# Power Transfer Efficiency Analysis for Omnidirectional Wireless Power Transfer System Using Three-Phase-Shifted Drive

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

**I**

_{1}is a current phasor and

**I**

_{1}is the RMS value of

**I**

_{1}. Figure 2 shows the lumped element circuit of the three-phase-shifted drive omnidirectional WPT system. According to the mutual inductance coupling theory, its circuit equations can be expressed in Equation (1).

_{i}is the reactance ωL

_{i}− 1/(ωC

_{i}), L

_{i}is the self-inductance, C

_{i}is the compensating capacitance,

**I**

_{i}is the current, and R

_{i}is the resistance in Resonator i (where i = 1, 2, 3, a); R is the load resistance; ω is the angular frequency of AC voltage source; M

_{ja}is the mutual inductance between transmitting coil j and receiving coil a (the mutual inductances of the three transmitting coils are zero theoretically and can be negligible in practice), R

_{sj}is the source resistance, and U

_{j}expressed in Equation (2) is the voltage source (where j = 1, 2, 3).

_{0}is the RMS value of the AC voltage sources; β

_{1}, β

_{2}and β

_{3}are the phase angles of the corresponding excitation source.

_{DC}is the DC voltage source, S

_{j}is the power switch (where j = 1, 2, 3, 4), Z is the equivalent load; in Figure 3b, t

_{d}is the dead-time of the drive pulses for power switch, α is the phase-shifted angle, the range of α is from 0 to 2π (Note: α ∊ [π, 2π] refer to the phase-shifted operation after the reverse of the DC voltage source polarity and the operating phase-shifted angle is π − α). Then, the parameters in Equation (2) can be expressed as:

_{1}= R

_{2}= R

_{3}= R

_{0}, and the resonators should be operated in resonance mode in order to maximize the power transfer capability. Thus, under the resonance mode (X

_{1}= X

_{2}= X

_{3}= X

_{a}= 0), the currents can be expressed into Equation (4) from Equations (1) and (2).

**A**= M

_{1a}

**u**

_{1}+ M

_{2a}

**u**

_{3}+ M

_{3a}

**u**

_{3}, $B=\frac{{\omega}^{2}}{{\omega}^{2}F+{R}_{0}\left({R}_{a}+R\right)}$, $F={M}_{1a}{}^{2}+{M}_{2a}{}^{2}+{M}_{3a}{}^{2}$.

_{1}, α

_{2}and α

_{3}are all set as 0, the following equation should be met:

_{i}is the ratio of mutual inductance, ∆I

_{i}= I

_{i}− I

_{0}, I

_{0}= U

_{0}/R

_{0}is the RMS value of no-load current which is a constant in a certain WPT system due to the negligible variety of the DC voltage U

_{0}and resistance R

_{0}, |∆I| = max{|∆I

_{i}|}, sgn(∆I) = sgn(∆I

_{i}), |M

_{0}| = max{|M

_{ia}|}, sgn(M

_{0}) = sgn(M

_{ia}) (where i = 1, 2, 3, a), |·| refers to calculate the absolute value of the parameter, max{·} refers to obtain the maximum value of the parameters set, sgn(·) refers to get the sign of the variable. Then, the defined system parameter M

_{B}can be identified as

_{i}(i = 1, 2 and 3) from Equation (5). So, all the mutual inductance signs can be artificially adjusted to the same by adding the phase-shifted angle of corresponding full-bridge inverter as 2π. Then, the maximum efficiency transfer, the maximum power transfer, and the genuine omnidirectional power transfer can be realized if the three currents are set as Equation (7), and the values of the angle θ and φ can be set as Equation (8).

_{A}= R

_{a}+ R is the total impedance of load resonant tank.

- (1)
- zero phase-shifted angle depends on ∆I, e.g., if ∆I = ∆I
_{2}, α_{2}= 0. - (2)
- the minimum source polarity inverse principle, e.g., if $\mathrm{sgn}\left({m}_{1}\right)=\mathrm{sgn}\left({m}_{2}\right)\ne \mathrm{sgn}\left({m}_{3}\right)$ , α
_{3}= α_{3}+ 2π. - (3)
- define the parameter α
_{dd}as:$${\alpha}_{add}=\{\begin{array}{ll}{\left[\begin{array}{ccc}2\mathsf{\pi}& 0& 0\end{array}\right]}^{{\rm T}},\hfill & \mathrm{sgn}\left({m}_{1}\right)\ne \mathrm{sgn}\left({m}_{2}\right)=\mathrm{sgn}\left({m}_{3}\right)\hfill \\ {\left[\begin{array}{ccc}0& 2\mathsf{\pi}& 0\end{array}\right]}^{{\rm T}},\hfill & \mathrm{sgn}\left({m}_{2}\right)\ne \mathrm{sgn}\left({m}_{3}\right)=\mathrm{sgn}\left({m}_{1}\right)\hfill \\ {\left[\begin{array}{ccc}0& 0& 2\mathsf{\pi}\end{array}\right]}^{{\rm T}},\hfill & \mathrm{sgn}\left({m}_{3}\right)\ne \mathrm{sgn}\left({m}_{1}\right)=\mathrm{sgn}\left({m}_{2}\right)\hfill \\ {\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{{\rm T}},\hfill & \mathrm{otherwise}\hfill \end{array}.$$

## 3. Mechanism of the Three-Phase-Shifted Drive Omnidirectional WPT System

## 4. Computer-Aided Analysis

#### 4.1. On the Tangent Plane of the Green Spherical Surface

#### 4.2. On the Green Spherical Surface with Different Postures

#### 4.3. In the Spherical Shell with Different Locations and Postures

#### 4.4. On the Green Spherical Surface with Different Locations and Postures

## 5. Practical Verification

_{1}= 15.6 nF, C

_{2}= 15.5 nF, C

_{3}= 15.5 nF and C

_{a}= 15.7 nF, respectively. The distance between the centers of the transmitting coil and the receiving coil is d = 0.4 m.

#### 5.1. On the Tangent Plane of the Green Spherical Surface

#### 5.2. On the Green Spherical Surface with Different Postures

#### 5.3. In the Spherical Shell with Different Locations and Postures

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**A typical topology of the excitation power source: (

**a**) the topology of inverter; (

**b**) the drive pulses of power switches.

**Figure 7.**The coordinate system and power transmission space of omnidirectional WPT system: (

**a**) center relationship between the transmitting coils and the receiving coil and (

**b**) space division.

**Figure 8.**The power transfer efficiency versus the receiving coil distributed on the different tangent plane of the green spherical surface (radius is 0.4 m).

**Figure 9.**The power transfer efficiency versus the receiving coil with different locations and postures on the green spherical surface (radius is 0.4 m).

**Figure 10.**The power transfer efficiency versus the receiving coil with different locations and postures in the spherical shell.

**Figure 15.**Practical measurements in the conditions Θ = 90°, Φ = 45°, θ′ = 0°, φ′ = 90°, ψ′ = 45° and d = 0.4 m. (

**a**) Locations and postures of system coils; (

**b**) phase-shifted drive pulses; (

**c**) input/output voltage and current waveforms.

**Figure 16.**The power transfer efficiency versus the receiving coil rotating along the dotted line at the location (0.28 m, 0.28 m and 0 m).

**Figure 17.**Practical measurements in the conditions Θ = 90°, Φ = 45°, θ′ = 0°, φ′ = 90°, ψ′ = 0° and d = 0.4 m. (

**a**) Locations and postures of system coils; (

**b**) phase-shifted drive pulses; (

**c**) input/output voltage and current waveforms.

**Figure 18.**The power transfer efficiencies versus the receiving coil moving and rotating in the spherical shell.

**Figure 19.**Practical measurements in the conditions Θ = 90°, Φ = 45°, θ′ = 90°, φ′ = 0°, ψ′ = 45° and d = 0.4 m. (

**a**) Locations and postures of system coils; (

**b**) phase-shifted drive pulses; (

**c**) input/output voltage and current waveforms.

Parameters | Symbol | Value |
---|---|---|

Inner radius | r_{i} | 100 mm |

Distance of turns | d_{t} | 10 mm |

Number of turns per layer | - | 10 |

Number of layers | - | 1 |

Diameter of the coils | d_{c} | 40 cm |

NO. | Θ (°) | Φ (°) |
---|---|---|

1 | −169 | −159 |

2 | 88 | 66 |

3 | 0 | −165 |

4 | −7 | −154 |

5 | 146 | 8 |

6 | 40 | −145 |

7 | 42 | 115 |

8 | 129 | 114 |

9 | 110 | 80 |

10 | 28 | −126 |

11 | −114 | 57 |

12 | −94 | 7 |

13 | 139 | 170 |

14 | −170 | 54 |

15 | −4 | 108 |

16 | −120 | −17 |

17 | 172 | −24 |

18 | 77 | 117 |

19 | 0 | −150 |

20 | −10 | −132 |

Location | Θ (°) | Φ (°) | NO. | θ′ (°) | φ′ (°) | ψ′ (°) | Location | Θ (°) | Φ (°) | NO. | θ′ (°) | φ′ (°) | ψ′ (°) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | 88 | 66 | 1 | 123 | 15 | 158 | D | 110 | 80 | 1 | 22 | 136 | 176 |

2 | 120 | 133 | 52 | 2 | 155 | 176 | 10 | ||||||

3 | −88 | −85 | −7 | 3 | 71 | −180 | −7 | ||||||

4 | 41 | −65 | 50 | 4 | 30 | 132 | 108 | ||||||

5 | 30 | −137 | 16 | 5 | 114 | 41 | −98 | ||||||

B | −7 | −154 | 1 | 123 | 15 | 158 | E | −120 | −17 | 1 | 22 | 136 | 176 |

2 | 120 | 133 | 52 | 2 | 155 | 176 | 10 | ||||||

3 | −88 | −85 | −7 | 3 | 71 | −180 | −7 | ||||||

4 | 41 | −65 | 50 | 4 | 30 | 132 | 108 | ||||||

5 | 30 | −137 | 16 | 5 | 114 | 41 | −98 | ||||||

C | 146 | 8 | 1 | 123 | 15 | 158 | F | 0 | −150 | 1 | 22 | 136 | 176 |

2 | 120 | 133 | 52 | 2 | 155 | 176 | 10 | ||||||

3 | −88 | −85 | −7 | 3 | 71 | −180 | −7 | ||||||

4 | 41 | −65 | 50 | 4 | 30 | 132 | 108 | ||||||

5 | 30 | −137 | 16 | 5 | 114 | 41 | −98 |

NO. | d | Θ (°) | Φ (°) | θ′ (°) | φ′ (°) | ψ′ (°) | NO. | d | Θ (°) | Φ (°) | θ′ (°) | φ′ (°) | ψ′ (°) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.49 | −17 | −142 | 60 | −137 | −172 | 11 | 0.36 | −36 | −28 | −47 | 174 | 72 |

2 | 0.41 | −24 | −46 | −116 | 32 | −27 | 12 | 0.39 | 10 | 17 | −14 | 83 | 50 |

3 | 0.37 | 117 | −109 | −134 | −99 | −67 | 13 | 0.41 | −30 | 159 | 173 | −56 | −168 |

4 | 0.32 | −150 | −4 | 180 | −42 | −122 | 14 | 0.39 | 56 | −30 | −124 | 30 | −155 |

5 | 0.42 | −132 | −58 | −118 | 30 | −116 | 15 | 0.48 | 46 | 174 | 128 | −141 | −65 |

6 | 0.46 | −118 | 163 | −168 | −89 | −28 | 16 | 0.40 | −75 | −71 | 52 | 146 | 11 |

7 | 0.38 | −39 | 151 | 22 | −75 | −146 | 17 | 0.49 | −25 | 72 | −45 | 137 | 56 |

8 | 0.32 | 119 | −161 | 137 | 42 | 35 | 18 | 0.43 | −174 | 60 | −111 | 114 | −33 |

9 | 0.35 | 109 | 86 | 61 | −84 | −10 | 19 | 0.49 | 174 | 14 | −26 | −86 | 115 |

10 | 0.33 | −158 | −83 | −111 | 117 | 71 | 20 | 0.35 | −120 | 71 | −6 | 34 | 79 |

Parameters | Symbol | Value |
---|---|---|

Self-inductance of coil i | L_{1} | 39.8 μH |

L_{2} | 39.8 μH | |

L_{3} | 39.9 μH | |

L_{a} | 39.3 μH | |

Self-resonance frequency (SRF) of resonator i | f_{1} | 201.98 kHz |

f_{2} | 202.63 kHz | |

f_{3} | 202.38 kHz | |

f_{a} | 202.62 kHz | |

Resistance of coil i | R_{1} | 0.32 Ω |

R_{2} | 0.33 Ω | |

R_{3} | 0.35 Ω | |

R_{a} | 0.22 Ω | |

Mutual inductance between coil i and coil j | M_{12} | 0.18 μH |

M_{23} | 0.25 μH | |

M_{13} | 0.20 μH |

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## Share and Cite

**MDPI and ACS Style**

Ye, Z.; Sun, Y.; Liu, X.; Wang, P.; Tang, C.; Tian, H. Power Transfer Efficiency Analysis for Omnidirectional Wireless Power Transfer System Using Three-Phase-Shifted Drive. *Energies* **2018**, *11*, 2159.
https://doi.org/10.3390/en11082159

**AMA Style**

Ye Z, Sun Y, Liu X, Wang P, Tang C, Tian H. Power Transfer Efficiency Analysis for Omnidirectional Wireless Power Transfer System Using Three-Phase-Shifted Drive. *Energies*. 2018; 11(8):2159.
https://doi.org/10.3390/en11082159

**Chicago/Turabian Style**

Ye, Zhaohong, Yue Sun, Xiufang Liu, Peiyue Wang, Chunsen Tang, and Hailin Tian. 2018. "Power Transfer Efficiency Analysis for Omnidirectional Wireless Power Transfer System Using Three-Phase-Shifted Drive" *Energies* 11, no. 8: 2159.
https://doi.org/10.3390/en11082159