A wireless power transfer (WPT) system transfers power in the form of a magnetic field without using connective wires [1–3]. As a result, it is considered a significantly more convenient system than conventional wired power transmission methods. Moreover, it is free from the danger of electric shocks. In this study, a convenient and safe WPT system is proposed for a wide variety of applications, ranging from mobile devices to electric vehicles (EVs) [4, 5]. Notably, other studies have also explored WPT systems for a wide variety of applications [1, 6, 7].

Among the many applications of WPT systems, EVs are the most noteworthy. SAE-J2954 [8], distributed by the Society of Automotive Engineers International, sets the standards for the wireless charging of EVs. The standard specifies the coil shapes, inductance values, and resonant system values of wireless charging systems for EVs. Before the release of SAE-J2954, the topology of coils and resonant circuits in WPT systems varied widely. However, the double-sided LCC topology, which was adopted as the main topology in SAE-J2954, is now the most common and widely used topology [8]. LCC, which stands for series inductor-parallel capacitor-series capacitor, refers to the compensation circuit of a WPT system. Since double-sided LCC topology offers many advantages, it is employed in a wide variety of applications, as well as EVs [9–11].

However, the WPT system, which has now reached the commercialization stage, still raises concerns due to the disadvantage of its leakage magnetic field [12, 13]. Owing to the nature of power transferred by a magnetic field, the magnitude of the leakage magnetic field of a WPT system may be larger than that of other electronic devices. Many previous studies have investigated the impact of leakage magnetic fields from WPT systems on the human body [14–18]. Currently, engineers designing WPT systems ensure minimal leakage of magnetic fields under specific conditions.

Many studies have been conducted to reduce the leakage magnetic field of WPT systems, with one of the main approaches being the addition of a shielding coil to generate a magnetic field with a phase opposite to that of the magnetic field generated by the WPT coil [13, 19–21]. One study [20] summarized various shielding methods to reduce the magnetic field generated in a WPT system, while others [19, 21] proposed an active shield method that uses an external power source. Additionally, in [13], a reactive shield coil method—a magnetic field shielding method that does not use an external power source—was proposed. However, all these methods require the addition of a shielding coil as well as a WPT coil, which presents significant disadvantages in terms of cost and space.

Another method for reducing the leakage magnetic field of WPT systems is reducing mutual magnetic fields using a number of coils [22] or three-phase coils [23]. However, in this method, a number of coils must be used on the transmitting (TX) or receiving (RX) sides. Furthermore, a study recommended using series inductors as shielding coils in a double-sided LCC topology [24, 25]. However, this method also suffers from disadvantages in terms of cost and space. Therefore, no previous study has been able to reduce the leakage magnetic field using the resonance value in WPT systems with double-sided LCC topology.

In this study, a novel method for selecting the compensation circuit value to minimize the leakage magnetic field in WPT systems with a double-sided LCC topology is proposed. More specifically, this study proposes a method for identifying a compensation circuit value that minimizes the vector sum of the current generated in the TX and RX coils in a rated state, where the output power is determined. When the sum of the current in the TX and RX coils is minimized, the leakage magnetic field is minimized as well. Furthermore, arithmetic–geometric mean inequality is employed to confirm that the current vector sum becomes minimal when the proposed method is applied. The proposed resonant circuit determination method is derived mathematically and then verified through simulations and experiments, confirming that the leakage magnetic field is minimized under the same input and output conditions.

This paper is structured as follows. In Section II, the proposed resonant condition for minimizing the leakage magnetic field is derived. These derivations are verified through simulations in Section III and through experimentation in Section IV. Section V presents a discussion of the results, after which Section VI presents the conclusion of the study.

Fig. 1 presents a WPT system with a double-sided LCC topology, which has been studied proven to be effective in various research and proven for its effectiveness [9–11]. Although WPT systems can be analyzed using a variety of methods, this study uses the first harmonic approximation (FHA) method [26–28], which analyzes systems in the phasor form, considering only its fundamental component.

The basic resonance method for double-sided LCC topology, which has already been proven in previous studies, can be expressed as Eqs. (1) and (2):

In Eqs. (1) and (2), ω_n denotes the resonant frequencies of the TX and RX sides. Notably, Fig. 1 includes a full-bridge (FB) inverter on the input side and an FB rectifier on the output side for more accurate modeling. For further analysis, a circuit diagram that considers only the AC components of a WPT system is shown in Fig. 2. The input AC voltage V_inv is related to the DC input V_in of the circuit based on Eq. (3), while the equivalent AC load resistance RL of the WPT system in Fig. 2 is related to the load resistance R_load at the rear of the rectifier based on Eq. (4).

In Eq. (3)V_inv is the fundamental square wave voltage, which is 4/π times larger than the input voltage V_in. Meanwhile, in Eq. (4), the difference is 8/π² times larger than the load conversion relationship of the FB rectifier.

Applying Kirchhoff’s voltage law to the circuit in Fig. 2 gives Eq. (5), where ω refers to the operating frequency of the FB inverter. If ω is equal to ω_n in Eqs. (1) and (2), that is, if the system satisfies the resonant condition, the current in each loop can be calculated using Eqs. (6)–(9).

As understood from Eqs. (7) and (9), for a double-sided LCC topology, the current (I₁) and output current (I_rect) of the TX coil are not related to the equivalent AC load resistance (R_L).

In power conversion systems, such as WPT systems, the rated state is very significant. In-depth analysis is required when the voltage and current of each input and output are in the rated state. In the WPT system shown in Figs. 1 and 2, the output power is determined in terms of the rated state required by the system. In other words, in the rated state, I_rect and R_L assume the rated value of the designed system. Furthermore, in a stationary WPT system, the mutual inductance (M) and operating frequency (ω) are constant. A representative example of such a situation is when an electric vehicle is being wirelessly charged at a charging station.

If the value of I_rect in Eq. (9) is constant, we can express the product of the mutual inductance and the parallel capacitors (C_P1, C_P2) on each side as Eq. (10), as noted below:

Furthermore, according to Eq. (10), the newly defined α is always a constant value in the rated state. Therefore, using the constant value of α, currents I₁ and I₂ of each coil can be calculated from Eqs. (11) and (12), as follows:

Eqs. (11) and (12) highlight that in the rated state, C_P2, which is the parallel capacitor on the RX side, determines the currents I₁ and I₂ in each coil.

Notably, the current in previously induced coils and the leakage magnetic field of a WPT system have been found to be closely related. According to [13], the magnetic field generated in the coil is proportional to the current in it. In other words, it is the coil that generates most of the magnetic field in a WPT system. Therefore, it is the leakage magnetic field from the coil that must be minimized. Fig. 3 illustrates the magnetic field leakage by TX and RX coils. If an air gap d between the coils is sufficiently small compared to the distance from the TX and RX coils to an arbitrary point P (r_TX ≪ d, r_RX ≪ d), the leakage magnetic field at that arbitrary observation point is proportional to the vector sum of the currents in the TX and RX coils. Therefore, to minimize the leakage magnetic field, the vector sum of the currents in the TX and RX coils must be minimized.

Meanwhile, as evident from Eqs. (11) and (12), the TX (I₁) and RX coil currents (I₂) are 90° out of phase. This phenomenon is represented graphically in Fig. 4, which shows that current I₂ of the RX coil is in phase with the input AC voltage V_inv, while current I₁ of the TX coil lags behind I₂ by 90°. For the leakage magnetic field to be minimized in the WPT system, the vector sum of I₁ and I₂, as indicated by the dotted line in Fig. 4, must be minimized. In other words, the magnitude of Eq. (13), which is the sum of Eqs. (11) and (12), must be minimal.

The condition to minimize the results of Eq. (13) can be attained by applying the arithmetic–geometric mean inequality, as in Eq. (14):

In other words, the minimum value of Eq. (13) is ω2αVinv2ωRL/M, which is produced when the values of I₁ and I₂ are the same (I₁=I₂). Therefore, using Eqs. (11) and (12), the conditions under which I₁ and I₂ are equal can be achieved. At this juncture, the value of the parallel capacitor C_P2 on the RX side can be obtained from Eqs. (15) and (16) as follows:

These equations indicate that, considering the rated load R_L, the vector sum of the currents of the TX and RX coils (I₁+ I₂) are minimal when the parallel capacitor C_P2 on the RX side attains the value calculated using Eq. (16) .

Meanwhile, if the parallel capacitor C_P1 on the TX side is obtained using Eqs. (10) and (16), C_P1 can be expressed as Eq. (17):

If C_P2 is determined according to the rated load, C_P1 becomes dependent on C_P2. Furthermore, once C_P1 and C_P2 are determined, L_S1 and L_S2 can be determined using Eqs. (1) and (2). Finally, C_S1 and C_S2 can be determined according to the remaining compensation circuit values and Eqs. (1) and (2). In other words, the proposed method can be employed to determine the resonant compensation circuit values that minimize the leakage magnetic field arising from TX and RX coils under specific rated load conditions.

The method proposed in this paper is studied under the assumption that mutual inductance (M) is constant. However, when misalignment occurs between coils, it could lead to cases in which C_P1 and C_P2 have to be changed. Various methods for changing the parameters of compensation circuits have been extensively studied using a tunable matching network (TMN) [29]. Therefore, in the case of applications where coil misalignment is inevitable, adding a TMN can be one way of accurately applying the method proposed in this paper.

To verify the effectiveness of the proposed method for selecting compensation circuit values, WPT coils were designed using a magnetic field analysis simulator based on the finite element method. As shown in Fig. 5, the designed coil system includes an aluminum case and magnetic materials (ferrite) for magnetic field shielding. Table 1 shows the simulation results for the coils depicted in Fig. 5, with the operating frequency of the system assumed to be 85 kHz. Table 2 shows the target values of the input and output of the proposed WPT system. Furthermore, equivalent load resistances, voltages, and currents are matched to the circuit elements in Figs. 1 and 2. Table 3 lists the compensation circuit element values calculated using the proposed method in a 200-W class WPT system with a double-sided LCC topology. The C_P1 and C_P2 values were determined using Eqs. (16) and (17), respectively, while the remaining circuit element values (L_S1, L_S2, C_S1, C_S2) were calculated using Eqs. (1) and (2).

Before the circuit simulation or the simulation of magnetic field measurement, comparison groups were selected to conduct a fair comparison with the values proposed in Table 3. Drawing on Eq. (16)C_P2 was considered the starting point for selecting the values for the resonant circuit. WPT systems with smaller and larger C_P2 values relative to the proposed C_P2 values listed in Table 3 were used as comparison groups. In particular, C_P2 values that were 20% to 60% smaller and larger than the proposed value were selected at intervals of 20%. The remaining compensation circuit values (C_P1, L_S1, L_S2, C_S1, C_S2) in Fig. 2 were obtained based on the C_P2 values for each case noted in Table 3 using Eqs. (1), (2), and (17).

Fig. 6 shows the current values of the TX and RX coils and the load DC obtained by implementing the WPT system in the circuit simulation program presented in Fig. 1. The x-axis of the graph represents the proposed C_P2 value (0%) and those of the comparison groups. It is observed that when the C_P2 value is smaller than the proposed value, the current in the TX coil (I₁) is larger than that in the RX coil (I₂). In contrast, as shown in Table 3, when selecting the C_P2 value using the proposed method, the currents of the TX and RX coils are the same. This satisfies the condition of the arithmetic–geometric mean inequality stated in Eq. (14)—that the currents of the two coils must be the same. Additionally, in cases where the C_P2 value of the comparison groups is greater than the proposed value, the RX current is larger than the TX current. Fig. 6 also demonstrates that the magnitude of the vector sum of the TX and RX coil currents is at a minimum when the proposed compensation circuit selection method is applied. Meanwhile, since the load condition is assumed to maintain the rated state, the current (R_load) delivered to the equivalent load resistor (R_load) remains constant, even when the resonant value changes.

Fig. 7 presents a phasor diagram tracing the changes in the current values of the TX and RX coils in response to changes in the parallel capacitor C_P2 on the RX side. As evident from Eqs. (6), (7), (16), and (17), the currents in the TX and RX coils depend on the value of the parallel capacitor C_P2 on the RX side. Therefore, the TX and RX current values vary depending on whether C_P2 is larger or smaller than the C_{P2_Proposed} value selected using the method proposed in this paper. In Fig. 7(a), since C_P2 is smaller than C_{P2_Proposed}, the current (I₁) of the TX coil is larger than that of the RX coil (I₂). Furthermore, Fig. 7(b) shows that when C_P2 is the same as C_{P2_Proposed}, I₁ and I₂ are identical. As previously highlighted by Eq. (14), this is the condition in which the vector sum of the TX and RX coil current values, I₁ and I₂, are at a minimum. Therefore, this case represents the C_P2 condition at which the leakage magnetic field is minimized. Lastly, Fig. 7(c) presents a case in which C_P2 is greater than C_{P2_Proposed}, leading to I₂ being greater than I₁.

Finally, magnetic field simulations were conducted to confirm whether the compensation circuit selected based on the proposed method minimized the leakage magnetic field of the WPT system. Fig. 8 shows the simulation setup for magnetic field measurement in the WPT coil system. Notably, the magnetic field was measured in a straight line from the center end of the WPT coil system. Fig. 9 presents the magnetic field magnitude based on the different C_P2 values of the magnetic fields measured in the simulation. When the magnetic field of each WPT system was simulated, the C_P2 values were found to be smaller (20%, 40%, and 60% smaller) and larger (20%, 40%, and 60% larger) than those of the proposed C_P2 value (C_{P2_Proposed}) plotted in Fig. 6. Furthermore, similar to the observations made when the proposed C_P2 (C_{P2_Proposed}) was used, the vector sum of the coil currents and the magnetic field generated by the WPT coil were both minimized. Overall, these results demonstrate the effectiveness of the proposed method for selecting a compensation circuit value that minimizes the leakage magnetic field of a WPT system.

Fig. 10 depicts the coil system fabricated for the experiment. Notably, it has the same design specifications as the coil designed in the simulation in Fig. 5. The TX and RX coils are identical, with the number of coil turns being 12 in 1 layer. Table 4 shows the self-inductance, equivalent series resistance, and mutual inductance measured for each TX and RX coil when they were positioned to have an air gap of 50 mm between them, as shown in Fig. 10.

Fig. 11 shows the overall setup for the WPT experiment based on the real manufactured coils. The power input is a DC power supply. The experimental setup is composed of an FB inverter, TX and RX coils, a resonant system board, an FB rectifier, and an electronic load. Notably, similar to the circuit simulation, all WPT experiments were conducted based on the input and output values listed in Table 2. In addition, consistent with the simulation, the compensation circuit values were selected using the method proposed in this paper, as shown in the second column of Table 5. As previously observed for the analysis and simulation, the decision of the parallel capacitor on the RX side (C_P2) is key to reducing the magnetic field in the WPT system. Therefore, the experimental comparison groups of −20%, −40%, +20%, and +40% were selected based on the C_P2 listed in Table 5. Fig. 11 displays the resonant boards manufactured by accounting for all the resonant values listed in Table 5. Similar to the simulation conducted in this study, the WPT coil system, input, and output were kept the same for all cases.

Fig. 12 shows the voltage and current waveforms (I₁, I₂) of the TX and RX coils based on changes in the compensation circuit values. In particular, Fig. 12(a) depicts the TX and RX coil current waveforms when the resonant values are determined using the proposed method. As expected, when the compensation circuit values obtained using the proposed method are used, the currents of the TX and RX coils are almost identical in value, while the vector sum is at a minimum. This can also be observed in Fig. 13, which traces the current values. Additionally, Fig. 12(b) and 12(c) show the current waveforms when the WPT system has a 40% smaller C_P2 value and a 40% larger C_P2 value than the proposed C_P2 value (C_{P2_Proposed}), respectively. Similar to the observations in Fig. 7, when the C_P2 value is smaller than C_{P2_Proposed}, the TX coil current (I₁) becomes larger than the RX coil current (I₂). In contrast, when C_P2 is greater than C_{P2_Proposed}, I₁ becomes smaller than I₂. In addition to the corresponding results, the results of the experiment conducted using the resonant values listed in Table 5 are shown in Fig. 13.

Table 6 presents the measured values of input power, output power, and power transfer efficiency (PTE) for each resonant value at which the experiment was performed. The design of the system ensured that the input and output powers were almost the same for all resonant values, with the highest PTE achieved when the proposed resonance method was adopted. The reason for the maximized PTE when the method proposed in Table 6 is applied is the minimization of the vector sum of the currents in the TX and RX coils, as shown in Fig. 7. Since the power loss in the coils is proportional to the square of the current, they are minimized when the vector sum of the currents is at its minimum. Notably, this can be considered another advantage of the proposed compensation circuit decision method.

The PTE in Table 6 refers to the DC-to-DC efficiency from the inverter to the rectifier in Fig. 1. In this paper, PTE was measured to be 82%–84%, which is not much different from an efficiency of 80%, according to the input side PFC (power factor correction circuits) specified in SAE-J2954 WPT1 [8] for electric vehicles. While the PFC generally shows an efficiency of over 95%, on converting the efficiency of WPT1 [8] into that from the inverter to the rectifier under the same conditions as in this paper, the efficiency was about 84.2%. Therefore, it may be inferred that the efficiency achieved by the experiment is an acceptable value.

Table 6, which presents the experimental results, also indicates that the final power delivered to the load was lower than the design goal of 200 W. This may be attributed to the fact that the various characteristics (loss, DCM, etc.) of the inverter and rectifier cannot be accounted for in the FHA method. To compensate for the actual power being lower than the target power, the output power in the actual design was adjusted by modifying the compensation circuit to reflect the characteristics of the power circuit in the simulation. This method is organized as a flowchart in Fig. 14.

Finally, the magnetic field was measured to determine whether a decline occurred in the leakage magnetic field of the WPT system when implementing the compensation circuit values selected using the proposed method. Fig. 15 presents the method used to measure the magnetic field. As described in Fig. 8, the leakage magnetic field was measured at the end of the coil, while the magnetic field was measured at 50 mm, 100 mm, and 150 mm from the end of the coil. Fig. 16 shows the magnetic field measurement results of the system in which the proposed method was applied and the comparison group. Similar to the results of the analysis and simulation, the lowest magnetic field value is attained by implementing the proposed compensation circuit selection method on the WPT system. It was further confirmed that the proposed method reduced the magnetic field by at least 8% and up to 43% compared to the comparison group.

This section describes the process of applying the proposed method to WPT systems for EVs based on WPT2/Z2 with a 7.7 kW input power, in compliance with SAE J2954 [8]. To apply the proposed method to WPT systems for EVs, Eqs. (13)–(16) need to be modified. Eq. (13) can be applied when the shapes of the TX and RX coils are the same. However, when the coils have different sizes, different numbers of turns, and different arrangements of the surrounding conductors or magnetic materials, Eq. (13) should be modified into Eq. (18), as noted below:

where A and B are the magnitude ratios of the magnetic field created by I₁ and I₂, respectively.

Fig. 17 presents the magnetic field distribution based on the excitation of the current in a WPT2/Z2 standard coil [8]. In particular, Fig. 17(a) and 17(b) show the magnetic field distributions when the current is excited only at I₁ and only at I₂, respectively. Drawing on Fig. 17, the magnitude ratios of the magnetic field created by I₁ and I₂ can be defined in terms of Eq. (19) as follows:

Furthermore, the condition for Eq. (18) to reach its minimum value can be obtained from Eq. (20) by applying the arithmetic-geometric mean inequality.

This value is produced as Eq. (21) when the values of AI₁ and BI₂ are the same (AI₁ = BI₂).

Finally, C_P2 can be calculated using Eq. (22), as noted below:

From the magnetic field simulation shown in Fig. 17, x was found to be x = 3.88.

Fig. 18 illustrates the method for extracting the magnetic field. In this study, the magnetic field was extracted 150 mm away from the end of the TX coil. In addition, Fig. 19 shows the magnetic field based on the distance from the WPT systems for EVs, with an output power of 6.6 kW maintained for all cases. Notably, the proposed method reduced the leakage magnetic field by up to 20% compared to other compensation circuits with different parameters. Therefore, the simulations verified the feasibility of applying this method to WPT systems for EVs.

In this study, a novel method for selecting compensation circuit values that minimize the leakage magnetic field in WPT systems with double-sided LCC topology was proposed. The method was mathematically derived based on the condition of minimizing the vector sum of the TX and RX coil current values. In particular, we confirmed that the currents of the TX and RX coils depend on the parallel capacitor value on the RX side. Subsequently, the condition under which the vector sum of the coil currents is minimal was identified using arithmetic–geometric mean inequality, while the method for determining the compensation circuit values under this condition was mathematically derived. The proposed method was proven to be effective by conducting simulations and experimentation. The experimental results showed that the leakage magnetic field of a WPT system with double-sided LCC topology that applied the proposed compensation circuit method was reduced by up to 43% compared to the system without the proposed method.