Folded Spiral Resonator with Double-Layered Structure for Near-Field Wireless Power Transfer

In this paper, a folded spiral resonator with a double-layered structure for near-field wireless power transfer is proposed. In near-field wireless power transfer, conjugate impedance matching is important to achieve high transfer efficiency. To achieve maximum available efficiency, it is common to connect a matching circuit to the antenna. However, the loss increases if a matching circuit is used. A coupling inductor with a resonant capacitor has the capability to adjust an imaginary part of the input impedance, whereas the folded spiral resonator has the capability to adjust both the imaginary and real parts of the input impedance. This resonator can achieve the maximum available efficiency without a matching circuit. This paper shows that the folded spiral resonator with a double-layered structure realizes high transfer efficiency compared to conventional models.

To clarify the characteristics of our research, we categorize these near-field WPT systems. Firstly, we designate "coupler" as a device exchanging power between circuit and air. In this definition, the "coupler" includes a coupling coil and antenna. However, a "resonator" is not always included in a "coupler"; when the resonator comprises a coupling coil and a resonance capacitor, only the coupling coil is included in the "coupler". To maximize power transmission [34], simultaneous conjugate matching both the transmitting (Tx) and receiving (Rx) side is necessary [35]. The real part of the impedance and the imaginary part of the impedance should be optimized in the WPT system. A "coupler" can be classified into three types: The most widely used coupling coil in power-electronics based WPT systems [14][15][16][17] is classified into this type. For power factor compensation, a resonant capacitor is used, which plays the role of impedance matching of only the imaginary part. Thus, an impedance transformer or inverter circuit is required to adjust the real part of the impedance.
(b) Coupler with the capability of coupling and imaginary-part impedance matching Some types of self-resonant resonators, such as open-end spiral coils [36][37][38], are classified into this type. At the resonant frequency, the imaginary part of the impedance becomes zero. However, to achieve maximum transmission efficiency, an impedance conversion circuit to adjust the real part of the impedance is necessary.
(c) Coupler with the capability of coupling and conjugate impedance matching The WPT system proposed by MIT [13] utilizes a self-resonant open-end spiral coil and a one-turn loop coil. This structure has the capability of impedance matching of both the real and imaginary parts. Therefore, no additional matching circuit is required. The so-called "four-coil WPT system" [39][40][41][42] is also included in this type.
For wireless-communication systems, a folded dipole antenna [43] is commonly used because of the impedance matching function. A folded dipole antenna is also used for far-field WPT or energy harvesting systems [44,45]. However, this antenna is not appropriate for the use in near-field WPT because far-field radiation becomes an undesired emission in near-field WPT. A folded spiral resonator for near-field WPT with a single-layer structure has been proposed [46]. This resonator can adjust the input impedance of the real part by varying the conductor radius. Thus, the maximum available efficiency can be achieved without a matching circuit. However, the folded spiral antenna with a single-layered structure has a disadvantage in that the range of tunable impedance is limited since the pitch of the spiral structure limits the distance between the folded structure.
In this paper, a folded spiral resonator with a double-layered structure is proposed. Since two wires of a folded structure are wound in different planes for the double-layered structure, the folded distance can be determined independently of the spiral pitch. Furthermore, the double-layered structure reduces the proximity effect because the wire pitch is extended when compared to the single-layered structure. Furthermore, current distribution of the single-layered structure is not ideal because the inner conductor is interrupted between the outer conductor. By using a double-layered structure this problem is avoided. The effectiveness of the proposed resonator is compared with a conventional resonator. Figure 1 shows the working principle of a folded dipole antenna for far-field communication. Compared to the dipole antenna, the folded dipole antenna has two wires: the radius of the outer wire, ρ 1 , and the radius of the inner wire, ρ 2 . The distance between the two wires (folded distance) is w. Let Z 2 be the input impedance of the dipole antenna (i.e., without the inner wire). The input impedance of the folded dipole antenna Z 1 is obtained by:

Principle of the Folded Structure
where v i is the current distribution ratio. The impedance amplification ratio 1/v i 2 can be adjusted by changing the ratio of ρ 1 and ρ 2 . Moreover, when the radius of the two conductors are different, 1/v 2 i can also be adjusted by changing w. Therefore, by changing ρ 1 , ρ 2 , and w, the real part of the input impedance is adjustable. A conventional folded dipole antenna radiates far-field emission, whereas a folded resonator for near-field WPT requires less far-field emission because it becomes an undesired emission. By adjusting the conductor radius or folded distance, a folded spiral resonator can achieve impedance matching. Figure 2a shows a conventional spiral resonator used for near-field WPT. The radius of the wire is . The pitch of the spiral is . This resonator is used as a self-resonant resonator. Figure 2b shows a folded spiral resonator with a single-layered structure. The folded principle shown in Figure 1 was applied to the conventional spiral resonator. The folded dipole antenna is intended to radiate far-field emission, and a folded spiral resonator is used in the near-field region because of its low-emission property. The radii of the outer and inner wires are . and respectively. The distance of the folded structure (i.e., between the outer and the inner wire) is . The pitch of the spiral structure is . In the single-layered structure, the outer and inner wires are wound in the same plane. Figure 2c shows a folded spiral resonator with a double-layered structure (bird's-eye view). A folded structure is applied to the spiral antenna in the vertical direction, thereby the distance between the wire elements is wider than the single-layered folded spiral resonator. Therefore, it is considered that the proximity effect of the double-layered folded spiral resonator can be reduced. Additionally, the single-layered structure has a problem in that the outer and the inner conductor lays in the same plane in turn. This causes a disturbance in current distribution. By using the double-layered structure, this problem can be avoided. The outer conductor is wound in the source layer, and the inner conductor is wound in the folded layer. Both ends of the spiral wire are connected by a shorting pin. Figure 2d shows a side view of the folded spiral resonator with a double-layered structure. The distance between the two layers (corresponding to the folded distance) is .
A conventional folded dipole antenna radiates far-field emission, whereas a folded resonator for near-field WPT requires less far-field emission because it becomes an undesired emission. By adjusting the conductor radius or folded distance, a folded spiral resonator can achieve impedance matching. Figure 2a shows a conventional spiral resonator used for near-field WPT. The radius of the wire is ρ 1 . The pitch of the spiral is p. This resonator is used as a self-resonant resonator. Figure 2b shows a folded spiral resonator with a single-layered structure. The folded principle shown in Figure 1 was applied to the conventional spiral resonator. The folded dipole antenna is intended to radiate far-field emission, and a folded spiral resonator is used in the near-field region because of its low-emission property. The radii of the outer and inner wires are ρ 1 . and ρ 2 respectively. The distance of the folded structure (i.e., between the outer and the inner wire) is w. The pitch of the spiral structure is p. In the single-layered structure, the outer and inner wires are wound in the same plane. Figure 2c shows a folded spiral resonator with a double-layered structure (bird's-eye view). A folded structure is applied to the spiral antenna in the vertical direction, thereby the distance between the wire elements is wider than the single-layered folded spiral resonator. Therefore, it is considered that the proximity effect of the double-layered folded spiral resonator can be reduced. Additionally, the single-layered structure has a problem in that the outer and the inner conductor lays in the same plane in turn. This causes a disturbance in current distribution. By using the double-layered structure, this problem can be avoided. The outer conductor is wound in the source layer, and the inner conductor is wound in the folded layer. Both ends of the spiral wire are connected by a shorting pin. Figure 2d shows a side view of the folded spiral resonator with a double-layered structure. The distance between the two layers (corresponding to the folded distance) is w.  A conventional folded dipole antenna radiates far-field emission, whereas a folded resonator for near-field WPT requires less far-field emission because it becomes an undesired emission. By adjusting the conductor radius or folded distance, a folded spiral resonator can achieve impedance matching. Figure 2a shows a conventional spiral resonator used for near-field WPT. The radius of the wire is . The pitch of the spiral is . This resonator is used as a self-resonant resonator. Figure 2b shows a folded spiral resonator with a single-layered structure. The folded principle shown in Figure 1 was applied to the conventional spiral resonator. The folded dipole antenna is intended to radiate far-field emission, and a folded spiral resonator is used in the near-field region because of its low-emission property. The radii of the outer and inner wires are . and respectively. The distance of the folded structure (i.e., between the outer and the inner wire) is . The pitch of the spiral structure is . In the single-layered structure, the outer and inner wires are wound in the same plane. Figure 2c shows a folded spiral resonator with a double-layered structure (bird's-eye view). A folded structure is applied to the spiral antenna in the vertical direction, thereby the distance between the wire elements is wider than the single-layered folded spiral resonator. Therefore, it is considered that the proximity effect of the double-layered folded spiral resonator can be reduced. Additionally, the single-layered structure has a problem in that the outer and the inner conductor lays in the same plane in turn. This causes a disturbance in current distribution. By using the double-layered structure, this problem can be avoided. The outer conductor is wound in the source layer, and the inner conductor is wound in the folded layer. Both ends of the spiral wire are connected by a shorting pin. Figure 2d shows a side view of the folded spiral resonator with a double-layered structure. The distance between the two layers (corresponding to the folded distance) is .

Simulation Models
To demonstrate the validity of the proposed structure, a method of moment (MoM) simulation using the commercial software FEKO was employed. Figure 3 shows the simulation model of the conventional spiral resonator, the folded spiral resonator with a single-layered structure, and the folded spiral resonator with a double-layered structure, respectively. The same structure was used for both transmitting (Tx) and receiving (Rx) resonators. The Tx resonator had Port 1 for feeding. The Rx resonator had Port 2 to connect the load. The source impedance and the load impedance were set to 50 Ω. The conductivity of copper (σ = 57.8 × 10 S/m) was used. The mechanical parameters of the resonators are described in Table 1. The inner radius , the number of turns N, the conductor radii , , the folded distance , and the gap between spiral conductors were adjusted, thereby the transfer efficiency | | reached a maximum at 6.78 MHz

Simulation Models
To demonstrate the validity of the proposed structure, a method of moment (MoM) simulation using the commercial software FEKO was employed. Figure 3 shows the simulation model of the conventional spiral resonator, the folded spiral resonator with a single-layered structure, and the folded spiral resonator with a double-layered structure, respectively. The same structure was used for both transmitting (Tx) and receiving (Rx) resonators. The Tx resonator had Port 1 for feeding. The Rx resonator had Port 2 to connect the load. The source impedance and the load impedance were set to 50 Ω. The conductivity of copper (σ = 57.8 × 10 6 S/m) was used.

Simulation Models
To demonstrate the validity of the proposed structure, a method of moment (MoM) simulation using the commercial software FEKO was employed. Figure 3 shows the simulation model of the conventional spiral resonator, the folded spiral resonator with a single-layered structure, and the folded spiral resonator with a double-layered structure, respectively. The same structure was used for both transmitting (Tx) and receiving (Rx) resonators. The Tx resonator had Port 1 for feeding. The Rx resonator had Port 2 to connect the load. The source impedance and the load impedance were set to 50 Ω. The conductivity of copper (σ = 57.8 × 10 S/m) was used. The mechanical parameters of the resonators are described in Table 1. The inner radius , the number of turns N, the conductor radii , , the folded distance , and the gap between spiral conductors were adjusted, thereby the transfer efficiency | | reached a maximum at 6.78 MHz  The mechanical parameters of the resonators are described in Table 1. The inner radius R in , the number of turns N, the conductor radii ρ 1 , ρ 2 , the folded distance w, and the gap between spiral conductors p were adjusted, thereby the transfer efficiency |S 21 | 2 reached a maximum at 6.78 MHz by Energies 2020, 13, 1581 5 of 13 using a genetic algorithm. The outer radius of all resonators were set to R out = 150 mm. The transfer distance was set to D = 0.4 m. The conductors were composed of wires.

Impedance Matching by the Folded Structure
To confirm impedance matching by adjusting the conductor radius, the input impedance and frequency characteristics are calculated when the inner conductor radius changes. The transfer distance D is set to 0.4 m.
The input impedance of the spiral resonator is shown in Figure 4a. The wire radius ρ does not affect the real part of the input impedance. The input impedance of the folded spiral resonator with single-layered and double-layered structures is shown in Figure 4b,c, respectively. It is confirmed that the real part of the impedance is adjustable by changing the wire radius.
Energies 2020, 13, x FOR PEER REVIEW 5 of 13 by using a genetic algorithm. The outer radius of all resonators were set to = 150 mm. The transfer distance was set to = 0.4 m. The conductors were composed of wires.

Impedance Matching by the Folded Structure
To confirm impedance matching by adjusting the conductor radius, the input impedance and frequency characteristics are calculated when the inner conductor radius changes. The transfer distance D is set to 0.4 m.
The input impedance of the spiral resonator is shown in Figure 4a. The wire radius does not affect the real part of the input impedance. The input impedance of the folded spiral resonator with single-layered and double-layered structures is shown in Figure 4b,c, respectively. It is confirmed that the real part of the impedance is adjustable by changing the wire radius.   The transfer efficiency is calculated as a function of the wire radius. Figure 5 shows the wire radius characteristics | 21 | 2 and maximum transfer efficiency max at 6.78 MHz. | 21 | 2 corresponds to a transmission efficiency for 50 Ω load impedance. max is obtained from the Z parameters through the S parameters [34].
where the matrix R is the real part of the Z parameter matrix, which is calculated from the S parameter matrix. |R| shows the determinant of the matrix R. From these results, it is found that the folded spiral resonator with a double-layered structure has the highest transmission efficiency.

Comparison
In this section, the frequency characteristics of transmission efficiency are considered. Figure 6 shows the frequency characteristics of max and | 21 | 2 . From the result, the folded spiral resonator with a double-layered structure has the highest max and | 21 | 2 at 6.78 MHz. The transfer efficiency is calculated as a function of the wire radius. Figure 5 shows the wire radius characteristics |S 21 | 2 and maximum transfer efficiency η max at 6.78 MHz. |S 21 | 2 corresponds to a transmission efficiency for 50 Ω load impedance. η max is obtained from the Z parameters through the S parameters [34].
where the matrix R is the real part of the Z parameter matrix, which is calculated from the S parameter matrix. |R| shows the determinant of the matrix R. From these results, it is found that the folded spiral resonator with a double-layered structure has the highest transmission efficiency.
Energies 2020, 13, x FOR PEER REVIEW 6 of 13 (c) The transfer efficiency is calculated as a function of the wire radius. Figure 5 shows the wire radius characteristics | 21 | 2 and maximum transfer efficiency max at 6.78 MHz. | 21 | 2 corresponds to a transmission efficiency for 50 Ω load impedance. max is obtained from the Z parameters through the S parameters [34].
where the matrix R is the real part of the Z parameter matrix, which is calculated from the S parameter matrix. |R| shows the determinant of the matrix R. From these results, it is found that the folded spiral resonator with a double-layered structure has the highest transmission efficiency.

Comparison
In this section, the frequency characteristics of transmission efficiency are considered. Figure 6 shows the frequency characteristics of max and | 21 | 2 . From the result, the folded spiral resonator with a double-layered structure has the highest max and | 21 | 2 at 6.78 MHz.

Comparison
In this section, the frequency characteristics of transmission efficiency are considered. Figure 6 shows the frequency characteristics of η max and |S 21 | 2 . From the result, the folded spiral resonator with a double-layered structure has the highest η max and |S 21 | 2 at 6.78 MHz.   To investigate the improvement mechanism of transmission efficiency, the Q factor is calculated by using the following equation: where is the resonant frequency. and are the low-frequency and high-frequency at 1/√2 of the peak reflection coefficient, respectively.
The coupling coefficient k is also calculated using the Q factor and kQ product. The kQ product is obtained from the Z parameters [     To investigate the improvement mechanism of transmission efficiency, the Q factor is calculated by using the following equation: where is the resonant frequency. and are the low-frequency and high-frequency at 1/√2 of the peak reflection coefficient, respectively.
The coupling coefficient k is also calculated using the Q factor and kQ product. The kQ product is obtained from the Z parameters [  To investigate the improvement mechanism of transmission efficiency, the Q factor is calculated by using the following equation: where f 0 is the resonant frequency. f 1 and f 2 are the low-frequency and high-frequency at 1/ √ 2 of the peak reflection coefficient, respectively.
The coupling coefficient k is also calculated using the Q factor and kQ product. The kQ product is obtained from the Z parameters [34]: where |R| is the determinant of the real part of the Z parameter matrix. The transfer distance D is set to 0.3 m. The calculated Q factor and the coupling coefficient k are shown in Figure 8. It is verified that the Q factor of the folded spiral resonator with a double-layered Energies 2020, 13, 1581 8 of 13 structure is increased by a factor of four compared to the others. The coupling coefficient k of the folded spiral resonator with a double-layered structure is almost identical to that of the spiral resonator. The coupling coefficient of the folded spiral resonator with a single-layered strucuture is small. This is because the current distribution of the single-layered structure is disturbed because the inner and the outer conductors are laid in an identical plane in turn.
To investigate undesired far-field emission, the radiation pattern is shown in Figure 9. In this calculation, the Tx and Rx resonators are treated as a transmitting antenna (i.e., the Rx resonator can be considered as a parasitic element of the Tx resonator) to evaluate the far-field radiation as an antenna gain. In the near-field WPT system, far-field radiation becomes an undesired emission, and a lower gain is desired. From this figure, it is confirmed that the far-field radiation of the folded spiral antenna with a double-layered structure is 20 dB less than a conventional spiral resonator. where | | is the determinant of the real part of the Z parameter matrix. The transfer distance D is set to 0.3 m. The calculated Q factor and the coupling coefficient k are shown in Figure 8. It is verified that the Q factor of the folded spiral resonator with a double-layered structure is increased by a factor of four compared to the others. The coupling coefficient k of the folded spiral resonator with a double-layered structure is almost identical to that of the spiral resonator. The coupling coefficient of the folded spiral resonator with a single-layered strucuture is small. This is because the current distribution of the single-layered structure is disturbed because the inner and the outer conductors are laid in an identical plane in turn.
To investigate undesired far-field emission, the radiation pattern is shown in Figure 9. In this calculation, the Tx and Rx resonators are treated as a transmitting antenna (i.e., the Rx resonator can be considered as a parasitic element of the Tx resonator) to evaluate the far-field radiation as an antenna gain. In the near-field WPT system, far-field radiation becomes an undesired emission, and a lower gain is desired. From this figure, it is confirmed that the far-field radiation of the folded spiral antenna with a double-layered structure is 20 dB less than a conventional spiral resonator.

Comparison with the Conventional Structure
To verify the effectiveness of the folded spiral resonator with a double-layered structure, the transmission efficiency is compared with the conventional structure shown in Figure 10: where | | is the determinant of the real part of the Z parameter matrix. The transfer distance D is set to 0.3 m. The calculated Q factor and the coupling coefficient k are shown in Figure 8. It is verified that the Q factor of the folded spiral resonator with a double-layered structure is increased by a factor of four compared to the others. The coupling coefficient k of the folded spiral resonator with a double-layered structure is almost identical to that of the spiral resonator. The coupling coefficient of the folded spiral resonator with a single-layered strucuture is small. This is because the current distribution of the single-layered structure is disturbed because the inner and the outer conductors are laid in an identical plane in turn.
To investigate undesired far-field emission, the radiation pattern is shown in Figure 9. In this calculation, the Tx and Rx resonators are treated as a transmitting antenna (i.e., the Rx resonator can be considered as a parasitic element of the Tx resonator) to evaluate the far-field radiation as an antenna gain. In the near-field WPT system, far-field radiation becomes an undesired emission, and a lower gain is desired. From this figure, it is confirmed that the far-field radiation of the folded spiral antenna with a double-layered structure is 20 dB less than a conventional spiral resonator.

Comparison with the Conventional Structure
To verify the effectiveness of the folded spiral resonator with a double-layered structure, the transmission efficiency is compared with the conventional structure shown in Figure 10:

Comparison with the Conventional Structure
To verify the effectiveness of the folded spiral resonator with a double-layered structure, the transmission efficiency is compared with the conventional structure shown in Figure 10: (a) Short-type single-layered spiral A short-type single-layered spiral is shown in Figure 10a. This structure is a conventional inductor. For resonance, a resonant capacitor is used.
Energies 2020, 13, 1581 9 of 13 (b) Short-type double-layered spiral A short-type double-layered spiral coupler is shown in Figure 10b. This structure consists of two spirals in a series connection. This structure also acts as a conventional inductor. A resonant capacitor is used.
(c) Open-type double-layered self-resonant spiral An open-type double-layered spiral coupler is shown in Figure 10c. This structure consists of two open-end spirals. This structure is a self-resonant type, so a resonant capacitor is not necessary.
(d) Folded spiral with a double-layered structure The proposed structure is shown in Figure 10d.
Energies 2020, 13, x FOR PEER REVIEW 9 of 13 (a) Short-type single-layered spiral A short-type single-layered spiral is shown in Figure 10a. This structure is a conventional inductor. For resonance, a resonant capacitor is used. (b) Short-type double-layered spiral A short-type double-layered spiral coupler is shown in Figure 10b. This structure consists of two spirals in a series connection. This structure also acts as a conventional inductor. A resonant capacitor is used. (c) Open-type double-layered self-resonant spiral An open-type double-layered spiral coupler is shown in Figure 10c. This structure consists of two open-end spirals. This structure is a self-resonant type, so a resonant capacitor is not necessary. (d) Folded spiral with a double-layered structure The proposed structure is shown in Figure 10d.

Consideration Model
The configuration of the numerical simulation is shown in Figure 11. The transfer distance is set to 300 mm. A voltage source with an output impedance of 50 Ω is connected to Port1. A 50 Ω load is connected to Port2. The conductivity of copper σ = 5.813 × 10 7 S/m is assumed. Figure 11. Consideration model (transmitter (Tx) and receiver (Rx)). Figure 10. Single models: (a) short-type single-layered spiral, (b) short-type double-layered spiral, (c) Open-type double-layered self-resonant spiral, and (d) folded spiral resonator with double-layered structure.

Consideration Model
The configuration of the numerical simulation is shown in Figure 11. The transfer distance D is set to 300 mm. A voltage source with an output impedance of 50 Ω is connected to Port1. A 50 Ω load is connected to Port2. The conductivity of copper σ = 5.813 × 10 7 S/m is assumed.
Energies 2020, 13, x FOR PEER REVIEW 9 of 13 (a) Short-type single-layered spiral A short-type single-layered spiral is shown in Figure 10a. This structure is a conventional inductor. For resonance, a resonant capacitor is used. (b) Short-type double-layered spiral A short-type double-layered spiral coupler is shown in Figure 10b. This structure consists of two spirals in a series connection. This structure also acts as a conventional inductor. A resonant capacitor is used. (c) Open-type double-layered self-resonant spiral An open-type double-layered spiral coupler is shown in Figure 10c. This structure consists of two open-end spirals. This structure is a self-resonant type, so a resonant capacitor is not necessary. (d) Folded spiral with a double-layered structure The proposed structure is shown in Figure 10d.

Consideration Model
The configuration of the numerical simulation is shown in Figure 11. The transfer distance is set to 300 mm. A voltage source with an output impedance of 50 Ω is connected to Port1. A 50 Ω load is connected to Port2. The conductivity of copper σ = 5.813 × 10 7 S/m is assumed. Figure 11. Consideration model (transmitter (Tx) and receiver (Rx)). Figure 11. Consideration model (transmitter (Tx) and receiver (Rx)).

R out R in Port
Energies 2020, 13, 1581 10 of 13 The couplers are optimally designed to compare the characteristics of the couplers with unified external dimensions. The outer diameter R out is fixed to 150 mm. The height h of the single couplers is fixed to 5 mm. The inner diameter R in and the number of turns N are designed to maximize η max . The optimization frequency is set to 6.78 MHz. The optimized parameters are shown in Table 2. The frequency characteristics of the maximum transmission efficiency η max are shown in Figure 12a. At the resonant frequency of 6.78 MHz, the double-layered folded spiral achieves its highest efficiency η max : 96.97%.
The transfer distance characteristics of η max at 6.78 MHz are shown in Figure 12b. The transfer distance of the double-layered folded spiral at which η max achieves 60% is 1.19 times larger than the open-type double-layered spiral.
Energies 2020, 13, x FOR PEER REVIEW 10 of 13 The couplers are optimally designed to compare the characteristics of the couplers with unified external dimensions. The outer diameter is fixed to 150 mm. The height ℎ of the single couplers is fixed to 5 mm. The inner diameter and the number of turns are designed to maximize . The optimization frequency is set to 6.78 MHz. The optimized parameters are shown in Table 2. The frequency characteristics of the maximum transmission efficiency are shown in Figure  12a. At the resonant frequency of 6.78 MHz, the double-layered folded spiral achieves its highest efficiency : 96.97%.
The transfer distance characteristics of at 6.78 MHz are shown in Figure 12b. The transfer distance of the double-layered folded spiral at which achieves 60% is 1.19 times larger than the open-type double-layered spiral. To consider the difference between of these couplers, the kQ product, the coupling coefficient , and the Q factor are also shown in Table 2. The Q factor of the double-layered folded spiral resonator presents the highest value. Therefore, it is found that the double-layered folded spiral presents the highest efficiency because of its high Q factor.

Conclusions
A folded spiral resonator with a double-layered structure for near-field WPT is proposed. It is verified that the transmission distance over 60% increased by 1.82 and 1.69 times compared to the conventional spiral resonator and the folded spiral resonator with a single-layered structure, 6 6.5  To consider the difference between η max of these couplers, the kQ product, the coupling coefficient k, and the Q factor are also shown in Table 2. The Q factor of the double-layered folded spiral resonator presents the highest value. Therefore, it is found that the double-layered folded spiral presents the highest efficiency because of its high Q factor.

Conclusions
A folded spiral resonator with a double-layered structure for near-field WPT is proposed. It is verified that the transmission distance over 60% η max increased by 1.82 and 1.69 times compared to the conventional spiral resonator and the folded spiral resonator with a single-layered structure, respectively. Compared to the conventional spiral resonator, this is caused by the impedance matching Energies 2020, 13, 1581 11 of 13 effect of the real part of the folded structure. Compared to the folded spiral resonator with a single-layered structure, this is caused by winding the outer conductor and the inner conductor in different planes.
The experimental validation of this is future work.
Author Contributions: The authors have contributed equally to the theoretical development, computational calculation and writing of the paper. All authors have read and agreed to the published version of the manuscript.