### I. Introduction

### II. Two-Coil Magnetic Resonance WPT System with Transformers

*V*

_{s}), the source resistance (

*R*

_{s}), the load resistance (

*R*

_{L}), and capacitors for resonance (

*C*

_{1},

*C*

_{2},

*C*

_{3},

*C*

_{4}). Helical coils were used as the transmitting (

*T*

_{x}) and receiving (

*R*

_{x}) coils. When an electric power with the same resonance frequency as the LC circuit is applied to the transformers (

*TR*

_{1},

*TR*

_{2}), a magnetic field is generated from the transmitting coil. The generated magnetic field transmits electric power to the receiving coil. Then the receiving part of the coil operates analogous to the transmitting part. The receiving part of the coil is modeled by substituting the power source of the transmitting part to the load.

### III. Circuit Model

*M*

_{tr1}and

*M*

_{tr2}are the mutual inductances of transformers 1 and 2, respectively. The coupling coefficient of the transmitting and receiving coils is

*k*

_{23}.

*V*

_{s}), source impedance (

*R*

_{s}), primary leakage inductance of the transformer 1 (

*L*

_{tr1-p}), and internal resistance of transformer 1 (

*R*

_{1}). The second part of the transmitting circuit is composed of the secondary leakage inductance of transformer 1 (

*L*

_{tr1-s}), the inductance of the transmitting coil (

*L*

_{1}), and the composite resistance of the internal resistance of the transformer and the coil resistance (

*R*

_{2}).

*C*

_{1}, is determined by the sum of the primary leakage inductance (

*L*

_{tr1-p}) and mutual inductance (

*M*

_{tr1}). The capacitance,

*C*

_{2}, is determined by sum of the secondary leakage inductance (

*L*

_{tr1-s}), mutual inductance (

*M*

_{tr1}), and inductance of the transmitting coil (

*L*

_{1}). The receiving circuit is composed similarly to the transmitting circuit.

##### (1)

*S*

_{21}function can be calculated using Eq. (4) [13].

##### (3)

*S*

_{21}parameter is calculated using Eq. (4) and is represented in Eq. (5) [14]. (

*S*

_{21})

^{2}represents the maximum power ratio and transferring efficiency of the WPT.

##### (4)

*k*

_{23}) and frequency. The red-colored area represents a highly efficient operation. The figure shows a frequency split, a characteristic of a mid-distance WPT system. The values of the circuit elements used in calculating the transfer function are listed in Table 1.

*k*

_{critical}represents the specific distance for high-efficiency operation when the system is in its resonance [12]. When

*k*

_{23}is larger than

*k*

_{critical}, the WPT system is over-coupled and has two frequencies with a high efficiency. As the coupling coefficient decreases, the frequency for a high

*S*

_{21}converges to the critical point. When

*k*

_{23}is smaller than

*k*

_{critical}, the transferred power decreases rapidly with distance.

*k*

_{critical}is necessary for efficient power transmission and is described in the next section.

### IV. Derivation of Critical Coupling Coefficient

*k*

_{critical}, which provides a high transmission efficiency to the proposed WPT system, can be deduced. When the system is in resonance, the system impedance is represented as Eq. (6).

*Q*

_{tr1-p}=

*Q*

_{tr2-p}=

*Q*

_{P},

*Q*

_{tr1-s}=

*Q*

_{tr2-s}=

*Q*

_{S}, and

*Q*

_{1}=

*Q*

_{2}=

*Q*

_{coil}. Because the coupling coefficient of the transformer is approximately 1, the coupling coefficient is substituted to

*k*

_{tr1}=

*k*

_{tr2}

*≈*1. The modified equation is given as \Eq. (9). Under resonance, the voltage gain in Eq. (9) is the same as in Eq. (3).

##### (8)

*k*

_{critical}, can be derived by differentiating Eq. (9).

*k*

_{critical}in Eq. (10) represents

*k*

_{23}at which the transmitting and receiving coil transmits maximum power. The distance between the coils is inversely proportional to

*k*

_{23}.

*k*

_{critical}is attained when the inductance of the transformer is small, and the inductances of the transmitting and receiving coils are large. Fig. 4 shows the change in

*k*

_{critical}when the inductances of the transmitting and receiving coils are fixed, and the transformer inductance is 80, 110, and 140 μH. Depending on the transformer’s inductor, the red part of Fig. 4, which represents a high-efficient operation, varies. Although the transmitting distance is lengthened, when the transformer’s inductance decreases, the transmitting efficiency is reduced due to the reduced

*k*

_{critical}. The values of the circuit elements used in the calculation is shown in Table 1. The transformer’s inductance and capacitance are calculated based on the resonance frequency.

### V. Experiments and Simulation

### 1. Comparison of the Proposed WPT System with the Conventional Two-Coil and Four-Coil WPT Systems

*C*

_{2},

*C*

_{3}) [4]. The four-coil system consists of a two-coil system with the addition of a source coil and a load coil. In the four-coil system, the capacitors are connected in a series to compensate for the coil inductances [13]. The transmitting and receiving coils (with a diameter of 170 mm) are wound using a 0.05 mm enameled wire. The capacitance for the resonance operation is calculated using Eq. (7). The two-coil WPT system with transformers consists of a source transformer, transmitting and receiving coils, and a load transformer. Transformers with a ferrite core, which have a high permeability and high magnetic flux density, are used in the proposed WPT system to minimize the losses due to the frequency. To minimize the system volume, a PQ-type core is used in the transformers, which are wound using the same enameled wire used in the transmitting and receiving coils.

*S*

_{21}parameter was measured. The values of the circuit elements used in the simulation and experiments are shown in Table 2.

*S*

_{21}parameter for the proposed system is 0.76 at a coil distance of 4 cm. The maximum

*S*

_{21}parameter for the conventional two-coil and four-coil system is 0.93 at a coil distance of 0 cm and 0.79 at a coil distance of 6 cm, respectively. The results are shown in Fig. 6.

*S*

_{21}parameter of the proposed WPT system and the conventional two-coil system and four-coil system is 0.9 at 5 cm, 0.93 at 1 cm, and 0.89 at 5 cm, respectively. The simulation results are shown in Fig. 7. The maximum

*S*

_{21}of both the conventional two-coil system and proposed system is 1 cm shorter compared to that in the simulation. However, the maximum

*S*

_{21}of the four-coil system is 1 cm farther compared to that of the simulation. Additionally, the experimental efficiency was measured as less than that of the simulation when the transmitting distance was greater. The difference between the experiment and the simulation results is presumed to be the measuring error of the inductance and capacitance.

### 2. Transmitting Distance Change Depending on the Various Transformer Inductances

*k*

_{critical}with the change in the transformer inductance. The three types of transformers were analyzed using the same transmitting and receiving coils.

*k*

_{critical}varies with the change in the transformer inductance. Fig. 8 shows the change in the

*S*

_{21}parameter with the transformer inductance. The maximum

*S*

_{21}for the transformer with 80 μH (Tr-1) is 0.76 at 4 cm. The maximum

*S*

_{21}for the transformer with 140 μH (Tr-3) is 0.87 at 0 cm.

*S*

_{21}for the transformer with 80 μH is 0.9 at 5 cm. The maximum

*S*

_{21}for the transformer with 140 μH is 0.95 at 0 cm.

*k*

_{critical}and

*S*

_{21}values were mostly the same,

*S*

_{21}demonstrated a greater difference between the experiments and simulations. This difference tended to increase with the transmission distance. Due to the operation frequency limits of the transformer, the proposed system is operated at a lower frequency compared to the conventional magnetic resonant WPT system. The lower operating frequency and radiation losses are considered to be the cause of these differences.

### VI. Conclusion

*k*

_{critical}, thereby giving a high efficiency to the proposed WPT. The system parameter for the optimum distance could be found from the experiments.

*k*

_{critical}decreases as the transformer inductance decreases, and the decrease in

*k*

_{critical}leads to an increased transmission distance. Transformers with different inductances were used in the experiments and simulations to verify the hypothesis. A maximum

*S*

_{21}of 0.84 at 2 cm and 0. 76 at 4 cm were obtained with inductance values of 110 μH and 80 μH, respectively. Despite some slight errors, the simulations and experiments can be considered as having agreement with the results. These results will be studied in future work to obtain a WPT system with constant power transferring efficiencies regardless of the change in the transmitting distance.