# Optimal Efficiency Tracking Control Scheme Based on Power Stabilization for a Wireless Power Transfer System with Multiple Receivers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Structure and Theoretical Analysis

#### 2.1. System Structure

_{t}and C

_{t}are the self-inductance and the resonant capacitance of the transmitter. L

_{r}

_{1}to L

_{rn}, C

_{r}

_{1}to C

_{rn}, and R

_{o}

_{1}to R

_{on}are the inductances, resonant capacitance, and load resistance of the multiple receivers, respectively. M

_{11}to M

_{1n}are the mutual inductance between transmitter and the multiple receivers. η

_{Amplifier}, η

_{Linked}, and η

_{dc-regulator}denote the efficiency of the power amplifier, transformation, and DC regulator, respectively.

#### 2.2. Optimal Load Condition for Reflected Impedance Match

#### 2.2.1. System with Single Receiver

_{L}

_{1}denotes the equivalent AC load of the system. The impedances Z

_{in}and Z

_{out}viewed from the transmitter toward the load and viewed from the receiver toward the source are expressed as:

_{1}in terms of I

_{t}:

_{t}and receiver q

_{r}

_{1}are expressed separately:

_{L}

_{1}for a maximum η is expressed as:

#### 2.2.2. System with Multiple Receivers

_{j}(j = 1, 2, ⋯, n) is the mutual inductance between the transmitter and receivers; therefore, the coupling coefficient can be expressed as:

_{R}

_{1}to Z

_{Rn}are the reflected impedance looking from the transmitter to the receivers, with the expression of the loss coefficients in a multiple-receiver system given as:

_{Rj}(j = 1, 2, …, n) can be derived using the expression of loss coefficients from Equation (20) and coupling coefficients from Equation (19):

_{t}flowing through the impedance of each receiver, the power division ratio can be derived using the ratio of impedance as follows:

_{i}can be derived as:

#### 2.3. Analysis of Load Transformation

_{L}when the coupling coefficient of the system is fixed. From previous research [13], we found that the DC-DC converters can be used to transform equivalent load resistance. In Figure 4, the equivalent resistance R

_{Li}looking from the rectifier was adjusted by the duty cycle D of the converter. The three equations in Figure 4 represent the process of impedance. For the Buck converter, assuming that the loss can be ignored, it was found that:

_{o}and U

_{in}are the input and output voltages. The equivalent input resistance for the converter looking forward rectifier can be expressed as:

_{oi}is regulated to the required R

_{Li}calculated by Equation (36) and then the power of the receiver can be fixed. The characteristics of other DC-DC converters used to regulate the equivalent input resistance are listed in Table 1.

_{o}, when the equivalent input resistance is the optimal load.

_{Li}depends not only on the range of R

_{o}, but also the available duty cycle D, affected by the accuracy of the controller and MOSFET driver. From the viewpoint of previous research, the boost and single-ended primary inductance converter (SEPIC) are suitable and easy to control, as their input currents are continuous.

## 3. Analysis of the Power Division and Efficiency Evaluation Method

_{in}, the power ratio between any two receivers is expressed as:

_{%_opt}(≤η

_{_opt}). It is obvious that there will be an optimal total reflected impedance that can reach the maximum system efficiency in a constant reflected impedance ratio condition.

_{oi}and η are the functions of D

_{i}and R

_{oi}. It is obvious that the system efficiency is related to the transmitter input current and the output current of the receiver load, respectively. In order to achieve optimal efficiency, the transmitter and the receivers should work cooperatively. Assuming that the load R

_{oi}is stable in the meantime, the calculation of the load power can be realized by measuring the current I

_{oi}flowing into the load. Based on the constant power charging of the receivers, the optimal efficiency η

_{_opt}is tracked by measuring the minimum P

_{dc}at the transmitter. In this paper, the perturbation and observation (P&O) algorithm was introduced to design the control scheme of the system. The method is widely used in applications of photovoltaic power generation to track the maximum power points, and its correctness and practicability have already been effectively verified [25]. A detailed analysis on the proposed control method is illustrated in the following.

## 4. Analysis of the Proposed Control Scheme

#### 4.1. Structure of Proposed Control Scheme

#### 4.2. Operating Principle Analysis

_{dc}of the power amplifier, the optimal efficiency of the system tracking under the stable output power division can be obtained. However, the premise behind the system working properly is that the system must meet the power requirement of all loads when the system is initialized. The detail description of Figure 7 is as follows:

- In the initialization phase, the initial output power of the transmitter is determined by the number of loads and power requirements. The duty cycle D is adjusted by the PI controlled DC-DC converter so that the current I
_{o}at the receiving end reaches the required current. - The input power P
_{dc}of the transmitter is recorded after initialization and the corresponding change of the input power is calculated in real time by increasing or decreasing the DC input current I_{dc}of the power amplifier. When the change of I_{dc}is ΔI_{dc}, the new value of Idc is expressed as I_{dc}_{1}= I_{dc}_{0}− ΔI_{dc}or I_{dc}_{1}= I_{dc}_{0}+ ΔI_{dc}. The new value of the input power is expressed as P_{dc}_{1}= P_{dc}_{0}+ ΔP_{dc}or P_{dc}_{1}= P_{dc}_{0}+ ΔP_{dc}. As the transmitter’s power varies, the output current I_{o}of the receiver will also change due to the variation in power received by the load. - After storing the new current I
_{dc}_{1}and input power P_{dc}_{1}of the transmitter, the receiver adjusts D again to restore the output current I_{o}to the set value. - Finally, the new input power P
_{dc1}obtained by adjusting the I_{dc}is compared with the P_{dc}. If the P_{dc}_{1}is less than the P_{dc}, the current I_{dc}should continue to be adjusted in the forward direction, and steps 2 and 3 repeated until the P_{dc}_{1}increases. Otherwise, if P_{dc}_{1}is greater than P_{dc}, the adjustment direction of I_{dc}should be reversed and steps 2 and 3 repeated until the P_{dc}_{1}no longer decreases. At this point, the maximum efficiency of the system is tracked.

## 5. Experimental Verification

#### 5.1. Experimental Setup

_{t}of 2.13 μH, and an equivalent series resistance R

_{t}of 0.53 Ω. The receiving coil had a size of 63 mm × 45 mm, an inductance L

_{r}of 3.73 μH, and the equivalent series resistance R

_{r}

_{1}of 0.76 Ω. As shown in Figure 9, the coupling coefficient of the two coils at different axial distances and plane distances were simulated by Ansoft Maxwell 14.

#### 5.2. Experiment Results of the Proposed Control Method

_{1}, k

_{2}, and k

_{3}) and those of mutual inductance (M

_{1}, M

_{2}, and M

_{3}). Table 3 presents the variable parameters of the DC load and frequency.

_{L1}and R

_{L2}. The optimal point was R

_{L}

_{1}= R

_{L}

_{2}= 17 Ω, and η

_{_opt}= 93%. The maximum efficiency η

_{%_opt}was 92% and 91% when the power distribution ratio was fixed at 2:1 and 4:3, respectively. The actual efficiency achieved through maximum efficiency tracking was 84.3% and 83.5%, respectively.

_{L}

_{1}and R

_{L}

_{2}. The optimal efficiency η

_{_opt}= 92% and the optimal point was R

_{L}

_{1}= R

_{L}

_{2}= 18 Ω. The maximum efficiency with a fixed power distribution ratio at 2:1 and 4:3 was 86% and 88%, respectively. The measured maximum efficiency η

_{%_opt}with the maximum efficiency tracking was 77.8% and 78.2%.

_{L}

_{1}, R

_{L}

_{2}in Table 3(C). The distance between the transfer coil and the receiving coils changed, and the coupling coefficient and mutual inductance are shown in Table 3(C). The optimal efficiency reached 0.886, when the R

_{L}

_{1}= R

_{L}

_{2}= 12 Ω. The efficiency was 87.4% and 88.2% when the power distribution ratio was 2:1 and 4:3, respectively. This was nearly the same as the optimal efficiency. The actual measured efficiency with maximum efficiency tracking was 78.6% and 79.4%.

_{system}in Figure 1). Compared with Figure 3, to more easily analyze the relationship between the power distribution of the multiple receivers and the optimization of transmission efficiency, the losses in the DC-DC converters and the switching loss of the power amplifier were not taken into account in the calculation.

#### 5.3. Discussion of System Loss

_{o}denotes the output power of the receiver, and η is the efficiency of the system. It can be seen that the loss mainly came from the conduction loss, followed by the loss of the SEPIC and the loss of the power amplifier. The magnetic coupling loss was caused by power dissipation in the internal resistance of the coupled coils and resonant capacitor. Compared with the loss of the power amplifier, the loss of the SEPIC included not only the switching loss of the MOSFET, but also the internal resistance loss of the converter inductance. Therefore, in practice, low distributed parameters and low resistivity switching devices are needed to decrease these losses; for example, using a gallium nitride (GaN) MOSFET can effectively reduce the switching loss [27].

#### 5.4. Comparison to Previous Methods

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Converter Type | Voltage Gain (U_{out}/U_{in}) | Load Resistance Transformation (R_{Li}/R_{oi}) | The Range of R_{Li} |
---|---|---|---|

Boost | $\frac{1}{(1-D)}$ | $\frac{8{(1-D)}^{2}}{{\pi}^{2}}$ | 0~R_{oi} |

Buck | $D$ | $\frac{8}{{\left(\pi D\right)}^{2}}$ | R_{oi}~+∞ |

Buck-Boost | $\frac{-D}{(1-D)}$ | $\frac{8{(1-D)}^{2}}{{(\pi D)}^{2}}$ | 0~+∞ |

Single-Ended Primary Inductance Converter (SEPIC) | $\frac{D}{(1-D)}$ | $\frac{8{(1-D)}^{2}}{{(\pi D)}^{2}}$ | 0~+∞ |

Symbol | Quantity | Value |
---|---|---|

L_{t} | Primary-side coil inductance | 2.13 μH |

R_{t} | Primary-side coil resistance | 0.53 Ω |

L_{ri} | Secondary-side coil inductance | 3.73 μH |

R_{ri} | Secondary-side coil resistance | 0.76 Ω |

f_{s} | System operating frequency | 6.78 MHz |

I_{o}_{i} | Charging current | 0.1 A~2 A |

**Table 3.**Power division ratio and efficiency with different receiver positions in a two-receiver system.

Symbol | (A) Position 1 | (B) Position 2 | (C) Position 3 | |||
---|---|---|---|---|---|---|

k_{1} | 0.089 | 0.048 | 0.051 | |||

k_{2} | 0.062 | 0.089 | 0.048 | |||

M_{1} | 0.251 | 0.135 | 0.144 | |||

M_{2} | 0.175 | 0.251 | 0.135 | |||

Ƞ_{_opt} | 93% | 92% | 88.6% | |||

P_{1}:P_{2} | 2:1 | 4:3 | 2:1 | 4:3 | 2:1 | 4:3 |

Ƞ_{%_opt} | 92% | 91% | 86% | 88% | 87.4% | 88.2% |

Ƞ_{mea} | 84.3% | 83.5% | 77.8% | 78.2% | 78.6% | 79.4% |

**Table 4.**System parameters of the proposed system scheme and the A4WP system in an efficiency comparison process.

Symbol | A4WP | Designed Prototype |
---|---|---|

P_{1}:P_{2} | 1:1 | 1:1 |

k_{1} | 0.064 | 0.065 |

k_{2} | 0.058 | 0.056 |

M_{1} | 0.18 | 0.183 |

M_{2} | 0.163 | 0.158 |

I_{oi} | 0.1 A~2 A | 0.1 A~2 A |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Song, K.; Li, Z.; Jiang, J.; Zhu, C.
Optimal Efficiency Tracking Control Scheme Based on Power Stabilization for a Wireless Power Transfer System with Multiple Receivers. *Energies* **2018**, *11*, 1232.
https://doi.org/10.3390/en11051232

**AMA Style**

Li Y, Song K, Li Z, Jiang J, Zhu C.
Optimal Efficiency Tracking Control Scheme Based on Power Stabilization for a Wireless Power Transfer System with Multiple Receivers. *Energies*. 2018; 11(5):1232.
https://doi.org/10.3390/en11051232

**Chicago/Turabian Style**

Li, Yang, Kai Song, Zhenjie Li, Jinhai Jiang, and Chunbo Zhu.
2018. "Optimal Efficiency Tracking Control Scheme Based on Power Stabilization for a Wireless Power Transfer System with Multiple Receivers" *Energies* 11, no. 5: 1232.
https://doi.org/10.3390/en11051232