### I. Introduction

### II. Theoretical Analysis

*ω*

_{0}using capacitors in a series. The Tx coils are connected to full-bridge converters (high-frequency sources), as illustrated in Fig. 1. The coupled mode theory model of the proposed WPT system can be expressed as in [24]:

##### (1)

*a*

*=*

_{i}*A*

_{s}*e*

^{−}

*denotes the energy modes of the transmitting and receiving circuits,*

^{jωt}*ω*

*denotes the coil’s intrinsic frequency,*

_{i}*τ*

*denotes the coil’s loss,*

_{i}*K*

*=*

_{ij}*K*

*(*

_{ji}*i*=

*j*:1, 2, …,

*n*) denotes the coupling coefficient between the coils, and

*F*

*denotes the DC power supply.*

_{i}##### (2)

*K*

*and inversely proportional to the coil’s loss and intrinsic frequency, so the resonant system reduces the loss. When the coils’ position change,*

_{ij}*K*

*also changes, causing fluctuations in transmission performance. When the Tx coils are perpendicular, they contain only supply power and coupled power. Therefore, they should be kept nearly perpendicular to reduce coupling effects. Relay coils, which have only coupled power, can increase the transmission range. By optimizing their structure, transmission performance can increase. The Rx coils have only coupled power. Transmission performance stability is achieved when the power load is constant. When an Rx coil changes positions,*

_{ij}*K*

*should remain the same to maintain a constant power load. Thus, to achieve transfer performance stability,*

_{ij}*K*

*should remain constant. The coupling coefficient is expressed as*

_{ij}*L*

*denotes the coil’s inductance, and*

_{i}*M*denotes the coupler’s mutual inductance. Eq. (3) indicates that transmission performance is stable when

*M*is constant.

*M*is related to the coupler’s physical structure and spatial position. The coils are coupled. To reduce the calculations and analysis, we introduce magnetic induction as

*I*denotes the coil’s current, and

*N*

*denotes the turns of the Rx. Eq. (4) indicates that*

_{R}*M*remains stable when the magnetic induction (

*B*) density distribution of the Tx is uniform.

*B*is calculated as

*I*denotes the coil’s current,

*dl*denotes the line element along the wire,

*r*denotes the distance between the source points and space points, which is (

*x*

_{0},

*y*

_{0},

*z*

_{0}). We substitute Eq. (5) with Eq. (4) as

*S̃*denotes the area of the Rx. When the Rx moves, the total

*B*shows little change in the Rx area, so the transmission performance is stable. Eq. (6) can be divided into three components:

##### (7)

*x*

*,*

_{i}*y*

*, and*

_{i}*z*

*denote the range of the wire’s position change. The three components of Eq. (7) allow us to clearly observe changes in the magnetic field within a certain area. The quasi-omnidirectional Tx consists of multiple curved coils. The coordinates of the points on the Tx coil-x, coil-y, and coil-z can be expressed as*

_{i}##### (8)

##### (9)

##### (11)

*B*is constant. To find the optimal combination, we simulate and analyze the combinations of different numbers of curved coils.

### III. The Proposed WPT Tx

*B*more uniform, many coils degrade transmission performance. We need to analyze the magnetic field distribution of coils with different curved angles and determine the arrangement and number of coils according to the characteristics of the magnetic field. The coil’s curved angles range from 90° to 180° An angle cannot be less than 90° because the current’s direction is opposite within 90°, which will reduce the absolute value of

*B*. As shown in Fig. 2, the coil angle changes from 180° to 90° by 10° each time.

*θ*= 90° and

*ϕ*= 0°–360° (

*θ*movement), the

*B*value of the NAA increases, so the mutual inductance increases. The opposite occurs in the WAA. When the central point of the Rx is around

*θ*= 0°–360° and

*ϕ*= 90° (

*ϕ*movement), the curved coils are equally spaced. As the curved angle narrows, the mutual inductance increases in the front of the curved coil (FCC) and decreases in the reverse of the curved coil (RCC). Therefore, we need to analyze the

*B*value in both directions.

*B*indicates that the transmission performance of the 90° curved coil is better than that of the other coils. The variation of the WAA’s

*B*is smaller than that of the NAA. Therefore, the WAA can be used only to enhance transmission performance. However, the 90° curved coil also covers the smallest area, so good transmission performance is achieved in a limited area.

*ϕ*movement divides the space into two equal parts, and the

*B*value of the FCC is greater than that of the RCC. This change is consistent with the 180° coil in the Tx design, but the transmission performance of the FCC is enhanced.

The curved coil can enhance the transmission performance of the front area.

As the curved angle widens, the area of high transmission performance shrinks.

The

*ϕ*movement divides the space evenly, but the*θ*movement does not.

*θ*movement.

*ϕ*movement.

*θ*movement, and the 90° coil is the optimal coil for the

*ϕ*movement. Therefore, we design two curved-coil Txs, CTx1 and CTx2, as shown in Fig. 6. CTx1 is composed of four 90° coils. Two opposite coils form a pair, and the current of the opposite coils is reversed, canceling out the magnetic field. The magnetic field of the 90° coil is small in the RCC, so its influence is reduced. Two pair are perpendicular to each other, as shown in Fig. 6(a). CTx2 is composed of four 160° coils. To expand the effective charging area and reduce the angle of the same current, the coils are combined irregularly. In each pair, one side of one coil overlaps with one side of the other. Moreover, one side of one pair overlap with one side of the other, as shown in Fig. 6(b).

### IV. Experimental Verification

*B*distribution of CTx1 is sparse but relatively uniform, whereas that of CTx2 is denser but has two sparse areas. CTx2 has better transmission performance than CTx1, but it also has a dead zone. The reason for this is that the direction of the rotating magnetic field is parallel to the Rx at the dihedral angle of the same current. The transmission performance of CTx1 is relatively stable. The simulated distributions of the magnetic field vectors generated by the Tx coils agree well with the theoretical analysis presented above.

*xoy*plane. All coils are tuned to the design frequency using NPO capacitors. The experimental frequency is set to 100 kHz, which is slightly different from the design frequency due to the use of a compensation capacitor. For the power source, a full-bridge inverter is built using four MOSFETs (IRLL024NPbF). The load resistance is connected to the Rx coil. The input voltage (

*V*

*) is 5 V, and the load is 1 Ω. The input current (*

_{in}*I*

*), VRMS, and IRMS of the load are recorded at 10° intervals to calculate the transmission power and efficiency, as shown in Fig. 9. The Rx is manually rotated counterclockwise around the origin of coordinate in the*

_{in}*xoy*plane from

*θ*= 0° to

*θ*= 360° on coordinate paper, with the distance between the Rx and the center fixed at 50 mm.

*B*of Tx to be parallel to the Rx, so the system cannot transmit power.

### V. Conclusion

*θ*movement, while the 90° coil is the optimal coil for the

*ϕ*movement. CTx2 has an average efficiency of 63% with 4.45 W at 30°–60° and 150°–330° but it also has two dead zones. CTx1 can deliver power of 4.41 W to the Rx coil arranged around it with an efficiency of approximately 27% at an operating frequency of 100 kHz and an optimal separation of 50 mm. By adding a magnetic shielding material, the efficiency can rise to 32%. Low-cost fabrication without using a current control methodology was implemented to verify the practical design of the quasi-omnidirectional curved-coil Tx with a single Rx.