### I. Introduction

### II. Optimization of a MISO System for Maximum Efficiency

*k*is the coupling coefficient [10, 11]. A positive

*k*means the same magnetic flux direction crossing the Tx and Rx loops. A negative

*k*means that the magnetic flux originating from the Tx loop flows into the Rx loop in an opposite direction. A circle is located above the Tx loop where

*k*is zero or very small. Based on the equivalent circuit analysis in [11], efficiency (

*η*) can be expressed as

*P*

*is the total input power,*

_{in}*P*

*is the power supplied to the load with*

_{L}*R*

*, and*

_{L}*b*is the normalized load resistance defined by

*b*=

*R*

*/*

_{L}*R*

_{L}_{, opt}, where

*F*is the figure of merit given by

*b*= 1, the efficiency touches an upper bound (maximum). When

*b*> 1 or

*b*< 1, the system becomes under-coupled or over-coupled, respectively, and the efficiencies fall lower than the maximum.

*θ*as shown. For the MISO WPT system in Fig. 2, a circuit equation can be written as

##### (2)

*Z*matrix at the resonant design frequency are

*R*

*’s are the loss resistances of the loops, and*

_{i}*R*

*is the load resistance of the receiver. The remaining elements of [*

_{L}*Z*], mutual impedances, are given by

*k*

*is the coupling coefficient,*

_{ij}*L*

*is the self-inductance of the loops (or coils), and*

_{i}*M*

*is the mutual inductance. The column matrix [*

_{ij}*I*] can be obtained by [

*Z*]

^{−1}[

*V*]. The total input power and the received power at the load resistance are

*x*voltage (

*V*

_{1},

*V*

_{2}, …,

*V*

*) magnitudes and phases and*

_{N}*R*

_{L}. Finding their analytical solutions considering all the system details given in (2) is usually difficult. Thus, some GA techniques, which are not easily available in industries, must be used to maximize the system efficiency in practice. However, in the case of small mutual impedances or inductances between Tx’s, some convenient solutions for a maximum efficiency can be found.

*N*= 2) under the assumption of a negligible coupling between them. For this system, the current flowing on the Rx loop can be analytically obtained as

*γ*= 1 +

*F*

_{1}

^{2}+

*F*

_{2}

^{2}+

*β*, and

*β*=

*R*

*/*

_{L}*R*

*. The current flowing on each Tx loop at the resonant freuency can be expressed as*

_{0}*φ*is the difference in the excitation phase (

*φ*

_{1}–

*φ*

_{2}). The total input power and received power are given by

##### (16)

*m*= |

*V*

_{2}/

*V*

_{1}|, efficiency (16) is clearly a function of three variables (

*m*,

*φ*, and

*β*) given by

*m*is a real number. Maximum efficiency is mostly not obtained with the use of

*V*

_{1}=

*V*

_{2}, which is usually applied in practice. When inserting these solutions into (17), the maximum efficiency is expressed as

*N*= 2). For a MISO system with

*N*transmitters, we define the system figure of merit as

*b*= 1. However, note that this is true only when the couplings between the Tx’s are negligible.

*m*and

*β*for the case of

*F*

_{1}= 10,

*F*

_{2}= 100, and

*R*

_{0}=

*R*

_{1}=

*R*

_{2}= 0.01 Ω. Fig. 3 shows that

*m*and

*β*must have specific values to achieve maximum efficiency. When the voltage ratio

*m*=

*V*

_{2}/

*V*

_{1}= 10 and the normalized load impedance

*V*

_{2}=

*V*

_{1}) in a MISO may result in a much lower efficiency than the achievable maximum.

### III. Circuit and Electromagnetic Simulations

*r*

_{1}=

*r*

_{2}= 6 cm,

*r*

_{0}= 5 cm,

*d*= 10 cm,

*c*= 0 cm, and

*g*= 20 cm in Fig. 4. The Tx and Rx loops are made of copper rings with a radius of 1 mm. In this case,

*R*

_{1}=

*R*

_{2}= 0.0408 Ω,

*R*

_{0}= 0.034 Ω,

*L*

_{1}=

*L*

_{2}= 0.335 μH, and

*L*

_{0}= 0.266 μH. We choose

*C*

_{1}=

*C*

_{2}= 1.65 nF and

*C*

_{0}= 2.07 nF for resonance at 6.78 MHz. The quality factors

*Q*

_{1}(or

*Q*

_{2}) and

*Q*

_{0}are 348.6 and 334.2, respectively. The coupling coefficient (

*k*

_{21}) [11] between the two Tx’s is −0.0064. The minus sign implies that the magnetic flux generated from one Tx loop crosses the other in an opposite direction. Thus, the figure of merit

*F*

_{12}= −2.23. The GA in MATLAB (optimization toolbox) is applied to find the optimum magnitude and phase of

*V*

_{2}/

*V*

_{1}and the Rx load resistance (

*R*

_{L}) through which the efficiency reaches a maximum. EM simulation was performed by ANSYS HFSS using copper loops and lumped C components for resonance. From the EM simulation, we obtain the scattering matrix and coupling coefficients between all copper loops.

*F*

_{1},

*F*

_{2}) and optimum load resistances using the GA and Eq. (21) as a function of the misalignment angle

*θ*at 0°–90°. While

*θ*= 30°. The negative

*F*

_{2}comes from the downward magnetic flux on Rx0 when

*θ*> 30°. The magnitude of

*F*

_{12}is shown to be smaller than that of

*F*

_{1}.

*F*

_{2}changing sign from positive to negative at

*θ*= 30° (

*F*

_{2}= 0) can be understood by observing the direction of the magnetic flux. The optimum load resistance (

*R*

*) (21) is almost the same as the result obtained from the GA even though*

_{L, opt}*F*

_{12}(= −2.23) is not considered in (21).

*m*=

*V*

_{2}/

*V*

_{1}) as a function of the misalignment angle

*θ*using the GA and (24). The magnitudes and phases of

*V*

_{2}/

*V*

_{1}using the GA and (24) are in agreement except at around

*θ*= 30°. The discrepancies around

*θ*= 30° comes from the fact that, while the assumption made for (24) is the smaller mutual coupling (|

*F*

_{1}| > |

*F*

_{12}|) and |

*F*

_{2}| > |

*F*

_{12}|), |

*F*

_{2}| is much smaller than |

*F*

_{12}| at around

*θ*= 30°.

*V*

_{2}/

*V*

_{1}| (GA) and ∠

*V*

_{2}/

*V*

_{1}(GA) in Fig. 5(b) also agrees with them. These efficiencies are shown to be higher than those of the SISO systems. The MISO (21, 24, 8) is also shown to be higher than the SISO in which |

*F*

_{12}| (= 2.23) is roughly smaller than the system figure of merit

*F*. As |

*F*

_{12}| >

*F*, near

*θ*= 30°, we need to optimize the system parameters considering the mutual couplings. However, the evaluation of the maximum MISO efficiency is shown to have been exactly conducted with (27) not depending on the information of mutual coupling. The common practice of using the same Tx voltages (

*V*

_{2}=

*V*

_{1}) is against the solution (22) and leads to much lower efficiencies than the achievable maximum as shown by MISO (

*V*

_{2}/

*V*

_{1}= 1) in Fig. 6.