### I. Introduction

### II. Efficiency Bound Calculation Algorithm

### 1. System Model and Estimation of CIS

*N*

_{t}^{2}and

*N*

_{r}^{2}antennas, where

*N*

*and*

_{t}*N*

*are the number of elements per one side of the Tx and the Rx, respectively. In general, the edge effect in which the radiation of the edge element differs from that of the center element occurs in a finite array. However, in the case of an array antenna consisting of hundreds of antennas, the edge effect is quite small in the radiation characteristics of the array because the ratio of the number of edge elements to the total number of elements is small enough to ignore the edge effect. For example, if the size of the array antenna is 16 × 16, the ratio is 14%. In this study, as the Tx and the Rx are assumed to be large arrays, the edge effect can be ignored, and the AEP can be used for the calculation of the CSI. By using the AEP, the magnitude and phase of the CSI can be determined in closed form. In the farfield region of each element of the Tx and the Rx, the magnitude of the CSI is calculated from the Friis transmission equation between element*

_{r}*t*of the Tx and element

*r*of the Rx array, as follows:

*G*

*(*

_{t}*θ*

*,*

_{t}*ϕ*

*) and*

_{t}*G*

*(*

_{r}*θ*

*,*

_{r}*ϕ*

*) are the gain of the elements in angles*

_{r}*θ*and

*ϕ*with respect to the Tx and Rx normal planes, respectively,

*λ*is the wavelength of the operating frequency, and

*d*

*is the distance between those elements. The magnitude of the CSI in the near field is expressed as [10]:*

_{tr}*τ*

^{2}=

*A*

_{t}*A*

*cos*

_{r}*θ*

*cos*

_{t}*θ*

*/(*

_{r}*λd*

*)*

_{tr}^{2}, and

*A*

*and*

_{t}*A*

*are the aperture areas of the elements of the Tx and the Rx. cos*

_{r}*θ*

*and cos*

_{t}*θ*

*are contained in*

_{r}*τ*

^{2}to consider angles

*θ*and

*ϕ*with respect to the Tx and Rx normal planes. The phase of the CSI is calculated by converting the distance between each element to phase terms, as follows

*φ*=2

*πd*

*/*

_{tr}*λ*. The PTE results obtained using the estimated CSI and the actual CSI are compared in Section III. In this study, the actual CSI is defined as a scattering parameter between the Tx and Rx elements obtained through a full EM simulation.

### 2. Optimization Problem for Efficiency Bound Calculation

*x*

*∈ ℂ*

_{t}

^{N}^{t2×1}and

*H*denotes the conjugate transpose. Let the CSI matrix between each element of the Tx and the Rx be denoted by C∈ℂ

^{Nt2}×

^{Nr2}, where C is calculated using the method proposed in Section II-1. The sum of power received by each element can be expressed as

*P*

*(S) = ||C*

_{R}*x*

*||*

_{t}^{2}= tr(C

*CS) [6].*

^{H}*x*

*||*

_{t}^{2}=tr(S)≤

*P*

*, where*

_{t}*P*

*is the limited transmit power of the MPT system. The aforementioned design problem can be formulated as:*

_{t}*P*

*/*

_{r}*P*

*.*

_{t}### III. Numerical Results and Discussion

### 1. PTE Variation with Distance

*d*between the Tx and the Rx was varied under the condition of fixed physical sizes of the Tx and the Rx, as shown in Fig. 3. These were fixed to 0.15 m × 0.15 m and 0.06 m × 0.06 m, respectively. The number of antenna elements in the array was set according to physical size, the spacing between elements, and the operating frequency. As we designed the single-element antenna structure with a size of 0.6 wavelength, the number of (Tx, Rx) elements was determined as (11 × 11, 4 × 4) and (26 × 26, 10 × 10) at 10 GHz and 24 GHz, respectively. The PTE bound of the proposed method was compared to those in [11] and [13], as shown in Fig. 4.

*PTE*

*–*

_{estimated CSI}*PTE*

*)/*

_{actual CSI}*PTE*

*. The average errors of 10 GHz and 24 GHz are 5% and 3.9%, respectively. These results demonstrate that the proposed method accurately reflects the practical MPT system. The small discrepancy between the results using the proposed CSI and the actual CSI is a result of the limited number of mesh cells in the EM simulation and the difference between the AEPs and the actual radiation pattern of the antenna elements in the finite array. More than 2.4 billion finite-difference timedomain (FDTD) mesh cells are required to obtain an accurate actual CSI in the 24 GHz scenario. However, the number of mesh cells that can be simulated is limited to 2.4 billion. Therefore, we reduced this even if the actual CSI accuracy was lower.*

_{actual CSI}*λ*

^{2}for an array with a physical size A. Conversely, in the case of the near-field region, the difference between PTEs at 10 GHz and 24 GHz is reduced because the distance is close, and most transmit power is transferred to Rx in both cases. It is noticeable that the higher the frequency, the larger the PTE for the Tx and Rx arrays of the same size. This suggests that frequency selection is important to determine the size of the Tx and the Rx when the desired MPT system specifications, such as PTE and range, are provided.

### 2. PTE Variation with a Tilted Angle

*θ*of the Rx at a fixed distance of 1 m from the Tx, as shown in Fig. 4. The physical sizes and numbers of the Tx and Rx elements were the same as those in Section III-1. The PTE bounds of [11] are the uppermost bound, as with the results in the previous section. At an angle of 50° and a frequency of 24 GHz, the PTEs of the proposed method and [11] are 58.5% and 30%, respectively. This means that the PTE bound of [11] is larger than that of the proposed method by a 95% ratio. The PTEs using the proposed method agree well with those using the actual CSI. These results indicate that the proposed method can provide a tighter PTE bound than the other methods can and that it can be applied to the design of practical MPT systems.