**) two-port network connected to source and load. (

**b**) The transmitter and receiver are equivalent circuits.

**Figure 3.**The efficiency of CPT system: (

**a**) Maximum efficiency solution. (

**b**) Maximum power solution. (

**c**) Conjugate-image solution.

**Figure 4.**The measurement setup [14].

**Figure 5.**The measured $\psi $ and $\chi $ versus the separation distance and the frequency: (

**a**) The $\psi $ coefficient. (

**b**) The $\chi $ coefficient.

**Figure 6.**The efficiency and normalized power versus the separation distance and the frequency: (a) Maximum efficiency solution. (b) Maximum power solution. (c) Conjugate-image solution.

**Figure 7.**The efficiency and normalized power versus $\chi $ for lossy and lossless solutions: (

**a**) Maximum efficiency solution. (

**b**) Maximum power solution.

Maximum Efficiency | Maximum Power | Conjugate-Image | |
---|---|---|---|

${g}_{\mathrm{s}}$ | 0 | 0 | ${g}_{1}{\theta}_{\mathrm{G}}$ |

${b}_{\mathrm{s}}$ | −${b}_{1}$ | −${b}_{1}$ | ${g}_{1}{\theta}_{\mathrm{B}}$ |

${g}_{\mathrm{L}}$ | ${g}_{2}{\theta}_{\mathrm{G}}$ | ${g}_{2}\left({\theta}_{\mathrm{G}}^{2}+{\theta}_{\mathrm{B}}^{2}\right)$ | ${g}_{2}{\theta}_{\mathrm{G}}$ |

${b}_{\mathrm{L}}$ | ${g}_{2}{\theta}_{\mathrm{B}}$−${b}_{2}$ | $2{g}_{2}{\theta}_{\mathrm{B}}$−${b}_{2}$ | ${g}_{2}{\theta}_{\mathrm{B}}$−${b}_{2}$ |

Lossy System | |||

${P}_{L}$ | $4{P}_{\mathrm{s}},\mathrm{max}\frac{\left(\phantom{\rule{0.166667em}{0ex}}{\psi}^{2}+{\chi}^{2}\phantom{\rule{0.166667em}{0ex}}\right)\phantom{\rule{0.166667em}{0ex}}{\theta}_{\mathrm{G}}}{{\left[\phantom{\rule{0.166667em}{0ex}}{\theta}_{\mathrm{G}}\left(\phantom{\rule{0.166667em}{0ex}}1+{\theta}_{\mathrm{G}}\phantom{\rule{0.166667em}{0ex}}\right)+{\theta}_{\mathrm{B}}^{2}\phantom{\rule{0.166667em}{0ex}}\right]}^{2}+{\theta}_{\mathrm{B}}^{2}}$ | ${P}_{\mathrm{s}},\mathrm{max}\frac{{\psi}^{2}+{\chi}^{2}}{{\theta}_{\mathrm{G}}^{2}+{\theta}_{\mathrm{B}}^{2}}$ | ${P}_{\mathrm{s}},\mathrm{max}\frac{{\psi}^{2}+{\chi}^{2}}{{\theta}_{\mathrm{G}}\phantom{\rule{0.166667em}{0ex}}\left[\phantom{\rule{0.166667em}{0ex}}{\left(\phantom{\rule{0.166667em}{0ex}}1+{\theta}_{\mathrm{G}}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}+{\theta}_{\mathrm{B}}^{2}\phantom{\rule{0.166667em}{0ex}}\right]}$ |

$\eta $ | $\frac{{\psi}^{2}+{\chi}^{2}}{{\left(\phantom{\rule{0.166667em}{0ex}}1+{\theta}_{\mathrm{G}}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}+{\theta}_{\mathrm{B}}^{2}}$ | $\frac{1}{2}\frac{{\psi}^{2}+{\chi}^{2}}{\left(\phantom{\rule{0.166667em}{0ex}}1+{\theta}_{\mathrm{G}}^{2}+{\theta}_{\mathrm{B}}^{2}\phantom{\rule{0.166667em}{0ex}}\right)}$ | $\frac{1}{2}\frac{{\psi}^{2}+{\chi}^{2}}{{\left(\phantom{\rule{0.166667em}{0ex}}1+{\theta}_{\mathrm{G}}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}+{\theta}_{\mathrm{B}}^{2}}$ |

Lossless System | |||

${P}_{L}$ | $4{P}_{\mathrm{s}},\mathrm{max}\frac{{\chi}^{2}}{\sqrt{1+{\chi}^{2}}{\left(\phantom{\rule{0.166667em}{0ex}}1+\sqrt{1+{\chi}^{2}}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}}$ | ${P}_{\mathrm{s}},\mathrm{max}\frac{{\chi}^{2}}{1+{\chi}^{2}}$ | ${P}_{\mathrm{s}},\mathrm{max}\frac{{\chi}^{2}}{\sqrt{1+{\chi}^{2}}{\left(\phantom{\rule{0.166667em}{0ex}}1+\sqrt{1+{\chi}^{2}}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}}$ |

$\eta $ | $\frac{{\chi}^{2}}{\phantom{\rule{0.166667em}{0ex}}{\left(\phantom{\rule{0.166667em}{0ex}}1+\sqrt{1+\chi}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}}$ | $\frac{1}{2}\frac{{\chi}^{2}}{\phantom{\rule{0.166667em}{0ex}}\left(\phantom{\rule{0.166667em}{0ex}}2+{\chi}^{2}\phantom{\rule{0.166667em}{0ex}}\right)}$ | $\frac{1}{2}\frac{{\chi}^{2}}{\phantom{\rule{0.166667em}{0ex}}{\left(\phantom{\rule{0.166667em}{0ex}}1+\sqrt{1+\chi}\phantom{\rule{0.166667em}{0ex}}\right)}^{2}}$ |

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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mahdi, H.; Hoff, B.; Østrem, T.
Optimal Solutions for Underwater Capacitive Power Transfer. *Sensors* **2021**, *21*, 8233.
https://doi.org/10.3390/s21248233

**AMA Style**

Mahdi H, Hoff B, Østrem T.
Optimal Solutions for Underwater Capacitive Power Transfer. *Sensors*. 2021; 21(24):8233.
https://doi.org/10.3390/s21248233

**Chicago/Turabian Style**

Mahdi, Hussein, Bjarte Hoff, and Trond Østrem.
2021. "Optimal Solutions for Underwater Capacitive Power Transfer" *Sensors* 21, no. 24: 8233.
https://doi.org/10.3390/s21248233