Magnetically coupled resonance (MCR) wireless power transfer (WPT) system is a promising technology in electric energy transmission. But, if its system parameters are designed unreasonably, output power and transmission efficiency will be low. Therefore, optimized parameters design of MCR WPT has important research value. In the MCR WPT system with designated coil structure, the main parameters affecting output power and transmission efficiency are the distance between the coils, the resonance frequency and the resistance of the load. Based on the established mathematical model and the differential evolution algorithm, the change of output power and transmission efficiency with parameters can be simulated. From the simulation results, it can be seen that output power and transmission efficiency of the two-coil MCR WPT system and four-coil one with designated coil structure are improved. The simulation results confirm the validity of the optimization method for MCR WPT system with designated coil structure.

## I. INTRODUCTION

Recently, wireless power transfer (WPT) is a promising technology to be applied in many modern applications ranging from electronic equipment to electric vehicles.^{1,2} Two popular technologies for WPT are magnetically coupled inductive (MCI) WPT system and magnetically coupled resonance (MCR) one. But, the transmission efficiency of MCI WPT system drops rapidly with transmission distance and is not suitable for mid-range wireless energy transmission.^{3} MCR WPT system possesses the characteristics of high efficiency, non-radiant energy and good penetrability, which can light up a 60W bulb with a transmission efficiency of about 40% at distance about 2m by scientists in Massachusetts Institute of Technology.^{4} So far, MCR WPT system is considered to be one of the most potential technologies for mid-range wireless energy transmission applications. In order to successfully apply wireless power transfer to engineering practice, the research on improving output power and transmission efficiency for MCR WPT system are very important.

The parameters affecting output power and transmission efficiency are the frequency, the turn number of the coils, the vertical distance, the horizontal distance and the angle between the coils.^{5,6} So far, tuning capacitor and impedance matching are usually used to improve the transmission efficiency of wireless power transfer system.^{7–9} However, they did not take into account how to design the parameters, such as the distance between the source coil and the transmitting one, the distance between the receiving coil and the load one, the resonance frequency and the resistance of the load, to improve output power and transmission efficiency of MCR WPT system with designated coil structure. Therefore, it is necessary to design the parameters reasonably to improve output power and transmission efficiency of MCR WPT system.

In MCR WPT system, the two-coil system is suitable for short-range applications, while the four-coil system is suitable for mid-range applications.^{3} In this paper, based on the established mathematical model and the differential evolution algorithm, an optimization method for output power and transmission efficiency of the two-coil MCR WPT system and four-coil one is proposed. The optimal parameters such as the distance between the coils, the resonance frequency and the resistance of the load are obtained by means of differential evolution algorithm. By comparing and analyzing simulation results, output power and transmission efficiency of MCR WPT system with designated coil structure are improved.

## II. OPTIMAL MATHEMATICAL MODEL AND SIMULATION ANALYSIS FOR TWO-COIL MCR WPT

In order to enhance the capacity of energy transmission, transmitting coil and receiving one should have the same resonance frequency. The structure and equivalent circuit of the two-coil MCR WPT system are shown in Fig. 1, where *U*_{S} and *R*_{S} are the voltage and equivalent internal resistance of the power supply. *I*_{1} and *I*_{2}, *R*_{1} and *R*_{2}, *L*_{1} and *L*_{2}, *r*_{1} and *r*_{2}, *n*_{1} and *n*_{2} are current, resistance, inductance, radius and turn number of the transmitting coil and the receiving one, respectively. *R*_{L} is the resistance of the load. *C*_{1} and *C*_{2} are the resonance capacitor of *LC* circuits, respectively. *M* and *d* are the mutual inductance and the distance between the transmitting coil and receiving one, respectively. The transmitting coil and receiving one are made by winding copper enameled wire with the diameter of *a* into spiral structure. *μ*_{0} is the permeability of vacuum. *σ* is the conductivity of copper. Parameters of the two-coil MCR WPT system are shown in Table I.

Parameters . | Transmitting coil . | Receiving coil . |
---|---|---|

Inductance (L_{1}, L_{2}) | $ \mu 0 r 1 n 1 2 ln 8 r 1 a \u2212 2 $ | $ \mu 0 r 2 n 2 2 ln 8 r 2 a \u2212 2 $ |

Resonance capacitor (C_{1},C_{2}) | $ 1 \omega 2 L 1 $ | $ 1 \omega 2 L 2 $ |

Resistance (R_{1}, R_{2}) | $ \omega \mu 0 2 \sigma n 1 r 1 a $ | $ \omega \mu 0 2 \sigma n 2 r 2 a $ |

Parameters . | Transmitting coil . | Receiving coil . |
---|---|---|

Inductance (L_{1}, L_{2}) | $ \mu 0 r 1 n 1 2 ln 8 r 1 a \u2212 2 $ | $ \mu 0 r 2 n 2 2 ln 8 r 2 a \u2212 2 $ |

Resonance capacitor (C_{1},C_{2}) | $ 1 \omega 2 L 1 $ | $ 1 \omega 2 L 2 $ |

Resistance (R_{1}, R_{2}) | $ \omega \mu 0 2 \sigma n 1 r 1 a $ | $ \omega \mu 0 2 \sigma n 2 r 2 a $ |

According to Kirchhoff voltage law and equivalent circuit shown in Fig. 1, we can obtain

where the transmitting coil is coaxial with the receiving coil, the relation between the distance and mutual inductance of MCR WPT system is defined as^{10}

When the two-coil MCR WPT system is at resonance, the output power *P* and transmission efficiency *η* can be expressed as

Equation (3) shows that output power and transmission efficiency are both depend on the distance *d*, the resonance frequency *f* and the resistance *R*_{L}. However, differential evolution algorithm has strong global convergence and robustness, which can solve complex optimization problems that some mathematical programming methods are difficult to solve.^{11} In this paper, differential evolution optimization algorithm is used to optimize the parameters to improve the output power and transmission efficiency of MCR WPT system with designated coil structure.

In differential evolution algorithm, each individual can be expressed as *x*_{i,G} (*i*=1, 2, …, *NP*), where *i*, *G*, *NP* are sequence of individuals in a population, evolution generation and population size, respectively. The parameters of the differential evolution algorithm are listed as follows: initial mutation operator *F*_{0}= 0.4, crossover operator *CR* = 0.1, *Np* =50 and *G*= 200. From the given boundary constraints, the initial population is defined as

where the boundary of the parameter variable is *x*_{j}^{(L)} < *x*_{j} < *x*_{j}^{(U)} (*j*=1, 2, …, D), rand[0,1] is the uniform random number on [0,1]. So the initial variable is *x*_{1}, *x*_{2}, …, *x*_{j}, and the range of change for each variable *x*_{j} is [*x*_{j}^{(L)}, *x*_{j}^{(U)}]. When *x*_{j}^{(L)}=0, *x*_{j}^{(U)}=1, the optimization *R*_{L} defines the domain as [0,200] and the unit is Ω. *f* is defined as [0,20] and the unit is MHz. When *R*_{L}=200*x*_{1}, *f*=20*x*_{2}, equation (3) combining with the differential evolution algorithm can be expressed as

In this paper, the specifications are as follows: *R*_{S}=50Ω, *r*_{1}=*r*_{2}=0.1m, *n*_{1}=*n*_{2}=6, *U*_{S}=10V, a=1mm. Output power and transmission efficiency with different *d*, *f* and *R*_{L} are shown in Fig. 2. Fig. 2(a) shows that output power and transmission efficiency decline sharply with distance at a certain distance. To improve the values of output power and transmission efficiency at a certain distance, *f* and *R*_{L} can be optimized by differential evolution algorithm. Without loss of generality, this paper focus on the both optimal parameters to achieve the improvement of output power and transmission efficiency at *d*=0.2m, which is as depicted in Fig. 2(d). Similarly, Fig. 2(b) and Fig. 2(c) show that output power and transmission efficiency are both on the low side at *d*=0.05m, *f*=5MHz and at *R*_{L}=200Ω, *d*=0.05m. Fig. 2(e) and Fig. 2(f) achieve the improvement of output power and transmission efficiency at *f*=5MHz and *R*_{L}=200Ω by optimizing *f* and *R*_{L}, respectively.

From Fig. 2, it can be seen that output power and transmission efficiency can be improved at *d*=0.2m, *f*=5MHz and *R*_{L}=200Ω by optimizing parameters reasonably with the differential evolution algorithm. When *R*_{L}=7.06Ω, *f*=20MHz and *R*_{L}=1.06Ω, *f*=20MHz, output power and transmission efficiency can achieve maximum at *d*=0.2m, respectively. When *R*_{L}=200Ω and *R*_{L}=16.82Ω, output power and transmission efficiency can achieve maximum at *d*=0.05m, *f*=5MHz, respectively. When *f*=1.68MHz and *f*=20MHz, output power and transmission efficiency can achieve maximum at *R*_{L}=200Ω, *d*=0.05m, respectively. Therefore, parameters optimization has obvious advantages for the optimization of output power and transmission efficiency with designated coil structure.

## III. OPTIMAL MATHEMATICAL MODEL AND SIMULATION ANALYSIS FOR FOUR-COIL MCR WPT

André Kurs pointed that the four-coil WPT system is suitable for mid-range wireless energy transmission.^{4} The structure and equivalent circuit of the four-coil MCR WPT system is shown in Fig. 3, where *U*_{S} and *R*_{S} are the voltage and equivalent internal resistance of the power supply. *I*_{1}, *I*_{2}, *I*_{3} and *I*_{4}, *R*_{1}, *R*_{2}, *R*_{3} and *R*_{4}, *L*_{1}, *L*_{2}, *L*_{3} and *L*_{4}, *r*_{1}, *r*_{2}, *r*_{3} and *r*_{4}, *n*_{1}, *n*_{2}, *n*_{3} and *n*_{4} are current, resistance inductance, radius and turn number of the source coil, the transmitting coil, the receiving coil and the load coil, respectively. *C*_{1}, *C*_{2}, *C*_{3} and *C*_{4} are the resonance capacitor of *LC* circuits, respectively. *R*_{L} is the resistance of the load. *M*_{12}, *M*_{23}, *M*_{34} and *d*_{12}, *d*_{23}, *d*_{34} are mutual inductance and the distance between source coil and the transmitting one, the transmitting coil and the receiving one, receiving coil and load one. Parameters of the four-coil MCR WPT system are shown in Table II.

Parameters . | Source coil . | Transmitting coil . | Receiving coil . | Load coil . |
---|---|---|---|---|

Inductance | $ \mu 0 r 1 n 1 2 ln 8 r 1 a \u2212 2 $ | $ \mu 0 r 2 n 2 2 ln 8 r 2 a \u2212 2 $ | $ \mu 0 r 3 n 3 2 ln 8 r 3 a \u2212 2 $ | $ \mu 0 r 4 n 4 2 ln 8 r 4 a \u2212 2 $ |

(L_{1}, L_{2}, L_{3}, L_{4}) | ||||

Resonance capacitor | $ 1 \omega 2 L 1 $ | $ 1 \omega 2 L 2 $ | $ 1 \omega 2 L 3 $ | $ 1 \omega 2 L 4 $ |

(C_{1}, C_{2}, C_{3}, C_{4}) | ||||

Resistance (R_{1}, R_{2}, R_{3}, R_{4}) | $ \omega \mu 0 2 \sigma n 1 r 1 a $ | $ \omega \mu 0 2 \sigma n 2 r 2 a $ | $ \omega \mu 0 2 \sigma n 3 r 3 a $ | $ \omega \mu 0 2 \sigma n 4 r 4 a $ |

Parameters . | Source coil . | Transmitting coil . | Receiving coil . | Load coil . |
---|---|---|---|---|

Inductance | $ \mu 0 r 1 n 1 2 ln 8 r 1 a \u2212 2 $ | $ \mu 0 r 2 n 2 2 ln 8 r 2 a \u2212 2 $ | $ \mu 0 r 3 n 3 2 ln 8 r 3 a \u2212 2 $ | $ \mu 0 r 4 n 4 2 ln 8 r 4 a \u2212 2 $ |

(L_{1}, L_{2}, L_{3}, L_{4}) | ||||

Resonance capacitor | $ 1 \omega 2 L 1 $ | $ 1 \omega 2 L 2 $ | $ 1 \omega 2 L 3 $ | $ 1 \omega 2 L 4 $ |

(C_{1}, C_{2}, C_{3}, C_{4}) | ||||

Resistance (R_{1}, R_{2}, R_{3}, R_{4}) | $ \omega \mu 0 2 \sigma n 1 r 1 a $ | $ \omega \mu 0 2 \sigma n 2 r 2 a $ | $ \omega \mu 0 2 \sigma n 3 r 3 a $ | $ \omega \mu 0 2 \sigma n 4 r 4 a $ |

When all coils work at resonance, output power and transmission efficiency are expressed as

Combining output power *P* and transmission efficiency *η* with the differential evolution algorithm, optimized parameters can be expressed as

Substituting (7) into (6), optimization mathematical model of output power and transmission efficiency can be obtained, where *r*_{1}=*r*_{2}=*r*_{3}=*r*_{4}=0.1m, *n*_{1}=*n*_{4}=1, *n*_{2}=*n*_{3}=6, *R*_{S}=50Ω, and *U*_{S}=10V. Then, four variables *d*_{12}, *d*_{34}, *R*_{L} and *f* can be optimized by the optimization mathematical model, which is named four parameters optimization. With the limited conditions, not all parameters can meet the requirements in practice. Without loss of generality, three variables *d*_{12}, *d*_{34} and *R*_{L} can be optimized for a given resonance frequency *f*=1MHz, which is named three parameters optimization. Fig. 4 shows the curves of output power and transmission efficiency at optimized transmission distance *d*_{23}=0.2m and *d*_{23}=0.3m obtained by three parameters optimization and four parameters optimization.

From Fig. 4(a) and Fig. 4(b), when optimized transmission distance is *d*_{23}=0.2m, it can be seen that output power and transmission efficiency of four parameters optimization are *P*≈0.46W, *η*≈0.93 at *d*_{23}=0.2m and *P*≈0.10W, *η*≈0.67 at *d*_{23}=0.3m, respectively. From Fig. 4(c) and Fig. 4(d), when optimized transmission distance is *d*_{23}=0.3m, it is known that output power and transmission efficiency of four parameters optimization are *P*≈0.17W, *η*≈0.88 at *d*_{23}=0.2m and *P*≈0.38W, *η*≈0.77 at *d*_{23}=0.3m, respectively. At *d*_{23}=0.2m, *P*≈0.46W>*P*≈0.17W, *η*≈0.93>*η*≈0.88, i.e. output power and transmission efficiency with designated coil structure are improved at the optimized transmission distance. Besides, when there is an additional constraint, output power and transmission efficiency of three parameters optimization is lower than that of four parameters optimization.

## IV. CONCLUSION

In the paper, based on the mathematical model and the differential evolution algorithm, output power and transmission efficiency of the two-coil MCR WPT system and four-coil one are improved at resonance. Thus, the proposed optimization method is verified. Nonetheless, different coil structures would need further investigation.

## ACKNOWLEDGMENTS

This work was supported in part by the National Natural Science Foundation of China under Grant 51777054 and State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology.