### I. Introduction

*θ*,

*φ*).

*r*,

*θ*,

*φ*). For far- and near-field transmissions, the feeding phases of the Tx array should be properly controlled such that the fields from the array elements are coherently added in the direction and at the position of the Rx, respectively, to achieve maximum efficiency. This coherent reception at the Rx position is also possible via the retrodirective method, in which a pilot RF signal from the Rx is necessary [8]. Power transfer efficiency formulas in the near field for the case of coherent feeding (or reception at the Rx position) are rarely found in the literature, although some corrections have been made to the Friis equation [1] in the near field for broadside arrays using in-phase feeding [9, 10].

### II. Theory

*P*

*is the transmitted power at Tx,*

_{t}*P*

*is the received power at Rx,*

_{r}*G*

*is the Tx antenna gain,*

_{t}*G*

*is the Rx antenna gain,*

_{r}*R*is the distance between the Tx and Rx, and λ is the wavelength in free space. The efficiency (1) is estimated as a function of a direction, say

*η*(

*θ*,

*φ*). For (1) to be accurate, the distance,

*R*, should satisfy

*R*≥ 2

*D*

^{2}/

*λ*, where

*D*is the largest linear dimension of either of the antennas [1]. In the near field (

*R*≪ 2

*D*

^{2}/

*λ*, there is an ambiguity regarding

*R*, which is defined here as the distance between the center of the Tx array antenna and Rx as shown in Fig. 1. Thus, we need to derive an appropriate formula to address this situation under the assumption that the electric fields from the transmitting antenna elements are coherently added at a receiving antenna. This coherent feeding is possible by controlling the phases of the Tx array elements when the Rx position is known. Another approach is the retrodirective array [5], in which radiation elements reverse the phases of a pilot incoming wave from the Rx, and as a result, the fields at the Rx position are coherently added regardless of the Rx position. Fig. 1 shows a configuration of an Rx antenna in the near range of a Tx array antenna with its elements spaced at

*d*= λ/2. The Tx antenna consists of

*N*elements, which transfer power to the receiving antenna. For the delivery of maximum power, the electric field must be constructively (or coherently) added at the Rx position by properly controlling the Tx element phases. When the powers of the Tx radiation elements are

*P*

_{1},

*P*

_{2}, …, and

*P*

*, respectively, and the distances between the Rx antenna and each Tx element are*

_{N}*R*

_{1},

*R*

_{2}, …, and

*R*

*, respectively, the electric field,*

_{N}*E*

*, generated from the*

_{i}*i*

*Tx array element is obtained as follows:*

^{th}*G*

_{t}_{0}(

*θ*,

*φ*) is the gain of each element, based on the direction to Rx, and

*η*

_{0}is the intrinsic impedance in free space. The total electric field at the Rx position should be obtained as a vector sum, which is cumbersome and too much time-consuming for EM-simulations. However, for a quick estimation near the broadside direction, it is here approximated as follows:

*θ*,

*φ*), may be different for each index,

*i*, in general. The power flux density at the position of the Rx antenna is given by the following:

*P*

*is expressed as follows:*

_{r}*A*

*is the effective aperture area of the Rx antenna and*

_{er}*G*

*is the gain of the Rx antenna based on its directions, but here, we use the gain of the broadside for simplicity. The efficiency,*

_{r}*η*, is defined as the ratio of the received power,

*P*

*, to the Tx input power,*

_{r}*P*

*, and is given by the following equation:*

_{t}*G*

*, may be approximated as*

_{t}*G*

*=*

_{t}*NG*

_{t}_{0}. Furthermore, for a uniform array (

*P*

*=*

_{i}*P*

_{0}for

*i*= 1, 2, …,

*N*), (6) can be simplified to the following:

*R*

*is larger than the shortest of (*

_{mean}*R*

_{1},

*R*

_{2}, ···,

*R*

*) and smaller than the arithmetic mean of (*

_{N}*R*

_{1},

*R*

_{2}, ···,

*R*

*). In the far field, since*

_{N}*R*

_{1}≈

*R*

_{2}≈ ··· ≈

*R*

*=*

_{N}*R*and

*R*

*=*

_{mean}*R*, (9) approaches the Friis transmission equation (1).

*R*/ λ increases from 0.5 to 8 for 1 × 8, 8 × 8, 1 × 16, and 16 × 16 Tx array antennas. The spacing between the array elements is assumed to be λ/2. A single Rx is assumed. In (8), all of the distances from each Tx element to the Rx antenna are considered. For the largest array of 16 elements, the far-field distance, (2

*D*

^{2}/

*λ*) is 128λ. This means that when

*R*> 128λ,

*R*

*is very close to R. It is noted that when*

_{mean}*R*= 16λ,

*R*

*≈ 16.32 λ (2% larger than*

_{mean}*R*). This discrepancy is shown to become larger as

*R*/λ decreases. When

*R*/λ is 2 for the case of the 16 × 16 array,

*R*

*/λ is 3.5, which results in 1/3 the transfer efficiency in (9) since (3.5/2)*

_{mean}^{2}≈ 3. To validate the efficiency formula (9), we first examine the method of evaluating the efficiencies from the

*S*-parameter through EM-simulations. Since the fields from the Tx array elements are controlled to be added in-phase at the receiver, the efficiency based on EM-simulations is given by the following:

*R*

*is the antenna resistance of the Tx and Rx antenna elements, which are assumed to be the same, and*

_{A}*S*

_{0}

*is the*

_{i}*S-*parameter from the

*i*

*Tx element to the receiver terminal. Thus, (10) is simplified to*

^{th}### III. Validation of Theory

*R*/λ increases up to 16. The Tx array antenna consists of 8 × 8 half-wave dipole elements. The input impedance of each dipole element is 73 Ω, and their antenna gain is 1.69 at 2.4 GHz. The spacing between adjacent antennas is λ/2 (62.5 mm). For this case, 2

*D*

^{2}/

*λ*= 32

*λ*. While the efficiencies based on the Friis formula (1) show large discrepancies with those of EM-simulations, especially when

*R*/λ is smaller than 4, the efficiencies based on the proposed formula (6) are in good agreement with the simulations in all ranges (

*R*/λ ≧ 1). The proposed formula (6) is also shown to converge to (1) in the far field.

### IV. Measurement

*G*

*= 29.7). The linearly polarized Rx antenna is also a patch type, and its gain is 6.9 dB (*

_{t}*G*

*= 4.8). Fig. 4 shows the entire system of a transmitting system. The schematic of a 1 × 8 Tx system and its fabricated photo are showed in Fig. 4(a) and (b). The system consists of a 2.4 GHz RF source, power dividers, attenuators, power amplifiers, patch radiation elements, and MCU. The magnitudes and phases for a coherent reception at Rx positions are controlled by an MCU.*

_{r}*R*/λ increases from 1 to 16.

*R*/λ increases from 1 to 16. In the case of the efficiency of (9),

*G*

*has been obtained by taking into account the gain of each element in the direction of the receiver.*

_{t}*R*/λ decreases.