Curved-Retrodirective Beamforming System to Improve Microwave Power Transmission Efficiency in the Fresnel Region

This article presents a curved-retrodirective beamforming (RDB) system for improving microwave power transmission efficiency in the Fresnel region. Since microwave power transmission in the far-field region has very low efficiency, studies on the Fresnel region are being actively conducted. In these studies, an RDB technique is popular. The RDB system with a subarray structure was a realistic structure that reduced system complexity. However, the transmission efficiency is lowered because the beamwidth of the transmitter antenna element is narrow. To solve this problem, this article proposes a curved-RDB system that can focus the microwave power on the receiver. The proposed system uses the peak gain of the transmitter antenna element by using tilted beams to improve transmission efficiency. The system design method that can maximize the transmission efficiency is also presented depending on the given conditions, such as transmission distance, characteristics of the transmitter and receiver antenna. The simulation showed a reduction in power leakage compared to the conventional system. The fabrication and measurement validated the efficiency improvement of the proposed system for Internet of Things devices in the Fresnel region.


I. INTRODUCTION
R ECENTLY, with the increase and development of mobile devices and the Internet of Things (IoT) devices, the demand for technology to transmit power wirelessly is increasing. Microwave power transmission is useful for supplying power in situations where wiring is inconvenient, dangerous, or infeasible. The power supply to the wireline limits the mobility of the device and can cause short circuits and contact problems due to corrosion. A battery can use to compensate for this wiring problem. However, the weight and volume of the system increase, and the battery needs to be maintained and replaced. For this reason, wireless power transmission technology has been continuously studied, and applied to various systems [1], [2]. Wireless power transmission technology can be divided into nonradiative and radiative methods [3]. The radiation method using the microwave is a technology capable of transmitting power over a long distance. However, since it uses microwaves, the power and efficiency decrease as the transmission distance increases. For these reasons, it does not apply to various industries, and technology maturity is not high [4], [5], [6]. For the practical implementation of various IoT strategies, it is necessary to increase the transmission efficiency of microwave power transmission. As shown in Fig. 1, there are five types of efficiency in microwave power transmission systems in IoT devices. First, the dc power supply to RF conversion efficiency includes the dc to RF conversion efficiency and the RF active element efficiency [7], [8], [9]. Second, the efficiency of the Tx antenna includes the matching efficiency and the radiation efficiency [10]. Third, efficiency in air includes beamforming efficiency and path loss [11], [12], [13], [14], [15], [16], [17]. Fourth, the efficiency of the Rx antenna includes the matching efficiency and the radiation efficiency. Finally, the received RF to dc rectification efficiency includes the efficiency of the rectifier and the efficiency of the dc or RF combining [18]. The overall efficiency of the microwave power transmission system is calculated as the product of these five efficiencies [19], [20]. Among them, the improvement of the efficiency in the air is discussed in this article.
Early microwave power transmission has been studied in the far-field region. The receiver at the far-field region sends a pilot signal to the transmitter and the transmitter sends the directed microwave power to the position of the receiver using the beamforming technique as shown in Fig. 2(a). However, microwave power transmission in the far-field region has a very low efficiency compared to the nonradiative methods. To overcome this limitation, microwave power transmission has been studied in the Fresnel region to improve efficiency. A well-known microwave power transmission technique in the Fresnel region is retrodirective beamforming (RDB). Its receiver sends a pilot signal to the transmitter. Then the transmitter focuses the microwave power on the position of the receiver using phase differences of the pilot signal arriving at each transmit antenna element as shown in Fig. 2(b) [21], [22], [23], [24], [25], [26], [27], [28], [29]. Table I summarizes some previous studies of microwave power transmission by country [5], [24], [30], [31]. The operating frequency is mainly set at 0.915, 2.4, or 5.8 GHz. These studies were conducted in the Fresnel region to increase efficiency, which is closer than the far-field area calculated from the antenna size. In addition, an array antenna was used to increase efficiency. However, the number of array antennas cannot increase infinitely in reality. Therefore, Yi et al. [30], Yi et al. [32], Wang et al. [33], and Chen et al. [34] adopted a subarray structure and reduced the system complexity. In addition, the subarray RDB system reduces the number of active components, channels, heat, and cost. However, since the subarray RDB system has a narrow beam, it can reduce the efficiency of microwave power transmission in the Fresnel region. Detailed problem analysis is covered in Section II. To solve this problem, we propose a curved-RDB system that can efficiently focus the microwave power on the position of the receiver.
The contributions of this article are summarized as follows.
1) The conventional RDB system is mathematically designed and the problem that the received power is saturated with the number of Tx antennas increasing is presented. 2) To solve the problem, a mathematically modified equation is proposed. To improve the performance, the proposed equation is compared with the conventional equation, discussed, and evaluated through simulation in several cases. 3) An optimal method with three steps to design our system is proposed. In addition, nine examples designed with the proposed method are presented as guidelines. 4) We prove through an electromagnetic (EM) simulation and measurement that the proposed system has higher efficiency and lower leakage power than the conventional system. 5) The proposed system with better performance shows that the initial charging time and recharging interval for the IoT sensor operation are improved compared to the conventional system. The optimal system configuration for RDB can vary according to various conditions, such as the characteristics of the Tx and Rx antennas and the position between the TRx. In this article, we analyze nine scenarios with various microwave power transmission conditions of TRx and present a system design method for optimal microwave power transmission. Our approach is not limited to these nine examples and can be applied to microwave power transmission in various situations if the process presented in Section II is followed. It is expected that the approach of this article will be a guideline for designing an optimal microwave power transmission system in the future.
The configuration of this article is as follows. In Section II, the system analysis and design method of the proposed curved-RDB system are presented. The comparison of the beamforming characteristic of the conventional RDB system and the proposed curved-RDB system are also presented using the EM simulator. Section III presents the simulation and experimental results of the proposed system. The system configuration and experimental environment are described and the experimental results are compared with the simulation results. The discussion and conclusion of this work are given in Section IV. The notations in this article are listed in Table II.

A. System Analysis
In free space, the received power from the transmitter can be expressed by the Friis equation. Since the Friis equation is assumed to be in the far-field region, the emitted wave is considered as a plane wave. Because the received power is inversely proportional to the square of the transmission distance, the efficiency of microwave power transmission decreases rapidly as the transmission distance increases. The power transmitted from the Tx array antenna which is composed of N t antenna elements to the Rx antenna can be expressed as follows [35]: In (1), P t , G t , N t , G r , λ 0 , and r denote the total transmit power, Tx antenna element gain, number of Tx antenna elements, Rx antenna gain, wavelength of the operating frequency in the air, and center to center distance from Tx array antenna to Rx antenna, respectively. Ray tracing is a useful method for beamforming analysis of microwave power transmission. The array antenna in the early microwave power transmission used the same phase value, which was analyzed as a large antenna. When each phase value is used differently, it can not be analyzed as one antenna. Each antenna element can be operated independently, so it need to be analyzed through the rays of each antenna element. For ray tracing analysis, the Rx antenna should be located in the Fresnel region for the Tx array antenna and in the far-field region for each Tx antenna element. Fig. 3 shows the configuration of the conventional RDB system and the proposed curved-RDB system using tilted beams. The tilted beam means that the direction of the main beam of the Tx antenna element is electrically or mechanically inclined toward the center of the Rx antenna. The mechanically tilted beam was adopted in this article [36]. The proposed curved-RDB system can use the peak gain of the Tx antenna element by using tilted beams, so an improvement of the power transmission efficiency can be expected. The sources of the conventional RDB system and the proposed curved-RDB system are assumed to be continuous wave (CW) and the input power of the N t transmit antennas is P t /N t when the total transmit power is P t . In Fig. 3(a), the r n and θ n denote the distance and angle between the nth Tx antenna element and the center of the Rx antenna, respectively. In Fig. 3(b), all Tx antenna elements are arranged in an arc shape with respect to the center of the Rx antenna. The r tilt denotes the distance between tilted Tx antenna element and the center of the Rx antenna. The difference in the arrangement structure of each system results in different distances (r 0 = r tilt ) between the center of the systems and the receiver. Conventional RDB system has different distances between each Tx antenna and Rx antenna, but the proposed curved-RDB system has the same distance between all Tx antennas and Rx antenna. Therefore, between the two systems, the distance between the center of the system and the Rx antenna is different. A fair comparison of the performance of the two systems requires a distance with the same path loss as shown in (2) from the distances of each Tx antenna and Rx antenna. Equation (2) means that the sum of the losses for the distance between the Tx antennas and Rx antenna of the conventional RDB system and the proposed curved-RDB system is the same. When the number of Tx array antenna is odd, r 1 is the same as r 0 . The r tilt , where r 0 < r tilt < r N t , can be expressed as (3) 1 The received signal from the nth Tx antenna element to the Rx antenna in the Fresnel region can be expressed as (4). G t (θ n ) and G r (θ n ) are the gain of the nth Tx antenna element and Rx antenna at the angle θ n , respectively. The strength of the received signal is expressed as the product of the distance factor and the gain factor The sum of signal strengths received from each Tx antenna element is the strength of the total received signal, which is expressed as follows: Therefore, the power received from the Tx array antenna to Rx antenna is obtained as (6). To calculate the received power in the Fresnel region, (1) can be modified as follows [5]: In order to compare the improvement of the power transmission efficiency of the proposed curved-RDB system with the conventional RDB system, simulations were conducted in three cases to examine the received power. In case 1, the distance r 0 , the Tx antenna element gain G t , and Rx antenna gain G r were set as 5 m, 16 dBi, and 16 dBi, respectively. To obtain the gain of 16 dBi, 4×4 patch antenna array is used. In case 2, the r 0 , G t , and G r were set as 5 m, 16 dBi, and 10 dBi, respectively. To obtain the gain of 10 dBi, 2×2 patch antenna array is used. In case 3, the r 0 , G t , and G r were set as 3 m, 16 dBi, and 16 dBi, respectively. In the simulation, the value of r tilt was calculated based on (3).
In all cases, it can be seen that in both systems, the received power increases and then becomes saturated as the number of Tx antenna elements (N t ) increases. This phenomenon is related to the beamwidth of the Rx antenna. As the number of N t increases, the area occupied by the Tx array antenna is widened. When the area exceeds the half-power beamwidth (HPBW) of the Rx antenna, the received power saturates rapidly. Based on these characteristics, the maximum number of N t in which the Tx array antenna can be arranged in an area corresponding to the HPBW of the Rx antenna is determined as the optimal number. Meanwhile, when the number of N t is the same, the proposed curved-RDB system has overall higher received power than the conventional RDB system. This is because the proposed system uses tilted beams. Since the main beams of all Tx antenna elements are directed toward the center of the Rx antenna, the microwave power can be transmitted with a gain of G t,max . In the conventional system, the microwave power is transmitted with a gain of G t (θ n ) depending on the angle between each Tx antenna element and the Rx antenna. Therefore, a gain loss occurs according to the θ n . However, the proposed system has the advantage that the microwave power can be transmitted with maximum gain without gain loss. The received power of the conventional RDB system is expressed as (6), and the received power of the proposed curved-RDB system can be expressed as follows: Comparing the results of Fig. 4(a) and (b) reveal the change in the received power with regard to the Rx antenna gain under the condition of the same transmit distance and Tx antenna element gain. As the gain of the Rx antenna decreases from 16 to 10 dBi, the amount of the received power decreases. But the optimal number increases because the HPBW of the Rx antenna becomes wider. Comparing the results of Fig. 4(a) and (c) reveals the change in the received power according to the transmission distance under the condition of the same Tx antenna element gain and Rx antenna gain. As the transmission distance decreases from 5 to 3 m, the amount of the received power increases. But the optimal number decreases because the area where the Tx array antenna can be located within the HPBW of the Rx antenna becomes narrower. Therefore, an optimal number for efficient power reception can be obtained according to the given condition, and the proposed curved-RDB system can receive more microwave power than the conventional RDB system.

B. Sector Area Derivation of the Tx Array Antenna
The design of the proposed curved-RDB system consists of three steps. In step 1, the area of a sector where the Tx array antenna can be arranged is determined by the HPBW of the Rx antenna. The size of the arc in which the Tx array antenna can be arranged is determined by the distance between Tx array antenna and Rx antenna in step 2. Finally in step 3, the number of antenna elements of the Tx array is determined with regard to the Tx antenna element gain.
As the parameters, such as desired Rx power and Rx size are determined depending on an application, the characteristics of the Rx antenna, such as gain and HPBW can be calculated. Since the Rx antenna can receive most of the transmit power through the area of HPBW, the Tx array antenna is placed within this area of the sector. Fig. 5 shows the received power according to the HPBW of the Rx antenna when the distance between the Tx array antenna and Rx antenna r tilt is fixed. The transmission distance, which is the distance between TRx antennas, is set to 20λ 0 . The gain and HPBW of the Tx antenna element are set to 16 dBi and 32 • , respectively. The gain and HPBW of the Rx antenna are set to 10, 13, and 16 dBi, and 64 • , 45 • , and 32 • , respectively. The gains of Tx antenna element and Rx antenna are calculated using (8) from the effective aperture of the antenna. It is assumed that each antenna has the maximum gain in the effective aperture with an aperture efficiency of 1. G max is the maximum gain value of the antenna, A e is the effective aperture, and D is the antenna size. Therefore, it is possible to model antenna gain and HPBW according to the antenna size through simple calculation. In Fig. 6, the maximum antenna gain and HPBW with regard to the antenna size are introduced [37]. Based on As described in the last section, it can be seen from Fig. 5 that the received power is saturated as the N t exceeds the optimal number. The narrower the HPBW of the Rx antenna, the smaller the optimal number that can reach the maximum received power. When the gain and HPBW of the Rx antennas are 10, 13, and 16 dBi, and 64 • , 45 • , and 32 • , the optimal numbers of the Tx array antennas are 16, 8, and 4, respectively. The optimal number is determined by the HPBW of the Rx antenna. Therefore, the area of a sector where the Tx array antenna can be arranged is determined by the HPBW of the Rx antenna.

C. Arc Size Derivation of the Tx Array Antenna
The arc size of the Tx array antenna can be determined by the transmission distance r tilt in the derived sector area in step 1. When the Rx antenna characteristics are the same, the greater the distance between the TRx antennas, the more Tx antenna elements can be arranged. As the r tilt increases, the optimal number increases and vice versa. In Fig. 7, the transmission distance is set to three types as 20λ 0 , 40λ 0 , and 60λ 0 . The gain and HPBW of the Tx antenna element and Rx antenna are set to 16 dBi and 32 • , respectively. Fig. 7 shows the received power according to the transmission distance. It can be seen that the arc size and the optimal number of the Tx array antenna increase as the transmission distance increases. When the transmission distances are 20λ 0 , 40λ 0 , and 60λ 0 , the optimal numbers of the Tx array antenna are 4, 8, and 12, respectively. Therefore, The size of the arc in which the Tx array antenna can be arranged is determined by the transmission distance.

D. Determination of the Number of the Tx Array Antenna
In step 3, the number of Tx array antenna is determined from the gain of the Tx antenna element. In the previous steps,  the sector area and arc size were derived according to the characteristics of the Rx antenna and the transmission distance. In this step, the number of Tx array antenna is determined according to the characteristics of the Tx antenna element. In Fig. 8, the transmission distance, the gain, and HPBW of the Rx antenna are set to 20λ 0 , 16 dBi, and 32 • , respectively. The gain and HPBW of the Tx antenna element are set to three types as 10, 13, and 16 dBi, and 64 • , 45 • , and 32 • , respectively. Fig. 8 shows the received power according to the Tx antenna element gain. It can be seen that the optimal number of Tx array antenna decrease as the Tx antenna element gain increases. When the Tx antenna element gains are 10, 13, and 16 dBi, the optimal numbers of the Tx array antenna are 16, 8, and 4, respectively. In all simulations, the Rx antenna is located in the Fresnel region for the Tx array antenna and in the far-field region for each Tx antenna element. Therefore, the number of Tx array antenna can be determined according to the Tx antenna element gain. Through the above steps, the Tx array antenna specifications, such as the sector area, arc size, and the optimal number can be determined with regard to the Rx antenna characteristic. The three steps design method of the practical perspective is also expressed from a theoretical perspective. The gain and HPBW of the Rx antenna are calculated as (8), (9), [32]. Arc size, which is inversely proportional to the Rx antenna dimension D r , is calculated by the characteristics of the receiver antenna in (10). The Tx antenna dimension D t is calculated as (11) from the Tx element gain, where an aperture efficiency ap is defined as the ratio of the maximum effective area of the antenna to the physical area depending on the type of antenna [32] and assumed to be 0.5 in this article. As a result, the optimal number of Tx antenna (N opt ) is expressed by the conditions, such as transmission distance, characteristics of the transmitter, and receiver antenna in (12) Arc size = π r · HPBW 180 = rλ 0 D r (10) For example, the characteristics of the Rx antenna can be obtained assuming that the size of the IoT device is the aperture size of the Rx antenna. In Table III, the sizes of the IoT devices are considered as the aperture size of the Rx antenna. The curved-RDB system design guideline for the IoT devices when the transmission distance is 1 or 2 m and Tx antenna element gain is 10, 13, or 16 dBi are presented in Table III. Various optimal numbers can be derived according to the size the IoT device, Tx antenna element gain, and transmission distance. Based on the system design process, the simulation and the implementation configuration for microwave power transmission are established. In the following section, the design, simulation, and measurement of the proposed curved-RDB system are presented.

III. SIMULATION AND MEASUREMENT RESULTS
Section III is a section to verify the method presented in Section II through simulation and measurement. Therefore, Section III was carried out by selecting one of the guidelines presented in Section II. Five major parameters, which are Rx size, Rx maximum gain, Tx element gain, distance, and the optimal number of Tx, were determined through three steps in Section II. In this article, simulation and measurement results are presented for the case where the Rx maximum gain of 16 dBi, the Tx element gain of 16 dBi, the transmission distance of 1 m, and the optimal number of 4 in Table III. Other cases in the guidelines can be selected and utilized depending on the size of the IoT device.

A. System Configuration
This section describes the simulation environment for microwave power transmission using the proposed curved-RDB system. In order to satisfy the simulation configuration established above, a unit antenna design with a gain of 16 dBi and an HPBW of 32 • is required. A unit antenna was designed as shown in Fig. 9(a). The unit antenna consists of a 4×4 array in which microstrip patch antennas operating on 5.8 GHz are arranged with a spacing of 31.01 mm. The patch antennas are designed on a substrate TLY-5A with a dielectric constant and thickness of 2.17 and 1.575 mm, respectively. The T-junction power divider was designed for power distribution of the 4×4 array. Parameter values on the figure are shown in the caption. Fig. 9(b) shows the simulated return loss of the unit antenna. The simulated 10-dB impedance bandwidth of the unit antenna is 5.75-5.84 GHz. The simulated radiation pattern of the unit antenna at 5.8 GHz is shown in Fig. 9(c). The simulated peak gain and HPBW of the unit antenna are 16.31 dBi and 30 • , respectively. As a result of the simulation, it can be seen that the unit antenna satisfies the design goals of a gain of 16 dBi and an HPBW of 32 • .
The Tx array antenna with the optimal number of 4 is composed of an array of the four-unit antennas. The Rx antenna is composed of one unit antenna. For comparison as shown in Fig. 3, a conventional RDB system was constructed by arranging the unit antennas linearly, and a proposed curved-RDB system was constructed by arranging the unit antennas in an arc shape. The transmission distance r is set as 1 m. For a fair comparison of the received power between two systems, r tilt is set as 1.017 m according to (3). The transmit power is set as 30 dBm.

B. Comparison of the Conventional and Proposed System
The microwave power transmission simulation was conducted using the system configuration previously designed. In the proposed curved-RDB system, the Tx array antenna with four-unit antennas arranged in an arc shape, and the Rx antenna composed of one unit antenna are located at a distance of 1.017 m. In the conventional RDB system, the Tx array antenna with four-unit antennas arranged linearly, and the Rx antenna composed of one unit antenna are located at a distance of 1 m. Fig. 10 shows the power flow distribution of both beamforming systems. In Fig. 10(a), the Tx array antenna  of the conventional RDB system is located on the left side and the small triangle denotes the position of the Rx antenna. It can be seen that some of the transmitted power is concentrated to the receiver, but there is a large power leakage in the vicinity of the Tx array antenna. Since the Rx antenna is located outside the HPBW range of each Tx antenna element, the maximum power cannot be transmitted toward the receiver direction(θ n ) from each Tx antenna element. For this reason, the power leakage occurs in the Fresnel region in the conventional RDB system. In Fig. 10(b), the Tx array antenna of the proposed curved-RDB system is located on the left side and the small triangle denotes the position of the Rx antenna. It can be seen that most of the transmitted power is concentrated to the receiver and the power leakage is reduced. Since the main beam of all Tx antenna elements are directed toward the receiver, the maximum power can be transmitted toward the receiver. For this reason, the proposed curved-RDB system can increase the power transmission efficiency by minimizing the power leakage in the Fresnel region. Fig. 11 shows the simulated power density of both beamforming systems on the x-axis at the position of the receiver and z-axis. The red dashed lines are the results of the conventional RDB system and the black solid lines are the results of the proposed curved-RDB system. The focal width and power density of the two systems can be compared. The focal width is defined as a 13.5% range of the power density at the focal point, i.e., the position of the receiver [10], [38]. The focal widths of the conventional RDB system and the proposed curved-RDB system are 170 and 112 mm, respectively. It means that the proposed system has a 34% narrower focal width than the conventional system and can transmit power more intensively to the receiver. In Fig. 11(b), it has a higher power density at the position of the receiver. The conventional RDB system has 72.01 W/m 2 at 1 m and the proposed curved-RDB system has 94.97 W/m 2 at 1.017 m. Also, in the conventional system, it can be seen that the power leakage occurs in the vicinity of the Tx array antenna rather than the position of the receiver. Therefore, the proposed system can transmit power more intensively to the position of the receiver, thereby increasing the power transmission efficiency.

C. Experimental Results
Fabrication and experiments are carried out to validate the increase in power transmission efficiency of the proposed curved-RDB system. The prototype unit antenna is fabricated as shown in Fig. 12(a). The prototype unit antenna consists of a 4×4 array in which microstrip patch antennas operating on 5.8 GHz are arranged with a spacing of 31.01 mm as designed in Fig. 9(a). The patch antennas are printed on the substrate TLY-5A. The T-junction power divider was implemented for the power distribution of the 4×4 array. A network analyzer (Keysight E8362B) and an anechoic chamber were used for measurement. Fig. 12(b) shows the simulated and measured return loss of the unit antenna. The red dash lines are the simulated results and the black solid lines are the measured results. The measured 10-dB impedance bandwidth of the unit antenna is 5.76-5.85 GHz. The simulated and measured radiation pattern of the unit antenna at 5.8 GHz is shown in Fig. 12(c). The measured peak gain and HPBW of the unit antenna are 16.08 dBi and 30 • , respectively. The measured return loss and The Tx array antenna with the optimal number of 4 is implemented using four-unit antennas. The Rx antenna is implemented using one unit antenna. Fig. 13(a) shows the photographs of the conventional RDB system and proposed curved-RDB system. The conventional RDB system was fabricated by arranging the four-unit antennas linearly, and the proposed curved-RDB system was fabricated by arranging the four-unit antennas in an arc shape. For arranging and fixing the unit antennas, plastic jigs are designed and fabricated by polylactic acid (PLA) using the 3-D printer. The proposed curved-RDB system uses a jig for the curved array instead of the existing jigs to improve efficiency. Other than this, the proposed system can utilize the same system as shown in Fig. 1. Fig. 13(b) shows the measurement environment for microwave power transmission. The experiment was conducted by configuring the Tx array antenna, the Rx antenna, and 4-way power divider, which is a T-junction type, as shown in the figure using the near-field planar scanner of Songdo IoT Technical Support Center. The transmission distances of the conventional RDB system and proposed curved-RDB system is set as 1 and 1.017 m, respectively. A signal generator (Keysight E8257D) and power amplifier (Qorvo QPA1019) were used to apply microwave power to the Tx array antenna, and the power received by the Rx antenna was measured using a spectrum analyzer (Keysight E4405B). The power transmission efficiency (η) is compared to evaluate the performance of the conventional RDB system and the proposed curved-RDB system. The power transmission efficiency (η) in [5] and [33] is defined as the ratio of the RF power at the receiver to the total radiated power by the transmitter in (13). The transmit power (P t ) was set as 30 dBm at 5.8 GHz The measured received power in Fig. 14(a) is the result of measuring the received power while moving the Rx antenna in the x-axis from −300 to 300 mm at 10-mm intervals. The simulated received power in Fig. 14(a) is the result of simulating the received power using the EM simulator under the same condition. The simulated and measured received power of the conventional RDB system at 1 m are 18.57 and 17.61 dBm, respectively. The power transmission efficiencies (η conv ) of the conventional system calculated using these results are 7.19% and 5.77%, respectively. The simulated and measured received power of the proposed curved-RDB system at 1.017 m are 19.78 and 19.47 dBm, respectively. The power transmission efficiencies (η prop ) of the proposed system calculated using these results are 9.51% and 8.85%, respectively. The measurement results showed reasonable agreement with the simulation results, and the proposed system received greater power than the conventional system. The measured received power increased from 17.61 to 19.47 dBm, and the power transmission efficiency increased by 53.4% from 5.77% to 8.85%. In addition, as discussed in the previous section, the proposed system can receive microwave power more intensively because the focal width is narrower.
The measured received power in Fig. 14(b) is the result of measuring the received power while moving the Rx antenna in the z-axis from 0 to 1500 mm at 50-mm intervals. The simulated received power in Fig. 14(b) is the result of simulating the received power using the EM simulator under the same condition. The results similar to the power density in Fig. 11(b) was obtained. A large power leakage occurs near the Tx array antenna in the conventional system, but the proposed system can concentrate more power to the receiver. Therefore, the power transmission efficiency improvement of the proposed curved-RDB system was validated experimentally.

D. Comparison of Charging Performance
In this section, the charging performances of the curved-RDB system were evaluated through experiments. Fig. 15(a) shows the configuration of a receiver with Rx antenna, impedance matching network, Greinacher circuit, power management unit (PMU), storage capacitor (C3), and Bluetooth low energy (BLE) module. BLE(CC2541) was adopted as the IoT device in this experiment. Fig. 15(b) shows the prototype of receiver. The experimental setup was described in Fig. 15(c). A signal generator (Keysight E8257D) were used to excite microwave power to the Tx array antenna, and the power received by the Rx antenna was rectified from RF to dc. Then, the output power of the Greinacher circuit with SMS7630 is stored in the storage capacitor (FG0H103ZF) and extracted by the PMU (AEM30940) to power the BLE when needed. Fig. 16 shows the output voltage measurement results of the Greinacher circuit according to the Tx power. The cold start of the PMU requires a voltage of 0.2 V or higher. The PMU operates from −5.61 and −7.47 dBm of signal generator power of the conventional and the curved-RDB system, respectively. Since the power density of the curved-RDB system is higher than that of the conventional system, higher voltage, and power can be obtained from the same receiver with the same Tx power. Fig. 17 shows the waveforms by the oscilloscope (LeCroy 104MXi) with a signal generator power of 12 dBm, where V cap and V sup represent the voltage of the storage capacitor and the supply voltage to the BLE, respectively. The higher voltage measured in Fig. 16 makes the charging speed faster and the recharging interval shorter. The initial charging time and charging interval of the curved-RDB system were 232 and 42 s, respectively, which were shorter than 411 and 77 s of the conventional RDB system. Therefore, it was experimentally evaluated that the operational efficiency of the IoT device increases with the improvement of the power transmission efficiency of the proposed curved RDB system.

IV. CONCLUSION
This article presented the curved-RDB system and the design guideline consisting of three steps for improving microwave power transmission efficiency. In the conventional RDB system, there is a problem in that the received power is saturated despite the increase in the number of Tx antenna elements. An optimal design method for improving microwave power transmission efficiency was proposed by modifying the equation of the conventional RDB system. The optimal configuration for the RDB system can vary depending on various conditions. Nine scenarios of the conditions were analyzed, and a system design method was presented for optimal microwave power transmission. Naturally, our approach can be applied to various situations in addition to the nine scenarios. The approach in this article will be a guideline for optimal design in the future. One of the scenarios was simulated and experimented with. The proposed curved-RDB system has narrower focal width than the conventional system and can transmit microwave power more intensively to the receiver. The fabrication and measurement were conducted and showed reasonable agreement with the simulation. The received power of the proposed system increased compared to the conventional system, achieving efficiency improvement. In addition, this article greatly reduces the large power density generated at an undesired position by the conventional RDB system. Therefore, the power density directed at people and animals in undesired positions can be reduced. This consideration would be a good solution for wireless power transmission applications for various IoT devices.