Simulation-Assisted Design Process of a 22 kW Wireless Power Transfer System Using Three-Phase Coil Coupling for EVs

Abstract

The objective of this paper is to study a 22 kW high-power wireless power transfer (WPT) system for battery charging in electric vehicles (EVs). The proposed WPT system consists of a three-phase half-bridge LC–LC (i.e., primary LC/secondary LC) resonant power converter and a three-phase sandwich wound coil set (transmitter, Tx; receiver, Rx). To transfer power effectively with a 250 mm air gap, the WPT system uses three-phase, sandwich-wound Tx/Rx coils to minimize the magnetic flux leakage effect and increase the power transfer efficiency (PTE). Furthermore, the relationship of the coupling coefficient between the Tx/Rx coils is complicated, as the coupling coefficient is not only dominated by the coupling strength of the primary and secondary sides but also relates to the primary or secondary three-phase magnetic coupling effects. In order to analyze the proposed three-phase WPT system, a detailed equivalent circuit model is derived for a better understanding. To give a design reference, a novel coil design method that can achieve high conversion efficiency for a high-power WPT system was developed based on a simulation-assisted design procedure. A pair of magnetically coupled Tx and Rx coils and the circuit parameters of the three-phase half-bridge LC–LC resonant converter for a 22 kW WPT system are adjusted through PSIM and CST STUDIO SUITE™ simulation to execute the derivation of the design formulas. Finally, the system achieved a PTE of 93.47% at an 85 kHz operating frequency with a 170 mm air gap between the coils. The results verify the feasibility of a simulation-assisted design in which the developed coils can comply with a high-power and high-efficiency WPT system in addition to a size reduction.

1. Introduction

Electric vehicles (EVs) have the potential to serve as a solution to reducing petroleum consumption and as an alternative for replacing conventional fuel vehicles globally [1,2]. Major vehicle manufacturers are developing pure battery EVs, hybrid EVs (HEVs), and plug-in hybrid EVs (PHEVs) as their next-generation products [3,4]. Although EVs can help reduce global oil and gas depletion and environmental issues, the technology gap remains. EVs have limited mileage and battery capacity, longer charging periods, and higher overall costs. They also require charging cables, galvanic isolation components, battery chargers, and battery safety qualifications [5]. Furthermore, humidity, metallic particles, and air exposure can cause oxidation or rust on the charging gun for EVs due to long-term usage. Rust may cause overheating and potential fire risks when executing high-current transmissions during EV battery charging. Additionally, as the demand for higher battery capacity continues to grow, so does the size of the charging guns.

Generally, wireless charging, inductive power, non-contact charging, and contactless charging refer to the same technology [6,7,8,9,10,11,12,13,14,15,16]. Wireless charging is the technology of electrically charging battery-powered devices without the need for a wired electrical connection. When compared to charging by wire, the WPT system is safe, efficient, and can reduce the loss of power transfer elements by preventing aging and oxidation by means of enhancing equipment durability [10]. Currently, three-phase topologies are considered to be most suitable for high-power WPT applications [8,9,13,14,15,16]. In the literature presented above, we do not see a complete and clarified design process for constructing cost-effective three-phase Tx/Rx coils due to the more complicated relationship of the coupling coefficient between the Tx/Rx coils.

Figure 1 is a schematic diagram of a WPT system for battery charging in an EV; it consists of an inverter, TX and RX coils, a rectifier, and a battery. The installation of a WPT system in EVs poses specific challenges, including limited storage space, restricted size of the coils, a potential decrease in efficiency, issues of misalignment tolerance, and increased overall costs.

This paper developed a 22 kW high-power WPT system with a three-phase, half-bridge LC–LC resonant power converter and three-phase WPT sandwich-wound coils. The system operates at around a frequency of 85 kHz and complies with the SAE J2954 Z3 grade [17], based on inductive power transfer technology with magnetic resonance. Furthermore, in this paper, we proposed a novel coil design method to attain high conversion efficiency; PSIM and CST STUDIO SUITE™ simulation processes were used to design and adjust the parameters of the pair of magnetically coupled coils for the 22 kW WPT system, and then the design formulas were derived. Finally, the system achieved a PTE of 93.47% at a frequency of 85 kHz and with a 170 mm air gap between the Tx/Rx coils. The results indicate the effectiveness of the developed coil design method. The complete procedures for a high-power WPT system with high-efficiency and reduced coil size, which can demonstrate an overall performance that complies with the simulation-assisted design, were validated.

2. Circuit Topology

Figure 2 shows the circuit topology of the proposed three-phase high-power WPT system, which consists of an inverter with power MOSFETs (Q₁~Q₆), Tx/Rx coils with inductances (L_1a/L_2a, L_1b/L_2b, L_1c/L_2c), primary/secondary capacitors (C_1a/C_2a, C_1b/C_2b, C_1c/C_2c), the output rectifier with power diodes (D₁~D₆), the output filter capacitor (C_o), and load resistance (R_L).

In Figure 2, the circuit topology consists of three half-bridge LC–LC resonant power modules connected in parallel. Note that the phase difference of the control signals is 120° for each power module.

A novel sandwich-wound overlapping structure design is adopted in this paper to reduce the size of the three-phase Tx/Rx coils, which improves the power density of the WPT system. However, the relationship of the coupling coefficient between the three-phase Tx/Rx coils is more complicated, as the coupling coefficient is dominated by the coupling strength of the primary and secondary sides and relates to the primary or secondary three-phase magnetic coupling effects. To obtain a better understanding of the three-phase coil design concept, an equivalent circuit model is derived by reflecting the secondary side impedance to the primary side to analyze the proposed three-phase circuit topology. The circuit parameters are derived based on the specifications shown in

Table 1. Specifications of the 22 kW WPT system.

Inverter input voltage (Vin)	750 V
Rectifier output voltage (Vo)	450 V
Output power (Po)	22 kW
Operating frequency	85 kHz
Transmission distance (air gap)	170 mm~250 mm

and were followed by simulations to verify the proposed feasibility.

As shown in Figure 3, the load resistance (R_o) is reflected into the equivalent resistance (R_La, R_Lb, R_Lc) of the corresponding three-phase coils. Assuming that the three-phase coils are balanced and the rectifier circuit has no power loss, one can obtain the following equation [18]:

$$R_{La}=R_{Lb}=R_{Lc}=\frac{6}{{\pi }^2}{R}_L$$

(1)

The parameters of the secondary side circuit can be reflected to the primary side to simplify the analysis; thus, the circuit in Figure 4a is equivalent to the single loop circuit shown in Figure 4b.

In Figure 4a, the primary side voltages (V_L1a, V_L1b, V_L1c) and the secondary side voltages (V_L2a, V_L2b, V_L2c) of the three-phase coils can be obtained according to Kirchhoff’s voltage law. The circuit equation relationship of the primary side can be expressed as Equation (2) and on the secondary side as Equation (3).

$$\left[\begin{array}{c}{V}_{L1a}\\ V_{L1b}\\ V_{L1c}\end{array}\right]=\left[\begin{array}{c}{V}_{ina}\\ V_{inb}\\ V_{inc}\end{array}\right]+\left[\begin{array}{ccc}-\frac{1}{s{C}_{1a}}& -s{M}_{1a1b}& -s{M}_{1a1c}\\ -s{M}_{1a1b}& -\frac{1}{s{C}_{1b}}& -s{M}_{1b1c}\\ -s{M}_{1a1c}& -s{M}_{1b1c}& -\frac{1}{s{C}_{1c}}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]$$

(2)

$$\left[\begin{array}{c}{V}_{L2a}\\ V_{L2b}\\ V_{L2c}\end{array}\right]=\left[\begin{array}{ccc}-R_{La}-\frac{1}{s{C}_{1a}}& -s{M}_{2a2b}& -s{M}_{2a2c}\\ -s{M}_{2a2b}& -R_{Lb}-\frac{1}{s{C}_{1b}}& -s{M}_{2b2c}\\ -s{M}_{2a2c}& -s{M}_{2b2c}& -R_{Lc}-\frac{1}{s{C}_{1c}}\end{array}\right]\left[\begin{array}{c}{I}_{2a}\\ I_{2b}\\ I_{2c}\end{array}\right]$$

(3)

The primary side voltages (V_L1a, V_L1b, V_L1c) and the secondary side voltages (V_L2a, V_L2b, V_L2c) can also be obtained by considering the primary or secondary three-phase magnetic coupling effects as shown in Equations (4) and (5):

$$\left[\begin{array}{c}{V}_{L1a}\\ V_{L1b}\\ V_{L1c}\end{array}\right]=\left[\begin{array}{c}s{L}_{1a}{I}_{1a}\\ s{L}_{1b}{I}_{1b}\\ s{L}_{1c}{I}_{1c}\end{array}\right]+\left[\begin{array}{ccc}s{M}_{1a2a}& s{M}_{1a2b}& s{M}_{1a2c}\\ s{M}_{1b2a}& s{M}_{1b2b}& s{M}_{1b2c}\\ s{M}_{1c2a}& s{M}_{1c2b}& s{M}_{1c2c}\end{array}\right]\left[\begin{array}{c}{I}_{2a}\\ I_{2b}\\ I_{2c}\end{array}\right]$$

(4)

$$\left[\begin{array}{c}{V}_{L2a}\\ V_{L2b}\\ V_{L2c}\end{array}\right]=\left[\begin{array}{c}s{L}_{2a}{I}_{2a}\\ s{L}_{2b}{I}_{2b}\\ s{L}_{2c}{I}_{2c}\end{array}\right]+\left[\begin{array}{ccc}s{M}_{1a2a}& s{M}_{1b2a}& s{M}_{1c2a}\\ s{M}_{1a2b}& s{M}_{1b2b}& s{M}_{1c2b}\\ s{M}_{1a2c}& s{M}_{1b2c}& s{M}_{1c2c}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]$$

(5)

where Equation (2) is equal to Equation (4); Equation (3) is equal to Equation (5).

Rearranging Equations (4) and (5) leads to the following:

$$\left[\begin{array}{c}{V}_{ina}\\ V_{inb}\\ V_{inc}\end{array}\right]+\left[\begin{array}{ccc}-\frac{1}{s{C}_{1a}}& -s{M}_{1a1b}& -s{M}_{1a1c}\\ -s{M}_{1a1b}& -\frac{1}{s{C}_{1b}}& -s{M}_{1b1c}\\ -s{M}_{1a1c}& -s{M}_{1b1c}& -\frac{1}{s{C}_{1c}}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]=\left[\begin{array}{c}s{L}_{1a}{I}_{1a}\\ s{L}_{1b}{I}_{1b}\\ s{L}_{1c}{I}_{1c}\end{array}\right]+\left[\begin{array}{ccc}s{M}_{1a2a}& s{M}_{1a2b}& s{M}_{1a2c}\\ s{M}_{1b2a}& s{M}_{1b2b}& s{M}_{1b2c}\\ s{M}_{1c2a}& s{M}_{1c2b}& s{M}_{1c2c}\end{array}\right]\left[\begin{array}{c}{I}_{2a}\\ I_{2b}\\ I_{2c}\end{array}\right]$$

(6)

$$\left[\begin{array}{c}s{L}_{2a}{I}_{2a}\\ s{L}_{2b}{I}_{2b}\\ s{L}_{2c}{I}_{2c}\end{array}\right]+\left[\begin{array}{ccc}s{M}_{1a2a}& s{M}_{1b2a}& s{M}_{1c2a}\\ s{M}_{1a2b}& s{M}_{1b2b}& s{M}_{1c2b}\\ s{M}_{1a2c}& s{M}_{1b2c}& s{M}_{1c2c}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]=\left[\begin{array}{ccc}-R_{La}-\frac{1}{s{C}_{2a}}& -s{M}_{2a2b}& -s{M}_{2a2c}\\ -s{M}_{2a2b}& -R_{Lb}-\frac{1}{s{C}_{2b}}& -s{M}_{2b2c}\\ -s{M}_{2a2c}& -s{M}_{2b2c}& -R_{Lc}-\frac{1}{s{C}_{2c}}\end{array}\right]\left[\begin{array}{c}{I}_{2a}\\ I_{2b}\\ I_{2c}\end{array}\right]$$

(7)

From Equation (6), if one eliminates the coupling factors caused by secondary side currents (I_2a, I_2b, I_2c), the relationship of the equivalent circuit from the secondary side mapped to the primary side can be derived accordingly. In Equation (7), one can see the relationship between the primary and secondary side currents. As the circuit is designed as a series resonant WPT, the secondary side coil inductors are resonated with the secondary side capacitors, which can be derived as Equations (8)–(10) [19]. It is notable that the parameters of the coil inductors are dominated by the coupling strength of the primary and secondary sides and relate to the primary or secondary three-phase magnetic coupling effects. The coil inductors should be selected with fine-tuning at first through the CST STUDIO SUITE™ simulation (referring to Section 3).

$$C_{2a}=\frac{1}{\left(L_{2a}-M_{2a2b}-M_{2a2c}+M_{2\mathrm{b}2c}\right){\omega }^2}$$

(8)

$$C_{2b}=\frac{1}{\left(L_{2b}-M_{2a2b}-M_{2\mathrm{b}2c}+M_{2a2c}\right){\omega }^2}$$

(9)

$$C_{2c}=\frac{1}{\left(L_{2c}-M_{2a2c}-M_{2\mathrm{b}2c}+M_{2a2b}\right){\omega }^2}$$

(10)

Once the secondary-side resonant parameters are confirmed, substituting Equations (8)–(10) into Equation (6) yields the relationship between the primary side and secondary side currents, as seen below:

$$I_{2a}=\frac{s{\alpha }_{1a2a}{I}_{1a}+s{\alpha }_{1b2a}{I}_{1b}+s{\alpha }_{1c2a}{I}_{1c}}{R_{Labc}}$$

(11)

$$I_{2b}=\frac{s{\alpha }_{1a2b}{I}_{1a}+s{\alpha }_{1b2b}{I}_{1b}+s{\alpha }_{1c2b}{I}_{1c}}{R_{Labc}}$$

(12)

$$I_{2c}=\frac{s{\alpha }_{1a2c}{I}_{1a}+s{\alpha }_{1b2c}{I}_{1b}+s{\alpha }_{1c2c}{I}_{1c}}{R_{Labc}}$$

(13)

where

$$R_{Labc}=R_{La}{R}_{Lb}+R_{La}{R}_{Lc}+R_{Lb}{R}_{Lc}$$

(14)

$$\begin{array}{l}\left\{\begin{array}{l}{\alpha }_{1a2a}=M_{1a2c}{R}_{Lb}+M_{1a2b}{R}_{Lc}-M_{1a2a}\left(R_{Lb}+R_{Lc}\right);{\alpha }_{1b2a}=M_{1b2c}{R}_{Lb}+M_{1b2b}{R}_{Lc}-M_{1b2a}\left(R_{Lb}+R_{Lc}\right);\\ {\alpha }_{1c2a}=M_{1c2c}{R}_{Lb}+M_{1c2b}{R}_{Lc}-M_{1c2a}\left(R_{Lb}+R_{Lc}\right)\end{array}\right\}\\ \left\{\begin{array}{l}{\alpha }_{1a2b}=M_{1a2c}{R}_{La}+M_{1a2a}{R}_{Lc}-M_{1a2b}\left(R_{La}+R_{Lc}\right);{\alpha }_{1b2b}=M_{1b2c}{R}_{La}+M_{1b2a}{R}_{Lc}-M_{1b2b}\left(R_{La}+R_{Lc}\right);\\ {\alpha }_{1c2b}=M_{1c2c}{R}_{La}+M_{1c2a}{R}_{Lc}-M_{1c2b}\left(R_{La}+R_{Lc}\right)\end{array}\right\}\\ \left\{\begin{array}{l}{\alpha }_{1a2c}=M_{1a2b}{R}_{La}+M_{1a2a}{R}_{Lb}-M_{1a2c}\left(R_{La}+R_{Lb}\right);{\alpha }_{1b2c}=M_{1b2b}{R}_{La}+M_{1b2a}{R}_{Lb}-M_{1b2c}\left(R_{La}+R_{Lb}\right);\\ {\alpha }_{1c2c}=M_{1c2b}{R}_{La}+M_{1c2a}{R}_{Lb}-M_{1c2c}\left(R_{La}+R_{Lb}\right)\end{array}\right\}\end{array}$$

(15)

Once the relationship between the primary and secondary side currents is obtained, Equations (11)–(13) can be substituted into (6) to calculate the circuit parameters. Then, Equation (7) is arranged to obtain (16):

$$\left[\begin{array}{c}{V}_{ina}\\ V_{inb}\\ V_{inc}\end{array}\right]=\left[\begin{array}{ccc}s{L}_{1a}+\frac{1}{s{C}_{1a}}& s{M}_{1a1b}& s{M}_{1a1c}\\ s{M}_{1a1b}& s{L}_{1b}+\frac{1}{s{C}_{1b}}& s{M}_{1b1c}\\ s{M}_{1a1c}& s{M}_{1b1c}& s{L}_{1c}+\frac{1}{s{C}_{1c}}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]+\left[\begin{array}{ccc}\frac{s^2{\beta }_{1a1a}}{R_{Labc}}& \frac{s^2{\beta }_{1a1b}}{R_{Labc}}& \frac{s^2{\beta }_{1a1c}}{R_{Labc}}\\ \frac{s^2{\beta }_{1b1a}}{R_{Labc}}& \frac{s^2{\beta }_{1b1b}}{R_{Labc}}& \frac{s^2{\beta }_{1b1c}}{R_{Labc}}\\ \frac{s^2{\beta }_{1c1a}}{R_{Labc}}& \frac{s^2{\beta }_{1c1b}}{R_{Labc}}& \frac{s^2{\beta }_{1c1c}}{R_{Labc}}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]$$

(16)

where

$$\begin{array}{l}\left\{\begin{array}{l}{\beta }_{1a1a}=M_{1a2a}{\alpha }_{1a2a}+M_{1a2b}{\alpha }_{1a2b}-M_{1a2c}{\alpha }_{1a2c};{\beta }_{1a1b}=M_{1a2a}{\alpha }_{1b2a}+M_{1a2b}{\alpha }_{1b2b}-M_{1a2c}{\alpha }_{1b2c}\\ {\beta }_{1a1c}=M_{1a2a}{\alpha }_{1c2a}+M_{1a2b}{\alpha }_{1c2b}-M_{1a2c}{\alpha }_{1c2c}\end{array}\right\}\\ \left\{\begin{array}{l}{\beta }_{1b1a}=M_{1b2a}{\alpha }_{1a2a}+M_{1b2b}{\alpha }_{1a2b}-M_{1b2c}{\alpha }_{1a2c};{\beta }_{1b1b}=M_{1b2a}{\alpha }_{1b2a}+M_{1b2b}{\alpha }_{1b2b}-M_{1b2c}{\alpha }_{1b2c}\\ {\beta }_{1b1c}=M_{1b2a}{\alpha }_{1c2a}+M_{1b2b}{\alpha }_{1c2b}-M_{1b2c}{\alpha }_{1c2c}\end{array}\right\}\\ \left\{\begin{array}{l}{\beta }_{1c1a}=M_{1c2a}{\alpha }_{1a2a}+M_{1c2b}{\alpha }_{1a2b}-M_{1c2c}{\alpha }_{1a2c};{\beta }_{1c1b}=M_{1c2a}{\alpha }_{1b2a}+M_{1c2b}{\alpha }_{1b2b}-M_{1c2c}{\alpha }_{1b2c}\\ {\beta }_{1c1c}=M_{1c2a}{\alpha }_{1c2a}+M_{1c2b}{\alpha }_{1c2b}-M_{1c2c}{\alpha }_{1c2c}\end{array}\right\}\end{array}$$

(17)

Similarly, the primary side resonant capacitors can be derived with Equation (16), as seen below:

$$C_{1a}=\frac{1}{\left(L_{1a}-M_{1a1b}-M_{1a1c}+M_{1\mathrm{b}1c}\right){\omega }^2}$$

(18)

$$C_{1b}=\frac{1}{\left(L_{1b}-M_{1a1b}-M_{1\mathrm{b}1c}+M_{1a1c}\right){\omega }^2}$$

(19)

$$C_{1c}=\frac{1}{\left(L_{1c}-M_{1a1c}-M_{1\mathrm{b}1c}+M_{1a1b}\right){\omega }^2}$$

(20)

Once the primary side resonant capacitances are obtained, the capacitance parameters Equations (18)–(20) can be substituted into Equation (16) and derived as

$$\left[\begin{array}{c}{V}_{ina}\\ V_{inb}\\ V_{inc}\end{array}\right]=\left[\begin{array}{ccc}-s{M}_{1b1c}& -s{M}_{1a1c}& -s{M}_{1a1b}\\ -s{M}_{1b1c}& -s{M}_{1a1c}& -s{M}_{1a1b}\\ -s{M}_{1b1c}& -s{M}_{1a1c}& -s{M}_{1a1b}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]+\left[\begin{array}{ccc}\frac{s^2{\beta }_{1a1a}}{R_{Labc}}& \frac{s^2{\beta }_{1a1b}}{R_{Labc}}& \frac{s^2{\beta }_{1a1c}}{R_{Labc}}\\ \frac{s^2{\beta }_{1b1a}}{R_{Labc}}& \frac{s^2{\beta }_{1b1b}}{R_{Labc}}& \frac{s^2{\beta }_{1b1c}}{R_{Labc}}\\ \frac{s^2{\beta }_{1c1a}}{R_{Labc}}& \frac{s^2{\beta }_{1c1b}}{R_{Labc}}& \frac{s^2{\beta }_{1c1c}}{R_{Labc}}\end{array}\right]\left[\begin{array}{c}{I}_{1a}\\ I_{1b}\\ I_{1c}\end{array}\right]$$

(21)

Referring to Figure 4b and Equation (21), to calculate the resistances (R_1a, R_1b, R_1c) of the equivalent circuit, the source voltage (V_inb) is subtracted from the source voltage (V_ina), the source voltage (V_inc) is subtracted from the source voltage (V_inb), and the source voltage (V_ina) is subtracted from the source voltage (V_inc), summarized below:

$$V_{ina}-V_{inb}=I_{1a}\frac{s^2}{R_{Labc}}\left({\beta }_{1a1a}-{\beta }_{1a1c}-{\beta }_{1b1a}+{\beta }_{1b1c}\right)-I_{1b}\frac{s^2}{R_{Labc}}\left({\beta }_{1b1b}-{\beta }_{1b1c}-{\beta }_{1a1b}+{\beta }_{1a1c}\right)$$

(22)

$$V_{inb}-V_{inc}=I_{1b}\frac{s^2}{R_{Labc}}\left({\beta }_{1b1b}-{\beta }_{1b1a}-{\beta }_{1c1b}+{\beta }_{1c1a}\right)-I_{1c}\frac{s^2}{R_{Labc}}\left({\beta }_{1c1c}-{\beta }_{1c1a}-{\beta }_{1b1c}+{\beta }_{1b1a}\right)$$

(23)

$$V_{inc}-V_{ina}=I_{1c}\frac{s^2}{R_{Labc}}\left({\beta }_{1c1c}-{\beta }_{1c1b}-{\beta }_{1a1c}+{\beta }_{1a1b}\right)-I_{1a}\frac{s^2}{R_{Labc}}\left({\beta }_{1a1a}-{\beta }_{1a1b}-{\beta }_{1c1a}+{\beta }_{1c1b}\right)$$

(24)

From Equations (22)–(24), the equivalent resistance in Equations (25)–(27) can be derived as seen below:

$$R_{1a}=\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1a1a}-{\beta }_{1a1c}-{\beta }_{1b1a}+{\beta }_{1b1c}\right)=\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1a1a}-{\beta }_{1a1b}-{\beta }_{1c1a}+{\beta }_{1c1b}\right)$$

(25)

$$R_{1b}=\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1b1b}-{\beta }_{1b1c}-{\beta }_{1a1b}+{\beta }_{1a1c}\right)=\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1b1b}-{\beta }_{1b1a}-{\beta }_{1c1b}+{\beta }_{1c1a}\right)$$

(26)

$$R_{1c}=\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1c1c}-{\beta }_{1c1a}-{\beta }_{1b1c}+{\beta }_{1b1a}\right)=\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1c1c}-{\beta }_{1c1b}-{\beta }_{1a1c}+{\beta }_{1a1b}\right)$$

(27)

Once the equivalent resistance reflected to the primary side is obtained, Figure 2 is equivalent to the circuit diagram, as seen in Figure 5. The output power to the primary side coil of the inverter module can be expressed as Equations (28)–(31):

$$P_{1a}=\frac{2}{{\pi }^2}\frac{V_{in}^2}{R_{1a}}=\frac{2}{{\pi }^2}\frac{V_{in}^2}{\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1a1a}-{\beta }_{1a1c}-{\beta }_{1b1a}+{\beta }_{1b1c}\right)}$$

(28)

$$P_{1b}=\frac{2}{{\pi }^2}\frac{V_{in}^2}{R_{1b}}=\frac{2}{{\pi }^2}\frac{V_{in}^2}{\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1b1b}-{\beta }_{1b1c}-{\beta }_{1a1b}+{\beta }_{1a1c}\right)}$$

(29)

$$P_{1c}=\frac{2}{{\pi }^2}\frac{V_{in}^2}{R_{1c}}=\frac{2}{{\pi }^2}\frac{V_{in}^2}{\frac{-{\omega }^2}{R_{Labc}}\left({\beta }_{1c1c}-{\beta }_{1c1a}-{\beta }_{1b1c}+{\beta }_{1b1a}\right)}$$

(30)

$$P_t=\frac{6}{{\pi }^2}\frac{V_{in}^2}{\frac{-{\omega }^2}{1.11R_L^2}\left({\beta }_{1a1a}-{\beta }_{1a1c}-{\beta }_{1b1a}+{\beta }_{1b1c}\right)}$$

(31)

In summary, all the equations are discussed in the previous description; once the required power, output voltage, coil parameters, and operating frequency are confirmed, the parameters of the circuit can be calculated by Equations (18)–(20) and (31).

3. Coil Design

The coil design was processed in advance using PSIM and CST STUDIO SUITE™ for self-inductance (i.e., coil inductor), coupling coefficient, misalignment effects, and voltage gain for the WPT converter design in order to reach the specifications shown in Table 1 with higher efficiency. Figure 6 shows the proposed sandwich wound coils, which overlap at an angle difference of 120°; it provides the following features: (i) compact size to provide good reliability and light weight; and (ii) a high coupling coefficient and better tolerance of misalignment between primary and secondary coils to enhance efficiency and keep the variation in parameters small for reasonable control.

In Figure 6, the Tx coils and Rx coils are equal in design size; the corresponding design specifications are shown in

Table 2. Coil specifications and design parameters.

Coil Type	Tx/Rx Coils
Size	600 mm × 450 mm
Number of turns	14
Wire thickness	8 mm
Magnetic material specification
Magnetic material	Ferrite FAT100 xiaoshuang(Crown Ferrite Enterprise Co.)
Magnetic material size	882 m × 763 mm
Magnetic material thickness	5 mm
Magnetic field shielding plates
Shielding plate material	Aluminum
Shielding plate size	890 mm × 780 mm
Shielding plate thickness	6 mm

. The main difference is that the overlapping order is different from the arrangement of the magnetic materials.

To make the three sets of coils approach balance in design, the sub-coils on the primary and secondary side coupling have the same coupling distance; hence, the lowermost coil of the primary side corresponds to the uppermost coil of the secondary side and the uppermost coil of the primary side corresponds to the lowermost coil of the secondary side. Furthermore, the arrangements of the Tx and Rx coils are different; as a result, L_1a corresponds to L_2a, L_1b corresponds to L_2b, and L_1c corresponds to L_2c. To avoid imbalance of the inductance of the Tx/Rx coils, one arranges the magnetic material at different heights instead of placing it under the coils, as shown in Figure 7. Thus, the coils can achieve similar inductance, as verified by simulations.

Based on the coil drawing schematic shown in Figure 6 and Figure 7, the simulation results using CST STUDIO SUITE™ are listed in

Table 3. CST STUDIO SUITE™ simulation results of the Tx and Rx coils.

Items	Tx Coils			Rx Coils
Self-inductance (μH)	L1a	L1b	L1c	L2a	L2b	L2c
	131	131.1	130.3	130.9	130.6	130.7
Mutual inductance (μH)	M1b1a	M1c1b	M1a1c	M1b1a	M1c1b	M1a1c
	33.26	33.69	35.6	35.67	33.64	33.07

. It can be seen that the simulated results for the inductance of the Tx and Rx coils approach the balance condition.

Figure 8 shows the 3D Tx/Rx coupling schematic diagram with CST STUDIO SUITE™, where L_1a corresponds to L_2a, L_1b corresponds to L_2b, and L_1c corresponds to L_2c. According to the SAE J2954 Z3 grade [17], the air gap and power transmission distance between the Rx and Tx coil coupling are 170 mm to 250 mm. Figure 9 shows the mutual inductance values under different air gap distances for the primary and secondary side coil coupling. In Figure 9, M_1a2a is the mutual inductance of coils L_1a and L_2a, which has a positive alignment relationship and is larger than M_1a2b (i.e., the mutual inductance of coils L_1a and L_2b) and M_1a2c (i.e., the mutual inductance of coils L_1a and L_2c). When misalignment occurs, the conversion efficiency will decrease due to the reduction in the coils’ magnetic coupling.

Similarly, M_1b2b of coils L_1b and L_2b is larger than M_1a2b (i.e., the mutual inductance of coils L_1a and L_2b) and M_1c2b (i.e., the mutual inductance of coils L_1c and L_2b). Once the self-inductance and mutual inductance parameters of the Tx/Rx coils are selected, these are substituted into Equations (18)–(20) and (31) to calculate the resonant capacitors (referring to Section 2).

4. Results

To validate the derived equation of the proposed WPT system, the circuit schematic was drawn with PSIM software as shown in Figure 10; the calculated parameters are listed in

Table 4. PSIM circuit simulation parameters of the proposed 22 kW WPT system.

Primary coils (μH)	L1a	L1b	L1c	M1b1a	M1c1b	M1a1c
	131	131.1	130.3	33.26	33.69	35.6
Secondary coils (μH)	L2a	L2b	L2c	M2b2a	M2c2b	M2a2c
	130.9	130.6	130.7	35.67	33.64	33.07
Output resistor RL	Simulation (9.2 Ω)			Calculation (9 Ω)
Mutual inductance (μH)	M1a2a		M1b2a		M1c2a
	29.96		13.13		12.02
Primary side capacitors (nF)	Simulation			Calculation
	C1a	C1b	C1c	C1a	C1b	C1c
	36.6	35.2	37.2	39	34.5	36.5
Secondary side capacitors (nF)	Simulation			Calculation
	C1a	C1b	C1c	C1a	C1b	C1c
	36.6	37.2	35.2	37	37	35

.

In Figure 10, a half-bridge leg is used as the inverter module, and a three-phase diode circuit is used as the secondary-side rectifier. Furthermore, the coil coupling circuit includes three-phase, sandwich-wound Tx/Rx coils; the series resonant capacitors; and the reflected load resistances. To control the inverter module, as shown in Figure 11, one sets the pulse-width modulation (PWM) signal at 50% duty cycle and 85 kHz. The phase difference between the upper power switch and lower power switch of the inverter module is 180°; the phase difference of the three inverter modules is 120°.

The simulated and calculated circuit parameters are summarized and listed in Table 4. One can see that the primary side capacitances are slightly different from the calculation results because of the parasitic capacitance effect of the power switch of the inverter module. Therefore, the primary side capacitances need to be modified slightly to make the power switch operate with zero voltage switching (ZVS). Figure 12 shows the ZVS waveforms of the power switch for the inverter; the drain voltages and conducting currents of the three half-bridge power modules are measured, respectively. From Figure 12, it can be seen that the switch current is nearly zero when the switch is turned on, which shows that the ZVS condition in this instant is provided, and the efficiency can be improved.

Figure 13 shows the waveforms of the input voltage (V_in) and the output voltage (V_o); when the input voltage is about 750 V, the WPT system can regulate the output voltage to 449.75 V, close to the specification voltage shown in Table 1. Figure 14 shows the conversion efficiency of the WPT system with a 170 mm air gap; it indicates that the input power is 24.05 kW, the output power is 22.48 kW, and the efficiency is 93.47%, thus achieving a high-efficiency WPT design. Based on the simulation-assisted design results of Figure 6, the realized prototype of the Tx/Rx coils was built, as shown in Figure 15 and Figure 16. The measured results of the coil prototypes are also listed in

Table 5. Measured results of the realized Tx/Rx coils.

Primary coils	L1a	161.79 μH	Secondary coils		L2a	158.69 μH
	L1b	152.86 μH			L2b	160.25 μH
	L1c	158.83 μH			L2c	164.07 μH
	M1b1a	35.56 μH			M2b2a	41.20 μH
	M1c1b	39.52 μH			M2c2b	41.25 μH
	M1a1c	37.30 μH			M2a2c	37.82 μH
Mutual inductances	M1a2a	36.40 μH	M1b2a	16.89 μH	M1c2a	14.76 μH
	M1a2b	13.84 μH	M1b2b	36.56 μH	M1c2b	14.81 μH
	M1a2c	14.81 μH	M1b2c	16.81 μH	M1c2c	37.84 μH

; the tolerances of parameters are limited to 7~18%, compared with the designed points.

5. Conclusions

This paper designed a three-phase 22 kW high-power WPT system through PSIM and CST STUDIO SUITE™ simulation. The WPT system used three-phase, sandwich-wound Tx/Rx coils to minimize the magnetic flux leakage effect and increase the PTE in order to transfer power effectively across the 170~250 mm air gap. A detailed equivalent circuit model has been derived for selecting the primary- and secondary-side coil inductors and capacitors. In this paper, the system simulation achieved a PTE of 93.47% at an operating frequency of 85 kHz with a 170 mm air gap between the coils. The results verify the feasibility of the simulation-assisted design. The developed coils can comply with a high-power and high-efficiency WPT system, as well as a size reduction. Based on the simulation-assisted design results, the realized prototype of the Tx/Rx coils was built; the tolerances of parameters are limited to 7~18% compared with the designed points. This paper shows promising results for the further implementation of the high-power WPT system.