# Scaling Rules at Constant Frequency for Resonant Inductive Power Transfer Systems for Electric Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Circuit Model of Inductive Power Transfer System

## 3. Derivation of Scaling Rules

- The working frequency ${\omega}_{0}$ and the current density remain constant in the scaling process. This assumption allows for considering the same exploitation of the conductors material. Moreover, keeping the frequency fixed allows for operating with similar characteristics of the power electronics drivers and control.
- The number of turns of transmitter and receiver coils does not change in the scaling process. This assumption guarantees that the re-sized structure changes only in the dimensions while the constructive characteristics are left unchanged.

#### 3.1. Scaling of Self and Mutual Inductance

#### 3.2. Scaling of Currents, Voltages and Power

#### 3.3. Scaling of the Compensation Capacitances

## 4. Scaling-Rules’ Limits of Validity

#### 4.1. Coil Resistances

#### 4.2. Capacitors’ ESR

## 5. Power Losses and Efficiency

## 6. Experimental Validation

#### 6.1. Power and Losses

#### 6.2. Compensation Capacitors’ Technology

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Scaling of Inductances in Complex Environment

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

Transmitter radius | $400\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Transmitter wire diameter | $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Receiver radius | $350\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Receiver wire diameter | $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Coils distance | $200\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Ferrite plate inner rad. | $100\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Ferrite plate outer rad. | $450\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Ferrite plate thickness | $20\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Aluminium plate radius | $500\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Aluminium plate thickness | $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Aluminium-Ferrite plates dist. | $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Ferrite magnetic permeability | 2000 |

Ferrite conductivity | $0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{S}/\mathrm{m}$ |

Aluminium conductivity | $34\phantom{\rule{3.33333pt}{0ex}}\mathrm{MS}/\mathrm{m}$ |

**Figure A4.**Mutual inductance and transmitter self-inductance versus the scaling factor for ferrite thickness proportional to ${\gamma}^{2}$.

**Figure A5.**Mean magnetic flux density in the ferrite versus the scaling factor for ferrite thickness proportional to ${\gamma}^{2}$.

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**Figure 1.**Circuit model of two coupled inductors with the sinusoidal voltage source of amplitude ${V}_{1}$ and the equivalent load ${R}_{\mathrm{L}}$.

**Figure 3.**Tested inductive power transfer (IPT) systems. Starting system (

**a**) and down-scaled one (

**b**).

**Figure 4.**Coil resistances, equivalent load and total impedance of the starting system versus scaling factor.

**Figure 6.**Comparison of theoretical and actual scaling ratios for self-inductances, mutual inductance and coils currents obtained downstream of the down-scaling.

**Figure 7.**Adopted compensation capacitors. High-voltage film capacitor in yellow. Ceramic capacitor in orange.

Compensation Topology | Capacitor ${\mathit{C}}_{1}$ |
---|---|

series-series | $\frac{1}{{\omega}_{0}^{2}{L}_{1}}$ |

series-parallel | $\frac{1}{{\omega}_{0}^{2}\left(\right)open="("\; close=")">{L}_{1}-\frac{{M}^{2}}{{L}_{2}}}$ |

parallel-series | $\frac{{L}_{1}}{{\left(\right)}^{\frac{{\omega}_{0}^{2}{M}^{2}}{{R}_{\mathrm{L}}}}2}$ |

parallel-parallel | $\frac{{L}_{1}-\frac{{M}^{2}}{{L}_{2}}}{{\left(\right)}^{\frac{{M}^{2}{R}_{L}}{{L}_{2}^{2}}}2}$ |

Parameter | Dimension |
---|---|

Coils inner width | $50\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ |

Coils inner length | $25\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ |

Coils distance | $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ |

Number of turns | 9 |

Litz wire diameter | $4\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ |

Single ferrite bar | $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}\times 2.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}\times 2.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ |

Ferrite core | $20\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}\times 20\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}\times 5\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ |

Parameter | Original System | Down-Scaled System | Reference Rule | Expected Scaling Ratio | Measured Scaling Ratio |
---|---|---|---|---|---|

${L}_{1}$ | $90.8\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{H}$ | $43.2\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{H}$ | Equation (16) | 0.5 | 0.48 |

${L}_{2}$ | $93.4\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{H}$ | $44.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{H}$ | Equation (16) | 0.5 | 0.48 |

M | $20.9\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{H}$ | $11.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{H}$ | Equation (16) | 0.5 | 0.53 |

${R}_{1,2}$ | $0.13\phantom{\rule{3.33333pt}{0ex}}\Omega $ | $0.25\phantom{\rule{3.33333pt}{0ex}}\Omega $ | Equation (23) | 0.5 | 0.52 |

Parameter | Original System | Down-Scaled System | Reference Rule | Expected Scaling Ratio | Measured Scaling Ratio |
---|---|---|---|---|---|

${V}_{\mathrm{DClink}}$ | $233.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ | 28.7$\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ | Equation (20) | 0.125 | 0.123 |

${I}_{1pk}$ | $7.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ | $1.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ | Equation (17) | 0.25 | 0.27 |

${I}_{2pk}$ | $19.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ | $4.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ | Equation (17) | 0.25 | 0.23 |

Parameter | Original System | Down-Scaled System | Reference Rule | Expected Scaling Ratio | Measured Scaling Ratio |
---|---|---|---|---|---|

${P}_{2}$ | $942.11\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}$ | 29.12$\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}$ | Equation (19) | 0.0312 | 0.0309 |

${P}_{\mathrm{J}}$ | $29.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}$ | $2.94\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}$ | Equation (24) | 0.125 | 0.101 |

${P}_{\mathrm{Fe}}$ | $209.49\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}$ | $4.86\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}$ | Equation (31) | 0.0191 | 0.0232 |

Efficiency | Original | Down-Scaled |
---|---|---|

$\eta $ | $0.798$ | $0.789$ |

**Table 7.**Values of capacitance and equivalent series resistance (ESR) of two samples of the adopted compensation capacitors.

IPT System | Technology | Capacitance | ESR |
---|---|---|---|

Original | Film | $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{F}$ | $42\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\Omega $ |

Down-scaled | Ceramic | $1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{F}$ | $34\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\Omega $ |

© 2018 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**

Cirimele, V.; Freschi, F.; Guglielmi, P.
Scaling Rules at Constant Frequency for Resonant Inductive Power Transfer Systems for Electric Vehicles. *Energies* **2018**, *11*, 1754.
https://doi.org/10.3390/en11071754

**AMA Style**

Cirimele V, Freschi F, Guglielmi P.
Scaling Rules at Constant Frequency for Resonant Inductive Power Transfer Systems for Electric Vehicles. *Energies*. 2018; 11(7):1754.
https://doi.org/10.3390/en11071754

**Chicago/Turabian Style**

Cirimele, Vincenzo, Fabio Freschi, and Paolo Guglielmi.
2018. "Scaling Rules at Constant Frequency for Resonant Inductive Power Transfer Systems for Electric Vehicles" *Energies* 11, no. 7: 1754.
https://doi.org/10.3390/en11071754