# Coupling-Independent Capacitive Wireless Power Transfer Using Frequency Bifurcation

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## Abstract

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## 1. Introduction

- optimize the efficiency of the system, e.g., for the wireless charging of high power applications as electric vehicles.
- optimize the power transfer to the load, e.g., charging transcutaneous biomedical implants.

- In a fixed frequency design, an impedance compensation network is added to realize optimum efficiency or power transfer at the operating frequency. The value of its components are independent of the coupling between transmitter and receiver, but for each different value of the coupling, another optimal load value applies. This approach has the advantage of realizing optimum efficiency or power transfer at a fixed operating frequency. However, at fluctuating coupling, the load value has to change to retain the optimal conditions.
- In a frequency agile design, optimum efficiency or power transfer can be achieved for a fixed load value, even at fluctuating coupling. With the same impedance compensation network as in the fixed frequency design, a constant efficiency or power transfer can be realized by changing the operating frequency, depending on the coupling between transmitter and receiver.

- determination of the power and transducer gain for a general CPT system;
- analytical calculation of the bifurcation conditions and frequencies, necessary to determine the optimal operating frequency;
- analytical computation of the optimal solution for achieving a practically coupling-independent CPT link;
- illustration of the similarities to IPT.

## 2. Methodology

- The power gain ${G}_{P}$, defined as the ratio between the power ${P}_{L}$ dissipated by the load and the input power ${P}_{in}$ of the network. This definition corresponds with the efficiency definition often applied in the context of WPT [32]. Maximizing ${G}_{P}$ corresponds with maximizing the efficiency of the system.
- The transducer gain ${G}_{T}$ is defined as the ratio between the power ${P}_{L}$ dissipated by the load and the maximum available power ${P}_{AG}$ of the generator. For a fixed ${P}_{AG}$, maximizing ${G}_{T}$ corresponds to maximizing the amount of power transferred to the load.

## 3. Discussion

## 4. Comparison to IPT

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Power Gain Expressions for a Two-Port as Function of the Admittance Parameters

#### Appendix A.1. Equivalent Circuit

**Figure A1.**Equivalent circuit of a two-port network with load admittance ${Y}_{L}$ and current source ${I}_{S}$ with internal admittance ${Y}_{S}$. The input power ${P}_{in}$ is indicated.

#### Appendix A.2. Power Gain

#### Appendix A.3. Transducer Gain

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**Figure 1.**Equivalent circuit representation of a general capacitive wireless power transfer (CPT) system, indicating the input admittance ${Y}_{in}$, output admittance ${Y}_{out}$, and input power ${P}_{in}$.

**Figure 2.**The main resonant frequency u = 1 and the secondary frequencies ${u}_{\pm}$ as function of the coupling coefficient k for the example configuration. Frequency bifurcation occurs for a coupling factor k higher than the bifurcation coupling ${k}_{B}$. The solid lines indicate the results for the approximate admittance matrix ${\mathit{Y}}_{a}$, the dashed lines for the exact admittance matrix $\mathit{Y}$.

**Figure 3.**The gains ${G}_{P}$ and ${G}_{T}$ at the main resonant frequency u = 1 and the secondary resonant frequencies ${u}_{\pm}$ (for $k>{k}_{B}$), as function of the coupling coefficient k for the example configuration. The solid lines indicate the results for the approximate admittance matrix ${\mathit{Y}}_{a}$, the dashed line for the exact admittance matrix $\mathit{Y}$.

**Figure 4.**Equivalent circuit representation of a general inductive wireless power transfer (IPT) system.

**Figure 5.**The continuous lines show the gains ${G}_{P,IPT}$ and ${G}_{T,IPT}$ at the main resonant frequency u = 1 as function of the coupling coefficient ${k}_{IPT}$ for the example IPT configuration. The dashed lines indicate the values at the secondary resonant frequencies ${u}_{\pm}$, valid for ${k}_{IPT}>{k}_{B}$.

${g}_{S}={G}_{S}/\left({\omega}_{0}{C}_{1}\right)=1/{Q}_{S}$ | ${g}_{L}={G}_{L}/\left({\omega}_{0}{C}_{2}\right)=1/{Q}_{L}$ |

${r}_{L1}={R}_{L1}/\left({\omega}_{0}{L}_{1}\right)=1/{Q}_{L1}$ | ${r}_{L2}={R}_{L2}/\left({\omega}_{0}{L}_{2}\right)=1/{Q}_{L2}$ |

${r}_{C1}={\omega}_{0}{C}_{1}{R}_{C1}=1/{Q}_{C1}$ | ${r}_{C2}={\omega}_{0}{C}_{2}{R}_{C2}=1/{Q}_{C2}$ |

${g}_{in}={G}_{in}/\left({\omega}_{0}{C}_{1}\right)$ | ${g}_{out}={G}_{out}/\left({\omega}_{0}{C}_{2}\right)$ |

${b}_{in}={B}_{in}/\left({\omega}_{0}{C}_{1}\right)$ | ${b}_{out}={B}_{out}/\left({\omega}_{0}{C}_{2}\right)$ |

${r}_{1}={R}_{1}/\left({\omega}_{0}{L}_{1}\right)$ | ${r}_{2}={R}_{2}/\left({\omega}_{0}{L}_{2}\right)$ |

${r}_{S}={R}_{S}/\left({\omega}_{0}{L}_{1}\right)$ | ${r}_{L}={R}_{L}/\left({\omega}_{0}{L}_{2}\right)$ |

${r}_{Tx}={r}_{1}+{r}_{S}$ | ${r}_{Rx}={r}_{2}+{r}_{L}$ |

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**MDPI and ACS Style**

Minnaert, B.; Mastri, F.; Stevens, N.; Costanzo, A.; Mongiardo, M. Coupling-Independent Capacitive Wireless Power Transfer Using Frequency Bifurcation. *Energies* **2018**, *11*, 1912.
https://doi.org/10.3390/en11071912

**AMA Style**

Minnaert B, Mastri F, Stevens N, Costanzo A, Mongiardo M. Coupling-Independent Capacitive Wireless Power Transfer Using Frequency Bifurcation. *Energies*. 2018; 11(7):1912.
https://doi.org/10.3390/en11071912

**Chicago/Turabian Style**

Minnaert, Ben, Franco Mastri, Nobby Stevens, Alessandra Costanzo, and Mauro Mongiardo. 2018. "Coupling-Independent Capacitive Wireless Power Transfer Using Frequency Bifurcation" *Energies* 11, no. 7: 1912.
https://doi.org/10.3390/en11071912